ACTA ACUSTICA UNITED WITH ACUSTICAVol 101 (2015) 394 ndash 407

DOI 103813AAA918835

Post-Classification of Nominally IdenticalSteel-String Guitars Using Bridge Admittances

Hossein Mansour Vincent Freacuteour Charalampos Saitis Gary P ScavoneComputational Acoustic Modeling Laboratory Centre for Interdisciplinary Research in Music Media and Tech-nology (CIRMMT) Schulich School of Music McGill University 555 Sherbrooke Street West MontreacutealQueacutebec H3A 1E3 Canada hosseinmansourmailmcgillca

SummaryDynamic and acoustical measurements were conducted on 18 nominally identical acoustic guitars coming off thesame production line and post-classified by the manufacturer as either bassy (ie with a more prominent bassresponse) mid-even (ie well-rounded and sounding even from string to string) or treble (ie with a brightersound that cuts through the band) One goal was to find features of the guitar admittances that could be used toautomatically classify them according to these categories A second goal was to investigate whether experiencedguitarists agreed with the classifications provided by the manufacturer Physical properties were investigatedindependently and in conjunction with perceptual assessments by musicians collected during a classificationtask Despite very low agreement across guitarists as well as between musicians and the manufacturer resultsshowed that the bassy guitars had a lower frequency for their breathing mode This suggests a lower stiffness-to-weight ratio for the respective guitar bodies which may be caused by small variations in the plate thickness orwood properties The guitars characterized as treble in this study tended to have lower averaged mobility in the600ndash2000 Hz range which might suggest a weaker string-to-body coupling at those frequencies andor a longerdecay for higher partials though these characteristics were not confirmedPACS no 4366Jh 4375Gh

1 Introduction

This paper presents the results of a recent study investigat-ing possible links between measurable characteristics of18 nominally identical acoustic guitars and their perceiveddifferences as classified by the manufacturer The guitarswere built in a modern factory with very precise control ofgeometrical conditions But within a given guitar modelthe manufacturer noted variations of sounding character-istics that were assumed to be due to natural variations inthe wood used for their construction

The study commenced with the manufacturer providing18 guitars from the same production line (model GodinSeagull Maritime SWS) and post-classified by a singleemployee at the factory as either bassy (ie with a moreprominent bass response) mid-even (ie well-roundedand sounding even from string to string) or treble (iewith a brighter sound that cuts through the band) One ofthe goals of the work was to identify measurable featuresof the guitars that could be used to automatically clas-sify them Another goal of the study was to investigate thelevel of agreement among guitar players in classifying theguitars into these categories The measurement techniquesused were chosen with an eye toward eventual implemen-

Received 30 March 2014accepted 01 December 2014

tation in a factory setting where vibroacoustic isolationwould likely be limited and the ease and repeatability ofthe measurements would be critical

The measurements performed included input admit-tance at the bridge radiativity and long-term average spec-tra (LTAS) from live acoustical recordings A few wire-breaking tests were conducted but a complete set of mea-surements across all guitars was not possible in the lim-ited project time frame Overall results showed that gui-tars categorized as bassy by the manufacturer had a lowerbreathing mode frequency The treble guitars demon-strated a lower averaged mobility in the frequency rangeof 600ndash2000 Hz

A perceptual experiment was conducted in which 13 ex-perienced guitarists were asked to categorize the 18 gui-tars into the three above-mentioned categories In generalthere was very low agreement among the players in termsof the classifications That said there was more agreementon the bassy classifications than the other two categories

This paper is organized as follows Section 2 providesbackground on measurement and analysis results previ-ously described in the literature Section 3 reports the re-sults of the physical measurements conducted in this studyand analyses related to the guitar categorizations Sec-tion 4 describes the perceptual experiment conducted andits results Finally the overall results are considered in

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Mansour et al Post-classification of steel string guitars ACTA ACUSTICA UNITED WITH ACUSTICAVol 101 (2015)

Section 5 with thoughts on possible future work that couldbe performed to help clarify the results

2 Background

The energy of a vibrating string is primarily transmittedto the soundboard of the guitar through the bridge Themechanical input admittance at the bridge (or bridge mo-bility) is defined in the frequency domain and in a givendirection as the ratio of the velocity to an input force driv-ing the structure at a given location It provides a descrip-tion of the ability of a point of the sound board to be dis-placed as a function of frequency Therefore this quantityrelates to the ability of the sound board to move and to ra-diate acoustic energy while it also describes the amount ofenergy absorbed from the string which ultimately affectsthe decay rate of string vibration Consequently measure-ments of the bridge mobility are likely to provide relevantinformation with regards to the vibrational characteristicsof an instrument

The top plate of the guitar constitutes the primary radi-ating element of the instrument and thus it plays a veryimportant role in the characteristics of the radiated soundThe analysis of the modal shapes of the lowest frequencyguitar modes reveals that the higher mobility of the topplate occurs on its lower bout the upper bout presentingless mobility [1] In the amplitude curve of the frequencyresponse of the instrument measured at the bridge of theinstrument these low-frequency modes can be easily iden-tified and are commonly designated in reference to theirphysical origins

Different discrete models have been proposed to rep-resent the low frequency and weakly damped modal re-sponse of the guitar body Christensen [2] presented asimple two-degree of freedom model that allows a de-scription of the coupling between the soundbox and theHelmholtz resonance and subsequently a three-degree offreedom model including an additional mass to representthe back plate of the guitar [3 4] That model identifiesthree coupled modes which are the result of the stronginteraction between the first bending modes of the topand back plates as well as the Helmholtz resonance ofthe cavity The first of the three is the breathing modeof the soundbox (hereafter T(11)1) in which the top andback plate move out-of-phase This leads to the expansionand contraction of the soundbox resulting in the air beingsucked in or pushed out through the sound hole respec-tively In the second mode the two plates move in-phasewhile the ribs move in the opposite direction Changingthe boundary condition around the ribs will thus affect thefrequency of this mode fairly strongly (as much as 30 Hz[5]) The third mode (hereafter T(11)2 or anti-breathingmode) is the same as the first but instead the air is beingpushed out when the soundbox expands and vice versaTypically there is an anti-resonance between the breath-ing and the anti-breathing modes at the frequency wherethe pure Helmholtz mode would fall if the soundbox was

rigid Among the three the first and the third modes in-volve a change in the volume of the cavity and thus radiatewell while the second mode does not radiate well Morerecently Popp proposed a four-degree of freedom model[6] By including a fourth oscillator representing the mo-tion of guitar sides and neck this model allows a moreprecise exploration of the influence of rib mass on the lo-cation of the low-frequency resonances There are a fewother modes below 400 Hz that involve a coupling betweenthe top and back plates above that frequency however topplate back plate and air cavity behave relatively indepen-dently

For most structures the bandwidth of individual modesincreases roughly linearly with frequency while the aver-age spacing between adjacent modes remains almost con-stant This results in an increasingly stronger interactionbetween close modes At higher frequencies the modesoverlap to define the response of the system which makesit more meaningful to think about statistical features ratherthan individual modes In this regard Modal Overlap Fac-tor (MOF) is a useful criterion to find the strength of theinteraction between close modes MOF is the ratio of thebandwidth of individual resonance peaks to the averagespacing of adjacent resonances and is expected to grow athigher frequencies [7] Common MOF values for charac-terizing the different frequency ranges are MOF lt 03 forlow frequencies 03 lt MOF lt 1 for mid-frequencies andMOF gt 1 for high frequencies [8] In the higher frequencyrange quantities such as MOF averaged admittance aver-age peak spacing and typical peak-to-valley heights pro-vide useful information that can potentially allow for thedifferentiation between two nearly identical instruments

Recent approaches involve Statistical Energy Analysis(SEA) of the frequency response over a broader frequencyrange [9] and high-resolution modal analysis based on themodeling of the impulse response of a structure as a sum ofcomplex damped sinusoids [8] Using the latter approachElie et al proposed a few macro dynamic features fordiscriminating between different classes of classical gui-tars Although the low-quality (ldquoindustrialrdquo) guitars werecarved out of a different material (plywood) than the high-quality (ldquohand-maderdquo) instruments (made with spruce orred cedar) the results illustrate the potential of that ap-proach to associate global physical quantities of a guitarwith construction details

Long-term average spectra (LTAS) offers an alternativeapproach to the analysis of sound from complex sourcessuch as musical instruments [10] LTAS is the mean of suc-cessive short-term spectra computed over the segments ofa signal This method reduces the noise in the estimatedpower spectra in exchange for reducing the frequency res-olution [11] thereby highlighting longer term aspects ofperceived sound quality LTAS has been used extensivelyfor studying perceived differences in speech and singingvoices (eg [12]) but less often for quantifying instrumen-tal sound quality [10 13]

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ACTA ACUSTICA UNITED WITH ACUSTICA Mansour et al Post-classification of steel string guitarsVol 101 (2015)

3 Physical measurements

31 Method

311 Input admittanceAs mentioned in the previous section the mechanical ad-mittance measured at the bridge provides useful informa-tion regarding the ability of the sound board to be excitedby the strings We chose to perform measurement of thebridge admittance (or mobility) as well as the radiativity(transfer function between the input force at the bridge andthe acoustic pressure radiated at a given point outside theinstrument) on the set of guitars provided by the manufac-turer As shown in Figure 1 the guitar was held verticallyby means of a wooden stand the guitar was maintainedby two rigid bars covered with foam holding the ribs justbelow the upper bout at the level where the body of the in-strument is the narrowest without exerting too much pres-sure on the ribs The strings were properly tuned and thendamped using a clamp with foam fixed between the 9thand 10th frets The part of the strings above the nut wasalso damped using a piece of cardboard

Energy was injected into the system at the bridge be-tween the low E and A strings using an impact hammer(PCB Model 086E80) mounted on a pendulum allowingrelatively repeatable impacts normal to the surface of thebridge Point mobility measurements require collocatedexcitation and acquisition of the velocity The velocity washence measured as close as possible from the impact pointbetween the E and A strings using a velocimeter (Poly-tec LDV-100) pointing normal to the surface of the bridgeThe particular choice of the bass side of the bridge wasmade due to the fact that some information about the sym-metric and anti-symmetric modes of the body (with re-spect to the longitudinal axis of the instrument) might belost if the measurements were made near the middle ofthe bridge and the lowest string is the one that is moststrongly coupled to the body hence measuring the bridgeadmittance underneath that string can potentially providethe most relevant information about the body A micro-phone (Bruumlel amp Kjaeligr Type 4190-L-001) was placed onemeter from the instrument at the level of the neck to bodyjunction pointing towards the center of the sound holeThe measurements took place in a semi-anechoic roomwith dimensions of 54 times 64 times 36 m3 After appropri-ate pre-amplification of the acquired signals acquisitionwas performed using a National Instruments data acquisi-tion card running at 441 kHz and a Matlab interface Eachmeasurement was of three-second duration after the be-ginning of the impact For each guitar final measurementswere averaged over five repetitions At each acquisitionthe coherence was calculated and displayed to the opera-tor allowing any erroneous measurement to be discardedand repeated if necessary

312 Modal fittingMode fitting was performed using an analysis method andalgorithm developed by Woodhouse [9] This method usesthe Matlab function invfreqs allowing rational fraction

Figure 1 Experimental setup for input admittance and radiativitymeasurements

polynomial approximation It first involves modal extrac-tion through pole-residue fitting This first step is followedby an optimization procedure allowing selection of thebest sets of complex and real residues by minimizing themean of the modulus-squared deviation between measure-ment and reconstruction

313 Recordings and LTAS

For the LTAS all 18 guitars were recorded by a profes-sional guitar player The choice of musical input for therecorded sample sounds can have a significant influenceon the LTAS obtained On each of the guitars we recordeda sequence of 8 strummed chords selected to excite a widerange of frequencies E (E2 B2 E3 G 3 B3 E4) rarr F m7(F 2 E3 F 3 A3 C 4) rarr Emaj7 (E2 B2 D 3 G 3 B3 E4) rarrAE (E2 A2 E3 A3 C 4) rarr F m7 (F 2 C 3 E3 A3 E4) rarrEmaj7 (E2 G 3 B3 D 4 G 4) rarr AE (E3 A3 C 4 E4 A4) rarrE (E3 G 3 B3 E4 G 4) The musician was instructed to playmezzoforte at a tempo of 80 beats per minute

The recordings took place in the same room as the ad-mittance measurements A 12-inch free-field microphone(Bruumlel amp Kjaeligr Type 4190-L-001 with Type 2669-L pream-plifier) with a sound quality conditioning amplifier (Bruumlelamp Kjaeligr Type 2672) was used The microphone was posi-tioned 75 cm from the played guitar facing directly at itstop side between the center of the sound hole and the fin-gerboard The recorded phrases were 30 seconds long andsaved in 16-bit stereo 441 kHz WAV format

396

Mansour et al Post-classification of steel string guitars ACTA ACUSTICA UNITED WITH ACUSTICAVol 101 (2015)

LTAS were obtained using the Matlab function pwelchsymmetric four-term Blackman-Harris windows an 8192-point window size and 50 overlap between the windowsSubsequently the results were smoothed in the frequencydomain using a 500 Hz sliding rectangular window

32 Variation among the instruments

We first evaluated the amount of variation among the 18guitars The precise manufacturing process in the produc-tion of these instruments made it almost impossible to seeany appreciable geometrical difference among the individ-ual instruments except a slight variation in the design oftheir necks (some were intentionally cut more flat and oth-ers more rounded) Therefore the biggest uncertainty fac-tor would be the variation in material properties Althoughthe instruments of our study were all made with sprucetops and mahogany backs and ribs some variation of tim-bre is inevitable

Figure 2 shows the averaged admittance over the pool of18 instruments plusminus one standard deviation A sam-ple instrument from the pool and an outlier are includedfor comparison The outlier was a full-size acoustic guitarfrom the same manufacturer but of a different model Thestandard deviation for all frequency points from 70 Hz to6 kHz averages to 275 dB when calculated for all 18 in-struments The level of variation is slightly higher if welook at the lower range of frequencies (379 dB for the fre-quency range of 70 Hz to 1 kHz) As a reference we calcu-lated the standard deviation for 4 repeated measurementsmade on the same instrument removed and put back on thefixture before each repetition The standard deviation aver-aged to 054 dB for the range 70 Hz to 6 kHz and 094 dBfor the range 70 Hz to 600 Hz Note the different samplesize when comparing these numbers with their equivalentsreported earlier for the 18 instruments

The same analysis can be applied to the instrumentswithin the same perceptual categories (ie bassy mid-even and treble) Figure 3 shows the average admittancefor the instruments of the different categories The av-erage standard deviations within the categories of bassymid-even and treble are respectively 263 dB 236 dB and215 dB for the frequency range of 70 Hz to 6 kHz and329 dB 231 dB 242 dB for the frequency range of 70 Hzto 1 kHz The fact that the variation within each categoryis less than the variation among all the instruments lendssome credibility to the classification made by the manufac-turer Another observation in Figure 3 is that the mid-evenand treble instruments follow almost the same trend below600 Hz while bassy instruments are considerably distinctin the frequency range of 80 Hz to 100 Hz and again from300 Hz to 350 Hz showing a shift of the main resonancestowards lower frequencies

33 General properties of the acoustic guitars

331 Real-imaginary parts

Figure 4 shows the real and imaginary parts of the av-eraged admittances using an averaging bandwidth of

70 100 200 500 1000 2000 6000minus65

minus60

minus55

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minus25

minus20

Frequency (Hz)

Averagedadmittance

(dB)

A sample instrumentfrom the pool

An outlier

Averaged plusmn one STD

Figure 2 Gray shading shows the averaged admittance over thepool of 18 instruments plusminus one standard deviation Thesolid black line is a sample instrument from the pool and dashedline is an outlier The dB reference is 1 msN

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minus60

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minus50

minus45

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minus30

minus25

Frequency (Hz)

Averagedadmittance

(dB)

Averaged bassyAveraged midminusevenAveraged treble

Figure 3 Averaged admittance over the bassy instruments (so-lid) mid-even instruments (dash-dot) and treble instruments(dashed) The dB reference is 1 msN

1000 Hz One can see that the acoustic guitar does not rep-resent the properties of an infinite plate (ie positive realpart with roughly constant magnitude and imaginary partmuch smaller) at least not as well as a classical guitar does[9 8] The magnitude of the imaginary part is smaller thanthe real part but still large enough to be non-negligibleThe real part stays positive over all frequencies which isa property of any passive system The average amplitudeof the real part is smaller by almost a factor of two thanthe same measure for a typical classical guitar it reachesthe typical value for a classical guitar at around 3 kHz andagain drops at higher frequencies A broad peak in the realpart and a tendency in the imaginary part towards negativevalues creates a hill-like feature [14] at around 2800 Hz forthe majority of the instruments with a similar pattern thatrepeats less strongly at around 4 kHz These hills are quiteunexpected as an acoustic guitar does not have a bridge

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ACTA ACUSTICA UNITED WITH ACUSTICA Mansour et al Post-classification of steel string guitarsVol 101 (2015)

500 1000 1500 2000 2500 3000 3500 4000 45000002

0004

0006

0008

Re(ltYgt)(msN)

500 1000 1500 2000 2500 3000 3500 4000 4500minus0002

minus0001

0

0001

0002

0003

Frequency (Hz)

Im(ltYgt)(msN)

A sample instrumentAveraged plusmn one STD

(a)

(b)

Figure 4 Band-averaged real (a) and imaginary (b) parts of theadmittance for a sample instrument (solid) and averaged overall instruments of the pool plusminus one standard deviation(shaded) Note that the vertical scales are different for the realand imaginary parts

like a violin nor has it a feature like f-holes to create suchlocal resonances A possible explanation is that the cross-bracing of the steel string guitars unlike the fan-bracing ofmost classical guitars circumvents the bridge and formsan isolated area around it and this may contribute to suchlocal resonances

332 Modal spacing and MOFIt is useful to find a practical upper limit for the range ofmode fitting (ie where the MOF reaches one) and to get asense of the average spacing between the modes Figure 5shows the MOF and the average mode spacing for a sam-ple instrument from the pool and the average plusminusone standard deviation over all instruments An averagingbandwidth of 500 Hz is used in this case to give a betterprecision at lower frequencies The statistical features in-cluding MOF are calculated based on variance theory formore details see equations (17) to (19) of [9] As expectedthe general trend of the overlap factor is rising although itfalls a bit between 1500 Hz and 2000 Hz it reaches thevalue of one at around 610 Hz The average mode spac-ing goes unrealistically high (as much as 150 Hz at around200 Hz) at lower frequencies The mode spacing stays al-most constant at around 20 Hz for the frequency range of600ndash1200 Hz The mode spacing drops to around 8 Hz athigher frequencies and then stays almost unchanged

333 Mode fittingUsing the method described in Section 2 a mode fittingprocess was performed on all 18 instruments of the poolup to 600 Hz [9] The fitting program identified 20 to 24modes for different instruments amongst which a maxi-mum of 16 modes were manually selected for each instru-ment The criterion was to keep the stronger modes thatwere also identified for most of the instruments

The general shape of the admittance follows the pre-dictions of [6] the first neck bending at around 65 Hz

600 1000 1500 2000 2500 3000 35000512

510

40

Modaloverlapfactor

600 1000 1500 2000 2500 3000 35000

10

20

30

40

50

Frequency (Hz)

Modalspacing(Hz)

A sample instrumentAveraged plusmn one STD

(b)

MOF = 1atF asymp 610 Hz

(a)

Figure 5 Modal overlap factor (a) and the average mode spacing(b) for a sample instrument from the pool (solid) and the averageplusminus one standard deviation over all instruments (shaded)An averaging bandwidth of 500 Hz is used

the T(11)1 or the breathing mode at around 100 Hz andT(11)2 or anti-breathing mode at around 200 Hz Modesare categorized as (mn) where m and n are the number ofldquohalf-wavesrdquo across and along the plate and T refers to thetop plate As pointed out by Popp [6] the in-phase motionof the top and back plates coincides in frequency with theanti-breathing mode at around 200 Hz

For our measurements the guitars were held in a fixturewith two arms that pressed against the sides of the ribsThis was intended to create a boundary condition simi-lar to that when held by a player Popp argues that a con-siderable amount of mass would be needed to immobilizethe ribs therefore our measurement condition is betweenthe fixed and free-free cases but perhaps more similar tothe latter After the measurements were performed we ob-served a peak (or a double peak for some instruments)at around 125 Hz where one normally expects to see theanti-resonance representing the pure Helmholtz mode [2]When we repeated the measurements on some of the gui-tars using a free-free boundary condition we noticed thatthe anti-resonance appeared as expected This suggests thepeak is either a property of the fixture rather than the in-struments themselves or it is the (11) in-phase motion ofthe top and back plates shifted downward due to the massof the fixture added to the ribs The latter would requirea 60 Hz downward shift of that frequency which seemsunlikely No effort was made to repeat all the measure-ments with a free-free boundary condition partly becausethe measurements were made before subsequent data anal-ysis and partly because free-free measurements are lesspractical and reproducible in a factoryrsquos production line(due to for example variations in accelerometer place-ment and potential damage to a new guitar surface finishwhen attaching and detaching an accelerometer)

The second neck bending mode was evident as a smallpeak at around 225 Hz for all instruments There weretwo other small peaks at around 250 Hz and 290 Hz which

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Mansour et al Post-classification of steel string guitars ACTA ACUSTICA UNITED WITH ACUSTICAVol 101 (2015)

were probably related to the torsion and lateral vibrationsof the neck The next important modes were T(21) andT(12) For Martin D-28 acoustic guitars these modes arearound 100 Hz apart [5] hence relatively distinct how-ever for the guitars of our study they both appear at around350 Hz and sometimes switch places We measured the ad-mittance at the bass side of the bridge where T(21) hasits antinode therefore for each guitar the stronger modein that region is associated to T(21) In addition to theHelmholtz mode there are two other cavity modes in thefrequency range of our study which are A1 at 430 Hz andA2 at around 500 Hz [5] Approximate mode shapes ofthe first four corpus modes (exclusive of local resonancessuch as neck bending and neck twisting) for the guitars ofour study are illustrated in Figure 6 the frequencies areaveraged and vary from one instrument to another

The average frequency amplitude and Q-factors calcu-lated for the 18 guitars were used to reconstruct the ad-mittance of a ldquomodally-averagedrdquo guitar that representsthe model that we studied Figure 7 shows this ldquomodally-averaged admittancerdquo reconstructed up to 600 Hz (dashedline) compared with a sample measured admittance aswell as the raw averaged admittance previously shown inFigure 2 It can be seen that the peaks are excessivelydamped for the raw average while the admittance of theldquomodally averagedrdquo guitar shows a better resemblance tothe sample measured admittance close to the resonancesand anti-resonances This modal average thus better rep-resents the admittance of a guitar coming off this partic-ular production line It can be overlaid with the measuredadmittance of a new guitar to highlight its significant fea-tures

If needed the admittance of the ldquomodally averagedrdquoguitar can be reconstructed for extended frequencies eitherby simple averaging over different instruments or moreprecisely by using a ldquostatistical fitrdquo method as describedin [16] In short a random number generator can give thefrequency of the high-frequency modes with the correctdensity and spacing statistics as well as damping factorsand modal masses with approximately correct statisticaldistribution Such a statistical fit was not pursued in thisstudy

334 LTAS

Figure 8 shows the LTAS of a sample instrument fromthe pool (solid black line) as well as the average over allinstruments plusminus one standard deviation (shaded)The first few dominant modes identified in an admittancehave demonstrated themselves in the form of broad peaksin the lower frequencies of the LTAS below about 400 HzSome instruments such as the depicted sample show a peakat around 25 kHz which may correspond to the hill-likefeature earlier identified in band-averaged mobility curves(see Figure 4) Beyond 300 Hz the LTAS drops almostlinearly with the average slope of 16 dBoctave The av-erage standard deviation for the frequency range of 70 Hzto 6 kHz for all instruments is 256 dB which indicates theplayer was able to play consistently on all instruments It is

Figure 6 The four first corpus mode shapes of an acoustic guitartaken from [15]

70 100 200 500minus65

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minus30

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minus20

Frequency (Hz)

Averagedadmittance

(dB)

A sample measured admittanceModallyminusaveraged guitarAveragedminusoverminusallminusinstruments

Figure 7 Raw averaged admittance over all instruments of thepool (dash-dot) reconstructed ldquomodally-averagedrdquo admittance(dashed see text for details) and a sample admittance for com-parison (solid) The dB reference is 1 msN

worth mentioning that this number is not directly compa-rable to the averaged standard deviation for the measuredadmittance as the LTAS is significantly smoothed in thefrequency domain in comparison

34 Investigating different features for classification

341 Modal propertiesAveraged value and standard deviation of frequency am-plitude and Q factor of the first four corpus modes are

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calculated for the three perceptual classes and comparedin Figure 9 In all three figures the values are normalizedto the average of the same modal value taken over all in-struments and the error bars show one standard deviationon each side In case of a missing mode for one instru-ment its frequency and Q factor were eliminated from thecalculation of the average and standard deviation but theamplitude was considered to be zero (as that mode wasnegligible or non-existent) The admittance was not com-pensated for added damping from strings [9] partly be-cause the exact same strings were used on all instrumentshence the comparison was fair and partly because the ex-act amount of damping added by the strings is unknownand the assumption of perfect damping at the strings mayresult in over-compensation of damping [9]

As was also evident from Figure 3 Figure 9a shows thatthe corpus modes of the bassy instruments occur at signif-icantly lower frequencies than the mid-even or treble onesAnother feature that distinguishes bassy instruments fromthe mid-even or treble ones can be seen in Figure 9b theamplitude of the first corpus mode T(11)1 is higher forthe bassy instruments The Q factor of the first four cor-pus modes did not show any noticeable difference amongdifferent perceptual classes

342 Averaged admittance and radiativity

To obtain a more clear comparison of different categoriesin the statistical range (ie at higher frequencies) the ad-mittance and radiativity were smoothed in the frequencydomain using a 500 Hz sliding rectangular window andthen averaged for different categories (see Figure 10a andFigure 10b) The proper way to calculate radiativity wouldhave been to measure the averaged sound pressure over anarray of microphones around the instruments That saidour results using only a single microphone provide a rea-sonable approximation because they were smoothed in thefrequency domain and averaged over different instrumentsSince we have already found distinctive features for the in-struments of the bassy class in the subsequent figures theplusminus standard deviation is eliminated for the bassyclass to help clarity

It can be seen in Figure 10a that the band-averaged ad-mittance of the mid-even instruments is greater than orequal to that of the treble instruments over a wide rangeof frequencies particularly in the range 600 Hz to 2 kHzAlthough this dominance does not appear significant formost of the frequencies (ie the averages for categoriesoverlap when plotted with plusminus one standard devia-tion) the mean level of admittance from 600 Hz to 2 kHzcan distinguish mid-even from treble instruments at leastwithin the pool of our 18 instruments This result will befurther discussed in Section 35 The band-averaged admit-tance of the bassy instruments falls between the two othercategories over most of the frequencies

Band-averaged radiativity does not reveal a significantdifference among different categories except a lower av-erage for the bassy guitars in the range from 2 kHz to275 kHz (see Figure 10b) However this difference is

70 100 200 300 500 1000 2000 6000minus180

minus170

minus160

minus150

minus140

minus130

minus120

minus110

minus100

minus90

minus80

Frequency (Hz)

LTAS(dBHz)

A sample instrumentAveraged plusmn one STD

Figure 8 Long-term average spectra for a sample instrumentfrom the pool (solid line) and the average plusminus one stan-dard deviation over all instruments (shaded)

08

09

1

11

Frequency

(norm) (a)

BassyMidminusevenTreble

0

05

1

15

2

25

Amplitude

(norm)

(b)

05

1

15

2

Qminusfactor(norm)

Guitar corpus modes

(c)

T(21)T(11)2

T(11)1 T(12)

Figure 9 Normalized average frequency (a) amplitude (b) andQ factor (c) for the first four corpus modes for bassy guitars (darkgray bars blue in online version) mid-even guitars (light graybars green in online version) and treble guitars (medium graybras red in online version) The error bars show one standarddeviation on each side

likely not important as the standard deviation of the bassyinstruments in this frequency range was found to be muchhigher than the other two categories

343 Damping trend

A method based on time-frequency analysis first intro-duced in [17] and most recently applied in [9] was usedhere to find the trend of the modal damping up to the fre-quency of 6 kHz As mentioned earlier the admittancesare not compensated for added damping from strings Fig-

400

Mansour et al Post-classification of steel string guitars ACTA ACUSTICA UNITED WITH ACUSTICAVol 101 (2015)

500 1000 2000 6000minus54

minus52

minus50

minus48

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Bandminusaveraged

admittance

(dB) Bandminusaveraged bassy

Bandminusaveraged midminusevenBandminusaveraged trebleMidminuseven plusmn one STDTreble plusmn one STD

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minus20

minus18

minus16

minus14

minus12

minus10

minus8

minus6

Frequency (Hz)

Bandminusaveraged

radiativity

(dB)

(a)

(b)

Figure 10 Averaged admittance (a) and radiativity (b) takenover bassy instruments (solid) mid-even instruments (dash-dot)plusminus one standard deviation (light gray shade green in on-line version) and treble instruments (dashed) plusminus onestandard deviation (darker gray shade red in online version)The dB reference is 1 msN

ure 11 shows the averaged damping trend taken over thethree perceptual categories

The Q factor for all classes rises more sharply from Q asymp30 at 100 Hz to Qasymp 60 at 600 Hz where it reaches a plateauand rises more gradually afterwards However the generalbehavior is not far from linear on a log-log scale As itcan be seen in Figure 11a and 11b the value of Q factorover different frequencies and its trend is similar for trebleand mid-even classes which make them indistinguishableby looking at the damping trend From the power-law fitresults the average power for bassy category is 008 andfor both categories of mid-even and treble is 016 This re-sults in an average-over-all-instruments of 013 comparedto 026 and -009 reported earlier for the violins and theclassic guitars respectively in [9] Among the instrumentsof the pool only one showed a negative power fit

344 Harmonic features

Plucking a guitar string excites multiple harmonics simul-taneously which makes it possible that a weighted sum ofthe admittance over some range of harmonics rather thanthe admittance at the fundamental only is a better way ofdistinguishing categories of instruments Forgetting aboutlosses for the moment the amplitude of the transverse

200 300 500 1000 2000 5000

30

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Frequency (Hz)

Qminusfactortrend

200 300 500 1000 2000 5000

20

30

40

50

70

100

Bandminusaveraged

Qminusfactor

Averaged bassy

Averaged midminuseven

Averaged treble

Averaged midminuseven plusmn STD

Averaged treble plusmn STD

(b)

(a)

Figure 11 Effective Q-factors averaged for the guitars of differ-ent categories plusminus one standard deviation (solid line forbassy dash-dot line and light gray green shade for mid-evenand dashed line and darker gray red shade for treble) band-average (a) and regression line for power-law behavior (b) Notethat vertical axes are different

force that the nth harmonic of a plucked string applies tothe bridge can be found from [5]

Fn =2dT0

nπL0

β

1 minus βsin(βnπ) (1)

where L0 is the length of the string T0 is its tension d isthe displacement at the plucking point and β is the rela-tive distance of the plucking point from the bridge Thisformula does not take into account the possible strongcoupling of the plucked string to the body and to otherstrings which should be a reasonable approximation forthe intended purpose and for most of the notes played ona guitar [16] The magnitude of the power transfer fromthe nth harmonic of the string to the body is then equalto the bridge velocity multiplied by the applied force orequivalently

Pn = Fnvn = F 2nGω = A

sin(βnπ)n

2

Gω (2)

in which Gω is the real part of the bridge admittance (alsocalled conductance) and A is a constant assuming a par-ticular string is being played with a constant pluckingstrength and at a fixed relative distance from the bridgeand again neglecting the losses This power transfer to thebody is largely responsible for the damping of the string

401

ACTA ACUSTICA UNITED WITH ACUSTICA Mansour et al Post-classification of steel string guitarsVol 101 (2015)

mode in question as well as the radiated sound at that fre-quency Based on this idea two features are calculated forall semitones played on a guitar (from E2 to E6) and aver-aged over the three perceptual categories For both cases βis arbitrarily chosen as 03 which is a reasonable numberfor a normal playing condition

Figure 12a shows the sum of the transferred power bythe first 5 harmonics of each played note here calledldquopower transfer to the bodyrdquo The choice of 5 harmonics isan arbitrary limit for which the individual harmonics of thestring could be perceived individually and within whichmost of the energy transfer to the body happens in a nor-mal playing condition The frequency range that is consid-ered in this fashion would be up to 5 times the highest notestudied (ie E6) or equivalently 66 kHz this is about thelimit up to which the measured admittances show a reason-ably good coherence The unknown constant A in equation(2) could only cause an offset in the results therefore itis considered to be 1 for convenience The energy trans-fer to the body represents the strength of the stringbodycoupling for a guitar A large value for this number wouldresult in a strong and more percussive (banjo-like) soundand a small value would result in a weaker sound with alonger decay (more similar to an electric guitar without anamplifier)

Figure 12b shows a slightly different use of the trans-ferred energy by each harmonic It is calculated by divid-ing the sum of the energy transferred by the 2nd to 5th har-monic to the energy transferred by the fundamental aloneAs well as specifying the ratio of higher harmonic energyto that at the fundamental this feature defines how thesound evolves once the instrument is plucked Extremelylarge or small values for a particular note mean that thenote changes its character when it decays while an aver-age value may be correlated to a more stable sound

Neither graph shows any clear distinction between themid-even and treble categories The bassy category doeshowever stand out most of the patterns in both graphs areshifted downward by a few semitones

345 LTAS and radiativity

The LTAS results averaged over different categories areillustrated in Figure 13a The results for the mid-even in-struments are greater than the other two groups over al-most all frequencies Figure 13b is the same as Figure 13abut the LTAS for each individual instrument is normalizedby its area before further processing After normalizationall categories behave similarly over different frequenciesmeaning mid-even instruments were simply louder thanthe other instruments rather than being stronger in a par-ticular range of frequencies

The band-averaged admittance of the mid-even instru-ments earlier presented in Figure 10 was slightly greaterthan the treble instruments in some frequency rangeshowever the significance of the stronger LTAS of mid-even instruments especially beyond 3 kHz can hardly bejustified by that alone A possible explanation is that themid-even instruments were played systematically harder

E2 100 E3 250 E4 500 E5 1000 E6

minus60

minus55

minus50

minus45

minus40

minus35

minus30

Pow

ertransfertothebody

(dB)

E2 100 E3 250 E4 500 E5 1000 E6

minus20

minus10

0

10

Harmonicstofundam

entalratio(dB)

Fundamental frequency (Hz)

Averaged bassyAveraged midminusevenAveraged treble

(a)

(b)

Figure 12 Power transfer to the body (a) and harmonics to fun-damental ratio (b) calculated for each played note over the op-erational range of a guitar (E2 to E6) the vertical lines indicateoctaves The values are averaged for the guitars of different cat-egories (solid line for bassy dash-dot line and light gray green(online version) shade for mid-even and dashed line and darkergray red (online version) shade for treble)

compared to the treble instruments There are at least tworeasons as to why this might have been the case First theguitars were recorded per category starting with the trebleand ending with the mid-even ones It is therefore possiblethat the musician felt more confident in the course of therecording session Second the mid-even guitars had beendescribed by the manufacturer as having a ldquomore stablerdquosound Such a sense might have allowed the performer toplay those instruments more freely Nevertheless we areaware that the frequency distribution obtained using LTAScan be strongly dependent on the nature of the input sig-nal depending on the sequence played and used as an in-put certain frequency bands may be less excited than oth-ers which may attenuate differences in some frequencyregions of interest

35 Classification results

According to our results finding a feature that can distin-guish bassy instruments is a relatively easy task As wasshown in Figure 9a the frequencies of the first 4 modesfor the bassy instruments are around 5 lower than thosefor the two other classes The frequency of the first modecan be determined without much ambiguity or even theneed for mode extraction All of the other 3 low-frequencymodes have close adjacent modes making it harder andsometimes impossible to find their correct modal parame-ters

The amplitude of the first mode is another indicator fora bassy instrument that can be used for more assurance

402

Mansour et al Post-classification of steel string guitars ACTA ACUSTICA UNITED WITH ACUSTICAVol 101 (2015)

500 1000 2000 6000minus180

minus170

minus160

minus150

minus140

minus130

minus120

LTAS(dBHz)

500 1000 2000 6000minus120

minus110

minus100

minus90

minus80

minus70

minus60

Frequency (Hz)

Normalized

LTAS(dBHz)

BassyMidminusevenTrebleTreble plusmn one STDMidminuseven plusmn one STD

(b)

(a)

Figure 13 Non-normalized LTAS (a) and normalized LTAS (b)averaged for the guitars of different categories plusminus onestandard deviation (solid line for bassy dash-dot line and lightgray green (online version) shade for mid-even and dashed lineand darker gray red (online version) shade for treble)

(see Figure 9b) Figure 14a shows a 2-D plane with thenormalized frequency and amplitude of the first mode onits horizontal and vertical axes respectively It can be seenthat the bassy instruments are quite distinct from the mid-even and treble ones Moreover the guitar with the lowestbreathing mode (B3 fT (11)1 = 91 Hz) was described bythe factory employee as extra-bassy

It should be emphasized that the frequency of the firstpeak works particularly well as a feature not becausethe ldquobassinessrdquo of the instrument is defined by the notesaround this modersquos frequency (only the first three noteson the E2 string are below that frequency) but becauseit represents the overall stiffness-to-weight ratio of thesoundbox that effectively scales the frequency of all bodymodes It is noteworthy that although T(11)1 has a strongHelmholtz component the amount by which its frequencydeviates from the frequency of the pure Helmholtz reso-nance is strongly dependent on the frequency separationof the pure Helmholtz and pure body breathing modesThe frequency of the pure Helmholtz mode is assumed thesame for all instruments of the pool given their very simi-lar geometry (both for the sound box and the sound hole)

The in-plane dimensions of the instruments should havea very small variation percentage given the precise con-trol of geometrical conditions in the factory Moreover alower frequency for the first mode is usually accompaniedby a higher amplitude for that mode which would not nor-mally be the case if such deviation was caused by using a

90 92 94 96 98 100 102 10404

05

06

07

08

09

1

Normalized

T(11) 1am

plitude

BassyMid EvenTreble

90 92 94 96 98 100 102 104minus50

minus49

minus48

minus47

minus46

minus45

B1B2

B3

B6

B4B5

ME6ME1

ME2

ME3ME4

ME5

T2

T3

T4

T5 T6

T1

Frequency of the T(11)1mode (Hz)

Averagedmobility(600minus2000Hz)(dB)

(b)

(a)

Figure 14 Classification of the acoustic guitars (a) by the fre-quency and amplitude of their first corpus mode T(11)1 onlyto distinguish bassy instruments from the others and (b) by thefrequency of their first corpus mode T(11)1 and their averagedmobility from 600 Hz to 2 kHz (B = bassy ME = mid-even T =treble) The dB reference is 1 msN

denser wood this leaves variations in the plate thicknessand the equivalent stiffness (ie very roughly the geomet-ric mean of the Youngrsquos moduli in long-grain and cross-grain directions [14]) as those most likely responsible forthe observed shift in mode frequencies Mode frequenciesof a flat plate scale with Eρ(tL2) where E and ρ arethe elastic constant and density t the thickness and L acharacteristic scaling length If either of the two factors issolely responsible to make a 5 shift it would require ei-ther a 5 variation (asymp 0125 mm) of the plate thicknessor a 10 variation in the equivalent stiffness Given thetight tolerances in cutting the plates and their mechanizedsanding it thus appears that variations in equivalent stiff-ness are the primary reason that some of the instrumentsin the pool sound more bassy In the manufacturing pro-cess a wood slab is sliced to produce several plates Thelateral direction of each plate hence represents a particularcombination of the slabrsquos radial and tangential directions(due to the growth rings of the tree) This can introduce asignificant variation in the platesrsquo stiffness in the lateral di-rection which produces variations of the effective stiffnessof the plates

Finding a feature that distinguishes mid-even from tre-ble instruments is less obvious We found that the aver-aged admittances of the treble instruments in this study areslightly lower than the mid-even instruments in the rangeof 600 Hz to 2 kHz Although this difference is not signif-icant over the mentioned range (ie average plusminus

403

ACTA ACUSTICA UNITED WITH ACUSTICA Mansour et al Post-classification of steel string guitarsVol 101 (2015)

one standard deviation of the two categories overlap) itbecomes significant if the average level of the admittancefor the whole range is compared (ie the average valueof all frequency points of the admittance from 600 Hz to2 kHz) Figure 14b shows a 2-D plane with the frequencyof the first mode on the horizontal axis and the average ad-mittance for the range of 600 Hz to 2 kHz The differencefor the mean of the mid-even and treble categories is about2 dB It seems likely that the weaker admittance of the tre-ble category at higher frequencies would contribute to alonger decay for the higher partials This may contributeto a sharper after-sound of the treble instruments that isknown to allow them to cut through the band more effec-tively This hypothesis could be tested by recording theplucked sound of the instruments and comparing the decayrate of the partials among different categories Due to thelimited project time frame this was not investigated fur-ther The mid-even guitars also show stronger LTAS thanthe treble instruments though this result might have beendue to playing conditions (see Section 345)

We applied the same classification (frequency ofT(11)1 versus average mobility between 600 Hz and2 kHz) to 6 acoustic guitars from a different model by thesame manufacturer (Simon amp Patrick) Unlike the Seagullguitars which were all nominally identical some of theSampP instruments had a slightly different bracing patterndesigned to exemplify the properties of a particular per-ceptual category The manufacturerrsquos classification of the6 guitars was not provided to us prior to our own mea-surements though we were told that there were 2 guitarsof each category The resulting classification based on thediscussed features is depicted in Figure 15 The two gui-tars designed to be more bassy were successfully identi-fied from the frequency of their breathing mode but dis-tinguishing between the mid-even and treble instrumentsfrom the averaged admittance criterion was inconclusive

4 Perceptual evaluation

As previously discussed in Section 1 the investigatedacoustic guitars were post-classified by the manufacturerbased on their experience of what type of instrument mightbetter fit a certain playing stylemdashfor example an amateurmusician that plays mostly strummed chords may prefera bassy guitar while a more experienced guitarist mayopt for a treble instrument We therefore conducted a per-ceptual experiment to investigate the level of agreementamong guitar players of varying experience and musicalbackground in classifying the guitars into the categoriesconsidered by the manufacturer

41 Method

The experiment took place in an acoustically dry roomwith dimensions of 60 times 78 times 32 m3 and a reverberationtime of about 03 s to help minimize the effects of roomreflections on the direct sound from the guitars To avoidthe potential problems of using a common pick across all

85 86 87 88 89 90 91 92minus505

minus50

minus495

minus49

minus485

minus48B1

SampP

B2SampP

ME1SampP

ME2SampP

T1SampP

T2SampP

Frequency of the T(11)1mode (Hz)

Averagedmobility(600minus2000Hz)(dB)

BassyMid EvenTreble

Figure 15 Classification of the six SampP guitars based on thefrequency of the T(11)1 mode versus averaged mobility from600 Hz to 2 kHz (B= bassy ME=mid-even T= treble) The dBreference is 1 msN

participants (eg musicians being uncomfortable with apick they are not familiar with) participants were asked tocarry out the classification task using their own pick(s)

411 Participants

Thirteen guitar players took part in the experiment (1 fe-male 12 males average age = 25 yrs SD = 5 yrs range =19ndash36 yrs) They had at least 6 years of acoustic guitar ex-perience (average years of training = 11 yrs SD = 4 yrsrange = 6ndash20 yrs average hours of practice per week =14 hrs SD = 10 hrs range = 3ndash40 hrs) as well as vary-ing experience with different types of guitars [nylon-string(85) acoustic-electric (62) acoustic bass (15) elec-tric (100) and electric bass (54)] Participants re-ported playing a wide range of musical styles [classical(77) jazz (85) pop-rock (92) country folk blue-grass (66) hard rock heavy metal (46) flamenco(15) latin (46) blues (77) avant-garde contempo-rary (54) and funk alternative (46)] both in a bandand as soloists Seven guitarists (54) reported that theytypically or most often use a pick whereas 4 players (31)reported they do not 2 guitarists (15) reported using apick occasionally rather than typically Nine participantshad higher-level degrees in music performance or othermusic-related areas [BMus (music theory classical gui-tar) BFA (jazz studies) MMus (guitar performance) MA(digital musics) PhD (music theory) Diplocircme drsquoEacutetat deProfesseur de Musique (France)] All guitar players werepaid for their participation

412 Procedure

The experimental session lasted 2 hours and was organizedin two phases In the first phase participants were pre-sented with the guitars randomly ordered on the floor (de-termined by computer calculations) They were asked toplay all instruments for up to 20 minutes in order to famil-iarize themselves with the set and use this time to tune theguitars as needed In the second phase participants were

404

Mansour et al Post-classification of steel string guitars ACTA ACUSTICA UNITED WITH ACUSTICAVol 101 (2015)

given up to 20 minutes to classify each of the guitars as ei-ther bassy mid-even or treble by placing them in speciallymarked areas of the room At the end of the second phaseparticipants were given the opportunity to revise their ini-tial judgements if they so wanted until they were satis-fied with their classification No playing constraint wasimposed on the evaluation process (eg specific repertoireor technique) Participants were instead encouraged to fol-low their own strategy with the suggestion to use bothstrumming-oriented songs (eg Sitting Waiting Wishingby Jack Johnson) and fingerpicking-oriented pieces (egDust in the Wind by Kansas) Participants were instructedto maximize evaluation speed and accuracy Upon com-pleting the session participants talked about their classi-fications and overall experience through written responsesto the following questions ldquoFor each class of guitar whatsound characteristics did you look for Can you describehow you evaluated the guitars in terms of these character-istics (eg musical material technique style)rdquo (Q1) ldquoTowhat extend do you agree with the given classification ofguitarsrdquo (Q2) and ldquoCan you suggest a classification thatbetter reflects how you evaluate the sound characteristicsof guitarsrdquo (Q3)

42 Results

Inter-individual consistency was measured as the Fleissrsquokappa coefficient between guitar classifications from dif-ferent participants using a Bayesian approach to inference[18] Fleissrsquo kappa is a generalization of Cohenrsquos kappastatistic κ a chance-corrected measure of agreement be-tween two assessors to multiple raters [19] It ranges fromminus1 to 1 κ (AB) = 1 and minus1 if A = B and A = minusBrespectively and κ (AB) = 0 in case of no associationbetween A and B (ie exactly what would be expectedby chance) We implemented Gibbs sampling through thesoftware WinBUGS (Windows Bayesian inference UsingGibbs Sampling [20]) with 50000 observations generatedfrom the posterior distribution with a burn-in of 1000 ob-servations A uniformly distributed multicategorical priorwas assumed with total information across all possiblecells (ie from different combinations of participants) be-ing equivalent to a single prior observation (ie a priorsample size equivalent of 1)

Fleissrsquos kappa was estimated with a posterior mean of01 and a 95 credible interval [-02 05] indicating avery low amount of agreement between the participantsin their classifications Figure 16 shows the proportion oftimes a guitar was classified as either bassy mid-even ortreble (the guitars are arranged in three groups accord-ing to the initial factory classification) Whereas someagreement concerning the bassy guitars can be observedmdashindeed the extra bassy guitar (B3) was most often clas-sified as bassymdashthe perceptions for the other two classeslargely vary across different musicians In particular manyguitars that had been characterized as treble by the fac-tory employee were perceived as mid-even Several mu-sicians specifically reported encountering difficulties withthe mid-even category In answering Q3 a player observed

B1 B2 B3 B4 B5 B60

25

50

75

100

classifiedas

(a)BassyMidminusevenTreble

ME1 ME2 ME3 ME4 ME5 ME60

25

50

75

100

classifiedas

(b)

T1 T2 T3 T4 T5 T60

25

50

75

100

classifiedas

Guitars

(c)

Figure 16 Perceptual evaluation Proportion of times a guitarwas classified into one of the categories by the participants Gui-tars are labelled and grouped according to the factory classifica-tion (B = bassy ME = mid-even T = treble) The color scheme(in the online version) corresponds to the classifications providedby the guitar players

that ldquothere is a difference [between] a guitar with domi-nant mid tones and one with an even soundrdquo

Overall participants expressed no strong difficulties incarrying out the classification task On a scale from 1(strongly disagree) to 5 (strongly agree) all performers se-lected 4 when asked ldquoTo what extend do you agree withthe given classification of guitarsrdquo This indicates that the3 categories considered by the manufacturer are gener-ally meaningful to musicians However verbal responsesto questions Q1 and Q3 reveal that participants often as-sociated bassy with more prominent E2 and A2 strings(ie they focused on the low-frequency end of the gui-tar) and similarly treble with better emphasized B3 andE4 strings (ie they paid attention to the high-frequencyrange) As one guitar player noted ldquoI feel that descriptionslike warm neutral and bright would be easier to classifydue to the nature of the instrument Bass leads my mindto bass strings and to heed the low end instead of the en-tire range from the highest to lowest notes Same for tre-ble vs bright rdquo In particular 4 participants used the wordldquotwangrdquo to describe guitars they classified as treble whileanother guitarist reported that a treble guitar ldquosounds bestfor finger picking and light strumming as heavy playingcan get quite ear-piercing and harsh This last commentseems to echo the second hypothesis for the origin of thestronger LTAS shown by the mid-even guitars (see Sec-tion 345)

Two participants reported looking out for problematicnotes as a more meaningful way of finding bassy guitarsThis likely points toward the ldquoveeringrdquo phenomenon as-sociated with a stronger admittance at lower frequencies

405

ACTA ACUSTICA UNITED WITH ACUSTICA Mansour et al Post-classification of steel string guitarsVol 101 (2015)

[16] Further one musician referred to the tactile feedbackreceived from a vibrating body associating bassy guitarswith more felt vibrations than the other two categoriesThis is a perspective shared by the manufacturer as wellIn fact body vibrations can provide tactile and proprio-ceptive cues that contribute to the perception of the radi-ated sound so that the musician can assess their interactionwith the instrument cross-modally [21 22]

5 Discussion

A set of 18 steel string guitars of the same model man-ufactured with high geometrical tolerances in a modernfactory were evaluated in this study These guitars wereconsidered by the manufacturer to possess subtle yet per-ceivable variations that could be classified as either bassymid-even or treble The results of this study showed thatthe instruments classified as bassy had a consistently lowerT(11)1 breathing mode (see Figure 9a 14b and 15) Thissuggests a lower stiffness-to-weight ratio for the bodies ofthese guitars which is most likely the result of variationsin equivalent stiffness (see Section 35)

It was further shown that whenever the frequency ofT(11)1 was shifted downward or upward for a particu-lar guitar the frequencies of its higher modes consistentlychanged in a similar way (see Figure 9 also the slightlyshifted bumps at around 600 750 and 1200 Hz in Fig-ure 12a) This would not necessarily be the case if theguitars were of different models (ie with varying geome-tries) or of different types of wood (eg plywood versusspruce) This is an important result as there is a practi-cal advantage to having higher-frequency modes move ac-cording to the first corpus mode across guitars of the samemodel T(11)1 is a low-frequency resonance that is suffi-ciently separated from the adjacent modes and hence eas-ily identifiable Measuring its frequency across differentinstruments is easy and affordable in a small workshopenvironment One can simply hit the guitar with a reg-ular hammer and recordanalyse the sound using an in-expensive microphone and a laptop reminiscent of thetraditional tap-tone method used by luthiers Moreoversuch measurements can be very reliable provided that theboundary conditions of the instrument (ie the installedfixture that holds the instrument) as well as the air tem-perature pressure and humidity are kept consistent Theseparameters constitute the biggest source of uncertainty infrequency-based features and can easily be controlled in afactory space or small workshop

Finding a reliable differentiator between the mid-evenand treble guitars was less obvious Although we ob-served that the averaged mobility curves of the mid-eveninstruments tended to be slightly higher in the vicinityof 600ndash2000 Hz than the treble guitars further measure-ments would have been necessary to determine whetherthe weaker admittance of the treble instruments at higherfrequencies would contribute to a longer decay for thehigher partials (see Section 35) In addition the differ-ence in averaged mobility magnitude between the two cat-

egories was only about 2 dB which is close to the marginof error for the measurement system Such a small differ-ence in an amplitude-based feature may be easily obscuredby variations in the exact angle of the hammer location ofhitting and measuring or calibration of the sensors

We invited guitar players of varying musical back-ground and experience to classify the guitars into thesecategories We observed very poor agreement among theparticipants but it is common in recent perceptual studiesof musical instruments that musicians do not agree witheach other [23 24 25] This seems a result of them lis-tening to (or feeling) different aspects of the instrumentand not necessarily all attending to the same features Theemployee that post-classified the guitars at the factory wasthe person that tests every guitar coming off the productionline thus he must have a lot of experience evaluating thesound of these instruments Assuming he did his classifi-cation carefully then there likely were particular things hewas listening for The participants in our perceptual exper-iment would not necessarily listen for the same featuresso it is not too surprising they largely disagreed on theirclassifications That said the differences between the tre-ble and mid-even guitars were extremely subtle and evencompany representatives often mis-categorized them dur-ing informal playing tests And this together with the factthat our measurements showed no consistent differencesbetween the two categories would seem to lead us to ques-tion whether the distinction between mid-even and trebletruly exists at all Another possible explanation is that dif-ferent musicians may follow different processes to assessthose features considered essential for the evaluation of aninstrument [23 26] Accordingly guitar players may agreeof what particular characteristics they look for in a tre-ble guitar but the perception of the same characteristicswidely varies across individuals

6 Conclusions

Efforts to understand how the tonal quality of an instru-ment relates to its measurable physical properties have of-ten been inconclusive (eg [27]) In the light of some re-cent studies the concept of ldquogoodrdquo versus ldquobadrdquo instru-ments is likely too general and it would be more advanta-geous to instead consider specific perceptual attributes ofan instrument [23 28] In the long run it may be possibleto derive a multidimensional timbre space where each di-mension is explained by some measurable quantities andin which instruments with similar qualities are likely tocluster In this study more specific sound characteristicsof instruments were thus targeted rather than their overalltonal quality

From bridge admittance measurements on 18 steelstring guitars from the same production line and post-classified by the manufacturer as either bassy mid-evenor treble (see Section 1 for details) it was shown that itis possible to discriminate between bassy guitars and theother two categories based on the frequency of the breath-ing mode However finding a similarly reliable feature to

406

Mansour et al Post-classification of steel string guitars ACTA ACUSTICA UNITED WITH ACUSTICAVol 101 (2015)

distinguish mid-even from treble instruments is less triv-ial at this point The treble guitars demonstrated a loweraveraged mobility in the frequency range of 600ndash2000 Hzthough more empirical investigation would be necessarybefore drawing any conclusions Results from a percep-tual experiment wherein guitar players were asked to clas-sify the guitars into these categories showed almost noagreement between guitarists in their judgements as wellas between the latter and the classification by the manufac-turer Despite the overall lack of consensus guitar playerstended to agree more on the bassy instruments than theother two categories

Some essential additions to this study are a more care-ful analysis on the perceptual definition of mid-even ver-sus treble and analysing the decay rate of the partials fora plucked sound to see how it correlates with the admit-tance Analysis of this kind has been performed for syn-thesized acoustic guitar sounds (see Figure 7 in [16]) but asimilar analysis on real guitar sounds will involve difficul-ties caused by the double set of harmonics [29] as well aslongitudinal string vibrations (also dubbed ldquophantom par-tialsrdquo) [30]

Acknowledgements

This project was funded by a Natural Sciences and En-gineering Research Council of Canada Engage grant incollaboration with Godin Guitars The authors wish tothank Robert Godin for this opportunity We also thankLawrence Joseph (statistics) and Garry Antonio (guitarperformance) for constructive comments We are grate-ful to Jim Woodhouse for sharing his analysis code Thefirst author would like to acknowledge the Government ofCanada for a Vanier Canada Graduate Scholarship

References

[1] E V Jansson A study of acoustical and hologram inter-ferometric measurements of the top plate vibrations of aguitar Acust Acta Acust 25 (1971) 95ndash100

[2] O Christensen B B Vistisen Simple model for low-frequency guitar function J Acoust Soc Am 68 (1980)758ndash766

[3] O Christensen Quantitative models for low frequency gui-tar function J Guitar Acoustics 6 (1982) 10ndash25

[4] H Wright The acoustics and psychoacoustics of the guitarDissertation Dept of Physics and Astronomy Universityof Wales Cardiff UK 1996

[5] N H Fletcher T D Rossing The physics of musical in-struments Second ed Springer New York 1998

[6] J E Popp Four mass coupled oscillator guitar model JAcoust Soc Am 131 (2012) 829ndash836

[7] J Woodhouse Body vibration of the violinmdashwhat can amaker except to control Catgut Acoust Soc J (Series II)4 (2002) 43ndash49

[8] B Elie F Gautier B David Macro parameters describingthe mechanical behavior of classical guitars J Acoust SocAm 132 (2012) 4013ndash4024

[9] J Woodhouse R S Langley Interpreting the input admit-tance of violins and guitars Acust Acta Acust 98 (2012)611ndash628

[10] A Gabrielsson E V Jansson Long-time-average-spectraand rated qualities of twenty-two violins Acustica 42(1979) 47ndash55

[11] P D Welch The use of Fast Fourier Transform for the es-timation of power spectra A method based on time averag-ing over short modified periodograms IEEE Trans Audioand Electroacoust AU-15 (1967) 70ndash73

[12] J Sundberg The singerrsquos formant revisited J Voice 4(1995) 106ndash119

[13] A Buen On timbre parameters and sound levels ofrecorded old violins J Violin Soc Am VSA Papers XXI(2007) 57ndash68

[14] J Woodhouse On the ldquobridge hillrdquo of the violin AcustActa Acust 91 (2005) 155ndash165

[15] B Richardson The acoustical development of the guitarCatgut Acoustical Society Journal 2 (1994) 1ndash10

[16] J Woodhouse On the synthesis of guitar plucks AcustActa Acust 90 (2004) 928ndash944

[17] C H Hodges J Power J Woodhouse The use of the sono-gram in structural acoustics and an application to the vibra-tions of cylindrical shells J Sound Vib 101 (1985) 203ndash218

[18] L D Broemeling Bayesian methods for measures ofagreement Chapman amp HallCRC New York 2009 (Bio-statistics)

[19] J L Fleiss Measuring nominal scale agreement amongmany raters Psychological Bulletin 76 (1971) 378ndash382

[20] D Lunn C Jackson N Best A Thomas D Spiegelhal-ter The BUGS Book A Practical Introduction to BayesianAnalysis Chapman amp HallCRC New York 2012 (Textsin Statistical Science)

[21] C Saitis Evaluating violin quality Player reliability andverbalization Dissertation Dept of Music ResearchMcGill University Montreal Quebec Canada 2013

[22] I Wollman Perception bimodale des violonistes en situa-tion de jeu Influence des retours auditif et vibrotactile surlrsquoeacutevaluation du violon Dissertation Universiteacute Pierre etMarie Curie Paris France 2013

[23] C Saitis B L Giordano C Fritz G P Scavone Percep-tual evaluation of violins A quantitative analysis of pref-erence judgements by experienced players J Acoust SocAm 132 (2012) 4002ndash4012

[24] C Fritz J Curtin J Poitevineau P Morrel-Samuels F-CTao Player preferences among new and old violins ProcNat Acad Sci USA 109 (2012) 760ndash763

[25] P Eveno J-P Dalmont R Causseacute G P Scavone Anacoustic and perceptual evaluation of saxophone pad ldquores-onatorsrdquo Acta Acustica united with Acustica 101 (2015)246ndash255

[26] C Fritz J Woodhouse F P-H Cheng I Cross A FBlackwell B C J Moore Perceptual studies of violinbody damping and vibrato J Acoust Soc Am 127 (2010)513ndash524

[27] G Bissinger Structural acoustics of good and bad violinsJ Acoust Soc Am 124 (2008) 1764ndash1773

[28] C Saitis C Fritz C Guastavino G P Scavone Con-ceptualization of violin quality by experienced performersProc Stockholm Music Acoust Conf Stockholm Swe-den 2013 123ndash128

[29] J Woodhouse Plucked guitar transients Comparison ofmeasurements and synthesis Acust Acta Acust 90 (2004)945ndash965

[30] B Bank L Sujbert Generation of longitudinal vibrationsin piano strings From physics to sound synthesis JAcoust Soc Am 117 (2004) 2268ndash2278

407

Mansour et al Post-classification of steel string guitars ACTA ACUSTICA UNITED WITH ACUSTICAVol 101 (2015)

Section 5 with thoughts on possible future work that couldbe performed to help clarify the results

2 Background

The energy of a vibrating string is primarily transmittedto the soundboard of the guitar through the bridge Themechanical input admittance at the bridge (or bridge mo-bility) is defined in the frequency domain and in a givendirection as the ratio of the velocity to an input force driv-ing the structure at a given location It provides a descrip-tion of the ability of a point of the sound board to be dis-placed as a function of frequency Therefore this quantityrelates to the ability of the sound board to move and to ra-diate acoustic energy while it also describes the amount ofenergy absorbed from the string which ultimately affectsthe decay rate of string vibration Consequently measure-ments of the bridge mobility are likely to provide relevantinformation with regards to the vibrational characteristicsof an instrument

The top plate of the guitar constitutes the primary radi-ating element of the instrument and thus it plays a veryimportant role in the characteristics of the radiated soundThe analysis of the modal shapes of the lowest frequencyguitar modes reveals that the higher mobility of the topplate occurs on its lower bout the upper bout presentingless mobility [1] In the amplitude curve of the frequencyresponse of the instrument measured at the bridge of theinstrument these low-frequency modes can be easily iden-tified and are commonly designated in reference to theirphysical origins

Different discrete models have been proposed to rep-resent the low frequency and weakly damped modal re-sponse of the guitar body Christensen [2] presented asimple two-degree of freedom model that allows a de-scription of the coupling between the soundbox and theHelmholtz resonance and subsequently a three-degree offreedom model including an additional mass to representthe back plate of the guitar [3 4] That model identifiesthree coupled modes which are the result of the stronginteraction between the first bending modes of the topand back plates as well as the Helmholtz resonance ofthe cavity The first of the three is the breathing modeof the soundbox (hereafter T(11)1) in which the top andback plate move out-of-phase This leads to the expansionand contraction of the soundbox resulting in the air beingsucked in or pushed out through the sound hole respec-tively In the second mode the two plates move in-phasewhile the ribs move in the opposite direction Changingthe boundary condition around the ribs will thus affect thefrequency of this mode fairly strongly (as much as 30 Hz[5]) The third mode (hereafter T(11)2 or anti-breathingmode) is the same as the first but instead the air is beingpushed out when the soundbox expands and vice versaTypically there is an anti-resonance between the breath-ing and the anti-breathing modes at the frequency wherethe pure Helmholtz mode would fall if the soundbox was

rigid Among the three the first and the third modes in-volve a change in the volume of the cavity and thus radiatewell while the second mode does not radiate well Morerecently Popp proposed a four-degree of freedom model[6] By including a fourth oscillator representing the mo-tion of guitar sides and neck this model allows a moreprecise exploration of the influence of rib mass on the lo-cation of the low-frequency resonances There are a fewother modes below 400 Hz that involve a coupling betweenthe top and back plates above that frequency however topplate back plate and air cavity behave relatively indepen-dently

For most structures the bandwidth of individual modesincreases roughly linearly with frequency while the aver-age spacing between adjacent modes remains almost con-stant This results in an increasingly stronger interactionbetween close modes At higher frequencies the modesoverlap to define the response of the system which makesit more meaningful to think about statistical features ratherthan individual modes In this regard Modal Overlap Fac-tor (MOF) is a useful criterion to find the strength of theinteraction between close modes MOF is the ratio of thebandwidth of individual resonance peaks to the averagespacing of adjacent resonances and is expected to grow athigher frequencies [7] Common MOF values for charac-terizing the different frequency ranges are MOF lt 03 forlow frequencies 03 lt MOF lt 1 for mid-frequencies andMOF gt 1 for high frequencies [8] In the higher frequencyrange quantities such as MOF averaged admittance aver-age peak spacing and typical peak-to-valley heights pro-vide useful information that can potentially allow for thedifferentiation between two nearly identical instruments

Recent approaches involve Statistical Energy Analysis(SEA) of the frequency response over a broader frequencyrange [9] and high-resolution modal analysis based on themodeling of the impulse response of a structure as a sum ofcomplex damped sinusoids [8] Using the latter approachElie et al proposed a few macro dynamic features fordiscriminating between different classes of classical gui-tars Although the low-quality (ldquoindustrialrdquo) guitars werecarved out of a different material (plywood) than the high-quality (ldquohand-maderdquo) instruments (made with spruce orred cedar) the results illustrate the potential of that ap-proach to associate global physical quantities of a guitarwith construction details

Long-term average spectra (LTAS) offers an alternativeapproach to the analysis of sound from complex sourcessuch as musical instruments [10] LTAS is the mean of suc-cessive short-term spectra computed over the segments ofa signal This method reduces the noise in the estimatedpower spectra in exchange for reducing the frequency res-olution [11] thereby highlighting longer term aspects ofperceived sound quality LTAS has been used extensivelyfor studying perceived differences in speech and singingvoices (eg [12]) but less often for quantifying instrumen-tal sound quality [10 13]

395

ACTA ACUSTICA UNITED WITH ACUSTICA Mansour et al Post-classification of steel string guitarsVol 101 (2015)

3 Physical measurements

31 Method

311 Input admittanceAs mentioned in the previous section the mechanical ad-mittance measured at the bridge provides useful informa-tion regarding the ability of the sound board to be excitedby the strings We chose to perform measurement of thebridge admittance (or mobility) as well as the radiativity(transfer function between the input force at the bridge andthe acoustic pressure radiated at a given point outside theinstrument) on the set of guitars provided by the manufac-turer As shown in Figure 1 the guitar was held verticallyby means of a wooden stand the guitar was maintainedby two rigid bars covered with foam holding the ribs justbelow the upper bout at the level where the body of the in-strument is the narrowest without exerting too much pres-sure on the ribs The strings were properly tuned and thendamped using a clamp with foam fixed between the 9thand 10th frets The part of the strings above the nut wasalso damped using a piece of cardboard

Energy was injected into the system at the bridge be-tween the low E and A strings using an impact hammer(PCB Model 086E80) mounted on a pendulum allowingrelatively repeatable impacts normal to the surface of thebridge Point mobility measurements require collocatedexcitation and acquisition of the velocity The velocity washence measured as close as possible from the impact pointbetween the E and A strings using a velocimeter (Poly-tec LDV-100) pointing normal to the surface of the bridgeThe particular choice of the bass side of the bridge wasmade due to the fact that some information about the sym-metric and anti-symmetric modes of the body (with re-spect to the longitudinal axis of the instrument) might belost if the measurements were made near the middle ofthe bridge and the lowest string is the one that is moststrongly coupled to the body hence measuring the bridgeadmittance underneath that string can potentially providethe most relevant information about the body A micro-phone (Bruumlel amp Kjaeligr Type 4190-L-001) was placed onemeter from the instrument at the level of the neck to bodyjunction pointing towards the center of the sound holeThe measurements took place in a semi-anechoic roomwith dimensions of 54 times 64 times 36 m3 After appropri-ate pre-amplification of the acquired signals acquisitionwas performed using a National Instruments data acquisi-tion card running at 441 kHz and a Matlab interface Eachmeasurement was of three-second duration after the be-ginning of the impact For each guitar final measurementswere averaged over five repetitions At each acquisitionthe coherence was calculated and displayed to the opera-tor allowing any erroneous measurement to be discardedand repeated if necessary

312 Modal fittingMode fitting was performed using an analysis method andalgorithm developed by Woodhouse [9] This method usesthe Matlab function invfreqs allowing rational fraction

Figure 1 Experimental setup for input admittance and radiativitymeasurements

polynomial approximation It first involves modal extrac-tion through pole-residue fitting This first step is followedby an optimization procedure allowing selection of thebest sets of complex and real residues by minimizing themean of the modulus-squared deviation between measure-ment and reconstruction

313 Recordings and LTAS

For the LTAS all 18 guitars were recorded by a profes-sional guitar player The choice of musical input for therecorded sample sounds can have a significant influenceon the LTAS obtained On each of the guitars we recordeda sequence of 8 strummed chords selected to excite a widerange of frequencies E (E2 B2 E3 G 3 B3 E4) rarr F m7(F 2 E3 F 3 A3 C 4) rarr Emaj7 (E2 B2 D 3 G 3 B3 E4) rarrAE (E2 A2 E3 A3 C 4) rarr F m7 (F 2 C 3 E3 A3 E4) rarrEmaj7 (E2 G 3 B3 D 4 G 4) rarr AE (E3 A3 C 4 E4 A4) rarrE (E3 G 3 B3 E4 G 4) The musician was instructed to playmezzoforte at a tempo of 80 beats per minute

The recordings took place in the same room as the ad-mittance measurements A 12-inch free-field microphone(Bruumlel amp Kjaeligr Type 4190-L-001 with Type 2669-L pream-plifier) with a sound quality conditioning amplifier (Bruumlelamp Kjaeligr Type 2672) was used The microphone was posi-tioned 75 cm from the played guitar facing directly at itstop side between the center of the sound hole and the fin-gerboard The recorded phrases were 30 seconds long andsaved in 16-bit stereo 441 kHz WAV format

396

Mansour et al Post-classification of steel string guitars ACTA ACUSTICA UNITED WITH ACUSTICAVol 101 (2015)

LTAS were obtained using the Matlab function pwelchsymmetric four-term Blackman-Harris windows an 8192-point window size and 50 overlap between the windowsSubsequently the results were smoothed in the frequencydomain using a 500 Hz sliding rectangular window

32 Variation among the instruments

We first evaluated the amount of variation among the 18guitars The precise manufacturing process in the produc-tion of these instruments made it almost impossible to seeany appreciable geometrical difference among the individ-ual instruments except a slight variation in the design oftheir necks (some were intentionally cut more flat and oth-ers more rounded) Therefore the biggest uncertainty fac-tor would be the variation in material properties Althoughthe instruments of our study were all made with sprucetops and mahogany backs and ribs some variation of tim-bre is inevitable

Figure 2 shows the averaged admittance over the pool of18 instruments plusminus one standard deviation A sam-ple instrument from the pool and an outlier are includedfor comparison The outlier was a full-size acoustic guitarfrom the same manufacturer but of a different model Thestandard deviation for all frequency points from 70 Hz to6 kHz averages to 275 dB when calculated for all 18 in-struments The level of variation is slightly higher if welook at the lower range of frequencies (379 dB for the fre-quency range of 70 Hz to 1 kHz) As a reference we calcu-lated the standard deviation for 4 repeated measurementsmade on the same instrument removed and put back on thefixture before each repetition The standard deviation aver-aged to 054 dB for the range 70 Hz to 6 kHz and 094 dBfor the range 70 Hz to 600 Hz Note the different samplesize when comparing these numbers with their equivalentsreported earlier for the 18 instruments

The same analysis can be applied to the instrumentswithin the same perceptual categories (ie bassy mid-even and treble) Figure 3 shows the average admittancefor the instruments of the different categories The av-erage standard deviations within the categories of bassymid-even and treble are respectively 263 dB 236 dB and215 dB for the frequency range of 70 Hz to 6 kHz and329 dB 231 dB 242 dB for the frequency range of 70 Hzto 1 kHz The fact that the variation within each categoryis less than the variation among all the instruments lendssome credibility to the classification made by the manufac-turer Another observation in Figure 3 is that the mid-evenand treble instruments follow almost the same trend below600 Hz while bassy instruments are considerably distinctin the frequency range of 80 Hz to 100 Hz and again from300 Hz to 350 Hz showing a shift of the main resonancestowards lower frequencies

33 General properties of the acoustic guitars

331 Real-imaginary parts

Figure 4 shows the real and imaginary parts of the av-eraged admittances using an averaging bandwidth of

70 100 200 500 1000 2000 6000minus65

minus60

minus55

minus50

minus45

minus40

minus35

minus30

minus25

minus20

Frequency (Hz)

Averagedadmittance

(dB)

A sample instrumentfrom the pool

An outlier

Averaged plusmn one STD

Figure 2 Gray shading shows the averaged admittance over thepool of 18 instruments plusminus one standard deviation Thesolid black line is a sample instrument from the pool and dashedline is an outlier The dB reference is 1 msN

100 200 300 500 1000 2000 6000

minus60

minus55

minus50

minus45

minus40

minus35

minus30

minus25

Frequency (Hz)

Averagedadmittance

(dB)

Averaged bassyAveraged midminusevenAveraged treble

Figure 3 Averaged admittance over the bassy instruments (so-lid) mid-even instruments (dash-dot) and treble instruments(dashed) The dB reference is 1 msN

1000 Hz One can see that the acoustic guitar does not rep-resent the properties of an infinite plate (ie positive realpart with roughly constant magnitude and imaginary partmuch smaller) at least not as well as a classical guitar does[9 8] The magnitude of the imaginary part is smaller thanthe real part but still large enough to be non-negligibleThe real part stays positive over all frequencies which isa property of any passive system The average amplitudeof the real part is smaller by almost a factor of two thanthe same measure for a typical classical guitar it reachesthe typical value for a classical guitar at around 3 kHz andagain drops at higher frequencies A broad peak in the realpart and a tendency in the imaginary part towards negativevalues creates a hill-like feature [14] at around 2800 Hz forthe majority of the instruments with a similar pattern thatrepeats less strongly at around 4 kHz These hills are quiteunexpected as an acoustic guitar does not have a bridge

397

ACTA ACUSTICA UNITED WITH ACUSTICA Mansour et al Post-classification of steel string guitarsVol 101 (2015)

500 1000 1500 2000 2500 3000 3500 4000 45000002

0004

0006

0008

Re(ltYgt)(msN)

500 1000 1500 2000 2500 3000 3500 4000 4500minus0002

minus0001

0

0001

0002

0003

Frequency (Hz)

Im(ltYgt)(msN)

A sample instrumentAveraged plusmn one STD

(a)

(b)

Figure 4 Band-averaged real (a) and imaginary (b) parts of theadmittance for a sample instrument (solid) and averaged overall instruments of the pool plusminus one standard deviation(shaded) Note that the vertical scales are different for the realand imaginary parts

like a violin nor has it a feature like f-holes to create suchlocal resonances A possible explanation is that the cross-bracing of the steel string guitars unlike the fan-bracing ofmost classical guitars circumvents the bridge and formsan isolated area around it and this may contribute to suchlocal resonances

332 Modal spacing and MOFIt is useful to find a practical upper limit for the range ofmode fitting (ie where the MOF reaches one) and to get asense of the average spacing between the modes Figure 5shows the MOF and the average mode spacing for a sam-ple instrument from the pool and the average plusminusone standard deviation over all instruments An averagingbandwidth of 500 Hz is used in this case to give a betterprecision at lower frequencies The statistical features in-cluding MOF are calculated based on variance theory formore details see equations (17) to (19) of [9] As expectedthe general trend of the overlap factor is rising although itfalls a bit between 1500 Hz and 2000 Hz it reaches thevalue of one at around 610 Hz The average mode spac-ing goes unrealistically high (as much as 150 Hz at around200 Hz) at lower frequencies The mode spacing stays al-most constant at around 20 Hz for the frequency range of600ndash1200 Hz The mode spacing drops to around 8 Hz athigher frequencies and then stays almost unchanged

333 Mode fittingUsing the method described in Section 2 a mode fittingprocess was performed on all 18 instruments of the poolup to 600 Hz [9] The fitting program identified 20 to 24modes for different instruments amongst which a maxi-mum of 16 modes were manually selected for each instru-ment The criterion was to keep the stronger modes thatwere also identified for most of the instruments

The general shape of the admittance follows the pre-dictions of [6] the first neck bending at around 65 Hz

600 1000 1500 2000 2500 3000 35000512

510

40

Modaloverlapfactor

600 1000 1500 2000 2500 3000 35000

10

20

30

40

50

Frequency (Hz)

Modalspacing(Hz)

A sample instrumentAveraged plusmn one STD

(b)

MOF = 1atF asymp 610 Hz

(a)

Figure 5 Modal overlap factor (a) and the average mode spacing(b) for a sample instrument from the pool (solid) and the averageplusminus one standard deviation over all instruments (shaded)An averaging bandwidth of 500 Hz is used

the T(11)1 or the breathing mode at around 100 Hz andT(11)2 or anti-breathing mode at around 200 Hz Modesare categorized as (mn) where m and n are the number ofldquohalf-wavesrdquo across and along the plate and T refers to thetop plate As pointed out by Popp [6] the in-phase motionof the top and back plates coincides in frequency with theanti-breathing mode at around 200 Hz

For our measurements the guitars were held in a fixturewith two arms that pressed against the sides of the ribsThis was intended to create a boundary condition simi-lar to that when held by a player Popp argues that a con-siderable amount of mass would be needed to immobilizethe ribs therefore our measurement condition is betweenthe fixed and free-free cases but perhaps more similar tothe latter After the measurements were performed we ob-served a peak (or a double peak for some instruments)at around 125 Hz where one normally expects to see theanti-resonance representing the pure Helmholtz mode [2]When we repeated the measurements on some of the gui-tars using a free-free boundary condition we noticed thatthe anti-resonance appeared as expected This suggests thepeak is either a property of the fixture rather than the in-struments themselves or it is the (11) in-phase motion ofthe top and back plates shifted downward due to the massof the fixture added to the ribs The latter would requirea 60 Hz downward shift of that frequency which seemsunlikely No effort was made to repeat all the measure-ments with a free-free boundary condition partly becausethe measurements were made before subsequent data anal-ysis and partly because free-free measurements are lesspractical and reproducible in a factoryrsquos production line(due to for example variations in accelerometer place-ment and potential damage to a new guitar surface finishwhen attaching and detaching an accelerometer)

The second neck bending mode was evident as a smallpeak at around 225 Hz for all instruments There weretwo other small peaks at around 250 Hz and 290 Hz which

398

Mansour et al Post-classification of steel string guitars ACTA ACUSTICA UNITED WITH ACUSTICAVol 101 (2015)

were probably related to the torsion and lateral vibrationsof the neck The next important modes were T(21) andT(12) For Martin D-28 acoustic guitars these modes arearound 100 Hz apart [5] hence relatively distinct how-ever for the guitars of our study they both appear at around350 Hz and sometimes switch places We measured the ad-mittance at the bass side of the bridge where T(21) hasits antinode therefore for each guitar the stronger modein that region is associated to T(21) In addition to theHelmholtz mode there are two other cavity modes in thefrequency range of our study which are A1 at 430 Hz andA2 at around 500 Hz [5] Approximate mode shapes ofthe first four corpus modes (exclusive of local resonancessuch as neck bending and neck twisting) for the guitars ofour study are illustrated in Figure 6 the frequencies areaveraged and vary from one instrument to another

The average frequency amplitude and Q-factors calcu-lated for the 18 guitars were used to reconstruct the ad-mittance of a ldquomodally-averagedrdquo guitar that representsthe model that we studied Figure 7 shows this ldquomodally-averaged admittancerdquo reconstructed up to 600 Hz (dashedline) compared with a sample measured admittance aswell as the raw averaged admittance previously shown inFigure 2 It can be seen that the peaks are excessivelydamped for the raw average while the admittance of theldquomodally averagedrdquo guitar shows a better resemblance tothe sample measured admittance close to the resonancesand anti-resonances This modal average thus better rep-resents the admittance of a guitar coming off this partic-ular production line It can be overlaid with the measuredadmittance of a new guitar to highlight its significant fea-tures

If needed the admittance of the ldquomodally averagedrdquoguitar can be reconstructed for extended frequencies eitherby simple averaging over different instruments or moreprecisely by using a ldquostatistical fitrdquo method as describedin [16] In short a random number generator can give thefrequency of the high-frequency modes with the correctdensity and spacing statistics as well as damping factorsand modal masses with approximately correct statisticaldistribution Such a statistical fit was not pursued in thisstudy

334 LTAS

Figure 8 shows the LTAS of a sample instrument fromthe pool (solid black line) as well as the average over allinstruments plusminus one standard deviation (shaded)The first few dominant modes identified in an admittancehave demonstrated themselves in the form of broad peaksin the lower frequencies of the LTAS below about 400 HzSome instruments such as the depicted sample show a peakat around 25 kHz which may correspond to the hill-likefeature earlier identified in band-averaged mobility curves(see Figure 4) Beyond 300 Hz the LTAS drops almostlinearly with the average slope of 16 dBoctave The av-erage standard deviation for the frequency range of 70 Hzto 6 kHz for all instruments is 256 dB which indicates theplayer was able to play consistently on all instruments It is

Figure 6 The four first corpus mode shapes of an acoustic guitartaken from [15]

70 100 200 500minus65

minus60

minus55

minus50

minus45

minus40

minus35

minus30

minus25

minus20

Frequency (Hz)

Averagedadmittance

(dB)

A sample measured admittanceModallyminusaveraged guitarAveragedminusoverminusallminusinstruments

Figure 7 Raw averaged admittance over all instruments of thepool (dash-dot) reconstructed ldquomodally-averagedrdquo admittance(dashed see text for details) and a sample admittance for com-parison (solid) The dB reference is 1 msN

worth mentioning that this number is not directly compa-rable to the averaged standard deviation for the measuredadmittance as the LTAS is significantly smoothed in thefrequency domain in comparison

34 Investigating different features for classification

341 Modal propertiesAveraged value and standard deviation of frequency am-plitude and Q factor of the first four corpus modes are

399

ACTA ACUSTICA UNITED WITH ACUSTICA Mansour et al Post-classification of steel string guitarsVol 101 (2015)

calculated for the three perceptual classes and comparedin Figure 9 In all three figures the values are normalizedto the average of the same modal value taken over all in-struments and the error bars show one standard deviationon each side In case of a missing mode for one instru-ment its frequency and Q factor were eliminated from thecalculation of the average and standard deviation but theamplitude was considered to be zero (as that mode wasnegligible or non-existent) The admittance was not com-pensated for added damping from strings [9] partly be-cause the exact same strings were used on all instrumentshence the comparison was fair and partly because the ex-act amount of damping added by the strings is unknownand the assumption of perfect damping at the strings mayresult in over-compensation of damping [9]

As was also evident from Figure 3 Figure 9a shows thatthe corpus modes of the bassy instruments occur at signif-icantly lower frequencies than the mid-even or treble onesAnother feature that distinguishes bassy instruments fromthe mid-even or treble ones can be seen in Figure 9b theamplitude of the first corpus mode T(11)1 is higher forthe bassy instruments The Q factor of the first four cor-pus modes did not show any noticeable difference amongdifferent perceptual classes

342 Averaged admittance and radiativity

To obtain a more clear comparison of different categoriesin the statistical range (ie at higher frequencies) the ad-mittance and radiativity were smoothed in the frequencydomain using a 500 Hz sliding rectangular window andthen averaged for different categories (see Figure 10a andFigure 10b) The proper way to calculate radiativity wouldhave been to measure the averaged sound pressure over anarray of microphones around the instruments That saidour results using only a single microphone provide a rea-sonable approximation because they were smoothed in thefrequency domain and averaged over different instrumentsSince we have already found distinctive features for the in-struments of the bassy class in the subsequent figures theplusminus standard deviation is eliminated for the bassyclass to help clarity

It can be seen in Figure 10a that the band-averaged ad-mittance of the mid-even instruments is greater than orequal to that of the treble instruments over a wide rangeof frequencies particularly in the range 600 Hz to 2 kHzAlthough this dominance does not appear significant formost of the frequencies (ie the averages for categoriesoverlap when plotted with plusminus one standard devia-tion) the mean level of admittance from 600 Hz to 2 kHzcan distinguish mid-even from treble instruments at leastwithin the pool of our 18 instruments This result will befurther discussed in Section 35 The band-averaged admit-tance of the bassy instruments falls between the two othercategories over most of the frequencies

Band-averaged radiativity does not reveal a significantdifference among different categories except a lower av-erage for the bassy guitars in the range from 2 kHz to275 kHz (see Figure 10b) However this difference is

70 100 200 300 500 1000 2000 6000minus180

minus170

minus160

minus150

minus140

minus130

minus120

minus110

minus100

minus90

minus80

Frequency (Hz)

LTAS(dBHz)

A sample instrumentAveraged plusmn one STD

Figure 8 Long-term average spectra for a sample instrumentfrom the pool (solid line) and the average plusminus one stan-dard deviation over all instruments (shaded)

08

09

1

11

Frequency

(norm) (a)

BassyMidminusevenTreble

0

05

1

15

2

25

Amplitude

(norm)

(b)

05

1

15

2

Qminusfactor(norm)

Guitar corpus modes

(c)

T(21)T(11)2

T(11)1 T(12)

Figure 9 Normalized average frequency (a) amplitude (b) andQ factor (c) for the first four corpus modes for bassy guitars (darkgray bars blue in online version) mid-even guitars (light graybars green in online version) and treble guitars (medium graybras red in online version) The error bars show one standarddeviation on each side

likely not important as the standard deviation of the bassyinstruments in this frequency range was found to be muchhigher than the other two categories

343 Damping trend

A method based on time-frequency analysis first intro-duced in [17] and most recently applied in [9] was usedhere to find the trend of the modal damping up to the fre-quency of 6 kHz As mentioned earlier the admittancesare not compensated for added damping from strings Fig-

400

Mansour et al Post-classification of steel string guitars ACTA ACUSTICA UNITED WITH ACUSTICAVol 101 (2015)

500 1000 2000 6000minus54

minus52

minus50

minus48

minus46

minus44

minus42

Bandminusaveraged

admittance

(dB) Bandminusaveraged bassy

Bandminusaveraged midminusevenBandminusaveraged trebleMidminuseven plusmn one STDTreble plusmn one STD

500 1000 2000 6000minus22

minus20

minus18

minus16

minus14

minus12

minus10

minus8

minus6

Frequency (Hz)

Bandminusaveraged

radiativity

(dB)

(a)

(b)

Figure 10 Averaged admittance (a) and radiativity (b) takenover bassy instruments (solid) mid-even instruments (dash-dot)plusminus one standard deviation (light gray shade green in on-line version) and treble instruments (dashed) plusminus onestandard deviation (darker gray shade red in online version)The dB reference is 1 msN

ure 11 shows the averaged damping trend taken over thethree perceptual categories

The Q factor for all classes rises more sharply from Q asymp30 at 100 Hz to Qasymp 60 at 600 Hz where it reaches a plateauand rises more gradually afterwards However the generalbehavior is not far from linear on a log-log scale As itcan be seen in Figure 11a and 11b the value of Q factorover different frequencies and its trend is similar for trebleand mid-even classes which make them indistinguishableby looking at the damping trend From the power-law fitresults the average power for bassy category is 008 andfor both categories of mid-even and treble is 016 This re-sults in an average-over-all-instruments of 013 comparedto 026 and -009 reported earlier for the violins and theclassic guitars respectively in [9] Among the instrumentsof the pool only one showed a negative power fit

344 Harmonic features

Plucking a guitar string excites multiple harmonics simul-taneously which makes it possible that a weighted sum ofthe admittance over some range of harmonics rather thanthe admittance at the fundamental only is a better way ofdistinguishing categories of instruments Forgetting aboutlosses for the moment the amplitude of the transverse

200 300 500 1000 2000 5000

30

40

50

70

Frequency (Hz)

Qminusfactortrend

200 300 500 1000 2000 5000

20

30

40

50

70

100

Bandminusaveraged

Qminusfactor

Averaged bassy

Averaged midminuseven

Averaged treble

Averaged midminuseven plusmn STD

Averaged treble plusmn STD

(b)

(a)

Figure 11 Effective Q-factors averaged for the guitars of differ-ent categories plusminus one standard deviation (solid line forbassy dash-dot line and light gray green shade for mid-evenand dashed line and darker gray red shade for treble) band-average (a) and regression line for power-law behavior (b) Notethat vertical axes are different

force that the nth harmonic of a plucked string applies tothe bridge can be found from [5]

Fn =2dT0

nπL0

β

1 minus βsin(βnπ) (1)

where L0 is the length of the string T0 is its tension d isthe displacement at the plucking point and β is the rela-tive distance of the plucking point from the bridge Thisformula does not take into account the possible strongcoupling of the plucked string to the body and to otherstrings which should be a reasonable approximation forthe intended purpose and for most of the notes played ona guitar [16] The magnitude of the power transfer fromthe nth harmonic of the string to the body is then equalto the bridge velocity multiplied by the applied force orequivalently

Pn = Fnvn = F 2nGω = A

sin(βnπ)n

2

Gω (2)

in which Gω is the real part of the bridge admittance (alsocalled conductance) and A is a constant assuming a par-ticular string is being played with a constant pluckingstrength and at a fixed relative distance from the bridgeand again neglecting the losses This power transfer to thebody is largely responsible for the damping of the string

401

ACTA ACUSTICA UNITED WITH ACUSTICA Mansour et al Post-classification of steel string guitarsVol 101 (2015)

mode in question as well as the radiated sound at that fre-quency Based on this idea two features are calculated forall semitones played on a guitar (from E2 to E6) and aver-aged over the three perceptual categories For both cases βis arbitrarily chosen as 03 which is a reasonable numberfor a normal playing condition

Figure 12a shows the sum of the transferred power bythe first 5 harmonics of each played note here calledldquopower transfer to the bodyrdquo The choice of 5 harmonics isan arbitrary limit for which the individual harmonics of thestring could be perceived individually and within whichmost of the energy transfer to the body happens in a nor-mal playing condition The frequency range that is consid-ered in this fashion would be up to 5 times the highest notestudied (ie E6) or equivalently 66 kHz this is about thelimit up to which the measured admittances show a reason-ably good coherence The unknown constant A in equation(2) could only cause an offset in the results therefore itis considered to be 1 for convenience The energy trans-fer to the body represents the strength of the stringbodycoupling for a guitar A large value for this number wouldresult in a strong and more percussive (banjo-like) soundand a small value would result in a weaker sound with alonger decay (more similar to an electric guitar without anamplifier)

Figure 12b shows a slightly different use of the trans-ferred energy by each harmonic It is calculated by divid-ing the sum of the energy transferred by the 2nd to 5th har-monic to the energy transferred by the fundamental aloneAs well as specifying the ratio of higher harmonic energyto that at the fundamental this feature defines how thesound evolves once the instrument is plucked Extremelylarge or small values for a particular note mean that thenote changes its character when it decays while an aver-age value may be correlated to a more stable sound

Neither graph shows any clear distinction between themid-even and treble categories The bassy category doeshowever stand out most of the patterns in both graphs areshifted downward by a few semitones

345 LTAS and radiativity

The LTAS results averaged over different categories areillustrated in Figure 13a The results for the mid-even in-struments are greater than the other two groups over al-most all frequencies Figure 13b is the same as Figure 13abut the LTAS for each individual instrument is normalizedby its area before further processing After normalizationall categories behave similarly over different frequenciesmeaning mid-even instruments were simply louder thanthe other instruments rather than being stronger in a par-ticular range of frequencies

The band-averaged admittance of the mid-even instru-ments earlier presented in Figure 10 was slightly greaterthan the treble instruments in some frequency rangeshowever the significance of the stronger LTAS of mid-even instruments especially beyond 3 kHz can hardly bejustified by that alone A possible explanation is that themid-even instruments were played systematically harder

E2 100 E3 250 E4 500 E5 1000 E6

minus60

minus55

minus50

minus45

minus40

minus35

minus30

Pow

ertransfertothebody

(dB)

E2 100 E3 250 E4 500 E5 1000 E6

minus20

minus10

0

10

Harmonicstofundam

entalratio(dB)

Fundamental frequency (Hz)

Averaged bassyAveraged midminusevenAveraged treble

(a)

(b)

Figure 12 Power transfer to the body (a) and harmonics to fun-damental ratio (b) calculated for each played note over the op-erational range of a guitar (E2 to E6) the vertical lines indicateoctaves The values are averaged for the guitars of different cat-egories (solid line for bassy dash-dot line and light gray green(online version) shade for mid-even and dashed line and darkergray red (online version) shade for treble)

compared to the treble instruments There are at least tworeasons as to why this might have been the case First theguitars were recorded per category starting with the trebleand ending with the mid-even ones It is therefore possiblethat the musician felt more confident in the course of therecording session Second the mid-even guitars had beendescribed by the manufacturer as having a ldquomore stablerdquosound Such a sense might have allowed the performer toplay those instruments more freely Nevertheless we areaware that the frequency distribution obtained using LTAScan be strongly dependent on the nature of the input sig-nal depending on the sequence played and used as an in-put certain frequency bands may be less excited than oth-ers which may attenuate differences in some frequencyregions of interest

35 Classification results

According to our results finding a feature that can distin-guish bassy instruments is a relatively easy task As wasshown in Figure 9a the frequencies of the first 4 modesfor the bassy instruments are around 5 lower than thosefor the two other classes The frequency of the first modecan be determined without much ambiguity or even theneed for mode extraction All of the other 3 low-frequencymodes have close adjacent modes making it harder andsometimes impossible to find their correct modal parame-ters

The amplitude of the first mode is another indicator fora bassy instrument that can be used for more assurance

402

Mansour et al Post-classification of steel string guitars ACTA ACUSTICA UNITED WITH ACUSTICAVol 101 (2015)

500 1000 2000 6000minus180

minus170

minus160

minus150

minus140

minus130

minus120

LTAS(dBHz)

500 1000 2000 6000minus120

minus110

minus100

minus90

minus80

minus70

minus60

Frequency (Hz)

Normalized

LTAS(dBHz)

BassyMidminusevenTrebleTreble plusmn one STDMidminuseven plusmn one STD

(b)

(a)

Figure 13 Non-normalized LTAS (a) and normalized LTAS (b)averaged for the guitars of different categories plusminus onestandard deviation (solid line for bassy dash-dot line and lightgray green (online version) shade for mid-even and dashed lineand darker gray red (online version) shade for treble)

(see Figure 9b) Figure 14a shows a 2-D plane with thenormalized frequency and amplitude of the first mode onits horizontal and vertical axes respectively It can be seenthat the bassy instruments are quite distinct from the mid-even and treble ones Moreover the guitar with the lowestbreathing mode (B3 fT (11)1 = 91 Hz) was described bythe factory employee as extra-bassy

It should be emphasized that the frequency of the firstpeak works particularly well as a feature not becausethe ldquobassinessrdquo of the instrument is defined by the notesaround this modersquos frequency (only the first three noteson the E2 string are below that frequency) but becauseit represents the overall stiffness-to-weight ratio of thesoundbox that effectively scales the frequency of all bodymodes It is noteworthy that although T(11)1 has a strongHelmholtz component the amount by which its frequencydeviates from the frequency of the pure Helmholtz reso-nance is strongly dependent on the frequency separationof the pure Helmholtz and pure body breathing modesThe frequency of the pure Helmholtz mode is assumed thesame for all instruments of the pool given their very simi-lar geometry (both for the sound box and the sound hole)

The in-plane dimensions of the instruments should havea very small variation percentage given the precise con-trol of geometrical conditions in the factory Moreover alower frequency for the first mode is usually accompaniedby a higher amplitude for that mode which would not nor-mally be the case if such deviation was caused by using a

90 92 94 96 98 100 102 10404

05

06

07

08

09

1

Normalized

T(11) 1am

plitude

BassyMid EvenTreble

90 92 94 96 98 100 102 104minus50

minus49

minus48

minus47

minus46

minus45

B1B2

B3

B6

B4B5

ME6ME1

ME2

ME3ME4

ME5

T2

T3

T4

T5 T6

T1

Frequency of the T(11)1mode (Hz)

Averagedmobility(600minus2000Hz)(dB)

(b)

(a)

Figure 14 Classification of the acoustic guitars (a) by the fre-quency and amplitude of their first corpus mode T(11)1 onlyto distinguish bassy instruments from the others and (b) by thefrequency of their first corpus mode T(11)1 and their averagedmobility from 600 Hz to 2 kHz (B = bassy ME = mid-even T =treble) The dB reference is 1 msN

denser wood this leaves variations in the plate thicknessand the equivalent stiffness (ie very roughly the geomet-ric mean of the Youngrsquos moduli in long-grain and cross-grain directions [14]) as those most likely responsible forthe observed shift in mode frequencies Mode frequenciesof a flat plate scale with Eρ(tL2) where E and ρ arethe elastic constant and density t the thickness and L acharacteristic scaling length If either of the two factors issolely responsible to make a 5 shift it would require ei-ther a 5 variation (asymp 0125 mm) of the plate thicknessor a 10 variation in the equivalent stiffness Given thetight tolerances in cutting the plates and their mechanizedsanding it thus appears that variations in equivalent stiff-ness are the primary reason that some of the instrumentsin the pool sound more bassy In the manufacturing pro-cess a wood slab is sliced to produce several plates Thelateral direction of each plate hence represents a particularcombination of the slabrsquos radial and tangential directions(due to the growth rings of the tree) This can introduce asignificant variation in the platesrsquo stiffness in the lateral di-rection which produces variations of the effective stiffnessof the plates

Finding a feature that distinguishes mid-even from tre-ble instruments is less obvious We found that the aver-aged admittances of the treble instruments in this study areslightly lower than the mid-even instruments in the rangeof 600 Hz to 2 kHz Although this difference is not signif-icant over the mentioned range (ie average plusminus

403

ACTA ACUSTICA UNITED WITH ACUSTICA Mansour et al Post-classification of steel string guitarsVol 101 (2015)

one standard deviation of the two categories overlap) itbecomes significant if the average level of the admittancefor the whole range is compared (ie the average valueof all frequency points of the admittance from 600 Hz to2 kHz) Figure 14b shows a 2-D plane with the frequencyof the first mode on the horizontal axis and the average ad-mittance for the range of 600 Hz to 2 kHz The differencefor the mean of the mid-even and treble categories is about2 dB It seems likely that the weaker admittance of the tre-ble category at higher frequencies would contribute to alonger decay for the higher partials This may contributeto a sharper after-sound of the treble instruments that isknown to allow them to cut through the band more effec-tively This hypothesis could be tested by recording theplucked sound of the instruments and comparing the decayrate of the partials among different categories Due to thelimited project time frame this was not investigated fur-ther The mid-even guitars also show stronger LTAS thanthe treble instruments though this result might have beendue to playing conditions (see Section 345)

We applied the same classification (frequency ofT(11)1 versus average mobility between 600 Hz and2 kHz) to 6 acoustic guitars from a different model by thesame manufacturer (Simon amp Patrick) Unlike the Seagullguitars which were all nominally identical some of theSampP instruments had a slightly different bracing patterndesigned to exemplify the properties of a particular per-ceptual category The manufacturerrsquos classification of the6 guitars was not provided to us prior to our own mea-surements though we were told that there were 2 guitarsof each category The resulting classification based on thediscussed features is depicted in Figure 15 The two gui-tars designed to be more bassy were successfully identi-fied from the frequency of their breathing mode but dis-tinguishing between the mid-even and treble instrumentsfrom the averaged admittance criterion was inconclusive

4 Perceptual evaluation

As previously discussed in Section 1 the investigatedacoustic guitars were post-classified by the manufacturerbased on their experience of what type of instrument mightbetter fit a certain playing stylemdashfor example an amateurmusician that plays mostly strummed chords may prefera bassy guitar while a more experienced guitarist mayopt for a treble instrument We therefore conducted a per-ceptual experiment to investigate the level of agreementamong guitar players of varying experience and musicalbackground in classifying the guitars into the categoriesconsidered by the manufacturer

41 Method

The experiment took place in an acoustically dry roomwith dimensions of 60 times 78 times 32 m3 and a reverberationtime of about 03 s to help minimize the effects of roomreflections on the direct sound from the guitars To avoidthe potential problems of using a common pick across all

85 86 87 88 89 90 91 92minus505

minus50

minus495

minus49

minus485

minus48B1

SampP

B2SampP

ME1SampP

ME2SampP

T1SampP

T2SampP

Frequency of the T(11)1mode (Hz)

Averagedmobility(600minus2000Hz)(dB)

BassyMid EvenTreble

Figure 15 Classification of the six SampP guitars based on thefrequency of the T(11)1 mode versus averaged mobility from600 Hz to 2 kHz (B= bassy ME=mid-even T= treble) The dBreference is 1 msN

participants (eg musicians being uncomfortable with apick they are not familiar with) participants were asked tocarry out the classification task using their own pick(s)

411 Participants

Thirteen guitar players took part in the experiment (1 fe-male 12 males average age = 25 yrs SD = 5 yrs range =19ndash36 yrs) They had at least 6 years of acoustic guitar ex-perience (average years of training = 11 yrs SD = 4 yrsrange = 6ndash20 yrs average hours of practice per week =14 hrs SD = 10 hrs range = 3ndash40 hrs) as well as vary-ing experience with different types of guitars [nylon-string(85) acoustic-electric (62) acoustic bass (15) elec-tric (100) and electric bass (54)] Participants re-ported playing a wide range of musical styles [classical(77) jazz (85) pop-rock (92) country folk blue-grass (66) hard rock heavy metal (46) flamenco(15) latin (46) blues (77) avant-garde contempo-rary (54) and funk alternative (46)] both in a bandand as soloists Seven guitarists (54) reported that theytypically or most often use a pick whereas 4 players (31)reported they do not 2 guitarists (15) reported using apick occasionally rather than typically Nine participantshad higher-level degrees in music performance or othermusic-related areas [BMus (music theory classical gui-tar) BFA (jazz studies) MMus (guitar performance) MA(digital musics) PhD (music theory) Diplocircme drsquoEacutetat deProfesseur de Musique (France)] All guitar players werepaid for their participation

412 Procedure

The experimental session lasted 2 hours and was organizedin two phases In the first phase participants were pre-sented with the guitars randomly ordered on the floor (de-termined by computer calculations) They were asked toplay all instruments for up to 20 minutes in order to famil-iarize themselves with the set and use this time to tune theguitars as needed In the second phase participants were

404

Mansour et al Post-classification of steel string guitars ACTA ACUSTICA UNITED WITH ACUSTICAVol 101 (2015)

given up to 20 minutes to classify each of the guitars as ei-ther bassy mid-even or treble by placing them in speciallymarked areas of the room At the end of the second phaseparticipants were given the opportunity to revise their ini-tial judgements if they so wanted until they were satis-fied with their classification No playing constraint wasimposed on the evaluation process (eg specific repertoireor technique) Participants were instead encouraged to fol-low their own strategy with the suggestion to use bothstrumming-oriented songs (eg Sitting Waiting Wishingby Jack Johnson) and fingerpicking-oriented pieces (egDust in the Wind by Kansas) Participants were instructedto maximize evaluation speed and accuracy Upon com-pleting the session participants talked about their classi-fications and overall experience through written responsesto the following questions ldquoFor each class of guitar whatsound characteristics did you look for Can you describehow you evaluated the guitars in terms of these character-istics (eg musical material technique style)rdquo (Q1) ldquoTowhat extend do you agree with the given classification ofguitarsrdquo (Q2) and ldquoCan you suggest a classification thatbetter reflects how you evaluate the sound characteristicsof guitarsrdquo (Q3)

42 Results

Inter-individual consistency was measured as the Fleissrsquokappa coefficient between guitar classifications from dif-ferent participants using a Bayesian approach to inference[18] Fleissrsquo kappa is a generalization of Cohenrsquos kappastatistic κ a chance-corrected measure of agreement be-tween two assessors to multiple raters [19] It ranges fromminus1 to 1 κ (AB) = 1 and minus1 if A = B and A = minusBrespectively and κ (AB) = 0 in case of no associationbetween A and B (ie exactly what would be expectedby chance) We implemented Gibbs sampling through thesoftware WinBUGS (Windows Bayesian inference UsingGibbs Sampling [20]) with 50000 observations generatedfrom the posterior distribution with a burn-in of 1000 ob-servations A uniformly distributed multicategorical priorwas assumed with total information across all possiblecells (ie from different combinations of participants) be-ing equivalent to a single prior observation (ie a priorsample size equivalent of 1)

Fleissrsquos kappa was estimated with a posterior mean of01 and a 95 credible interval [-02 05] indicating avery low amount of agreement between the participantsin their classifications Figure 16 shows the proportion oftimes a guitar was classified as either bassy mid-even ortreble (the guitars are arranged in three groups accord-ing to the initial factory classification) Whereas someagreement concerning the bassy guitars can be observedmdashindeed the extra bassy guitar (B3) was most often clas-sified as bassymdashthe perceptions for the other two classeslargely vary across different musicians In particular manyguitars that had been characterized as treble by the fac-tory employee were perceived as mid-even Several mu-sicians specifically reported encountering difficulties withthe mid-even category In answering Q3 a player observed

B1 B2 B3 B4 B5 B60

25

50

75

100

classifiedas

(a)BassyMidminusevenTreble

ME1 ME2 ME3 ME4 ME5 ME60

25

50

75

100

classifiedas

(b)

T1 T2 T3 T4 T5 T60

25

50

75

100

classifiedas

Guitars

(c)

Figure 16 Perceptual evaluation Proportion of times a guitarwas classified into one of the categories by the participants Gui-tars are labelled and grouped according to the factory classifica-tion (B = bassy ME = mid-even T = treble) The color scheme(in the online version) corresponds to the classifications providedby the guitar players

that ldquothere is a difference [between] a guitar with domi-nant mid tones and one with an even soundrdquo

Overall participants expressed no strong difficulties incarrying out the classification task On a scale from 1(strongly disagree) to 5 (strongly agree) all performers se-lected 4 when asked ldquoTo what extend do you agree withthe given classification of guitarsrdquo This indicates that the3 categories considered by the manufacturer are gener-ally meaningful to musicians However verbal responsesto questions Q1 and Q3 reveal that participants often as-sociated bassy with more prominent E2 and A2 strings(ie they focused on the low-frequency end of the gui-tar) and similarly treble with better emphasized B3 andE4 strings (ie they paid attention to the high-frequencyrange) As one guitar player noted ldquoI feel that descriptionslike warm neutral and bright would be easier to classifydue to the nature of the instrument Bass leads my mindto bass strings and to heed the low end instead of the en-tire range from the highest to lowest notes Same for tre-ble vs bright rdquo In particular 4 participants used the wordldquotwangrdquo to describe guitars they classified as treble whileanother guitarist reported that a treble guitar ldquosounds bestfor finger picking and light strumming as heavy playingcan get quite ear-piercing and harsh This last commentseems to echo the second hypothesis for the origin of thestronger LTAS shown by the mid-even guitars (see Sec-tion 345)

Two participants reported looking out for problematicnotes as a more meaningful way of finding bassy guitarsThis likely points toward the ldquoveeringrdquo phenomenon as-sociated with a stronger admittance at lower frequencies

405

ACTA ACUSTICA UNITED WITH ACUSTICA Mansour et al Post-classification of steel string guitarsVol 101 (2015)

[16] Further one musician referred to the tactile feedbackreceived from a vibrating body associating bassy guitarswith more felt vibrations than the other two categoriesThis is a perspective shared by the manufacturer as wellIn fact body vibrations can provide tactile and proprio-ceptive cues that contribute to the perception of the radi-ated sound so that the musician can assess their interactionwith the instrument cross-modally [21 22]

5 Discussion

A set of 18 steel string guitars of the same model man-ufactured with high geometrical tolerances in a modernfactory were evaluated in this study These guitars wereconsidered by the manufacturer to possess subtle yet per-ceivable variations that could be classified as either bassymid-even or treble The results of this study showed thatthe instruments classified as bassy had a consistently lowerT(11)1 breathing mode (see Figure 9a 14b and 15) Thissuggests a lower stiffness-to-weight ratio for the bodies ofthese guitars which is most likely the result of variationsin equivalent stiffness (see Section 35)

It was further shown that whenever the frequency ofT(11)1 was shifted downward or upward for a particu-lar guitar the frequencies of its higher modes consistentlychanged in a similar way (see Figure 9 also the slightlyshifted bumps at around 600 750 and 1200 Hz in Fig-ure 12a) This would not necessarily be the case if theguitars were of different models (ie with varying geome-tries) or of different types of wood (eg plywood versusspruce) This is an important result as there is a practi-cal advantage to having higher-frequency modes move ac-cording to the first corpus mode across guitars of the samemodel T(11)1 is a low-frequency resonance that is suffi-ciently separated from the adjacent modes and hence eas-ily identifiable Measuring its frequency across differentinstruments is easy and affordable in a small workshopenvironment One can simply hit the guitar with a reg-ular hammer and recordanalyse the sound using an in-expensive microphone and a laptop reminiscent of thetraditional tap-tone method used by luthiers Moreoversuch measurements can be very reliable provided that theboundary conditions of the instrument (ie the installedfixture that holds the instrument) as well as the air tem-perature pressure and humidity are kept consistent Theseparameters constitute the biggest source of uncertainty infrequency-based features and can easily be controlled in afactory space or small workshop

Finding a reliable differentiator between the mid-evenand treble guitars was less obvious Although we ob-served that the averaged mobility curves of the mid-eveninstruments tended to be slightly higher in the vicinityof 600ndash2000 Hz than the treble guitars further measure-ments would have been necessary to determine whetherthe weaker admittance of the treble instruments at higherfrequencies would contribute to a longer decay for thehigher partials (see Section 35) In addition the differ-ence in averaged mobility magnitude between the two cat-

egories was only about 2 dB which is close to the marginof error for the measurement system Such a small differ-ence in an amplitude-based feature may be easily obscuredby variations in the exact angle of the hammer location ofhitting and measuring or calibration of the sensors

We invited guitar players of varying musical back-ground and experience to classify the guitars into thesecategories We observed very poor agreement among theparticipants but it is common in recent perceptual studiesof musical instruments that musicians do not agree witheach other [23 24 25] This seems a result of them lis-tening to (or feeling) different aspects of the instrumentand not necessarily all attending to the same features Theemployee that post-classified the guitars at the factory wasthe person that tests every guitar coming off the productionline thus he must have a lot of experience evaluating thesound of these instruments Assuming he did his classifi-cation carefully then there likely were particular things hewas listening for The participants in our perceptual exper-iment would not necessarily listen for the same featuresso it is not too surprising they largely disagreed on theirclassifications That said the differences between the tre-ble and mid-even guitars were extremely subtle and evencompany representatives often mis-categorized them dur-ing informal playing tests And this together with the factthat our measurements showed no consistent differencesbetween the two categories would seem to lead us to ques-tion whether the distinction between mid-even and trebletruly exists at all Another possible explanation is that dif-ferent musicians may follow different processes to assessthose features considered essential for the evaluation of aninstrument [23 26] Accordingly guitar players may agreeof what particular characteristics they look for in a tre-ble guitar but the perception of the same characteristicswidely varies across individuals

6 Conclusions

Efforts to understand how the tonal quality of an instru-ment relates to its measurable physical properties have of-ten been inconclusive (eg [27]) In the light of some re-cent studies the concept of ldquogoodrdquo versus ldquobadrdquo instru-ments is likely too general and it would be more advanta-geous to instead consider specific perceptual attributes ofan instrument [23 28] In the long run it may be possibleto derive a multidimensional timbre space where each di-mension is explained by some measurable quantities andin which instruments with similar qualities are likely tocluster In this study more specific sound characteristicsof instruments were thus targeted rather than their overalltonal quality

From bridge admittance measurements on 18 steelstring guitars from the same production line and post-classified by the manufacturer as either bassy mid-evenor treble (see Section 1 for details) it was shown that itis possible to discriminate between bassy guitars and theother two categories based on the frequency of the breath-ing mode However finding a similarly reliable feature to

406

Mansour et al Post-classification of steel string guitars ACTA ACUSTICA UNITED WITH ACUSTICAVol 101 (2015)

distinguish mid-even from treble instruments is less triv-ial at this point The treble guitars demonstrated a loweraveraged mobility in the frequency range of 600ndash2000 Hzthough more empirical investigation would be necessarybefore drawing any conclusions Results from a percep-tual experiment wherein guitar players were asked to clas-sify the guitars into these categories showed almost noagreement between guitarists in their judgements as wellas between the latter and the classification by the manufac-turer Despite the overall lack of consensus guitar playerstended to agree more on the bassy instruments than theother two categories

Some essential additions to this study are a more care-ful analysis on the perceptual definition of mid-even ver-sus treble and analysing the decay rate of the partials fora plucked sound to see how it correlates with the admit-tance Analysis of this kind has been performed for syn-thesized acoustic guitar sounds (see Figure 7 in [16]) but asimilar analysis on real guitar sounds will involve difficul-ties caused by the double set of harmonics [29] as well aslongitudinal string vibrations (also dubbed ldquophantom par-tialsrdquo) [30]

Acknowledgements

This project was funded by a Natural Sciences and En-gineering Research Council of Canada Engage grant incollaboration with Godin Guitars The authors wish tothank Robert Godin for this opportunity We also thankLawrence Joseph (statistics) and Garry Antonio (guitarperformance) for constructive comments We are grate-ful to Jim Woodhouse for sharing his analysis code Thefirst author would like to acknowledge the Government ofCanada for a Vanier Canada Graduate Scholarship

References

[1] E V Jansson A study of acoustical and hologram inter-ferometric measurements of the top plate vibrations of aguitar Acust Acta Acust 25 (1971) 95ndash100

[2] O Christensen B B Vistisen Simple model for low-frequency guitar function J Acoust Soc Am 68 (1980)758ndash766

[3] O Christensen Quantitative models for low frequency gui-tar function J Guitar Acoustics 6 (1982) 10ndash25

[4] H Wright The acoustics and psychoacoustics of the guitarDissertation Dept of Physics and Astronomy Universityof Wales Cardiff UK 1996

[5] N H Fletcher T D Rossing The physics of musical in-struments Second ed Springer New York 1998

[6] J E Popp Four mass coupled oscillator guitar model JAcoust Soc Am 131 (2012) 829ndash836

[7] J Woodhouse Body vibration of the violinmdashwhat can amaker except to control Catgut Acoust Soc J (Series II)4 (2002) 43ndash49

[8] B Elie F Gautier B David Macro parameters describingthe mechanical behavior of classical guitars J Acoust SocAm 132 (2012) 4013ndash4024

[9] J Woodhouse R S Langley Interpreting the input admit-tance of violins and guitars Acust Acta Acust 98 (2012)611ndash628

[10] A Gabrielsson E V Jansson Long-time-average-spectraand rated qualities of twenty-two violins Acustica 42(1979) 47ndash55

[11] P D Welch The use of Fast Fourier Transform for the es-timation of power spectra A method based on time averag-ing over short modified periodograms IEEE Trans Audioand Electroacoust AU-15 (1967) 70ndash73

[12] J Sundberg The singerrsquos formant revisited J Voice 4(1995) 106ndash119

[13] A Buen On timbre parameters and sound levels ofrecorded old violins J Violin Soc Am VSA Papers XXI(2007) 57ndash68

[14] J Woodhouse On the ldquobridge hillrdquo of the violin AcustActa Acust 91 (2005) 155ndash165

[15] B Richardson The acoustical development of the guitarCatgut Acoustical Society Journal 2 (1994) 1ndash10

[16] J Woodhouse On the synthesis of guitar plucks AcustActa Acust 90 (2004) 928ndash944

[17] C H Hodges J Power J Woodhouse The use of the sono-gram in structural acoustics and an application to the vibra-tions of cylindrical shells J Sound Vib 101 (1985) 203ndash218

[18] L D Broemeling Bayesian methods for measures ofagreement Chapman amp HallCRC New York 2009 (Bio-statistics)

[19] J L Fleiss Measuring nominal scale agreement amongmany raters Psychological Bulletin 76 (1971) 378ndash382

[20] D Lunn C Jackson N Best A Thomas D Spiegelhal-ter The BUGS Book A Practical Introduction to BayesianAnalysis Chapman amp HallCRC New York 2012 (Textsin Statistical Science)

[21] C Saitis Evaluating violin quality Player reliability andverbalization Dissertation Dept of Music ResearchMcGill University Montreal Quebec Canada 2013

[22] I Wollman Perception bimodale des violonistes en situa-tion de jeu Influence des retours auditif et vibrotactile surlrsquoeacutevaluation du violon Dissertation Universiteacute Pierre etMarie Curie Paris France 2013

[23] C Saitis B L Giordano C Fritz G P Scavone Percep-tual evaluation of violins A quantitative analysis of pref-erence judgements by experienced players J Acoust SocAm 132 (2012) 4002ndash4012

[24] C Fritz J Curtin J Poitevineau P Morrel-Samuels F-CTao Player preferences among new and old violins ProcNat Acad Sci USA 109 (2012) 760ndash763

[25] P Eveno J-P Dalmont R Causseacute G P Scavone Anacoustic and perceptual evaluation of saxophone pad ldquores-onatorsrdquo Acta Acustica united with Acustica 101 (2015)246ndash255

[26] C Fritz J Woodhouse F P-H Cheng I Cross A FBlackwell B C J Moore Perceptual studies of violinbody damping and vibrato J Acoust Soc Am 127 (2010)513ndash524

[27] G Bissinger Structural acoustics of good and bad violinsJ Acoust Soc Am 124 (2008) 1764ndash1773

[28] C Saitis C Fritz C Guastavino G P Scavone Con-ceptualization of violin quality by experienced performersProc Stockholm Music Acoust Conf Stockholm Swe-den 2013 123ndash128

[29] J Woodhouse Plucked guitar transients Comparison ofmeasurements and synthesis Acust Acta Acust 90 (2004)945ndash965

[30] B Bank L Sujbert Generation of longitudinal vibrationsin piano strings From physics to sound synthesis JAcoust Soc Am 117 (2004) 2268ndash2278

407

ACTA ACUSTICA UNITED WITH ACUSTICA Mansour et al Post-classification of steel string guitarsVol 101 (2015)

3 Physical measurements

31 Method

311 Input admittanceAs mentioned in the previous section the mechanical ad-mittance measured at the bridge provides useful informa-tion regarding the ability of the sound board to be excitedby the strings We chose to perform measurement of thebridge admittance (or mobility) as well as the radiativity(transfer function between the input force at the bridge andthe acoustic pressure radiated at a given point outside theinstrument) on the set of guitars provided by the manufac-turer As shown in Figure 1 the guitar was held verticallyby means of a wooden stand the guitar was maintainedby two rigid bars covered with foam holding the ribs justbelow the upper bout at the level where the body of the in-strument is the narrowest without exerting too much pres-sure on the ribs The strings were properly tuned and thendamped using a clamp with foam fixed between the 9thand 10th frets The part of the strings above the nut wasalso damped using a piece of cardboard

Energy was injected into the system at the bridge be-tween the low E and A strings using an impact hammer(PCB Model 086E80) mounted on a pendulum allowingrelatively repeatable impacts normal to the surface of thebridge Point mobility measurements require collocatedexcitation and acquisition of the velocity The velocity washence measured as close as possible from the impact pointbetween the E and A strings using a velocimeter (Poly-tec LDV-100) pointing normal to the surface of the bridgeThe particular choice of the bass side of the bridge wasmade due to the fact that some information about the sym-metric and anti-symmetric modes of the body (with re-spect to the longitudinal axis of the instrument) might belost if the measurements were made near the middle ofthe bridge and the lowest string is the one that is moststrongly coupled to the body hence measuring the bridgeadmittance underneath that string can potentially providethe most relevant information about the body A micro-phone (Bruumlel amp Kjaeligr Type 4190-L-001) was placed onemeter from the instrument at the level of the neck to bodyjunction pointing towards the center of the sound holeThe measurements took place in a semi-anechoic roomwith dimensions of 54 times 64 times 36 m3 After appropri-ate pre-amplification of the acquired signals acquisitionwas performed using a National Instruments data acquisi-tion card running at 441 kHz and a Matlab interface Eachmeasurement was of three-second duration after the be-ginning of the impact For each guitar final measurementswere averaged over five repetitions At each acquisitionthe coherence was calculated and displayed to the opera-tor allowing any erroneous measurement to be discardedand repeated if necessary

312 Modal fittingMode fitting was performed using an analysis method andalgorithm developed by Woodhouse [9] This method usesthe Matlab function invfreqs allowing rational fraction

Figure 1 Experimental setup for input admittance and radiativitymeasurements

polynomial approximation It first involves modal extrac-tion through pole-residue fitting This first step is followedby an optimization procedure allowing selection of thebest sets of complex and real residues by minimizing themean of the modulus-squared deviation between measure-ment and reconstruction

313 Recordings and LTAS

For the LTAS all 18 guitars were recorded by a profes-sional guitar player The choice of musical input for therecorded sample sounds can have a significant influenceon the LTAS obtained On each of the guitars we recordeda sequence of 8 strummed chords selected to excite a widerange of frequencies E (E2 B2 E3 G 3 B3 E4) rarr F m7(F 2 E3 F 3 A3 C 4) rarr Emaj7 (E2 B2 D 3 G 3 B3 E4) rarrAE (E2 A2 E3 A3 C 4) rarr F m7 (F 2 C 3 E3 A3 E4) rarrEmaj7 (E2 G 3 B3 D 4 G 4) rarr AE (E3 A3 C 4 E4 A4) rarrE (E3 G 3 B3 E4 G 4) The musician was instructed to playmezzoforte at a tempo of 80 beats per minute

The recordings took place in the same room as the ad-mittance measurements A 12-inch free-field microphone(Bruumlel amp Kjaeligr Type 4190-L-001 with Type 2669-L pream-plifier) with a sound quality conditioning amplifier (Bruumlelamp Kjaeligr Type 2672) was used The microphone was posi-tioned 75 cm from the played guitar facing directly at itstop side between the center of the sound hole and the fin-gerboard The recorded phrases were 30 seconds long andsaved in 16-bit stereo 441 kHz WAV format

396

Mansour et al Post-classification of steel string guitars ACTA ACUSTICA UNITED WITH ACUSTICAVol 101 (2015)

LTAS were obtained using the Matlab function pwelchsymmetric four-term Blackman-Harris windows an 8192-point window size and 50 overlap between the windowsSubsequently the results were smoothed in the frequencydomain using a 500 Hz sliding rectangular window

32 Variation among the instruments

We first evaluated the amount of variation among the 18guitars The precise manufacturing process in the produc-tion of these instruments made it almost impossible to seeany appreciable geometrical difference among the individ-ual instruments except a slight variation in the design oftheir necks (some were intentionally cut more flat and oth-ers more rounded) Therefore the biggest uncertainty fac-tor would be the variation in material properties Althoughthe instruments of our study were all made with sprucetops and mahogany backs and ribs some variation of tim-bre is inevitable

Figure 2 shows the averaged admittance over the pool of18 instruments plusminus one standard deviation A sam-ple instrument from the pool and an outlier are includedfor comparison The outlier was a full-size acoustic guitarfrom the same manufacturer but of a different model Thestandard deviation for all frequency points from 70 Hz to6 kHz averages to 275 dB when calculated for all 18 in-struments The level of variation is slightly higher if welook at the lower range of frequencies (379 dB for the fre-quency range of 70 Hz to 1 kHz) As a reference we calcu-lated the standard deviation for 4 repeated measurementsmade on the same instrument removed and put back on thefixture before each repetition The standard deviation aver-aged to 054 dB for the range 70 Hz to 6 kHz and 094 dBfor the range 70 Hz to 600 Hz Note the different samplesize when comparing these numbers with their equivalentsreported earlier for the 18 instruments

The same analysis can be applied to the instrumentswithin the same perceptual categories (ie bassy mid-even and treble) Figure 3 shows the average admittancefor the instruments of the different categories The av-erage standard deviations within the categories of bassymid-even and treble are respectively 263 dB 236 dB and215 dB for the frequency range of 70 Hz to 6 kHz and329 dB 231 dB 242 dB for the frequency range of 70 Hzto 1 kHz The fact that the variation within each categoryis less than the variation among all the instruments lendssome credibility to the classification made by the manufac-turer Another observation in Figure 3 is that the mid-evenand treble instruments follow almost the same trend below600 Hz while bassy instruments are considerably distinctin the frequency range of 80 Hz to 100 Hz and again from300 Hz to 350 Hz showing a shift of the main resonancestowards lower frequencies

33 General properties of the acoustic guitars

331 Real-imaginary parts

Figure 4 shows the real and imaginary parts of the av-eraged admittances using an averaging bandwidth of

70 100 200 500 1000 2000 6000minus65

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minus20

Frequency (Hz)

Averagedadmittance

(dB)

A sample instrumentfrom the pool

An outlier

Averaged plusmn one STD

Figure 2 Gray shading shows the averaged admittance over thepool of 18 instruments plusminus one standard deviation Thesolid black line is a sample instrument from the pool and dashedline is an outlier The dB reference is 1 msN

100 200 300 500 1000 2000 6000

minus60

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minus50

minus45

minus40

minus35

minus30

minus25

Frequency (Hz)

Averagedadmittance

(dB)

Averaged bassyAveraged midminusevenAveraged treble

Figure 3 Averaged admittance over the bassy instruments (so-lid) mid-even instruments (dash-dot) and treble instruments(dashed) The dB reference is 1 msN

1000 Hz One can see that the acoustic guitar does not rep-resent the properties of an infinite plate (ie positive realpart with roughly constant magnitude and imaginary partmuch smaller) at least not as well as a classical guitar does[9 8] The magnitude of the imaginary part is smaller thanthe real part but still large enough to be non-negligibleThe real part stays positive over all frequencies which isa property of any passive system The average amplitudeof the real part is smaller by almost a factor of two thanthe same measure for a typical classical guitar it reachesthe typical value for a classical guitar at around 3 kHz andagain drops at higher frequencies A broad peak in the realpart and a tendency in the imaginary part towards negativevalues creates a hill-like feature [14] at around 2800 Hz forthe majority of the instruments with a similar pattern thatrepeats less strongly at around 4 kHz These hills are quiteunexpected as an acoustic guitar does not have a bridge

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ACTA ACUSTICA UNITED WITH ACUSTICA Mansour et al Post-classification of steel string guitarsVol 101 (2015)

500 1000 1500 2000 2500 3000 3500 4000 45000002

0004

0006

0008

Re(ltYgt)(msN)

500 1000 1500 2000 2500 3000 3500 4000 4500minus0002

minus0001

0

0001

0002

0003

Frequency (Hz)

Im(ltYgt)(msN)

A sample instrumentAveraged plusmn one STD

(a)

(b)

Figure 4 Band-averaged real (a) and imaginary (b) parts of theadmittance for a sample instrument (solid) and averaged overall instruments of the pool plusminus one standard deviation(shaded) Note that the vertical scales are different for the realand imaginary parts

like a violin nor has it a feature like f-holes to create suchlocal resonances A possible explanation is that the cross-bracing of the steel string guitars unlike the fan-bracing ofmost classical guitars circumvents the bridge and formsan isolated area around it and this may contribute to suchlocal resonances

332 Modal spacing and MOFIt is useful to find a practical upper limit for the range ofmode fitting (ie where the MOF reaches one) and to get asense of the average spacing between the modes Figure 5shows the MOF and the average mode spacing for a sam-ple instrument from the pool and the average plusminusone standard deviation over all instruments An averagingbandwidth of 500 Hz is used in this case to give a betterprecision at lower frequencies The statistical features in-cluding MOF are calculated based on variance theory formore details see equations (17) to (19) of [9] As expectedthe general trend of the overlap factor is rising although itfalls a bit between 1500 Hz and 2000 Hz it reaches thevalue of one at around 610 Hz The average mode spac-ing goes unrealistically high (as much as 150 Hz at around200 Hz) at lower frequencies The mode spacing stays al-most constant at around 20 Hz for the frequency range of600ndash1200 Hz The mode spacing drops to around 8 Hz athigher frequencies and then stays almost unchanged

333 Mode fittingUsing the method described in Section 2 a mode fittingprocess was performed on all 18 instruments of the poolup to 600 Hz [9] The fitting program identified 20 to 24modes for different instruments amongst which a maxi-mum of 16 modes were manually selected for each instru-ment The criterion was to keep the stronger modes thatwere also identified for most of the instruments

The general shape of the admittance follows the pre-dictions of [6] the first neck bending at around 65 Hz

600 1000 1500 2000 2500 3000 35000512

510

40

Modaloverlapfactor

600 1000 1500 2000 2500 3000 35000

10

20

30

40

50

Frequency (Hz)

Modalspacing(Hz)

A sample instrumentAveraged plusmn one STD

(b)

MOF = 1atF asymp 610 Hz

(a)

Figure 5 Modal overlap factor (a) and the average mode spacing(b) for a sample instrument from the pool (solid) and the averageplusminus one standard deviation over all instruments (shaded)An averaging bandwidth of 500 Hz is used

the T(11)1 or the breathing mode at around 100 Hz andT(11)2 or anti-breathing mode at around 200 Hz Modesare categorized as (mn) where m and n are the number ofldquohalf-wavesrdquo across and along the plate and T refers to thetop plate As pointed out by Popp [6] the in-phase motionof the top and back plates coincides in frequency with theanti-breathing mode at around 200 Hz

For our measurements the guitars were held in a fixturewith two arms that pressed against the sides of the ribsThis was intended to create a boundary condition simi-lar to that when held by a player Popp argues that a con-siderable amount of mass would be needed to immobilizethe ribs therefore our measurement condition is betweenthe fixed and free-free cases but perhaps more similar tothe latter After the measurements were performed we ob-served a peak (or a double peak for some instruments)at around 125 Hz where one normally expects to see theanti-resonance representing the pure Helmholtz mode [2]When we repeated the measurements on some of the gui-tars using a free-free boundary condition we noticed thatthe anti-resonance appeared as expected This suggests thepeak is either a property of the fixture rather than the in-struments themselves or it is the (11) in-phase motion ofthe top and back plates shifted downward due to the massof the fixture added to the ribs The latter would requirea 60 Hz downward shift of that frequency which seemsunlikely No effort was made to repeat all the measure-ments with a free-free boundary condition partly becausethe measurements were made before subsequent data anal-ysis and partly because free-free measurements are lesspractical and reproducible in a factoryrsquos production line(due to for example variations in accelerometer place-ment and potential damage to a new guitar surface finishwhen attaching and detaching an accelerometer)

The second neck bending mode was evident as a smallpeak at around 225 Hz for all instruments There weretwo other small peaks at around 250 Hz and 290 Hz which

398

Mansour et al Post-classification of steel string guitars ACTA ACUSTICA UNITED WITH ACUSTICAVol 101 (2015)

were probably related to the torsion and lateral vibrationsof the neck The next important modes were T(21) andT(12) For Martin D-28 acoustic guitars these modes arearound 100 Hz apart [5] hence relatively distinct how-ever for the guitars of our study they both appear at around350 Hz and sometimes switch places We measured the ad-mittance at the bass side of the bridge where T(21) hasits antinode therefore for each guitar the stronger modein that region is associated to T(21) In addition to theHelmholtz mode there are two other cavity modes in thefrequency range of our study which are A1 at 430 Hz andA2 at around 500 Hz [5] Approximate mode shapes ofthe first four corpus modes (exclusive of local resonancessuch as neck bending and neck twisting) for the guitars ofour study are illustrated in Figure 6 the frequencies areaveraged and vary from one instrument to another

The average frequency amplitude and Q-factors calcu-lated for the 18 guitars were used to reconstruct the ad-mittance of a ldquomodally-averagedrdquo guitar that representsthe model that we studied Figure 7 shows this ldquomodally-averaged admittancerdquo reconstructed up to 600 Hz (dashedline) compared with a sample measured admittance aswell as the raw averaged admittance previously shown inFigure 2 It can be seen that the peaks are excessivelydamped for the raw average while the admittance of theldquomodally averagedrdquo guitar shows a better resemblance tothe sample measured admittance close to the resonancesand anti-resonances This modal average thus better rep-resents the admittance of a guitar coming off this partic-ular production line It can be overlaid with the measuredadmittance of a new guitar to highlight its significant fea-tures

If needed the admittance of the ldquomodally averagedrdquoguitar can be reconstructed for extended frequencies eitherby simple averaging over different instruments or moreprecisely by using a ldquostatistical fitrdquo method as describedin [16] In short a random number generator can give thefrequency of the high-frequency modes with the correctdensity and spacing statistics as well as damping factorsand modal masses with approximately correct statisticaldistribution Such a statistical fit was not pursued in thisstudy

334 LTAS

Figure 8 shows the LTAS of a sample instrument fromthe pool (solid black line) as well as the average over allinstruments plusminus one standard deviation (shaded)The first few dominant modes identified in an admittancehave demonstrated themselves in the form of broad peaksin the lower frequencies of the LTAS below about 400 HzSome instruments such as the depicted sample show a peakat around 25 kHz which may correspond to the hill-likefeature earlier identified in band-averaged mobility curves(see Figure 4) Beyond 300 Hz the LTAS drops almostlinearly with the average slope of 16 dBoctave The av-erage standard deviation for the frequency range of 70 Hzto 6 kHz for all instruments is 256 dB which indicates theplayer was able to play consistently on all instruments It is

Figure 6 The four first corpus mode shapes of an acoustic guitartaken from [15]

70 100 200 500minus65

minus60

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minus40

minus35

minus30

minus25

minus20

Frequency (Hz)

Averagedadmittance

(dB)

A sample measured admittanceModallyminusaveraged guitarAveragedminusoverminusallminusinstruments

Figure 7 Raw averaged admittance over all instruments of thepool (dash-dot) reconstructed ldquomodally-averagedrdquo admittance(dashed see text for details) and a sample admittance for com-parison (solid) The dB reference is 1 msN

worth mentioning that this number is not directly compa-rable to the averaged standard deviation for the measuredadmittance as the LTAS is significantly smoothed in thefrequency domain in comparison

34 Investigating different features for classification

341 Modal propertiesAveraged value and standard deviation of frequency am-plitude and Q factor of the first four corpus modes are

399

ACTA ACUSTICA UNITED WITH ACUSTICA Mansour et al Post-classification of steel string guitarsVol 101 (2015)

calculated for the three perceptual classes and comparedin Figure 9 In all three figures the values are normalizedto the average of the same modal value taken over all in-struments and the error bars show one standard deviationon each side In case of a missing mode for one instru-ment its frequency and Q factor were eliminated from thecalculation of the average and standard deviation but theamplitude was considered to be zero (as that mode wasnegligible or non-existent) The admittance was not com-pensated for added damping from strings [9] partly be-cause the exact same strings were used on all instrumentshence the comparison was fair and partly because the ex-act amount of damping added by the strings is unknownand the assumption of perfect damping at the strings mayresult in over-compensation of damping [9]

As was also evident from Figure 3 Figure 9a shows thatthe corpus modes of the bassy instruments occur at signif-icantly lower frequencies than the mid-even or treble onesAnother feature that distinguishes bassy instruments fromthe mid-even or treble ones can be seen in Figure 9b theamplitude of the first corpus mode T(11)1 is higher forthe bassy instruments The Q factor of the first four cor-pus modes did not show any noticeable difference amongdifferent perceptual classes

342 Averaged admittance and radiativity

To obtain a more clear comparison of different categoriesin the statistical range (ie at higher frequencies) the ad-mittance and radiativity were smoothed in the frequencydomain using a 500 Hz sliding rectangular window andthen averaged for different categories (see Figure 10a andFigure 10b) The proper way to calculate radiativity wouldhave been to measure the averaged sound pressure over anarray of microphones around the instruments That saidour results using only a single microphone provide a rea-sonable approximation because they were smoothed in thefrequency domain and averaged over different instrumentsSince we have already found distinctive features for the in-struments of the bassy class in the subsequent figures theplusminus standard deviation is eliminated for the bassyclass to help clarity

It can be seen in Figure 10a that the band-averaged ad-mittance of the mid-even instruments is greater than orequal to that of the treble instruments over a wide rangeof frequencies particularly in the range 600 Hz to 2 kHzAlthough this dominance does not appear significant formost of the frequencies (ie the averages for categoriesoverlap when plotted with plusminus one standard devia-tion) the mean level of admittance from 600 Hz to 2 kHzcan distinguish mid-even from treble instruments at leastwithin the pool of our 18 instruments This result will befurther discussed in Section 35 The band-averaged admit-tance of the bassy instruments falls between the two othercategories over most of the frequencies

Band-averaged radiativity does not reveal a significantdifference among different categories except a lower av-erage for the bassy guitars in the range from 2 kHz to275 kHz (see Figure 10b) However this difference is

70 100 200 300 500 1000 2000 6000minus180

minus170

minus160

minus150

minus140

minus130

minus120

minus110

minus100

minus90

minus80

Frequency (Hz)

LTAS(dBHz)

A sample instrumentAveraged plusmn one STD

Figure 8 Long-term average spectra for a sample instrumentfrom the pool (solid line) and the average plusminus one stan-dard deviation over all instruments (shaded)

08

09

1

11

Frequency

(norm) (a)

BassyMidminusevenTreble

0

05

1

15

2

25

Amplitude

(norm)

(b)

05

1

15

2

Qminusfactor(norm)

Guitar corpus modes

(c)

T(21)T(11)2

T(11)1 T(12)

Figure 9 Normalized average frequency (a) amplitude (b) andQ factor (c) for the first four corpus modes for bassy guitars (darkgray bars blue in online version) mid-even guitars (light graybars green in online version) and treble guitars (medium graybras red in online version) The error bars show one standarddeviation on each side

likely not important as the standard deviation of the bassyinstruments in this frequency range was found to be muchhigher than the other two categories

343 Damping trend

A method based on time-frequency analysis first intro-duced in [17] and most recently applied in [9] was usedhere to find the trend of the modal damping up to the fre-quency of 6 kHz As mentioned earlier the admittancesare not compensated for added damping from strings Fig-

400

Mansour et al Post-classification of steel string guitars ACTA ACUSTICA UNITED WITH ACUSTICAVol 101 (2015)

500 1000 2000 6000minus54

minus52

minus50

minus48

minus46

minus44

minus42

Bandminusaveraged

admittance

(dB) Bandminusaveraged bassy

Bandminusaveraged midminusevenBandminusaveraged trebleMidminuseven plusmn one STDTreble plusmn one STD

500 1000 2000 6000minus22

minus20

minus18

minus16

minus14

minus12

minus10

minus8

minus6

Frequency (Hz)

Bandminusaveraged

radiativity

(dB)

(a)

(b)

Figure 10 Averaged admittance (a) and radiativity (b) takenover bassy instruments (solid) mid-even instruments (dash-dot)plusminus one standard deviation (light gray shade green in on-line version) and treble instruments (dashed) plusminus onestandard deviation (darker gray shade red in online version)The dB reference is 1 msN

ure 11 shows the averaged damping trend taken over thethree perceptual categories

The Q factor for all classes rises more sharply from Q asymp30 at 100 Hz to Qasymp 60 at 600 Hz where it reaches a plateauand rises more gradually afterwards However the generalbehavior is not far from linear on a log-log scale As itcan be seen in Figure 11a and 11b the value of Q factorover different frequencies and its trend is similar for trebleand mid-even classes which make them indistinguishableby looking at the damping trend From the power-law fitresults the average power for bassy category is 008 andfor both categories of mid-even and treble is 016 This re-sults in an average-over-all-instruments of 013 comparedto 026 and -009 reported earlier for the violins and theclassic guitars respectively in [9] Among the instrumentsof the pool only one showed a negative power fit

344 Harmonic features

Plucking a guitar string excites multiple harmonics simul-taneously which makes it possible that a weighted sum ofthe admittance over some range of harmonics rather thanthe admittance at the fundamental only is a better way ofdistinguishing categories of instruments Forgetting aboutlosses for the moment the amplitude of the transverse

200 300 500 1000 2000 5000

30

40

50

70

Frequency (Hz)

Qminusfactortrend

200 300 500 1000 2000 5000

20

30

40

50

70

100

Bandminusaveraged

Qminusfactor

Averaged bassy

Averaged midminuseven

Averaged treble

Averaged midminuseven plusmn STD

Averaged treble plusmn STD

(b)

(a)

Figure 11 Effective Q-factors averaged for the guitars of differ-ent categories plusminus one standard deviation (solid line forbassy dash-dot line and light gray green shade for mid-evenand dashed line and darker gray red shade for treble) band-average (a) and regression line for power-law behavior (b) Notethat vertical axes are different

force that the nth harmonic of a plucked string applies tothe bridge can be found from [5]

Fn =2dT0

nπL0

β

1 minus βsin(βnπ) (1)

where L0 is the length of the string T0 is its tension d isthe displacement at the plucking point and β is the rela-tive distance of the plucking point from the bridge Thisformula does not take into account the possible strongcoupling of the plucked string to the body and to otherstrings which should be a reasonable approximation forthe intended purpose and for most of the notes played ona guitar [16] The magnitude of the power transfer fromthe nth harmonic of the string to the body is then equalto the bridge velocity multiplied by the applied force orequivalently

Pn = Fnvn = F 2nGω = A

sin(βnπ)n

2

Gω (2)

in which Gω is the real part of the bridge admittance (alsocalled conductance) and A is a constant assuming a par-ticular string is being played with a constant pluckingstrength and at a fixed relative distance from the bridgeand again neglecting the losses This power transfer to thebody is largely responsible for the damping of the string

401

ACTA ACUSTICA UNITED WITH ACUSTICA Mansour et al Post-classification of steel string guitarsVol 101 (2015)

mode in question as well as the radiated sound at that fre-quency Based on this idea two features are calculated forall semitones played on a guitar (from E2 to E6) and aver-aged over the three perceptual categories For both cases βis arbitrarily chosen as 03 which is a reasonable numberfor a normal playing condition

Figure 12a shows the sum of the transferred power bythe first 5 harmonics of each played note here calledldquopower transfer to the bodyrdquo The choice of 5 harmonics isan arbitrary limit for which the individual harmonics of thestring could be perceived individually and within whichmost of the energy transfer to the body happens in a nor-mal playing condition The frequency range that is consid-ered in this fashion would be up to 5 times the highest notestudied (ie E6) or equivalently 66 kHz this is about thelimit up to which the measured admittances show a reason-ably good coherence The unknown constant A in equation(2) could only cause an offset in the results therefore itis considered to be 1 for convenience The energy trans-fer to the body represents the strength of the stringbodycoupling for a guitar A large value for this number wouldresult in a strong and more percussive (banjo-like) soundand a small value would result in a weaker sound with alonger decay (more similar to an electric guitar without anamplifier)

Figure 12b shows a slightly different use of the trans-ferred energy by each harmonic It is calculated by divid-ing the sum of the energy transferred by the 2nd to 5th har-monic to the energy transferred by the fundamental aloneAs well as specifying the ratio of higher harmonic energyto that at the fundamental this feature defines how thesound evolves once the instrument is plucked Extremelylarge or small values for a particular note mean that thenote changes its character when it decays while an aver-age value may be correlated to a more stable sound

Neither graph shows any clear distinction between themid-even and treble categories The bassy category doeshowever stand out most of the patterns in both graphs areshifted downward by a few semitones

345 LTAS and radiativity

The LTAS results averaged over different categories areillustrated in Figure 13a The results for the mid-even in-struments are greater than the other two groups over al-most all frequencies Figure 13b is the same as Figure 13abut the LTAS for each individual instrument is normalizedby its area before further processing After normalizationall categories behave similarly over different frequenciesmeaning mid-even instruments were simply louder thanthe other instruments rather than being stronger in a par-ticular range of frequencies

The band-averaged admittance of the mid-even instru-ments earlier presented in Figure 10 was slightly greaterthan the treble instruments in some frequency rangeshowever the significance of the stronger LTAS of mid-even instruments especially beyond 3 kHz can hardly bejustified by that alone A possible explanation is that themid-even instruments were played systematically harder

E2 100 E3 250 E4 500 E5 1000 E6

minus60

minus55

minus50

minus45

minus40

minus35

minus30

Pow

ertransfertothebody

(dB)

E2 100 E3 250 E4 500 E5 1000 E6

minus20

minus10

0

10

Harmonicstofundam

entalratio(dB)

Fundamental frequency (Hz)

Averaged bassyAveraged midminusevenAveraged treble

(a)

(b)

Figure 12 Power transfer to the body (a) and harmonics to fun-damental ratio (b) calculated for each played note over the op-erational range of a guitar (E2 to E6) the vertical lines indicateoctaves The values are averaged for the guitars of different cat-egories (solid line for bassy dash-dot line and light gray green(online version) shade for mid-even and dashed line and darkergray red (online version) shade for treble)

compared to the treble instruments There are at least tworeasons as to why this might have been the case First theguitars were recorded per category starting with the trebleand ending with the mid-even ones It is therefore possiblethat the musician felt more confident in the course of therecording session Second the mid-even guitars had beendescribed by the manufacturer as having a ldquomore stablerdquosound Such a sense might have allowed the performer toplay those instruments more freely Nevertheless we areaware that the frequency distribution obtained using LTAScan be strongly dependent on the nature of the input sig-nal depending on the sequence played and used as an in-put certain frequency bands may be less excited than oth-ers which may attenuate differences in some frequencyregions of interest

35 Classification results

According to our results finding a feature that can distin-guish bassy instruments is a relatively easy task As wasshown in Figure 9a the frequencies of the first 4 modesfor the bassy instruments are around 5 lower than thosefor the two other classes The frequency of the first modecan be determined without much ambiguity or even theneed for mode extraction All of the other 3 low-frequencymodes have close adjacent modes making it harder andsometimes impossible to find their correct modal parame-ters

The amplitude of the first mode is another indicator fora bassy instrument that can be used for more assurance

402

Mansour et al Post-classification of steel string guitars ACTA ACUSTICA UNITED WITH ACUSTICAVol 101 (2015)

500 1000 2000 6000minus180

minus170

minus160

minus150

minus140

minus130

minus120

LTAS(dBHz)

500 1000 2000 6000minus120

minus110

minus100

minus90

minus80

minus70

minus60

Frequency (Hz)

Normalized

LTAS(dBHz)

BassyMidminusevenTrebleTreble plusmn one STDMidminuseven plusmn one STD

(b)

(a)

Figure 13 Non-normalized LTAS (a) and normalized LTAS (b)averaged for the guitars of different categories plusminus onestandard deviation (solid line for bassy dash-dot line and lightgray green (online version) shade for mid-even and dashed lineand darker gray red (online version) shade for treble)

(see Figure 9b) Figure 14a shows a 2-D plane with thenormalized frequency and amplitude of the first mode onits horizontal and vertical axes respectively It can be seenthat the bassy instruments are quite distinct from the mid-even and treble ones Moreover the guitar with the lowestbreathing mode (B3 fT (11)1 = 91 Hz) was described bythe factory employee as extra-bassy

It should be emphasized that the frequency of the firstpeak works particularly well as a feature not becausethe ldquobassinessrdquo of the instrument is defined by the notesaround this modersquos frequency (only the first three noteson the E2 string are below that frequency) but becauseit represents the overall stiffness-to-weight ratio of thesoundbox that effectively scales the frequency of all bodymodes It is noteworthy that although T(11)1 has a strongHelmholtz component the amount by which its frequencydeviates from the frequency of the pure Helmholtz reso-nance is strongly dependent on the frequency separationof the pure Helmholtz and pure body breathing modesThe frequency of the pure Helmholtz mode is assumed thesame for all instruments of the pool given their very simi-lar geometry (both for the sound box and the sound hole)

The in-plane dimensions of the instruments should havea very small variation percentage given the precise con-trol of geometrical conditions in the factory Moreover alower frequency for the first mode is usually accompaniedby a higher amplitude for that mode which would not nor-mally be the case if such deviation was caused by using a

90 92 94 96 98 100 102 10404

05

06

07

08

09

1

Normalized

T(11) 1am

plitude

BassyMid EvenTreble

90 92 94 96 98 100 102 104minus50

minus49

minus48

minus47

minus46

minus45

B1B2

B3

B6

B4B5

ME6ME1

ME2

ME3ME4

ME5

T2

T3

T4

T5 T6

T1

Frequency of the T(11)1mode (Hz)

Averagedmobility(600minus2000Hz)(dB)

(b)

(a)

Figure 14 Classification of the acoustic guitars (a) by the fre-quency and amplitude of their first corpus mode T(11)1 onlyto distinguish bassy instruments from the others and (b) by thefrequency of their first corpus mode T(11)1 and their averagedmobility from 600 Hz to 2 kHz (B = bassy ME = mid-even T =treble) The dB reference is 1 msN

denser wood this leaves variations in the plate thicknessand the equivalent stiffness (ie very roughly the geomet-ric mean of the Youngrsquos moduli in long-grain and cross-grain directions [14]) as those most likely responsible forthe observed shift in mode frequencies Mode frequenciesof a flat plate scale with Eρ(tL2) where E and ρ arethe elastic constant and density t the thickness and L acharacteristic scaling length If either of the two factors issolely responsible to make a 5 shift it would require ei-ther a 5 variation (asymp 0125 mm) of the plate thicknessor a 10 variation in the equivalent stiffness Given thetight tolerances in cutting the plates and their mechanizedsanding it thus appears that variations in equivalent stiff-ness are the primary reason that some of the instrumentsin the pool sound more bassy In the manufacturing pro-cess a wood slab is sliced to produce several plates Thelateral direction of each plate hence represents a particularcombination of the slabrsquos radial and tangential directions(due to the growth rings of the tree) This can introduce asignificant variation in the platesrsquo stiffness in the lateral di-rection which produces variations of the effective stiffnessof the plates

Finding a feature that distinguishes mid-even from tre-ble instruments is less obvious We found that the aver-aged admittances of the treble instruments in this study areslightly lower than the mid-even instruments in the rangeof 600 Hz to 2 kHz Although this difference is not signif-icant over the mentioned range (ie average plusminus

403

ACTA ACUSTICA UNITED WITH ACUSTICA Mansour et al Post-classification of steel string guitarsVol 101 (2015)

one standard deviation of the two categories overlap) itbecomes significant if the average level of the admittancefor the whole range is compared (ie the average valueof all frequency points of the admittance from 600 Hz to2 kHz) Figure 14b shows a 2-D plane with the frequencyof the first mode on the horizontal axis and the average ad-mittance for the range of 600 Hz to 2 kHz The differencefor the mean of the mid-even and treble categories is about2 dB It seems likely that the weaker admittance of the tre-ble category at higher frequencies would contribute to alonger decay for the higher partials This may contributeto a sharper after-sound of the treble instruments that isknown to allow them to cut through the band more effec-tively This hypothesis could be tested by recording theplucked sound of the instruments and comparing the decayrate of the partials among different categories Due to thelimited project time frame this was not investigated fur-ther The mid-even guitars also show stronger LTAS thanthe treble instruments though this result might have beendue to playing conditions (see Section 345)

We applied the same classification (frequency ofT(11)1 versus average mobility between 600 Hz and2 kHz) to 6 acoustic guitars from a different model by thesame manufacturer (Simon amp Patrick) Unlike the Seagullguitars which were all nominally identical some of theSampP instruments had a slightly different bracing patterndesigned to exemplify the properties of a particular per-ceptual category The manufacturerrsquos classification of the6 guitars was not provided to us prior to our own mea-surements though we were told that there were 2 guitarsof each category The resulting classification based on thediscussed features is depicted in Figure 15 The two gui-tars designed to be more bassy were successfully identi-fied from the frequency of their breathing mode but dis-tinguishing between the mid-even and treble instrumentsfrom the averaged admittance criterion was inconclusive

4 Perceptual evaluation

As previously discussed in Section 1 the investigatedacoustic guitars were post-classified by the manufacturerbased on their experience of what type of instrument mightbetter fit a certain playing stylemdashfor example an amateurmusician that plays mostly strummed chords may prefera bassy guitar while a more experienced guitarist mayopt for a treble instrument We therefore conducted a per-ceptual experiment to investigate the level of agreementamong guitar players of varying experience and musicalbackground in classifying the guitars into the categoriesconsidered by the manufacturer

41 Method

The experiment took place in an acoustically dry roomwith dimensions of 60 times 78 times 32 m3 and a reverberationtime of about 03 s to help minimize the effects of roomreflections on the direct sound from the guitars To avoidthe potential problems of using a common pick across all

85 86 87 88 89 90 91 92minus505

minus50

minus495

minus49

minus485

minus48B1

SampP

B2SampP

ME1SampP

ME2SampP

T1SampP

T2SampP

Frequency of the T(11)1mode (Hz)

Averagedmobility(600minus2000Hz)(dB)

BassyMid EvenTreble

Figure 15 Classification of the six SampP guitars based on thefrequency of the T(11)1 mode versus averaged mobility from600 Hz to 2 kHz (B= bassy ME=mid-even T= treble) The dBreference is 1 msN

participants (eg musicians being uncomfortable with apick they are not familiar with) participants were asked tocarry out the classification task using their own pick(s)

411 Participants

Thirteen guitar players took part in the experiment (1 fe-male 12 males average age = 25 yrs SD = 5 yrs range =19ndash36 yrs) They had at least 6 years of acoustic guitar ex-perience (average years of training = 11 yrs SD = 4 yrsrange = 6ndash20 yrs average hours of practice per week =14 hrs SD = 10 hrs range = 3ndash40 hrs) as well as vary-ing experience with different types of guitars [nylon-string(85) acoustic-electric (62) acoustic bass (15) elec-tric (100) and electric bass (54)] Participants re-ported playing a wide range of musical styles [classical(77) jazz (85) pop-rock (92) country folk blue-grass (66) hard rock heavy metal (46) flamenco(15) latin (46) blues (77) avant-garde contempo-rary (54) and funk alternative (46)] both in a bandand as soloists Seven guitarists (54) reported that theytypically or most often use a pick whereas 4 players (31)reported they do not 2 guitarists (15) reported using apick occasionally rather than typically Nine participantshad higher-level degrees in music performance or othermusic-related areas [BMus (music theory classical gui-tar) BFA (jazz studies) MMus (guitar performance) MA(digital musics) PhD (music theory) Diplocircme drsquoEacutetat deProfesseur de Musique (France)] All guitar players werepaid for their participation

412 Procedure

The experimental session lasted 2 hours and was organizedin two phases In the first phase participants were pre-sented with the guitars randomly ordered on the floor (de-termined by computer calculations) They were asked toplay all instruments for up to 20 minutes in order to famil-iarize themselves with the set and use this time to tune theguitars as needed In the second phase participants were

404

Mansour et al Post-classification of steel string guitars ACTA ACUSTICA UNITED WITH ACUSTICAVol 101 (2015)

given up to 20 minutes to classify each of the guitars as ei-ther bassy mid-even or treble by placing them in speciallymarked areas of the room At the end of the second phaseparticipants were given the opportunity to revise their ini-tial judgements if they so wanted until they were satis-fied with their classification No playing constraint wasimposed on the evaluation process (eg specific repertoireor technique) Participants were instead encouraged to fol-low their own strategy with the suggestion to use bothstrumming-oriented songs (eg Sitting Waiting Wishingby Jack Johnson) and fingerpicking-oriented pieces (egDust in the Wind by Kansas) Participants were instructedto maximize evaluation speed and accuracy Upon com-pleting the session participants talked about their classi-fications and overall experience through written responsesto the following questions ldquoFor each class of guitar whatsound characteristics did you look for Can you describehow you evaluated the guitars in terms of these character-istics (eg musical material technique style)rdquo (Q1) ldquoTowhat extend do you agree with the given classification ofguitarsrdquo (Q2) and ldquoCan you suggest a classification thatbetter reflects how you evaluate the sound characteristicsof guitarsrdquo (Q3)

42 Results

Inter-individual consistency was measured as the Fleissrsquokappa coefficient between guitar classifications from dif-ferent participants using a Bayesian approach to inference[18] Fleissrsquo kappa is a generalization of Cohenrsquos kappastatistic κ a chance-corrected measure of agreement be-tween two assessors to multiple raters [19] It ranges fromminus1 to 1 κ (AB) = 1 and minus1 if A = B and A = minusBrespectively and κ (AB) = 0 in case of no associationbetween A and B (ie exactly what would be expectedby chance) We implemented Gibbs sampling through thesoftware WinBUGS (Windows Bayesian inference UsingGibbs Sampling [20]) with 50000 observations generatedfrom the posterior distribution with a burn-in of 1000 ob-servations A uniformly distributed multicategorical priorwas assumed with total information across all possiblecells (ie from different combinations of participants) be-ing equivalent to a single prior observation (ie a priorsample size equivalent of 1)

Fleissrsquos kappa was estimated with a posterior mean of01 and a 95 credible interval [-02 05] indicating avery low amount of agreement between the participantsin their classifications Figure 16 shows the proportion oftimes a guitar was classified as either bassy mid-even ortreble (the guitars are arranged in three groups accord-ing to the initial factory classification) Whereas someagreement concerning the bassy guitars can be observedmdashindeed the extra bassy guitar (B3) was most often clas-sified as bassymdashthe perceptions for the other two classeslargely vary across different musicians In particular manyguitars that had been characterized as treble by the fac-tory employee were perceived as mid-even Several mu-sicians specifically reported encountering difficulties withthe mid-even category In answering Q3 a player observed

B1 B2 B3 B4 B5 B60

25

50

75

100

classifiedas

(a)BassyMidminusevenTreble

ME1 ME2 ME3 ME4 ME5 ME60

25

50

75

100

classifiedas

(b)

T1 T2 T3 T4 T5 T60

25

50

75

100

classifiedas

Guitars

(c)

Figure 16 Perceptual evaluation Proportion of times a guitarwas classified into one of the categories by the participants Gui-tars are labelled and grouped according to the factory classifica-tion (B = bassy ME = mid-even T = treble) The color scheme(in the online version) corresponds to the classifications providedby the guitar players

that ldquothere is a difference [between] a guitar with domi-nant mid tones and one with an even soundrdquo

Overall participants expressed no strong difficulties incarrying out the classification task On a scale from 1(strongly disagree) to 5 (strongly agree) all performers se-lected 4 when asked ldquoTo what extend do you agree withthe given classification of guitarsrdquo This indicates that the3 categories considered by the manufacturer are gener-ally meaningful to musicians However verbal responsesto questions Q1 and Q3 reveal that participants often as-sociated bassy with more prominent E2 and A2 strings(ie they focused on the low-frequency end of the gui-tar) and similarly treble with better emphasized B3 andE4 strings (ie they paid attention to the high-frequencyrange) As one guitar player noted ldquoI feel that descriptionslike warm neutral and bright would be easier to classifydue to the nature of the instrument Bass leads my mindto bass strings and to heed the low end instead of the en-tire range from the highest to lowest notes Same for tre-ble vs bright rdquo In particular 4 participants used the wordldquotwangrdquo to describe guitars they classified as treble whileanother guitarist reported that a treble guitar ldquosounds bestfor finger picking and light strumming as heavy playingcan get quite ear-piercing and harsh This last commentseems to echo the second hypothesis for the origin of thestronger LTAS shown by the mid-even guitars (see Sec-tion 345)

Two participants reported looking out for problematicnotes as a more meaningful way of finding bassy guitarsThis likely points toward the ldquoveeringrdquo phenomenon as-sociated with a stronger admittance at lower frequencies

405

ACTA ACUSTICA UNITED WITH ACUSTICA Mansour et al Post-classification of steel string guitarsVol 101 (2015)

[16] Further one musician referred to the tactile feedbackreceived from a vibrating body associating bassy guitarswith more felt vibrations than the other two categoriesThis is a perspective shared by the manufacturer as wellIn fact body vibrations can provide tactile and proprio-ceptive cues that contribute to the perception of the radi-ated sound so that the musician can assess their interactionwith the instrument cross-modally [21 22]

5 Discussion

A set of 18 steel string guitars of the same model man-ufactured with high geometrical tolerances in a modernfactory were evaluated in this study These guitars wereconsidered by the manufacturer to possess subtle yet per-ceivable variations that could be classified as either bassymid-even or treble The results of this study showed thatthe instruments classified as bassy had a consistently lowerT(11)1 breathing mode (see Figure 9a 14b and 15) Thissuggests a lower stiffness-to-weight ratio for the bodies ofthese guitars which is most likely the result of variationsin equivalent stiffness (see Section 35)

It was further shown that whenever the frequency ofT(11)1 was shifted downward or upward for a particu-lar guitar the frequencies of its higher modes consistentlychanged in a similar way (see Figure 9 also the slightlyshifted bumps at around 600 750 and 1200 Hz in Fig-ure 12a) This would not necessarily be the case if theguitars were of different models (ie with varying geome-tries) or of different types of wood (eg plywood versusspruce) This is an important result as there is a practi-cal advantage to having higher-frequency modes move ac-cording to the first corpus mode across guitars of the samemodel T(11)1 is a low-frequency resonance that is suffi-ciently separated from the adjacent modes and hence eas-ily identifiable Measuring its frequency across differentinstruments is easy and affordable in a small workshopenvironment One can simply hit the guitar with a reg-ular hammer and recordanalyse the sound using an in-expensive microphone and a laptop reminiscent of thetraditional tap-tone method used by luthiers Moreoversuch measurements can be very reliable provided that theboundary conditions of the instrument (ie the installedfixture that holds the instrument) as well as the air tem-perature pressure and humidity are kept consistent Theseparameters constitute the biggest source of uncertainty infrequency-based features and can easily be controlled in afactory space or small workshop

Finding a reliable differentiator between the mid-evenand treble guitars was less obvious Although we ob-served that the averaged mobility curves of the mid-eveninstruments tended to be slightly higher in the vicinityof 600ndash2000 Hz than the treble guitars further measure-ments would have been necessary to determine whetherthe weaker admittance of the treble instruments at higherfrequencies would contribute to a longer decay for thehigher partials (see Section 35) In addition the differ-ence in averaged mobility magnitude between the two cat-

egories was only about 2 dB which is close to the marginof error for the measurement system Such a small differ-ence in an amplitude-based feature may be easily obscuredby variations in the exact angle of the hammer location ofhitting and measuring or calibration of the sensors

We invited guitar players of varying musical back-ground and experience to classify the guitars into thesecategories We observed very poor agreement among theparticipants but it is common in recent perceptual studiesof musical instruments that musicians do not agree witheach other [23 24 25] This seems a result of them lis-tening to (or feeling) different aspects of the instrumentand not necessarily all attending to the same features Theemployee that post-classified the guitars at the factory wasthe person that tests every guitar coming off the productionline thus he must have a lot of experience evaluating thesound of these instruments Assuming he did his classifi-cation carefully then there likely were particular things hewas listening for The participants in our perceptual exper-iment would not necessarily listen for the same featuresso it is not too surprising they largely disagreed on theirclassifications That said the differences between the tre-ble and mid-even guitars were extremely subtle and evencompany representatives often mis-categorized them dur-ing informal playing tests And this together with the factthat our measurements showed no consistent differencesbetween the two categories would seem to lead us to ques-tion whether the distinction between mid-even and trebletruly exists at all Another possible explanation is that dif-ferent musicians may follow different processes to assessthose features considered essential for the evaluation of aninstrument [23 26] Accordingly guitar players may agreeof what particular characteristics they look for in a tre-ble guitar but the perception of the same characteristicswidely varies across individuals

6 Conclusions

Efforts to understand how the tonal quality of an instru-ment relates to its measurable physical properties have of-ten been inconclusive (eg [27]) In the light of some re-cent studies the concept of ldquogoodrdquo versus ldquobadrdquo instru-ments is likely too general and it would be more advanta-geous to instead consider specific perceptual attributes ofan instrument [23 28] In the long run it may be possibleto derive a multidimensional timbre space where each di-mension is explained by some measurable quantities andin which instruments with similar qualities are likely tocluster In this study more specific sound characteristicsof instruments were thus targeted rather than their overalltonal quality

From bridge admittance measurements on 18 steelstring guitars from the same production line and post-classified by the manufacturer as either bassy mid-evenor treble (see Section 1 for details) it was shown that itis possible to discriminate between bassy guitars and theother two categories based on the frequency of the breath-ing mode However finding a similarly reliable feature to

406

Mansour et al Post-classification of steel string guitars ACTA ACUSTICA UNITED WITH ACUSTICAVol 101 (2015)

distinguish mid-even from treble instruments is less triv-ial at this point The treble guitars demonstrated a loweraveraged mobility in the frequency range of 600ndash2000 Hzthough more empirical investigation would be necessarybefore drawing any conclusions Results from a percep-tual experiment wherein guitar players were asked to clas-sify the guitars into these categories showed almost noagreement between guitarists in their judgements as wellas between the latter and the classification by the manufac-turer Despite the overall lack of consensus guitar playerstended to agree more on the bassy instruments than theother two categories

Some essential additions to this study are a more care-ful analysis on the perceptual definition of mid-even ver-sus treble and analysing the decay rate of the partials fora plucked sound to see how it correlates with the admit-tance Analysis of this kind has been performed for syn-thesized acoustic guitar sounds (see Figure 7 in [16]) but asimilar analysis on real guitar sounds will involve difficul-ties caused by the double set of harmonics [29] as well aslongitudinal string vibrations (also dubbed ldquophantom par-tialsrdquo) [30]

Acknowledgements

This project was funded by a Natural Sciences and En-gineering Research Council of Canada Engage grant incollaboration with Godin Guitars The authors wish tothank Robert Godin for this opportunity We also thankLawrence Joseph (statistics) and Garry Antonio (guitarperformance) for constructive comments We are grate-ful to Jim Woodhouse for sharing his analysis code Thefirst author would like to acknowledge the Government ofCanada for a Vanier Canada Graduate Scholarship

References

[1] E V Jansson A study of acoustical and hologram inter-ferometric measurements of the top plate vibrations of aguitar Acust Acta Acust 25 (1971) 95ndash100

[2] O Christensen B B Vistisen Simple model for low-frequency guitar function J Acoust Soc Am 68 (1980)758ndash766

[3] O Christensen Quantitative models for low frequency gui-tar function J Guitar Acoustics 6 (1982) 10ndash25

[4] H Wright The acoustics and psychoacoustics of the guitarDissertation Dept of Physics and Astronomy Universityof Wales Cardiff UK 1996

[5] N H Fletcher T D Rossing The physics of musical in-struments Second ed Springer New York 1998

[6] J E Popp Four mass coupled oscillator guitar model JAcoust Soc Am 131 (2012) 829ndash836

[7] J Woodhouse Body vibration of the violinmdashwhat can amaker except to control Catgut Acoust Soc J (Series II)4 (2002) 43ndash49

[8] B Elie F Gautier B David Macro parameters describingthe mechanical behavior of classical guitars J Acoust SocAm 132 (2012) 4013ndash4024

[9] J Woodhouse R S Langley Interpreting the input admit-tance of violins and guitars Acust Acta Acust 98 (2012)611ndash628

[10] A Gabrielsson E V Jansson Long-time-average-spectraand rated qualities of twenty-two violins Acustica 42(1979) 47ndash55

[11] P D Welch The use of Fast Fourier Transform for the es-timation of power spectra A method based on time averag-ing over short modified periodograms IEEE Trans Audioand Electroacoust AU-15 (1967) 70ndash73

[12] J Sundberg The singerrsquos formant revisited J Voice 4(1995) 106ndash119

[13] A Buen On timbre parameters and sound levels ofrecorded old violins J Violin Soc Am VSA Papers XXI(2007) 57ndash68

[14] J Woodhouse On the ldquobridge hillrdquo of the violin AcustActa Acust 91 (2005) 155ndash165

[15] B Richardson The acoustical development of the guitarCatgut Acoustical Society Journal 2 (1994) 1ndash10

[16] J Woodhouse On the synthesis of guitar plucks AcustActa Acust 90 (2004) 928ndash944

[17] C H Hodges J Power J Woodhouse The use of the sono-gram in structural acoustics and an application to the vibra-tions of cylindrical shells J Sound Vib 101 (1985) 203ndash218

[18] L D Broemeling Bayesian methods for measures ofagreement Chapman amp HallCRC New York 2009 (Bio-statistics)

[19] J L Fleiss Measuring nominal scale agreement amongmany raters Psychological Bulletin 76 (1971) 378ndash382

[20] D Lunn C Jackson N Best A Thomas D Spiegelhal-ter The BUGS Book A Practical Introduction to BayesianAnalysis Chapman amp HallCRC New York 2012 (Textsin Statistical Science)

[21] C Saitis Evaluating violin quality Player reliability andverbalization Dissertation Dept of Music ResearchMcGill University Montreal Quebec Canada 2013

[22] I Wollman Perception bimodale des violonistes en situa-tion de jeu Influence des retours auditif et vibrotactile surlrsquoeacutevaluation du violon Dissertation Universiteacute Pierre etMarie Curie Paris France 2013

[23] C Saitis B L Giordano C Fritz G P Scavone Percep-tual evaluation of violins A quantitative analysis of pref-erence judgements by experienced players J Acoust SocAm 132 (2012) 4002ndash4012

[24] C Fritz J Curtin J Poitevineau P Morrel-Samuels F-CTao Player preferences among new and old violins ProcNat Acad Sci USA 109 (2012) 760ndash763

[25] P Eveno J-P Dalmont R Causseacute G P Scavone Anacoustic and perceptual evaluation of saxophone pad ldquores-onatorsrdquo Acta Acustica united with Acustica 101 (2015)246ndash255

[26] C Fritz J Woodhouse F P-H Cheng I Cross A FBlackwell B C J Moore Perceptual studies of violinbody damping and vibrato J Acoust Soc Am 127 (2010)513ndash524

[27] G Bissinger Structural acoustics of good and bad violinsJ Acoust Soc Am 124 (2008) 1764ndash1773

[28] C Saitis C Fritz C Guastavino G P Scavone Con-ceptualization of violin quality by experienced performersProc Stockholm Music Acoust Conf Stockholm Swe-den 2013 123ndash128

[29] J Woodhouse Plucked guitar transients Comparison ofmeasurements and synthesis Acust Acta Acust 90 (2004)945ndash965

[30] B Bank L Sujbert Generation of longitudinal vibrationsin piano strings From physics to sound synthesis JAcoust Soc Am 117 (2004) 2268ndash2278

407

Mansour et al Post-classification of steel string guitars ACTA ACUSTICA UNITED WITH ACUSTICAVol 101 (2015)

LTAS were obtained using the Matlab function pwelchsymmetric four-term Blackman-Harris windows an 8192-point window size and 50 overlap between the windowsSubsequently the results were smoothed in the frequencydomain using a 500 Hz sliding rectangular window

32 Variation among the instruments

We first evaluated the amount of variation among the 18guitars The precise manufacturing process in the produc-tion of these instruments made it almost impossible to seeany appreciable geometrical difference among the individ-ual instruments except a slight variation in the design oftheir necks (some were intentionally cut more flat and oth-ers more rounded) Therefore the biggest uncertainty fac-tor would be the variation in material properties Althoughthe instruments of our study were all made with sprucetops and mahogany backs and ribs some variation of tim-bre is inevitable

Figure 2 shows the averaged admittance over the pool of18 instruments plusminus one standard deviation A sam-ple instrument from the pool and an outlier are includedfor comparison The outlier was a full-size acoustic guitarfrom the same manufacturer but of a different model Thestandard deviation for all frequency points from 70 Hz to6 kHz averages to 275 dB when calculated for all 18 in-struments The level of variation is slightly higher if welook at the lower range of frequencies (379 dB for the fre-quency range of 70 Hz to 1 kHz) As a reference we calcu-lated the standard deviation for 4 repeated measurementsmade on the same instrument removed and put back on thefixture before each repetition The standard deviation aver-aged to 054 dB for the range 70 Hz to 6 kHz and 094 dBfor the range 70 Hz to 600 Hz Note the different samplesize when comparing these numbers with their equivalentsreported earlier for the 18 instruments

The same analysis can be applied to the instrumentswithin the same perceptual categories (ie bassy mid-even and treble) Figure 3 shows the average admittancefor the instruments of the different categories The av-erage standard deviations within the categories of bassymid-even and treble are respectively 263 dB 236 dB and215 dB for the frequency range of 70 Hz to 6 kHz and329 dB 231 dB 242 dB for the frequency range of 70 Hzto 1 kHz The fact that the variation within each categoryis less than the variation among all the instruments lendssome credibility to the classification made by the manufac-turer Another observation in Figure 3 is that the mid-evenand treble instruments follow almost the same trend below600 Hz while bassy instruments are considerably distinctin the frequency range of 80 Hz to 100 Hz and again from300 Hz to 350 Hz showing a shift of the main resonancestowards lower frequencies

33 General properties of the acoustic guitars

331 Real-imaginary parts

Figure 4 shows the real and imaginary parts of the av-eraged admittances using an averaging bandwidth of

70 100 200 500 1000 2000 6000minus65

minus60

minus55

minus50

minus45

minus40

minus35

minus30

minus25

minus20

Frequency (Hz)

Averagedadmittance

(dB)

A sample instrumentfrom the pool

An outlier

Averaged plusmn one STD

Figure 2 Gray shading shows the averaged admittance over thepool of 18 instruments plusminus one standard deviation Thesolid black line is a sample instrument from the pool and dashedline is an outlier The dB reference is 1 msN

100 200 300 500 1000 2000 6000

minus60

minus55

minus50

minus45

minus40

minus35

minus30

minus25

Frequency (Hz)

Averagedadmittance

(dB)

Averaged bassyAveraged midminusevenAveraged treble

Figure 3 Averaged admittance over the bassy instruments (so-lid) mid-even instruments (dash-dot) and treble instruments(dashed) The dB reference is 1 msN

1000 Hz One can see that the acoustic guitar does not rep-resent the properties of an infinite plate (ie positive realpart with roughly constant magnitude and imaginary partmuch smaller) at least not as well as a classical guitar does[9 8] The magnitude of the imaginary part is smaller thanthe real part but still large enough to be non-negligibleThe real part stays positive over all frequencies which isa property of any passive system The average amplitudeof the real part is smaller by almost a factor of two thanthe same measure for a typical classical guitar it reachesthe typical value for a classical guitar at around 3 kHz andagain drops at higher frequencies A broad peak in the realpart and a tendency in the imaginary part towards negativevalues creates a hill-like feature [14] at around 2800 Hz forthe majority of the instruments with a similar pattern thatrepeats less strongly at around 4 kHz These hills are quiteunexpected as an acoustic guitar does not have a bridge

397

ACTA ACUSTICA UNITED WITH ACUSTICA Mansour et al Post-classification of steel string guitarsVol 101 (2015)

500 1000 1500 2000 2500 3000 3500 4000 45000002

0004

0006

0008

Re(ltYgt)(msN)

500 1000 1500 2000 2500 3000 3500 4000 4500minus0002

minus0001

0

0001

0002

0003

Frequency (Hz)

Im(ltYgt)(msN)

A sample instrumentAveraged plusmn one STD

(a)

(b)

Figure 4 Band-averaged real (a) and imaginary (b) parts of theadmittance for a sample instrument (solid) and averaged overall instruments of the pool plusminus one standard deviation(shaded) Note that the vertical scales are different for the realand imaginary parts

like a violin nor has it a feature like f-holes to create suchlocal resonances A possible explanation is that the cross-bracing of the steel string guitars unlike the fan-bracing ofmost classical guitars circumvents the bridge and formsan isolated area around it and this may contribute to suchlocal resonances

332 Modal spacing and MOFIt is useful to find a practical upper limit for the range ofmode fitting (ie where the MOF reaches one) and to get asense of the average spacing between the modes Figure 5shows the MOF and the average mode spacing for a sam-ple instrument from the pool and the average plusminusone standard deviation over all instruments An averagingbandwidth of 500 Hz is used in this case to give a betterprecision at lower frequencies The statistical features in-cluding MOF are calculated based on variance theory formore details see equations (17) to (19) of [9] As expectedthe general trend of the overlap factor is rising although itfalls a bit between 1500 Hz and 2000 Hz it reaches thevalue of one at around 610 Hz The average mode spac-ing goes unrealistically high (as much as 150 Hz at around200 Hz) at lower frequencies The mode spacing stays al-most constant at around 20 Hz for the frequency range of600ndash1200 Hz The mode spacing drops to around 8 Hz athigher frequencies and then stays almost unchanged

333 Mode fittingUsing the method described in Section 2 a mode fittingprocess was performed on all 18 instruments of the poolup to 600 Hz [9] The fitting program identified 20 to 24modes for different instruments amongst which a maxi-mum of 16 modes were manually selected for each instru-ment The criterion was to keep the stronger modes thatwere also identified for most of the instruments

The general shape of the admittance follows the pre-dictions of [6] the first neck bending at around 65 Hz

600 1000 1500 2000 2500 3000 35000512

510

40

Modaloverlapfactor

600 1000 1500 2000 2500 3000 35000

10

20

30

40

50

Frequency (Hz)

Modalspacing(Hz)

A sample instrumentAveraged plusmn one STD

(b)

MOF = 1atF asymp 610 Hz

(a)

Figure 5 Modal overlap factor (a) and the average mode spacing(b) for a sample instrument from the pool (solid) and the averageplusminus one standard deviation over all instruments (shaded)An averaging bandwidth of 500 Hz is used

the T(11)1 or the breathing mode at around 100 Hz andT(11)2 or anti-breathing mode at around 200 Hz Modesare categorized as (mn) where m and n are the number ofldquohalf-wavesrdquo across and along the plate and T refers to thetop plate As pointed out by Popp [6] the in-phase motionof the top and back plates coincides in frequency with theanti-breathing mode at around 200 Hz

For our measurements the guitars were held in a fixturewith two arms that pressed against the sides of the ribsThis was intended to create a boundary condition simi-lar to that when held by a player Popp argues that a con-siderable amount of mass would be needed to immobilizethe ribs therefore our measurement condition is betweenthe fixed and free-free cases but perhaps more similar tothe latter After the measurements were performed we ob-served a peak (or a double peak for some instruments)at around 125 Hz where one normally expects to see theanti-resonance representing the pure Helmholtz mode [2]When we repeated the measurements on some of the gui-tars using a free-free boundary condition we noticed thatthe anti-resonance appeared as expected This suggests thepeak is either a property of the fixture rather than the in-struments themselves or it is the (11) in-phase motion ofthe top and back plates shifted downward due to the massof the fixture added to the ribs The latter would requirea 60 Hz downward shift of that frequency which seemsunlikely No effort was made to repeat all the measure-ments with a free-free boundary condition partly becausethe measurements were made before subsequent data anal-ysis and partly because free-free measurements are lesspractical and reproducible in a factoryrsquos production line(due to for example variations in accelerometer place-ment and potential damage to a new guitar surface finishwhen attaching and detaching an accelerometer)

The second neck bending mode was evident as a smallpeak at around 225 Hz for all instruments There weretwo other small peaks at around 250 Hz and 290 Hz which

398

Mansour et al Post-classification of steel string guitars ACTA ACUSTICA UNITED WITH ACUSTICAVol 101 (2015)

were probably related to the torsion and lateral vibrationsof the neck The next important modes were T(21) andT(12) For Martin D-28 acoustic guitars these modes arearound 100 Hz apart [5] hence relatively distinct how-ever for the guitars of our study they both appear at around350 Hz and sometimes switch places We measured the ad-mittance at the bass side of the bridge where T(21) hasits antinode therefore for each guitar the stronger modein that region is associated to T(21) In addition to theHelmholtz mode there are two other cavity modes in thefrequency range of our study which are A1 at 430 Hz andA2 at around 500 Hz [5] Approximate mode shapes ofthe first four corpus modes (exclusive of local resonancessuch as neck bending and neck twisting) for the guitars ofour study are illustrated in Figure 6 the frequencies areaveraged and vary from one instrument to another

The average frequency amplitude and Q-factors calcu-lated for the 18 guitars were used to reconstruct the ad-mittance of a ldquomodally-averagedrdquo guitar that representsthe model that we studied Figure 7 shows this ldquomodally-averaged admittancerdquo reconstructed up to 600 Hz (dashedline) compared with a sample measured admittance aswell as the raw averaged admittance previously shown inFigure 2 It can be seen that the peaks are excessivelydamped for the raw average while the admittance of theldquomodally averagedrdquo guitar shows a better resemblance tothe sample measured admittance close to the resonancesand anti-resonances This modal average thus better rep-resents the admittance of a guitar coming off this partic-ular production line It can be overlaid with the measuredadmittance of a new guitar to highlight its significant fea-tures

If needed the admittance of the ldquomodally averagedrdquoguitar can be reconstructed for extended frequencies eitherby simple averaging over different instruments or moreprecisely by using a ldquostatistical fitrdquo method as describedin [16] In short a random number generator can give thefrequency of the high-frequency modes with the correctdensity and spacing statistics as well as damping factorsand modal masses with approximately correct statisticaldistribution Such a statistical fit was not pursued in thisstudy

334 LTAS

Figure 8 shows the LTAS of a sample instrument fromthe pool (solid black line) as well as the average over allinstruments plusminus one standard deviation (shaded)The first few dominant modes identified in an admittancehave demonstrated themselves in the form of broad peaksin the lower frequencies of the LTAS below about 400 HzSome instruments such as the depicted sample show a peakat around 25 kHz which may correspond to the hill-likefeature earlier identified in band-averaged mobility curves(see Figure 4) Beyond 300 Hz the LTAS drops almostlinearly with the average slope of 16 dBoctave The av-erage standard deviation for the frequency range of 70 Hzto 6 kHz for all instruments is 256 dB which indicates theplayer was able to play consistently on all instruments It is

Figure 6 The four first corpus mode shapes of an acoustic guitartaken from [15]

70 100 200 500minus65

minus60

minus55

minus50

minus45

minus40

minus35

minus30

minus25

minus20

Frequency (Hz)

Averagedadmittance

(dB)

A sample measured admittanceModallyminusaveraged guitarAveragedminusoverminusallminusinstruments

Figure 7 Raw averaged admittance over all instruments of thepool (dash-dot) reconstructed ldquomodally-averagedrdquo admittance(dashed see text for details) and a sample admittance for com-parison (solid) The dB reference is 1 msN

worth mentioning that this number is not directly compa-rable to the averaged standard deviation for the measuredadmittance as the LTAS is significantly smoothed in thefrequency domain in comparison

34 Investigating different features for classification

341 Modal propertiesAveraged value and standard deviation of frequency am-plitude and Q factor of the first four corpus modes are

399

ACTA ACUSTICA UNITED WITH ACUSTICA Mansour et al Post-classification of steel string guitarsVol 101 (2015)

calculated for the three perceptual classes and comparedin Figure 9 In all three figures the values are normalizedto the average of the same modal value taken over all in-struments and the error bars show one standard deviationon each side In case of a missing mode for one instru-ment its frequency and Q factor were eliminated from thecalculation of the average and standard deviation but theamplitude was considered to be zero (as that mode wasnegligible or non-existent) The admittance was not com-pensated for added damping from strings [9] partly be-cause the exact same strings were used on all instrumentshence the comparison was fair and partly because the ex-act amount of damping added by the strings is unknownand the assumption of perfect damping at the strings mayresult in over-compensation of damping [9]

As was also evident from Figure 3 Figure 9a shows thatthe corpus modes of the bassy instruments occur at signif-icantly lower frequencies than the mid-even or treble onesAnother feature that distinguishes bassy instruments fromthe mid-even or treble ones can be seen in Figure 9b theamplitude of the first corpus mode T(11)1 is higher forthe bassy instruments The Q factor of the first four cor-pus modes did not show any noticeable difference amongdifferent perceptual classes

342 Averaged admittance and radiativity

To obtain a more clear comparison of different categoriesin the statistical range (ie at higher frequencies) the ad-mittance and radiativity were smoothed in the frequencydomain using a 500 Hz sliding rectangular window andthen averaged for different categories (see Figure 10a andFigure 10b) The proper way to calculate radiativity wouldhave been to measure the averaged sound pressure over anarray of microphones around the instruments That saidour results using only a single microphone provide a rea-sonable approximation because they were smoothed in thefrequency domain and averaged over different instrumentsSince we have already found distinctive features for the in-struments of the bassy class in the subsequent figures theplusminus standard deviation is eliminated for the bassyclass to help clarity

It can be seen in Figure 10a that the band-averaged ad-mittance of the mid-even instruments is greater than orequal to that of the treble instruments over a wide rangeof frequencies particularly in the range 600 Hz to 2 kHzAlthough this dominance does not appear significant formost of the frequencies (ie the averages for categoriesoverlap when plotted with plusminus one standard devia-tion) the mean level of admittance from 600 Hz to 2 kHzcan distinguish mid-even from treble instruments at leastwithin the pool of our 18 instruments This result will befurther discussed in Section 35 The band-averaged admit-tance of the bassy instruments falls between the two othercategories over most of the frequencies

Band-averaged radiativity does not reveal a significantdifference among different categories except a lower av-erage for the bassy guitars in the range from 2 kHz to275 kHz (see Figure 10b) However this difference is

70 100 200 300 500 1000 2000 6000minus180

minus170

minus160

minus150

minus140

minus130

minus120

minus110

minus100

minus90

minus80

Frequency (Hz)

LTAS(dBHz)

A sample instrumentAveraged plusmn one STD

Figure 8 Long-term average spectra for a sample instrumentfrom the pool (solid line) and the average plusminus one stan-dard deviation over all instruments (shaded)

08

09

1

11

Frequency

(norm) (a)

BassyMidminusevenTreble

0

05

1

15

2

25

Amplitude

(norm)

(b)

05

1

15

2

Qminusfactor(norm)

Guitar corpus modes

(c)

T(21)T(11)2

T(11)1 T(12)

Figure 9 Normalized average frequency (a) amplitude (b) andQ factor (c) for the first four corpus modes for bassy guitars (darkgray bars blue in online version) mid-even guitars (light graybars green in online version) and treble guitars (medium graybras red in online version) The error bars show one standarddeviation on each side

likely not important as the standard deviation of the bassyinstruments in this frequency range was found to be muchhigher than the other two categories

343 Damping trend

A method based on time-frequency analysis first intro-duced in [17] and most recently applied in [9] was usedhere to find the trend of the modal damping up to the fre-quency of 6 kHz As mentioned earlier the admittancesare not compensated for added damping from strings Fig-

400

Mansour et al Post-classification of steel string guitars ACTA ACUSTICA UNITED WITH ACUSTICAVol 101 (2015)

500 1000 2000 6000minus54

minus52

minus50

minus48

minus46

minus44

minus42

Bandminusaveraged

admittance

(dB) Bandminusaveraged bassy

Bandminusaveraged midminusevenBandminusaveraged trebleMidminuseven plusmn one STDTreble plusmn one STD

500 1000 2000 6000minus22

minus20

minus18

minus16

minus14

minus12

minus10

minus8

minus6

Frequency (Hz)

Bandminusaveraged

radiativity

(dB)

(a)

(b)

Figure 10 Averaged admittance (a) and radiativity (b) takenover bassy instruments (solid) mid-even instruments (dash-dot)plusminus one standard deviation (light gray shade green in on-line version) and treble instruments (dashed) plusminus onestandard deviation (darker gray shade red in online version)The dB reference is 1 msN

ure 11 shows the averaged damping trend taken over thethree perceptual categories

The Q factor for all classes rises more sharply from Q asymp30 at 100 Hz to Qasymp 60 at 600 Hz where it reaches a plateauand rises more gradually afterwards However the generalbehavior is not far from linear on a log-log scale As itcan be seen in Figure 11a and 11b the value of Q factorover different frequencies and its trend is similar for trebleand mid-even classes which make them indistinguishableby looking at the damping trend From the power-law fitresults the average power for bassy category is 008 andfor both categories of mid-even and treble is 016 This re-sults in an average-over-all-instruments of 013 comparedto 026 and -009 reported earlier for the violins and theclassic guitars respectively in [9] Among the instrumentsof the pool only one showed a negative power fit

344 Harmonic features

Plucking a guitar string excites multiple harmonics simul-taneously which makes it possible that a weighted sum ofthe admittance over some range of harmonics rather thanthe admittance at the fundamental only is a better way ofdistinguishing categories of instruments Forgetting aboutlosses for the moment the amplitude of the transverse

200 300 500 1000 2000 5000

30

40

50

70

Frequency (Hz)

Qminusfactortrend

200 300 500 1000 2000 5000

20

30

40

50

70

100

Bandminusaveraged

Qminusfactor

Averaged bassy

Averaged midminuseven

Averaged treble

Averaged midminuseven plusmn STD

Averaged treble plusmn STD

(b)

(a)

Figure 11 Effective Q-factors averaged for the guitars of differ-ent categories plusminus one standard deviation (solid line forbassy dash-dot line and light gray green shade for mid-evenand dashed line and darker gray red shade for treble) band-average (a) and regression line for power-law behavior (b) Notethat vertical axes are different

force that the nth harmonic of a plucked string applies tothe bridge can be found from [5]

Fn =2dT0

nπL0

β

1 minus βsin(βnπ) (1)

where L0 is the length of the string T0 is its tension d isthe displacement at the plucking point and β is the rela-tive distance of the plucking point from the bridge Thisformula does not take into account the possible strongcoupling of the plucked string to the body and to otherstrings which should be a reasonable approximation forthe intended purpose and for most of the notes played ona guitar [16] The magnitude of the power transfer fromthe nth harmonic of the string to the body is then equalto the bridge velocity multiplied by the applied force orequivalently

Pn = Fnvn = F 2nGω = A

sin(βnπ)n

2

Gω (2)

in which Gω is the real part of the bridge admittance (alsocalled conductance) and A is a constant assuming a par-ticular string is being played with a constant pluckingstrength and at a fixed relative distance from the bridgeand again neglecting the losses This power transfer to thebody is largely responsible for the damping of the string

401

ACTA ACUSTICA UNITED WITH ACUSTICA Mansour et al Post-classification of steel string guitarsVol 101 (2015)

mode in question as well as the radiated sound at that fre-quency Based on this idea two features are calculated forall semitones played on a guitar (from E2 to E6) and aver-aged over the three perceptual categories For both cases βis arbitrarily chosen as 03 which is a reasonable numberfor a normal playing condition

Figure 12a shows the sum of the transferred power bythe first 5 harmonics of each played note here calledldquopower transfer to the bodyrdquo The choice of 5 harmonics isan arbitrary limit for which the individual harmonics of thestring could be perceived individually and within whichmost of the energy transfer to the body happens in a nor-mal playing condition The frequency range that is consid-ered in this fashion would be up to 5 times the highest notestudied (ie E6) or equivalently 66 kHz this is about thelimit up to which the measured admittances show a reason-ably good coherence The unknown constant A in equation(2) could only cause an offset in the results therefore itis considered to be 1 for convenience The energy trans-fer to the body represents the strength of the stringbodycoupling for a guitar A large value for this number wouldresult in a strong and more percussive (banjo-like) soundand a small value would result in a weaker sound with alonger decay (more similar to an electric guitar without anamplifier)

Figure 12b shows a slightly different use of the trans-ferred energy by each harmonic It is calculated by divid-ing the sum of the energy transferred by the 2nd to 5th har-monic to the energy transferred by the fundamental aloneAs well as specifying the ratio of higher harmonic energyto that at the fundamental this feature defines how thesound evolves once the instrument is plucked Extremelylarge or small values for a particular note mean that thenote changes its character when it decays while an aver-age value may be correlated to a more stable sound

Neither graph shows any clear distinction between themid-even and treble categories The bassy category doeshowever stand out most of the patterns in both graphs areshifted downward by a few semitones

345 LTAS and radiativity

The LTAS results averaged over different categories areillustrated in Figure 13a The results for the mid-even in-struments are greater than the other two groups over al-most all frequencies Figure 13b is the same as Figure 13abut the LTAS for each individual instrument is normalizedby its area before further processing After normalizationall categories behave similarly over different frequenciesmeaning mid-even instruments were simply louder thanthe other instruments rather than being stronger in a par-ticular range of frequencies

The band-averaged admittance of the mid-even instru-ments earlier presented in Figure 10 was slightly greaterthan the treble instruments in some frequency rangeshowever the significance of the stronger LTAS of mid-even instruments especially beyond 3 kHz can hardly bejustified by that alone A possible explanation is that themid-even instruments were played systematically harder

E2 100 E3 250 E4 500 E5 1000 E6

minus60

minus55

minus50

minus45

minus40

minus35

minus30

Pow

ertransfertothebody

(dB)

E2 100 E3 250 E4 500 E5 1000 E6

minus20

minus10

0

10

Harmonicstofundam

entalratio(dB)

Fundamental frequency (Hz)

Averaged bassyAveraged midminusevenAveraged treble

(a)

(b)

Figure 12 Power transfer to the body (a) and harmonics to fun-damental ratio (b) calculated for each played note over the op-erational range of a guitar (E2 to E6) the vertical lines indicateoctaves The values are averaged for the guitars of different cat-egories (solid line for bassy dash-dot line and light gray green(online version) shade for mid-even and dashed line and darkergray red (online version) shade for treble)

compared to the treble instruments There are at least tworeasons as to why this might have been the case First theguitars were recorded per category starting with the trebleand ending with the mid-even ones It is therefore possiblethat the musician felt more confident in the course of therecording session Second the mid-even guitars had beendescribed by the manufacturer as having a ldquomore stablerdquosound Such a sense might have allowed the performer toplay those instruments more freely Nevertheless we areaware that the frequency distribution obtained using LTAScan be strongly dependent on the nature of the input sig-nal depending on the sequence played and used as an in-put certain frequency bands may be less excited than oth-ers which may attenuate differences in some frequencyregions of interest

35 Classification results

According to our results finding a feature that can distin-guish bassy instruments is a relatively easy task As wasshown in Figure 9a the frequencies of the first 4 modesfor the bassy instruments are around 5 lower than thosefor the two other classes The frequency of the first modecan be determined without much ambiguity or even theneed for mode extraction All of the other 3 low-frequencymodes have close adjacent modes making it harder andsometimes impossible to find their correct modal parame-ters

The amplitude of the first mode is another indicator fora bassy instrument that can be used for more assurance

402

Mansour et al Post-classification of steel string guitars ACTA ACUSTICA UNITED WITH ACUSTICAVol 101 (2015)

500 1000 2000 6000minus180

minus170

minus160

minus150

minus140

minus130

minus120

LTAS(dBHz)

500 1000 2000 6000minus120

minus110

minus100

minus90

minus80

minus70

minus60

Frequency (Hz)

Normalized

LTAS(dBHz)

BassyMidminusevenTrebleTreble plusmn one STDMidminuseven plusmn one STD

(b)

(a)

Figure 13 Non-normalized LTAS (a) and normalized LTAS (b)averaged for the guitars of different categories plusminus onestandard deviation (solid line for bassy dash-dot line and lightgray green (online version) shade for mid-even and dashed lineand darker gray red (online version) shade for treble)

(see Figure 9b) Figure 14a shows a 2-D plane with thenormalized frequency and amplitude of the first mode onits horizontal and vertical axes respectively It can be seenthat the bassy instruments are quite distinct from the mid-even and treble ones Moreover the guitar with the lowestbreathing mode (B3 fT (11)1 = 91 Hz) was described bythe factory employee as extra-bassy

It should be emphasized that the frequency of the firstpeak works particularly well as a feature not becausethe ldquobassinessrdquo of the instrument is defined by the notesaround this modersquos frequency (only the first three noteson the E2 string are below that frequency) but becauseit represents the overall stiffness-to-weight ratio of thesoundbox that effectively scales the frequency of all bodymodes It is noteworthy that although T(11)1 has a strongHelmholtz component the amount by which its frequencydeviates from the frequency of the pure Helmholtz reso-nance is strongly dependent on the frequency separationof the pure Helmholtz and pure body breathing modesThe frequency of the pure Helmholtz mode is assumed thesame for all instruments of the pool given their very simi-lar geometry (both for the sound box and the sound hole)

The in-plane dimensions of the instruments should havea very small variation percentage given the precise con-trol of geometrical conditions in the factory Moreover alower frequency for the first mode is usually accompaniedby a higher amplitude for that mode which would not nor-mally be the case if such deviation was caused by using a

90 92 94 96 98 100 102 10404

05

06

07

08

09

1

Normalized

T(11) 1am

plitude

BassyMid EvenTreble

90 92 94 96 98 100 102 104minus50

minus49

minus48

minus47

minus46

minus45

B1B2

B3

B6

B4B5

ME6ME1

ME2

ME3ME4

ME5

T2

T3

T4

T5 T6

T1

Frequency of the T(11)1mode (Hz)

Averagedmobility(600minus2000Hz)(dB)

(b)

(a)

Figure 14 Classification of the acoustic guitars (a) by the fre-quency and amplitude of their first corpus mode T(11)1 onlyto distinguish bassy instruments from the others and (b) by thefrequency of their first corpus mode T(11)1 and their averagedmobility from 600 Hz to 2 kHz (B = bassy ME = mid-even T =treble) The dB reference is 1 msN

denser wood this leaves variations in the plate thicknessand the equivalent stiffness (ie very roughly the geomet-ric mean of the Youngrsquos moduli in long-grain and cross-grain directions [14]) as those most likely responsible forthe observed shift in mode frequencies Mode frequenciesof a flat plate scale with Eρ(tL2) where E and ρ arethe elastic constant and density t the thickness and L acharacteristic scaling length If either of the two factors issolely responsible to make a 5 shift it would require ei-ther a 5 variation (asymp 0125 mm) of the plate thicknessor a 10 variation in the equivalent stiffness Given thetight tolerances in cutting the plates and their mechanizedsanding it thus appears that variations in equivalent stiff-ness are the primary reason that some of the instrumentsin the pool sound more bassy In the manufacturing pro-cess a wood slab is sliced to produce several plates Thelateral direction of each plate hence represents a particularcombination of the slabrsquos radial and tangential directions(due to the growth rings of the tree) This can introduce asignificant variation in the platesrsquo stiffness in the lateral di-rection which produces variations of the effective stiffnessof the plates

Finding a feature that distinguishes mid-even from tre-ble instruments is less obvious We found that the aver-aged admittances of the treble instruments in this study areslightly lower than the mid-even instruments in the rangeof 600 Hz to 2 kHz Although this difference is not signif-icant over the mentioned range (ie average plusminus

403

ACTA ACUSTICA UNITED WITH ACUSTICA Mansour et al Post-classification of steel string guitarsVol 101 (2015)

one standard deviation of the two categories overlap) itbecomes significant if the average level of the admittancefor the whole range is compared (ie the average valueof all frequency points of the admittance from 600 Hz to2 kHz) Figure 14b shows a 2-D plane with the frequencyof the first mode on the horizontal axis and the average ad-mittance for the range of 600 Hz to 2 kHz The differencefor the mean of the mid-even and treble categories is about2 dB It seems likely that the weaker admittance of the tre-ble category at higher frequencies would contribute to alonger decay for the higher partials This may contributeto a sharper after-sound of the treble instruments that isknown to allow them to cut through the band more effec-tively This hypothesis could be tested by recording theplucked sound of the instruments and comparing the decayrate of the partials among different categories Due to thelimited project time frame this was not investigated fur-ther The mid-even guitars also show stronger LTAS thanthe treble instruments though this result might have beendue to playing conditions (see Section 345)

We applied the same classification (frequency ofT(11)1 versus average mobility between 600 Hz and2 kHz) to 6 acoustic guitars from a different model by thesame manufacturer (Simon amp Patrick) Unlike the Seagullguitars which were all nominally identical some of theSampP instruments had a slightly different bracing patterndesigned to exemplify the properties of a particular per-ceptual category The manufacturerrsquos classification of the6 guitars was not provided to us prior to our own mea-surements though we were told that there were 2 guitarsof each category The resulting classification based on thediscussed features is depicted in Figure 15 The two gui-tars designed to be more bassy were successfully identi-fied from the frequency of their breathing mode but dis-tinguishing between the mid-even and treble instrumentsfrom the averaged admittance criterion was inconclusive

4 Perceptual evaluation

As previously discussed in Section 1 the investigatedacoustic guitars were post-classified by the manufacturerbased on their experience of what type of instrument mightbetter fit a certain playing stylemdashfor example an amateurmusician that plays mostly strummed chords may prefera bassy guitar while a more experienced guitarist mayopt for a treble instrument We therefore conducted a per-ceptual experiment to investigate the level of agreementamong guitar players of varying experience and musicalbackground in classifying the guitars into the categoriesconsidered by the manufacturer

41 Method

The experiment took place in an acoustically dry roomwith dimensions of 60 times 78 times 32 m3 and a reverberationtime of about 03 s to help minimize the effects of roomreflections on the direct sound from the guitars To avoidthe potential problems of using a common pick across all

85 86 87 88 89 90 91 92minus505

minus50

minus495

minus49

minus485

minus48B1

SampP

B2SampP

ME1SampP

ME2SampP

T1SampP

T2SampP

Frequency of the T(11)1mode (Hz)

Averagedmobility(600minus2000Hz)(dB)

BassyMid EvenTreble

Figure 15 Classification of the six SampP guitars based on thefrequency of the T(11)1 mode versus averaged mobility from600 Hz to 2 kHz (B= bassy ME=mid-even T= treble) The dBreference is 1 msN

participants (eg musicians being uncomfortable with apick they are not familiar with) participants were asked tocarry out the classification task using their own pick(s)

411 Participants

Thirteen guitar players took part in the experiment (1 fe-male 12 males average age = 25 yrs SD = 5 yrs range =19ndash36 yrs) They had at least 6 years of acoustic guitar ex-perience (average years of training = 11 yrs SD = 4 yrsrange = 6ndash20 yrs average hours of practice per week =14 hrs SD = 10 hrs range = 3ndash40 hrs) as well as vary-ing experience with different types of guitars [nylon-string(85) acoustic-electric (62) acoustic bass (15) elec-tric (100) and electric bass (54)] Participants re-ported playing a wide range of musical styles [classical(77) jazz (85) pop-rock (92) country folk blue-grass (66) hard rock heavy metal (46) flamenco(15) latin (46) blues (77) avant-garde contempo-rary (54) and funk alternative (46)] both in a bandand as soloists Seven guitarists (54) reported that theytypically or most often use a pick whereas 4 players (31)reported they do not 2 guitarists (15) reported using apick occasionally rather than typically Nine participantshad higher-level degrees in music performance or othermusic-related areas [BMus (music theory classical gui-tar) BFA (jazz studies) MMus (guitar performance) MA(digital musics) PhD (music theory) Diplocircme drsquoEacutetat deProfesseur de Musique (France)] All guitar players werepaid for their participation

412 Procedure

The experimental session lasted 2 hours and was organizedin two phases In the first phase participants were pre-sented with the guitars randomly ordered on the floor (de-termined by computer calculations) They were asked toplay all instruments for up to 20 minutes in order to famil-iarize themselves with the set and use this time to tune theguitars as needed In the second phase participants were

404

Mansour et al Post-classification of steel string guitars ACTA ACUSTICA UNITED WITH ACUSTICAVol 101 (2015)

given up to 20 minutes to classify each of the guitars as ei-ther bassy mid-even or treble by placing them in speciallymarked areas of the room At the end of the second phaseparticipants were given the opportunity to revise their ini-tial judgements if they so wanted until they were satis-fied with their classification No playing constraint wasimposed on the evaluation process (eg specific repertoireor technique) Participants were instead encouraged to fol-low their own strategy with the suggestion to use bothstrumming-oriented songs (eg Sitting Waiting Wishingby Jack Johnson) and fingerpicking-oriented pieces (egDust in the Wind by Kansas) Participants were instructedto maximize evaluation speed and accuracy Upon com-pleting the session participants talked about their classi-fications and overall experience through written responsesto the following questions ldquoFor each class of guitar whatsound characteristics did you look for Can you describehow you evaluated the guitars in terms of these character-istics (eg musical material technique style)rdquo (Q1) ldquoTowhat extend do you agree with the given classification ofguitarsrdquo (Q2) and ldquoCan you suggest a classification thatbetter reflects how you evaluate the sound characteristicsof guitarsrdquo (Q3)

42 Results

Inter-individual consistency was measured as the Fleissrsquokappa coefficient between guitar classifications from dif-ferent participants using a Bayesian approach to inference[18] Fleissrsquo kappa is a generalization of Cohenrsquos kappastatistic κ a chance-corrected measure of agreement be-tween two assessors to multiple raters [19] It ranges fromminus1 to 1 κ (AB) = 1 and minus1 if A = B and A = minusBrespectively and κ (AB) = 0 in case of no associationbetween A and B (ie exactly what would be expectedby chance) We implemented Gibbs sampling through thesoftware WinBUGS (Windows Bayesian inference UsingGibbs Sampling [20]) with 50000 observations generatedfrom the posterior distribution with a burn-in of 1000 ob-servations A uniformly distributed multicategorical priorwas assumed with total information across all possiblecells (ie from different combinations of participants) be-ing equivalent to a single prior observation (ie a priorsample size equivalent of 1)

Fleissrsquos kappa was estimated with a posterior mean of01 and a 95 credible interval [-02 05] indicating avery low amount of agreement between the participantsin their classifications Figure 16 shows the proportion oftimes a guitar was classified as either bassy mid-even ortreble (the guitars are arranged in three groups accord-ing to the initial factory classification) Whereas someagreement concerning the bassy guitars can be observedmdashindeed the extra bassy guitar (B3) was most often clas-sified as bassymdashthe perceptions for the other two classeslargely vary across different musicians In particular manyguitars that had been characterized as treble by the fac-tory employee were perceived as mid-even Several mu-sicians specifically reported encountering difficulties withthe mid-even category In answering Q3 a player observed

B1 B2 B3 B4 B5 B60

25

50

75

100

classifiedas

(a)BassyMidminusevenTreble

ME1 ME2 ME3 ME4 ME5 ME60

25

50

75

100

classifiedas

(b)

T1 T2 T3 T4 T5 T60

25

50

75

100

classifiedas

Guitars

(c)

Figure 16 Perceptual evaluation Proportion of times a guitarwas classified into one of the categories by the participants Gui-tars are labelled and grouped according to the factory classifica-tion (B = bassy ME = mid-even T = treble) The color scheme(in the online version) corresponds to the classifications providedby the guitar players

that ldquothere is a difference [between] a guitar with domi-nant mid tones and one with an even soundrdquo

Overall participants expressed no strong difficulties incarrying out the classification task On a scale from 1(strongly disagree) to 5 (strongly agree) all performers se-lected 4 when asked ldquoTo what extend do you agree withthe given classification of guitarsrdquo This indicates that the3 categories considered by the manufacturer are gener-ally meaningful to musicians However verbal responsesto questions Q1 and Q3 reveal that participants often as-sociated bassy with more prominent E2 and A2 strings(ie they focused on the low-frequency end of the gui-tar) and similarly treble with better emphasized B3 andE4 strings (ie they paid attention to the high-frequencyrange) As one guitar player noted ldquoI feel that descriptionslike warm neutral and bright would be easier to classifydue to the nature of the instrument Bass leads my mindto bass strings and to heed the low end instead of the en-tire range from the highest to lowest notes Same for tre-ble vs bright rdquo In particular 4 participants used the wordldquotwangrdquo to describe guitars they classified as treble whileanother guitarist reported that a treble guitar ldquosounds bestfor finger picking and light strumming as heavy playingcan get quite ear-piercing and harsh This last commentseems to echo the second hypothesis for the origin of thestronger LTAS shown by the mid-even guitars (see Sec-tion 345)

Two participants reported looking out for problematicnotes as a more meaningful way of finding bassy guitarsThis likely points toward the ldquoveeringrdquo phenomenon as-sociated with a stronger admittance at lower frequencies

405

ACTA ACUSTICA UNITED WITH ACUSTICA Mansour et al Post-classification of steel string guitarsVol 101 (2015)

[16] Further one musician referred to the tactile feedbackreceived from a vibrating body associating bassy guitarswith more felt vibrations than the other two categoriesThis is a perspective shared by the manufacturer as wellIn fact body vibrations can provide tactile and proprio-ceptive cues that contribute to the perception of the radi-ated sound so that the musician can assess their interactionwith the instrument cross-modally [21 22]

5 Discussion

A set of 18 steel string guitars of the same model man-ufactured with high geometrical tolerances in a modernfactory were evaluated in this study These guitars wereconsidered by the manufacturer to possess subtle yet per-ceivable variations that could be classified as either bassymid-even or treble The results of this study showed thatthe instruments classified as bassy had a consistently lowerT(11)1 breathing mode (see Figure 9a 14b and 15) Thissuggests a lower stiffness-to-weight ratio for the bodies ofthese guitars which is most likely the result of variationsin equivalent stiffness (see Section 35)

It was further shown that whenever the frequency ofT(11)1 was shifted downward or upward for a particu-lar guitar the frequencies of its higher modes consistentlychanged in a similar way (see Figure 9 also the slightlyshifted bumps at around 600 750 and 1200 Hz in Fig-ure 12a) This would not necessarily be the case if theguitars were of different models (ie with varying geome-tries) or of different types of wood (eg plywood versusspruce) This is an important result as there is a practi-cal advantage to having higher-frequency modes move ac-cording to the first corpus mode across guitars of the samemodel T(11)1 is a low-frequency resonance that is suffi-ciently separated from the adjacent modes and hence eas-ily identifiable Measuring its frequency across differentinstruments is easy and affordable in a small workshopenvironment One can simply hit the guitar with a reg-ular hammer and recordanalyse the sound using an in-expensive microphone and a laptop reminiscent of thetraditional tap-tone method used by luthiers Moreoversuch measurements can be very reliable provided that theboundary conditions of the instrument (ie the installedfixture that holds the instrument) as well as the air tem-perature pressure and humidity are kept consistent Theseparameters constitute the biggest source of uncertainty infrequency-based features and can easily be controlled in afactory space or small workshop

Finding a reliable differentiator between the mid-evenand treble guitars was less obvious Although we ob-served that the averaged mobility curves of the mid-eveninstruments tended to be slightly higher in the vicinityof 600ndash2000 Hz than the treble guitars further measure-ments would have been necessary to determine whetherthe weaker admittance of the treble instruments at higherfrequencies would contribute to a longer decay for thehigher partials (see Section 35) In addition the differ-ence in averaged mobility magnitude between the two cat-

egories was only about 2 dB which is close to the marginof error for the measurement system Such a small differ-ence in an amplitude-based feature may be easily obscuredby variations in the exact angle of the hammer location ofhitting and measuring or calibration of the sensors

We invited guitar players of varying musical back-ground and experience to classify the guitars into thesecategories We observed very poor agreement among theparticipants but it is common in recent perceptual studiesof musical instruments that musicians do not agree witheach other [23 24 25] This seems a result of them lis-tening to (or feeling) different aspects of the instrumentand not necessarily all attending to the same features Theemployee that post-classified the guitars at the factory wasthe person that tests every guitar coming off the productionline thus he must have a lot of experience evaluating thesound of these instruments Assuming he did his classifi-cation carefully then there likely were particular things hewas listening for The participants in our perceptual exper-iment would not necessarily listen for the same featuresso it is not too surprising they largely disagreed on theirclassifications That said the differences between the tre-ble and mid-even guitars were extremely subtle and evencompany representatives often mis-categorized them dur-ing informal playing tests And this together with the factthat our measurements showed no consistent differencesbetween the two categories would seem to lead us to ques-tion whether the distinction between mid-even and trebletruly exists at all Another possible explanation is that dif-ferent musicians may follow different processes to assessthose features considered essential for the evaluation of aninstrument [23 26] Accordingly guitar players may agreeof what particular characteristics they look for in a tre-ble guitar but the perception of the same characteristicswidely varies across individuals

6 Conclusions

Efforts to understand how the tonal quality of an instru-ment relates to its measurable physical properties have of-ten been inconclusive (eg [27]) In the light of some re-cent studies the concept of ldquogoodrdquo versus ldquobadrdquo instru-ments is likely too general and it would be more advanta-geous to instead consider specific perceptual attributes ofan instrument [23 28] In the long run it may be possibleto derive a multidimensional timbre space where each di-mension is explained by some measurable quantities andin which instruments with similar qualities are likely tocluster In this study more specific sound characteristicsof instruments were thus targeted rather than their overalltonal quality

From bridge admittance measurements on 18 steelstring guitars from the same production line and post-classified by the manufacturer as either bassy mid-evenor treble (see Section 1 for details) it was shown that itis possible to discriminate between bassy guitars and theother two categories based on the frequency of the breath-ing mode However finding a similarly reliable feature to

406

Mansour et al Post-classification of steel string guitars ACTA ACUSTICA UNITED WITH ACUSTICAVol 101 (2015)

distinguish mid-even from treble instruments is less triv-ial at this point The treble guitars demonstrated a loweraveraged mobility in the frequency range of 600ndash2000 Hzthough more empirical investigation would be necessarybefore drawing any conclusions Results from a percep-tual experiment wherein guitar players were asked to clas-sify the guitars into these categories showed almost noagreement between guitarists in their judgements as wellas between the latter and the classification by the manufac-turer Despite the overall lack of consensus guitar playerstended to agree more on the bassy instruments than theother two categories

Some essential additions to this study are a more care-ful analysis on the perceptual definition of mid-even ver-sus treble and analysing the decay rate of the partials fora plucked sound to see how it correlates with the admit-tance Analysis of this kind has been performed for syn-thesized acoustic guitar sounds (see Figure 7 in [16]) but asimilar analysis on real guitar sounds will involve difficul-ties caused by the double set of harmonics [29] as well aslongitudinal string vibrations (also dubbed ldquophantom par-tialsrdquo) [30]

Acknowledgements

This project was funded by a Natural Sciences and En-gineering Research Council of Canada Engage grant incollaboration with Godin Guitars The authors wish tothank Robert Godin for this opportunity We also thankLawrence Joseph (statistics) and Garry Antonio (guitarperformance) for constructive comments We are grate-ful to Jim Woodhouse for sharing his analysis code Thefirst author would like to acknowledge the Government ofCanada for a Vanier Canada Graduate Scholarship

References

[1] E V Jansson A study of acoustical and hologram inter-ferometric measurements of the top plate vibrations of aguitar Acust Acta Acust 25 (1971) 95ndash100

[2] O Christensen B B Vistisen Simple model for low-frequency guitar function J Acoust Soc Am 68 (1980)758ndash766

[3] O Christensen Quantitative models for low frequency gui-tar function J Guitar Acoustics 6 (1982) 10ndash25

[4] H Wright The acoustics and psychoacoustics of the guitarDissertation Dept of Physics and Astronomy Universityof Wales Cardiff UK 1996

[5] N H Fletcher T D Rossing The physics of musical in-struments Second ed Springer New York 1998

[6] J E Popp Four mass coupled oscillator guitar model JAcoust Soc Am 131 (2012) 829ndash836

[7] J Woodhouse Body vibration of the violinmdashwhat can amaker except to control Catgut Acoust Soc J (Series II)4 (2002) 43ndash49

[8] B Elie F Gautier B David Macro parameters describingthe mechanical behavior of classical guitars J Acoust SocAm 132 (2012) 4013ndash4024

[9] J Woodhouse R S Langley Interpreting the input admit-tance of violins and guitars Acust Acta Acust 98 (2012)611ndash628

[10] A Gabrielsson E V Jansson Long-time-average-spectraand rated qualities of twenty-two violins Acustica 42(1979) 47ndash55

[11] P D Welch The use of Fast Fourier Transform for the es-timation of power spectra A method based on time averag-ing over short modified periodograms IEEE Trans Audioand Electroacoust AU-15 (1967) 70ndash73

[12] J Sundberg The singerrsquos formant revisited J Voice 4(1995) 106ndash119

[13] A Buen On timbre parameters and sound levels ofrecorded old violins J Violin Soc Am VSA Papers XXI(2007) 57ndash68

[14] J Woodhouse On the ldquobridge hillrdquo of the violin AcustActa Acust 91 (2005) 155ndash165

[15] B Richardson The acoustical development of the guitarCatgut Acoustical Society Journal 2 (1994) 1ndash10

[16] J Woodhouse On the synthesis of guitar plucks AcustActa Acust 90 (2004) 928ndash944

[17] C H Hodges J Power J Woodhouse The use of the sono-gram in structural acoustics and an application to the vibra-tions of cylindrical shells J Sound Vib 101 (1985) 203ndash218

[18] L D Broemeling Bayesian methods for measures ofagreement Chapman amp HallCRC New York 2009 (Bio-statistics)

[19] J L Fleiss Measuring nominal scale agreement amongmany raters Psychological Bulletin 76 (1971) 378ndash382

[20] D Lunn C Jackson N Best A Thomas D Spiegelhal-ter The BUGS Book A Practical Introduction to BayesianAnalysis Chapman amp HallCRC New York 2012 (Textsin Statistical Science)

[21] C Saitis Evaluating violin quality Player reliability andverbalization Dissertation Dept of Music ResearchMcGill University Montreal Quebec Canada 2013

[22] I Wollman Perception bimodale des violonistes en situa-tion de jeu Influence des retours auditif et vibrotactile surlrsquoeacutevaluation du violon Dissertation Universiteacute Pierre etMarie Curie Paris France 2013

[23] C Saitis B L Giordano C Fritz G P Scavone Percep-tual evaluation of violins A quantitative analysis of pref-erence judgements by experienced players J Acoust SocAm 132 (2012) 4002ndash4012

[24] C Fritz J Curtin J Poitevineau P Morrel-Samuels F-CTao Player preferences among new and old violins ProcNat Acad Sci USA 109 (2012) 760ndash763

[25] P Eveno J-P Dalmont R Causseacute G P Scavone Anacoustic and perceptual evaluation of saxophone pad ldquores-onatorsrdquo Acta Acustica united with Acustica 101 (2015)246ndash255

[26] C Fritz J Woodhouse F P-H Cheng I Cross A FBlackwell B C J Moore Perceptual studies of violinbody damping and vibrato J Acoust Soc Am 127 (2010)513ndash524

[27] G Bissinger Structural acoustics of good and bad violinsJ Acoust Soc Am 124 (2008) 1764ndash1773

[28] C Saitis C Fritz C Guastavino G P Scavone Con-ceptualization of violin quality by experienced performersProc Stockholm Music Acoust Conf Stockholm Swe-den 2013 123ndash128

[29] J Woodhouse Plucked guitar transients Comparison ofmeasurements and synthesis Acust Acta Acust 90 (2004)945ndash965

[30] B Bank L Sujbert Generation of longitudinal vibrationsin piano strings From physics to sound synthesis JAcoust Soc Am 117 (2004) 2268ndash2278

407

ACTA ACUSTICA UNITED WITH ACUSTICA Mansour et al Post-classification of steel string guitarsVol 101 (2015)

500 1000 1500 2000 2500 3000 3500 4000 45000002

0004

0006

0008

Re(ltYgt)(msN)

500 1000 1500 2000 2500 3000 3500 4000 4500minus0002

minus0001

0

0001

0002

0003

Frequency (Hz)

Im(ltYgt)(msN)

A sample instrumentAveraged plusmn one STD

(a)

(b)

Figure 4 Band-averaged real (a) and imaginary (b) parts of theadmittance for a sample instrument (solid) and averaged overall instruments of the pool plusminus one standard deviation(shaded) Note that the vertical scales are different for the realand imaginary parts

like a violin nor has it a feature like f-holes to create suchlocal resonances A possible explanation is that the cross-bracing of the steel string guitars unlike the fan-bracing ofmost classical guitars circumvents the bridge and formsan isolated area around it and this may contribute to suchlocal resonances

332 Modal spacing and MOFIt is useful to find a practical upper limit for the range ofmode fitting (ie where the MOF reaches one) and to get asense of the average spacing between the modes Figure 5shows the MOF and the average mode spacing for a sam-ple instrument from the pool and the average plusminusone standard deviation over all instruments An averagingbandwidth of 500 Hz is used in this case to give a betterprecision at lower frequencies The statistical features in-cluding MOF are calculated based on variance theory formore details see equations (17) to (19) of [9] As expectedthe general trend of the overlap factor is rising although itfalls a bit between 1500 Hz and 2000 Hz it reaches thevalue of one at around 610 Hz The average mode spac-ing goes unrealistically high (as much as 150 Hz at around200 Hz) at lower frequencies The mode spacing stays al-most constant at around 20 Hz for the frequency range of600ndash1200 Hz The mode spacing drops to around 8 Hz athigher frequencies and then stays almost unchanged

333 Mode fittingUsing the method described in Section 2 a mode fittingprocess was performed on all 18 instruments of the poolup to 600 Hz [9] The fitting program identified 20 to 24modes for different instruments amongst which a maxi-mum of 16 modes were manually selected for each instru-ment The criterion was to keep the stronger modes thatwere also identified for most of the instruments

The general shape of the admittance follows the pre-dictions of [6] the first neck bending at around 65 Hz

600 1000 1500 2000 2500 3000 35000512

510

40

Modaloverlapfactor

600 1000 1500 2000 2500 3000 35000

10

20

30

40

50

Frequency (Hz)

Modalspacing(Hz)

A sample instrumentAveraged plusmn one STD

(b)

MOF = 1atF asymp 610 Hz

(a)

Figure 5 Modal overlap factor (a) and the average mode spacing(b) for a sample instrument from the pool (solid) and the averageplusminus one standard deviation over all instruments (shaded)An averaging bandwidth of 500 Hz is used

the T(11)1 or the breathing mode at around 100 Hz andT(11)2 or anti-breathing mode at around 200 Hz Modesare categorized as (mn) where m and n are the number ofldquohalf-wavesrdquo across and along the plate and T refers to thetop plate As pointed out by Popp [6] the in-phase motionof the top and back plates coincides in frequency with theanti-breathing mode at around 200 Hz

For our measurements the guitars were held in a fixturewith two arms that pressed against the sides of the ribsThis was intended to create a boundary condition simi-lar to that when held by a player Popp argues that a con-siderable amount of mass would be needed to immobilizethe ribs therefore our measurement condition is betweenthe fixed and free-free cases but perhaps more similar tothe latter After the measurements were performed we ob-served a peak (or a double peak for some instruments)at around 125 Hz where one normally expects to see theanti-resonance representing the pure Helmholtz mode [2]When we repeated the measurements on some of the gui-tars using a free-free boundary condition we noticed thatthe anti-resonance appeared as expected This suggests thepeak is either a property of the fixture rather than the in-struments themselves or it is the (11) in-phase motion ofthe top and back plates shifted downward due to the massof the fixture added to the ribs The latter would requirea 60 Hz downward shift of that frequency which seemsunlikely No effort was made to repeat all the measure-ments with a free-free boundary condition partly becausethe measurements were made before subsequent data anal-ysis and partly because free-free measurements are lesspractical and reproducible in a factoryrsquos production line(due to for example variations in accelerometer place-ment and potential damage to a new guitar surface finishwhen attaching and detaching an accelerometer)

The second neck bending mode was evident as a smallpeak at around 225 Hz for all instruments There weretwo other small peaks at around 250 Hz and 290 Hz which

398

Mansour et al Post-classification of steel string guitars ACTA ACUSTICA UNITED WITH ACUSTICAVol 101 (2015)

were probably related to the torsion and lateral vibrationsof the neck The next important modes were T(21) andT(12) For Martin D-28 acoustic guitars these modes arearound 100 Hz apart [5] hence relatively distinct how-ever for the guitars of our study they both appear at around350 Hz and sometimes switch places We measured the ad-mittance at the bass side of the bridge where T(21) hasits antinode therefore for each guitar the stronger modein that region is associated to T(21) In addition to theHelmholtz mode there are two other cavity modes in thefrequency range of our study which are A1 at 430 Hz andA2 at around 500 Hz [5] Approximate mode shapes ofthe first four corpus modes (exclusive of local resonancessuch as neck bending and neck twisting) for the guitars ofour study are illustrated in Figure 6 the frequencies areaveraged and vary from one instrument to another

The average frequency amplitude and Q-factors calcu-lated for the 18 guitars were used to reconstruct the ad-mittance of a ldquomodally-averagedrdquo guitar that representsthe model that we studied Figure 7 shows this ldquomodally-averaged admittancerdquo reconstructed up to 600 Hz (dashedline) compared with a sample measured admittance aswell as the raw averaged admittance previously shown inFigure 2 It can be seen that the peaks are excessivelydamped for the raw average while the admittance of theldquomodally averagedrdquo guitar shows a better resemblance tothe sample measured admittance close to the resonancesand anti-resonances This modal average thus better rep-resents the admittance of a guitar coming off this partic-ular production line It can be overlaid with the measuredadmittance of a new guitar to highlight its significant fea-tures

If needed the admittance of the ldquomodally averagedrdquoguitar can be reconstructed for extended frequencies eitherby simple averaging over different instruments or moreprecisely by using a ldquostatistical fitrdquo method as describedin [16] In short a random number generator can give thefrequency of the high-frequency modes with the correctdensity and spacing statistics as well as damping factorsand modal masses with approximately correct statisticaldistribution Such a statistical fit was not pursued in thisstudy

334 LTAS

Figure 8 shows the LTAS of a sample instrument fromthe pool (solid black line) as well as the average over allinstruments plusminus one standard deviation (shaded)The first few dominant modes identified in an admittancehave demonstrated themselves in the form of broad peaksin the lower frequencies of the LTAS below about 400 HzSome instruments such as the depicted sample show a peakat around 25 kHz which may correspond to the hill-likefeature earlier identified in band-averaged mobility curves(see Figure 4) Beyond 300 Hz the LTAS drops almostlinearly with the average slope of 16 dBoctave The av-erage standard deviation for the frequency range of 70 Hzto 6 kHz for all instruments is 256 dB which indicates theplayer was able to play consistently on all instruments It is

Figure 6 The four first corpus mode shapes of an acoustic guitartaken from [15]

70 100 200 500minus65

minus60

minus55

minus50

minus45

minus40

minus35

minus30

minus25

minus20

Frequency (Hz)

Averagedadmittance

(dB)

A sample measured admittanceModallyminusaveraged guitarAveragedminusoverminusallminusinstruments

Figure 7 Raw averaged admittance over all instruments of thepool (dash-dot) reconstructed ldquomodally-averagedrdquo admittance(dashed see text for details) and a sample admittance for com-parison (solid) The dB reference is 1 msN

worth mentioning that this number is not directly compa-rable to the averaged standard deviation for the measuredadmittance as the LTAS is significantly smoothed in thefrequency domain in comparison

34 Investigating different features for classification

341 Modal propertiesAveraged value and standard deviation of frequency am-plitude and Q factor of the first four corpus modes are

399

ACTA ACUSTICA UNITED WITH ACUSTICA Mansour et al Post-classification of steel string guitarsVol 101 (2015)

calculated for the three perceptual classes and comparedin Figure 9 In all three figures the values are normalizedto the average of the same modal value taken over all in-struments and the error bars show one standard deviationon each side In case of a missing mode for one instru-ment its frequency and Q factor were eliminated from thecalculation of the average and standard deviation but theamplitude was considered to be zero (as that mode wasnegligible or non-existent) The admittance was not com-pensated for added damping from strings [9] partly be-cause the exact same strings were used on all instrumentshence the comparison was fair and partly because the ex-act amount of damping added by the strings is unknownand the assumption of perfect damping at the strings mayresult in over-compensation of damping [9]

As was also evident from Figure 3 Figure 9a shows thatthe corpus modes of the bassy instruments occur at signif-icantly lower frequencies than the mid-even or treble onesAnother feature that distinguishes bassy instruments fromthe mid-even or treble ones can be seen in Figure 9b theamplitude of the first corpus mode T(11)1 is higher forthe bassy instruments The Q factor of the first four cor-pus modes did not show any noticeable difference amongdifferent perceptual classes

342 Averaged admittance and radiativity

To obtain a more clear comparison of different categoriesin the statistical range (ie at higher frequencies) the ad-mittance and radiativity were smoothed in the frequencydomain using a 500 Hz sliding rectangular window andthen averaged for different categories (see Figure 10a andFigure 10b) The proper way to calculate radiativity wouldhave been to measure the averaged sound pressure over anarray of microphones around the instruments That saidour results using only a single microphone provide a rea-sonable approximation because they were smoothed in thefrequency domain and averaged over different instrumentsSince we have already found distinctive features for the in-struments of the bassy class in the subsequent figures theplusminus standard deviation is eliminated for the bassyclass to help clarity

It can be seen in Figure 10a that the band-averaged ad-mittance of the mid-even instruments is greater than orequal to that of the treble instruments over a wide rangeof frequencies particularly in the range 600 Hz to 2 kHzAlthough this dominance does not appear significant formost of the frequencies (ie the averages for categoriesoverlap when plotted with plusminus one standard devia-tion) the mean level of admittance from 600 Hz to 2 kHzcan distinguish mid-even from treble instruments at leastwithin the pool of our 18 instruments This result will befurther discussed in Section 35 The band-averaged admit-tance of the bassy instruments falls between the two othercategories over most of the frequencies

Band-averaged radiativity does not reveal a significantdifference among different categories except a lower av-erage for the bassy guitars in the range from 2 kHz to275 kHz (see Figure 10b) However this difference is

70 100 200 300 500 1000 2000 6000minus180

minus170

minus160

minus150

minus140

minus130

minus120

minus110

minus100

minus90

minus80

Frequency (Hz)

LTAS(dBHz)

A sample instrumentAveraged plusmn one STD

Figure 8 Long-term average spectra for a sample instrumentfrom the pool (solid line) and the average plusminus one stan-dard deviation over all instruments (shaded)

08

09

1

11

Frequency

(norm) (a)

BassyMidminusevenTreble

0

05

1

15

2

25

Amplitude

(norm)

(b)

05

1

15

2

Qminusfactor(norm)

Guitar corpus modes

(c)

T(21)T(11)2

T(11)1 T(12)

Figure 9 Normalized average frequency (a) amplitude (b) andQ factor (c) for the first four corpus modes for bassy guitars (darkgray bars blue in online version) mid-even guitars (light graybars green in online version) and treble guitars (medium graybras red in online version) The error bars show one standarddeviation on each side

likely not important as the standard deviation of the bassyinstruments in this frequency range was found to be muchhigher than the other two categories

343 Damping trend

A method based on time-frequency analysis first intro-duced in [17] and most recently applied in [9] was usedhere to find the trend of the modal damping up to the fre-quency of 6 kHz As mentioned earlier the admittancesare not compensated for added damping from strings Fig-

400

Mansour et al Post-classification of steel string guitars ACTA ACUSTICA UNITED WITH ACUSTICAVol 101 (2015)

500 1000 2000 6000minus54

minus52

minus50

minus48

minus46

minus44

minus42

Bandminusaveraged

admittance

(dB) Bandminusaveraged bassy

Bandminusaveraged midminusevenBandminusaveraged trebleMidminuseven plusmn one STDTreble plusmn one STD

500 1000 2000 6000minus22

minus20

minus18

minus16

minus14

minus12

minus10

minus8

minus6

Frequency (Hz)

Bandminusaveraged

radiativity

(dB)

(a)

(b)

Figure 10 Averaged admittance (a) and radiativity (b) takenover bassy instruments (solid) mid-even instruments (dash-dot)plusminus one standard deviation (light gray shade green in on-line version) and treble instruments (dashed) plusminus onestandard deviation (darker gray shade red in online version)The dB reference is 1 msN

ure 11 shows the averaged damping trend taken over thethree perceptual categories

The Q factor for all classes rises more sharply from Q asymp30 at 100 Hz to Qasymp 60 at 600 Hz where it reaches a plateauand rises more gradually afterwards However the generalbehavior is not far from linear on a log-log scale As itcan be seen in Figure 11a and 11b the value of Q factorover different frequencies and its trend is similar for trebleand mid-even classes which make them indistinguishableby looking at the damping trend From the power-law fitresults the average power for bassy category is 008 andfor both categories of mid-even and treble is 016 This re-sults in an average-over-all-instruments of 013 comparedto 026 and -009 reported earlier for the violins and theclassic guitars respectively in [9] Among the instrumentsof the pool only one showed a negative power fit

344 Harmonic features

Plucking a guitar string excites multiple harmonics simul-taneously which makes it possible that a weighted sum ofthe admittance over some range of harmonics rather thanthe admittance at the fundamental only is a better way ofdistinguishing categories of instruments Forgetting aboutlosses for the moment the amplitude of the transverse

200 300 500 1000 2000 5000

30

40

50

70

Frequency (Hz)

Qminusfactortrend

200 300 500 1000 2000 5000

20

30

40

50

70

100

Bandminusaveraged

Qminusfactor

Averaged bassy

Averaged midminuseven

Averaged treble

Averaged midminuseven plusmn STD

Averaged treble plusmn STD

(b)

(a)

Figure 11 Effective Q-factors averaged for the guitars of differ-ent categories plusminus one standard deviation (solid line forbassy dash-dot line and light gray green shade for mid-evenand dashed line and darker gray red shade for treble) band-average (a) and regression line for power-law behavior (b) Notethat vertical axes are different

force that the nth harmonic of a plucked string applies tothe bridge can be found from [5]

Fn =2dT0

nπL0

β

1 minus βsin(βnπ) (1)

where L0 is the length of the string T0 is its tension d isthe displacement at the plucking point and β is the rela-tive distance of the plucking point from the bridge Thisformula does not take into account the possible strongcoupling of the plucked string to the body and to otherstrings which should be a reasonable approximation forthe intended purpose and for most of the notes played ona guitar [16] The magnitude of the power transfer fromthe nth harmonic of the string to the body is then equalto the bridge velocity multiplied by the applied force orequivalently

Pn = Fnvn = F 2nGω = A

sin(βnπ)n

2

Gω (2)

in which Gω is the real part of the bridge admittance (alsocalled conductance) and A is a constant assuming a par-ticular string is being played with a constant pluckingstrength and at a fixed relative distance from the bridgeand again neglecting the losses This power transfer to thebody is largely responsible for the damping of the string

401

ACTA ACUSTICA UNITED WITH ACUSTICA Mansour et al Post-classification of steel string guitarsVol 101 (2015)

mode in question as well as the radiated sound at that fre-quency Based on this idea two features are calculated forall semitones played on a guitar (from E2 to E6) and aver-aged over the three perceptual categories For both cases βis arbitrarily chosen as 03 which is a reasonable numberfor a normal playing condition

Figure 12a shows the sum of the transferred power bythe first 5 harmonics of each played note here calledldquopower transfer to the bodyrdquo The choice of 5 harmonics isan arbitrary limit for which the individual harmonics of thestring could be perceived individually and within whichmost of the energy transfer to the body happens in a nor-mal playing condition The frequency range that is consid-ered in this fashion would be up to 5 times the highest notestudied (ie E6) or equivalently 66 kHz this is about thelimit up to which the measured admittances show a reason-ably good coherence The unknown constant A in equation(2) could only cause an offset in the results therefore itis considered to be 1 for convenience The energy trans-fer to the body represents the strength of the stringbodycoupling for a guitar A large value for this number wouldresult in a strong and more percussive (banjo-like) soundand a small value would result in a weaker sound with alonger decay (more similar to an electric guitar without anamplifier)

Figure 12b shows a slightly different use of the trans-ferred energy by each harmonic It is calculated by divid-ing the sum of the energy transferred by the 2nd to 5th har-monic to the energy transferred by the fundamental aloneAs well as specifying the ratio of higher harmonic energyto that at the fundamental this feature defines how thesound evolves once the instrument is plucked Extremelylarge or small values for a particular note mean that thenote changes its character when it decays while an aver-age value may be correlated to a more stable sound

Neither graph shows any clear distinction between themid-even and treble categories The bassy category doeshowever stand out most of the patterns in both graphs areshifted downward by a few semitones

345 LTAS and radiativity

The LTAS results averaged over different categories areillustrated in Figure 13a The results for the mid-even in-struments are greater than the other two groups over al-most all frequencies Figure 13b is the same as Figure 13abut the LTAS for each individual instrument is normalizedby its area before further processing After normalizationall categories behave similarly over different frequenciesmeaning mid-even instruments were simply louder thanthe other instruments rather than being stronger in a par-ticular range of frequencies

The band-averaged admittance of the mid-even instru-ments earlier presented in Figure 10 was slightly greaterthan the treble instruments in some frequency rangeshowever the significance of the stronger LTAS of mid-even instruments especially beyond 3 kHz can hardly bejustified by that alone A possible explanation is that themid-even instruments were played systematically harder

E2 100 E3 250 E4 500 E5 1000 E6

minus60

minus55

minus50

minus45

minus40

minus35

minus30

Pow

ertransfertothebody

(dB)

E2 100 E3 250 E4 500 E5 1000 E6

minus20

minus10

0

10

Harmonicstofundam

entalratio(dB)

Fundamental frequency (Hz)

Averaged bassyAveraged midminusevenAveraged treble

(a)

(b)

Figure 12 Power transfer to the body (a) and harmonics to fun-damental ratio (b) calculated for each played note over the op-erational range of a guitar (E2 to E6) the vertical lines indicateoctaves The values are averaged for the guitars of different cat-egories (solid line for bassy dash-dot line and light gray green(online version) shade for mid-even and dashed line and darkergray red (online version) shade for treble)

compared to the treble instruments There are at least tworeasons as to why this might have been the case First theguitars were recorded per category starting with the trebleand ending with the mid-even ones It is therefore possiblethat the musician felt more confident in the course of therecording session Second the mid-even guitars had beendescribed by the manufacturer as having a ldquomore stablerdquosound Such a sense might have allowed the performer toplay those instruments more freely Nevertheless we areaware that the frequency distribution obtained using LTAScan be strongly dependent on the nature of the input sig-nal depending on the sequence played and used as an in-put certain frequency bands may be less excited than oth-ers which may attenuate differences in some frequencyregions of interest

35 Classification results

According to our results finding a feature that can distin-guish bassy instruments is a relatively easy task As wasshown in Figure 9a the frequencies of the first 4 modesfor the bassy instruments are around 5 lower than thosefor the two other classes The frequency of the first modecan be determined without much ambiguity or even theneed for mode extraction All of the other 3 low-frequencymodes have close adjacent modes making it harder andsometimes impossible to find their correct modal parame-ters

The amplitude of the first mode is another indicator fora bassy instrument that can be used for more assurance

402

Mansour et al Post-classification of steel string guitars ACTA ACUSTICA UNITED WITH ACUSTICAVol 101 (2015)

500 1000 2000 6000minus180

minus170

minus160

minus150

minus140

minus130

minus120

LTAS(dBHz)

500 1000 2000 6000minus120

minus110

minus100

minus90

minus80

minus70

minus60

Frequency (Hz)

Normalized

LTAS(dBHz)

BassyMidminusevenTrebleTreble plusmn one STDMidminuseven plusmn one STD

(b)

(a)

Figure 13 Non-normalized LTAS (a) and normalized LTAS (b)averaged for the guitars of different categories plusminus onestandard deviation (solid line for bassy dash-dot line and lightgray green (online version) shade for mid-even and dashed lineand darker gray red (online version) shade for treble)

(see Figure 9b) Figure 14a shows a 2-D plane with thenormalized frequency and amplitude of the first mode onits horizontal and vertical axes respectively It can be seenthat the bassy instruments are quite distinct from the mid-even and treble ones Moreover the guitar with the lowestbreathing mode (B3 fT (11)1 = 91 Hz) was described bythe factory employee as extra-bassy

It should be emphasized that the frequency of the firstpeak works particularly well as a feature not becausethe ldquobassinessrdquo of the instrument is defined by the notesaround this modersquos frequency (only the first three noteson the E2 string are below that frequency) but becauseit represents the overall stiffness-to-weight ratio of thesoundbox that effectively scales the frequency of all bodymodes It is noteworthy that although T(11)1 has a strongHelmholtz component the amount by which its frequencydeviates from the frequency of the pure Helmholtz reso-nance is strongly dependent on the frequency separationof the pure Helmholtz and pure body breathing modesThe frequency of the pure Helmholtz mode is assumed thesame for all instruments of the pool given their very simi-lar geometry (both for the sound box and the sound hole)

The in-plane dimensions of the instruments should havea very small variation percentage given the precise con-trol of geometrical conditions in the factory Moreover alower frequency for the first mode is usually accompaniedby a higher amplitude for that mode which would not nor-mally be the case if such deviation was caused by using a

90 92 94 96 98 100 102 10404

05

06

07

08

09

1

Normalized

T(11) 1am

plitude

BassyMid EvenTreble

90 92 94 96 98 100 102 104minus50

minus49

minus48

minus47

minus46

minus45

B1B2

B3

B6

B4B5

ME6ME1

ME2

ME3ME4

ME5

T2

T3

T4

T5 T6

T1

Frequency of the T(11)1mode (Hz)

Averagedmobility(600minus2000Hz)(dB)

(b)

(a)

Figure 14 Classification of the acoustic guitars (a) by the fre-quency and amplitude of their first corpus mode T(11)1 onlyto distinguish bassy instruments from the others and (b) by thefrequency of their first corpus mode T(11)1 and their averagedmobility from 600 Hz to 2 kHz (B = bassy ME = mid-even T =treble) The dB reference is 1 msN

denser wood this leaves variations in the plate thicknessand the equivalent stiffness (ie very roughly the geomet-ric mean of the Youngrsquos moduli in long-grain and cross-grain directions [14]) as those most likely responsible forthe observed shift in mode frequencies Mode frequenciesof a flat plate scale with Eρ(tL2) where E and ρ arethe elastic constant and density t the thickness and L acharacteristic scaling length If either of the two factors issolely responsible to make a 5 shift it would require ei-ther a 5 variation (asymp 0125 mm) of the plate thicknessor a 10 variation in the equivalent stiffness Given thetight tolerances in cutting the plates and their mechanizedsanding it thus appears that variations in equivalent stiff-ness are the primary reason that some of the instrumentsin the pool sound more bassy In the manufacturing pro-cess a wood slab is sliced to produce several plates Thelateral direction of each plate hence represents a particularcombination of the slabrsquos radial and tangential directions(due to the growth rings of the tree) This can introduce asignificant variation in the platesrsquo stiffness in the lateral di-rection which produces variations of the effective stiffnessof the plates

Finding a feature that distinguishes mid-even from tre-ble instruments is less obvious We found that the aver-aged admittances of the treble instruments in this study areslightly lower than the mid-even instruments in the rangeof 600 Hz to 2 kHz Although this difference is not signif-icant over the mentioned range (ie average plusminus

403

ACTA ACUSTICA UNITED WITH ACUSTICA Mansour et al Post-classification of steel string guitarsVol 101 (2015)

one standard deviation of the two categories overlap) itbecomes significant if the average level of the admittancefor the whole range is compared (ie the average valueof all frequency points of the admittance from 600 Hz to2 kHz) Figure 14b shows a 2-D plane with the frequencyof the first mode on the horizontal axis and the average ad-mittance for the range of 600 Hz to 2 kHz The differencefor the mean of the mid-even and treble categories is about2 dB It seems likely that the weaker admittance of the tre-ble category at higher frequencies would contribute to alonger decay for the higher partials This may contributeto a sharper after-sound of the treble instruments that isknown to allow them to cut through the band more effec-tively This hypothesis could be tested by recording theplucked sound of the instruments and comparing the decayrate of the partials among different categories Due to thelimited project time frame this was not investigated fur-ther The mid-even guitars also show stronger LTAS thanthe treble instruments though this result might have beendue to playing conditions (see Section 345)

We applied the same classification (frequency ofT(11)1 versus average mobility between 600 Hz and2 kHz) to 6 acoustic guitars from a different model by thesame manufacturer (Simon amp Patrick) Unlike the Seagullguitars which were all nominally identical some of theSampP instruments had a slightly different bracing patterndesigned to exemplify the properties of a particular per-ceptual category The manufacturerrsquos classification of the6 guitars was not provided to us prior to our own mea-surements though we were told that there were 2 guitarsof each category The resulting classification based on thediscussed features is depicted in Figure 15 The two gui-tars designed to be more bassy were successfully identi-fied from the frequency of their breathing mode but dis-tinguishing between the mid-even and treble instrumentsfrom the averaged admittance criterion was inconclusive

4 Perceptual evaluation

As previously discussed in Section 1 the investigatedacoustic guitars were post-classified by the manufacturerbased on their experience of what type of instrument mightbetter fit a certain playing stylemdashfor example an amateurmusician that plays mostly strummed chords may prefera bassy guitar while a more experienced guitarist mayopt for a treble instrument We therefore conducted a per-ceptual experiment to investigate the level of agreementamong guitar players of varying experience and musicalbackground in classifying the guitars into the categoriesconsidered by the manufacturer

41 Method

The experiment took place in an acoustically dry roomwith dimensions of 60 times 78 times 32 m3 and a reverberationtime of about 03 s to help minimize the effects of roomreflections on the direct sound from the guitars To avoidthe potential problems of using a common pick across all

85 86 87 88 89 90 91 92minus505

minus50

minus495

minus49

minus485

minus48B1

SampP

B2SampP

ME1SampP

ME2SampP

T1SampP

T2SampP

Frequency of the T(11)1mode (Hz)

Averagedmobility(600minus2000Hz)(dB)

BassyMid EvenTreble

Figure 15 Classification of the six SampP guitars based on thefrequency of the T(11)1 mode versus averaged mobility from600 Hz to 2 kHz (B= bassy ME=mid-even T= treble) The dBreference is 1 msN

participants (eg musicians being uncomfortable with apick they are not familiar with) participants were asked tocarry out the classification task using their own pick(s)

411 Participants

Thirteen guitar players took part in the experiment (1 fe-male 12 males average age = 25 yrs SD = 5 yrs range =19ndash36 yrs) They had at least 6 years of acoustic guitar ex-perience (average years of training = 11 yrs SD = 4 yrsrange = 6ndash20 yrs average hours of practice per week =14 hrs SD = 10 hrs range = 3ndash40 hrs) as well as vary-ing experience with different types of guitars [nylon-string(85) acoustic-electric (62) acoustic bass (15) elec-tric (100) and electric bass (54)] Participants re-ported playing a wide range of musical styles [classical(77) jazz (85) pop-rock (92) country folk blue-grass (66) hard rock heavy metal (46) flamenco(15) latin (46) blues (77) avant-garde contempo-rary (54) and funk alternative (46)] both in a bandand as soloists Seven guitarists (54) reported that theytypically or most often use a pick whereas 4 players (31)reported they do not 2 guitarists (15) reported using apick occasionally rather than typically Nine participantshad higher-level degrees in music performance or othermusic-related areas [BMus (music theory classical gui-tar) BFA (jazz studies) MMus (guitar performance) MA(digital musics) PhD (music theory) Diplocircme drsquoEacutetat deProfesseur de Musique (France)] All guitar players werepaid for their participation

412 Procedure

The experimental session lasted 2 hours and was organizedin two phases In the first phase participants were pre-sented with the guitars randomly ordered on the floor (de-termined by computer calculations) They were asked toplay all instruments for up to 20 minutes in order to famil-iarize themselves with the set and use this time to tune theguitars as needed In the second phase participants were

404

Mansour et al Post-classification of steel string guitars ACTA ACUSTICA UNITED WITH ACUSTICAVol 101 (2015)

given up to 20 minutes to classify each of the guitars as ei-ther bassy mid-even or treble by placing them in speciallymarked areas of the room At the end of the second phaseparticipants were given the opportunity to revise their ini-tial judgements if they so wanted until they were satis-fied with their classification No playing constraint wasimposed on the evaluation process (eg specific repertoireor technique) Participants were instead encouraged to fol-low their own strategy with the suggestion to use bothstrumming-oriented songs (eg Sitting Waiting Wishingby Jack Johnson) and fingerpicking-oriented pieces (egDust in the Wind by Kansas) Participants were instructedto maximize evaluation speed and accuracy Upon com-pleting the session participants talked about their classi-fications and overall experience through written responsesto the following questions ldquoFor each class of guitar whatsound characteristics did you look for Can you describehow you evaluated the guitars in terms of these character-istics (eg musical material technique style)rdquo (Q1) ldquoTowhat extend do you agree with the given classification ofguitarsrdquo (Q2) and ldquoCan you suggest a classification thatbetter reflects how you evaluate the sound characteristicsof guitarsrdquo (Q3)

42 Results

Inter-individual consistency was measured as the Fleissrsquokappa coefficient between guitar classifications from dif-ferent participants using a Bayesian approach to inference[18] Fleissrsquo kappa is a generalization of Cohenrsquos kappastatistic κ a chance-corrected measure of agreement be-tween two assessors to multiple raters [19] It ranges fromminus1 to 1 κ (AB) = 1 and minus1 if A = B and A = minusBrespectively and κ (AB) = 0 in case of no associationbetween A and B (ie exactly what would be expectedby chance) We implemented Gibbs sampling through thesoftware WinBUGS (Windows Bayesian inference UsingGibbs Sampling [20]) with 50000 observations generatedfrom the posterior distribution with a burn-in of 1000 ob-servations A uniformly distributed multicategorical priorwas assumed with total information across all possiblecells (ie from different combinations of participants) be-ing equivalent to a single prior observation (ie a priorsample size equivalent of 1)

Fleissrsquos kappa was estimated with a posterior mean of01 and a 95 credible interval [-02 05] indicating avery low amount of agreement between the participantsin their classifications Figure 16 shows the proportion oftimes a guitar was classified as either bassy mid-even ortreble (the guitars are arranged in three groups accord-ing to the initial factory classification) Whereas someagreement concerning the bassy guitars can be observedmdashindeed the extra bassy guitar (B3) was most often clas-sified as bassymdashthe perceptions for the other two classeslargely vary across different musicians In particular manyguitars that had been characterized as treble by the fac-tory employee were perceived as mid-even Several mu-sicians specifically reported encountering difficulties withthe mid-even category In answering Q3 a player observed

B1 B2 B3 B4 B5 B60

25

50

75

100

classifiedas

(a)BassyMidminusevenTreble

ME1 ME2 ME3 ME4 ME5 ME60

25

50

75

100

classifiedas

(b)

T1 T2 T3 T4 T5 T60

25

50

75

100

classifiedas

Guitars

(c)

Figure 16 Perceptual evaluation Proportion of times a guitarwas classified into one of the categories by the participants Gui-tars are labelled and grouped according to the factory classifica-tion (B = bassy ME = mid-even T = treble) The color scheme(in the online version) corresponds to the classifications providedby the guitar players

that ldquothere is a difference [between] a guitar with domi-nant mid tones and one with an even soundrdquo

Overall participants expressed no strong difficulties incarrying out the classification task On a scale from 1(strongly disagree) to 5 (strongly agree) all performers se-lected 4 when asked ldquoTo what extend do you agree withthe given classification of guitarsrdquo This indicates that the3 categories considered by the manufacturer are gener-ally meaningful to musicians However verbal responsesto questions Q1 and Q3 reveal that participants often as-sociated bassy with more prominent E2 and A2 strings(ie they focused on the low-frequency end of the gui-tar) and similarly treble with better emphasized B3 andE4 strings (ie they paid attention to the high-frequencyrange) As one guitar player noted ldquoI feel that descriptionslike warm neutral and bright would be easier to classifydue to the nature of the instrument Bass leads my mindto bass strings and to heed the low end instead of the en-tire range from the highest to lowest notes Same for tre-ble vs bright rdquo In particular 4 participants used the wordldquotwangrdquo to describe guitars they classified as treble whileanother guitarist reported that a treble guitar ldquosounds bestfor finger picking and light strumming as heavy playingcan get quite ear-piercing and harsh This last commentseems to echo the second hypothesis for the origin of thestronger LTAS shown by the mid-even guitars (see Sec-tion 345)

Two participants reported looking out for problematicnotes as a more meaningful way of finding bassy guitarsThis likely points toward the ldquoveeringrdquo phenomenon as-sociated with a stronger admittance at lower frequencies

405

ACTA ACUSTICA UNITED WITH ACUSTICA Mansour et al Post-classification of steel string guitarsVol 101 (2015)

[16] Further one musician referred to the tactile feedbackreceived from a vibrating body associating bassy guitarswith more felt vibrations than the other two categoriesThis is a perspective shared by the manufacturer as wellIn fact body vibrations can provide tactile and proprio-ceptive cues that contribute to the perception of the radi-ated sound so that the musician can assess their interactionwith the instrument cross-modally [21 22]

5 Discussion

A set of 18 steel string guitars of the same model man-ufactured with high geometrical tolerances in a modernfactory were evaluated in this study These guitars wereconsidered by the manufacturer to possess subtle yet per-ceivable variations that could be classified as either bassymid-even or treble The results of this study showed thatthe instruments classified as bassy had a consistently lowerT(11)1 breathing mode (see Figure 9a 14b and 15) Thissuggests a lower stiffness-to-weight ratio for the bodies ofthese guitars which is most likely the result of variationsin equivalent stiffness (see Section 35)

It was further shown that whenever the frequency ofT(11)1 was shifted downward or upward for a particu-lar guitar the frequencies of its higher modes consistentlychanged in a similar way (see Figure 9 also the slightlyshifted bumps at around 600 750 and 1200 Hz in Fig-ure 12a) This would not necessarily be the case if theguitars were of different models (ie with varying geome-tries) or of different types of wood (eg plywood versusspruce) This is an important result as there is a practi-cal advantage to having higher-frequency modes move ac-cording to the first corpus mode across guitars of the samemodel T(11)1 is a low-frequency resonance that is suffi-ciently separated from the adjacent modes and hence eas-ily identifiable Measuring its frequency across differentinstruments is easy and affordable in a small workshopenvironment One can simply hit the guitar with a reg-ular hammer and recordanalyse the sound using an in-expensive microphone and a laptop reminiscent of thetraditional tap-tone method used by luthiers Moreoversuch measurements can be very reliable provided that theboundary conditions of the instrument (ie the installedfixture that holds the instrument) as well as the air tem-perature pressure and humidity are kept consistent Theseparameters constitute the biggest source of uncertainty infrequency-based features and can easily be controlled in afactory space or small workshop

Finding a reliable differentiator between the mid-evenand treble guitars was less obvious Although we ob-served that the averaged mobility curves of the mid-eveninstruments tended to be slightly higher in the vicinityof 600ndash2000 Hz than the treble guitars further measure-ments would have been necessary to determine whetherthe weaker admittance of the treble instruments at higherfrequencies would contribute to a longer decay for thehigher partials (see Section 35) In addition the differ-ence in averaged mobility magnitude between the two cat-

egories was only about 2 dB which is close to the marginof error for the measurement system Such a small differ-ence in an amplitude-based feature may be easily obscuredby variations in the exact angle of the hammer location ofhitting and measuring or calibration of the sensors

We invited guitar players of varying musical back-ground and experience to classify the guitars into thesecategories We observed very poor agreement among theparticipants but it is common in recent perceptual studiesof musical instruments that musicians do not agree witheach other [23 24 25] This seems a result of them lis-tening to (or feeling) different aspects of the instrumentand not necessarily all attending to the same features Theemployee that post-classified the guitars at the factory wasthe person that tests every guitar coming off the productionline thus he must have a lot of experience evaluating thesound of these instruments Assuming he did his classifi-cation carefully then there likely were particular things hewas listening for The participants in our perceptual exper-iment would not necessarily listen for the same featuresso it is not too surprising they largely disagreed on theirclassifications That said the differences between the tre-ble and mid-even guitars were extremely subtle and evencompany representatives often mis-categorized them dur-ing informal playing tests And this together with the factthat our measurements showed no consistent differencesbetween the two categories would seem to lead us to ques-tion whether the distinction between mid-even and trebletruly exists at all Another possible explanation is that dif-ferent musicians may follow different processes to assessthose features considered essential for the evaluation of aninstrument [23 26] Accordingly guitar players may agreeof what particular characteristics they look for in a tre-ble guitar but the perception of the same characteristicswidely varies across individuals

6 Conclusions

Efforts to understand how the tonal quality of an instru-ment relates to its measurable physical properties have of-ten been inconclusive (eg [27]) In the light of some re-cent studies the concept of ldquogoodrdquo versus ldquobadrdquo instru-ments is likely too general and it would be more advanta-geous to instead consider specific perceptual attributes ofan instrument [23 28] In the long run it may be possibleto derive a multidimensional timbre space where each di-mension is explained by some measurable quantities andin which instruments with similar qualities are likely tocluster In this study more specific sound characteristicsof instruments were thus targeted rather than their overalltonal quality

From bridge admittance measurements on 18 steelstring guitars from the same production line and post-classified by the manufacturer as either bassy mid-evenor treble (see Section 1 for details) it was shown that itis possible to discriminate between bassy guitars and theother two categories based on the frequency of the breath-ing mode However finding a similarly reliable feature to

406

Mansour et al Post-classification of steel string guitars ACTA ACUSTICA UNITED WITH ACUSTICAVol 101 (2015)

distinguish mid-even from treble instruments is less triv-ial at this point The treble guitars demonstrated a loweraveraged mobility in the frequency range of 600ndash2000 Hzthough more empirical investigation would be necessarybefore drawing any conclusions Results from a percep-tual experiment wherein guitar players were asked to clas-sify the guitars into these categories showed almost noagreement between guitarists in their judgements as wellas between the latter and the classification by the manufac-turer Despite the overall lack of consensus guitar playerstended to agree more on the bassy instruments than theother two categories

Some essential additions to this study are a more care-ful analysis on the perceptual definition of mid-even ver-sus treble and analysing the decay rate of the partials fora plucked sound to see how it correlates with the admit-tance Analysis of this kind has been performed for syn-thesized acoustic guitar sounds (see Figure 7 in [16]) but asimilar analysis on real guitar sounds will involve difficul-ties caused by the double set of harmonics [29] as well aslongitudinal string vibrations (also dubbed ldquophantom par-tialsrdquo) [30]

Acknowledgements

This project was funded by a Natural Sciences and En-gineering Research Council of Canada Engage grant incollaboration with Godin Guitars The authors wish tothank Robert Godin for this opportunity We also thankLawrence Joseph (statistics) and Garry Antonio (guitarperformance) for constructive comments We are grate-ful to Jim Woodhouse for sharing his analysis code Thefirst author would like to acknowledge the Government ofCanada for a Vanier Canada Graduate Scholarship

References

[1] E V Jansson A study of acoustical and hologram inter-ferometric measurements of the top plate vibrations of aguitar Acust Acta Acust 25 (1971) 95ndash100

[2] O Christensen B B Vistisen Simple model for low-frequency guitar function J Acoust Soc Am 68 (1980)758ndash766

[3] O Christensen Quantitative models for low frequency gui-tar function J Guitar Acoustics 6 (1982) 10ndash25

[4] H Wright The acoustics and psychoacoustics of the guitarDissertation Dept of Physics and Astronomy Universityof Wales Cardiff UK 1996

[5] N H Fletcher T D Rossing The physics of musical in-struments Second ed Springer New York 1998

[6] J E Popp Four mass coupled oscillator guitar model JAcoust Soc Am 131 (2012) 829ndash836

[7] J Woodhouse Body vibration of the violinmdashwhat can amaker except to control Catgut Acoust Soc J (Series II)4 (2002) 43ndash49

[8] B Elie F Gautier B David Macro parameters describingthe mechanical behavior of classical guitars J Acoust SocAm 132 (2012) 4013ndash4024

[9] J Woodhouse R S Langley Interpreting the input admit-tance of violins and guitars Acust Acta Acust 98 (2012)611ndash628

[10] A Gabrielsson E V Jansson Long-time-average-spectraand rated qualities of twenty-two violins Acustica 42(1979) 47ndash55

[11] P D Welch The use of Fast Fourier Transform for the es-timation of power spectra A method based on time averag-ing over short modified periodograms IEEE Trans Audioand Electroacoust AU-15 (1967) 70ndash73

[12] J Sundberg The singerrsquos formant revisited J Voice 4(1995) 106ndash119

[13] A Buen On timbre parameters and sound levels ofrecorded old violins J Violin Soc Am VSA Papers XXI(2007) 57ndash68

[14] J Woodhouse On the ldquobridge hillrdquo of the violin AcustActa Acust 91 (2005) 155ndash165

[15] B Richardson The acoustical development of the guitarCatgut Acoustical Society Journal 2 (1994) 1ndash10

[16] J Woodhouse On the synthesis of guitar plucks AcustActa Acust 90 (2004) 928ndash944

[17] C H Hodges J Power J Woodhouse The use of the sono-gram in structural acoustics and an application to the vibra-tions of cylindrical shells J Sound Vib 101 (1985) 203ndash218

[18] L D Broemeling Bayesian methods for measures ofagreement Chapman amp HallCRC New York 2009 (Bio-statistics)

[19] J L Fleiss Measuring nominal scale agreement amongmany raters Psychological Bulletin 76 (1971) 378ndash382

[20] D Lunn C Jackson N Best A Thomas D Spiegelhal-ter The BUGS Book A Practical Introduction to BayesianAnalysis Chapman amp HallCRC New York 2012 (Textsin Statistical Science)

[21] C Saitis Evaluating violin quality Player reliability andverbalization Dissertation Dept of Music ResearchMcGill University Montreal Quebec Canada 2013

[22] I Wollman Perception bimodale des violonistes en situa-tion de jeu Influence des retours auditif et vibrotactile surlrsquoeacutevaluation du violon Dissertation Universiteacute Pierre etMarie Curie Paris France 2013

[23] C Saitis B L Giordano C Fritz G P Scavone Percep-tual evaluation of violins A quantitative analysis of pref-erence judgements by experienced players J Acoust SocAm 132 (2012) 4002ndash4012

[24] C Fritz J Curtin J Poitevineau P Morrel-Samuels F-CTao Player preferences among new and old violins ProcNat Acad Sci USA 109 (2012) 760ndash763

[25] P Eveno J-P Dalmont R Causseacute G P Scavone Anacoustic and perceptual evaluation of saxophone pad ldquores-onatorsrdquo Acta Acustica united with Acustica 101 (2015)246ndash255

[26] C Fritz J Woodhouse F P-H Cheng I Cross A FBlackwell B C J Moore Perceptual studies of violinbody damping and vibrato J Acoust Soc Am 127 (2010)513ndash524

[27] G Bissinger Structural acoustics of good and bad violinsJ Acoust Soc Am 124 (2008) 1764ndash1773

[28] C Saitis C Fritz C Guastavino G P Scavone Con-ceptualization of violin quality by experienced performersProc Stockholm Music Acoust Conf Stockholm Swe-den 2013 123ndash128

[29] J Woodhouse Plucked guitar transients Comparison ofmeasurements and synthesis Acust Acta Acust 90 (2004)945ndash965

[30] B Bank L Sujbert Generation of longitudinal vibrationsin piano strings From physics to sound synthesis JAcoust Soc Am 117 (2004) 2268ndash2278

407

Mansour et al Post-classification of steel string guitars ACTA ACUSTICA UNITED WITH ACUSTICAVol 101 (2015)

were probably related to the torsion and lateral vibrationsof the neck The next important modes were T(21) andT(12) For Martin D-28 acoustic guitars these modes arearound 100 Hz apart [5] hence relatively distinct how-ever for the guitars of our study they both appear at around350 Hz and sometimes switch places We measured the ad-mittance at the bass side of the bridge where T(21) hasits antinode therefore for each guitar the stronger modein that region is associated to T(21) In addition to theHelmholtz mode there are two other cavity modes in thefrequency range of our study which are A1 at 430 Hz andA2 at around 500 Hz [5] Approximate mode shapes ofthe first four corpus modes (exclusive of local resonancessuch as neck bending and neck twisting) for the guitars ofour study are illustrated in Figure 6 the frequencies areaveraged and vary from one instrument to another

The average frequency amplitude and Q-factors calcu-lated for the 18 guitars were used to reconstruct the ad-mittance of a ldquomodally-averagedrdquo guitar that representsthe model that we studied Figure 7 shows this ldquomodally-averaged admittancerdquo reconstructed up to 600 Hz (dashedline) compared with a sample measured admittance aswell as the raw averaged admittance previously shown inFigure 2 It can be seen that the peaks are excessivelydamped for the raw average while the admittance of theldquomodally averagedrdquo guitar shows a better resemblance tothe sample measured admittance close to the resonancesand anti-resonances This modal average thus better rep-resents the admittance of a guitar coming off this partic-ular production line It can be overlaid with the measuredadmittance of a new guitar to highlight its significant fea-tures

If needed the admittance of the ldquomodally averagedrdquoguitar can be reconstructed for extended frequencies eitherby simple averaging over different instruments or moreprecisely by using a ldquostatistical fitrdquo method as describedin [16] In short a random number generator can give thefrequency of the high-frequency modes with the correctdensity and spacing statistics as well as damping factorsand modal masses with approximately correct statisticaldistribution Such a statistical fit was not pursued in thisstudy

334 LTAS

Figure 8 shows the LTAS of a sample instrument fromthe pool (solid black line) as well as the average over allinstruments plusminus one standard deviation (shaded)The first few dominant modes identified in an admittancehave demonstrated themselves in the form of broad peaksin the lower frequencies of the LTAS below about 400 HzSome instruments such as the depicted sample show a peakat around 25 kHz which may correspond to the hill-likefeature earlier identified in band-averaged mobility curves(see Figure 4) Beyond 300 Hz the LTAS drops almostlinearly with the average slope of 16 dBoctave The av-erage standard deviation for the frequency range of 70 Hzto 6 kHz for all instruments is 256 dB which indicates theplayer was able to play consistently on all instruments It is

Figure 6 The four first corpus mode shapes of an acoustic guitartaken from [15]

70 100 200 500minus65

minus60

minus55

minus50

minus45

minus40

minus35

minus30

minus25

minus20

Frequency (Hz)

Averagedadmittance

(dB)

A sample measured admittanceModallyminusaveraged guitarAveragedminusoverminusallminusinstruments

Figure 7 Raw averaged admittance over all instruments of thepool (dash-dot) reconstructed ldquomodally-averagedrdquo admittance(dashed see text for details) and a sample admittance for com-parison (solid) The dB reference is 1 msN

worth mentioning that this number is not directly compa-rable to the averaged standard deviation for the measuredadmittance as the LTAS is significantly smoothed in thefrequency domain in comparison

34 Investigating different features for classification

341 Modal propertiesAveraged value and standard deviation of frequency am-plitude and Q factor of the first four corpus modes are

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ACTA ACUSTICA UNITED WITH ACUSTICA Mansour et al Post-classification of steel string guitarsVol 101 (2015)

calculated for the three perceptual classes and comparedin Figure 9 In all three figures the values are normalizedto the average of the same modal value taken over all in-struments and the error bars show one standard deviationon each side In case of a missing mode for one instru-ment its frequency and Q factor were eliminated from thecalculation of the average and standard deviation but theamplitude was considered to be zero (as that mode wasnegligible or non-existent) The admittance was not com-pensated for added damping from strings [9] partly be-cause the exact same strings were used on all instrumentshence the comparison was fair and partly because the ex-act amount of damping added by the strings is unknownand the assumption of perfect damping at the strings mayresult in over-compensation of damping [9]

As was also evident from Figure 3 Figure 9a shows thatthe corpus modes of the bassy instruments occur at signif-icantly lower frequencies than the mid-even or treble onesAnother feature that distinguishes bassy instruments fromthe mid-even or treble ones can be seen in Figure 9b theamplitude of the first corpus mode T(11)1 is higher forthe bassy instruments The Q factor of the first four cor-pus modes did not show any noticeable difference amongdifferent perceptual classes

342 Averaged admittance and radiativity

To obtain a more clear comparison of different categoriesin the statistical range (ie at higher frequencies) the ad-mittance and radiativity were smoothed in the frequencydomain using a 500 Hz sliding rectangular window andthen averaged for different categories (see Figure 10a andFigure 10b) The proper way to calculate radiativity wouldhave been to measure the averaged sound pressure over anarray of microphones around the instruments That saidour results using only a single microphone provide a rea-sonable approximation because they were smoothed in thefrequency domain and averaged over different instrumentsSince we have already found distinctive features for the in-struments of the bassy class in the subsequent figures theplusminus standard deviation is eliminated for the bassyclass to help clarity

It can be seen in Figure 10a that the band-averaged ad-mittance of the mid-even instruments is greater than orequal to that of the treble instruments over a wide rangeof frequencies particularly in the range 600 Hz to 2 kHzAlthough this dominance does not appear significant formost of the frequencies (ie the averages for categoriesoverlap when plotted with plusminus one standard devia-tion) the mean level of admittance from 600 Hz to 2 kHzcan distinguish mid-even from treble instruments at leastwithin the pool of our 18 instruments This result will befurther discussed in Section 35 The band-averaged admit-tance of the bassy instruments falls between the two othercategories over most of the frequencies

Band-averaged radiativity does not reveal a significantdifference among different categories except a lower av-erage for the bassy guitars in the range from 2 kHz to275 kHz (see Figure 10b) However this difference is

70 100 200 300 500 1000 2000 6000minus180

minus170

minus160

minus150

minus140

minus130

minus120

minus110

minus100

minus90

minus80

Frequency (Hz)

LTAS(dBHz)

A sample instrumentAveraged plusmn one STD

Figure 8 Long-term average spectra for a sample instrumentfrom the pool (solid line) and the average plusminus one stan-dard deviation over all instruments (shaded)

08

09

1

11

Frequency

(norm) (a)

BassyMidminusevenTreble

0

05

1

15

2

25

Amplitude

(norm)

(b)

05

1

15

2

Qminusfactor(norm)

Guitar corpus modes

(c)

T(21)T(11)2

T(11)1 T(12)

Figure 9 Normalized average frequency (a) amplitude (b) andQ factor (c) for the first four corpus modes for bassy guitars (darkgray bars blue in online version) mid-even guitars (light graybars green in online version) and treble guitars (medium graybras red in online version) The error bars show one standarddeviation on each side

likely not important as the standard deviation of the bassyinstruments in this frequency range was found to be muchhigher than the other two categories

343 Damping trend

A method based on time-frequency analysis first intro-duced in [17] and most recently applied in [9] was usedhere to find the trend of the modal damping up to the fre-quency of 6 kHz As mentioned earlier the admittancesare not compensated for added damping from strings Fig-

400

Mansour et al Post-classification of steel string guitars ACTA ACUSTICA UNITED WITH ACUSTICAVol 101 (2015)

500 1000 2000 6000minus54

minus52

minus50

minus48

minus46

minus44

minus42

Bandminusaveraged

admittance

(dB) Bandminusaveraged bassy

Bandminusaveraged midminusevenBandminusaveraged trebleMidminuseven plusmn one STDTreble plusmn one STD

500 1000 2000 6000minus22

minus20

minus18

minus16

minus14

minus12

minus10

minus8

minus6

Frequency (Hz)

Bandminusaveraged

radiativity

(dB)

(a)

(b)

Figure 10 Averaged admittance (a) and radiativity (b) takenover bassy instruments (solid) mid-even instruments (dash-dot)plusminus one standard deviation (light gray shade green in on-line version) and treble instruments (dashed) plusminus onestandard deviation (darker gray shade red in online version)The dB reference is 1 msN

ure 11 shows the averaged damping trend taken over thethree perceptual categories

The Q factor for all classes rises more sharply from Q asymp30 at 100 Hz to Qasymp 60 at 600 Hz where it reaches a plateauand rises more gradually afterwards However the generalbehavior is not far from linear on a log-log scale As itcan be seen in Figure 11a and 11b the value of Q factorover different frequencies and its trend is similar for trebleand mid-even classes which make them indistinguishableby looking at the damping trend From the power-law fitresults the average power for bassy category is 008 andfor both categories of mid-even and treble is 016 This re-sults in an average-over-all-instruments of 013 comparedto 026 and -009 reported earlier for the violins and theclassic guitars respectively in [9] Among the instrumentsof the pool only one showed a negative power fit

344 Harmonic features

Plucking a guitar string excites multiple harmonics simul-taneously which makes it possible that a weighted sum ofthe admittance over some range of harmonics rather thanthe admittance at the fundamental only is a better way ofdistinguishing categories of instruments Forgetting aboutlosses for the moment the amplitude of the transverse

200 300 500 1000 2000 5000

30

40

50

70

Frequency (Hz)

Qminusfactortrend

200 300 500 1000 2000 5000

20

30

40

50

70

100

Bandminusaveraged

Qminusfactor

Averaged bassy

Averaged midminuseven

Averaged treble

Averaged midminuseven plusmn STD

Averaged treble plusmn STD

(b)

(a)

Figure 11 Effective Q-factors averaged for the guitars of differ-ent categories plusminus one standard deviation (solid line forbassy dash-dot line and light gray green shade for mid-evenand dashed line and darker gray red shade for treble) band-average (a) and regression line for power-law behavior (b) Notethat vertical axes are different

force that the nth harmonic of a plucked string applies tothe bridge can be found from [5]

Fn =2dT0

nπL0

β

1 minus βsin(βnπ) (1)

where L0 is the length of the string T0 is its tension d isthe displacement at the plucking point and β is the rela-tive distance of the plucking point from the bridge Thisformula does not take into account the possible strongcoupling of the plucked string to the body and to otherstrings which should be a reasonable approximation forthe intended purpose and for most of the notes played ona guitar [16] The magnitude of the power transfer fromthe nth harmonic of the string to the body is then equalto the bridge velocity multiplied by the applied force orequivalently

Pn = Fnvn = F 2nGω = A

sin(βnπ)n

2

Gω (2)

in which Gω is the real part of the bridge admittance (alsocalled conductance) and A is a constant assuming a par-ticular string is being played with a constant pluckingstrength and at a fixed relative distance from the bridgeand again neglecting the losses This power transfer to thebody is largely responsible for the damping of the string

401

ACTA ACUSTICA UNITED WITH ACUSTICA Mansour et al Post-classification of steel string guitarsVol 101 (2015)

mode in question as well as the radiated sound at that fre-quency Based on this idea two features are calculated forall semitones played on a guitar (from E2 to E6) and aver-aged over the three perceptual categories For both cases βis arbitrarily chosen as 03 which is a reasonable numberfor a normal playing condition

Figure 12a shows the sum of the transferred power bythe first 5 harmonics of each played note here calledldquopower transfer to the bodyrdquo The choice of 5 harmonics isan arbitrary limit for which the individual harmonics of thestring could be perceived individually and within whichmost of the energy transfer to the body happens in a nor-mal playing condition The frequency range that is consid-ered in this fashion would be up to 5 times the highest notestudied (ie E6) or equivalently 66 kHz this is about thelimit up to which the measured admittances show a reason-ably good coherence The unknown constant A in equation(2) could only cause an offset in the results therefore itis considered to be 1 for convenience The energy trans-fer to the body represents the strength of the stringbodycoupling for a guitar A large value for this number wouldresult in a strong and more percussive (banjo-like) soundand a small value would result in a weaker sound with alonger decay (more similar to an electric guitar without anamplifier)

Figure 12b shows a slightly different use of the trans-ferred energy by each harmonic It is calculated by divid-ing the sum of the energy transferred by the 2nd to 5th har-monic to the energy transferred by the fundamental aloneAs well as specifying the ratio of higher harmonic energyto that at the fundamental this feature defines how thesound evolves once the instrument is plucked Extremelylarge or small values for a particular note mean that thenote changes its character when it decays while an aver-age value may be correlated to a more stable sound

Neither graph shows any clear distinction between themid-even and treble categories The bassy category doeshowever stand out most of the patterns in both graphs areshifted downward by a few semitones

345 LTAS and radiativity

The LTAS results averaged over different categories areillustrated in Figure 13a The results for the mid-even in-struments are greater than the other two groups over al-most all frequencies Figure 13b is the same as Figure 13abut the LTAS for each individual instrument is normalizedby its area before further processing After normalizationall categories behave similarly over different frequenciesmeaning mid-even instruments were simply louder thanthe other instruments rather than being stronger in a par-ticular range of frequencies

The band-averaged admittance of the mid-even instru-ments earlier presented in Figure 10 was slightly greaterthan the treble instruments in some frequency rangeshowever the significance of the stronger LTAS of mid-even instruments especially beyond 3 kHz can hardly bejustified by that alone A possible explanation is that themid-even instruments were played systematically harder

E2 100 E3 250 E4 500 E5 1000 E6

minus60

minus55

minus50

minus45

minus40

minus35

minus30

Pow

ertransfertothebody

(dB)

E2 100 E3 250 E4 500 E5 1000 E6

minus20

minus10

0

10

Harmonicstofundam

entalratio(dB)

Fundamental frequency (Hz)

Averaged bassyAveraged midminusevenAveraged treble

(a)

(b)

Figure 12 Power transfer to the body (a) and harmonics to fun-damental ratio (b) calculated for each played note over the op-erational range of a guitar (E2 to E6) the vertical lines indicateoctaves The values are averaged for the guitars of different cat-egories (solid line for bassy dash-dot line and light gray green(online version) shade for mid-even and dashed line and darkergray red (online version) shade for treble)

compared to the treble instruments There are at least tworeasons as to why this might have been the case First theguitars were recorded per category starting with the trebleand ending with the mid-even ones It is therefore possiblethat the musician felt more confident in the course of therecording session Second the mid-even guitars had beendescribed by the manufacturer as having a ldquomore stablerdquosound Such a sense might have allowed the performer toplay those instruments more freely Nevertheless we areaware that the frequency distribution obtained using LTAScan be strongly dependent on the nature of the input sig-nal depending on the sequence played and used as an in-put certain frequency bands may be less excited than oth-ers which may attenuate differences in some frequencyregions of interest

35 Classification results

According to our results finding a feature that can distin-guish bassy instruments is a relatively easy task As wasshown in Figure 9a the frequencies of the first 4 modesfor the bassy instruments are around 5 lower than thosefor the two other classes The frequency of the first modecan be determined without much ambiguity or even theneed for mode extraction All of the other 3 low-frequencymodes have close adjacent modes making it harder andsometimes impossible to find their correct modal parame-ters

The amplitude of the first mode is another indicator fora bassy instrument that can be used for more assurance

402

Mansour et al Post-classification of steel string guitars ACTA ACUSTICA UNITED WITH ACUSTICAVol 101 (2015)

500 1000 2000 6000minus180

minus170

minus160

minus150

minus140

minus130

minus120

LTAS(dBHz)

500 1000 2000 6000minus120

minus110

minus100

minus90

minus80

minus70

minus60

Frequency (Hz)

Normalized

LTAS(dBHz)

BassyMidminusevenTrebleTreble plusmn one STDMidminuseven plusmn one STD

(b)

(a)

Figure 13 Non-normalized LTAS (a) and normalized LTAS (b)averaged for the guitars of different categories plusminus onestandard deviation (solid line for bassy dash-dot line and lightgray green (online version) shade for mid-even and dashed lineand darker gray red (online version) shade for treble)

(see Figure 9b) Figure 14a shows a 2-D plane with thenormalized frequency and amplitude of the first mode onits horizontal and vertical axes respectively It can be seenthat the bassy instruments are quite distinct from the mid-even and treble ones Moreover the guitar with the lowestbreathing mode (B3 fT (11)1 = 91 Hz) was described bythe factory employee as extra-bassy

It should be emphasized that the frequency of the firstpeak works particularly well as a feature not becausethe ldquobassinessrdquo of the instrument is defined by the notesaround this modersquos frequency (only the first three noteson the E2 string are below that frequency) but becauseit represents the overall stiffness-to-weight ratio of thesoundbox that effectively scales the frequency of all bodymodes It is noteworthy that although T(11)1 has a strongHelmholtz component the amount by which its frequencydeviates from the frequency of the pure Helmholtz reso-nance is strongly dependent on the frequency separationof the pure Helmholtz and pure body breathing modesThe frequency of the pure Helmholtz mode is assumed thesame for all instruments of the pool given their very simi-lar geometry (both for the sound box and the sound hole)

The in-plane dimensions of the instruments should havea very small variation percentage given the precise con-trol of geometrical conditions in the factory Moreover alower frequency for the first mode is usually accompaniedby a higher amplitude for that mode which would not nor-mally be the case if such deviation was caused by using a

90 92 94 96 98 100 102 10404

05

06

07

08

09

1

Normalized

T(11) 1am

plitude

BassyMid EvenTreble

90 92 94 96 98 100 102 104minus50

minus49

minus48

minus47

minus46

minus45

B1B2

B3

B6

B4B5

ME6ME1

ME2

ME3ME4

ME5

T2

T3

T4

T5 T6

T1

Frequency of the T(11)1mode (Hz)

Averagedmobility(600minus2000Hz)(dB)

(b)

(a)

Figure 14 Classification of the acoustic guitars (a) by the fre-quency and amplitude of their first corpus mode T(11)1 onlyto distinguish bassy instruments from the others and (b) by thefrequency of their first corpus mode T(11)1 and their averagedmobility from 600 Hz to 2 kHz (B = bassy ME = mid-even T =treble) The dB reference is 1 msN

denser wood this leaves variations in the plate thicknessand the equivalent stiffness (ie very roughly the geomet-ric mean of the Youngrsquos moduli in long-grain and cross-grain directions [14]) as those most likely responsible forthe observed shift in mode frequencies Mode frequenciesof a flat plate scale with Eρ(tL2) where E and ρ arethe elastic constant and density t the thickness and L acharacteristic scaling length If either of the two factors issolely responsible to make a 5 shift it would require ei-ther a 5 variation (asymp 0125 mm) of the plate thicknessor a 10 variation in the equivalent stiffness Given thetight tolerances in cutting the plates and their mechanizedsanding it thus appears that variations in equivalent stiff-ness are the primary reason that some of the instrumentsin the pool sound more bassy In the manufacturing pro-cess a wood slab is sliced to produce several plates Thelateral direction of each plate hence represents a particularcombination of the slabrsquos radial and tangential directions(due to the growth rings of the tree) This can introduce asignificant variation in the platesrsquo stiffness in the lateral di-rection which produces variations of the effective stiffnessof the plates

Finding a feature that distinguishes mid-even from tre-ble instruments is less obvious We found that the aver-aged admittances of the treble instruments in this study areslightly lower than the mid-even instruments in the rangeof 600 Hz to 2 kHz Although this difference is not signif-icant over the mentioned range (ie average plusminus

403

ACTA ACUSTICA UNITED WITH ACUSTICA Mansour et al Post-classification of steel string guitarsVol 101 (2015)

one standard deviation of the two categories overlap) itbecomes significant if the average level of the admittancefor the whole range is compared (ie the average valueof all frequency points of the admittance from 600 Hz to2 kHz) Figure 14b shows a 2-D plane with the frequencyof the first mode on the horizontal axis and the average ad-mittance for the range of 600 Hz to 2 kHz The differencefor the mean of the mid-even and treble categories is about2 dB It seems likely that the weaker admittance of the tre-ble category at higher frequencies would contribute to alonger decay for the higher partials This may contributeto a sharper after-sound of the treble instruments that isknown to allow them to cut through the band more effec-tively This hypothesis could be tested by recording theplucked sound of the instruments and comparing the decayrate of the partials among different categories Due to thelimited project time frame this was not investigated fur-ther The mid-even guitars also show stronger LTAS thanthe treble instruments though this result might have beendue to playing conditions (see Section 345)

We applied the same classification (frequency ofT(11)1 versus average mobility between 600 Hz and2 kHz) to 6 acoustic guitars from a different model by thesame manufacturer (Simon amp Patrick) Unlike the Seagullguitars which were all nominally identical some of theSampP instruments had a slightly different bracing patterndesigned to exemplify the properties of a particular per-ceptual category The manufacturerrsquos classification of the6 guitars was not provided to us prior to our own mea-surements though we were told that there were 2 guitarsof each category The resulting classification based on thediscussed features is depicted in Figure 15 The two gui-tars designed to be more bassy were successfully identi-fied from the frequency of their breathing mode but dis-tinguishing between the mid-even and treble instrumentsfrom the averaged admittance criterion was inconclusive

4 Perceptual evaluation

As previously discussed in Section 1 the investigatedacoustic guitars were post-classified by the manufacturerbased on their experience of what type of instrument mightbetter fit a certain playing stylemdashfor example an amateurmusician that plays mostly strummed chords may prefera bassy guitar while a more experienced guitarist mayopt for a treble instrument We therefore conducted a per-ceptual experiment to investigate the level of agreementamong guitar players of varying experience and musicalbackground in classifying the guitars into the categoriesconsidered by the manufacturer

41 Method

The experiment took place in an acoustically dry roomwith dimensions of 60 times 78 times 32 m3 and a reverberationtime of about 03 s to help minimize the effects of roomreflections on the direct sound from the guitars To avoidthe potential problems of using a common pick across all

85 86 87 88 89 90 91 92minus505

minus50

minus495

minus49

minus485

minus48B1

SampP

B2SampP

ME1SampP

ME2SampP

T1SampP

T2SampP

Frequency of the T(11)1mode (Hz)

Averagedmobility(600minus2000Hz)(dB)

BassyMid EvenTreble

Figure 15 Classification of the six SampP guitars based on thefrequency of the T(11)1 mode versus averaged mobility from600 Hz to 2 kHz (B= bassy ME=mid-even T= treble) The dBreference is 1 msN

participants (eg musicians being uncomfortable with apick they are not familiar with) participants were asked tocarry out the classification task using their own pick(s)

411 Participants

Thirteen guitar players took part in the experiment (1 fe-male 12 males average age = 25 yrs SD = 5 yrs range =19ndash36 yrs) They had at least 6 years of acoustic guitar ex-perience (average years of training = 11 yrs SD = 4 yrsrange = 6ndash20 yrs average hours of practice per week =14 hrs SD = 10 hrs range = 3ndash40 hrs) as well as vary-ing experience with different types of guitars [nylon-string(85) acoustic-electric (62) acoustic bass (15) elec-tric (100) and electric bass (54)] Participants re-ported playing a wide range of musical styles [classical(77) jazz (85) pop-rock (92) country folk blue-grass (66) hard rock heavy metal (46) flamenco(15) latin (46) blues (77) avant-garde contempo-rary (54) and funk alternative (46)] both in a bandand as soloists Seven guitarists (54) reported that theytypically or most often use a pick whereas 4 players (31)reported they do not 2 guitarists (15) reported using apick occasionally rather than typically Nine participantshad higher-level degrees in music performance or othermusic-related areas [BMus (music theory classical gui-tar) BFA (jazz studies) MMus (guitar performance) MA(digital musics) PhD (music theory) Diplocircme drsquoEacutetat deProfesseur de Musique (France)] All guitar players werepaid for their participation

412 Procedure

The experimental session lasted 2 hours and was organizedin two phases In the first phase participants were pre-sented with the guitars randomly ordered on the floor (de-termined by computer calculations) They were asked toplay all instruments for up to 20 minutes in order to famil-iarize themselves with the set and use this time to tune theguitars as needed In the second phase participants were

404

Mansour et al Post-classification of steel string guitars ACTA ACUSTICA UNITED WITH ACUSTICAVol 101 (2015)

given up to 20 minutes to classify each of the guitars as ei-ther bassy mid-even or treble by placing them in speciallymarked areas of the room At the end of the second phaseparticipants were given the opportunity to revise their ini-tial judgements if they so wanted until they were satis-fied with their classification No playing constraint wasimposed on the evaluation process (eg specific repertoireor technique) Participants were instead encouraged to fol-low their own strategy with the suggestion to use bothstrumming-oriented songs (eg Sitting Waiting Wishingby Jack Johnson) and fingerpicking-oriented pieces (egDust in the Wind by Kansas) Participants were instructedto maximize evaluation speed and accuracy Upon com-pleting the session participants talked about their classi-fications and overall experience through written responsesto the following questions ldquoFor each class of guitar whatsound characteristics did you look for Can you describehow you evaluated the guitars in terms of these character-istics (eg musical material technique style)rdquo (Q1) ldquoTowhat extend do you agree with the given classification ofguitarsrdquo (Q2) and ldquoCan you suggest a classification thatbetter reflects how you evaluate the sound characteristicsof guitarsrdquo (Q3)

42 Results

Inter-individual consistency was measured as the Fleissrsquokappa coefficient between guitar classifications from dif-ferent participants using a Bayesian approach to inference[18] Fleissrsquo kappa is a generalization of Cohenrsquos kappastatistic κ a chance-corrected measure of agreement be-tween two assessors to multiple raters [19] It ranges fromminus1 to 1 κ (AB) = 1 and minus1 if A = B and A = minusBrespectively and κ (AB) = 0 in case of no associationbetween A and B (ie exactly what would be expectedby chance) We implemented Gibbs sampling through thesoftware WinBUGS (Windows Bayesian inference UsingGibbs Sampling [20]) with 50000 observations generatedfrom the posterior distribution with a burn-in of 1000 ob-servations A uniformly distributed multicategorical priorwas assumed with total information across all possiblecells (ie from different combinations of participants) be-ing equivalent to a single prior observation (ie a priorsample size equivalent of 1)

Fleissrsquos kappa was estimated with a posterior mean of01 and a 95 credible interval [-02 05] indicating avery low amount of agreement between the participantsin their classifications Figure 16 shows the proportion oftimes a guitar was classified as either bassy mid-even ortreble (the guitars are arranged in three groups accord-ing to the initial factory classification) Whereas someagreement concerning the bassy guitars can be observedmdashindeed the extra bassy guitar (B3) was most often clas-sified as bassymdashthe perceptions for the other two classeslargely vary across different musicians In particular manyguitars that had been characterized as treble by the fac-tory employee were perceived as mid-even Several mu-sicians specifically reported encountering difficulties withthe mid-even category In answering Q3 a player observed

B1 B2 B3 B4 B5 B60

25

50

75

100

classifiedas

(a)BassyMidminusevenTreble

ME1 ME2 ME3 ME4 ME5 ME60

25

50

75

100

classifiedas

(b)

T1 T2 T3 T4 T5 T60

25

50

75

100

classifiedas

Guitars

(c)

Figure 16 Perceptual evaluation Proportion of times a guitarwas classified into one of the categories by the participants Gui-tars are labelled and grouped according to the factory classifica-tion (B = bassy ME = mid-even T = treble) The color scheme(in the online version) corresponds to the classifications providedby the guitar players

that ldquothere is a difference [between] a guitar with domi-nant mid tones and one with an even soundrdquo

Overall participants expressed no strong difficulties incarrying out the classification task On a scale from 1(strongly disagree) to 5 (strongly agree) all performers se-lected 4 when asked ldquoTo what extend do you agree withthe given classification of guitarsrdquo This indicates that the3 categories considered by the manufacturer are gener-ally meaningful to musicians However verbal responsesto questions Q1 and Q3 reveal that participants often as-sociated bassy with more prominent E2 and A2 strings(ie they focused on the low-frequency end of the gui-tar) and similarly treble with better emphasized B3 andE4 strings (ie they paid attention to the high-frequencyrange) As one guitar player noted ldquoI feel that descriptionslike warm neutral and bright would be easier to classifydue to the nature of the instrument Bass leads my mindto bass strings and to heed the low end instead of the en-tire range from the highest to lowest notes Same for tre-ble vs bright rdquo In particular 4 participants used the wordldquotwangrdquo to describe guitars they classified as treble whileanother guitarist reported that a treble guitar ldquosounds bestfor finger picking and light strumming as heavy playingcan get quite ear-piercing and harsh This last commentseems to echo the second hypothesis for the origin of thestronger LTAS shown by the mid-even guitars (see Sec-tion 345)

Two participants reported looking out for problematicnotes as a more meaningful way of finding bassy guitarsThis likely points toward the ldquoveeringrdquo phenomenon as-sociated with a stronger admittance at lower frequencies

405

ACTA ACUSTICA UNITED WITH ACUSTICA Mansour et al Post-classification of steel string guitarsVol 101 (2015)

[16] Further one musician referred to the tactile feedbackreceived from a vibrating body associating bassy guitarswith more felt vibrations than the other two categoriesThis is a perspective shared by the manufacturer as wellIn fact body vibrations can provide tactile and proprio-ceptive cues that contribute to the perception of the radi-ated sound so that the musician can assess their interactionwith the instrument cross-modally [21 22]

5 Discussion

A set of 18 steel string guitars of the same model man-ufactured with high geometrical tolerances in a modernfactory were evaluated in this study These guitars wereconsidered by the manufacturer to possess subtle yet per-ceivable variations that could be classified as either bassymid-even or treble The results of this study showed thatthe instruments classified as bassy had a consistently lowerT(11)1 breathing mode (see Figure 9a 14b and 15) Thissuggests a lower stiffness-to-weight ratio for the bodies ofthese guitars which is most likely the result of variationsin equivalent stiffness (see Section 35)

It was further shown that whenever the frequency ofT(11)1 was shifted downward or upward for a particu-lar guitar the frequencies of its higher modes consistentlychanged in a similar way (see Figure 9 also the slightlyshifted bumps at around 600 750 and 1200 Hz in Fig-ure 12a) This would not necessarily be the case if theguitars were of different models (ie with varying geome-tries) or of different types of wood (eg plywood versusspruce) This is an important result as there is a practi-cal advantage to having higher-frequency modes move ac-cording to the first corpus mode across guitars of the samemodel T(11)1 is a low-frequency resonance that is suffi-ciently separated from the adjacent modes and hence eas-ily identifiable Measuring its frequency across differentinstruments is easy and affordable in a small workshopenvironment One can simply hit the guitar with a reg-ular hammer and recordanalyse the sound using an in-expensive microphone and a laptop reminiscent of thetraditional tap-tone method used by luthiers Moreoversuch measurements can be very reliable provided that theboundary conditions of the instrument (ie the installedfixture that holds the instrument) as well as the air tem-perature pressure and humidity are kept consistent Theseparameters constitute the biggest source of uncertainty infrequency-based features and can easily be controlled in afactory space or small workshop

Finding a reliable differentiator between the mid-evenand treble guitars was less obvious Although we ob-served that the averaged mobility curves of the mid-eveninstruments tended to be slightly higher in the vicinityof 600ndash2000 Hz than the treble guitars further measure-ments would have been necessary to determine whetherthe weaker admittance of the treble instruments at higherfrequencies would contribute to a longer decay for thehigher partials (see Section 35) In addition the differ-ence in averaged mobility magnitude between the two cat-

egories was only about 2 dB which is close to the marginof error for the measurement system Such a small differ-ence in an amplitude-based feature may be easily obscuredby variations in the exact angle of the hammer location ofhitting and measuring or calibration of the sensors

We invited guitar players of varying musical back-ground and experience to classify the guitars into thesecategories We observed very poor agreement among theparticipants but it is common in recent perceptual studiesof musical instruments that musicians do not agree witheach other [23 24 25] This seems a result of them lis-tening to (or feeling) different aspects of the instrumentand not necessarily all attending to the same features Theemployee that post-classified the guitars at the factory wasthe person that tests every guitar coming off the productionline thus he must have a lot of experience evaluating thesound of these instruments Assuming he did his classifi-cation carefully then there likely were particular things hewas listening for The participants in our perceptual exper-iment would not necessarily listen for the same featuresso it is not too surprising they largely disagreed on theirclassifications That said the differences between the tre-ble and mid-even guitars were extremely subtle and evencompany representatives often mis-categorized them dur-ing informal playing tests And this together with the factthat our measurements showed no consistent differencesbetween the two categories would seem to lead us to ques-tion whether the distinction between mid-even and trebletruly exists at all Another possible explanation is that dif-ferent musicians may follow different processes to assessthose features considered essential for the evaluation of aninstrument [23 26] Accordingly guitar players may agreeof what particular characteristics they look for in a tre-ble guitar but the perception of the same characteristicswidely varies across individuals

6 Conclusions

Efforts to understand how the tonal quality of an instru-ment relates to its measurable physical properties have of-ten been inconclusive (eg [27]) In the light of some re-cent studies the concept of ldquogoodrdquo versus ldquobadrdquo instru-ments is likely too general and it would be more advanta-geous to instead consider specific perceptual attributes ofan instrument [23 28] In the long run it may be possibleto derive a multidimensional timbre space where each di-mension is explained by some measurable quantities andin which instruments with similar qualities are likely tocluster In this study more specific sound characteristicsof instruments were thus targeted rather than their overalltonal quality

From bridge admittance measurements on 18 steelstring guitars from the same production line and post-classified by the manufacturer as either bassy mid-evenor treble (see Section 1 for details) it was shown that itis possible to discriminate between bassy guitars and theother two categories based on the frequency of the breath-ing mode However finding a similarly reliable feature to

406

Mansour et al Post-classification of steel string guitars ACTA ACUSTICA UNITED WITH ACUSTICAVol 101 (2015)

distinguish mid-even from treble instruments is less triv-ial at this point The treble guitars demonstrated a loweraveraged mobility in the frequency range of 600ndash2000 Hzthough more empirical investigation would be necessarybefore drawing any conclusions Results from a percep-tual experiment wherein guitar players were asked to clas-sify the guitars into these categories showed almost noagreement between guitarists in their judgements as wellas between the latter and the classification by the manufac-turer Despite the overall lack of consensus guitar playerstended to agree more on the bassy instruments than theother two categories

Some essential additions to this study are a more care-ful analysis on the perceptual definition of mid-even ver-sus treble and analysing the decay rate of the partials fora plucked sound to see how it correlates with the admit-tance Analysis of this kind has been performed for syn-thesized acoustic guitar sounds (see Figure 7 in [16]) but asimilar analysis on real guitar sounds will involve difficul-ties caused by the double set of harmonics [29] as well aslongitudinal string vibrations (also dubbed ldquophantom par-tialsrdquo) [30]

Acknowledgements

This project was funded by a Natural Sciences and En-gineering Research Council of Canada Engage grant incollaboration with Godin Guitars The authors wish tothank Robert Godin for this opportunity We also thankLawrence Joseph (statistics) and Garry Antonio (guitarperformance) for constructive comments We are grate-ful to Jim Woodhouse for sharing his analysis code Thefirst author would like to acknowledge the Government ofCanada for a Vanier Canada Graduate Scholarship

References

[1] E V Jansson A study of acoustical and hologram inter-ferometric measurements of the top plate vibrations of aguitar Acust Acta Acust 25 (1971) 95ndash100

[2] O Christensen B B Vistisen Simple model for low-frequency guitar function J Acoust Soc Am 68 (1980)758ndash766

[3] O Christensen Quantitative models for low frequency gui-tar function J Guitar Acoustics 6 (1982) 10ndash25

[4] H Wright The acoustics and psychoacoustics of the guitarDissertation Dept of Physics and Astronomy Universityof Wales Cardiff UK 1996

[5] N H Fletcher T D Rossing The physics of musical in-struments Second ed Springer New York 1998

[6] J E Popp Four mass coupled oscillator guitar model JAcoust Soc Am 131 (2012) 829ndash836

[7] J Woodhouse Body vibration of the violinmdashwhat can amaker except to control Catgut Acoust Soc J (Series II)4 (2002) 43ndash49

[8] B Elie F Gautier B David Macro parameters describingthe mechanical behavior of classical guitars J Acoust SocAm 132 (2012) 4013ndash4024

[9] J Woodhouse R S Langley Interpreting the input admit-tance of violins and guitars Acust Acta Acust 98 (2012)611ndash628

[10] A Gabrielsson E V Jansson Long-time-average-spectraand rated qualities of twenty-two violins Acustica 42(1979) 47ndash55

[11] P D Welch The use of Fast Fourier Transform for the es-timation of power spectra A method based on time averag-ing over short modified periodograms IEEE Trans Audioand Electroacoust AU-15 (1967) 70ndash73

[12] J Sundberg The singerrsquos formant revisited J Voice 4(1995) 106ndash119

[13] A Buen On timbre parameters and sound levels ofrecorded old violins J Violin Soc Am VSA Papers XXI(2007) 57ndash68

[14] J Woodhouse On the ldquobridge hillrdquo of the violin AcustActa Acust 91 (2005) 155ndash165

[15] B Richardson The acoustical development of the guitarCatgut Acoustical Society Journal 2 (1994) 1ndash10

[16] J Woodhouse On the synthesis of guitar plucks AcustActa Acust 90 (2004) 928ndash944

[17] C H Hodges J Power J Woodhouse The use of the sono-gram in structural acoustics and an application to the vibra-tions of cylindrical shells J Sound Vib 101 (1985) 203ndash218

[18] L D Broemeling Bayesian methods for measures ofagreement Chapman amp HallCRC New York 2009 (Bio-statistics)

[19] J L Fleiss Measuring nominal scale agreement amongmany raters Psychological Bulletin 76 (1971) 378ndash382

[20] D Lunn C Jackson N Best A Thomas D Spiegelhal-ter The BUGS Book A Practical Introduction to BayesianAnalysis Chapman amp HallCRC New York 2012 (Textsin Statistical Science)

[21] C Saitis Evaluating violin quality Player reliability andverbalization Dissertation Dept of Music ResearchMcGill University Montreal Quebec Canada 2013

[22] I Wollman Perception bimodale des violonistes en situa-tion de jeu Influence des retours auditif et vibrotactile surlrsquoeacutevaluation du violon Dissertation Universiteacute Pierre etMarie Curie Paris France 2013

[23] C Saitis B L Giordano C Fritz G P Scavone Percep-tual evaluation of violins A quantitative analysis of pref-erence judgements by experienced players J Acoust SocAm 132 (2012) 4002ndash4012

[24] C Fritz J Curtin J Poitevineau P Morrel-Samuels F-CTao Player preferences among new and old violins ProcNat Acad Sci USA 109 (2012) 760ndash763

[25] P Eveno J-P Dalmont R Causseacute G P Scavone Anacoustic and perceptual evaluation of saxophone pad ldquores-onatorsrdquo Acta Acustica united with Acustica 101 (2015)246ndash255

[26] C Fritz J Woodhouse F P-H Cheng I Cross A FBlackwell B C J Moore Perceptual studies of violinbody damping and vibrato J Acoust Soc Am 127 (2010)513ndash524

[27] G Bissinger Structural acoustics of good and bad violinsJ Acoust Soc Am 124 (2008) 1764ndash1773

[28] C Saitis C Fritz C Guastavino G P Scavone Con-ceptualization of violin quality by experienced performersProc Stockholm Music Acoust Conf Stockholm Swe-den 2013 123ndash128

[29] J Woodhouse Plucked guitar transients Comparison ofmeasurements and synthesis Acust Acta Acust 90 (2004)945ndash965

[30] B Bank L Sujbert Generation of longitudinal vibrationsin piano strings From physics to sound synthesis JAcoust Soc Am 117 (2004) 2268ndash2278

407

ACTA ACUSTICA UNITED WITH ACUSTICA Mansour et al Post-classification of steel string guitarsVol 101 (2015)

calculated for the three perceptual classes and comparedin Figure 9 In all three figures the values are normalizedto the average of the same modal value taken over all in-struments and the error bars show one standard deviationon each side In case of a missing mode for one instru-ment its frequency and Q factor were eliminated from thecalculation of the average and standard deviation but theamplitude was considered to be zero (as that mode wasnegligible or non-existent) The admittance was not com-pensated for added damping from strings [9] partly be-cause the exact same strings were used on all instrumentshence the comparison was fair and partly because the ex-act amount of damping added by the strings is unknownand the assumption of perfect damping at the strings mayresult in over-compensation of damping [9]

As was also evident from Figure 3 Figure 9a shows thatthe corpus modes of the bassy instruments occur at signif-icantly lower frequencies than the mid-even or treble onesAnother feature that distinguishes bassy instruments fromthe mid-even or treble ones can be seen in Figure 9b theamplitude of the first corpus mode T(11)1 is higher forthe bassy instruments The Q factor of the first four cor-pus modes did not show any noticeable difference amongdifferent perceptual classes

342 Averaged admittance and radiativity

To obtain a more clear comparison of different categoriesin the statistical range (ie at higher frequencies) the ad-mittance and radiativity were smoothed in the frequencydomain using a 500 Hz sliding rectangular window andthen averaged for different categories (see Figure 10a andFigure 10b) The proper way to calculate radiativity wouldhave been to measure the averaged sound pressure over anarray of microphones around the instruments That saidour results using only a single microphone provide a rea-sonable approximation because they were smoothed in thefrequency domain and averaged over different instrumentsSince we have already found distinctive features for the in-struments of the bassy class in the subsequent figures theplusminus standard deviation is eliminated for the bassyclass to help clarity

It can be seen in Figure 10a that the band-averaged ad-mittance of the mid-even instruments is greater than orequal to that of the treble instruments over a wide rangeof frequencies particularly in the range 600 Hz to 2 kHzAlthough this dominance does not appear significant formost of the frequencies (ie the averages for categoriesoverlap when plotted with plusminus one standard devia-tion) the mean level of admittance from 600 Hz to 2 kHzcan distinguish mid-even from treble instruments at leastwithin the pool of our 18 instruments This result will befurther discussed in Section 35 The band-averaged admit-tance of the bassy instruments falls between the two othercategories over most of the frequencies

Band-averaged radiativity does not reveal a significantdifference among different categories except a lower av-erage for the bassy guitars in the range from 2 kHz to275 kHz (see Figure 10b) However this difference is

70 100 200 300 500 1000 2000 6000minus180

minus170

minus160

minus150

minus140

minus130

minus120

minus110

minus100

minus90

minus80

Frequency (Hz)

LTAS(dBHz)

A sample instrumentAveraged plusmn one STD

Figure 8 Long-term average spectra for a sample instrumentfrom the pool (solid line) and the average plusminus one stan-dard deviation over all instruments (shaded)

08

09

1

11

Frequency

(norm) (a)

BassyMidminusevenTreble

0

05

1

15

2

25

Amplitude

(norm)

(b)

05

1

15

2

Qminusfactor(norm)

Guitar corpus modes

(c)

T(21)T(11)2

T(11)1 T(12)

Figure 9 Normalized average frequency (a) amplitude (b) andQ factor (c) for the first four corpus modes for bassy guitars (darkgray bars blue in online version) mid-even guitars (light graybars green in online version) and treble guitars (medium graybras red in online version) The error bars show one standarddeviation on each side

likely not important as the standard deviation of the bassyinstruments in this frequency range was found to be muchhigher than the other two categories

343 Damping trend

A method based on time-frequency analysis first intro-duced in [17] and most recently applied in [9] was usedhere to find the trend of the modal damping up to the fre-quency of 6 kHz As mentioned earlier the admittancesare not compensated for added damping from strings Fig-

400

Mansour et al Post-classification of steel string guitars ACTA ACUSTICA UNITED WITH ACUSTICAVol 101 (2015)

500 1000 2000 6000minus54

minus52

minus50

minus48

minus46

minus44

minus42

Bandminusaveraged

admittance

(dB) Bandminusaveraged bassy

Bandminusaveraged midminusevenBandminusaveraged trebleMidminuseven plusmn one STDTreble plusmn one STD

500 1000 2000 6000minus22

minus20

minus18

minus16

minus14

minus12

minus10

minus8

minus6

Frequency (Hz)

Bandminusaveraged

radiativity

(dB)

(a)

(b)

Figure 10 Averaged admittance (a) and radiativity (b) takenover bassy instruments (solid) mid-even instruments (dash-dot)plusminus one standard deviation (light gray shade green in on-line version) and treble instruments (dashed) plusminus onestandard deviation (darker gray shade red in online version)The dB reference is 1 msN

ure 11 shows the averaged damping trend taken over thethree perceptual categories

The Q factor for all classes rises more sharply from Q asymp30 at 100 Hz to Qasymp 60 at 600 Hz where it reaches a plateauand rises more gradually afterwards However the generalbehavior is not far from linear on a log-log scale As itcan be seen in Figure 11a and 11b the value of Q factorover different frequencies and its trend is similar for trebleand mid-even classes which make them indistinguishableby looking at the damping trend From the power-law fitresults the average power for bassy category is 008 andfor both categories of mid-even and treble is 016 This re-sults in an average-over-all-instruments of 013 comparedto 026 and -009 reported earlier for the violins and theclassic guitars respectively in [9] Among the instrumentsof the pool only one showed a negative power fit

344 Harmonic features

Plucking a guitar string excites multiple harmonics simul-taneously which makes it possible that a weighted sum ofthe admittance over some range of harmonics rather thanthe admittance at the fundamental only is a better way ofdistinguishing categories of instruments Forgetting aboutlosses for the moment the amplitude of the transverse

200 300 500 1000 2000 5000

30

40

50

70

Frequency (Hz)

Qminusfactortrend

200 300 500 1000 2000 5000

20

30

40

50

70

100

Bandminusaveraged

Qminusfactor

Averaged bassy

Averaged midminuseven

Averaged treble

Averaged midminuseven plusmn STD

Averaged treble plusmn STD

(b)

(a)

Figure 11 Effective Q-factors averaged for the guitars of differ-ent categories plusminus one standard deviation (solid line forbassy dash-dot line and light gray green shade for mid-evenand dashed line and darker gray red shade for treble) band-average (a) and regression line for power-law behavior (b) Notethat vertical axes are different

force that the nth harmonic of a plucked string applies tothe bridge can be found from [5]

Fn =2dT0

nπL0

β

1 minus βsin(βnπ) (1)

where L0 is the length of the string T0 is its tension d isthe displacement at the plucking point and β is the rela-tive distance of the plucking point from the bridge Thisformula does not take into account the possible strongcoupling of the plucked string to the body and to otherstrings which should be a reasonable approximation forthe intended purpose and for most of the notes played ona guitar [16] The magnitude of the power transfer fromthe nth harmonic of the string to the body is then equalto the bridge velocity multiplied by the applied force orequivalently

Pn = Fnvn = F 2nGω = A

sin(βnπ)n

2

Gω (2)

in which Gω is the real part of the bridge admittance (alsocalled conductance) and A is a constant assuming a par-ticular string is being played with a constant pluckingstrength and at a fixed relative distance from the bridgeand again neglecting the losses This power transfer to thebody is largely responsible for the damping of the string

401

ACTA ACUSTICA UNITED WITH ACUSTICA Mansour et al Post-classification of steel string guitarsVol 101 (2015)

mode in question as well as the radiated sound at that fre-quency Based on this idea two features are calculated forall semitones played on a guitar (from E2 to E6) and aver-aged over the three perceptual categories For both cases βis arbitrarily chosen as 03 which is a reasonable numberfor a normal playing condition

Figure 12a shows the sum of the transferred power bythe first 5 harmonics of each played note here calledldquopower transfer to the bodyrdquo The choice of 5 harmonics isan arbitrary limit for which the individual harmonics of thestring could be perceived individually and within whichmost of the energy transfer to the body happens in a nor-mal playing condition The frequency range that is consid-ered in this fashion would be up to 5 times the highest notestudied (ie E6) or equivalently 66 kHz this is about thelimit up to which the measured admittances show a reason-ably good coherence The unknown constant A in equation(2) could only cause an offset in the results therefore itis considered to be 1 for convenience The energy trans-fer to the body represents the strength of the stringbodycoupling for a guitar A large value for this number wouldresult in a strong and more percussive (banjo-like) soundand a small value would result in a weaker sound with alonger decay (more similar to an electric guitar without anamplifier)

Figure 12b shows a slightly different use of the trans-ferred energy by each harmonic It is calculated by divid-ing the sum of the energy transferred by the 2nd to 5th har-monic to the energy transferred by the fundamental aloneAs well as specifying the ratio of higher harmonic energyto that at the fundamental this feature defines how thesound evolves once the instrument is plucked Extremelylarge or small values for a particular note mean that thenote changes its character when it decays while an aver-age value may be correlated to a more stable sound

Neither graph shows any clear distinction between themid-even and treble categories The bassy category doeshowever stand out most of the patterns in both graphs areshifted downward by a few semitones

345 LTAS and radiativity

The LTAS results averaged over different categories areillustrated in Figure 13a The results for the mid-even in-struments are greater than the other two groups over al-most all frequencies Figure 13b is the same as Figure 13abut the LTAS for each individual instrument is normalizedby its area before further processing After normalizationall categories behave similarly over different frequenciesmeaning mid-even instruments were simply louder thanthe other instruments rather than being stronger in a par-ticular range of frequencies

The band-averaged admittance of the mid-even instru-ments earlier presented in Figure 10 was slightly greaterthan the treble instruments in some frequency rangeshowever the significance of the stronger LTAS of mid-even instruments especially beyond 3 kHz can hardly bejustified by that alone A possible explanation is that themid-even instruments were played systematically harder

E2 100 E3 250 E4 500 E5 1000 E6

minus60

minus55

minus50

minus45

minus40

minus35

minus30

Pow

ertransfertothebody

(dB)

E2 100 E3 250 E4 500 E5 1000 E6

minus20

minus10

0

10

Harmonicstofundam

entalratio(dB)

Fundamental frequency (Hz)

Averaged bassyAveraged midminusevenAveraged treble

(a)

(b)

Figure 12 Power transfer to the body (a) and harmonics to fun-damental ratio (b) calculated for each played note over the op-erational range of a guitar (E2 to E6) the vertical lines indicateoctaves The values are averaged for the guitars of different cat-egories (solid line for bassy dash-dot line and light gray green(online version) shade for mid-even and dashed line and darkergray red (online version) shade for treble)

compared to the treble instruments There are at least tworeasons as to why this might have been the case First theguitars were recorded per category starting with the trebleand ending with the mid-even ones It is therefore possiblethat the musician felt more confident in the course of therecording session Second the mid-even guitars had beendescribed by the manufacturer as having a ldquomore stablerdquosound Such a sense might have allowed the performer toplay those instruments more freely Nevertheless we areaware that the frequency distribution obtained using LTAScan be strongly dependent on the nature of the input sig-nal depending on the sequence played and used as an in-put certain frequency bands may be less excited than oth-ers which may attenuate differences in some frequencyregions of interest

35 Classification results

According to our results finding a feature that can distin-guish bassy instruments is a relatively easy task As wasshown in Figure 9a the frequencies of the first 4 modesfor the bassy instruments are around 5 lower than thosefor the two other classes The frequency of the first modecan be determined without much ambiguity or even theneed for mode extraction All of the other 3 low-frequencymodes have close adjacent modes making it harder andsometimes impossible to find their correct modal parame-ters

The amplitude of the first mode is another indicator fora bassy instrument that can be used for more assurance

402

Mansour et al Post-classification of steel string guitars ACTA ACUSTICA UNITED WITH ACUSTICAVol 101 (2015)

500 1000 2000 6000minus180

minus170

minus160

minus150

minus140

minus130

minus120

LTAS(dBHz)

500 1000 2000 6000minus120

minus110

minus100

minus90

minus80

minus70

minus60

Frequency (Hz)

Normalized

LTAS(dBHz)

BassyMidminusevenTrebleTreble plusmn one STDMidminuseven plusmn one STD

(b)

(a)

Figure 13 Non-normalized LTAS (a) and normalized LTAS (b)averaged for the guitars of different categories plusminus onestandard deviation (solid line for bassy dash-dot line and lightgray green (online version) shade for mid-even and dashed lineand darker gray red (online version) shade for treble)

(see Figure 9b) Figure 14a shows a 2-D plane with thenormalized frequency and amplitude of the first mode onits horizontal and vertical axes respectively It can be seenthat the bassy instruments are quite distinct from the mid-even and treble ones Moreover the guitar with the lowestbreathing mode (B3 fT (11)1 = 91 Hz) was described bythe factory employee as extra-bassy

It should be emphasized that the frequency of the firstpeak works particularly well as a feature not becausethe ldquobassinessrdquo of the instrument is defined by the notesaround this modersquos frequency (only the first three noteson the E2 string are below that frequency) but becauseit represents the overall stiffness-to-weight ratio of thesoundbox that effectively scales the frequency of all bodymodes It is noteworthy that although T(11)1 has a strongHelmholtz component the amount by which its frequencydeviates from the frequency of the pure Helmholtz reso-nance is strongly dependent on the frequency separationof the pure Helmholtz and pure body breathing modesThe frequency of the pure Helmholtz mode is assumed thesame for all instruments of the pool given their very simi-lar geometry (both for the sound box and the sound hole)

The in-plane dimensions of the instruments should havea very small variation percentage given the precise con-trol of geometrical conditions in the factory Moreover alower frequency for the first mode is usually accompaniedby a higher amplitude for that mode which would not nor-mally be the case if such deviation was caused by using a

90 92 94 96 98 100 102 10404

05

06

07

08

09

1

Normalized

T(11) 1am

plitude

BassyMid EvenTreble

90 92 94 96 98 100 102 104minus50

minus49

minus48

minus47

minus46

minus45

B1B2

B3

B6

B4B5

ME6ME1

ME2

ME3ME4

ME5

T2

T3

T4

T5 T6

T1

Frequency of the T(11)1mode (Hz)

Averagedmobility(600minus2000Hz)(dB)

(b)

(a)

Figure 14 Classification of the acoustic guitars (a) by the fre-quency and amplitude of their first corpus mode T(11)1 onlyto distinguish bassy instruments from the others and (b) by thefrequency of their first corpus mode T(11)1 and their averagedmobility from 600 Hz to 2 kHz (B = bassy ME = mid-even T =treble) The dB reference is 1 msN

denser wood this leaves variations in the plate thicknessand the equivalent stiffness (ie very roughly the geomet-ric mean of the Youngrsquos moduli in long-grain and cross-grain directions [14]) as those most likely responsible forthe observed shift in mode frequencies Mode frequenciesof a flat plate scale with Eρ(tL2) where E and ρ arethe elastic constant and density t the thickness and L acharacteristic scaling length If either of the two factors issolely responsible to make a 5 shift it would require ei-ther a 5 variation (asymp 0125 mm) of the plate thicknessor a 10 variation in the equivalent stiffness Given thetight tolerances in cutting the plates and their mechanizedsanding it thus appears that variations in equivalent stiff-ness are the primary reason that some of the instrumentsin the pool sound more bassy In the manufacturing pro-cess a wood slab is sliced to produce several plates Thelateral direction of each plate hence represents a particularcombination of the slabrsquos radial and tangential directions(due to the growth rings of the tree) This can introduce asignificant variation in the platesrsquo stiffness in the lateral di-rection which produces variations of the effective stiffnessof the plates

Finding a feature that distinguishes mid-even from tre-ble instruments is less obvious We found that the aver-aged admittances of the treble instruments in this study areslightly lower than the mid-even instruments in the rangeof 600 Hz to 2 kHz Although this difference is not signif-icant over the mentioned range (ie average plusminus

403

ACTA ACUSTICA UNITED WITH ACUSTICA Mansour et al Post-classification of steel string guitarsVol 101 (2015)

one standard deviation of the two categories overlap) itbecomes significant if the average level of the admittancefor the whole range is compared (ie the average valueof all frequency points of the admittance from 600 Hz to2 kHz) Figure 14b shows a 2-D plane with the frequencyof the first mode on the horizontal axis and the average ad-mittance for the range of 600 Hz to 2 kHz The differencefor the mean of the mid-even and treble categories is about2 dB It seems likely that the weaker admittance of the tre-ble category at higher frequencies would contribute to alonger decay for the higher partials This may contributeto a sharper after-sound of the treble instruments that isknown to allow them to cut through the band more effec-tively This hypothesis could be tested by recording theplucked sound of the instruments and comparing the decayrate of the partials among different categories Due to thelimited project time frame this was not investigated fur-ther The mid-even guitars also show stronger LTAS thanthe treble instruments though this result might have beendue to playing conditions (see Section 345)

We applied the same classification (frequency ofT(11)1 versus average mobility between 600 Hz and2 kHz) to 6 acoustic guitars from a different model by thesame manufacturer (Simon amp Patrick) Unlike the Seagullguitars which were all nominally identical some of theSampP instruments had a slightly different bracing patterndesigned to exemplify the properties of a particular per-ceptual category The manufacturerrsquos classification of the6 guitars was not provided to us prior to our own mea-surements though we were told that there were 2 guitarsof each category The resulting classification based on thediscussed features is depicted in Figure 15 The two gui-tars designed to be more bassy were successfully identi-fied from the frequency of their breathing mode but dis-tinguishing between the mid-even and treble instrumentsfrom the averaged admittance criterion was inconclusive

4 Perceptual evaluation

As previously discussed in Section 1 the investigatedacoustic guitars were post-classified by the manufacturerbased on their experience of what type of instrument mightbetter fit a certain playing stylemdashfor example an amateurmusician that plays mostly strummed chords may prefera bassy guitar while a more experienced guitarist mayopt for a treble instrument We therefore conducted a per-ceptual experiment to investigate the level of agreementamong guitar players of varying experience and musicalbackground in classifying the guitars into the categoriesconsidered by the manufacturer

41 Method

The experiment took place in an acoustically dry roomwith dimensions of 60 times 78 times 32 m3 and a reverberationtime of about 03 s to help minimize the effects of roomreflections on the direct sound from the guitars To avoidthe potential problems of using a common pick across all

85 86 87 88 89 90 91 92minus505

minus50

minus495

minus49

minus485

minus48B1

SampP

B2SampP

ME1SampP

ME2SampP

T1SampP

T2SampP

Frequency of the T(11)1mode (Hz)

Averagedmobility(600minus2000Hz)(dB)

BassyMid EvenTreble

Figure 15 Classification of the six SampP guitars based on thefrequency of the T(11)1 mode versus averaged mobility from600 Hz to 2 kHz (B= bassy ME=mid-even T= treble) The dBreference is 1 msN

participants (eg musicians being uncomfortable with apick they are not familiar with) participants were asked tocarry out the classification task using their own pick(s)

411 Participants

Thirteen guitar players took part in the experiment (1 fe-male 12 males average age = 25 yrs SD = 5 yrs range =19ndash36 yrs) They had at least 6 years of acoustic guitar ex-perience (average years of training = 11 yrs SD = 4 yrsrange = 6ndash20 yrs average hours of practice per week =14 hrs SD = 10 hrs range = 3ndash40 hrs) as well as vary-ing experience with different types of guitars [nylon-string(85) acoustic-electric (62) acoustic bass (15) elec-tric (100) and electric bass (54)] Participants re-ported playing a wide range of musical styles [classical(77) jazz (85) pop-rock (92) country folk blue-grass (66) hard rock heavy metal (46) flamenco(15) latin (46) blues (77) avant-garde contempo-rary (54) and funk alternative (46)] both in a bandand as soloists Seven guitarists (54) reported that theytypically or most often use a pick whereas 4 players (31)reported they do not 2 guitarists (15) reported using apick occasionally rather than typically Nine participantshad higher-level degrees in music performance or othermusic-related areas [BMus (music theory classical gui-tar) BFA (jazz studies) MMus (guitar performance) MA(digital musics) PhD (music theory) Diplocircme drsquoEacutetat deProfesseur de Musique (France)] All guitar players werepaid for their participation

412 Procedure

The experimental session lasted 2 hours and was organizedin two phases In the first phase participants were pre-sented with the guitars randomly ordered on the floor (de-termined by computer calculations) They were asked toplay all instruments for up to 20 minutes in order to famil-iarize themselves with the set and use this time to tune theguitars as needed In the second phase participants were

404

Mansour et al Post-classification of steel string guitars ACTA ACUSTICA UNITED WITH ACUSTICAVol 101 (2015)

given up to 20 minutes to classify each of the guitars as ei-ther bassy mid-even or treble by placing them in speciallymarked areas of the room At the end of the second phaseparticipants were given the opportunity to revise their ini-tial judgements if they so wanted until they were satis-fied with their classification No playing constraint wasimposed on the evaluation process (eg specific repertoireor technique) Participants were instead encouraged to fol-low their own strategy with the suggestion to use bothstrumming-oriented songs (eg Sitting Waiting Wishingby Jack Johnson) and fingerpicking-oriented pieces (egDust in the Wind by Kansas) Participants were instructedto maximize evaluation speed and accuracy Upon com-pleting the session participants talked about their classi-fications and overall experience through written responsesto the following questions ldquoFor each class of guitar whatsound characteristics did you look for Can you describehow you evaluated the guitars in terms of these character-istics (eg musical material technique style)rdquo (Q1) ldquoTowhat extend do you agree with the given classification ofguitarsrdquo (Q2) and ldquoCan you suggest a classification thatbetter reflects how you evaluate the sound characteristicsof guitarsrdquo (Q3)

42 Results

Inter-individual consistency was measured as the Fleissrsquokappa coefficient between guitar classifications from dif-ferent participants using a Bayesian approach to inference[18] Fleissrsquo kappa is a generalization of Cohenrsquos kappastatistic κ a chance-corrected measure of agreement be-tween two assessors to multiple raters [19] It ranges fromminus1 to 1 κ (AB) = 1 and minus1 if A = B and A = minusBrespectively and κ (AB) = 0 in case of no associationbetween A and B (ie exactly what would be expectedby chance) We implemented Gibbs sampling through thesoftware WinBUGS (Windows Bayesian inference UsingGibbs Sampling [20]) with 50000 observations generatedfrom the posterior distribution with a burn-in of 1000 ob-servations A uniformly distributed multicategorical priorwas assumed with total information across all possiblecells (ie from different combinations of participants) be-ing equivalent to a single prior observation (ie a priorsample size equivalent of 1)

Fleissrsquos kappa was estimated with a posterior mean of01 and a 95 credible interval [-02 05] indicating avery low amount of agreement between the participantsin their classifications Figure 16 shows the proportion oftimes a guitar was classified as either bassy mid-even ortreble (the guitars are arranged in three groups accord-ing to the initial factory classification) Whereas someagreement concerning the bassy guitars can be observedmdashindeed the extra bassy guitar (B3) was most often clas-sified as bassymdashthe perceptions for the other two classeslargely vary across different musicians In particular manyguitars that had been characterized as treble by the fac-tory employee were perceived as mid-even Several mu-sicians specifically reported encountering difficulties withthe mid-even category In answering Q3 a player observed

B1 B2 B3 B4 B5 B60

25

50

75

100

classifiedas

(a)BassyMidminusevenTreble

ME1 ME2 ME3 ME4 ME5 ME60

25

50

75

100

classifiedas

(b)

T1 T2 T3 T4 T5 T60

25

50

75

100

classifiedas

Guitars

(c)

Figure 16 Perceptual evaluation Proportion of times a guitarwas classified into one of the categories by the participants Gui-tars are labelled and grouped according to the factory classifica-tion (B = bassy ME = mid-even T = treble) The color scheme(in the online version) corresponds to the classifications providedby the guitar players

that ldquothere is a difference [between] a guitar with domi-nant mid tones and one with an even soundrdquo

Overall participants expressed no strong difficulties incarrying out the classification task On a scale from 1(strongly disagree) to 5 (strongly agree) all performers se-lected 4 when asked ldquoTo what extend do you agree withthe given classification of guitarsrdquo This indicates that the3 categories considered by the manufacturer are gener-ally meaningful to musicians However verbal responsesto questions Q1 and Q3 reveal that participants often as-sociated bassy with more prominent E2 and A2 strings(ie they focused on the low-frequency end of the gui-tar) and similarly treble with better emphasized B3 andE4 strings (ie they paid attention to the high-frequencyrange) As one guitar player noted ldquoI feel that descriptionslike warm neutral and bright would be easier to classifydue to the nature of the instrument Bass leads my mindto bass strings and to heed the low end instead of the en-tire range from the highest to lowest notes Same for tre-ble vs bright rdquo In particular 4 participants used the wordldquotwangrdquo to describe guitars they classified as treble whileanother guitarist reported that a treble guitar ldquosounds bestfor finger picking and light strumming as heavy playingcan get quite ear-piercing and harsh This last commentseems to echo the second hypothesis for the origin of thestronger LTAS shown by the mid-even guitars (see Sec-tion 345)

Two participants reported looking out for problematicnotes as a more meaningful way of finding bassy guitarsThis likely points toward the ldquoveeringrdquo phenomenon as-sociated with a stronger admittance at lower frequencies

405

ACTA ACUSTICA UNITED WITH ACUSTICA Mansour et al Post-classification of steel string guitarsVol 101 (2015)

[16] Further one musician referred to the tactile feedbackreceived from a vibrating body associating bassy guitarswith more felt vibrations than the other two categoriesThis is a perspective shared by the manufacturer as wellIn fact body vibrations can provide tactile and proprio-ceptive cues that contribute to the perception of the radi-ated sound so that the musician can assess their interactionwith the instrument cross-modally [21 22]

5 Discussion

A set of 18 steel string guitars of the same model man-ufactured with high geometrical tolerances in a modernfactory were evaluated in this study These guitars wereconsidered by the manufacturer to possess subtle yet per-ceivable variations that could be classified as either bassymid-even or treble The results of this study showed thatthe instruments classified as bassy had a consistently lowerT(11)1 breathing mode (see Figure 9a 14b and 15) Thissuggests a lower stiffness-to-weight ratio for the bodies ofthese guitars which is most likely the result of variationsin equivalent stiffness (see Section 35)

It was further shown that whenever the frequency ofT(11)1 was shifted downward or upward for a particu-lar guitar the frequencies of its higher modes consistentlychanged in a similar way (see Figure 9 also the slightlyshifted bumps at around 600 750 and 1200 Hz in Fig-ure 12a) This would not necessarily be the case if theguitars were of different models (ie with varying geome-tries) or of different types of wood (eg plywood versusspruce) This is an important result as there is a practi-cal advantage to having higher-frequency modes move ac-cording to the first corpus mode across guitars of the samemodel T(11)1 is a low-frequency resonance that is suffi-ciently separated from the adjacent modes and hence eas-ily identifiable Measuring its frequency across differentinstruments is easy and affordable in a small workshopenvironment One can simply hit the guitar with a reg-ular hammer and recordanalyse the sound using an in-expensive microphone and a laptop reminiscent of thetraditional tap-tone method used by luthiers Moreoversuch measurements can be very reliable provided that theboundary conditions of the instrument (ie the installedfixture that holds the instrument) as well as the air tem-perature pressure and humidity are kept consistent Theseparameters constitute the biggest source of uncertainty infrequency-based features and can easily be controlled in afactory space or small workshop

Finding a reliable differentiator between the mid-evenand treble guitars was less obvious Although we ob-served that the averaged mobility curves of the mid-eveninstruments tended to be slightly higher in the vicinityof 600ndash2000 Hz than the treble guitars further measure-ments would have been necessary to determine whetherthe weaker admittance of the treble instruments at higherfrequencies would contribute to a longer decay for thehigher partials (see Section 35) In addition the differ-ence in averaged mobility magnitude between the two cat-

egories was only about 2 dB which is close to the marginof error for the measurement system Such a small differ-ence in an amplitude-based feature may be easily obscuredby variations in the exact angle of the hammer location ofhitting and measuring or calibration of the sensors

We invited guitar players of varying musical back-ground and experience to classify the guitars into thesecategories We observed very poor agreement among theparticipants but it is common in recent perceptual studiesof musical instruments that musicians do not agree witheach other [23 24 25] This seems a result of them lis-tening to (or feeling) different aspects of the instrumentand not necessarily all attending to the same features Theemployee that post-classified the guitars at the factory wasthe person that tests every guitar coming off the productionline thus he must have a lot of experience evaluating thesound of these instruments Assuming he did his classifi-cation carefully then there likely were particular things hewas listening for The participants in our perceptual exper-iment would not necessarily listen for the same featuresso it is not too surprising they largely disagreed on theirclassifications That said the differences between the tre-ble and mid-even guitars were extremely subtle and evencompany representatives often mis-categorized them dur-ing informal playing tests And this together with the factthat our measurements showed no consistent differencesbetween the two categories would seem to lead us to ques-tion whether the distinction between mid-even and trebletruly exists at all Another possible explanation is that dif-ferent musicians may follow different processes to assessthose features considered essential for the evaluation of aninstrument [23 26] Accordingly guitar players may agreeof what particular characteristics they look for in a tre-ble guitar but the perception of the same characteristicswidely varies across individuals

6 Conclusions

Efforts to understand how the tonal quality of an instru-ment relates to its measurable physical properties have of-ten been inconclusive (eg [27]) In the light of some re-cent studies the concept of ldquogoodrdquo versus ldquobadrdquo instru-ments is likely too general and it would be more advanta-geous to instead consider specific perceptual attributes ofan instrument [23 28] In the long run it may be possibleto derive a multidimensional timbre space where each di-mension is explained by some measurable quantities andin which instruments with similar qualities are likely tocluster In this study more specific sound characteristicsof instruments were thus targeted rather than their overalltonal quality

From bridge admittance measurements on 18 steelstring guitars from the same production line and post-classified by the manufacturer as either bassy mid-evenor treble (see Section 1 for details) it was shown that itis possible to discriminate between bassy guitars and theother two categories based on the frequency of the breath-ing mode However finding a similarly reliable feature to

406

Mansour et al Post-classification of steel string guitars ACTA ACUSTICA UNITED WITH ACUSTICAVol 101 (2015)

distinguish mid-even from treble instruments is less triv-ial at this point The treble guitars demonstrated a loweraveraged mobility in the frequency range of 600ndash2000 Hzthough more empirical investigation would be necessarybefore drawing any conclusions Results from a percep-tual experiment wherein guitar players were asked to clas-sify the guitars into these categories showed almost noagreement between guitarists in their judgements as wellas between the latter and the classification by the manufac-turer Despite the overall lack of consensus guitar playerstended to agree more on the bassy instruments than theother two categories

Some essential additions to this study are a more care-ful analysis on the perceptual definition of mid-even ver-sus treble and analysing the decay rate of the partials fora plucked sound to see how it correlates with the admit-tance Analysis of this kind has been performed for syn-thesized acoustic guitar sounds (see Figure 7 in [16]) but asimilar analysis on real guitar sounds will involve difficul-ties caused by the double set of harmonics [29] as well aslongitudinal string vibrations (also dubbed ldquophantom par-tialsrdquo) [30]

Acknowledgements

This project was funded by a Natural Sciences and En-gineering Research Council of Canada Engage grant incollaboration with Godin Guitars The authors wish tothank Robert Godin for this opportunity We also thankLawrence Joseph (statistics) and Garry Antonio (guitarperformance) for constructive comments We are grate-ful to Jim Woodhouse for sharing his analysis code Thefirst author would like to acknowledge the Government ofCanada for a Vanier Canada Graduate Scholarship

References

[1] E V Jansson A study of acoustical and hologram inter-ferometric measurements of the top plate vibrations of aguitar Acust Acta Acust 25 (1971) 95ndash100

[2] O Christensen B B Vistisen Simple model for low-frequency guitar function J Acoust Soc Am 68 (1980)758ndash766

[3] O Christensen Quantitative models for low frequency gui-tar function J Guitar Acoustics 6 (1982) 10ndash25

[4] H Wright The acoustics and psychoacoustics of the guitarDissertation Dept of Physics and Astronomy Universityof Wales Cardiff UK 1996

[5] N H Fletcher T D Rossing The physics of musical in-struments Second ed Springer New York 1998

[6] J E Popp Four mass coupled oscillator guitar model JAcoust Soc Am 131 (2012) 829ndash836

[7] J Woodhouse Body vibration of the violinmdashwhat can amaker except to control Catgut Acoust Soc J (Series II)4 (2002) 43ndash49

[8] B Elie F Gautier B David Macro parameters describingthe mechanical behavior of classical guitars J Acoust SocAm 132 (2012) 4013ndash4024

[9] J Woodhouse R S Langley Interpreting the input admit-tance of violins and guitars Acust Acta Acust 98 (2012)611ndash628

[10] A Gabrielsson E V Jansson Long-time-average-spectraand rated qualities of twenty-two violins Acustica 42(1979) 47ndash55

[11] P D Welch The use of Fast Fourier Transform for the es-timation of power spectra A method based on time averag-ing over short modified periodograms IEEE Trans Audioand Electroacoust AU-15 (1967) 70ndash73

[12] J Sundberg The singerrsquos formant revisited J Voice 4(1995) 106ndash119

[13] A Buen On timbre parameters and sound levels ofrecorded old violins J Violin Soc Am VSA Papers XXI(2007) 57ndash68

[14] J Woodhouse On the ldquobridge hillrdquo of the violin AcustActa Acust 91 (2005) 155ndash165

[15] B Richardson The acoustical development of the guitarCatgut Acoustical Society Journal 2 (1994) 1ndash10

[16] J Woodhouse On the synthesis of guitar plucks AcustActa Acust 90 (2004) 928ndash944

[17] C H Hodges J Power J Woodhouse The use of the sono-gram in structural acoustics and an application to the vibra-tions of cylindrical shells J Sound Vib 101 (1985) 203ndash218

[18] L D Broemeling Bayesian methods for measures ofagreement Chapman amp HallCRC New York 2009 (Bio-statistics)

[19] J L Fleiss Measuring nominal scale agreement amongmany raters Psychological Bulletin 76 (1971) 378ndash382

[20] D Lunn C Jackson N Best A Thomas D Spiegelhal-ter The BUGS Book A Practical Introduction to BayesianAnalysis Chapman amp HallCRC New York 2012 (Textsin Statistical Science)

[21] C Saitis Evaluating violin quality Player reliability andverbalization Dissertation Dept of Music ResearchMcGill University Montreal Quebec Canada 2013

[22] I Wollman Perception bimodale des violonistes en situa-tion de jeu Influence des retours auditif et vibrotactile surlrsquoeacutevaluation du violon Dissertation Universiteacute Pierre etMarie Curie Paris France 2013

[23] C Saitis B L Giordano C Fritz G P Scavone Percep-tual evaluation of violins A quantitative analysis of pref-erence judgements by experienced players J Acoust SocAm 132 (2012) 4002ndash4012

[24] C Fritz J Curtin J Poitevineau P Morrel-Samuels F-CTao Player preferences among new and old violins ProcNat Acad Sci USA 109 (2012) 760ndash763

[25] P Eveno J-P Dalmont R Causseacute G P Scavone Anacoustic and perceptual evaluation of saxophone pad ldquores-onatorsrdquo Acta Acustica united with Acustica 101 (2015)246ndash255

[26] C Fritz J Woodhouse F P-H Cheng I Cross A FBlackwell B C J Moore Perceptual studies of violinbody damping and vibrato J Acoust Soc Am 127 (2010)513ndash524

[27] G Bissinger Structural acoustics of good and bad violinsJ Acoust Soc Am 124 (2008) 1764ndash1773

[28] C Saitis C Fritz C Guastavino G P Scavone Con-ceptualization of violin quality by experienced performersProc Stockholm Music Acoust Conf Stockholm Swe-den 2013 123ndash128

[29] J Woodhouse Plucked guitar transients Comparison ofmeasurements and synthesis Acust Acta Acust 90 (2004)945ndash965

[30] B Bank L Sujbert Generation of longitudinal vibrationsin piano strings From physics to sound synthesis JAcoust Soc Am 117 (2004) 2268ndash2278

407

Mansour et al Post-classification of steel string guitars ACTA ACUSTICA UNITED WITH ACUSTICAVol 101 (2015)

500 1000 2000 6000minus54

minus52

minus50

minus48

minus46

minus44

minus42

Bandminusaveraged

admittance

(dB) Bandminusaveraged bassy

Bandminusaveraged midminusevenBandminusaveraged trebleMidminuseven plusmn one STDTreble plusmn one STD

500 1000 2000 6000minus22

minus20

minus18

minus16

minus14

minus12

minus10

minus8

minus6

Frequency (Hz)

Bandminusaveraged

radiativity

(dB)

(a)

(b)

Figure 10 Averaged admittance (a) and radiativity (b) takenover bassy instruments (solid) mid-even instruments (dash-dot)plusminus one standard deviation (light gray shade green in on-line version) and treble instruments (dashed) plusminus onestandard deviation (darker gray shade red in online version)The dB reference is 1 msN

ure 11 shows the averaged damping trend taken over thethree perceptual categories

The Q factor for all classes rises more sharply from Q asymp30 at 100 Hz to Qasymp 60 at 600 Hz where it reaches a plateauand rises more gradually afterwards However the generalbehavior is not far from linear on a log-log scale As itcan be seen in Figure 11a and 11b the value of Q factorover different frequencies and its trend is similar for trebleand mid-even classes which make them indistinguishableby looking at the damping trend From the power-law fitresults the average power for bassy category is 008 andfor both categories of mid-even and treble is 016 This re-sults in an average-over-all-instruments of 013 comparedto 026 and -009 reported earlier for the violins and theclassic guitars respectively in [9] Among the instrumentsof the pool only one showed a negative power fit

344 Harmonic features

Plucking a guitar string excites multiple harmonics simul-taneously which makes it possible that a weighted sum ofthe admittance over some range of harmonics rather thanthe admittance at the fundamental only is a better way ofdistinguishing categories of instruments Forgetting aboutlosses for the moment the amplitude of the transverse

200 300 500 1000 2000 5000

30

40

50

70

Frequency (Hz)

Qminusfactortrend

200 300 500 1000 2000 5000

20

30

40

50

70

100

Bandminusaveraged

Qminusfactor

Averaged bassy

Averaged midminuseven

Averaged treble

Averaged midminuseven plusmn STD

Averaged treble plusmn STD

(b)

(a)

Figure 11 Effective Q-factors averaged for the guitars of differ-ent categories plusminus one standard deviation (solid line forbassy dash-dot line and light gray green shade for mid-evenand dashed line and darker gray red shade for treble) band-average (a) and regression line for power-law behavior (b) Notethat vertical axes are different

force that the nth harmonic of a plucked string applies tothe bridge can be found from [5]

Fn =2dT0

nπL0

β

1 minus βsin(βnπ) (1)

where L0 is the length of the string T0 is its tension d isthe displacement at the plucking point and β is the rela-tive distance of the plucking point from the bridge Thisformula does not take into account the possible strongcoupling of the plucked string to the body and to otherstrings which should be a reasonable approximation forthe intended purpose and for most of the notes played ona guitar [16] The magnitude of the power transfer fromthe nth harmonic of the string to the body is then equalto the bridge velocity multiplied by the applied force orequivalently

Pn = Fnvn = F 2nGω = A

sin(βnπ)n

2

Gω (2)

in which Gω is the real part of the bridge admittance (alsocalled conductance) and A is a constant assuming a par-ticular string is being played with a constant pluckingstrength and at a fixed relative distance from the bridgeand again neglecting the losses This power transfer to thebody is largely responsible for the damping of the string

401

ACTA ACUSTICA UNITED WITH ACUSTICA Mansour et al Post-classification of steel string guitarsVol 101 (2015)

mode in question as well as the radiated sound at that fre-quency Based on this idea two features are calculated forall semitones played on a guitar (from E2 to E6) and aver-aged over the three perceptual categories For both cases βis arbitrarily chosen as 03 which is a reasonable numberfor a normal playing condition

Figure 12a shows the sum of the transferred power bythe first 5 harmonics of each played note here calledldquopower transfer to the bodyrdquo The choice of 5 harmonics isan arbitrary limit for which the individual harmonics of thestring could be perceived individually and within whichmost of the energy transfer to the body happens in a nor-mal playing condition The frequency range that is consid-ered in this fashion would be up to 5 times the highest notestudied (ie E6) or equivalently 66 kHz this is about thelimit up to which the measured admittances show a reason-ably good coherence The unknown constant A in equation(2) could only cause an offset in the results therefore itis considered to be 1 for convenience The energy trans-fer to the body represents the strength of the stringbodycoupling for a guitar A large value for this number wouldresult in a strong and more percussive (banjo-like) soundand a small value would result in a weaker sound with alonger decay (more similar to an electric guitar without anamplifier)

Figure 12b shows a slightly different use of the trans-ferred energy by each harmonic It is calculated by divid-ing the sum of the energy transferred by the 2nd to 5th har-monic to the energy transferred by the fundamental aloneAs well as specifying the ratio of higher harmonic energyto that at the fundamental this feature defines how thesound evolves once the instrument is plucked Extremelylarge or small values for a particular note mean that thenote changes its character when it decays while an aver-age value may be correlated to a more stable sound

Neither graph shows any clear distinction between themid-even and treble categories The bassy category doeshowever stand out most of the patterns in both graphs areshifted downward by a few semitones

345 LTAS and radiativity

The LTAS results averaged over different categories areillustrated in Figure 13a The results for the mid-even in-struments are greater than the other two groups over al-most all frequencies Figure 13b is the same as Figure 13abut the LTAS for each individual instrument is normalizedby its area before further processing After normalizationall categories behave similarly over different frequenciesmeaning mid-even instruments were simply louder thanthe other instruments rather than being stronger in a par-ticular range of frequencies

The band-averaged admittance of the mid-even instru-ments earlier presented in Figure 10 was slightly greaterthan the treble instruments in some frequency rangeshowever the significance of the stronger LTAS of mid-even instruments especially beyond 3 kHz can hardly bejustified by that alone A possible explanation is that themid-even instruments were played systematically harder

E2 100 E3 250 E4 500 E5 1000 E6

minus60

minus55

minus50

minus45

minus40

minus35

minus30

Pow

ertransfertothebody

(dB)

E2 100 E3 250 E4 500 E5 1000 E6

minus20

minus10

0

10

Harmonicstofundam

entalratio(dB)

Fundamental frequency (Hz)

Averaged bassyAveraged midminusevenAveraged treble

(a)

(b)

Figure 12 Power transfer to the body (a) and harmonics to fun-damental ratio (b) calculated for each played note over the op-erational range of a guitar (E2 to E6) the vertical lines indicateoctaves The values are averaged for the guitars of different cat-egories (solid line for bassy dash-dot line and light gray green(online version) shade for mid-even and dashed line and darkergray red (online version) shade for treble)

compared to the treble instruments There are at least tworeasons as to why this might have been the case First theguitars were recorded per category starting with the trebleand ending with the mid-even ones It is therefore possiblethat the musician felt more confident in the course of therecording session Second the mid-even guitars had beendescribed by the manufacturer as having a ldquomore stablerdquosound Such a sense might have allowed the performer toplay those instruments more freely Nevertheless we areaware that the frequency distribution obtained using LTAScan be strongly dependent on the nature of the input sig-nal depending on the sequence played and used as an in-put certain frequency bands may be less excited than oth-ers which may attenuate differences in some frequencyregions of interest

35 Classification results

According to our results finding a feature that can distin-guish bassy instruments is a relatively easy task As wasshown in Figure 9a the frequencies of the first 4 modesfor the bassy instruments are around 5 lower than thosefor the two other classes The frequency of the first modecan be determined without much ambiguity or even theneed for mode extraction All of the other 3 low-frequencymodes have close adjacent modes making it harder andsometimes impossible to find their correct modal parame-ters

The amplitude of the first mode is another indicator fora bassy instrument that can be used for more assurance

402

Mansour et al Post-classification of steel string guitars ACTA ACUSTICA UNITED WITH ACUSTICAVol 101 (2015)

500 1000 2000 6000minus180

minus170

minus160

minus150

minus140

minus130

minus120

LTAS(dBHz)

500 1000 2000 6000minus120

minus110

minus100

minus90

minus80

minus70

minus60

Frequency (Hz)

Normalized

LTAS(dBHz)

BassyMidminusevenTrebleTreble plusmn one STDMidminuseven plusmn one STD

(b)

(a)

Figure 13 Non-normalized LTAS (a) and normalized LTAS (b)averaged for the guitars of different categories plusminus onestandard deviation (solid line for bassy dash-dot line and lightgray green (online version) shade for mid-even and dashed lineand darker gray red (online version) shade for treble)

(see Figure 9b) Figure 14a shows a 2-D plane with thenormalized frequency and amplitude of the first mode onits horizontal and vertical axes respectively It can be seenthat the bassy instruments are quite distinct from the mid-even and treble ones Moreover the guitar with the lowestbreathing mode (B3 fT (11)1 = 91 Hz) was described bythe factory employee as extra-bassy

It should be emphasized that the frequency of the firstpeak works particularly well as a feature not becausethe ldquobassinessrdquo of the instrument is defined by the notesaround this modersquos frequency (only the first three noteson the E2 string are below that frequency) but becauseit represents the overall stiffness-to-weight ratio of thesoundbox that effectively scales the frequency of all bodymodes It is noteworthy that although T(11)1 has a strongHelmholtz component the amount by which its frequencydeviates from the frequency of the pure Helmholtz reso-nance is strongly dependent on the frequency separationof the pure Helmholtz and pure body breathing modesThe frequency of the pure Helmholtz mode is assumed thesame for all instruments of the pool given their very simi-lar geometry (both for the sound box and the sound hole)

The in-plane dimensions of the instruments should havea very small variation percentage given the precise con-trol of geometrical conditions in the factory Moreover alower frequency for the first mode is usually accompaniedby a higher amplitude for that mode which would not nor-mally be the case if such deviation was caused by using a

90 92 94 96 98 100 102 10404

05

06

07

08

09

1

Normalized

T(11) 1am

plitude

BassyMid EvenTreble

90 92 94 96 98 100 102 104minus50

minus49

minus48

minus47

minus46

minus45

B1B2

B3

B6

B4B5

ME6ME1

ME2

ME3ME4

ME5

T2

T3

T4

T5 T6

T1

Frequency of the T(11)1mode (Hz)

Averagedmobility(600minus2000Hz)(dB)

(b)

(a)

Figure 14 Classification of the acoustic guitars (a) by the fre-quency and amplitude of their first corpus mode T(11)1 onlyto distinguish bassy instruments from the others and (b) by thefrequency of their first corpus mode T(11)1 and their averagedmobility from 600 Hz to 2 kHz (B = bassy ME = mid-even T =treble) The dB reference is 1 msN

denser wood this leaves variations in the plate thicknessand the equivalent stiffness (ie very roughly the geomet-ric mean of the Youngrsquos moduli in long-grain and cross-grain directions [14]) as those most likely responsible forthe observed shift in mode frequencies Mode frequenciesof a flat plate scale with Eρ(tL2) where E and ρ arethe elastic constant and density t the thickness and L acharacteristic scaling length If either of the two factors issolely responsible to make a 5 shift it would require ei-ther a 5 variation (asymp 0125 mm) of the plate thicknessor a 10 variation in the equivalent stiffness Given thetight tolerances in cutting the plates and their mechanizedsanding it thus appears that variations in equivalent stiff-ness are the primary reason that some of the instrumentsin the pool sound more bassy In the manufacturing pro-cess a wood slab is sliced to produce several plates Thelateral direction of each plate hence represents a particularcombination of the slabrsquos radial and tangential directions(due to the growth rings of the tree) This can introduce asignificant variation in the platesrsquo stiffness in the lateral di-rection which produces variations of the effective stiffnessof the plates

Finding a feature that distinguishes mid-even from tre-ble instruments is less obvious We found that the aver-aged admittances of the treble instruments in this study areslightly lower than the mid-even instruments in the rangeof 600 Hz to 2 kHz Although this difference is not signif-icant over the mentioned range (ie average plusminus

403

ACTA ACUSTICA UNITED WITH ACUSTICA Mansour et al Post-classification of steel string guitarsVol 101 (2015)

one standard deviation of the two categories overlap) itbecomes significant if the average level of the admittancefor the whole range is compared (ie the average valueof all frequency points of the admittance from 600 Hz to2 kHz) Figure 14b shows a 2-D plane with the frequencyof the first mode on the horizontal axis and the average ad-mittance for the range of 600 Hz to 2 kHz The differencefor the mean of the mid-even and treble categories is about2 dB It seems likely that the weaker admittance of the tre-ble category at higher frequencies would contribute to alonger decay for the higher partials This may contributeto a sharper after-sound of the treble instruments that isknown to allow them to cut through the band more effec-tively This hypothesis could be tested by recording theplucked sound of the instruments and comparing the decayrate of the partials among different categories Due to thelimited project time frame this was not investigated fur-ther The mid-even guitars also show stronger LTAS thanthe treble instruments though this result might have beendue to playing conditions (see Section 345)

We applied the same classification (frequency ofT(11)1 versus average mobility between 600 Hz and2 kHz) to 6 acoustic guitars from a different model by thesame manufacturer (Simon amp Patrick) Unlike the Seagullguitars which were all nominally identical some of theSampP instruments had a slightly different bracing patterndesigned to exemplify the properties of a particular per-ceptual category The manufacturerrsquos classification of the6 guitars was not provided to us prior to our own mea-surements though we were told that there were 2 guitarsof each category The resulting classification based on thediscussed features is depicted in Figure 15 The two gui-tars designed to be more bassy were successfully identi-fied from the frequency of their breathing mode but dis-tinguishing between the mid-even and treble instrumentsfrom the averaged admittance criterion was inconclusive

4 Perceptual evaluation

As previously discussed in Section 1 the investigatedacoustic guitars were post-classified by the manufacturerbased on their experience of what type of instrument mightbetter fit a certain playing stylemdashfor example an amateurmusician that plays mostly strummed chords may prefera bassy guitar while a more experienced guitarist mayopt for a treble instrument We therefore conducted a per-ceptual experiment to investigate the level of agreementamong guitar players of varying experience and musicalbackground in classifying the guitars into the categoriesconsidered by the manufacturer

41 Method

The experiment took place in an acoustically dry roomwith dimensions of 60 times 78 times 32 m3 and a reverberationtime of about 03 s to help minimize the effects of roomreflections on the direct sound from the guitars To avoidthe potential problems of using a common pick across all

85 86 87 88 89 90 91 92minus505

minus50

minus495

minus49

minus485

minus48B1

SampP

B2SampP

ME1SampP

ME2SampP

T1SampP

T2SampP

Frequency of the T(11)1mode (Hz)

Averagedmobility(600minus2000Hz)(dB)

BassyMid EvenTreble

Figure 15 Classification of the six SampP guitars based on thefrequency of the T(11)1 mode versus averaged mobility from600 Hz to 2 kHz (B= bassy ME=mid-even T= treble) The dBreference is 1 msN

participants (eg musicians being uncomfortable with apick they are not familiar with) participants were asked tocarry out the classification task using their own pick(s)

411 Participants

Thirteen guitar players took part in the experiment (1 fe-male 12 males average age = 25 yrs SD = 5 yrs range =19ndash36 yrs) They had at least 6 years of acoustic guitar ex-perience (average years of training = 11 yrs SD = 4 yrsrange = 6ndash20 yrs average hours of practice per week =14 hrs SD = 10 hrs range = 3ndash40 hrs) as well as vary-ing experience with different types of guitars [nylon-string(85) acoustic-electric (62) acoustic bass (15) elec-tric (100) and electric bass (54)] Participants re-ported playing a wide range of musical styles [classical(77) jazz (85) pop-rock (92) country folk blue-grass (66) hard rock heavy metal (46) flamenco(15) latin (46) blues (77) avant-garde contempo-rary (54) and funk alternative (46)] both in a bandand as soloists Seven guitarists (54) reported that theytypically or most often use a pick whereas 4 players (31)reported they do not 2 guitarists (15) reported using apick occasionally rather than typically Nine participantshad higher-level degrees in music performance or othermusic-related areas [BMus (music theory classical gui-tar) BFA (jazz studies) MMus (guitar performance) MA(digital musics) PhD (music theory) Diplocircme drsquoEacutetat deProfesseur de Musique (France)] All guitar players werepaid for their participation

412 Procedure

The experimental session lasted 2 hours and was organizedin two phases In the first phase participants were pre-sented with the guitars randomly ordered on the floor (de-termined by computer calculations) They were asked toplay all instruments for up to 20 minutes in order to famil-iarize themselves with the set and use this time to tune theguitars as needed In the second phase participants were

404

Mansour et al Post-classification of steel string guitars ACTA ACUSTICA UNITED WITH ACUSTICAVol 101 (2015)

given up to 20 minutes to classify each of the guitars as ei-ther bassy mid-even or treble by placing them in speciallymarked areas of the room At the end of the second phaseparticipants were given the opportunity to revise their ini-tial judgements if they so wanted until they were satis-fied with their classification No playing constraint wasimposed on the evaluation process (eg specific repertoireor technique) Participants were instead encouraged to fol-low their own strategy with the suggestion to use bothstrumming-oriented songs (eg Sitting Waiting Wishingby Jack Johnson) and fingerpicking-oriented pieces (egDust in the Wind by Kansas) Participants were instructedto maximize evaluation speed and accuracy Upon com-pleting the session participants talked about their classi-fications and overall experience through written responsesto the following questions ldquoFor each class of guitar whatsound characteristics did you look for Can you describehow you evaluated the guitars in terms of these character-istics (eg musical material technique style)rdquo (Q1) ldquoTowhat extend do you agree with the given classification ofguitarsrdquo (Q2) and ldquoCan you suggest a classification thatbetter reflects how you evaluate the sound characteristicsof guitarsrdquo (Q3)

42 Results

Inter-individual consistency was measured as the Fleissrsquokappa coefficient between guitar classifications from dif-ferent participants using a Bayesian approach to inference[18] Fleissrsquo kappa is a generalization of Cohenrsquos kappastatistic κ a chance-corrected measure of agreement be-tween two assessors to multiple raters [19] It ranges fromminus1 to 1 κ (AB) = 1 and minus1 if A = B and A = minusBrespectively and κ (AB) = 0 in case of no associationbetween A and B (ie exactly what would be expectedby chance) We implemented Gibbs sampling through thesoftware WinBUGS (Windows Bayesian inference UsingGibbs Sampling [20]) with 50000 observations generatedfrom the posterior distribution with a burn-in of 1000 ob-servations A uniformly distributed multicategorical priorwas assumed with total information across all possiblecells (ie from different combinations of participants) be-ing equivalent to a single prior observation (ie a priorsample size equivalent of 1)

Fleissrsquos kappa was estimated with a posterior mean of01 and a 95 credible interval [-02 05] indicating avery low amount of agreement between the participantsin their classifications Figure 16 shows the proportion oftimes a guitar was classified as either bassy mid-even ortreble (the guitars are arranged in three groups accord-ing to the initial factory classification) Whereas someagreement concerning the bassy guitars can be observedmdashindeed the extra bassy guitar (B3) was most often clas-sified as bassymdashthe perceptions for the other two classeslargely vary across different musicians In particular manyguitars that had been characterized as treble by the fac-tory employee were perceived as mid-even Several mu-sicians specifically reported encountering difficulties withthe mid-even category In answering Q3 a player observed

B1 B2 B3 B4 B5 B60

25

50

75

100

classifiedas

(a)BassyMidminusevenTreble

ME1 ME2 ME3 ME4 ME5 ME60

25

50

75

100

classifiedas

(b)

T1 T2 T3 T4 T5 T60

25

50

75

100

classifiedas

Guitars

(c)

Figure 16 Perceptual evaluation Proportion of times a guitarwas classified into one of the categories by the participants Gui-tars are labelled and grouped according to the factory classifica-tion (B = bassy ME = mid-even T = treble) The color scheme(in the online version) corresponds to the classifications providedby the guitar players

that ldquothere is a difference [between] a guitar with domi-nant mid tones and one with an even soundrdquo

Overall participants expressed no strong difficulties incarrying out the classification task On a scale from 1(strongly disagree) to 5 (strongly agree) all performers se-lected 4 when asked ldquoTo what extend do you agree withthe given classification of guitarsrdquo This indicates that the3 categories considered by the manufacturer are gener-ally meaningful to musicians However verbal responsesto questions Q1 and Q3 reveal that participants often as-sociated bassy with more prominent E2 and A2 strings(ie they focused on the low-frequency end of the gui-tar) and similarly treble with better emphasized B3 andE4 strings (ie they paid attention to the high-frequencyrange) As one guitar player noted ldquoI feel that descriptionslike warm neutral and bright would be easier to classifydue to the nature of the instrument Bass leads my mindto bass strings and to heed the low end instead of the en-tire range from the highest to lowest notes Same for tre-ble vs bright rdquo In particular 4 participants used the wordldquotwangrdquo to describe guitars they classified as treble whileanother guitarist reported that a treble guitar ldquosounds bestfor finger picking and light strumming as heavy playingcan get quite ear-piercing and harsh This last commentseems to echo the second hypothesis for the origin of thestronger LTAS shown by the mid-even guitars (see Sec-tion 345)

Two participants reported looking out for problematicnotes as a more meaningful way of finding bassy guitarsThis likely points toward the ldquoveeringrdquo phenomenon as-sociated with a stronger admittance at lower frequencies

405

ACTA ACUSTICA UNITED WITH ACUSTICA Mansour et al Post-classification of steel string guitarsVol 101 (2015)

[16] Further one musician referred to the tactile feedbackreceived from a vibrating body associating bassy guitarswith more felt vibrations than the other two categoriesThis is a perspective shared by the manufacturer as wellIn fact body vibrations can provide tactile and proprio-ceptive cues that contribute to the perception of the radi-ated sound so that the musician can assess their interactionwith the instrument cross-modally [21 22]

5 Discussion

A set of 18 steel string guitars of the same model man-ufactured with high geometrical tolerances in a modernfactory were evaluated in this study These guitars wereconsidered by the manufacturer to possess subtle yet per-ceivable variations that could be classified as either bassymid-even or treble The results of this study showed thatthe instruments classified as bassy had a consistently lowerT(11)1 breathing mode (see Figure 9a 14b and 15) Thissuggests a lower stiffness-to-weight ratio for the bodies ofthese guitars which is most likely the result of variationsin equivalent stiffness (see Section 35)

It was further shown that whenever the frequency ofT(11)1 was shifted downward or upward for a particu-lar guitar the frequencies of its higher modes consistentlychanged in a similar way (see Figure 9 also the slightlyshifted bumps at around 600 750 and 1200 Hz in Fig-ure 12a) This would not necessarily be the case if theguitars were of different models (ie with varying geome-tries) or of different types of wood (eg plywood versusspruce) This is an important result as there is a practi-cal advantage to having higher-frequency modes move ac-cording to the first corpus mode across guitars of the samemodel T(11)1 is a low-frequency resonance that is suffi-ciently separated from the adjacent modes and hence eas-ily identifiable Measuring its frequency across differentinstruments is easy and affordable in a small workshopenvironment One can simply hit the guitar with a reg-ular hammer and recordanalyse the sound using an in-expensive microphone and a laptop reminiscent of thetraditional tap-tone method used by luthiers Moreoversuch measurements can be very reliable provided that theboundary conditions of the instrument (ie the installedfixture that holds the instrument) as well as the air tem-perature pressure and humidity are kept consistent Theseparameters constitute the biggest source of uncertainty infrequency-based features and can easily be controlled in afactory space or small workshop

Finding a reliable differentiator between the mid-evenand treble guitars was less obvious Although we ob-served that the averaged mobility curves of the mid-eveninstruments tended to be slightly higher in the vicinityof 600ndash2000 Hz than the treble guitars further measure-ments would have been necessary to determine whetherthe weaker admittance of the treble instruments at higherfrequencies would contribute to a longer decay for thehigher partials (see Section 35) In addition the differ-ence in averaged mobility magnitude between the two cat-

egories was only about 2 dB which is close to the marginof error for the measurement system Such a small differ-ence in an amplitude-based feature may be easily obscuredby variations in the exact angle of the hammer location ofhitting and measuring or calibration of the sensors

We invited guitar players of varying musical back-ground and experience to classify the guitars into thesecategories We observed very poor agreement among theparticipants but it is common in recent perceptual studiesof musical instruments that musicians do not agree witheach other [23 24 25] This seems a result of them lis-tening to (or feeling) different aspects of the instrumentand not necessarily all attending to the same features Theemployee that post-classified the guitars at the factory wasthe person that tests every guitar coming off the productionline thus he must have a lot of experience evaluating thesound of these instruments Assuming he did his classifi-cation carefully then there likely were particular things hewas listening for The participants in our perceptual exper-iment would not necessarily listen for the same featuresso it is not too surprising they largely disagreed on theirclassifications That said the differences between the tre-ble and mid-even guitars were extremely subtle and evencompany representatives often mis-categorized them dur-ing informal playing tests And this together with the factthat our measurements showed no consistent differencesbetween the two categories would seem to lead us to ques-tion whether the distinction between mid-even and trebletruly exists at all Another possible explanation is that dif-ferent musicians may follow different processes to assessthose features considered essential for the evaluation of aninstrument [23 26] Accordingly guitar players may agreeof what particular characteristics they look for in a tre-ble guitar but the perception of the same characteristicswidely varies across individuals

6 Conclusions

Efforts to understand how the tonal quality of an instru-ment relates to its measurable physical properties have of-ten been inconclusive (eg [27]) In the light of some re-cent studies the concept of ldquogoodrdquo versus ldquobadrdquo instru-ments is likely too general and it would be more advanta-geous to instead consider specific perceptual attributes ofan instrument [23 28] In the long run it may be possibleto derive a multidimensional timbre space where each di-mension is explained by some measurable quantities andin which instruments with similar qualities are likely tocluster In this study more specific sound characteristicsof instruments were thus targeted rather than their overalltonal quality

From bridge admittance measurements on 18 steelstring guitars from the same production line and post-classified by the manufacturer as either bassy mid-evenor treble (see Section 1 for details) it was shown that itis possible to discriminate between bassy guitars and theother two categories based on the frequency of the breath-ing mode However finding a similarly reliable feature to

406

Mansour et al Post-classification of steel string guitars ACTA ACUSTICA UNITED WITH ACUSTICAVol 101 (2015)

distinguish mid-even from treble instruments is less triv-ial at this point The treble guitars demonstrated a loweraveraged mobility in the frequency range of 600ndash2000 Hzthough more empirical investigation would be necessarybefore drawing any conclusions Results from a percep-tual experiment wherein guitar players were asked to clas-sify the guitars into these categories showed almost noagreement between guitarists in their judgements as wellas between the latter and the classification by the manufac-turer Despite the overall lack of consensus guitar playerstended to agree more on the bassy instruments than theother two categories

Some essential additions to this study are a more care-ful analysis on the perceptual definition of mid-even ver-sus treble and analysing the decay rate of the partials fora plucked sound to see how it correlates with the admit-tance Analysis of this kind has been performed for syn-thesized acoustic guitar sounds (see Figure 7 in [16]) but asimilar analysis on real guitar sounds will involve difficul-ties caused by the double set of harmonics [29] as well aslongitudinal string vibrations (also dubbed ldquophantom par-tialsrdquo) [30]

Acknowledgements

This project was funded by a Natural Sciences and En-gineering Research Council of Canada Engage grant incollaboration with Godin Guitars The authors wish tothank Robert Godin for this opportunity We also thankLawrence Joseph (statistics) and Garry Antonio (guitarperformance) for constructive comments We are grate-ful to Jim Woodhouse for sharing his analysis code Thefirst author would like to acknowledge the Government ofCanada for a Vanier Canada Graduate Scholarship

References

[1] E V Jansson A study of acoustical and hologram inter-ferometric measurements of the top plate vibrations of aguitar Acust Acta Acust 25 (1971) 95ndash100

[2] O Christensen B B Vistisen Simple model for low-frequency guitar function J Acoust Soc Am 68 (1980)758ndash766

[3] O Christensen Quantitative models for low frequency gui-tar function J Guitar Acoustics 6 (1982) 10ndash25

[4] H Wright The acoustics and psychoacoustics of the guitarDissertation Dept of Physics and Astronomy Universityof Wales Cardiff UK 1996

[5] N H Fletcher T D Rossing The physics of musical in-struments Second ed Springer New York 1998

[6] J E Popp Four mass coupled oscillator guitar model JAcoust Soc Am 131 (2012) 829ndash836

[7] J Woodhouse Body vibration of the violinmdashwhat can amaker except to control Catgut Acoust Soc J (Series II)4 (2002) 43ndash49

[8] B Elie F Gautier B David Macro parameters describingthe mechanical behavior of classical guitars J Acoust SocAm 132 (2012) 4013ndash4024

[9] J Woodhouse R S Langley Interpreting the input admit-tance of violins and guitars Acust Acta Acust 98 (2012)611ndash628

[10] A Gabrielsson E V Jansson Long-time-average-spectraand rated qualities of twenty-two violins Acustica 42(1979) 47ndash55

[11] P D Welch The use of Fast Fourier Transform for the es-timation of power spectra A method based on time averag-ing over short modified periodograms IEEE Trans Audioand Electroacoust AU-15 (1967) 70ndash73

[12] J Sundberg The singerrsquos formant revisited J Voice 4(1995) 106ndash119

[13] A Buen On timbre parameters and sound levels ofrecorded old violins J Violin Soc Am VSA Papers XXI(2007) 57ndash68

[14] J Woodhouse On the ldquobridge hillrdquo of the violin AcustActa Acust 91 (2005) 155ndash165

[15] B Richardson The acoustical development of the guitarCatgut Acoustical Society Journal 2 (1994) 1ndash10

[16] J Woodhouse On the synthesis of guitar plucks AcustActa Acust 90 (2004) 928ndash944

[17] C H Hodges J Power J Woodhouse The use of the sono-gram in structural acoustics and an application to the vibra-tions of cylindrical shells J Sound Vib 101 (1985) 203ndash218

[18] L D Broemeling Bayesian methods for measures ofagreement Chapman amp HallCRC New York 2009 (Bio-statistics)

[19] J L Fleiss Measuring nominal scale agreement amongmany raters Psychological Bulletin 76 (1971) 378ndash382

[20] D Lunn C Jackson N Best A Thomas D Spiegelhal-ter The BUGS Book A Practical Introduction to BayesianAnalysis Chapman amp HallCRC New York 2012 (Textsin Statistical Science)

[21] C Saitis Evaluating violin quality Player reliability andverbalization Dissertation Dept of Music ResearchMcGill University Montreal Quebec Canada 2013

[22] I Wollman Perception bimodale des violonistes en situa-tion de jeu Influence des retours auditif et vibrotactile surlrsquoeacutevaluation du violon Dissertation Universiteacute Pierre etMarie Curie Paris France 2013

[23] C Saitis B L Giordano C Fritz G P Scavone Percep-tual evaluation of violins A quantitative analysis of pref-erence judgements by experienced players J Acoust SocAm 132 (2012) 4002ndash4012

[24] C Fritz J Curtin J Poitevineau P Morrel-Samuels F-CTao Player preferences among new and old violins ProcNat Acad Sci USA 109 (2012) 760ndash763

[25] P Eveno J-P Dalmont R Causseacute G P Scavone Anacoustic and perceptual evaluation of saxophone pad ldquores-onatorsrdquo Acta Acustica united with Acustica 101 (2015)246ndash255

[26] C Fritz J Woodhouse F P-H Cheng I Cross A FBlackwell B C J Moore Perceptual studies of violinbody damping and vibrato J Acoust Soc Am 127 (2010)513ndash524

[27] G Bissinger Structural acoustics of good and bad violinsJ Acoust Soc Am 124 (2008) 1764ndash1773

[28] C Saitis C Fritz C Guastavino G P Scavone Con-ceptualization of violin quality by experienced performersProc Stockholm Music Acoust Conf Stockholm Swe-den 2013 123ndash128

[29] J Woodhouse Plucked guitar transients Comparison ofmeasurements and synthesis Acust Acta Acust 90 (2004)945ndash965

[30] B Bank L Sujbert Generation of longitudinal vibrationsin piano strings From physics to sound synthesis JAcoust Soc Am 117 (2004) 2268ndash2278

407

ACTA ACUSTICA UNITED WITH ACUSTICA Mansour et al Post-classification of steel string guitarsVol 101 (2015)

mode in question as well as the radiated sound at that fre-quency Based on this idea two features are calculated forall semitones played on a guitar (from E2 to E6) and aver-aged over the three perceptual categories For both cases βis arbitrarily chosen as 03 which is a reasonable numberfor a normal playing condition

Figure 12a shows the sum of the transferred power bythe first 5 harmonics of each played note here calledldquopower transfer to the bodyrdquo The choice of 5 harmonics isan arbitrary limit for which the individual harmonics of thestring could be perceived individually and within whichmost of the energy transfer to the body happens in a nor-mal playing condition The frequency range that is consid-ered in this fashion would be up to 5 times the highest notestudied (ie E6) or equivalently 66 kHz this is about thelimit up to which the measured admittances show a reason-ably good coherence The unknown constant A in equation(2) could only cause an offset in the results therefore itis considered to be 1 for convenience The energy trans-fer to the body represents the strength of the stringbodycoupling for a guitar A large value for this number wouldresult in a strong and more percussive (banjo-like) soundand a small value would result in a weaker sound with alonger decay (more similar to an electric guitar without anamplifier)

Figure 12b shows a slightly different use of the trans-ferred energy by each harmonic It is calculated by divid-ing the sum of the energy transferred by the 2nd to 5th har-monic to the energy transferred by the fundamental aloneAs well as specifying the ratio of higher harmonic energyto that at the fundamental this feature defines how thesound evolves once the instrument is plucked Extremelylarge or small values for a particular note mean that thenote changes its character when it decays while an aver-age value may be correlated to a more stable sound

Neither graph shows any clear distinction between themid-even and treble categories The bassy category doeshowever stand out most of the patterns in both graphs areshifted downward by a few semitones

345 LTAS and radiativity

The LTAS results averaged over different categories areillustrated in Figure 13a The results for the mid-even in-struments are greater than the other two groups over al-most all frequencies Figure 13b is the same as Figure 13abut the LTAS for each individual instrument is normalizedby its area before further processing After normalizationall categories behave similarly over different frequenciesmeaning mid-even instruments were simply louder thanthe other instruments rather than being stronger in a par-ticular range of frequencies

The band-averaged admittance of the mid-even instru-ments earlier presented in Figure 10 was slightly greaterthan the treble instruments in some frequency rangeshowever the significance of the stronger LTAS of mid-even instruments especially beyond 3 kHz can hardly bejustified by that alone A possible explanation is that themid-even instruments were played systematically harder

E2 100 E3 250 E4 500 E5 1000 E6

minus60

minus55

minus50

minus45

minus40

minus35

minus30

Pow

ertransfertothebody

(dB)

E2 100 E3 250 E4 500 E5 1000 E6

minus20

minus10

0

10

Harmonicstofundam

entalratio(dB)

Fundamental frequency (Hz)

Averaged bassyAveraged midminusevenAveraged treble

(a)

(b)

Figure 12 Power transfer to the body (a) and harmonics to fun-damental ratio (b) calculated for each played note over the op-erational range of a guitar (E2 to E6) the vertical lines indicateoctaves The values are averaged for the guitars of different cat-egories (solid line for bassy dash-dot line and light gray green(online version) shade for mid-even and dashed line and darkergray red (online version) shade for treble)

compared to the treble instruments There are at least tworeasons as to why this might have been the case First theguitars were recorded per category starting with the trebleand ending with the mid-even ones It is therefore possiblethat the musician felt more confident in the course of therecording session Second the mid-even guitars had beendescribed by the manufacturer as having a ldquomore stablerdquosound Such a sense might have allowed the performer toplay those instruments more freely Nevertheless we areaware that the frequency distribution obtained using LTAScan be strongly dependent on the nature of the input sig-nal depending on the sequence played and used as an in-put certain frequency bands may be less excited than oth-ers which may attenuate differences in some frequencyregions of interest

35 Classification results

According to our results finding a feature that can distin-guish bassy instruments is a relatively easy task As wasshown in Figure 9a the frequencies of the first 4 modesfor the bassy instruments are around 5 lower than thosefor the two other classes The frequency of the first modecan be determined without much ambiguity or even theneed for mode extraction All of the other 3 low-frequencymodes have close adjacent modes making it harder andsometimes impossible to find their correct modal parame-ters

The amplitude of the first mode is another indicator fora bassy instrument that can be used for more assurance

402

Mansour et al Post-classification of steel string guitars ACTA ACUSTICA UNITED WITH ACUSTICAVol 101 (2015)

500 1000 2000 6000minus180

minus170

minus160

minus150

minus140

minus130

minus120

LTAS(dBHz)

500 1000 2000 6000minus120

minus110

minus100

minus90

minus80

minus70

minus60

Frequency (Hz)

Normalized

LTAS(dBHz)

BassyMidminusevenTrebleTreble plusmn one STDMidminuseven plusmn one STD

(b)

(a)

Figure 13 Non-normalized LTAS (a) and normalized LTAS (b)averaged for the guitars of different categories plusminus onestandard deviation (solid line for bassy dash-dot line and lightgray green (online version) shade for mid-even and dashed lineand darker gray red (online version) shade for treble)

(see Figure 9b) Figure 14a shows a 2-D plane with thenormalized frequency and amplitude of the first mode onits horizontal and vertical axes respectively It can be seenthat the bassy instruments are quite distinct from the mid-even and treble ones Moreover the guitar with the lowestbreathing mode (B3 fT (11)1 = 91 Hz) was described bythe factory employee as extra-bassy

It should be emphasized that the frequency of the firstpeak works particularly well as a feature not becausethe ldquobassinessrdquo of the instrument is defined by the notesaround this modersquos frequency (only the first three noteson the E2 string are below that frequency) but becauseit represents the overall stiffness-to-weight ratio of thesoundbox that effectively scales the frequency of all bodymodes It is noteworthy that although T(11)1 has a strongHelmholtz component the amount by which its frequencydeviates from the frequency of the pure Helmholtz reso-nance is strongly dependent on the frequency separationof the pure Helmholtz and pure body breathing modesThe frequency of the pure Helmholtz mode is assumed thesame for all instruments of the pool given their very simi-lar geometry (both for the sound box and the sound hole)

The in-plane dimensions of the instruments should havea very small variation percentage given the precise con-trol of geometrical conditions in the factory Moreover alower frequency for the first mode is usually accompaniedby a higher amplitude for that mode which would not nor-mally be the case if such deviation was caused by using a

90 92 94 96 98 100 102 10404

05

06

07

08

09

1

Normalized

T(11) 1am

plitude

BassyMid EvenTreble

90 92 94 96 98 100 102 104minus50

minus49

minus48

minus47

minus46

minus45

B1B2

B3

B6

B4B5

ME6ME1

ME2

ME3ME4

ME5

T2

T3

T4

T5 T6

T1

Frequency of the T(11)1mode (Hz)

Averagedmobility(600minus2000Hz)(dB)

(b)

(a)

Figure 14 Classification of the acoustic guitars (a) by the fre-quency and amplitude of their first corpus mode T(11)1 onlyto distinguish bassy instruments from the others and (b) by thefrequency of their first corpus mode T(11)1 and their averagedmobility from 600 Hz to 2 kHz (B = bassy ME = mid-even T =treble) The dB reference is 1 msN

denser wood this leaves variations in the plate thicknessand the equivalent stiffness (ie very roughly the geomet-ric mean of the Youngrsquos moduli in long-grain and cross-grain directions [14]) as those most likely responsible forthe observed shift in mode frequencies Mode frequenciesof a flat plate scale with Eρ(tL2) where E and ρ arethe elastic constant and density t the thickness and L acharacteristic scaling length If either of the two factors issolely responsible to make a 5 shift it would require ei-ther a 5 variation (asymp 0125 mm) of the plate thicknessor a 10 variation in the equivalent stiffness Given thetight tolerances in cutting the plates and their mechanizedsanding it thus appears that variations in equivalent stiff-ness are the primary reason that some of the instrumentsin the pool sound more bassy In the manufacturing pro-cess a wood slab is sliced to produce several plates Thelateral direction of each plate hence represents a particularcombination of the slabrsquos radial and tangential directions(due to the growth rings of the tree) This can introduce asignificant variation in the platesrsquo stiffness in the lateral di-rection which produces variations of the effective stiffnessof the plates

Finding a feature that distinguishes mid-even from tre-ble instruments is less obvious We found that the aver-aged admittances of the treble instruments in this study areslightly lower than the mid-even instruments in the rangeof 600 Hz to 2 kHz Although this difference is not signif-icant over the mentioned range (ie average plusminus

403

ACTA ACUSTICA UNITED WITH ACUSTICA Mansour et al Post-classification of steel string guitarsVol 101 (2015)

one standard deviation of the two categories overlap) itbecomes significant if the average level of the admittancefor the whole range is compared (ie the average valueof all frequency points of the admittance from 600 Hz to2 kHz) Figure 14b shows a 2-D plane with the frequencyof the first mode on the horizontal axis and the average ad-mittance for the range of 600 Hz to 2 kHz The differencefor the mean of the mid-even and treble categories is about2 dB It seems likely that the weaker admittance of the tre-ble category at higher frequencies would contribute to alonger decay for the higher partials This may contributeto a sharper after-sound of the treble instruments that isknown to allow them to cut through the band more effec-tively This hypothesis could be tested by recording theplucked sound of the instruments and comparing the decayrate of the partials among different categories Due to thelimited project time frame this was not investigated fur-ther The mid-even guitars also show stronger LTAS thanthe treble instruments though this result might have beendue to playing conditions (see Section 345)

We applied the same classification (frequency ofT(11)1 versus average mobility between 600 Hz and2 kHz) to 6 acoustic guitars from a different model by thesame manufacturer (Simon amp Patrick) Unlike the Seagullguitars which were all nominally identical some of theSampP instruments had a slightly different bracing patterndesigned to exemplify the properties of a particular per-ceptual category The manufacturerrsquos classification of the6 guitars was not provided to us prior to our own mea-surements though we were told that there were 2 guitarsof each category The resulting classification based on thediscussed features is depicted in Figure 15 The two gui-tars designed to be more bassy were successfully identi-fied from the frequency of their breathing mode but dis-tinguishing between the mid-even and treble instrumentsfrom the averaged admittance criterion was inconclusive

4 Perceptual evaluation

As previously discussed in Section 1 the investigatedacoustic guitars were post-classified by the manufacturerbased on their experience of what type of instrument mightbetter fit a certain playing stylemdashfor example an amateurmusician that plays mostly strummed chords may prefera bassy guitar while a more experienced guitarist mayopt for a treble instrument We therefore conducted a per-ceptual experiment to investigate the level of agreementamong guitar players of varying experience and musicalbackground in classifying the guitars into the categoriesconsidered by the manufacturer

41 Method

The experiment took place in an acoustically dry roomwith dimensions of 60 times 78 times 32 m3 and a reverberationtime of about 03 s to help minimize the effects of roomreflections on the direct sound from the guitars To avoidthe potential problems of using a common pick across all

85 86 87 88 89 90 91 92minus505

minus50

minus495

minus49

minus485

minus48B1

SampP

B2SampP

ME1SampP

ME2SampP

T1SampP

T2SampP

Frequency of the T(11)1mode (Hz)

Averagedmobility(600minus2000Hz)(dB)

BassyMid EvenTreble

Figure 15 Classification of the six SampP guitars based on thefrequency of the T(11)1 mode versus averaged mobility from600 Hz to 2 kHz (B= bassy ME=mid-even T= treble) The dBreference is 1 msN

participants (eg musicians being uncomfortable with apick they are not familiar with) participants were asked tocarry out the classification task using their own pick(s)

411 Participants

Thirteen guitar players took part in the experiment (1 fe-male 12 males average age = 25 yrs SD = 5 yrs range =19ndash36 yrs) They had at least 6 years of acoustic guitar ex-perience (average years of training = 11 yrs SD = 4 yrsrange = 6ndash20 yrs average hours of practice per week =14 hrs SD = 10 hrs range = 3ndash40 hrs) as well as vary-ing experience with different types of guitars [nylon-string(85) acoustic-electric (62) acoustic bass (15) elec-tric (100) and electric bass (54)] Participants re-ported playing a wide range of musical styles [classical(77) jazz (85) pop-rock (92) country folk blue-grass (66) hard rock heavy metal (46) flamenco(15) latin (46) blues (77) avant-garde contempo-rary (54) and funk alternative (46)] both in a bandand as soloists Seven guitarists (54) reported that theytypically or most often use a pick whereas 4 players (31)reported they do not 2 guitarists (15) reported using apick occasionally rather than typically Nine participantshad higher-level degrees in music performance or othermusic-related areas [BMus (music theory classical gui-tar) BFA (jazz studies) MMus (guitar performance) MA(digital musics) PhD (music theory) Diplocircme drsquoEacutetat deProfesseur de Musique (France)] All guitar players werepaid for their participation

412 Procedure

The experimental session lasted 2 hours and was organizedin two phases In the first phase participants were pre-sented with the guitars randomly ordered on the floor (de-termined by computer calculations) They were asked toplay all instruments for up to 20 minutes in order to famil-iarize themselves with the set and use this time to tune theguitars as needed In the second phase participants were

404

Mansour et al Post-classification of steel string guitars ACTA ACUSTICA UNITED WITH ACUSTICAVol 101 (2015)

given up to 20 minutes to classify each of the guitars as ei-ther bassy mid-even or treble by placing them in speciallymarked areas of the room At the end of the second phaseparticipants were given the opportunity to revise their ini-tial judgements if they so wanted until they were satis-fied with their classification No playing constraint wasimposed on the evaluation process (eg specific repertoireor technique) Participants were instead encouraged to fol-low their own strategy with the suggestion to use bothstrumming-oriented songs (eg Sitting Waiting Wishingby Jack Johnson) and fingerpicking-oriented pieces (egDust in the Wind by Kansas) Participants were instructedto maximize evaluation speed and accuracy Upon com-pleting the session participants talked about their classi-fications and overall experience through written responsesto the following questions ldquoFor each class of guitar whatsound characteristics did you look for Can you describehow you evaluated the guitars in terms of these character-istics (eg musical material technique style)rdquo (Q1) ldquoTowhat extend do you agree with the given classification ofguitarsrdquo (Q2) and ldquoCan you suggest a classification thatbetter reflects how you evaluate the sound characteristicsof guitarsrdquo (Q3)

42 Results

Inter-individual consistency was measured as the Fleissrsquokappa coefficient between guitar classifications from dif-ferent participants using a Bayesian approach to inference[18] Fleissrsquo kappa is a generalization of Cohenrsquos kappastatistic κ a chance-corrected measure of agreement be-tween two assessors to multiple raters [19] It ranges fromminus1 to 1 κ (AB) = 1 and minus1 if A = B and A = minusBrespectively and κ (AB) = 0 in case of no associationbetween A and B (ie exactly what would be expectedby chance) We implemented Gibbs sampling through thesoftware WinBUGS (Windows Bayesian inference UsingGibbs Sampling [20]) with 50000 observations generatedfrom the posterior distribution with a burn-in of 1000 ob-servations A uniformly distributed multicategorical priorwas assumed with total information across all possiblecells (ie from different combinations of participants) be-ing equivalent to a single prior observation (ie a priorsample size equivalent of 1)

Fleissrsquos kappa was estimated with a posterior mean of01 and a 95 credible interval [-02 05] indicating avery low amount of agreement between the participantsin their classifications Figure 16 shows the proportion oftimes a guitar was classified as either bassy mid-even ortreble (the guitars are arranged in three groups accord-ing to the initial factory classification) Whereas someagreement concerning the bassy guitars can be observedmdashindeed the extra bassy guitar (B3) was most often clas-sified as bassymdashthe perceptions for the other two classeslargely vary across different musicians In particular manyguitars that had been characterized as treble by the fac-tory employee were perceived as mid-even Several mu-sicians specifically reported encountering difficulties withthe mid-even category In answering Q3 a player observed

B1 B2 B3 B4 B5 B60

25

50

75

100

classifiedas

(a)BassyMidminusevenTreble

ME1 ME2 ME3 ME4 ME5 ME60

25

50

75

100

classifiedas

(b)

T1 T2 T3 T4 T5 T60

25

50

75

100

classifiedas

Guitars

(c)

Figure 16 Perceptual evaluation Proportion of times a guitarwas classified into one of the categories by the participants Gui-tars are labelled and grouped according to the factory classifica-tion (B = bassy ME = mid-even T = treble) The color scheme(in the online version) corresponds to the classifications providedby the guitar players

that ldquothere is a difference [between] a guitar with domi-nant mid tones and one with an even soundrdquo

Overall participants expressed no strong difficulties incarrying out the classification task On a scale from 1(strongly disagree) to 5 (strongly agree) all performers se-lected 4 when asked ldquoTo what extend do you agree withthe given classification of guitarsrdquo This indicates that the3 categories considered by the manufacturer are gener-ally meaningful to musicians However verbal responsesto questions Q1 and Q3 reveal that participants often as-sociated bassy with more prominent E2 and A2 strings(ie they focused on the low-frequency end of the gui-tar) and similarly treble with better emphasized B3 andE4 strings (ie they paid attention to the high-frequencyrange) As one guitar player noted ldquoI feel that descriptionslike warm neutral and bright would be easier to classifydue to the nature of the instrument Bass leads my mindto bass strings and to heed the low end instead of the en-tire range from the highest to lowest notes Same for tre-ble vs bright rdquo In particular 4 participants used the wordldquotwangrdquo to describe guitars they classified as treble whileanother guitarist reported that a treble guitar ldquosounds bestfor finger picking and light strumming as heavy playingcan get quite ear-piercing and harsh This last commentseems to echo the second hypothesis for the origin of thestronger LTAS shown by the mid-even guitars (see Sec-tion 345)

Two participants reported looking out for problematicnotes as a more meaningful way of finding bassy guitarsThis likely points toward the ldquoveeringrdquo phenomenon as-sociated with a stronger admittance at lower frequencies

405

ACTA ACUSTICA UNITED WITH ACUSTICA Mansour et al Post-classification of steel string guitarsVol 101 (2015)

[16] Further one musician referred to the tactile feedbackreceived from a vibrating body associating bassy guitarswith more felt vibrations than the other two categoriesThis is a perspective shared by the manufacturer as wellIn fact body vibrations can provide tactile and proprio-ceptive cues that contribute to the perception of the radi-ated sound so that the musician can assess their interactionwith the instrument cross-modally [21 22]

5 Discussion

A set of 18 steel string guitars of the same model man-ufactured with high geometrical tolerances in a modernfactory were evaluated in this study These guitars wereconsidered by the manufacturer to possess subtle yet per-ceivable variations that could be classified as either bassymid-even or treble The results of this study showed thatthe instruments classified as bassy had a consistently lowerT(11)1 breathing mode (see Figure 9a 14b and 15) Thissuggests a lower stiffness-to-weight ratio for the bodies ofthese guitars which is most likely the result of variationsin equivalent stiffness (see Section 35)

It was further shown that whenever the frequency ofT(11)1 was shifted downward or upward for a particu-lar guitar the frequencies of its higher modes consistentlychanged in a similar way (see Figure 9 also the slightlyshifted bumps at around 600 750 and 1200 Hz in Fig-ure 12a) This would not necessarily be the case if theguitars were of different models (ie with varying geome-tries) or of different types of wood (eg plywood versusspruce) This is an important result as there is a practi-cal advantage to having higher-frequency modes move ac-cording to the first corpus mode across guitars of the samemodel T(11)1 is a low-frequency resonance that is suffi-ciently separated from the adjacent modes and hence eas-ily identifiable Measuring its frequency across differentinstruments is easy and affordable in a small workshopenvironment One can simply hit the guitar with a reg-ular hammer and recordanalyse the sound using an in-expensive microphone and a laptop reminiscent of thetraditional tap-tone method used by luthiers Moreoversuch measurements can be very reliable provided that theboundary conditions of the instrument (ie the installedfixture that holds the instrument) as well as the air tem-perature pressure and humidity are kept consistent Theseparameters constitute the biggest source of uncertainty infrequency-based features and can easily be controlled in afactory space or small workshop

Finding a reliable differentiator between the mid-evenand treble guitars was less obvious Although we ob-served that the averaged mobility curves of the mid-eveninstruments tended to be slightly higher in the vicinityof 600ndash2000 Hz than the treble guitars further measure-ments would have been necessary to determine whetherthe weaker admittance of the treble instruments at higherfrequencies would contribute to a longer decay for thehigher partials (see Section 35) In addition the differ-ence in averaged mobility magnitude between the two cat-

egories was only about 2 dB which is close to the marginof error for the measurement system Such a small differ-ence in an amplitude-based feature may be easily obscuredby variations in the exact angle of the hammer location ofhitting and measuring or calibration of the sensors

We invited guitar players of varying musical back-ground and experience to classify the guitars into thesecategories We observed very poor agreement among theparticipants but it is common in recent perceptual studiesof musical instruments that musicians do not agree witheach other [23 24 25] This seems a result of them lis-tening to (or feeling) different aspects of the instrumentand not necessarily all attending to the same features Theemployee that post-classified the guitars at the factory wasthe person that tests every guitar coming off the productionline thus he must have a lot of experience evaluating thesound of these instruments Assuming he did his classifi-cation carefully then there likely were particular things hewas listening for The participants in our perceptual exper-iment would not necessarily listen for the same featuresso it is not too surprising they largely disagreed on theirclassifications That said the differences between the tre-ble and mid-even guitars were extremely subtle and evencompany representatives often mis-categorized them dur-ing informal playing tests And this together with the factthat our measurements showed no consistent differencesbetween the two categories would seem to lead us to ques-tion whether the distinction between mid-even and trebletruly exists at all Another possible explanation is that dif-ferent musicians may follow different processes to assessthose features considered essential for the evaluation of aninstrument [23 26] Accordingly guitar players may agreeof what particular characteristics they look for in a tre-ble guitar but the perception of the same characteristicswidely varies across individuals

6 Conclusions

Efforts to understand how the tonal quality of an instru-ment relates to its measurable physical properties have of-ten been inconclusive (eg [27]) In the light of some re-cent studies the concept of ldquogoodrdquo versus ldquobadrdquo instru-ments is likely too general and it would be more advanta-geous to instead consider specific perceptual attributes ofan instrument [23 28] In the long run it may be possibleto derive a multidimensional timbre space where each di-mension is explained by some measurable quantities andin which instruments with similar qualities are likely tocluster In this study more specific sound characteristicsof instruments were thus targeted rather than their overalltonal quality

From bridge admittance measurements on 18 steelstring guitars from the same production line and post-classified by the manufacturer as either bassy mid-evenor treble (see Section 1 for details) it was shown that itis possible to discriminate between bassy guitars and theother two categories based on the frequency of the breath-ing mode However finding a similarly reliable feature to

406

Mansour et al Post-classification of steel string guitars ACTA ACUSTICA UNITED WITH ACUSTICAVol 101 (2015)

distinguish mid-even from treble instruments is less triv-ial at this point The treble guitars demonstrated a loweraveraged mobility in the frequency range of 600ndash2000 Hzthough more empirical investigation would be necessarybefore drawing any conclusions Results from a percep-tual experiment wherein guitar players were asked to clas-sify the guitars into these categories showed almost noagreement between guitarists in their judgements as wellas between the latter and the classification by the manufac-turer Despite the overall lack of consensus guitar playerstended to agree more on the bassy instruments than theother two categories

Some essential additions to this study are a more care-ful analysis on the perceptual definition of mid-even ver-sus treble and analysing the decay rate of the partials fora plucked sound to see how it correlates with the admit-tance Analysis of this kind has been performed for syn-thesized acoustic guitar sounds (see Figure 7 in [16]) but asimilar analysis on real guitar sounds will involve difficul-ties caused by the double set of harmonics [29] as well aslongitudinal string vibrations (also dubbed ldquophantom par-tialsrdquo) [30]

Acknowledgements

This project was funded by a Natural Sciences and En-gineering Research Council of Canada Engage grant incollaboration with Godin Guitars The authors wish tothank Robert Godin for this opportunity We also thankLawrence Joseph (statistics) and Garry Antonio (guitarperformance) for constructive comments We are grate-ful to Jim Woodhouse for sharing his analysis code Thefirst author would like to acknowledge the Government ofCanada for a Vanier Canada Graduate Scholarship

References

[1] E V Jansson A study of acoustical and hologram inter-ferometric measurements of the top plate vibrations of aguitar Acust Acta Acust 25 (1971) 95ndash100

[2] O Christensen B B Vistisen Simple model for low-frequency guitar function J Acoust Soc Am 68 (1980)758ndash766

[3] O Christensen Quantitative models for low frequency gui-tar function J Guitar Acoustics 6 (1982) 10ndash25

[4] H Wright The acoustics and psychoacoustics of the guitarDissertation Dept of Physics and Astronomy Universityof Wales Cardiff UK 1996

[5] N H Fletcher T D Rossing The physics of musical in-struments Second ed Springer New York 1998

[6] J E Popp Four mass coupled oscillator guitar model JAcoust Soc Am 131 (2012) 829ndash836

[7] J Woodhouse Body vibration of the violinmdashwhat can amaker except to control Catgut Acoust Soc J (Series II)4 (2002) 43ndash49

[8] B Elie F Gautier B David Macro parameters describingthe mechanical behavior of classical guitars J Acoust SocAm 132 (2012) 4013ndash4024

[9] J Woodhouse R S Langley Interpreting the input admit-tance of violins and guitars Acust Acta Acust 98 (2012)611ndash628

[10] A Gabrielsson E V Jansson Long-time-average-spectraand rated qualities of twenty-two violins Acustica 42(1979) 47ndash55

[11] P D Welch The use of Fast Fourier Transform for the es-timation of power spectra A method based on time averag-ing over short modified periodograms IEEE Trans Audioand Electroacoust AU-15 (1967) 70ndash73

[12] J Sundberg The singerrsquos formant revisited J Voice 4(1995) 106ndash119

[13] A Buen On timbre parameters and sound levels ofrecorded old violins J Violin Soc Am VSA Papers XXI(2007) 57ndash68

[14] J Woodhouse On the ldquobridge hillrdquo of the violin AcustActa Acust 91 (2005) 155ndash165

[15] B Richardson The acoustical development of the guitarCatgut Acoustical Society Journal 2 (1994) 1ndash10

[16] J Woodhouse On the synthesis of guitar plucks AcustActa Acust 90 (2004) 928ndash944

[17] C H Hodges J Power J Woodhouse The use of the sono-gram in structural acoustics and an application to the vibra-tions of cylindrical shells J Sound Vib 101 (1985) 203ndash218

[18] L D Broemeling Bayesian methods for measures ofagreement Chapman amp HallCRC New York 2009 (Bio-statistics)

[19] J L Fleiss Measuring nominal scale agreement amongmany raters Psychological Bulletin 76 (1971) 378ndash382

[20] D Lunn C Jackson N Best A Thomas D Spiegelhal-ter The BUGS Book A Practical Introduction to BayesianAnalysis Chapman amp HallCRC New York 2012 (Textsin Statistical Science)

[21] C Saitis Evaluating violin quality Player reliability andverbalization Dissertation Dept of Music ResearchMcGill University Montreal Quebec Canada 2013

[22] I Wollman Perception bimodale des violonistes en situa-tion de jeu Influence des retours auditif et vibrotactile surlrsquoeacutevaluation du violon Dissertation Universiteacute Pierre etMarie Curie Paris France 2013

[23] C Saitis B L Giordano C Fritz G P Scavone Percep-tual evaluation of violins A quantitative analysis of pref-erence judgements by experienced players J Acoust SocAm 132 (2012) 4002ndash4012

[24] C Fritz J Curtin J Poitevineau P Morrel-Samuels F-CTao Player preferences among new and old violins ProcNat Acad Sci USA 109 (2012) 760ndash763

[25] P Eveno J-P Dalmont R Causseacute G P Scavone Anacoustic and perceptual evaluation of saxophone pad ldquores-onatorsrdquo Acta Acustica united with Acustica 101 (2015)246ndash255

[26] C Fritz J Woodhouse F P-H Cheng I Cross A FBlackwell B C J Moore Perceptual studies of violinbody damping and vibrato J Acoust Soc Am 127 (2010)513ndash524

[27] G Bissinger Structural acoustics of good and bad violinsJ Acoust Soc Am 124 (2008) 1764ndash1773

[28] C Saitis C Fritz C Guastavino G P Scavone Con-ceptualization of violin quality by experienced performersProc Stockholm Music Acoust Conf Stockholm Swe-den 2013 123ndash128

[29] J Woodhouse Plucked guitar transients Comparison ofmeasurements and synthesis Acust Acta Acust 90 (2004)945ndash965

[30] B Bank L Sujbert Generation of longitudinal vibrationsin piano strings From physics to sound synthesis JAcoust Soc Am 117 (2004) 2268ndash2278

407

Mansour et al Post-classification of steel string guitars ACTA ACUSTICA UNITED WITH ACUSTICAVol 101 (2015)

500 1000 2000 6000minus180

minus170

minus160

minus150

minus140

minus130

minus120

LTAS(dBHz)

500 1000 2000 6000minus120

minus110

minus100

minus90

minus80

minus70

minus60

Frequency (Hz)

Normalized

LTAS(dBHz)

BassyMidminusevenTrebleTreble plusmn one STDMidminuseven plusmn one STD

(b)

(a)

Figure 13 Non-normalized LTAS (a) and normalized LTAS (b)averaged for the guitars of different categories plusminus onestandard deviation (solid line for bassy dash-dot line and lightgray green (online version) shade for mid-even and dashed lineand darker gray red (online version) shade for treble)

(see Figure 9b) Figure 14a shows a 2-D plane with thenormalized frequency and amplitude of the first mode onits horizontal and vertical axes respectively It can be seenthat the bassy instruments are quite distinct from the mid-even and treble ones Moreover the guitar with the lowestbreathing mode (B3 fT (11)1 = 91 Hz) was described bythe factory employee as extra-bassy

It should be emphasized that the frequency of the firstpeak works particularly well as a feature not becausethe ldquobassinessrdquo of the instrument is defined by the notesaround this modersquos frequency (only the first three noteson the E2 string are below that frequency) but becauseit represents the overall stiffness-to-weight ratio of thesoundbox that effectively scales the frequency of all bodymodes It is noteworthy that although T(11)1 has a strongHelmholtz component the amount by which its frequencydeviates from the frequency of the pure Helmholtz reso-nance is strongly dependent on the frequency separationof the pure Helmholtz and pure body breathing modesThe frequency of the pure Helmholtz mode is assumed thesame for all instruments of the pool given their very simi-lar geometry (both for the sound box and the sound hole)

The in-plane dimensions of the instruments should havea very small variation percentage given the precise con-trol of geometrical conditions in the factory Moreover alower frequency for the first mode is usually accompaniedby a higher amplitude for that mode which would not nor-mally be the case if such deviation was caused by using a

90 92 94 96 98 100 102 10404

05

06

07

08

09

1

Normalized

T(11) 1am

plitude

BassyMid EvenTreble

90 92 94 96 98 100 102 104minus50

minus49

minus48

minus47

minus46

minus45

B1B2

B3

B6

B4B5

ME6ME1

ME2

ME3ME4

ME5

T2

T3

T4

T5 T6

T1

Frequency of the T(11)1mode (Hz)

Averagedmobility(600minus2000Hz)(dB)

(b)

(a)

Figure 14 Classification of the acoustic guitars (a) by the fre-quency and amplitude of their first corpus mode T(11)1 onlyto distinguish bassy instruments from the others and (b) by thefrequency of their first corpus mode T(11)1 and their averagedmobility from 600 Hz to 2 kHz (B = bassy ME = mid-even T =treble) The dB reference is 1 msN

denser wood this leaves variations in the plate thicknessand the equivalent stiffness (ie very roughly the geomet-ric mean of the Youngrsquos moduli in long-grain and cross-grain directions [14]) as those most likely responsible forthe observed shift in mode frequencies Mode frequenciesof a flat plate scale with Eρ(tL2) where E and ρ arethe elastic constant and density t the thickness and L acharacteristic scaling length If either of the two factors issolely responsible to make a 5 shift it would require ei-ther a 5 variation (asymp 0125 mm) of the plate thicknessor a 10 variation in the equivalent stiffness Given thetight tolerances in cutting the plates and their mechanizedsanding it thus appears that variations in equivalent stiff-ness are the primary reason that some of the instrumentsin the pool sound more bassy In the manufacturing pro-cess a wood slab is sliced to produce several plates Thelateral direction of each plate hence represents a particularcombination of the slabrsquos radial and tangential directions(due to the growth rings of the tree) This can introduce asignificant variation in the platesrsquo stiffness in the lateral di-rection which produces variations of the effective stiffnessof the plates

Finding a feature that distinguishes mid-even from tre-ble instruments is less obvious We found that the aver-aged admittances of the treble instruments in this study areslightly lower than the mid-even instruments in the rangeof 600 Hz to 2 kHz Although this difference is not signif-icant over the mentioned range (ie average plusminus

403

ACTA ACUSTICA UNITED WITH ACUSTICA Mansour et al Post-classification of steel string guitarsVol 101 (2015)

one standard deviation of the two categories overlap) itbecomes significant if the average level of the admittancefor the whole range is compared (ie the average valueof all frequency points of the admittance from 600 Hz to2 kHz) Figure 14b shows a 2-D plane with the frequencyof the first mode on the horizontal axis and the average ad-mittance for the range of 600 Hz to 2 kHz The differencefor the mean of the mid-even and treble categories is about2 dB It seems likely that the weaker admittance of the tre-ble category at higher frequencies would contribute to alonger decay for the higher partials This may contributeto a sharper after-sound of the treble instruments that isknown to allow them to cut through the band more effec-tively This hypothesis could be tested by recording theplucked sound of the instruments and comparing the decayrate of the partials among different categories Due to thelimited project time frame this was not investigated fur-ther The mid-even guitars also show stronger LTAS thanthe treble instruments though this result might have beendue to playing conditions (see Section 345)

We applied the same classification (frequency ofT(11)1 versus average mobility between 600 Hz and2 kHz) to 6 acoustic guitars from a different model by thesame manufacturer (Simon amp Patrick) Unlike the Seagullguitars which were all nominally identical some of theSampP instruments had a slightly different bracing patterndesigned to exemplify the properties of a particular per-ceptual category The manufacturerrsquos classification of the6 guitars was not provided to us prior to our own mea-surements though we were told that there were 2 guitarsof each category The resulting classification based on thediscussed features is depicted in Figure 15 The two gui-tars designed to be more bassy were successfully identi-fied from the frequency of their breathing mode but dis-tinguishing between the mid-even and treble instrumentsfrom the averaged admittance criterion was inconclusive

4 Perceptual evaluation

As previously discussed in Section 1 the investigatedacoustic guitars were post-classified by the manufacturerbased on their experience of what type of instrument mightbetter fit a certain playing stylemdashfor example an amateurmusician that plays mostly strummed chords may prefera bassy guitar while a more experienced guitarist mayopt for a treble instrument We therefore conducted a per-ceptual experiment to investigate the level of agreementamong guitar players of varying experience and musicalbackground in classifying the guitars into the categoriesconsidered by the manufacturer

41 Method

The experiment took place in an acoustically dry roomwith dimensions of 60 times 78 times 32 m3 and a reverberationtime of about 03 s to help minimize the effects of roomreflections on the direct sound from the guitars To avoidthe potential problems of using a common pick across all

85 86 87 88 89 90 91 92minus505

minus50

minus495

minus49

minus485

minus48B1

SampP

B2SampP

ME1SampP

ME2SampP

T1SampP

T2SampP

Frequency of the T(11)1mode (Hz)

Averagedmobility(600minus2000Hz)(dB)

BassyMid EvenTreble

Figure 15 Classification of the six SampP guitars based on thefrequency of the T(11)1 mode versus averaged mobility from600 Hz to 2 kHz (B= bassy ME=mid-even T= treble) The dBreference is 1 msN

participants (eg musicians being uncomfortable with apick they are not familiar with) participants were asked tocarry out the classification task using their own pick(s)

411 Participants

Thirteen guitar players took part in the experiment (1 fe-male 12 males average age = 25 yrs SD = 5 yrs range =19ndash36 yrs) They had at least 6 years of acoustic guitar ex-perience (average years of training = 11 yrs SD = 4 yrsrange = 6ndash20 yrs average hours of practice per week =14 hrs SD = 10 hrs range = 3ndash40 hrs) as well as vary-ing experience with different types of guitars [nylon-string(85) acoustic-electric (62) acoustic bass (15) elec-tric (100) and electric bass (54)] Participants re-ported playing a wide range of musical styles [classical(77) jazz (85) pop-rock (92) country folk blue-grass (66) hard rock heavy metal (46) flamenco(15) latin (46) blues (77) avant-garde contempo-rary (54) and funk alternative (46)] both in a bandand as soloists Seven guitarists (54) reported that theytypically or most often use a pick whereas 4 players (31)reported they do not 2 guitarists (15) reported using apick occasionally rather than typically Nine participantshad higher-level degrees in music performance or othermusic-related areas [BMus (music theory classical gui-tar) BFA (jazz studies) MMus (guitar performance) MA(digital musics) PhD (music theory) Diplocircme drsquoEacutetat deProfesseur de Musique (France)] All guitar players werepaid for their participation

412 Procedure

The experimental session lasted 2 hours and was organizedin two phases In the first phase participants were pre-sented with the guitars randomly ordered on the floor (de-termined by computer calculations) They were asked toplay all instruments for up to 20 minutes in order to famil-iarize themselves with the set and use this time to tune theguitars as needed In the second phase participants were

404

Mansour et al Post-classification of steel string guitars ACTA ACUSTICA UNITED WITH ACUSTICAVol 101 (2015)

given up to 20 minutes to classify each of the guitars as ei-ther bassy mid-even or treble by placing them in speciallymarked areas of the room At the end of the second phaseparticipants were given the opportunity to revise their ini-tial judgements if they so wanted until they were satis-fied with their classification No playing constraint wasimposed on the evaluation process (eg specific repertoireor technique) Participants were instead encouraged to fol-low their own strategy with the suggestion to use bothstrumming-oriented songs (eg Sitting Waiting Wishingby Jack Johnson) and fingerpicking-oriented pieces (egDust in the Wind by Kansas) Participants were instructedto maximize evaluation speed and accuracy Upon com-pleting the session participants talked about their classi-fications and overall experience through written responsesto the following questions ldquoFor each class of guitar whatsound characteristics did you look for Can you describehow you evaluated the guitars in terms of these character-istics (eg musical material technique style)rdquo (Q1) ldquoTowhat extend do you agree with the given classification ofguitarsrdquo (Q2) and ldquoCan you suggest a classification thatbetter reflects how you evaluate the sound characteristicsof guitarsrdquo (Q3)

42 Results

Inter-individual consistency was measured as the Fleissrsquokappa coefficient between guitar classifications from dif-ferent participants using a Bayesian approach to inference[18] Fleissrsquo kappa is a generalization of Cohenrsquos kappastatistic κ a chance-corrected measure of agreement be-tween two assessors to multiple raters [19] It ranges fromminus1 to 1 κ (AB) = 1 and minus1 if A = B and A = minusBrespectively and κ (AB) = 0 in case of no associationbetween A and B (ie exactly what would be expectedby chance) We implemented Gibbs sampling through thesoftware WinBUGS (Windows Bayesian inference UsingGibbs Sampling [20]) with 50000 observations generatedfrom the posterior distribution with a burn-in of 1000 ob-servations A uniformly distributed multicategorical priorwas assumed with total information across all possiblecells (ie from different combinations of participants) be-ing equivalent to a single prior observation (ie a priorsample size equivalent of 1)

Fleissrsquos kappa was estimated with a posterior mean of01 and a 95 credible interval [-02 05] indicating avery low amount of agreement between the participantsin their classifications Figure 16 shows the proportion oftimes a guitar was classified as either bassy mid-even ortreble (the guitars are arranged in three groups accord-ing to the initial factory classification) Whereas someagreement concerning the bassy guitars can be observedmdashindeed the extra bassy guitar (B3) was most often clas-sified as bassymdashthe perceptions for the other two classeslargely vary across different musicians In particular manyguitars that had been characterized as treble by the fac-tory employee were perceived as mid-even Several mu-sicians specifically reported encountering difficulties withthe mid-even category In answering Q3 a player observed

B1 B2 B3 B4 B5 B60

25

50

75

100

classifiedas

(a)BassyMidminusevenTreble

ME1 ME2 ME3 ME4 ME5 ME60

25

50

75

100

classifiedas

(b)

T1 T2 T3 T4 T5 T60

25

50

75

100

classifiedas

Guitars

(c)

Figure 16 Perceptual evaluation Proportion of times a guitarwas classified into one of the categories by the participants Gui-tars are labelled and grouped according to the factory classifica-tion (B = bassy ME = mid-even T = treble) The color scheme(in the online version) corresponds to the classifications providedby the guitar players

that ldquothere is a difference [between] a guitar with domi-nant mid tones and one with an even soundrdquo

Overall participants expressed no strong difficulties incarrying out the classification task On a scale from 1(strongly disagree) to 5 (strongly agree) all performers se-lected 4 when asked ldquoTo what extend do you agree withthe given classification of guitarsrdquo This indicates that the3 categories considered by the manufacturer are gener-ally meaningful to musicians However verbal responsesto questions Q1 and Q3 reveal that participants often as-sociated bassy with more prominent E2 and A2 strings(ie they focused on the low-frequency end of the gui-tar) and similarly treble with better emphasized B3 andE4 strings (ie they paid attention to the high-frequencyrange) As one guitar player noted ldquoI feel that descriptionslike warm neutral and bright would be easier to classifydue to the nature of the instrument Bass leads my mindto bass strings and to heed the low end instead of the en-tire range from the highest to lowest notes Same for tre-ble vs bright rdquo In particular 4 participants used the wordldquotwangrdquo to describe guitars they classified as treble whileanother guitarist reported that a treble guitar ldquosounds bestfor finger picking and light strumming as heavy playingcan get quite ear-piercing and harsh This last commentseems to echo the second hypothesis for the origin of thestronger LTAS shown by the mid-even guitars (see Sec-tion 345)

Two participants reported looking out for problematicnotes as a more meaningful way of finding bassy guitarsThis likely points toward the ldquoveeringrdquo phenomenon as-sociated with a stronger admittance at lower frequencies

405

ACTA ACUSTICA UNITED WITH ACUSTICA Mansour et al Post-classification of steel string guitarsVol 101 (2015)

[16] Further one musician referred to the tactile feedbackreceived from a vibrating body associating bassy guitarswith more felt vibrations than the other two categoriesThis is a perspective shared by the manufacturer as wellIn fact body vibrations can provide tactile and proprio-ceptive cues that contribute to the perception of the radi-ated sound so that the musician can assess their interactionwith the instrument cross-modally [21 22]

5 Discussion

A set of 18 steel string guitars of the same model man-ufactured with high geometrical tolerances in a modernfactory were evaluated in this study These guitars wereconsidered by the manufacturer to possess subtle yet per-ceivable variations that could be classified as either bassymid-even or treble The results of this study showed thatthe instruments classified as bassy had a consistently lowerT(11)1 breathing mode (see Figure 9a 14b and 15) Thissuggests a lower stiffness-to-weight ratio for the bodies ofthese guitars which is most likely the result of variationsin equivalent stiffness (see Section 35)

It was further shown that whenever the frequency ofT(11)1 was shifted downward or upward for a particu-lar guitar the frequencies of its higher modes consistentlychanged in a similar way (see Figure 9 also the slightlyshifted bumps at around 600 750 and 1200 Hz in Fig-ure 12a) This would not necessarily be the case if theguitars were of different models (ie with varying geome-tries) or of different types of wood (eg plywood versusspruce) This is an important result as there is a practi-cal advantage to having higher-frequency modes move ac-cording to the first corpus mode across guitars of the samemodel T(11)1 is a low-frequency resonance that is suffi-ciently separated from the adjacent modes and hence eas-ily identifiable Measuring its frequency across differentinstruments is easy and affordable in a small workshopenvironment One can simply hit the guitar with a reg-ular hammer and recordanalyse the sound using an in-expensive microphone and a laptop reminiscent of thetraditional tap-tone method used by luthiers Moreoversuch measurements can be very reliable provided that theboundary conditions of the instrument (ie the installedfixture that holds the instrument) as well as the air tem-perature pressure and humidity are kept consistent Theseparameters constitute the biggest source of uncertainty infrequency-based features and can easily be controlled in afactory space or small workshop

Finding a reliable differentiator between the mid-evenand treble guitars was less obvious Although we ob-served that the averaged mobility curves of the mid-eveninstruments tended to be slightly higher in the vicinityof 600ndash2000 Hz than the treble guitars further measure-ments would have been necessary to determine whetherthe weaker admittance of the treble instruments at higherfrequencies would contribute to a longer decay for thehigher partials (see Section 35) In addition the differ-ence in averaged mobility magnitude between the two cat-

egories was only about 2 dB which is close to the marginof error for the measurement system Such a small differ-ence in an amplitude-based feature may be easily obscuredby variations in the exact angle of the hammer location ofhitting and measuring or calibration of the sensors

We invited guitar players of varying musical back-ground and experience to classify the guitars into thesecategories We observed very poor agreement among theparticipants but it is common in recent perceptual studiesof musical instruments that musicians do not agree witheach other [23 24 25] This seems a result of them lis-tening to (or feeling) different aspects of the instrumentand not necessarily all attending to the same features Theemployee that post-classified the guitars at the factory wasthe person that tests every guitar coming off the productionline thus he must have a lot of experience evaluating thesound of these instruments Assuming he did his classifi-cation carefully then there likely were particular things hewas listening for The participants in our perceptual exper-iment would not necessarily listen for the same featuresso it is not too surprising they largely disagreed on theirclassifications That said the differences between the tre-ble and mid-even guitars were extremely subtle and evencompany representatives often mis-categorized them dur-ing informal playing tests And this together with the factthat our measurements showed no consistent differencesbetween the two categories would seem to lead us to ques-tion whether the distinction between mid-even and trebletruly exists at all Another possible explanation is that dif-ferent musicians may follow different processes to assessthose features considered essential for the evaluation of aninstrument [23 26] Accordingly guitar players may agreeof what particular characteristics they look for in a tre-ble guitar but the perception of the same characteristicswidely varies across individuals

6 Conclusions

Efforts to understand how the tonal quality of an instru-ment relates to its measurable physical properties have of-ten been inconclusive (eg [27]) In the light of some re-cent studies the concept of ldquogoodrdquo versus ldquobadrdquo instru-ments is likely too general and it would be more advanta-geous to instead consider specific perceptual attributes ofan instrument [23 28] In the long run it may be possibleto derive a multidimensional timbre space where each di-mension is explained by some measurable quantities andin which instruments with similar qualities are likely tocluster In this study more specific sound characteristicsof instruments were thus targeted rather than their overalltonal quality

From bridge admittance measurements on 18 steelstring guitars from the same production line and post-classified by the manufacturer as either bassy mid-evenor treble (see Section 1 for details) it was shown that itis possible to discriminate between bassy guitars and theother two categories based on the frequency of the breath-ing mode However finding a similarly reliable feature to

406

Mansour et al Post-classification of steel string guitars ACTA ACUSTICA UNITED WITH ACUSTICAVol 101 (2015)

distinguish mid-even from treble instruments is less triv-ial at this point The treble guitars demonstrated a loweraveraged mobility in the frequency range of 600ndash2000 Hzthough more empirical investigation would be necessarybefore drawing any conclusions Results from a percep-tual experiment wherein guitar players were asked to clas-sify the guitars into these categories showed almost noagreement between guitarists in their judgements as wellas between the latter and the classification by the manufac-turer Despite the overall lack of consensus guitar playerstended to agree more on the bassy instruments than theother two categories

Some essential additions to this study are a more care-ful analysis on the perceptual definition of mid-even ver-sus treble and analysing the decay rate of the partials fora plucked sound to see how it correlates with the admit-tance Analysis of this kind has been performed for syn-thesized acoustic guitar sounds (see Figure 7 in [16]) but asimilar analysis on real guitar sounds will involve difficul-ties caused by the double set of harmonics [29] as well aslongitudinal string vibrations (also dubbed ldquophantom par-tialsrdquo) [30]

Acknowledgements

This project was funded by a Natural Sciences and En-gineering Research Council of Canada Engage grant incollaboration with Godin Guitars The authors wish tothank Robert Godin for this opportunity We also thankLawrence Joseph (statistics) and Garry Antonio (guitarperformance) for constructive comments We are grate-ful to Jim Woodhouse for sharing his analysis code Thefirst author would like to acknowledge the Government ofCanada for a Vanier Canada Graduate Scholarship

References

[1] E V Jansson A study of acoustical and hologram inter-ferometric measurements of the top plate vibrations of aguitar Acust Acta Acust 25 (1971) 95ndash100

[2] O Christensen B B Vistisen Simple model for low-frequency guitar function J Acoust Soc Am 68 (1980)758ndash766

[3] O Christensen Quantitative models for low frequency gui-tar function J Guitar Acoustics 6 (1982) 10ndash25

[4] H Wright The acoustics and psychoacoustics of the guitarDissertation Dept of Physics and Astronomy Universityof Wales Cardiff UK 1996

[5] N H Fletcher T D Rossing The physics of musical in-struments Second ed Springer New York 1998

[6] J E Popp Four mass coupled oscillator guitar model JAcoust Soc Am 131 (2012) 829ndash836

[7] J Woodhouse Body vibration of the violinmdashwhat can amaker except to control Catgut Acoust Soc J (Series II)4 (2002) 43ndash49

[8] B Elie F Gautier B David Macro parameters describingthe mechanical behavior of classical guitars J Acoust SocAm 132 (2012) 4013ndash4024

[9] J Woodhouse R S Langley Interpreting the input admit-tance of violins and guitars Acust Acta Acust 98 (2012)611ndash628

[10] A Gabrielsson E V Jansson Long-time-average-spectraand rated qualities of twenty-two violins Acustica 42(1979) 47ndash55

[11] P D Welch The use of Fast Fourier Transform for the es-timation of power spectra A method based on time averag-ing over short modified periodograms IEEE Trans Audioand Electroacoust AU-15 (1967) 70ndash73

[12] J Sundberg The singerrsquos formant revisited J Voice 4(1995) 106ndash119

[13] A Buen On timbre parameters and sound levels ofrecorded old violins J Violin Soc Am VSA Papers XXI(2007) 57ndash68

[14] J Woodhouse On the ldquobridge hillrdquo of the violin AcustActa Acust 91 (2005) 155ndash165

[15] B Richardson The acoustical development of the guitarCatgut Acoustical Society Journal 2 (1994) 1ndash10

[16] J Woodhouse On the synthesis of guitar plucks AcustActa Acust 90 (2004) 928ndash944

[17] C H Hodges J Power J Woodhouse The use of the sono-gram in structural acoustics and an application to the vibra-tions of cylindrical shells J Sound Vib 101 (1985) 203ndash218

[18] L D Broemeling Bayesian methods for measures ofagreement Chapman amp HallCRC New York 2009 (Bio-statistics)

[19] J L Fleiss Measuring nominal scale agreement amongmany raters Psychological Bulletin 76 (1971) 378ndash382

[20] D Lunn C Jackson N Best A Thomas D Spiegelhal-ter The BUGS Book A Practical Introduction to BayesianAnalysis Chapman amp HallCRC New York 2012 (Textsin Statistical Science)

[21] C Saitis Evaluating violin quality Player reliability andverbalization Dissertation Dept of Music ResearchMcGill University Montreal Quebec Canada 2013

[22] I Wollman Perception bimodale des violonistes en situa-tion de jeu Influence des retours auditif et vibrotactile surlrsquoeacutevaluation du violon Dissertation Universiteacute Pierre etMarie Curie Paris France 2013

[23] C Saitis B L Giordano C Fritz G P Scavone Percep-tual evaluation of violins A quantitative analysis of pref-erence judgements by experienced players J Acoust SocAm 132 (2012) 4002ndash4012

[24] C Fritz J Curtin J Poitevineau P Morrel-Samuels F-CTao Player preferences among new and old violins ProcNat Acad Sci USA 109 (2012) 760ndash763

[25] P Eveno J-P Dalmont R Causseacute G P Scavone Anacoustic and perceptual evaluation of saxophone pad ldquores-onatorsrdquo Acta Acustica united with Acustica 101 (2015)246ndash255

[26] C Fritz J Woodhouse F P-H Cheng I Cross A FBlackwell B C J Moore Perceptual studies of violinbody damping and vibrato J Acoust Soc Am 127 (2010)513ndash524

[27] G Bissinger Structural acoustics of good and bad violinsJ Acoust Soc Am 124 (2008) 1764ndash1773

[28] C Saitis C Fritz C Guastavino G P Scavone Con-ceptualization of violin quality by experienced performersProc Stockholm Music Acoust Conf Stockholm Swe-den 2013 123ndash128

[29] J Woodhouse Plucked guitar transients Comparison ofmeasurements and synthesis Acust Acta Acust 90 (2004)945ndash965

[30] B Bank L Sujbert Generation of longitudinal vibrationsin piano strings From physics to sound synthesis JAcoust Soc Am 117 (2004) 2268ndash2278

407

ACTA ACUSTICA UNITED WITH ACUSTICA Mansour et al Post-classification of steel string guitarsVol 101 (2015)

one standard deviation of the two categories overlap) itbecomes significant if the average level of the admittancefor the whole range is compared (ie the average valueof all frequency points of the admittance from 600 Hz to2 kHz) Figure 14b shows a 2-D plane with the frequencyof the first mode on the horizontal axis and the average ad-mittance for the range of 600 Hz to 2 kHz The differencefor the mean of the mid-even and treble categories is about2 dB It seems likely that the weaker admittance of the tre-ble category at higher frequencies would contribute to alonger decay for the higher partials This may contributeto a sharper after-sound of the treble instruments that isknown to allow them to cut through the band more effec-tively This hypothesis could be tested by recording theplucked sound of the instruments and comparing the decayrate of the partials among different categories Due to thelimited project time frame this was not investigated fur-ther The mid-even guitars also show stronger LTAS thanthe treble instruments though this result might have beendue to playing conditions (see Section 345)

We applied the same classification (frequency ofT(11)1 versus average mobility between 600 Hz and2 kHz) to 6 acoustic guitars from a different model by thesame manufacturer (Simon amp Patrick) Unlike the Seagullguitars which were all nominally identical some of theSampP instruments had a slightly different bracing patterndesigned to exemplify the properties of a particular per-ceptual category The manufacturerrsquos classification of the6 guitars was not provided to us prior to our own mea-surements though we were told that there were 2 guitarsof each category The resulting classification based on thediscussed features is depicted in Figure 15 The two gui-tars designed to be more bassy were successfully identi-fied from the frequency of their breathing mode but dis-tinguishing between the mid-even and treble instrumentsfrom the averaged admittance criterion was inconclusive

4 Perceptual evaluation

As previously discussed in Section 1 the investigatedacoustic guitars were post-classified by the manufacturerbased on their experience of what type of instrument mightbetter fit a certain playing stylemdashfor example an amateurmusician that plays mostly strummed chords may prefera bassy guitar while a more experienced guitarist mayopt for a treble instrument We therefore conducted a per-ceptual experiment to investigate the level of agreementamong guitar players of varying experience and musicalbackground in classifying the guitars into the categoriesconsidered by the manufacturer

41 Method

The experiment took place in an acoustically dry roomwith dimensions of 60 times 78 times 32 m3 and a reverberationtime of about 03 s to help minimize the effects of roomreflections on the direct sound from the guitars To avoidthe potential problems of using a common pick across all

85 86 87 88 89 90 91 92minus505

minus50

minus495

minus49

minus485

minus48B1

SampP

B2SampP

ME1SampP

ME2SampP

T1SampP

T2SampP

Frequency of the T(11)1mode (Hz)

Averagedmobility(600minus2000Hz)(dB)

BassyMid EvenTreble

Figure 15 Classification of the six SampP guitars based on thefrequency of the T(11)1 mode versus averaged mobility from600 Hz to 2 kHz (B= bassy ME=mid-even T= treble) The dBreference is 1 msN

participants (eg musicians being uncomfortable with apick they are not familiar with) participants were asked tocarry out the classification task using their own pick(s)

411 Participants

Thirteen guitar players took part in the experiment (1 fe-male 12 males average age = 25 yrs SD = 5 yrs range =19ndash36 yrs) They had at least 6 years of acoustic guitar ex-perience (average years of training = 11 yrs SD = 4 yrsrange = 6ndash20 yrs average hours of practice per week =14 hrs SD = 10 hrs range = 3ndash40 hrs) as well as vary-ing experience with different types of guitars [nylon-string(85) acoustic-electric (62) acoustic bass (15) elec-tric (100) and electric bass (54)] Participants re-ported playing a wide range of musical styles [classical(77) jazz (85) pop-rock (92) country folk blue-grass (66) hard rock heavy metal (46) flamenco(15) latin (46) blues (77) avant-garde contempo-rary (54) and funk alternative (46)] both in a bandand as soloists Seven guitarists (54) reported that theytypically or most often use a pick whereas 4 players (31)reported they do not 2 guitarists (15) reported using apick occasionally rather than typically Nine participantshad higher-level degrees in music performance or othermusic-related areas [BMus (music theory classical gui-tar) BFA (jazz studies) MMus (guitar performance) MA(digital musics) PhD (music theory) Diplocircme drsquoEacutetat deProfesseur de Musique (France)] All guitar players werepaid for their participation

412 Procedure

The experimental session lasted 2 hours and was organizedin two phases In the first phase participants were pre-sented with the guitars randomly ordered on the floor (de-termined by computer calculations) They were asked toplay all instruments for up to 20 minutes in order to famil-iarize themselves with the set and use this time to tune theguitars as needed In the second phase participants were

404

Mansour et al Post-classification of steel string guitars ACTA ACUSTICA UNITED WITH ACUSTICAVol 101 (2015)

given up to 20 minutes to classify each of the guitars as ei-ther bassy mid-even or treble by placing them in speciallymarked areas of the room At the end of the second phaseparticipants were given the opportunity to revise their ini-tial judgements if they so wanted until they were satis-fied with their classification No playing constraint wasimposed on the evaluation process (eg specific repertoireor technique) Participants were instead encouraged to fol-low their own strategy with the suggestion to use bothstrumming-oriented songs (eg Sitting Waiting Wishingby Jack Johnson) and fingerpicking-oriented pieces (egDust in the Wind by Kansas) Participants were instructedto maximize evaluation speed and accuracy Upon com-pleting the session participants talked about their classi-fications and overall experience through written responsesto the following questions ldquoFor each class of guitar whatsound characteristics did you look for Can you describehow you evaluated the guitars in terms of these character-istics (eg musical material technique style)rdquo (Q1) ldquoTowhat extend do you agree with the given classification ofguitarsrdquo (Q2) and ldquoCan you suggest a classification thatbetter reflects how you evaluate the sound characteristicsof guitarsrdquo (Q3)

42 Results

Inter-individual consistency was measured as the Fleissrsquokappa coefficient between guitar classifications from dif-ferent participants using a Bayesian approach to inference[18] Fleissrsquo kappa is a generalization of Cohenrsquos kappastatistic κ a chance-corrected measure of agreement be-tween two assessors to multiple raters [19] It ranges fromminus1 to 1 κ (AB) = 1 and minus1 if A = B and A = minusBrespectively and κ (AB) = 0 in case of no associationbetween A and B (ie exactly what would be expectedby chance) We implemented Gibbs sampling through thesoftware WinBUGS (Windows Bayesian inference UsingGibbs Sampling [20]) with 50000 observations generatedfrom the posterior distribution with a burn-in of 1000 ob-servations A uniformly distributed multicategorical priorwas assumed with total information across all possiblecells (ie from different combinations of participants) be-ing equivalent to a single prior observation (ie a priorsample size equivalent of 1)

Fleissrsquos kappa was estimated with a posterior mean of01 and a 95 credible interval [-02 05] indicating avery low amount of agreement between the participantsin their classifications Figure 16 shows the proportion oftimes a guitar was classified as either bassy mid-even ortreble (the guitars are arranged in three groups accord-ing to the initial factory classification) Whereas someagreement concerning the bassy guitars can be observedmdashindeed the extra bassy guitar (B3) was most often clas-sified as bassymdashthe perceptions for the other two classeslargely vary across different musicians In particular manyguitars that had been characterized as treble by the fac-tory employee were perceived as mid-even Several mu-sicians specifically reported encountering difficulties withthe mid-even category In answering Q3 a player observed

B1 B2 B3 B4 B5 B60

25

50

75

100

classifiedas

(a)BassyMidminusevenTreble

ME1 ME2 ME3 ME4 ME5 ME60

25

50

75

100

classifiedas

(b)

T1 T2 T3 T4 T5 T60

25

50

75

100

classifiedas

Guitars

(c)

Figure 16 Perceptual evaluation Proportion of times a guitarwas classified into one of the categories by the participants Gui-tars are labelled and grouped according to the factory classifica-tion (B = bassy ME = mid-even T = treble) The color scheme(in the online version) corresponds to the classifications providedby the guitar players

that ldquothere is a difference [between] a guitar with domi-nant mid tones and one with an even soundrdquo

Overall participants expressed no strong difficulties incarrying out the classification task On a scale from 1(strongly disagree) to 5 (strongly agree) all performers se-lected 4 when asked ldquoTo what extend do you agree withthe given classification of guitarsrdquo This indicates that the3 categories considered by the manufacturer are gener-ally meaningful to musicians However verbal responsesto questions Q1 and Q3 reveal that participants often as-sociated bassy with more prominent E2 and A2 strings(ie they focused on the low-frequency end of the gui-tar) and similarly treble with better emphasized B3 andE4 strings (ie they paid attention to the high-frequencyrange) As one guitar player noted ldquoI feel that descriptionslike warm neutral and bright would be easier to classifydue to the nature of the instrument Bass leads my mindto bass strings and to heed the low end instead of the en-tire range from the highest to lowest notes Same for tre-ble vs bright rdquo In particular 4 participants used the wordldquotwangrdquo to describe guitars they classified as treble whileanother guitarist reported that a treble guitar ldquosounds bestfor finger picking and light strumming as heavy playingcan get quite ear-piercing and harsh This last commentseems to echo the second hypothesis for the origin of thestronger LTAS shown by the mid-even guitars (see Sec-tion 345)

Two participants reported looking out for problematicnotes as a more meaningful way of finding bassy guitarsThis likely points toward the ldquoveeringrdquo phenomenon as-sociated with a stronger admittance at lower frequencies

405

ACTA ACUSTICA UNITED WITH ACUSTICA Mansour et al Post-classification of steel string guitarsVol 101 (2015)

[16] Further one musician referred to the tactile feedbackreceived from a vibrating body associating bassy guitarswith more felt vibrations than the other two categoriesThis is a perspective shared by the manufacturer as wellIn fact body vibrations can provide tactile and proprio-ceptive cues that contribute to the perception of the radi-ated sound so that the musician can assess their interactionwith the instrument cross-modally [21 22]

5 Discussion

A set of 18 steel string guitars of the same model man-ufactured with high geometrical tolerances in a modernfactory were evaluated in this study These guitars wereconsidered by the manufacturer to possess subtle yet per-ceivable variations that could be classified as either bassymid-even or treble The results of this study showed thatthe instruments classified as bassy had a consistently lowerT(11)1 breathing mode (see Figure 9a 14b and 15) Thissuggests a lower stiffness-to-weight ratio for the bodies ofthese guitars which is most likely the result of variationsin equivalent stiffness (see Section 35)

It was further shown that whenever the frequency ofT(11)1 was shifted downward or upward for a particu-lar guitar the frequencies of its higher modes consistentlychanged in a similar way (see Figure 9 also the slightlyshifted bumps at around 600 750 and 1200 Hz in Fig-ure 12a) This would not necessarily be the case if theguitars were of different models (ie with varying geome-tries) or of different types of wood (eg plywood versusspruce) This is an important result as there is a practi-cal advantage to having higher-frequency modes move ac-cording to the first corpus mode across guitars of the samemodel T(11)1 is a low-frequency resonance that is suffi-ciently separated from the adjacent modes and hence eas-ily identifiable Measuring its frequency across differentinstruments is easy and affordable in a small workshopenvironment One can simply hit the guitar with a reg-ular hammer and recordanalyse the sound using an in-expensive microphone and a laptop reminiscent of thetraditional tap-tone method used by luthiers Moreoversuch measurements can be very reliable provided that theboundary conditions of the instrument (ie the installedfixture that holds the instrument) as well as the air tem-perature pressure and humidity are kept consistent Theseparameters constitute the biggest source of uncertainty infrequency-based features and can easily be controlled in afactory space or small workshop

Finding a reliable differentiator between the mid-evenand treble guitars was less obvious Although we ob-served that the averaged mobility curves of the mid-eveninstruments tended to be slightly higher in the vicinityof 600ndash2000 Hz than the treble guitars further measure-ments would have been necessary to determine whetherthe weaker admittance of the treble instruments at higherfrequencies would contribute to a longer decay for thehigher partials (see Section 35) In addition the differ-ence in averaged mobility magnitude between the two cat-

egories was only about 2 dB which is close to the marginof error for the measurement system Such a small differ-ence in an amplitude-based feature may be easily obscuredby variations in the exact angle of the hammer location ofhitting and measuring or calibration of the sensors

We invited guitar players of varying musical back-ground and experience to classify the guitars into thesecategories We observed very poor agreement among theparticipants but it is common in recent perceptual studiesof musical instruments that musicians do not agree witheach other [23 24 25] This seems a result of them lis-tening to (or feeling) different aspects of the instrumentand not necessarily all attending to the same features Theemployee that post-classified the guitars at the factory wasthe person that tests every guitar coming off the productionline thus he must have a lot of experience evaluating thesound of these instruments Assuming he did his classifi-cation carefully then there likely were particular things hewas listening for The participants in our perceptual exper-iment would not necessarily listen for the same featuresso it is not too surprising they largely disagreed on theirclassifications That said the differences between the tre-ble and mid-even guitars were extremely subtle and evencompany representatives often mis-categorized them dur-ing informal playing tests And this together with the factthat our measurements showed no consistent differencesbetween the two categories would seem to lead us to ques-tion whether the distinction between mid-even and trebletruly exists at all Another possible explanation is that dif-ferent musicians may follow different processes to assessthose features considered essential for the evaluation of aninstrument [23 26] Accordingly guitar players may agreeof what particular characteristics they look for in a tre-ble guitar but the perception of the same characteristicswidely varies across individuals

6 Conclusions

Efforts to understand how the tonal quality of an instru-ment relates to its measurable physical properties have of-ten been inconclusive (eg [27]) In the light of some re-cent studies the concept of ldquogoodrdquo versus ldquobadrdquo instru-ments is likely too general and it would be more advanta-geous to instead consider specific perceptual attributes ofan instrument [23 28] In the long run it may be possibleto derive a multidimensional timbre space where each di-mension is explained by some measurable quantities andin which instruments with similar qualities are likely tocluster In this study more specific sound characteristicsof instruments were thus targeted rather than their overalltonal quality

From bridge admittance measurements on 18 steelstring guitars from the same production line and post-classified by the manufacturer as either bassy mid-evenor treble (see Section 1 for details) it was shown that itis possible to discriminate between bassy guitars and theother two categories based on the frequency of the breath-ing mode However finding a similarly reliable feature to

406

Mansour et al Post-classification of steel string guitars ACTA ACUSTICA UNITED WITH ACUSTICAVol 101 (2015)

distinguish mid-even from treble instruments is less triv-ial at this point The treble guitars demonstrated a loweraveraged mobility in the frequency range of 600ndash2000 Hzthough more empirical investigation would be necessarybefore drawing any conclusions Results from a percep-tual experiment wherein guitar players were asked to clas-sify the guitars into these categories showed almost noagreement between guitarists in their judgements as wellas between the latter and the classification by the manufac-turer Despite the overall lack of consensus guitar playerstended to agree more on the bassy instruments than theother two categories

Some essential additions to this study are a more care-ful analysis on the perceptual definition of mid-even ver-sus treble and analysing the decay rate of the partials fora plucked sound to see how it correlates with the admit-tance Analysis of this kind has been performed for syn-thesized acoustic guitar sounds (see Figure 7 in [16]) but asimilar analysis on real guitar sounds will involve difficul-ties caused by the double set of harmonics [29] as well aslongitudinal string vibrations (also dubbed ldquophantom par-tialsrdquo) [30]

Acknowledgements

This project was funded by a Natural Sciences and En-gineering Research Council of Canada Engage grant incollaboration with Godin Guitars The authors wish tothank Robert Godin for this opportunity We also thankLawrence Joseph (statistics) and Garry Antonio (guitarperformance) for constructive comments We are grate-ful to Jim Woodhouse for sharing his analysis code Thefirst author would like to acknowledge the Government ofCanada for a Vanier Canada Graduate Scholarship

References

[1] E V Jansson A study of acoustical and hologram inter-ferometric measurements of the top plate vibrations of aguitar Acust Acta Acust 25 (1971) 95ndash100

[2] O Christensen B B Vistisen Simple model for low-frequency guitar function J Acoust Soc Am 68 (1980)758ndash766

[3] O Christensen Quantitative models for low frequency gui-tar function J Guitar Acoustics 6 (1982) 10ndash25

[4] H Wright The acoustics and psychoacoustics of the guitarDissertation Dept of Physics and Astronomy Universityof Wales Cardiff UK 1996

[5] N H Fletcher T D Rossing The physics of musical in-struments Second ed Springer New York 1998

[6] J E Popp Four mass coupled oscillator guitar model JAcoust Soc Am 131 (2012) 829ndash836

[7] J Woodhouse Body vibration of the violinmdashwhat can amaker except to control Catgut Acoust Soc J (Series II)4 (2002) 43ndash49

[8] B Elie F Gautier B David Macro parameters describingthe mechanical behavior of classical guitars J Acoust SocAm 132 (2012) 4013ndash4024

[9] J Woodhouse R S Langley Interpreting the input admit-tance of violins and guitars Acust Acta Acust 98 (2012)611ndash628

[10] A Gabrielsson E V Jansson Long-time-average-spectraand rated qualities of twenty-two violins Acustica 42(1979) 47ndash55

[11] P D Welch The use of Fast Fourier Transform for the es-timation of power spectra A method based on time averag-ing over short modified periodograms IEEE Trans Audioand Electroacoust AU-15 (1967) 70ndash73

[12] J Sundberg The singerrsquos formant revisited J Voice 4(1995) 106ndash119

[13] A Buen On timbre parameters and sound levels ofrecorded old violins J Violin Soc Am VSA Papers XXI(2007) 57ndash68

[14] J Woodhouse On the ldquobridge hillrdquo of the violin AcustActa Acust 91 (2005) 155ndash165

[15] B Richardson The acoustical development of the guitarCatgut Acoustical Society Journal 2 (1994) 1ndash10

[16] J Woodhouse On the synthesis of guitar plucks AcustActa Acust 90 (2004) 928ndash944

[17] C H Hodges J Power J Woodhouse The use of the sono-gram in structural acoustics and an application to the vibra-tions of cylindrical shells J Sound Vib 101 (1985) 203ndash218

[18] L D Broemeling Bayesian methods for measures ofagreement Chapman amp HallCRC New York 2009 (Bio-statistics)

[19] J L Fleiss Measuring nominal scale agreement amongmany raters Psychological Bulletin 76 (1971) 378ndash382

[20] D Lunn C Jackson N Best A Thomas D Spiegelhal-ter The BUGS Book A Practical Introduction to BayesianAnalysis Chapman amp HallCRC New York 2012 (Textsin Statistical Science)

[21] C Saitis Evaluating violin quality Player reliability andverbalization Dissertation Dept of Music ResearchMcGill University Montreal Quebec Canada 2013

[22] I Wollman Perception bimodale des violonistes en situa-tion de jeu Influence des retours auditif et vibrotactile surlrsquoeacutevaluation du violon Dissertation Universiteacute Pierre etMarie Curie Paris France 2013

[23] C Saitis B L Giordano C Fritz G P Scavone Percep-tual evaluation of violins A quantitative analysis of pref-erence judgements by experienced players J Acoust SocAm 132 (2012) 4002ndash4012

[24] C Fritz J Curtin J Poitevineau P Morrel-Samuels F-CTao Player preferences among new and old violins ProcNat Acad Sci USA 109 (2012) 760ndash763

[25] P Eveno J-P Dalmont R Causseacute G P Scavone Anacoustic and perceptual evaluation of saxophone pad ldquores-onatorsrdquo Acta Acustica united with Acustica 101 (2015)246ndash255

[26] C Fritz J Woodhouse F P-H Cheng I Cross A FBlackwell B C J Moore Perceptual studies of violinbody damping and vibrato J Acoust Soc Am 127 (2010)513ndash524

[27] G Bissinger Structural acoustics of good and bad violinsJ Acoust Soc Am 124 (2008) 1764ndash1773

[28] C Saitis C Fritz C Guastavino G P Scavone Con-ceptualization of violin quality by experienced performersProc Stockholm Music Acoust Conf Stockholm Swe-den 2013 123ndash128

[29] J Woodhouse Plucked guitar transients Comparison ofmeasurements and synthesis Acust Acta Acust 90 (2004)945ndash965

[30] B Bank L Sujbert Generation of longitudinal vibrationsin piano strings From physics to sound synthesis JAcoust Soc Am 117 (2004) 2268ndash2278

407

Mansour et al Post-classification of steel string guitars ACTA ACUSTICA UNITED WITH ACUSTICAVol 101 (2015)

given up to 20 minutes to classify each of the guitars as ei-ther bassy mid-even or treble by placing them in speciallymarked areas of the room At the end of the second phaseparticipants were given the opportunity to revise their ini-tial judgements if they so wanted until they were satis-fied with their classification No playing constraint wasimposed on the evaluation process (eg specific repertoireor technique) Participants were instead encouraged to fol-low their own strategy with the suggestion to use bothstrumming-oriented songs (eg Sitting Waiting Wishingby Jack Johnson) and fingerpicking-oriented pieces (egDust in the Wind by Kansas) Participants were instructedto maximize evaluation speed and accuracy Upon com-pleting the session participants talked about their classi-fications and overall experience through written responsesto the following questions ldquoFor each class of guitar whatsound characteristics did you look for Can you describehow you evaluated the guitars in terms of these character-istics (eg musical material technique style)rdquo (Q1) ldquoTowhat extend do you agree with the given classification ofguitarsrdquo (Q2) and ldquoCan you suggest a classification thatbetter reflects how you evaluate the sound characteristicsof guitarsrdquo (Q3)

42 Results

Inter-individual consistency was measured as the Fleissrsquokappa coefficient between guitar classifications from dif-ferent participants using a Bayesian approach to inference[18] Fleissrsquo kappa is a generalization of Cohenrsquos kappastatistic κ a chance-corrected measure of agreement be-tween two assessors to multiple raters [19] It ranges fromminus1 to 1 κ (AB) = 1 and minus1 if A = B and A = minusBrespectively and κ (AB) = 0 in case of no associationbetween A and B (ie exactly what would be expectedby chance) We implemented Gibbs sampling through thesoftware WinBUGS (Windows Bayesian inference UsingGibbs Sampling [20]) with 50000 observations generatedfrom the posterior distribution with a burn-in of 1000 ob-servations A uniformly distributed multicategorical priorwas assumed with total information across all possiblecells (ie from different combinations of participants) be-ing equivalent to a single prior observation (ie a priorsample size equivalent of 1)

Fleissrsquos kappa was estimated with a posterior mean of01 and a 95 credible interval [-02 05] indicating avery low amount of agreement between the participantsin their classifications Figure 16 shows the proportion oftimes a guitar was classified as either bassy mid-even ortreble (the guitars are arranged in three groups accord-ing to the initial factory classification) Whereas someagreement concerning the bassy guitars can be observedmdashindeed the extra bassy guitar (B3) was most often clas-sified as bassymdashthe perceptions for the other two classeslargely vary across different musicians In particular manyguitars that had been characterized as treble by the fac-tory employee were perceived as mid-even Several mu-sicians specifically reported encountering difficulties withthe mid-even category In answering Q3 a player observed

B1 B2 B3 B4 B5 B60

25

50

75

100

classifiedas

(a)BassyMidminusevenTreble

ME1 ME2 ME3 ME4 ME5 ME60

25

50

75

100

classifiedas

(b)

T1 T2 T3 T4 T5 T60

25

50

75

100

classifiedas

Guitars

(c)

Figure 16 Perceptual evaluation Proportion of times a guitarwas classified into one of the categories by the participants Gui-tars are labelled and grouped according to the factory classifica-tion (B = bassy ME = mid-even T = treble) The color scheme(in the online version) corresponds to the classifications providedby the guitar players

that ldquothere is a difference [between] a guitar with domi-nant mid tones and one with an even soundrdquo

Overall participants expressed no strong difficulties incarrying out the classification task On a scale from 1(strongly disagree) to 5 (strongly agree) all performers se-lected 4 when asked ldquoTo what extend do you agree withthe given classification of guitarsrdquo This indicates that the3 categories considered by the manufacturer are gener-ally meaningful to musicians However verbal responsesto questions Q1 and Q3 reveal that participants often as-sociated bassy with more prominent E2 and A2 strings(ie they focused on the low-frequency end of the gui-tar) and similarly treble with better emphasized B3 andE4 strings (ie they paid attention to the high-frequencyrange) As one guitar player noted ldquoI feel that descriptionslike warm neutral and bright would be easier to classifydue to the nature of the instrument Bass leads my mindto bass strings and to heed the low end instead of the en-tire range from the highest to lowest notes Same for tre-ble vs bright rdquo In particular 4 participants used the wordldquotwangrdquo to describe guitars they classified as treble whileanother guitarist reported that a treble guitar ldquosounds bestfor finger picking and light strumming as heavy playingcan get quite ear-piercing and harsh This last commentseems to echo the second hypothesis for the origin of thestronger LTAS shown by the mid-even guitars (see Sec-tion 345)

Two participants reported looking out for problematicnotes as a more meaningful way of finding bassy guitarsThis likely points toward the ldquoveeringrdquo phenomenon as-sociated with a stronger admittance at lower frequencies

405

ACTA ACUSTICA UNITED WITH ACUSTICA Mansour et al Post-classification of steel string guitarsVol 101 (2015)

[16] Further one musician referred to the tactile feedbackreceived from a vibrating body associating bassy guitarswith more felt vibrations than the other two categoriesThis is a perspective shared by the manufacturer as wellIn fact body vibrations can provide tactile and proprio-ceptive cues that contribute to the perception of the radi-ated sound so that the musician can assess their interactionwith the instrument cross-modally [21 22]

5 Discussion

A set of 18 steel string guitars of the same model man-ufactured with high geometrical tolerances in a modernfactory were evaluated in this study These guitars wereconsidered by the manufacturer to possess subtle yet per-ceivable variations that could be classified as either bassymid-even or treble The results of this study showed thatthe instruments classified as bassy had a consistently lowerT(11)1 breathing mode (see Figure 9a 14b and 15) Thissuggests a lower stiffness-to-weight ratio for the bodies ofthese guitars which is most likely the result of variationsin equivalent stiffness (see Section 35)

It was further shown that whenever the frequency ofT(11)1 was shifted downward or upward for a particu-lar guitar the frequencies of its higher modes consistentlychanged in a similar way (see Figure 9 also the slightlyshifted bumps at around 600 750 and 1200 Hz in Fig-ure 12a) This would not necessarily be the case if theguitars were of different models (ie with varying geome-tries) or of different types of wood (eg plywood versusspruce) This is an important result as there is a practi-cal advantage to having higher-frequency modes move ac-cording to the first corpus mode across guitars of the samemodel T(11)1 is a low-frequency resonance that is suffi-ciently separated from the adjacent modes and hence eas-ily identifiable Measuring its frequency across differentinstruments is easy and affordable in a small workshopenvironment One can simply hit the guitar with a reg-ular hammer and recordanalyse the sound using an in-expensive microphone and a laptop reminiscent of thetraditional tap-tone method used by luthiers Moreoversuch measurements can be very reliable provided that theboundary conditions of the instrument (ie the installedfixture that holds the instrument) as well as the air tem-perature pressure and humidity are kept consistent Theseparameters constitute the biggest source of uncertainty infrequency-based features and can easily be controlled in afactory space or small workshop

Finding a reliable differentiator between the mid-evenand treble guitars was less obvious Although we ob-served that the averaged mobility curves of the mid-eveninstruments tended to be slightly higher in the vicinityof 600ndash2000 Hz than the treble guitars further measure-ments would have been necessary to determine whetherthe weaker admittance of the treble instruments at higherfrequencies would contribute to a longer decay for thehigher partials (see Section 35) In addition the differ-ence in averaged mobility magnitude between the two cat-

egories was only about 2 dB which is close to the marginof error for the measurement system Such a small differ-ence in an amplitude-based feature may be easily obscuredby variations in the exact angle of the hammer location ofhitting and measuring or calibration of the sensors

We invited guitar players of varying musical back-ground and experience to classify the guitars into thesecategories We observed very poor agreement among theparticipants but it is common in recent perceptual studiesof musical instruments that musicians do not agree witheach other [23 24 25] This seems a result of them lis-tening to (or feeling) different aspects of the instrumentand not necessarily all attending to the same features Theemployee that post-classified the guitars at the factory wasthe person that tests every guitar coming off the productionline thus he must have a lot of experience evaluating thesound of these instruments Assuming he did his classifi-cation carefully then there likely were particular things hewas listening for The participants in our perceptual exper-iment would not necessarily listen for the same featuresso it is not too surprising they largely disagreed on theirclassifications That said the differences between the tre-ble and mid-even guitars were extremely subtle and evencompany representatives often mis-categorized them dur-ing informal playing tests And this together with the factthat our measurements showed no consistent differencesbetween the two categories would seem to lead us to ques-tion whether the distinction between mid-even and trebletruly exists at all Another possible explanation is that dif-ferent musicians may follow different processes to assessthose features considered essential for the evaluation of aninstrument [23 26] Accordingly guitar players may agreeof what particular characteristics they look for in a tre-ble guitar but the perception of the same characteristicswidely varies across individuals

6 Conclusions

Efforts to understand how the tonal quality of an instru-ment relates to its measurable physical properties have of-ten been inconclusive (eg [27]) In the light of some re-cent studies the concept of ldquogoodrdquo versus ldquobadrdquo instru-ments is likely too general and it would be more advanta-geous to instead consider specific perceptual attributes ofan instrument [23 28] In the long run it may be possibleto derive a multidimensional timbre space where each di-mension is explained by some measurable quantities andin which instruments with similar qualities are likely tocluster In this study more specific sound characteristicsof instruments were thus targeted rather than their overalltonal quality

From bridge admittance measurements on 18 steelstring guitars from the same production line and post-classified by the manufacturer as either bassy mid-evenor treble (see Section 1 for details) it was shown that itis possible to discriminate between bassy guitars and theother two categories based on the frequency of the breath-ing mode However finding a similarly reliable feature to

406

Mansour et al Post-classification of steel string guitars ACTA ACUSTICA UNITED WITH ACUSTICAVol 101 (2015)

distinguish mid-even from treble instruments is less triv-ial at this point The treble guitars demonstrated a loweraveraged mobility in the frequency range of 600ndash2000 Hzthough more empirical investigation would be necessarybefore drawing any conclusions Results from a percep-tual experiment wherein guitar players were asked to clas-sify the guitars into these categories showed almost noagreement between guitarists in their judgements as wellas between the latter and the classification by the manufac-turer Despite the overall lack of consensus guitar playerstended to agree more on the bassy instruments than theother two categories

Some essential additions to this study are a more care-ful analysis on the perceptual definition of mid-even ver-sus treble and analysing the decay rate of the partials fora plucked sound to see how it correlates with the admit-tance Analysis of this kind has been performed for syn-thesized acoustic guitar sounds (see Figure 7 in [16]) but asimilar analysis on real guitar sounds will involve difficul-ties caused by the double set of harmonics [29] as well aslongitudinal string vibrations (also dubbed ldquophantom par-tialsrdquo) [30]

Acknowledgements

This project was funded by a Natural Sciences and En-gineering Research Council of Canada Engage grant incollaboration with Godin Guitars The authors wish tothank Robert Godin for this opportunity We also thankLawrence Joseph (statistics) and Garry Antonio (guitarperformance) for constructive comments We are grate-ful to Jim Woodhouse for sharing his analysis code Thefirst author would like to acknowledge the Government ofCanada for a Vanier Canada Graduate Scholarship

References

[1] E V Jansson A study of acoustical and hologram inter-ferometric measurements of the top plate vibrations of aguitar Acust Acta Acust 25 (1971) 95ndash100

[2] O Christensen B B Vistisen Simple model for low-frequency guitar function J Acoust Soc Am 68 (1980)758ndash766

[3] O Christensen Quantitative models for low frequency gui-tar function J Guitar Acoustics 6 (1982) 10ndash25

[4] H Wright The acoustics and psychoacoustics of the guitarDissertation Dept of Physics and Astronomy Universityof Wales Cardiff UK 1996

[5] N H Fletcher T D Rossing The physics of musical in-struments Second ed Springer New York 1998

[6] J E Popp Four mass coupled oscillator guitar model JAcoust Soc Am 131 (2012) 829ndash836

[7] J Woodhouse Body vibration of the violinmdashwhat can amaker except to control Catgut Acoust Soc J (Series II)4 (2002) 43ndash49

[8] B Elie F Gautier B David Macro parameters describingthe mechanical behavior of classical guitars J Acoust SocAm 132 (2012) 4013ndash4024

[9] J Woodhouse R S Langley Interpreting the input admit-tance of violins and guitars Acust Acta Acust 98 (2012)611ndash628

[10] A Gabrielsson E V Jansson Long-time-average-spectraand rated qualities of twenty-two violins Acustica 42(1979) 47ndash55

[11] P D Welch The use of Fast Fourier Transform for the es-timation of power spectra A method based on time averag-ing over short modified periodograms IEEE Trans Audioand Electroacoust AU-15 (1967) 70ndash73

[12] J Sundberg The singerrsquos formant revisited J Voice 4(1995) 106ndash119

[13] A Buen On timbre parameters and sound levels ofrecorded old violins J Violin Soc Am VSA Papers XXI(2007) 57ndash68

[14] J Woodhouse On the ldquobridge hillrdquo of the violin AcustActa Acust 91 (2005) 155ndash165

[15] B Richardson The acoustical development of the guitarCatgut Acoustical Society Journal 2 (1994) 1ndash10

[16] J Woodhouse On the synthesis of guitar plucks AcustActa Acust 90 (2004) 928ndash944

[17] C H Hodges J Power J Woodhouse The use of the sono-gram in structural acoustics and an application to the vibra-tions of cylindrical shells J Sound Vib 101 (1985) 203ndash218

[18] L D Broemeling Bayesian methods for measures ofagreement Chapman amp HallCRC New York 2009 (Bio-statistics)

[19] J L Fleiss Measuring nominal scale agreement amongmany raters Psychological Bulletin 76 (1971) 378ndash382

[20] D Lunn C Jackson N Best A Thomas D Spiegelhal-ter The BUGS Book A Practical Introduction to BayesianAnalysis Chapman amp HallCRC New York 2012 (Textsin Statistical Science)

[21] C Saitis Evaluating violin quality Player reliability andverbalization Dissertation Dept of Music ResearchMcGill University Montreal Quebec Canada 2013

[22] I Wollman Perception bimodale des violonistes en situa-tion de jeu Influence des retours auditif et vibrotactile surlrsquoeacutevaluation du violon Dissertation Universiteacute Pierre etMarie Curie Paris France 2013

[23] C Saitis B L Giordano C Fritz G P Scavone Percep-tual evaluation of violins A quantitative analysis of pref-erence judgements by experienced players J Acoust SocAm 132 (2012) 4002ndash4012

[24] C Fritz J Curtin J Poitevineau P Morrel-Samuels F-CTao Player preferences among new and old violins ProcNat Acad Sci USA 109 (2012) 760ndash763

[25] P Eveno J-P Dalmont R Causseacute G P Scavone Anacoustic and perceptual evaluation of saxophone pad ldquores-onatorsrdquo Acta Acustica united with Acustica 101 (2015)246ndash255

[26] C Fritz J Woodhouse F P-H Cheng I Cross A FBlackwell B C J Moore Perceptual studies of violinbody damping and vibrato J Acoust Soc Am 127 (2010)513ndash524

[27] G Bissinger Structural acoustics of good and bad violinsJ Acoust Soc Am 124 (2008) 1764ndash1773

[28] C Saitis C Fritz C Guastavino G P Scavone Con-ceptualization of violin quality by experienced performersProc Stockholm Music Acoust Conf Stockholm Swe-den 2013 123ndash128

[29] J Woodhouse Plucked guitar transients Comparison ofmeasurements and synthesis Acust Acta Acust 90 (2004)945ndash965

[30] B Bank L Sujbert Generation of longitudinal vibrationsin piano strings From physics to sound synthesis JAcoust Soc Am 117 (2004) 2268ndash2278

407

ACTA ACUSTICA UNITED WITH ACUSTICA Mansour et al Post-classification of steel string guitarsVol 101 (2015)

[16] Further one musician referred to the tactile feedbackreceived from a vibrating body associating bassy guitarswith more felt vibrations than the other two categoriesThis is a perspective shared by the manufacturer as wellIn fact body vibrations can provide tactile and proprio-ceptive cues that contribute to the perception of the radi-ated sound so that the musician can assess their interactionwith the instrument cross-modally [21 22]

5 Discussion

A set of 18 steel string guitars of the same model man-ufactured with high geometrical tolerances in a modernfactory were evaluated in this study These guitars wereconsidered by the manufacturer to possess subtle yet per-ceivable variations that could be classified as either bassymid-even or treble The results of this study showed thatthe instruments classified as bassy had a consistently lowerT(11)1 breathing mode (see Figure 9a 14b and 15) Thissuggests a lower stiffness-to-weight ratio for the bodies ofthese guitars which is most likely the result of variationsin equivalent stiffness (see Section 35)

It was further shown that whenever the frequency ofT(11)1 was shifted downward or upward for a particu-lar guitar the frequencies of its higher modes consistentlychanged in a similar way (see Figure 9 also the slightlyshifted bumps at around 600 750 and 1200 Hz in Fig-ure 12a) This would not necessarily be the case if theguitars were of different models (ie with varying geome-tries) or of different types of wood (eg plywood versusspruce) This is an important result as there is a practi-cal advantage to having higher-frequency modes move ac-cording to the first corpus mode across guitars of the samemodel T(11)1 is a low-frequency resonance that is suffi-ciently separated from the adjacent modes and hence eas-ily identifiable Measuring its frequency across differentinstruments is easy and affordable in a small workshopenvironment One can simply hit the guitar with a reg-ular hammer and recordanalyse the sound using an in-expensive microphone and a laptop reminiscent of thetraditional tap-tone method used by luthiers Moreoversuch measurements can be very reliable provided that theboundary conditions of the instrument (ie the installedfixture that holds the instrument) as well as the air tem-perature pressure and humidity are kept consistent Theseparameters constitute the biggest source of uncertainty infrequency-based features and can easily be controlled in afactory space or small workshop

Finding a reliable differentiator between the mid-evenand treble guitars was less obvious Although we ob-served that the averaged mobility curves of the mid-eveninstruments tended to be slightly higher in the vicinityof 600ndash2000 Hz than the treble guitars further measure-ments would have been necessary to determine whetherthe weaker admittance of the treble instruments at higherfrequencies would contribute to a longer decay for thehigher partials (see Section 35) In addition the differ-ence in averaged mobility magnitude between the two cat-

egories was only about 2 dB which is close to the marginof error for the measurement system Such a small differ-ence in an amplitude-based feature may be easily obscuredby variations in the exact angle of the hammer location ofhitting and measuring or calibration of the sensors

We invited guitar players of varying musical back-ground and experience to classify the guitars into thesecategories We observed very poor agreement among theparticipants but it is common in recent perceptual studiesof musical instruments that musicians do not agree witheach other [23 24 25] This seems a result of them lis-tening to (or feeling) different aspects of the instrumentand not necessarily all attending to the same features Theemployee that post-classified the guitars at the factory wasthe person that tests every guitar coming off the productionline thus he must have a lot of experience evaluating thesound of these instruments Assuming he did his classifi-cation carefully then there likely were particular things hewas listening for The participants in our perceptual exper-iment would not necessarily listen for the same featuresso it is not too surprising they largely disagreed on theirclassifications That said the differences between the tre-ble and mid-even guitars were extremely subtle and evencompany representatives often mis-categorized them dur-ing informal playing tests And this together with the factthat our measurements showed no consistent differencesbetween the two categories would seem to lead us to ques-tion whether the distinction between mid-even and trebletruly exists at all Another possible explanation is that dif-ferent musicians may follow different processes to assessthose features considered essential for the evaluation of aninstrument [23 26] Accordingly guitar players may agreeof what particular characteristics they look for in a tre-ble guitar but the perception of the same characteristicswidely varies across individuals

6 Conclusions

Efforts to understand how the tonal quality of an instru-ment relates to its measurable physical properties have of-ten been inconclusive (eg [27]) In the light of some re-cent studies the concept of ldquogoodrdquo versus ldquobadrdquo instru-ments is likely too general and it would be more advanta-geous to instead consider specific perceptual attributes ofan instrument [23 28] In the long run it may be possibleto derive a multidimensional timbre space where each di-mension is explained by some measurable quantities andin which instruments with similar qualities are likely tocluster In this study more specific sound characteristicsof instruments were thus targeted rather than their overalltonal quality

From bridge admittance measurements on 18 steelstring guitars from the same production line and post-classified by the manufacturer as either bassy mid-evenor treble (see Section 1 for details) it was shown that itis possible to discriminate between bassy guitars and theother two categories based on the frequency of the breath-ing mode However finding a similarly reliable feature to

406

Mansour et al Post-classification of steel string guitars ACTA ACUSTICA UNITED WITH ACUSTICAVol 101 (2015)

distinguish mid-even from treble instruments is less triv-ial at this point The treble guitars demonstrated a loweraveraged mobility in the frequency range of 600ndash2000 Hzthough more empirical investigation would be necessarybefore drawing any conclusions Results from a percep-tual experiment wherein guitar players were asked to clas-sify the guitars into these categories showed almost noagreement between guitarists in their judgements as wellas between the latter and the classification by the manufac-turer Despite the overall lack of consensus guitar playerstended to agree more on the bassy instruments than theother two categories

Some essential additions to this study are a more care-ful analysis on the perceptual definition of mid-even ver-sus treble and analysing the decay rate of the partials fora plucked sound to see how it correlates with the admit-tance Analysis of this kind has been performed for syn-thesized acoustic guitar sounds (see Figure 7 in [16]) but asimilar analysis on real guitar sounds will involve difficul-ties caused by the double set of harmonics [29] as well aslongitudinal string vibrations (also dubbed ldquophantom par-tialsrdquo) [30]

Acknowledgements

This project was funded by a Natural Sciences and En-gineering Research Council of Canada Engage grant incollaboration with Godin Guitars The authors wish tothank Robert Godin for this opportunity We also thankLawrence Joseph (statistics) and Garry Antonio (guitarperformance) for constructive comments We are grate-ful to Jim Woodhouse for sharing his analysis code Thefirst author would like to acknowledge the Government ofCanada for a Vanier Canada Graduate Scholarship

References

[1] E V Jansson A study of acoustical and hologram inter-ferometric measurements of the top plate vibrations of aguitar Acust Acta Acust 25 (1971) 95ndash100

[2] O Christensen B B Vistisen Simple model for low-frequency guitar function J Acoust Soc Am 68 (1980)758ndash766

[3] O Christensen Quantitative models for low frequency gui-tar function J Guitar Acoustics 6 (1982) 10ndash25

[4] H Wright The acoustics and psychoacoustics of the guitarDissertation Dept of Physics and Astronomy Universityof Wales Cardiff UK 1996

[5] N H Fletcher T D Rossing The physics of musical in-struments Second ed Springer New York 1998

[6] J E Popp Four mass coupled oscillator guitar model JAcoust Soc Am 131 (2012) 829ndash836

[7] J Woodhouse Body vibration of the violinmdashwhat can amaker except to control Catgut Acoust Soc J (Series II)4 (2002) 43ndash49

[8] B Elie F Gautier B David Macro parameters describingthe mechanical behavior of classical guitars J Acoust SocAm 132 (2012) 4013ndash4024

[9] J Woodhouse R S Langley Interpreting the input admit-tance of violins and guitars Acust Acta Acust 98 (2012)611ndash628

[10] A Gabrielsson E V Jansson Long-time-average-spectraand rated qualities of twenty-two violins Acustica 42(1979) 47ndash55

[11] P D Welch The use of Fast Fourier Transform for the es-timation of power spectra A method based on time averag-ing over short modified periodograms IEEE Trans Audioand Electroacoust AU-15 (1967) 70ndash73

[12] J Sundberg The singerrsquos formant revisited J Voice 4(1995) 106ndash119

[13] A Buen On timbre parameters and sound levels ofrecorded old violins J Violin Soc Am VSA Papers XXI(2007) 57ndash68

[14] J Woodhouse On the ldquobridge hillrdquo of the violin AcustActa Acust 91 (2005) 155ndash165

[15] B Richardson The acoustical development of the guitarCatgut Acoustical Society Journal 2 (1994) 1ndash10

[16] J Woodhouse On the synthesis of guitar plucks AcustActa Acust 90 (2004) 928ndash944

[17] C H Hodges J Power J Woodhouse The use of the sono-gram in structural acoustics and an application to the vibra-tions of cylindrical shells J Sound Vib 101 (1985) 203ndash218

[18] L D Broemeling Bayesian methods for measures ofagreement Chapman amp HallCRC New York 2009 (Bio-statistics)

[19] J L Fleiss Measuring nominal scale agreement amongmany raters Psychological Bulletin 76 (1971) 378ndash382

[20] D Lunn C Jackson N Best A Thomas D Spiegelhal-ter The BUGS Book A Practical Introduction to BayesianAnalysis Chapman amp HallCRC New York 2012 (Textsin Statistical Science)

[21] C Saitis Evaluating violin quality Player reliability andverbalization Dissertation Dept of Music ResearchMcGill University Montreal Quebec Canada 2013

[22] I Wollman Perception bimodale des violonistes en situa-tion de jeu Influence des retours auditif et vibrotactile surlrsquoeacutevaluation du violon Dissertation Universiteacute Pierre etMarie Curie Paris France 2013

[23] C Saitis B L Giordano C Fritz G P Scavone Percep-tual evaluation of violins A quantitative analysis of pref-erence judgements by experienced players J Acoust SocAm 132 (2012) 4002ndash4012

[24] C Fritz J Curtin J Poitevineau P Morrel-Samuels F-CTao Player preferences among new and old violins ProcNat Acad Sci USA 109 (2012) 760ndash763

[25] P Eveno J-P Dalmont R Causseacute G P Scavone Anacoustic and perceptual evaluation of saxophone pad ldquores-onatorsrdquo Acta Acustica united with Acustica 101 (2015)246ndash255

[26] C Fritz J Woodhouse F P-H Cheng I Cross A FBlackwell B C J Moore Perceptual studies of violinbody damping and vibrato J Acoust Soc Am 127 (2010)513ndash524

[27] G Bissinger Structural acoustics of good and bad violinsJ Acoust Soc Am 124 (2008) 1764ndash1773

[28] C Saitis C Fritz C Guastavino G P Scavone Con-ceptualization of violin quality by experienced performersProc Stockholm Music Acoust Conf Stockholm Swe-den 2013 123ndash128

[29] J Woodhouse Plucked guitar transients Comparison ofmeasurements and synthesis Acust Acta Acust 90 (2004)945ndash965

[30] B Bank L Sujbert Generation of longitudinal vibrationsin piano strings From physics to sound synthesis JAcoust Soc Am 117 (2004) 2268ndash2278

407

Mansour et al Post-classification of steel string guitars ACTA ACUSTICA UNITED WITH ACUSTICAVol 101 (2015)

distinguish mid-even from treble instruments is less triv-ial at this point The treble guitars demonstrated a loweraveraged mobility in the frequency range of 600ndash2000 Hzthough more empirical investigation would be necessarybefore drawing any conclusions Results from a percep-tual experiment wherein guitar players were asked to clas-sify the guitars into these categories showed almost noagreement between guitarists in their judgements as wellas between the latter and the classification by the manufac-turer Despite the overall lack of consensus guitar playerstended to agree more on the bassy instruments than theother two categories

Some essential additions to this study are a more care-ful analysis on the perceptual definition of mid-even ver-sus treble and analysing the decay rate of the partials fora plucked sound to see how it correlates with the admit-tance Analysis of this kind has been performed for syn-thesized acoustic guitar sounds (see Figure 7 in [16]) but asimilar analysis on real guitar sounds will involve difficul-ties caused by the double set of harmonics [29] as well aslongitudinal string vibrations (also dubbed ldquophantom par-tialsrdquo) [30]

Acknowledgements

This project was funded by a Natural Sciences and En-gineering Research Council of Canada Engage grant incollaboration with Godin Guitars The authors wish tothank Robert Godin for this opportunity We also thankLawrence Joseph (statistics) and Garry Antonio (guitarperformance) for constructive comments We are grate-ful to Jim Woodhouse for sharing his analysis code Thefirst author would like to acknowledge the Government ofCanada for a Vanier Canada Graduate Scholarship

References

[1] E V Jansson A study of acoustical and hologram inter-ferometric measurements of the top plate vibrations of aguitar Acust Acta Acust 25 (1971) 95ndash100

[2] O Christensen B B Vistisen Simple model for low-frequency guitar function J Acoust Soc Am 68 (1980)758ndash766

[3] O Christensen Quantitative models for low frequency gui-tar function J Guitar Acoustics 6 (1982) 10ndash25

[4] H Wright The acoustics and psychoacoustics of the guitarDissertation Dept of Physics and Astronomy Universityof Wales Cardiff UK 1996

[5] N H Fletcher T D Rossing The physics of musical in-struments Second ed Springer New York 1998

[6] J E Popp Four mass coupled oscillator guitar model JAcoust Soc Am 131 (2012) 829ndash836

[7] J Woodhouse Body vibration of the violinmdashwhat can amaker except to control Catgut Acoust Soc J (Series II)4 (2002) 43ndash49

[8] B Elie F Gautier B David Macro parameters describingthe mechanical behavior of classical guitars J Acoust SocAm 132 (2012) 4013ndash4024

[9] J Woodhouse R S Langley Interpreting the input admit-tance of violins and guitars Acust Acta Acust 98 (2012)611ndash628

[10] A Gabrielsson E V Jansson Long-time-average-spectraand rated qualities of twenty-two violins Acustica 42(1979) 47ndash55

[11] P D Welch The use of Fast Fourier Transform for the es-timation of power spectra A method based on time averag-ing over short modified periodograms IEEE Trans Audioand Electroacoust AU-15 (1967) 70ndash73

[12] J Sundberg The singerrsquos formant revisited J Voice 4(1995) 106ndash119

[13] A Buen On timbre parameters and sound levels ofrecorded old violins J Violin Soc Am VSA Papers XXI(2007) 57ndash68

[14] J Woodhouse On the ldquobridge hillrdquo of the violin AcustActa Acust 91 (2005) 155ndash165

[15] B Richardson The acoustical development of the guitarCatgut Acoustical Society Journal 2 (1994) 1ndash10

[16] J Woodhouse On the synthesis of guitar plucks AcustActa Acust 90 (2004) 928ndash944

[17] C H Hodges J Power J Woodhouse The use of the sono-gram in structural acoustics and an application to the vibra-tions of cylindrical shells J Sound Vib 101 (1985) 203ndash218

[18] L D Broemeling Bayesian methods for measures ofagreement Chapman amp HallCRC New York 2009 (Bio-statistics)

[19] J L Fleiss Measuring nominal scale agreement amongmany raters Psychological Bulletin 76 (1971) 378ndash382

[20] D Lunn C Jackson N Best A Thomas D Spiegelhal-ter The BUGS Book A Practical Introduction to BayesianAnalysis Chapman amp HallCRC New York 2012 (Textsin Statistical Science)

[21] C Saitis Evaluating violin quality Player reliability andverbalization Dissertation Dept of Music ResearchMcGill University Montreal Quebec Canada 2013

[22] I Wollman Perception bimodale des violonistes en situa-tion de jeu Influence des retours auditif et vibrotactile surlrsquoeacutevaluation du violon Dissertation Universiteacute Pierre etMarie Curie Paris France 2013

[23] C Saitis B L Giordano C Fritz G P Scavone Percep-tual evaluation of violins A quantitative analysis of pref-erence judgements by experienced players J Acoust SocAm 132 (2012) 4002ndash4012

[24] C Fritz J Curtin J Poitevineau P Morrel-Samuels F-CTao Player preferences among new and old violins ProcNat Acad Sci USA 109 (2012) 760ndash763

[25] P Eveno J-P Dalmont R Causseacute G P Scavone Anacoustic and perceptual evaluation of saxophone pad ldquores-onatorsrdquo Acta Acustica united with Acustica 101 (2015)246ndash255

[26] C Fritz J Woodhouse F P-H Cheng I Cross A FBlackwell B C J Moore Perceptual studies of violinbody damping and vibrato J Acoust Soc Am 127 (2010)513ndash524

[27] G Bissinger Structural acoustics of good and bad violinsJ Acoust Soc Am 124 (2008) 1764ndash1773

[28] C Saitis C Fritz C Guastavino G P Scavone Con-ceptualization of violin quality by experienced performersProc Stockholm Music Acoust Conf Stockholm Swe-den 2013 123ndash128

[29] J Woodhouse Plucked guitar transients Comparison ofmeasurements and synthesis Acust Acta Acust 90 (2004)945ndash965

[30] B Bank L Sujbert Generation of longitudinal vibrationsin piano strings From physics to sound synthesis JAcoust Soc Am 117 (2004) 2268ndash2278

407