Dept. for Speech, Music and Hearing

Quarterly Progress andStatus Report

An analysis oflong-time-average-spectra of

twentytwo quality-ratedviolins

Gabrielsson, A. and Jansson, E. V.

journal: STL-QPSRvolume: 17number: 2-3year: 1976pages: 020-034

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

STL-QPSR 2-3/1976 20.

MUSICAL ACOUSTICS

A. AN ANALYSIS OF LONG-TIME-AVERAGE-SPECTRA O F TWENTYTWO QUALITY-RATED VIOLINS

A. ~ a b r i e l s s o n * and E. V. Jansson

Abstract

Long- Time-Average-Spectra (L TAS:es) were recorded of 22 quality- rated violins. The LTAS:es were analyzed by four different methods: weight functions, factor analysis (FA), multidimensional scaling (MDS), and separate correlation analysis. The average difference between the instruments rated highest and lowest was t r ied a s a function weighting tonal quality. This weight function explained 64 70 of the variance of the tonal quality-ratings. Factor analysis and multidimensional scaling in five factor s/dimensions gave approximately the same solutions. The so- lutions accounted for 74 70 and 44 '$0 respec t ive ly of the variance of the LTAS:es and for 74 70 and 69 70 respectively of the variance of the tonal quality-ratings. The variations in certain res t r ic ted frequency regions selected by correlation analysis accounted for 7 1-84 70 of the variance in the tonal quality-ratings. The different methods imply that "strong" f r e - quency components a r e favorable in a low frequency region and in a middle high frequency region, while "weak" frequency components a r e favorable in a high frequency region and in a l imited middle frequency region. The resu l t s seem reliable, a t l eas t for the selected instruments.

Introduction

Long-time-average- spectra (LTAS:es) of sca les played and recorded

in a reverberat ion chamber have proved to be reproducible and sufficiently

sensitive to display differences between different violins (Jansson, 1976).

The LTAS:es offer a simple method for surveyable analysis of the

sound f rom complex sources, a s for example the sustained and approxi-

mately invariant p a r t s of the tones of a violin. In this investigation the

LTAS:es a r e applied to the analysis of quality-rated violins.

The purpose of the investigation i s to extract main pa ramete r s de-

scribing differences between the physical sound of different violins. These

pa ramete r s may then be tested in experiments for their relation to pe r - I ceived tonal qualities. Thus, we a r e seeking the answers to two main

questions: Do suitable methods exist for analyzing the LTAS:es ? and

What information on the tonal quality can be extracted f rom such analysis?

In the following we shall descr ibe the recording of the violins and the mak-

ing of the LTAS:es. The LTAS:es a r e analyzed by averaging over groups,

* Department of P sychology, Univer sity of Upp sala

STL-QPSR 2-3/1976 21.

thereafter with more advanced statist ical methods a s factor analysis and

multidimensional scaling, and ultimately with correlation analysis. Fi-

nally the resu l t s a r e summarized and conclusions a r e drawn.

Recordings

The violins for this investigation were borrowed f rom the "1975 In-

strument Exhibition of the Scandinavian Violin Maker Associationf1. All

103 violins of this exhibition had been given ratings for tonal quality by a

jury containing two professional violin players (Slijjd & Ton, 1975). The in-

s t ruments were tested for equality in "loudness" and t imbre for a l l notes

and string@. In a f i r s t tes t two three-octave sca les were played slowly

and evenly, the f i r s t being an A flat major scale and the second an A

major scale. Fur thermore the instruments were tested for their ease

of playing. Both jurors gave their judgments a s players and l is teners .

rating being 80. Twentytwo violins were selected, which represented the

The violins got tonal quality rat ings f rom 72 to 36, the highest possible 1

different tonal quality ratings f rom the highest to the lowest, see Table

TABLE 111-A-I. Relation between the number of violins selected and their tonal quality rat ings, 80 being maximum rating.

Number

2

6 7

4

3

Tonal rating

70 - 80

60-69

50 - 59

.40 - 49

30- 39

Recordings of these instruments were made together with a "reference"

violin (H. Sundin 197 1) and four interesting instruments , one being an

ear ly Cremona violin, labelled Andr eas Guarneri . Cr emona 1640** and

one being labelled "Marcus Obbo fecit Napoli 1726".

* A. Pisuke, personal communication with one of the jurors af ter the completing of the scientific investigations. -

* Certificate f rom Hamma & Sohn, Stuttgart gives "Francesco Rugeri , Cr emona approx. 1690" a s maker and year of making.

STL-QPSR 2-3/1976 22.

All 27 violins were brought to the reverberat ion chamber of the Dept.

of Building Acoustics a t the Royal Institute of Technology in four rounds

on the same day. The recording procedure ear l ie r reported was used

(Jansson, 1976). A Briiel & K j z r 1/2 inch 4133 microphone was placed

a t the previously selected standard recording position and connected to a

Nagra I11 tape recorder run a t 19 cm/sec. The player was shown the

standard playing position, giving a diffusor to shield the microphone f rom

the player. He was instructed to play a set of three full tone sca les each

consisting of three octaves and with a short break between each scale to

le t the sound "die out". He was instructed to play the scales dbtachb

starting f rom the open G-string a s loud a s possible and with a tempo of

approximately one note every second. In the previous repor t this proce-

dure proved to work well (Jansson, 1976).

In the f i r s t recording round, the reference violin was recorded f i r s t , I thereafter eight violins f rom the exhibition and finally the reference violin

once more. In the second round the same procedure was repeated with

the reference and with eight "new" violins, and in the third round with I

six "new" violins f rom the exhibition. Finally, the same ~ r o c e d u r e was

repeated with the reference violin and with the four interesting violins.

Thus, recordings were obtained of each of the 22 quality-rated violins,

of each of the four especially interesting violins and of the eight t imes of

the "reference" violin. The player had not played and was not familiar

with any of the violins and their qualities except f rom his own Cremona-

violin.

Fir s t simple analysis

Long-time-average- spectra ( L T A S : ~ ~ ) were made of the recordings

by means of the computer program LTAVSPEC92. This program rey re -

sents an extension of that previously used. F i r s t the LTAS of the three I

successive sca les of each of the different single instruments was made,

and thereafter the "average" LTAS of a l l twentytwo violins together. Fi-

nally, 1, TAS:es were made of several instruments grocped together. The

groups were selected to represent approximately the same quality ratings.

The selected microphone gave a fair ly high noise level, but the level was

sufficiently low not to influence the LTAS:es.

20 BARK

Fig . 111-A- I . LTAS-diagram. Lower par t : LTAS (long-time average spectrum) of a violin with tonal qualily ra t ing 66 ( - ) and the average LTAS of 22 violins ( . . . . ). Upper par t : difference be tween the two LTAS:es, d i ss imi la r i ty 34. The bark frequency "average" sca le i s defined i n Fig . 111-A- 2.

20 BARK

I 1 I I I

1 I

I I 1 I kHz

I I

I I

I I

I

1) i2 d.4 063 0.92 127 1.72 2.32 3.15 4.4 6.4 2s

I 1 I I I I TONE

Fig . 111-A-2. LTAS-diagram. Lower part: LTAS of the eight violins with the highest tonal quality ratings ( - ) and the LTAS of the seven violins with the lowest ratings ( . . . . ). Upper part: Difference between the two LTAS:es, dissimilarity 22. The bark frequency-scale is defined in relation to the kHz frequency- scale and the equally tempered tone- scale - G, D , A , and E being the open strings of the violin.

. .

I

STL-QPSR 2-3/1976

An interesting fact i s indicated in the LTAS:es of Fig. 111-A- I. The

"average" LTAS of a l l 22 violins and the LTAS of this single violin a r e

not drast ical ly different, both LTAS:es display peaks a t 5, 8, and I I

Bark and a broad peak a t 13- 18 Bark. These four peaks were found in

most of the 22 violins. The lowest peak corresponds to the range of the

main wood resonance.

In Fig. 111-A-2 the average LTAS:es a r e plotted f a r the eight violins

with the highest quality-ratings and the seven violins with the lowest

quality-ratings. The peaks and dips a r e found i n approximately the same

regions a s before. The difference curve (the upper curve) indicates that

i t should be favorable for the tonal quality to have t ls t rongtf frequency I components a t low and medium high frequencies, and that "weaktf f r e -

quency components a r e favorable a t 10 Bark and above 16 Bark.

The difference curve indicates the sign and the magnitude of the dif-

ference a s function of frequency (Bark) and probably with some corre la -

tion with the differences in tonal quality. One might therefore t r y to use

the difference function a s a weight function for tonal quality. By multi-

plying this weight function with the difference function between the LTAS of

any single violin and the "averagetf L TAS of a l l 22 violins (the sum of the

respective weights t imes the respective differences in each filter band)

a score for each violin was obtained. A somewhat idealized function de-

rived f rom the difference function i s plotted in Fig. 111-A-3.together with

five variations of this idealized weight function.

The correlation between tonal quality-ratings and the calculated sco res

according to the different weight functions a r e given in Table 111-A-11.

The weight functions 0- 6 give approximately the same resu l t s , however

with slightly lower correlation coefficients for weight functions 2-5. This

indicates that the difference function is reasonably well correlated with

the tonal quality-ratings. The square of the correlation coefficient can I

be regarded a s displaying the proportion of variance accounted for by the

l inear regression, i. e. in our case 64 OJo. We now proceed to investigate

the possibility of using certain multivariate statist ical techniques a s an-

other way of analyzing differences i n L TAS:es.

Factor analysis and multidimensional sca l ing

Two commonly used methods in psychological s ta t is t ics a r e factor

analysis (FA) and multidimensional scaling (MDS) . FA was originally

........ ........... . . . . . . . . . . . . . . . . . \ ;

. . . . . . . . . . . . . . . . . . . . . . . . . . .

....... + . . . * .............. ................................

Z s! I- 0

............. z 4J \ .... . . . . . . . . . . . . . . . . . . . . . . . 3 U.

I-

..........................

...........................

1 I I 1 4 1 I I I I I

0 10 20 BARK

Fig. 111-A-3. Weight functions: Function 1 i s a simplification of the difference function of Fig . 111-A-2 and functions 2 - 6 r ep re sen t different var ia t ions of function 1.

0 10 20 BARK

Fig. 111-A-4. Graphical display of factor loadings within each of factors 1-5 (vertical) for different filter bands (hori- zontal). The curve for each factor represents the size and sign (positive or negative) of the factor load- ings for different filter bands. Shadowed part s indicate factor loadings - 7 0. 50 '(positive or negative).

STL-QPSR 2-3/1976 26.

mainly lying between those of factor 1 and factor 2, F 3 mainly a "high

middle frequencyt1 factor, and F 4 a "low middle frequencytr factor. Factor

5 (F5), finally, is represented by two widely separated filter bands. By extending the analysis to comprise seven factors, the variance accounted

for r i ses to 85 70. The five factors described above a r e also found here

with some minor modifications. The most obvious modification i s that

filter bands 16- 18, which had high loadings in factor 2 above, now con-

stitute the main loadings of the added factors 6 and 7,

In a more restricted FA including only the four special violins and the

eight different playings on the reference violin, four factors accounted

for 88 % of the variance. The f irst three factors were fairly similar to

factor s 1 - 3 above, while the fourth factor had no direct correspondence.

In a final analysis on the whole material (the 22 violins included in the

competition, the four special violins, and the eight playings of the re -

ference violin, in all 34 f'violins") a five factor solution accounted for

74 70 of the total variance. The f i rs t four factors were very similar to

factor s 1-4 described above, while factor 5 possibly represented a fusion

of the earlier mentioned F 5 and F7.

As a complement to the results above also the factor scores for the

different violins were computed. The factor scores represent the posi-

tions of the respective violins within each of the factors described above.

The detailed results a r e not given here, only some comments regarding

the factor scores in the las t FA including all the violins used and the

different playings of the reference violin, see Fig. 111-A-5. I t i s noted

that the dispersion of factor scores for the different playings of the re -

ference violin i s considerably smaller than the diaper sion of factor

scores for all other violins, that is, there a r e larger differences be-

tween different violins than between different playings on the same vio-

lin. It was also found that the factor scorsm for the very f i r s t playing on

the reference violin departed more o r l ess from those of the following

playings, which may suggest practice o r adaptation effects during re-

peated playing~. The Cremona violin got extreme factor scores in two

factors, while the Napoli violin got no outstanding factor scores in any

of the factors.

The factor scores of the violins may be used to pick out the two most

dissimilar violins in a certain factor (that i s , the violin with the highest

FACTOR SCORE

Fig. 111-A-5. Resul t of factor ana lys i s of the twentyseven violins. Spread in factor s co re s . The thick l ines m a r k spread for the twentytwo violins, the thin l ines m a r k the spread of the eight playings on the s a m e violin, the broken pa r t the distance to the f i r s t playing compared to the other seven playings,

- factor s co re s of the Cremona-violin a r e marked by c i r c l e s and those of the Nzpoli-violin by boxes.

STL-QPSR 2-3/1976 27.

factor score and the violin with the lowest factor score in that factor) and

compare their LTAS:es. This is i l lustrated in Fig. 111-A-6 for the case

of factors 1, 2, and 3. As seen there, there i s a considerable difference

between the two LTAS:es in the frequency regions emphasized by the

respective factor. The extreme violins in each of the factors might be

used a s stimuli in experiments aiming a t investigating the perceptual ef-

fects of specific differences in LTAS ( see Discussion).

Multidimensional sca l ing ------------- F o r comparison the LTAS values subjected to factor analysis were

a l so analyzed by a method for multidimensional scaling (MDS) called

INDSCAL (Individual Differences in scaling), Carrol l and Chang ( 1970).

In i t s most frequent application this method takes a s input individual r a t -

ings of perceived s imilar i t ies between certain given stimuli. The simi-

la r i t ies a r e interpreted a s distances between the stimuli represented a s

points in an Euclidean space of a certain dimensionality, the higher

similarity, the smaller distance and vice versa , and a solution i s sought

which gives a good fi t between the original s imilar i t ies and the distances

in the respective solution. The outstanding features with INDSCAL a r e

that the derived solution i s unique and requi res no further rotation of

axes. Fur thermore , the individual weights for the different resulting

dimensions a r e obtained (i. e. , different individuals may give different

weights to different dimensions).

In the present case INDSCAL was applied in the following way. The

measure of the "distance" between any two violins within the same fil ter

band (frequency region) was simply taken a s the absolute difference be-

tween the two corresponding LTAS values. Thus, the total input became

a distance ma t r ix for a l l pairwise combinations of the 22 violins for each

of the filter bands (i. e. , one ma t r ix for each fi l ter band). The expected

solution is a configuration of the violins in a space of certain dimensiona-

l i ty and weights for each fi l ter band within each dimension. By noting

which f i l ter bands obtain the highest weights for the respective dimen-

sions, the dimensions may be interpreted in t e r m s of frequency regions.

The main features of a five dimensional solution, accounting for 44 70 of the variance, a r e given in Fig. 111-A-7. There is an apparent s im-

i lar i ty between this solution and that obtained f rom FA on the 22 violins

(Fig. 111-AX,) a s r ega rds the f i r s t four dimensions. Dimension I (D I )

-

0 ,

10 20 BARK

0 10 20 BARK

Fig. III-A-6. Examples of ex t remes for single factor, maximum factor score ( -), minimum factor score ( . . . . ). (a) factor 1 (diss imilar i ty 57) , (b) factor 2 (diss imilar i ty 56), and (c) factor 3 (diss imilar i ty 50).

20 BARK

Fig . 111-A-7. Graphical display of weights in INDSCAL analysis with- in each of dimensions i - 5 (ver t ical) for different f i l ter bands (horizontal). The curve fo r each dimension r ep re sen t s the s ize of the weights for the different f i l ter bands. Shadowed p a r t s indicate weights s 0. 50. -

STL-QPSR 2-3/1976 29.

TABLE 111-A -11. Produc t moment correla t ion between tonal quality . ra t ings and calculated s co re s using weight function 0 ( the difference function of Fig. 111-A-2) and the weight functions 1 - 6 of Fig. 111-A-3.

Weight function

0

I

2

3

4

5

6

Correla t ion coefficient

. 8 0

. 7 8

. 6 8

. 6 7

. 6 7

. 6 7

. 7 8

TABLE 111-A-111. Correlation between tonal quality ra t ings and factor s c o r e s of violins in five fac tors given in Fig. 111-A-4.

Factor

I

2

3

4

5

Correla t ion coefficient

.08

. 28

. 4 0

. 0 7

-. 69

TABLE 111-A-IV. Correla t ion tonal quality ra t ings and positions in each of the five dimensions of the Indscal analysis given in Fig. 111-A-7.

Dimension

1

2

3

4

Correla t ion coefficient

. 3 1

. 3 6

-. 5 2

-. 16

STL-QPSR 2-3/1976

correlation of the five dimensions with the quality-ratings was 0.83 cor-

responding to aboat 69 70 variance accounted for.

I t may be noted that the correlat ions and proportions variance ac-

counted for given h e r e a r e somewhat higher than those obtained when

correlating the quality-ratings with scores for violins computed ac-

cording to a cer tain weight function (Table 111-A-11). These la t te r s co res

r e fe r red to the whole frequency region, while each of the factor s/dimen-

sions i n the FA/MDS mainly r e fe r red to cer tain l imited frequency regions.

In this way i t might be possible to get information about which frequency

regions a r e most important for evaluations of tonal quality o r other per -

ceptual aspects . However, there a r e certain discrepancies between cor-

responding correlat ions in Table 111-A-I11 (for FA) and 111-A-IV (for MDS),

which implies that the resu l t s should be regarded with caution. The dis-

crepancies a r e , of course, a consequence of cer tain differences between I

the FA and MDS analysis of the LTAS:es (compare Figs. 111-A-4 and

111-A-7). A s noted ear l ie r the MDS (INDSCAL) solution seemed somewhat

dubious . I Correlation between separate frequency regions and tonal quality-ratings

Another alternative to study which frequency regions may be important

for tonal quality i s , of course, to look a t the correlation between the LTAS

values of the violins in the different f i l te rs and their quality-rating scores.

These correlations appear in Table 111-A-V. Moderately high correla-

tions (0.41-0. 56) occur for fi l ter No. 3, 11, 13, 15, 19, 21, and 22. The

multiple correlation of fi l ter No. 19, 11, 15, and 13 with the quality-rat-

ings i s 0. 84, corresponding to about 7 1 7'0 variance accounted for . This

is s imilar to the multiple correlation of F5 , F3 , and F 2 with the quality-

ratings ( see above). In fact, these four f i l ter bands had high factor load-

ings i n any one or two of F5 , F3, and F 2 - f i l te r No. 19 in F 2 and F5,

fi l ter No. 11 in F5 , f i l ter band 15 in F 5 (and F4), and fi l ter No. 13 in F3.

Including a lso f i l te r No. 3, 21, and 22 (with high factor loadings i n Fi and

F2) r a i s e s the multiple correlation to 0.9 2 (about 84 7'0 variance accounted

for).

Discussion and conclusions

In the introduction the main questions behind this investigation were

presented. Do suitable methods exist for analyzing the LTAS:es of played

STL-QPSR 2-3/1976 32.

scales on violins ? What information on the tonal quality can be extracted

by means of this method? Four methods have been tested: weight functions,

factor analysis (FA), multidimensional scaling (MDS) , and separate corre-

lation analysis. But before answering the questions, let us summarize - the independent experimental findings.

The LTAS:es of the scales played on different instruments display con-

siderably larger dissimilarities among themselves than LTAS:es from dif-

ferent playings on the same instrument. I I

Factor analysis and multidimensional scaling resulted in fairly sim-

i lar solutions. Large par ts of the variances of the LTAS:es a r e accounted

for by five factors/dimensions (76 % and 44 % for the FA and MDS solu-

tions, respectively).

A score calculated by means of a weight function taking favorable and

unfavorable frequency regions into account gives a correlation with the I I

tonal quality ratings of 0.80, i. e. 64 70 of the variance in the tonal quality-

ratings a r e accounted for. The scores for violins derived from the FA and

the MDS in five factors/dimensions have approximately the same multiple

correlations with the tonal quality-ratings and account for about 74 O/o and

69 % of the variance, respectively. Similar resul ts a r e also obtained when

correlating the LTAS values in four to seven selected filter bands with the

tonal quality-ratings.

The weight function for calculating the tonal qualities contains weights

a t 2-5 and 10-22 Bark. The 4-5 factor s/dimensions represent in large a

splitting up of the weight function into 4-5 separate frequency regions.

The average dissimilarities between instruments of the highest and

the lowest quality ratings (Fig. 111-A-2) suggest that i t i s favorable with

"weak" frequency components around 10 Bark and above 17 Bark while

"strong" frequency components a r e favorable a t 2-5 and I I- I6 Bark. The

same tendency may be seen in the correlations between different fre-

quency regions and tonal quality-ratings (Table 111-A-V). I t i s also in-

dicated in the resul ts from the FA (Fig. 111-A-4 and Table 111-A-111 com-

bined). "Weak" frequency components around 10 Bark and above 16 Bark

a r e suggested favorable by the correlations of F 2 and F5 with the quality-

ratings (note the sign of the correlations) and " strongu frequency com-

ponents a t 2-6 Bark and 12-14 Bark by the correlation of F3 with the

quality-ratings. The INDSCAL results (Fig. 111-A-7 and Table 111-A-IV

STL-QPSR 2-3/1976 34.

In further r e sea rch there i s an urgent need of getting reliable data on

the "overall" tonal quality a s well a s on m o r e specific perceptual var iables

relevant for judgments about violins. A number of possibly relevant va r -

iables i s suggested in a recent study on adjectives commonly used by vio-

l inis ts and violin maker s to descr ibe the tonal quality of violins ( Gabriels-

son and Jansson, 1976).

Acknowledgments

Without the help of the Scandinavian Violin Makers ' Association and

the permission granted by i t s members this investigation would have been

impossible. Fur thermore , we want to thank L a r s F r y d t n for his ass i s t -

ance by playing all the recorded scales.

This work has been supported by the Swedish Humanistic Research

Council and the Swedish Natural Science Research Council.

References

CARROLL, J . D. and CHANG, J . J . (1970): "Analysis of individual dif- fe rences in multidimensional scaling via an N-way generalization of 'Eckert-Young' decomposition", Psychometrika, - 35, pp. 283-319.

DIXON, W. J . (ed. ) (197 1): Biomedical Computer P r o g r a m s , University of California P r e s s .

DIXON, W. J. (ed. ) ( 1972): Biomedical Computer P rograms . X-Series Supplement, University of California P r e s s .

GABRIELSSON, A. and JANSSON, E. ( 1976): "En sammanstallning av adjektiv karakter i serande upplevd klang frdn violinerr ' , CTM Report 4 (Dept. of Speech Communication, KTH).

GORSU CH, R. L. (1974): Factor Analysis, W. B. Saunders Co. , Phila- delphia.

HARMAN, H. H. ( 1967): Modern Fac tor Analysis ( rev . ed. ), University of Chicago P r e s s .

JANSSON, E . V. ( 1976): "Long-Time-Average-Spectra applied to analysis of music. P a r t 111: A simple method for surveyable analysis of complex sound sources by means of a reverberat ion chimberw, ~ c u s t i c a , 34; pp. - 275-280,

SLOJD OCH TON (1975); NO. 3-4.

TORGERSON, W. S. (1958): Theory and Methods of Scaling, John Wiley, New York.