Patrizia

PIANO AND RELATED INSTRUMENTS

Experimental and Computational Studies of the PianoN. Giordano, M. Jiang and S. Dietz

Department of Physics, Purdue University, West Lafayette, IN 47907-1396, USA

We review recent studies of the piano using the approach of physical modeling. This approach treatsall components of the instrumentusing first principles, i.e., Newton’s laws. In our work on the piano the modeling calculations have been tested and refined bycomparison with new experimental studies of hammers, the soundboard, and sound generation. The implications of this work for ourunderstanding of the piano are discussed.

INTRODUCTION

The generation of musical tones by “artificial” meanshas been an intriguing problem for some time. Initiallythis was attempted via analog methods, but in recentyears most attention has turned to synthesis techniquesemploying digital computers. The production of realis-tic sounds by such methods is not easy, since musicaltones are a complex and time dependent mixture of com-ponents which are usually not harmonically related. Mostapproaches to digital synthesis rely on waveforms, spec-tral components, or the like which are sampled from realinstruments. While this may lead to tones which sound“realistic,” such work often does not shed much insightinto the workings of the instrument.

In the past several years there has been increasinginterest in the physical modeling approach to synthe-sis. This approach is based on fundamental physics, i.e.,Newton’s laws. That is, Newton’s second law is used tocalculate the motion of all parts of the instrument (andif necessary the performer) and the sound that is gener-ated. This approach to modeling has the potential to teachus much about which aspects of an instrument are mostimportant for tone production. It could also give insightinto how radical changes in an instrument would affect itsperformance; e.g., in the case of a piano one might studyhow the use of a carbon fiber soundboard would affect thetone.

This paper will give an introduction to physical mod-eling work on the piano by our group. We will describe,albeit briefly, how our model has been constructed, testedand refined, and what is has taught us.

MODELING STRATEGY

Our piano model can be divided into several distinctsubsystems: (1) the hammers and strings, (2) the sound-board, and (3) the room. This separation is convenientsince to a good approximation, each of these subsystemssimply drives the succeeding one, with relatively little“feedback.” For example, the movement of the sound-

board drives the motion of the room air, but the effect ofthe resulting sound waves back on the soundboard can beneglected. Note that our modelingdoesallow for feed-back when it is important. For example, we do includethe “back-action” of the bridge motion on the strings, sothat sympathetic string vibrations can occur.

Our approach has been to construct and test com-putational models of each subsystem separately beforecombining them to form the “complete” model. Becauseof the large number of degrees of freedom, the nonlinear-ity found in the hammers, and the overall complexity ofthe problem, these are all computer based calculations.All of our calculations have employed explicit finitedifference-time domain solutions of the various equationsof motion [1]. For the hammer-string subsystem theseequations are the force-compression characteristic of thepiano hammer, Newton’s second law for the hammer,and the usual wave equation for a flexible string extendedto include the effects of string stiffness and dampingdue to dissipative processes internal to the string [1,2].The soundboard is modeled as a thin anisotropic plate(as appropriate for a material - wood - with a specificgrain direction) with position dependent stiffness moduli(due to the soundboard ribs) [3]. Sound generation andpropagation in the room containing the piano is treatedwith the usual three dimensional wave equation [4], withspecial care to model the measured frequency depen-dent reflection and absorption at the walls of the room [5].

The models of each of these subsystems have beentested and refined through comparison with experimen-tal studies. For example, the soundboard model hasbeen tested against measurements of the mechanicalimpedance of real soundboards [6], and the room modelhas been compared with studies of the (frequency depen-dent) generation of sound by vibrating soundboards [7].

WHAT CAN WE LEARN FROMPHYSICAL MODELING?

There are several conceivable motivations for the de-velopment of a physical model of a particular instrument.One is to produce a fast method for the synthesis of highquality tones, which could be used to replace the real in-strument. As might be expected, our model is very de-manding from a computational point of view. On a typ-ical fast personal computer, our current implementationproduces one second of sound in about one minute. How-ever, many aspects of the calculation are easily paralleliz-able, and several simplifications in the algorithm seemfeasible. We estimate that a real time calculation shouldbe possible with a computer that costs less than a new pi-ano [8]. While this is an interesting goal, we believe thata second motivating factor is more important. Physicalmodeling has the potential to provide qualitatively newinsights into the workings of an instrument. For example,in our initial modeling calculations we treated the ham-mer force law using a standard approach which had beenextensively studied and utilized in previous work (see, forexample, [2,9]). According to that approach the force ex-erted by a hammer on a string is given by the power law

Fh∼ zp , (1)

wherez is the amount that the hammer felt is compressedand the exponentp has a value in the range 2.5-5 (itcan be different for different hammers). While it is cer-tainly well appreciated that this simple reversible powerlaw form is not based on a first principles foundation, itwas thought (or at least hoped) that (1) would be ade-quate for describing real piano hammers. However, wehave found that tones calculated with (1) are very unre-alistic, as they have a decidedly “plucked” nature. Thisfinding motivated us to reexamine experimentally the na-ture ofFh in real hammers, and our results indicated thata more general functional form is needed to describe theforce exerted by a hammer on a string under realistic con-ditions [10]. We have discovered that the mathematicalapproach toFh suggested by Stulov [11] gives a better,but not yet perfect description of the measuredFh. Tonescalculated with the Stulov form forFh are, however, muchimproved. The message from this work is that our initialmodeling results with (1) indicated the need for a reap-praisal of the hammer force law. This stimulated new ex-periments, which in turn revealed important new aspectsof Fh. In this way, physical modeling has led to new in-sight into the instrument.

To date, physical models have been developed for onlya few percussion [12,13] and stringed instruments [14].Wind instruments pose many challenges, but seem quitefeasible. We are optimistic that physical modeling will

lead to important new perspectives into the function of avariety of musical instruments.

ACKNOWLEDGMENTS

We thank B. Martin and P. F. Muzikar for many help-ful discussions. This work was supported by the NSFthrough grant PHY-9988562.

REFERENCES

1. Chaigne, A., J. Acoustique5, 181-211 (1992).

2. Chaigne, A., and Askenfelt, A., J. Acoust. Soc. Am.95,1112-1118 (1994).

3. Giordano, N., J. Acoust. Soc. Am.102, 1159-1168 (1997).

4. Botteldooren, D., J. Acoust. Soc. Am.95, 2313-2319(1994);98, 3302-3308 (1995).

5. Beranek, L. L., J. Acoust. Soc. Am.12, 14-23 (1940).

6. Giordano, N., J. Acoust. Soc. Am.103, 2128-2133 (1998).

7. Giordano, N., J. Acoust. Soc. Am.103, 1648-1653 (1998).

8. Jiang, M.,Room Acoustics and Physical Modeling of thePiano,M.S. thesis, Purdue University, December 1999.

9. Hall, D. E., and Askenfelt, A., J. Acoust. Soc. Am.83,1627-1638 (1988). Boutillon, X., J. Acoust. Soc. Am.83,746-754 (1988). Russel, D., and Rossing, T., Acoustica84, 967-975 (1998). Giordano, N., and Winans, J. P., II,J. Acoust. Soc. Amer.107, 2248-2255 (2000).

10. Giordano, N., and Millis, J. P., to be published.

11. Stulov, A., J. Acoust. Soc. Am.97, 2577-2585 (1995).

12. Chaigne, A., and Doutaut, V., J. Acoust. Soc. Am.101,539-557 (1997); Doutaut, V., Matignon, D., and Chaigne,A., J. Acoust. Soc. Am.104, 1633-1647 (1998).

13. Rhaiuti, L., Chaigne, A., and Joly, P., J. Acoust. Soc. Am.105, 3345-3562 (1999).

14. Richardson, B. E., Walker, G. P., and Brooke, M., Proceed-ings of the Inst. of Acoustics12, 757-764 (1990); Brooke,M., and Richardson, B. E., J. Acoust. Soc. Am.89, 1878-1878 (1991).

Evaluation of the Elements of a Soundboard which Formthe Tonal Characteristics of a Grand Piano.

Blüthner-Haessler

Chairman of Julius Blüthner Pianofortefabrik

There is a necessity for a variety in the tonal characteristic of grand pianos. Since different interpretations of the existingmusic focus the interest of an audience not only the style of the artist but also the different tonal qualities of the usedinstrument kindle the interest and produce the variety that make people wanting to listen to music in a concert.It is therefore in the interest not only of the piano maker but also the artist to have offer differing instruments, their tonalcharacter varying from bright , loud to warm romantic.It has to be understood by the artists that these differences are more a matter of taste than of quality like the choice of whiteor red wine.

The primary element that influences the tonalcharacteristics is the material for the soundboardwhich is spruce selected to colour, grain density,and straightness . Due to the way a fir tree grows itdevelops annular rings, which have soft thinmembrane cells that grow in the early part of theyear alternating with hard horny cells developing inautumn.Thus a high elasticity at a low density is developedideal for a low inner impedance.The arching of the sound board is something whichwas introduced in thelater years. The first maker is not known for sure.Some believe him to be the Hungarian Berecshazywho carved a soundboard out of a log. It was themusic of Liszt which challenged the piano makersto design instruments that would withstand theattack of the pianist and produce the tone volumenecessary.The crown , i.e. “ arching “ of the soundboard isusually spherical . However Blüthner is using animproved design which consist in a cylindricalarching of the soundboard.This special design is more logical and permits tobalance the elasticity of the soundboard better. Thisis important since the soundboard presses againstthe stratum of the strings.The arching of the soundboard is done by gluingribs in perpendicular direction on the underside.To vary the stiffness of the soundboard twomethods are possible.

For a stiff sound board straight ribs are used thatare fixed on the sound board in a very hollow bed.They are bend at after the release they will try tostraighten out. This is only possible when thesoundboard is compressed. When the two opposedforces are balanced the board is arched.For a soft soundboard pre shaped ribs in form of abow are used. The sound board is completely dry tohave it’s least extension. The ribs are then glued onin a shallow bed. The inner tension comes when thetimber picks up the moisture of the surroundingatmosphere.The bridge acts as a lever which is set into motionby the vibrations of the strings. The up and downmovement of the string sets the bridge into aforward and backward motion before it also startsalso to move up and down. Therefore the width ofthe bridge and it height is important as well as thestraightness. To alleviate the necessary bend in thetreble Blüthnermoves the last notes forward to the keyboard byshortening the treble keys.It is also important to consider the position of thebridge pin that force the strings into a crookedposition. The pitch should be large enough to keepthe fixed on the bridge to transfer the forward andbackward motion, it should however also allow acertain slip when the instrument is tuned. When thisis not possible the bridge is pulled forward and thetension on the surface will be changed.Vibrations on the soundboard do not develop in asimple up and down movement. More so thesurface is divided up into fields if different shapes.

Each frequency produces it own field. This is verymuch like the sound figures which Chladnidescribes that form on a metal plate which is madeto vibrate. It shows knots an bows. The differentfrequencies form knot and bows that superimposeon each other forming a complicated system.Notes that produce high amplitudes will consumethe given energy faster than those of a smaller one.To have an evenness of tone volume is one of themajor problems in piano design.The form of the soundboard, the tapering towardsthe edges of ribs and board, the string tensions,their length are decisive but a scientific researchinto their interactions need still to be undertakenTo vibrate the soundboard has to be fixed to astrong foundation. This is the rim of the pianowhich usually is made of hard wood laminationsreinforced by back posts that maintain the shape.The better pianos have the strongest backs. Thefoundation shall be inert to the vibrations but shallreflect these back to the soundboard. Further attention has to be given to theconstruction of the iron frame. At the striking pointthe string length should be determined by a rathersturdy part of the frame that does not pick up thestrike of the hammer acting as a shock absorber.The further improvement of the piano is still amatter of concern to all those that are involved inits construction. I indicated a few points wherefurther studies are welcome The construction hashowever come to a certain point where pianomakers are looking to new challenges in thecompositions. Certainly electronic devices canyield one or the other improvement But theacoustic piano is a rather closed category and sincethe literature for this instrument is the largest byfar compared to other musical instruments it willhopefully remain more or less what it is now.

Among different components of a piano, thesoundboard is quite far the most crucial element. Thesoundboard amplifies vibrations generated by thekeyboard and consequently is demanded to providehigh efficiency, neutral response and excellent sustain.In order to maximise these characteristics, some studieshave been carried out involving different techniques:Transfer Function, Running Response and ModalAnalysis on a simplified structure first, secondly on apiano prototype and then on a regular piano model.In particular the attention has been focused on two mainareas: one involving the overall frequency domain inorder to increase efficiency and sustain and the otherfinalised to the low frequency range of the soundboardin order to equalise its frequency response.For the first target, a simplified structure, shown in theFigure 1, has been designed just for understanding theinfluence of the string load and their location over the

Acoustic Analysis of a Fazioli Piano

D. Baggioa , P. Faziolib

aResearch & Innovation, Electrolux Zanussi, 33170 Pordenone, ItalybFazioli Pianoforti, 33077 Sacile, Italy, E-Mail: info@faziolipianoforti.it

THE HANDCRAFTED PIANOEVOLUTION WITH THE HELP OF

THE MODERN ACOUSTIC ANALYSISTECHNIQUES

FIGURE 1. Simplified structure with bridge and string load

FIGURE 2. Response levels on the positions 1, 2 and 3 vs.different constructions (C1..C8)

It’s of primary importance, for a handcraft piano, to know the effects of each parameter on the quality of the sound in order tomanage all the involved variables for providing excellent performances of each manufactured piano. The scientific analysistechniques, hereafter presented, have been investigated and, based on the interesting results, have been, as a consequence, adopted.

soundboard as a function of different type ofsoundboard construction. A free dropping ball hasbeen used as a constant force of impact and theTransfer Function has been calculated in real timeusing the Fast Fourier Transform (FFT).Figure 2 shows one of the different response levels.As expected, the string load strongly affects theefficiency of the soundboard, but concurrently theresponse becomes more and more equalised.As a result, the best compromise has been found usingdifferent soundboard constructions.According to these conclusions, a benchmark has beencarried out on a regular piano in order to compare itsefficiency with the best reachable solution foundpreviously. The soundboard of the piano has been fullyinstrumented fixing many accelerometers on it and theregular keyboard has been used as source of impactforce. Like in the previous test, the force has been keptconstant using a special mechanism simulating thefinger impact. All signals have been acquired andstored simultaneously. Then the transfer function hasbeen calculated, using the FFT analysis, for eachaccelerometer signal and each of the 34 keysrepresenting the entire frequency range of the piano.

An instrumented hammer has been used for theexcitation of the soundboard and the real time responsehas been acquired applying the FFT analysis over eachacceleration co-ordinate. Figure 4 shows two of thevibration behaviours at resonance. Linking such aresults with the ones of the previous step on a regularpiano, it was possible to clearly visualise the shift of theresonance frequencies due to bridges, beamreinforcement and string load. According to theseresults, specific modifications have been decided andintroduced as the best compromise between theresponse linearity and efficiency and sustain.

REFERENCES

1. McCord, P., Morse, and Uno Ingard, K., TheoreticalAcoustics, Princeton University

2. Frey, A.R., and Coppens, A.B., Fundamentals of Acoustics,edited by L.E. Kinsler, John Wiley & Sons

3. Benade, A.H., Fundamentals of Musical Acoustics, DoverPublications

4. Beranek, L.L., Acoustical Measurements, AcousticalSociety of American Publications

5. Hartog, J.P.D., Mechanical Vibrations, edited by D.J.Hartog, Dover Publications

6. Harris, C.M., Shock and Vibration Handbook, edited byC.M. Harris, McGraw-Hill Professional Publishing

7. Beranek, L.L., Noise and Vibration Control, edited by L.L.Beranek, Acoustical Society of America Publications

8. Beranek, L.L.,Ver, I.L., Noise and Vibration ControlEngineering: Principles and Applications, edited by L.L.Beranek, I.L. Ver, John Wiley & Sons Publications

Figure 3 is an example of the relevant levels mapped ona surface for each frequency tone. As a conclusion, aclear difference has been noticed on some areas wherethe efficiency was out of the best compromise. Startingfrom these information a specific action plan has beenprepared and an intensive activity has been developedfor improving these critical areas. Of course, not onlythe soundboard construction: materials, compositionand shape has been modified, but also the beams andthe bridge location.For the second target, the behaviour of the piano at lowfrequencies has been investigated. Due to the largedimensions, particularly for the "Gran piano" model, thenatural resonance frequencies of the soundboard aremainly located on the low frequency range of thekeyboard tones. In this region the response is quiteirregular compared to the upper part of the keyboardtones. In order to have a more linear response, theresonance frequencies have to be smoothed, but at thesame time carefully deployed for the maximumefficiency. A visible understanding of the resonancefrequencies is provided by the Modal Analysis. Aprototype of soundboard with relevant beams, frame andbridges, but without strings has been prepared and set-upfixing many accelerometers on its surface.

FIGURE 4. First and second vibration mode of the soundboard

FIGURE 3. Sound power emitted by the soundboard

CONCLUSIONS

As a matter of fact, the sound generated by all themodels of the horizontal piano have been consistentlyimproved in terms of efficiency, linearity and sustainproviding to the pianist an easier control of the powergenerated and to the listener a well equilibrate, butpowerful sound over the full range of the keyboardtones. These analysis techniques are now part of thedevelopment procedure as a regular approach for acontinuous improvement of the existing pianos and forthe design of every new generation of piano.

A Comparison between Upright and Grand Pianos

T. Mori

Faculty of Integrated Arts and Sciences, The University of Tokushima, Minamijosanjima-cho 1-1, 770-8502Tokushima-shi, Japan

Recently, the factor to decide a piano quality is clarified by the advancement of the measurement technology. The piano, whichhas been made for a long time only by experience, comes to be designed and manufactured from a technological standpoint. Thepurpose of the research is to take up the main elements by which a piano quality is decided, and to show the indicator of itsconcrete improvement. It is discussed in this report, which physical parameter characterizes upright and grand pianos bycomparing two instruments, which have subjectively very different quality. The result can be applied also to the improvement ofthe electric keyboards

INTRODUCTION

The difference between upright and grand pianos isrelated to both the impression of the sound and thetouch. The difference of the sound is felt more at ownplaying than listening recorded one through loud-speaker or headphone. This shows that it is necessaryto think about all systems from the finger to listeneroverall.

VIBRATION ANALYSIS AND SOUNDRADIATION

The amplitude and decay time of a partial toneoriginates in the vibration of the string, soundboard,case, iron frame etc. This frequency-dependent vib-ration distribution can be determined, for example,with the aid of modal analysis [1]. One of the largestdifferences between upright and grand pianos is thefrequency of basic vibration. It was 105Hz and 130Hzrespectively by grand and upright piano used for thisreport. Fundamental tones between these frequencies(between about G#2 and C3) will be powerfully heardin the grand piano, because the vibration energy of thestring is efficiently transformed into the sound. Thisdifference was recognized well in subjective tests.

In an upright piano, the front cover located betweensoundboard and the player influence the soundradiation. Fig. 1 shows the transfer function from thebridge to the sound at the position of player's ear. Thepanel, which works as a filter, generally reduces thesound, while it grows with the front panel between 70Hz and 130 Hz; they are the eigenmodes of soundboardwith, and without the supporting post’s coupling. Thatmeans, in an upright piano, the sound radiation forwardis restricted with a front panel, and as a result, theacoustical short-circuit is prevented.

Fig.1. Transfer function from the bridge to listener’s position

ACTION

The action is one of the most important parts for theplayer, because he can influence the sound onlythrough it. It is decided how delicate nuance can beexpressed. A mathematics model as easy as possible isfirst constructed, by which we search for the parameter,which shows the most essential difference.

The rotation moment of the key, the hammer, andwippen is assumed to be IK, IH and IW respectively, andthe angular acceleration is assumed to be ωT,ωH(=αּωT), and ωW (=βּωT). The standardized entirerotation energy I (=α2 IH +β2IW+ IK) just before the letoff meets the equation

where static force necessary so that the action begins tomove is assumed to be FS, which reaches almost 0.5N(Experience value, corresponding to the weight of 50g,

-60

-50

-40

-30

-20

-10

0

0 100 200 300 400f / Hz

|H(f

)| / d

B

without front panels

standard

FFIω (C21

sin2

K − ) L=+ (1 )

common in upright and grand pianos). Fin is theconstant input force and C is a constant, whichoriginates in the spring and friction, etc. L is the Keytravel length before the let off at the front edge.This equation (1) is corresponding to the model ofDijksterhuis [6] when assuming that α and β areconstants (i.e. time and position-independent) andassuming C=0.Fig. 2 is the measurement result of the hammervelocity vH at striking point when input force Fin isvaried. The solid curve was computed using a leastsquares program based on the equation (1). The totalrotation moment just before the let off of the grandpiano action is (estimated from the curve fitting) about0.04 kgm2 and 0.02 kgm2 by grand and upright pianorespectively, while that of the electric piano is about0.003 kgm2. This electric piano action is extremelysensitive in pp and difficult to operate. That is, a littlechange in the input force becomes a big change in thekey velocity. This rotation moment is not measured bypiano manufacturing at the present in general, but onlystatic force Fin.

Fig.2. Measured key velocity of a upright, grand andelectrical piano

The damper acts also differently by varying input forceFin. The damper stops the vibration of the string in thegrand piano by using gravity, while the spring is usedin an upright piano. The entire curve moves right withthe damper by the upright piano, because the damperacts as added constant C in equation (1). It shows that alarger input force for the same hammer velocity isobtained when a damper is installed. On the other hand,the change in the hammer velocity is small by thedamper of the grand piano, where the damper acts asadded mass; I becomes larger. This is also one ofstructural differences of upright and grand piano action.Damper position on the stringThe grand piano damper is installed directly on thehammer striking point, while the upright piano damperdoes not work there, because it is installed from thesame direction as the hammer. The performances of thedamper of an upright piano and a grand piano can beeasily compared. If a lot of keys are pushed at the sametime, and released at once, the sound stops immediately

when a good damper system is installed (grand piano),while some partial sounds remain by not so gooddamper system (upright piano). The upright dampercannot work sometimes effectively because of itsposition, which causes so-called "flageolet-effect". Fig.3 shows sound spectrogram while damping the toneB2. The key was shortly played and immediatelyreleased by recording. The fifth partial tone decaysslower than the other partials by the upright. It isbecause that the distance between the edge of thedamper and the string end is just 1/5 of the stringlength. The partial tones are not damped effectively,whose vibration node meets the damper position.

Fig. 3. Sound spectrogram while damping the tone B2

SUMMARY

There is a lot of structural difference in upright andgrand piano also excluding these examples (the in-fluence of the inharmonicity and tuning, for example[2]). These physical parameters will be the im-provement indicator of upright pianos and electricpianos. Moreover, the measurable value can be used tothe judgment of a piano quality, which had relied ononly subjective opinion of pianists.

REFERENCES

1. Taro Mori and Ingolf Bork, "Modal analysis of anupright piano and its case", Proc. of InternationalSymposium on Musical Acoustics'972. Mori, T., Ein Vergleich der QualitätsbestimmendenFaktoren von Klavier und Flügel, Wissenschaftsverlag,Aachen, 2000

0

0.2

0.4

0.6

0 2 4 6 8 10F in/N

v K/(m

/s)

grand

upright

electronic piano

rel.

SPL

/dB

f /Hz0 500 1000

15 dB1 2 3 4 5 6

78

1 2 34 5 6 7 8

t/ s

t / s

0

0 500 1000

0,5

0,25

0

0,5

0,25

(a)

(b)

f /H z

Piano Hammers - Motion during String Contactand Shank bending

A. Askenfelt

Dept. of Speech, Music and Hearing, Royal Institute of Technology, SE-100 44 Stockholm, Sweden

The interaction between hammer and string is of vital importance for the tone quality in pianos. Recent studies, related tophysical modeling of pianos, indicate that the flexing of the hammer shank may influence the contact force significantly [1].The motion of the contact point on top of the hammer, before and during string contact, is revisited in view of these newresults. The influence of dynamic level, touch, and shank stiffness is investigated.

INTRODUCTION

Previous studies have shown that the shank of thepiano hammer bends during the acceleration towardsthe string [2,3]. The hammer can also be set invibrations for certain types of touch, mainly consistingof a mixture of the two lowest modes at about 50 and250 Hz [3]. The second mode features a pronouncedmotion of the hammer head in the string direction(horizontal). At string contact the boundary conditionschange. An example of the vertical and horizontalcomponents of the hammer motion during stringcontact is shown in Fig. 1, as measured at the woodenhammer molding. The horizontal motion involves afull period of vibration at approximately 800 Hz, withan estimated displacement motion at the top of thehammer of about 0.1 mm (p-p).

FIGURE 1. Vertical and horizontal acceleration of thehammer head during string contact (staccato, forte, C4). Thepositions of the accelerometers are marked with triangles. Aperiod of vibration at 800 Hz is included in the horizontalmotion for reference (dashed).

There are two mechanisms by which the touch couldinfluence the motion of the hammer at string contact.(1) The bending of the shank could change the initialangle of the hammer head, thus changing the strikingpoint slightly. (2) Vibrations in the hammer duringstring contact may occur, giving a rubbing motionagainst the string. The period of the major horizontal

vibrations in Fig. 1 is about 1.5 ms compared to thestring contact time of about 2 -3 ms in the midrange.The present study aimed at a direct measurement ofthe motion of the striking point at the top of a hammer,before and during string contact in two perpendiculardirections; in the striking direction (vertically), and inthe direction of the string (horizontally).

METHODThe motion of a target point on the side of the hammerhead, 2 mm below the striking point, was measured bymeans of an optical device (Zimmer-OHG Electro-optical displacement transducer), see Fig. 2. Thesmallest measurement range used was 5 x 5 mm,giving a resolution of 25 . 10-6 m. The bandwidth ofthe system was 200 kHz. The velocity of the hammerimmediately before string contact (initial hammervelocity vi) was measured by means of a slotted vaneon the hammer shank and a photogate. The contactbetween hammer and string was measured electrically.All measurements were made on a grand piano inconcert condition (Steinway & Sons, mod. C, 227 cm).

FIGURE 2. Striking geometry (Steinway & Sons. model C,note G#

3) and measuring ranges. All dimensions inmillimeters.

RESULTS

Comparisons of the hammer motion for a legato touch(finger resting on the key), a staccato touch (relaxedfinger hitting the key from above), and a strained

touch (strained finger hitting the key from above),respectively, at forte level are given in Fig. 3. Only thestaccato touch, giving a strong impulse at thebeginning of the key descent, excites the hammerresonances efficiently.

FIGURE 3. Comparison of the hammer motion for differenttypes of touch, showing the full hammer travel; verticalcomponent (lower panel) and horizontal component (upperpanel). Legato (full line), staccato (dashed) and strained(dotted); forte vi = 4.5 m/s, G#

3 = 208 Hz.

The influence of dynamic level on the hammer motionis illustrated in Fig. 4 (staccato), showing a close-uparound string contact.

FIGURE 4. Influence of dynamic level on hammer motion.Staccato, forte; vi = 3.7 m/s (full line); piano vi = 1.1 m/s (dashed).A predicted horizontal motion along a circular arc is shown dottedfor the forte case. The hammer-string contact time is indicated bythe horizontal bar between the two panels.

Looking at the motion in the vertical direction, thestriking point turned almost 1.5 mm higher in fortecompared to piano. This difference will be smaller forthe striking point than for a point on the woodenhammer molding due to the compression of the felt.

In the horizontal direction the motion stops shortlyafter (about 0.5 ms) the hammer has made contact withthe string, and a certain compression of the felt hasbeen reached. Compared to a predicted path along acircular arc, the maximum deviation was 0.3 mm(forte). This difference was interpreted as that thehammer is essentially stopped in its horizontal motionat string contact, and instead the shank bends or the

felt shears. Upon rebound the deflection persists untilthe shank has straightened or the felt expanded.

The striking point was shifted a little along the string(about 0.3 mm towards the bridge) for the blow at flevel compared to p. This indicates that the angle ofthe hammer head was different at string contact in pcompared to f. Also, a minor undulation in strikingpoint of about 0.05 mm is observed.

The influence of different types of touch on the detailsof the hammer motion at string contact was much smallerthan the difference evoked by the changes in dynamiclevel. A professional pianist did not produce anydeviating patterns compared to untrained subjects.

In order to study the influence of touch better, a soft,flexible shank was prepared by removing wood from theunderside of the shank (see Fig. 5). The difference inhorizontal displacement at string contact between legatoand strained touch was 0.1 - 0.2 mm. The hammer stayeddisplaced horizontally much longer with the soft shank,even increasing the horizontal displacement after stringcontact had ended. This indicates substantial flexing ofthe shank, during and after string contact.

FIGURE 5. Comparison of hammer motion for different types oftouch with a soft shank; forte. Legato (full line), staccato (dashed),and strained touch (dotted), all at forte level, vi = 3.5 m/s. Themaximum difference in vi between the three examples was 0.05 m/s.

CONCLUSIONSThe motion of the hammer before string contact can be

very different depending on the type of touch.The striking point is moved towards the bridge in forte

compared to piano (about 0.3 mm), indicating a strongerbending of the shank at loud dynamics. The influence oftouch on the shift in striking point is much smaller (0.1 mm).

The contact point between hammer and string moves onlymarginally during string contact (< 0.05 mm). The hammerstays esentially fixed at a horizontal position set shortly afterstring contact (0.5 ms).

REFERENCES1. N. Giordano JASA 107, 2248-2255 (2000).2. H. Suzuki, Vibration analysis of a hammer-shank system

CBS Internal Report (1983).3. Askenfelt A. & E.V. Jansson JASA. 90, 2383-2393

(1991).

Dynamical models of piano action

X. Boutillon

Laboratory for Musical Acoustics, CNRS, University of Paris 6, Ministry of Culture, 11 rue de Lourmel, 75015 Paris, France

Piano actions include those of the traditional grand- and upright-pianos, passive electronic keyboards, and experimental active servo-mechanisms. The lack of elaborate dynamical models limits the performance of electronic keyboards and, to a lesser extent, experimental mechanisms. Looking at the performance aspects, several important questions remain largely unanswered as of today: what are the relevant differences between the dynamics of grand- and upright-keys? What limits the controllability of the action at the piano nuance ? What causes the saturation at the forte level ? Why can be electronic keyboards so boring to play, regardless of the sound being generated ? Available experimental data and dynamical models is reviewed here. Suggestions for improvements of models are made.

THE MECHANICAL SYSTEM

The action of a grand piano comprises four main parts : the key K, the whippen W, the jack J, and the hammer H which all rotate around fixed point except the jack which rotates around a point attached to the whippen.

FIGURE 1. Piano action (after Gillespie [1]).

A felt is inserted between the key and the whippen. The repetition lever is not considered here since its role appears only after the hammer has struck the string. The motion of the key is limited by the key bed. The jacks pushes the hammer and begins to rotate when it reaches the escapement dolly E. At this point, its top surface (hard, covered with graphite) rubs against the felt surface of the knuckle of the hammer. Eventually, the hammer escapes and continues its motion, being subject to gravity only (and damping at the pivot). The action is also designed to allow fast repetition, but this aspect will not be considered here. The function of the action is double. (a) It sends the hammer towards the strings. (b) It generates a force on the finger of the pianist: this feedback is said to be

essential for a good playing control. This latter function is poorly rendered by the passive electronic keyboards. Active experimental servo-mechanisms seek to emulate this function so that the pianist would feel “at home” (if not at ease) with a non-traditional keyboard. Much efforts and literature has been devoted to this question: see for example [1] for a recent review. This presentation will mostly concentrate on details of the first function of the piano action.

A BRIEF LITERATURE REVIEW

Askenfelt and Jansson present in [2] a thorough kinematical study of the piano action. A comparison of the recorded key velocity between real and artificial playing (by a constant force, with or without an initial impulse) reveals that an oscillation occurs in real staccato playing that is not similar in artificial playing. It cannot therefore be attributed to the properties of the action. In other words, the hand reacts to the action and its dynamical characteristics are very significant in the temporal patterns of the applied force and velocity. This means that conclusions (on action design for example) from studies conducted with a prescribed pattern of force, velocity, or acceleration imposed on the key must be withdrawn with great care and or additional insight. Another very surprising finding in legato playing is that the hammer acceleration becomes negative (but less than gravity) while the force exerted by the jack is still positive and the contact between them is not lost. This calls for a detailed study of the jack-knuckle contact which has never been done. In the presented diagrams, the hammer never looses contact with the jack before the latter reaches the escapement dolly. This is contrary to

common belief and to the outcome of many dynamical models according to which the hammer could be launched by the flexibility of the action alone, before the jacks begins pivoting. This point could be clarified by extensive visualisations with a fast camera for example. This set of articles constitute a very solid basis for further dynamical modelling.

Dijksterhuis [3] presents calculations on an all-mass model: the key, the whippen (with the jack), and the hammer are put one on top of each other. Once any element reaches a block, escapement occurs. It is shown that different time-profiles of the force exerted onto the key may yield the same escapement velocity, although with a different delay after the beginning of the key strike. Since tones coming slightly in advance sound louder in a chord, this finding sheds an interesting light on the role of the pianist’s touch.

Oledzki [4] presents measurements and a dynamical model of an upright action. Measurements of the sound levels obtained at various levels of the constant force applied to the key reveal a saturation in the SPL when the force value exceeds 40 N. This might be due to the nonlinearity in hammer-string interaction (I showed in [5] that the sound pressure was indeed saturating when the hammer was striking the strings with velocities greater than 2m/s) or to nonlinearities in the action itself, or both. Oledzki’s dynamical model of action comprises two masses (one for the hammer, the other for all other parts) connected by a spring representing the internal flexibility of the action. Due to the disagreement between the features of the results of simulation and those of experimental observations, the author introduces a quadratic spring and a hammer mass which varies linearly with its position. The parameters are derived from static measurements. The final simulations results are not directly compared with experiments.

A recent article by Hayashi et al. [6] deals with the interesting problem of how to drive the action at piano nuance. It is well known that the gap between the escapement point of the hammer and the strings position causes the hammer to loose some or all of its velocity due to gravity. Not only this imposes a threshold on the escapement velocity for the hammer to produce any sound but at low levels, it also creates a steep variation of the impact velocity (or equivalently the sound level) as a function of the escapement velocity [7]. This steep variation has nothing to do with the properties of the action but the action itself might even worsen the situation to a yet unknown extent. However, the authors propose a velocity profile to be prescribed to the key, in order to allegedly

“produce stable soft tones” in the context of automatic piano playing.

In the dynamical model used by the authors, only the mass of the hammer is considered whereas the mass of the rest of the action (key, whippen, jack) is ignored. Under the restricted condition of imposing a motion to the key, this might seem acceptable as a first approximation but no finding about the dynamical reaction from the action onto the finger can be derived which constitutes a serious drawback of the model.. A spring connects the finger to the hammer and represents the overall flexibility of the action. Its definition as a piecewise linear spring is based on static measurements and it is incorporated in the dynamical model by means of two eigenfrequencies. The motion of each end of the spring is limited: one by the key bed, the other by the escapement dolly. In this model, the hammer necessarily looses contact as soon as the jack touches the escapement dolly. It is reported that for low values of the constant velocity prescribed to the key, the hammer looses contact before the jacks reaches this point. It should be noted that this has never been observed by other authors; moreover, one would imagine that this would happen at rather high levels of excitation. Despite all these shortcomings, the agreement between the simulations and the presented experimental results is amazingly good.

PROPOSED MODEL

The model that will be presented is a three-mass model: the key, the whippen, and the hammer. A non-linear spring links the key and the whippen and the contact between the jack and the knuckle is modelled by some friction law. Since piano technicians carefully adjust them, we have included dampers at each pivot. Simulation results will be presented. A fast-camera film of the motion of the different parts of a piano action will be displayed.

REFERENCES

1. Gillespie B., PhD thesis, Stanford University (1996). 2. Askenfelt A. and Jansson E.V., J. Acoust. Soc. Am. 88,

52-63 (1990) and 90, 2383-2393 (1991). 3. Dijkksterhuis P.R., Nederlandse Akoest. Genootschap 7,

50-65 (1965). 4. Oledzki A., Mechanism and Machine Theory 7, 373-385

(1972). 5. Boutillon X., J. Acoust. Soc. Am. 83, 746-754 (1988). 6 Hayashi E., Yamane M., and Mori H., J. Acoust. Soc.

Am. 105, 3534-3544 (1999). 7 Boutillon X., Grijalva R., J. Acoust. Soc. Am. 105 (2) Pt.

2, 1056 (1999).