GENERAL COMMENTS and CONCLUSIONS

The computer program which was written to convert the raw output of the subject results into a form appropriate to input to ANOVA also kept track of the responses for the comparison of identical sounds. There were 32 such pairs for each person, and ideally there would be roughly half ``maybe higher'' and half ``maybe lower'' responses. A heavier weighting toward higher or lower responses would indicate a bias toward one choice in the results. We include these data in Table 5.

It is interesting that the string players have a high number of ``definitely higher'' and ``definitely lower'' responses for these identical sounds. Since the other results from this group were slightly better than those of the MIT group, this is puzzling. Their earlier results show clearly that they perceive extremely small differences accurately, but they seem unable to recognize (or possibly to admit to) the situations in which certainty is not possible.

Although the principal goal of this study was the determination of the pitch center of musical notes played with vibrato, the simultaneous control experiment comparing unmodulated sounds provides an estimate of the JND for frequency for these subjects. From signal detection theory the 76%correct point on the psychometric curve corresponds to d prime = 1 (Moore, 1989), and the frequency separation at this point is an estimate of the JND. With our scale from 0 to 1 these would be the points .12 and .88 on the psychometric curves. Averaging over the values at these two points, we found 6 cents for the MIT group and 4.5 cents for the NEC group with an upper bound on the error of 2 cents. These values are slightly higher than previous values of 3.5 - 4 cents for pure tones summarized by Moore (1989), although they lie within our error range.

It is interesting to compare the JND to the control of a performer in repeating notes with the same frequency. The average of standard deviations of notes in Table 3 with respect to the same note (unmodulated) is 4.2 3.9 cents. This is almost identical to the average of the standard deviations for each of the individual notes (i.e. values taken from Table 2). Thus it is possible that ultimate control in intonation is limited by pitch perception rather than motor control in the production of the sound.

All of the data which were taken support the hypothesis that the mean frequency of a modulated sound is perceived when a subject is asked to compare it to an unmodulated sound. Further this modulated sound gives rise to the same psychometric curve as that of single frequency sound at its mean, i.e. for purposes of comparisons with a second sound it is equivalent to a single frequency sound having its mean frequency. Equivalently we can say that a human functioning as a frequency meter performs identically on an unmodulated sound and the mean of a frequency modulated sound.

ACKNOWLEDGEMENTS

A number of people at MIT were extremely generous with their time and effort and are responsable for making this study possible. First and foremost we are indebted to the violist for recording the sounds and for his participation in the listening experiment. We are very grateful to Adam Lindsay for running the listening test on most of the subjects. Without his time, effort, and expertise this work would not have been possible. Finally we would like to thank Alan Delapinasse, Dan Ellis, Adam Lindsay, Stephen Gilbert, and Barry Vercoe for giving their time as subjects in Experiment 2.

JCB would like to thank Larry Baldwin for helpful discussions on statistics and Daniel P.W. Ellis for the resampling software. She is also grateful to Caroline Palmer for comments on the initial design of the experiment and to Douglas Keefe for pointing out the problem with using ANOVA directly on the data. Susanne Winsberg was an invaluable source of information on logistic regression during a Sabbatical spent by JCB at IRCAM. None of these people is in any way responsible for any errors which might appear in the statistical analysis. Bill Hartmann and Steve McAdams were kind enough to read and make some extremely helpful suggestions on an earlier version of the manuscript. Finally JCB is very grateful to Wellesley College for Sabbatical leave support and for a grant for a portion of the payment to the subjects of Experiment 3.

1

REFERENCES

Aldrich, J.H., and Nelson, F.D. (1984). Linear Probablilty, Logit, and Probit Models Sage Publications, London.

Beauchamp, J.W. (1974). ``Time-variant spectra of violin tones," J.Acoust.Soc.Am. 56, 995-1004.

Brown, J.C. and Vaughn K.V. (1993). ``Pitch center of musical sounds with vibrato", J. Acoust. Soc. Am. 94, 1860.

d'Alessandro, C. and Castellengo, M. (1994). "The pitch of short-duration vibrato tones", J.Acoust.Soc.Am 95, 1617-1630.

Fletcher, H., Blackham, E.D. and Geertsen, O.N. (1965). ``Quality of Violin, Viola, Cello and Bass-Viol Tones," J. Acoust. Soc. Am. 37, 851-63.

Fletcher, H. and Sanders, L.C. (1967). ``Quality of Violin Vibrato Tones", J. Acoust. Soc. Am. 41, 1534-44.

Hake, H.W. and Rodwan, A.S. (1966). ``Perception and Recognition" in Methods and Instrumentation in Psychology Ed: J.B. Sidowski, McGraw Hill, New York.

Hirose, K. (1934). ``An experimental study on the principal pitch in the vibrato," Japan J. Psychol. 9, 793-845. (in Japanese)

Iwamiya, S., K. Kosygi, and O. Kitamura (1983). ``Perceived pitch of vibrato tones" J. Acoust. Soc. Japan 4, 73-82.

Iwamiya, S, and Fujiwara, K. (1985). "Perceived principal pitch of FM-AM tones as a function of the phase difference between frequency modulation and amplitude modulation" J. Acoust. Soc. Japan 6, 193-202.

Iwamiya, S, Miyakura, T. and Satoh,N (1989). ``Perceived Pitch of Complex FM-AM Tones" Proceedings of the First International Conference on Music Perception and Cognition, Kyota, Japan, 431-436.

Moore, B.C.J. (1989). An Introduction to the Psychology of Hearing, Academic Press, London, 135, 160.

Seashore, C.E. (1938). Psychology of Music McGraw Hill, New York pp. 20, 33-52.

Shonle, J.I. and K.E. Horan (1980). ``The pitch of vibrato tones," J. Acoust. Soc. Am. 67, 246-252.

Sundberg, J. (1978). ``Effects of the vibrato and the singing formant on pitch", Musicologica Slovatca 6, 51-69.

Tiffin, J. (1931). ``Some aspects of the psychophysics of the vibrato," Psychol. Rev. Monogr. 41 (4), 1153-200.

Ward, W.D. (1970). "Musical perception," in Foundations of Modern Auditory Theory, edited by J.V. Tobias (Academic Press, New York), 407-447.

FIGURES

Figure 1 Frequency vs time for the unmodulated notes , #, , and . Figure 2 Frequency vs time for frequency modulated notes , #, , and .

Figure 3 Log p/(1-p) as responses go from 8 ``maybe higher'' to 7 ``definitely higher'' and 1 ``maybe higher''. Figure 4. Fraction of responses higher plotted against target pitch where 0 corresponds to the mean of the vibrato for the most accurate MIT subject. Figure 5. Fraction of responses higher plotted against target pitch where 0 corresponds to the mean of the vibrato for MIT subjects. The average peak to peak amplitude is included below the curve for comparison. Figure 6. Logistic transformation of data in Figure 5. Figure 7. Average of all responses for a given note plotted against note given as midinote. Figure 8. Fraction of responses higher plotted against target pitch where 0 corresponds to the mean of the vibrato for string players. See Figure 5 for a comparison to the average peak to peak amplitude of the vibrato. Figure 9. Logistic transformation of data in Figure 8. Figure 10. Fraction of responses higher plotted against target pitch where 0 corresponds to the mean of the vibrato for the performer discussed in Experiment 1 and Experiment 4. TABLES

Table 1 Summary of previous work. The conclusion column indicates the position of the pitch center of the vibrato. Relative terms denote pitch height with respect to the mean of the vibrato.

Table 2 Properties of sounds used in the study. Columns 6-16 give the intensity in dB for harmonics with number corresponding to the labeling of the column.

Table 3 Summary of frequency tracking results on viola sounds. All frequencies are given in cents with respect to A 440.

Table 4 Pitch level corresponding to 50 %chosen higher. Columns 1 and 2 are the pitch level in cents for vibrato and non vibrato, and column 3 is their difference.

Table 5 Responses for comparison of same sound

Table of position of 50 vib nv v - nv

MIT .5 1 -.5 (wo kvv2)

afdl 1.5 .5 1 kvv -1 1 -2 dpwe 0 .5 -.5 sag -1 1 -2 bv 2 3 -1 atl 0 1 -1 kvv2 -.5 -1 .5

NEC or Prof Strings

kle 1 -1 2 fu 4 4 0 stet 1.5 2.5 -1 hag 0 -.5 .5

edP -.5 1.5 -2 mar .5 2.5 -1.5

from execu.hist.calc 2/14/94 -L10 is too large; 7 2's and 1 1 gives 4.143135 6 2's 2 1's gives 2.468100 diff -2.468100 + 4.143135 = 1.675035 (4.143135 + 1.675035) = 5.81817 which is about what it extrapolates to

We have not included all the individual

curves for each subject, but have summarized the data in Table 3. This is to show that the overall curves do not represent an average of individuals with very different perceptions? .

Note & Midi & Frequency(Hz) C_5# & 9.01 & 554.4 G_6 & 10.07 & 1568.0 ********from sf/sound/marcus/execu.hist.marcus.sndfiles has all the observations that are summarized in the Table They seem to be recorded in the order they appear except for d4.open etc which was done last but is put in the table parallel with a440

form ``old'' these were resampled 94 -rw-r-r- 1 brown 95904 Apr 12 1993 A5 from g5 136 -rw-r-r- 1 brown 125104 Apr 12 1993 Cs5 from ef5

184 -rw-r-r- 1 brown 176762 Apr 12 1993 Cs5

>>>>>>>>> from vibper/execu.hist.vibper sf/marcus/g5.nv g5.v exp/A5.aif exp/A5.v.aif

la/vibper/execu.hist 4/26/93 Amp(est by eye) straight-vib d4.v 15 ave in cents = -716.669 Std dev = 10.252 15 d4 ave in cents = -702.133 Std dev = 0.764

A5.v 27 ave in cents = 1194.282 Std dev = 18.035 3 A5 ave in cents = 1197.838 Std dev = 0.823

Cs5.v 25 ave in cents = 382.031 Std dev = 14.747 10 Cs5 ave in cents = 391.861 Std dev = 2.236 Cs5.2 ave in cents = 390.486 Std dev = 1.321

g6.v 15 ave in cents = 2209.072 Std dev = 11.436 6 g6.v.2 ave in cents = 2208.394 Std dev = 11.292 g6 ave in cents = 2215.030 Std dev = 2.627 g6.2 ave in cents = 2213.918 Std dev = 2.042

ORIGS (ident to p0) A5 A5.v Cs5 Cs5.v d4 d4.v g6 g6.v.2

lengths 93 -rw-r-r- 1 brown 94972 Apr 12 13:40 A5.aif 1.076 s 136 -rw-r-r- 1 brown 126774 Apr 12 13:40 A5.v.aif 136 -rw-r-r- 1 brown * 124172 Apr 12 13:40 Cs5.aif 160 -rw-r-r- 1 brown 154596 Apr 12 13:40 Cs5.v.aif 120 -rw-r-r- 1 brown 114510 Apr 12 13:40 d4.aif 120 -rw-r-r- 1 brown 112274 Apr 12 13:40 d4.v.aif 176 -rw-r-r- 1 brown * 168554 Apr 12 13:40 g6.aif 1.910 s 152 -rw-r-r- 1 brown 146658 Apr 12 13:40 g6.v.aif

>>>>>>>>

+++++++++/mas/disks/mc1/brown/vibper.txt<478> more execu.hist.vibper.txt 2/10/94 older stuff on this

8/15/94 better later

mechanism for musical sounds better perception - constantly changing and human perceptual apparatus very sensitive to changes.

natural sounds all constantly changing

2/26/93 (before)

execu.hist.fft try the A open string res -r 11025 D_4C_5A_5G_6D_4C_5A_5G_6D_4C_5A_5G_6 9500

Any comments on the following comments of Steve McAdams will be appreciated.

p. 4 l. 3-4 : what do you mean by "solid rational basis"? I think this statement needs justifying. I think adjustment methods are just fine!

p. 16, l. 14 : "upper bound" - I don't understand this formulation.