Dept. for Speech, Music and Hearing

Quarterly Progress andStatus Report

Fundamental frequencyadjustment in barbershop

singingHagerman, B. and Sundberg, J.

journal: STL-QPSRvolume: 21number: 1year: 1980pages: 028-042

http://www.speech.kth.se/qpsr

I n both quar te ts the exercises consisted of chord -sequences,

i .e. cadences, which are sunq on a CV-syl lablt-. I-icwvc?r, thc c-a-

dences d i f fered regarding L o t h chords and syll~~b1c.s . Quartet A

retained bas ica l ly the same chord frequencies '11.1 the t h e while

quartet B raised the frequencies by a half tone s t e p b e b e e n each

rendering. Quartet A repeatedly changa the sy l l ab le while quartet

B f i r s t used [ma] i n 1 1 versions of their cadence and then [m 1 i n

7 versions. The cadences are given i n Ekample 1.

QUARTET A QUARTET B

Example 1. Cadences sung by the two quar te ts . The symbols below the chords indicate their harmonic function.

i The recordings *re made i n an ordinary room. Each of the I

singers had a contact microphone (acceleramcter) glued t o the skin

of the trachea s m centirwters below the thyroid car t i lage . T k

s ignals picked up by these four contact micro~~horles wre recortlcd

on a four track tape recorder. This recording was used f o r meas-

uring fundarrental fretjucncy i n the s u b s c p n t analysis . '1% signal

fram an ordinary microphont? was recorcl~d on a second tape recmrder.

A fundamental frequency measurement c q u i p n t described else-

where was used (see Askenfelt, 1979) . Basically it measures the

periodici ty by means of a hardware fundamental frequency detector

which applies a double peak picking strategy. The rcsul t lng spare-

wave s ignal is fed t o t h e computer which cc>nwrts it i n t o frequency

data. Plots of the fundamntal frequency versus tirrre as *l.l a s

fundamental frequency histograms may be obtained. When the opera-

t o r has marked the nude frequency of each histogram peak by means

PROBABILITY ( '10, ARBITRARY SCALE)

INTERVAL MEAN IN [ma] - CHORDS (CENT)

~ i g . II-A-2. ~orrelaticn be- intaw11 mragas of 11 mmi ie rd of chords using the syllable [ma] and 7 renderings of the same chords using the syllable [m3 . The chords *re selected fran the cadence sung by quartet B.

STANDARD DEVIATION (CENT)

averages and standard deviations of these minor seconds are l i s ted

in Table 11-A-111. The average is sl ight ly greater than the Pytha-

gorean value of 90 cents and much n a r r w r than the 112 cents of

just intonation. The standard deviations a u n t t o 5.7 t o 7.7

cents. This is not much greater than the smallest standard devia-

t ions observed for h m n i c intervals. Thus, these singers seem

t o arrive a t a very high degree of accuracy even i n melodic inter-

vals, i .e . when the standard is merely stored as an internal refer-

ence.

Table 11-A-111. Interval average and standard deviations ( in cent) for lead singing the tones constituting the interval in succession.

Quartet Chords Mean SD

From the above it seems reasonable t o canclude tha t the inter-

val averages represent reliable information and tha t even small dif-

ferences betwen interval averages may be significant. It is then

interesting t o compare the interval averages with interval s izes

prescribed by the Pythagorean and the just scale. As fa r as the

seventh is concerned, a third version is relevant, namely the in-

terval betwen the fourth and the seventh pa r t i a l of a d n i c

spectrum. Table 11-A-IV lists a l l interval averages i n the material

including standard deviations, confidence intervals, and deviations

from just and Pythagorean intonation. Many interesting observations

can be made. The just and Pythagorean values, which agree for the

three simplest intervals (octave, f i f t h , and fourth) , f a l l within

the confidence intervals of the corresponding averages for these

simplest intervals with one exception. Eventhough mst of these

intervals wre sung saw cents n a r r m r than the ideal according t o

Pythagorean and just intonation, it seems justified t o assume that ' ,

these intervals wre sung in accordance with these theoretical values.

A l l major thirds show confidence intervals excluding the just version

and, with one single exception, also the Pythagorean.

STL-QPSR 1/1980 38.

The chords in the material analyzed can be divided into three

types :

(1) tonic chords

(2) major t r iads including seventh

( 3 ) minor t r iads w i t h o r without seventh

Disregarding for the m m n t the fact that the lead seems t o serve

as the c m n reference for the singers, w can ccanpute the intervals

relat ive t o the root of each chord. In tha t way, we may compare in-

tonation within and between each of the three types of chords mentioned.

This has been done in Table 11-A-V . The agreemnt k t w n the tvm

quartets is amazingly high in the case of the tonic chord. The octave

is pure, the f i f t h is 4 or 5 cents wider than pure and the third is

4 and 5 cents narrower than Pythagorean. This means tha t the third of

the major t o n i c is 18 and 17 cents wider than a pure third! I t seems

that the pure third is replaced by the qrthagorean third minus a small

correction i n the tonic chord. With respect to the major t r iads with

a seventh, there is a very high degree of agre-t in the case of the

third. Interestingly it is tuned t o a value which is 5 t o 11 cents

wider than the value of just intonation. The f i f t h shows less con-

sistency but is smaller than the f i f t h of the tonic chords in a l l cases.

The seventh of the subdominant seventh chord has a confidence interval

excluding a l l other versions of seventh. This suggests that a seventh

is performed narrowr when the chord has the function of a dminant

than when it has another hanmnic function. The seventh of the dani-

nantal t r iads exceeds the interval betwen par t ia ls 4 and 7 of a har-

monic spectrum by 8 to 16 cents, and a l l the confidence intervals

exclude t h i s hanmnic interval. The two minor chords are dissimilar

with respect to the c m n intervals: both confidence intervals ex-

clude the average of the other. Howver, the major th i rd between

the third and the f i f t h of these t r iads agree within one cent: 394

and 393 cents. This is very close to the values of the major thirds

of the major t r iads with seventh. W e may speculate that the major I

third has a sor t of key function in Barbershop intonation.

Discussion !

There are t m observations made above that require saw comrrents.

One is the magnitudes of the standard deviations, and the other is the

puzzling discrepancies betwen a l l k n m interval s izes and those per-

f o m d by our Barbershop singers.

T a b l e 11-A-V. Intervals (in c e n t ) relative to the root of the chords. Symbols as in Table 11-A-I.

(luartet

C h o r d

Wan

Conf . int . Mean

C o n f . int . Mean

Conf . int .

Major triad Major triad w i t h seventh Minor triad

A B A B B A B A A

T T D7 D7 Mod T7 S7 Sp7 Tp

STL-QPSR 1/1980 40.

W e found that the standard deviation of the lead, in performing

a melodic interval of a minor second, was 5.7 t o 7.7 cents. The 1

three smallest standard deviations for chord intervals *re found t o

be between 4.3, 5.1, and 5.2 cents (cf . Table 11-A-I). These num-

bers can be canpared with the difference l i m n for frequency. I f

musically m11 trained subjects repeatedly adjust the frequency of

a response tone to pitch agr-nt with a preceding standard tone, 1 the difference l i m n may be a s low a s 6 cents, which incidentally is

very close t o the standard deviations for the mlodic minor seconds.

When musically trained subjects adjusted tm synthesized vibrato

vowls t o different intervals, and thus had no beats t o use as a

cr i ter ion, the average intervals, and the average standard devia-

t ions shown in Table 11-A-VI were observed (Agren, unpublished thesis

w r k ) . With the major second as a possible exception the standard

deviation averages a l l f a l l within the standard deviations observed

for the melodic version of the minor second a s performed by the

leads (cf . Table 11-A-111) . Thus, the lmst standard deviations in

our Barbershop quartets are of the sarne order of magnitude as those

obtained i n psychoacoustic pitch matching and interval matching ex-

perimnts with musically w e l l trained subjects. F rm t h i s # con-

clude tha t the fundamental frequency control can be trained t o a

very high degree of s k i l l in singers. The accuracy with which a

subject adjusts the pitch of a synthetic tone by tmmmg a knob is

about the s m as the accuracy with which a singer can reproduce an

interval.

Table 11-A-VI. Averaged intervals and standard deviation (in cent) 1 from 8 subjects showing the lowst standard devia- t ion of a grmp of 17 musically t r ~ i n e d sub- jects who matched prescribed intervals betmen tm simultaneously sounding synthetic sung vibrato I

-1s. (According to Agren, unpublished thesis I

m r k . ) l ' I

SD

Interval Average &an Min Max

Octave 1203 6.2 3.6 8.8 Fif th 705 5.6 4 .O 6.8 Major third 400 5.7 4.4 9.4 Major second 20 3 8.7 5.6 13.7

Conclusions

The accuracy with which the fundamental frequencies are chosen

in Barbershop singing is extremely high and does not seem t o depend

on the v-1 t o any great extent. The lead serves a s the reference

to which the other singers adjust their fundamental frequencies so

as t o produce the desired chords. This is in a g r e m n t with Barber-

shop theorists. The nmber of camon par t ia ls (within a given fre-

quency range) between t~ tones constituting an interval tends t o af-

fec t the di f f icul ty with which the intervals is tuned. Thus, it

seems easier t o tune simple intervals, which share many par t ia ls than

t o tune intervals with few c m n part ials .

Most intervals in Barbershop singing deviate systematically from

the corresponding values according t o just and Fythagorean intonation.

The major third of the major t r i ad having the harmonic function of a

dominant may be interpreted a s a stretched version of a pure third,

while i n the tonic chord it can be regarded a s a flattened version

of a Pythagorean third. The major th i rd contained in a minor t r iad

shows the saw width a s the major th i rd of major t r iads including

seventh. These deviations from just intonation do not give rise t o

beats. The reason for t h i s muld be the f i n i t e degree of periodicity

of the tones produced by the singers. *

References

ASKENFELT, A. (1979) : "Automatic notation of played music", Farter Actis ?!usicaeU, pp. 109-120.

LARSSOtJ, B. (1977): "Pitch tracking of music signals", STL-QPSR 1/197'7, pp. 1-8.

Society for Preservation and Encouragmnt of Barbershop Quartet Singing i n America (Kenosha, W I , USA): Contest and Judging Handbook.

W e are indebted t o the quartets "Happiness Emporium" and "St&-

j w e t " for the i r expedient cooperation. This m r k was supported by

the FRN , NFR , and HSFR.

* This paper was presented a t the met ing of the Acoustical Socie- t y of the Nordic countries i n Finland, June 1980.