Dept. for Speech, Music and Hearing

Quarterly Progress andStatus Report

Long-time-average-spectra ofscales and spectra of single

tones from a violinAlonso Moral, J. and Jansson, E. V.

journal: STL-QPSRvolume: 19number: 1year: 1978pages: 030-039

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

STL-QPSR i/i978 30.

111. MUSICAL ACOUSTICS

A. LONG-TIME-AVERAGE-SPECTRA OF SCALES AND SPECTRA OF SINGLE TONES FROM A VIOLIN

J. Alonso oral" and E. Jansson

Abstract

Long-Time-Average-Spectra, LTAS:es, have proved to give repre- sentative records of violins. The fulltime scales recorded in a re- verberation chamber have, however, not been well suited for listen- ing tests. Therefore in this investigation the recordings were made in an anechoic chamber and the effects on the LTAS by reducing the number of tones played were tried. Furthermore, the relations were investigated between Single-Tone -Spectra, STS :es, and LTAS:es. The investigations show that a representative LTAS of a violin can be obtained from every third tone on the two middle strings. Further- more, i t shows that the spectrum level of all strings a r e approxi- mately equal to 12 Bark and thereafter it drops 4 dB from a string to the next lower and that the high frequency limit of each string drops one Bark from one string to the next lower string. The levels of the STS:es fall in general within f 5 dB of the LTAS of the same string.

Introduction

Long-Time-Average-Spectra, LTAS:es, recorded in a reverbera- (1 tion chamber have been proved to give reproducible records of violins .

Furthermore, i t has been shown that quality scores calculated by means

of LTAS:es correlate well with tonal quality ratings(2). In a pilot ex-

periment; however, i t was found that the recorded scales were not well

suited for listening tests. F i r s t the long reverberation of the tones

gives a dominating influence on the sound. Secondly only short frac-

tions of the scales could easily be remembered for comparisons in lis-

tening tests. Therefore the previous work has been followed up by the

present investigation.

If a few played tones give a representative LTAS, then these tones

describe acoustical properties of the instrument. Therefore in the

present work, we have investigated the effect on the LTAS caused by

reducing.the number of tones played. For future listening tests the

recordings were made in an anechoic chamber. The answers to the

two following questions were especially sought for:

(1) Can a representative LTAS of a violin be obtained with few played tone s ?

( 2 ) How differs the representative LTAS from the LTAS:es of single strings and the spectra of played single tones?

+ Graduate student at the Dept. of Speech Communication.

STL-QPSR 1/1978

In the following we shall discuss important parameters of LTAS:es

and single tone spectra, STS:es, describe the recordings of tones to

be analyzed, present the results of our analysis, and finally t ry to

answer the two questions.

Theory

A few tones played in succession give an impression of the tonal

quality of a musical instrument, as for instance the violin. From a

physical standpoint this i s difficult to understand.

(3) The function of our hearing can be modelled in the following way . The ear i s represented by 24 bandpass filters. The filters have the

bandwidths of one Bark and a r e speced one Bark apart. The output of the

different filters i s given somewhat different weights to account for the

variations in the sensitivity of the ea r a s function of frequency; in

general low weights at low and high frequencies, and high wights at

medium frequencies. The weights vary also a s a function of level in

each filter to account. for the variations in sensitivity a s a function of

sound level. The hearing threshold i s introduced a s a specific noise

level of each filter. Finally the filters have a specific high frequency

flank overlapping the higher filters to account for masking, i . e . the

fact that a strong partial in one critical band can mask a weak partial

in a higher band. The critical band model can be applied to the evalua-

tion of timbre, a s demonstrated by Plomp (4)(5)

The relation between the Bark frequency-scale and the Hz frequency-

scale i s given in Fig. 111-A- I . The estimated equal loudness contours,

the weights, added to the preemphasis used in our recordings, and the

masking limit (approx. - 10 d ~ / ~ a r k ) a r e also given in Fig. 111-A- 1.

We can see that our LTAS and STS should emphasize high frequency

components and suppress low frequency components compared with

horizontal, equal loudne s s lines.

The violin has a large number of sharp resonances, approx. 20-40

resonances below 5 kHz. These resonances shape the tone spectra

we listen to. The number of resonances i s approximately equal to the

number oof critical bands. This means that partials of different fre-

quencies but within the same critical band can have considerably dif-

ferent amplitudes. Thus, a representative LTAS of an instrument a s

10 BARK 20

Fig. 111-A- I. Preemphasis used in the playback (- -), preemphasis added to the 60 phone curve middle line (-), +20 dB upper (-), and -20 dB lower (-) as function of the Bark frequency scale. Approx. slope of the masking limit ( - --- ). For easy compari- son a kHz frequency scale and the equally tempered scale a r e plotted - G, D, A, and E being the open strings of the violin.

10 BARK 20

Fig. 111-A-2. Number of par t ia ls p e r Bark band a s function of the Bark frequency: a) fo r different th ree octave scales; complete violin, one octave chromatic scale each s t r ing ( -), G-major scale G3-G6 ( - - - - ) , full-tone scale G3-G6 ( - - - ), G-major t r iad G3-G6 ( - ' -) , b) for different s t r ings one octave, chromatic scales . G-string ( -), D-string ( - - - - ), A-string (- -), E-str ing (- . -1. c) for different one octave scales on the G-string, chromatic scale (-), G-major t r iad ( - -), every third chromatic tone (- -). d) for single played tones with fundamental frequency of 196, 294, 440, and 660 Hz, and complete violin ( c . f . Fig. 111-A-2a) (-) and the calculated density a f te r Eq. (1) ( - - - ).

STL-QPSR 1/1978

the lowest fundamental, f (in Hz), the number of octaves played m, 0 -

and the number of played tones per octave n. Empirically the density

of partials (per Bark) d N / d ~ a s a function of the frequency in Bark B

has been found approx. to follow

- i a = (0.02~fo+l.0)(2. 3.1n-~+0.25)(4.5-n -. 06)

for Bo 5 B L 18 Bark,

where B i s the frequency of the lowest fundamental in Bark, cf. Fig. 0

111-A-2d. The formula give,s an estimate within 1 40% of the densities

but in general considerably better, cf. Fig. ILI-A-2d. It should be noted

that the density distributions a r e approx. .proportional to the widths in

Hz of the Bark bands. The density of partials a t 9 Bark (1 kHz) i s ap-

proximately twice the density at 2 Bark (200 HZ).

Recordings were made in an anechoic chamber with a violin (Ca-

murat, &ole francaise, 1950 ~ i l b a o ) belonging to and played by one of

the authors (JAM). The microphone was placed at a distance of ap-

proximately 2 m to the righthand side of the player, slightly above and

slightly in front. For each string a one-octave chromatic scale was

played. Each tone was played three times in succession with a dura-

tion of approx. one sec and a short interval between the tones.

The recorded tones were analyzed by means of the LTAS program

previously. described, i. e. a preemphasis and a time constant of 13

msec of the smoothing f i l ters (1)(2). Spectra of each triad of tones,

STS:es, were recorded and stored for later comparisons, i. e. four

times thirteen STS :es. Thereafter LTAS:es were obtained by averag-

ing over different combinations of STS:es and were stored. Further-

more STS:es of each of the three tones on the open G-string and the

octave on the G-string were recorded to estimate the uncertainty limits

of the records.

STL-QPSR 1/1978

LTAS of different scales

The simplest way to obtain an LTAS for the complete violin i s in

our case to average over all tones played. Such an LTAS is shown

in Fig. III-A-3 together with the LTAS obtained according to the full-

tone scale used a s standard in a previous investigation(2). A corn-

parison between the two LTAS:es shows that they a r e closely the same

with a ) an average difference of 1 dB per filter and b) a 3 dB differ-

ence in one filter and 2 dB in six filters. The peaks, dips, low and

high frequency flanks a r e closely the same. The LTAS containing all

played tones gives equal weights to all strings. The three octave full-

tone scale gives unequal weights to all strings when a full octave i s

played on the E-string. The fulltone scale has also a lower density of

tones. Therefore it was decided to use the average over all the played

tones in spite of the overlappings of the different scales. A little cal-

culation shows that the partial densities d N / d ~ a r e moderately in-

fluenced by the different scales, the fulltone scale gives a 30% lower

density than the chromatic scale, and the average cver al l tones a

30% higher density (a min. of five partials per Bark a t 2 3ark).

Our LTAS i s elected a s most representative, the "complete" LTAS

(for one direction only) has a steep low frequency flank between 2 and

4 Bark, a peak at 4-6 Bark, a shallow dip at 7, a peak a t 9- 11, a dip

a t 12,. two prominent peaks at 15 and 17 with a sharp dip in between,

and a high frequency flank a t 18-21 Bark less steep in the high frequen-

cy end. ,

LTAS of different strings

The LTAS:es of the four strings a r e presented in Fig. III-A-4a-d.

Compared with our "complete" LTAS, we see that the G-string dis-

plays approximately the same peaks and dips, high and low frequency

flanks. The level of the G-string LTAS (a min. of five partials per

Bark a t 2 Bark) shows, however, that the low frequency partials a r e

in general stronger and the high frequency partials weaker, and further

more that the high frequency limits of the sound energy is lower. The

Fig. 111-A-3. Long-time-average- spectra of "complete violin" ( - ) and the full tone scale ( . ); dissimilarity 25. In the up- per part the differences between the two LTAS:es are plotted.

10 ' 20 BARK

Fig. 111-A-4. c) for the A-string, d) for the E-string, dissimilarities 76, 44, 43, and 114, respectively.

same results a r e obtained for the D-string (a min. of four partials

per Bark at 3 Bark) but less pronounced. The A-string LTAS (a min.

of three partials per Bark a t 4 Bark) has naturally a higher low fre-

quency limit but somewhat higher low frequency partials and approxi-

mately the same high frequency limit a s the "complete" LTAS. The

E-string (a min. of three partials per Bark at 6 Bark) finally, has a

high low frequency limit compared to the "complete" LTAS. The di s - similarity measures compared to the I t complete" LTAS a r e 76, 44, 43,

and 112. By excluding the filters containing no partials the dissimila-

rities a r e reduced to 72, 44, 24, and 52. Thus the A-string LTAS gives

a good approximation of the LTAS for the complete violin. The low fre-

quency part can be added by taking the LTAS for the combined D- and

A-string. By doing so we obtain a close approximation in the low fre-

quency range too, cf. Fig. 111-A-5, and a dissimilarity measure of 32.

Thus a representation LTAS may be obtained from the middle two strings

A m o r e detailed study i s suggested by Fig. 111-A-6. From the open

string to 12 Bark a r e the differences in al l fil ters less o r equal to $. 3

dB except from the open G-string only. From 13 Bark there a r e in the

large level differences of -7, - 4 . 0 and t4 dB for the G D- A- and E-

string respectively, i. e . in large, a discrepancy of 4 dB between the

adjacent strings. The lower part shows that the high frequency limits

a r e 18.5 Bark for the G- and D-string, 19.5 Bark for the A- and 22

Bark for the E-string. The high frequency limits may derive from the

position of the bow on the string. It fits also well with the second min-

imum predicted from the bowing position preferred by the player. But

a s no record was made of the bow position at the playing no definite

conclusion can be drawn.

Single tone spectra

Eight different STS:es of the three single notes played in succession

for the open G-string and four of the octaves of the G-string were re-

corded. Comparisons of the different STS:es gave a dissimilarity of

max. 52 and min. 25 for the open string and max. 77 and min. 44 for

the octave. The level discrepancies of the different Bark bands a r e

within f 2 dB in more than 7070 of al l cases. The dissimilarities between

Fig. 111-A- 5. . Long-time-average-spectra of the "complete violin" ( - ) and of the D- and A-string chromatic scales ( ). Dis similarity 32.

Fig. 111-A-6. LTAS of the chromatic scales of the G-string ( - -), D-string ( - -), A-string ( - - - ), and the E-string (- ) ; dissimilarities 76 , 44, 43, and 114 respectively. In the upper part the difference i s plotted between LTAS of the different strings and the LTAS of the "complete violin".

STL-QPSR 1/1978

different STS:es and LTAS of the corresponding string a r e shown in

Fig. In-A-7a-d. We can see that the dissimilarities increase with

the fundamental frequency of the played tones. This i s to be expected

a s the partial density decreases with increasing fundamental frequency.

The dissimilarities a r e larger than the ones obtained between STS:es

of the same tones. But the dissimilarity curves display strong local

maxima and minima. In many Bark bands there a r e no partials. The

"empty" Bark bands contribute much to the dissimilarity. These con-

tributions can be suppressed by multiplying with a weight function as

described in the theory part. If so i s done, we obtain the lower dissi-

milarity measures presented in Fig. 111-A-7a-d, i. e. we should ex-

pect an average discrepancy of 3.5 dB or less between each not empty

band of the LTAS and the STS.

An analysis of the variations within the different Bark bands for

tones of different fundamental frequencies, gave the following results:

from 2-4 Bark the variations in levels were f 5 dB, from 5-7 Bark

f 2 dB. At high frequencies 14.5-16.5 Bark the variations were f 7

dB, f 5 dB and f 1 dB for the cases of 1, 2, and 3 partials, respectively

within the single Bark band.

Another way i f comparing the STS:es and corresponding LTAS:es

i s shown in Fig. 111-A-8a-d. The average of the STS:es and the LTAS

i s not the same. This may seem contradictory but derives prelimi-

narily from the fact that the LTAS is the average of squared sound

pressures in linear measures, while the average STS i s the average

of the levels in dB. This means that the average STS i s lower o r equal

to the LTAS level. Secondly a t low frequencies the LTAS represents

the average taken over all STS:es, even those with no partials in the

low frequency bands. Thus the STS:es averaged over spectra with par-

tials in these bands will give a higher level. The filters have not in-

finitely steep flanks. The overlapping of the f i l ters gives a third ef-

fect of discrepancies. If the played spectra contain very strong par-

tials, these strong partials will give contributions in the neighbor

filter bands. Furthermore, if there exists no strong partial in the

neighbour filter band from any STS, the overlapping of strong partials

can give larger contributions than the partials in the same band. Thus

150

\ roo

c E u

Si r/)

50

Fig. 111-A-7. Dissimilar i t ies D between single tone spectra and LTAS of the chromatic scales on the corresponding s t r ing a s function of played tone. Upper line represents the diss imilar i ty cal- culated over al l f i l ters and the lower line over f i l te rs con- taining part ia ls of the STS; a ) G-string, b ) D -s t r ing, c ) A-str ing, d) E-str ing.

F i g . 111-A-7. c) A-str ing, d) E-string.

DIFFERENCE STS-LTAS d8

. DIFFERENCE STS-LTAS dB I d

I 6 I

ul V, 0

DIFFERENCE StS-LTAS dB

DIFFERENCE STS-LTAS dB

STL-QPSR 1/1978

the LTAS:es will give higher levels in these cases a s this effect i s

not encountered in calculating the average STS.

In Fig. 111-A-8 we can also see that the discrepancies between

min. and max. have the magnitudes of 10 dB somewhat increasing

with frequency but varying considerably locally. The large disc re - pancies a r e likely to derive from sharp peaks within the corresponding

Bark bands. The relations a r e however complex, as large discrepan-

cies do not fall in the same Bark bands for al l strings.

In the previous section it was shown that LTAS of the different

strings had specific simple relations in the high frequency range.

Two simple ways to describe the range of a spectrum exist, either to

give the number of partials o r to give the highest frequency of notice-

able sound energy. The recorded STS:es make i t possible to investigate

which description gives the best approximation for each string.

Table 111-A-I.

I I STS LTAS 1 String

G

. D

A

E

t3 dB limits for the G- D- and A-strings $6 dB limits for the E-string

Bark

18.5f0.5

i9.5f0.5

' 21 t i - 22 -1

kHz

5f0.5

660.5

The upper limits for spectral energy are,surnmarized in Table 111-A-I.

The table shows that the fix upper frequency limit for each string is a

fairly accurate description (rnin. max. deviations in the table) and that

the number of partials N varies considerably. The frequency limits

a r e higher for the higher strings; for the E-string slightly higher than

the highest fil ter. The limits a r e approximately 1 Bark lower for the

next lower string in consecut ive order.

It was also tried to see i f l e ss tones than the fulltone scales could

be used to make a representative LTAS for the different strings, see

Fig. 111-A-9. The figure shows that the dissimilarities between each

increase will decrease with decreasing density. The increase in dissi-

Bark N

milarity i s moderate up to the major triad except for the E-string. I

19

19

25- 12

19-11

-.- G-string --.*-- 0-string

4 6 8 10 12

PLAYED 'TONES/OCTAVE

Fig. 111-A-9. Dissimilarity D as function of number of played tones per octave for G-string ( - . -) , D-string ( - .. -), A-string ( - ), and E-string ( - - - )

STL-QPSR 1/1978

Thus every third chromatic tone should give a representative LTAS.

The partial density i s decreased to 0.3-0. 5 of that of the complete

LTAS, i. e. a min. of two tones per Bark. The LTAS :es made by the

suggested tones i s shown in Fig. 111-A- lOa-d and show small dissimi-

lari t ies with the LTAS of the chromatic scales. It was previously

found that the D- and A-string gave an LTAS representative of the

complete violin. By averaging over the STS:es giving LTAS:es of

the D- and A-string i t was found that this LTAS was representative

of the complete violin with moderate dissimilarity to the "complete"

LTAS, cf. Fig. 111-A- 1 I . The dissimilarity of 39 i s moderately

larger than the dissimilarity between the "complete" and the whole

tone scale LTAS of 25.

Conclusions

In this investigation the answers to the following two questions were

specifically sought for:

( i ) Can a representative LTAS of' a violin be obta.ined with few played tones ?

( 2 ) How differ spectra of single tones and LTAS:es of single strings from the representative LTAS?

Our investigation gives the following answers:

(1) An' LTAS of every third semitone on the D- string and A- string represents a close approximation of the most representative LTAS with a dissimilarity of only 39, i. e . an average difference of less than 1.5 d ~ / ~ a r k . Thus a representative LTAS can be obtained with few played tones.

(2) The different strings have closely the same LTAS up to 12 Bark, thereafter the highest strings have the higher level and the highest upper frequency limit and the lowest strings have the lowest level and the lowest upper frequency limit. The differ- ences between adjacent strings a r e approximately a level shift of 4 dB and a high frequency limit of 1 Bark. These differences may derive from the position of the bow on the string.

The different single tone spectra have on the average a somewhat

higher level at low frequencies and lower levels at high frequencies

compared to the LTAS of the same string.

The maximum level discrepancies in the different filters a r e of

the magnitude 5 dB. Large negative discrepancies a r e likely to be

caused by sharp peaks in the frequency response curve of the violin.

Fig: 111-A-10. Long-time-average-spectra of scales with every third chromatic tone ( - ) and chromatic scale on the a) G-string, b) D-string, c ) A-string, d) E-string; dissimilarities 28, 30 , 31 , and 34, respectively.

Sod8

1 0 10 20 BllRK 0 10 20 BARK .

f

- a Sod0

20 EARK

Fig. 111-A-10. c d.

soda L

Fig. 111-A-11. Long-time-average-spectra of the "complete violin" ( -) and of the D- and A-string scales with every third chromatic tone ( . o m ); dissimilarity 39.

STL-QPSR 1/1978

The experiments prove that a more stable level i s obtained with

many partizls in every Bark bands but that a min. of two partials may

give a representative LTAS.

The procedure of calculating dissimilarities works well when the

same critical bands contain spectral energy of the spectra to be com-

pared. Dissimilarities can also be calculated with a reference a s an 1 LTAS, when the partials of two played tones fall in different bands.

The auditory perception can hardly work in this way because the dis- 1 s imilarities calculated in this way a r e mainly a function of the reference

LTAS. It indicates however that the LTAS is a better description of

a violin than a few single tone spectra.

The results obtained in this investigation a r e , we believe, valid in

general. The results a r e still obtained with only one violin, which may

limit their general validity.

References:

(1) JANSSON, E. V. : "Long-Time-Average-Spectra applied to analysis of music. Pa r t 1x1: A simple method for surveyable analysis of complex sound sources by means of a reverbera- tion chamber", Acustica 34 (1976), pp. 275-280.

( 2 ) GABRIELSSON, A. and JANSSON, E. : "An analysis of Long- Time-Average-Spectra of tw:er$ytwo quality- rated violins", STL-QPSR 2-3/1976, pp. 20-34.

(3) ZWICKER, E. and FELDTKELLER, R. : Das Ohr a ls Nach- richtenempfiinger, S. Hirzel Verlag, Stuttgart 1967.

(4) PLOMP, R. : "Timbre a s a multidimensional attribute of complex tone$', pp. 397-4 1 1 in .Frequency Analysis and Periodicity Detection in eari in^ (eds. R. Plomp and G. F. Smoorenburg), A. W . Sijthoff, Leiden 1 970.

(5) PLOMP, R. : Aspects of Tone Sensation, Academic P re s s , London 1976.