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Pitch of Pure Tones Measured by Absolute Magnitude Estimation

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Pitch of Pure Tones Measured by Absolute Magnitude Estimation Pitch of Pure Tones Measured by Absolute Magnitude Estimation Andrzej Rakowski and Andrzej Mi4kiewicz Laboratory of Music Acoustics, Chopin Academy of Music, Ok61nik 2, 00-368 Warsaw, Poland Abstract: Pitch of pure tones was measured by absolute magnitude estimation. Results show that the function relating pitch to frequency in log-log coordinates is much steeper below about 250 Hz than at higher frequencies. me results also show that in part of the scale pitch is proportional to the cochlear frequency coordinates. INTRODUCTION The psychological scale of pitch was investigated by a number of authors with the use of various methods. Stevens’ mel scale was derived from bisection and fractionation (7, 8). Other studies employed equisection (3), eqd- appearing intervals (4), haIving and doubling (9), magnitude estimation (5, 6) and magnitude estimation with designated standmd (1). Recently, some criticism has been raised towards the validity and interpretation of the me] scale. In particular, G~enwood (3) argud that the mel scale was biased by a “hysteresis” effect. Rakowski (5) pointed at fundamental controversies in the interpretation of the psychological and musical scales of pitch by leadlng psychologists. The aim of the present study is to re-examine the pitch scale using the method of absolute magnitude estimation (10). PROCEDURE Pitch of 27 pure tones covering a frequency range from 31.5 Hz to 12.5 kHz in l/3-octave steps was measurd by absolute magnitude estimation. The tones were generatd by a Tucker-Davis System ~ and presented monaurally through a Beyer DT 911 earphone at a loudness level of 60 phons. Tone duration was 1500 ms with a 20-ms, squared-cosine rise and fall. Earphone calibration was 103.8 dB SPL for a IV input at 1 kHz. Tone levels corresponding to a 60-phon loudness level at each of the test frequencies were detetined in a preliminary experiment, by a two-interval, two-alternative forced choice, one-dowtione-up, Ioudness-bdance adaptive procedure. Twenty music students with normal hearing, 12 men and 8 women, agd 18–26 years served as listeners in the main experiment. None of them had absolute pitch. The listeners were tested individually in a sound-proof booth, They were required to assign a number to the pitch of the tone so that the subjective magnitudes of the two continua, pitch and number, appeared equal (10). The listener activated a single presentation of the tone by pressing a button on the response box and could listen to the tone at will before reporting the number through an intercom to the experimenter. Each listener completed one series of judgments of 27 frequencies presented in random order. RESULTS AND DISCUSSION The left panel of Fig. 1 shows the geometric means of 20 judgments as a function of tone frequency. The data m plotted in Iog-log coordinates. For comparison, the graph also includes the me] scale (8), and earlier data obtained for pure tones (6) and l/3-octave-band noises (5). The present results show that the pitch function is much steeper below about 250 Hz than at higher frequencies. This finding is in agreement with earlier data reported by Rakowski (5). Similarly, Stevens and Galanter’s pitch function (6) is concave downwards at low frequencies, however, that study did not extend below 100 Hz. In contrast, the mel scale does not exhibit such a pronounced differenm in steepness between the low and higher frequency ranges. The right panel of Fig. 1 presents the pitch estimates as a function of cochlear frequency coordinates proposed by Greenwood (2). The graph shows that the relation of pitch to cochlear distance is close to linear in the greater part of the basilw membrane’s length, except in its basal part, associated with frequencies higher than 5000 Hz, 161 5000 0- 300 1000 10 1 ❑ ❑ [,,1,,,,,,,,,,,,,,,,,,,,,< 0 l,, ,,1, ,m, l,, ,ml. ... i,, ,al, ,.1 20 100 1000 10000 0 10 20 30 Frequency (Hz) Distance from apex (mm) FIG~E 1. Left panel: Relation of pitch to frequency in log-log coordinates. Plotted are the geometric means of estimates obtained in the present study (circles), Stevens’ (8) mel scale (line), Stevens and Galanter’s (6) resul[s (triangles), and Rskowski’s (5) data (squares) obtained with stimuli of low pitch strength (1/3-octave bands of noise). Right panel: Pitch (present results) as a function of cochlear distance (2) plotted in linear coordinates. A straight line has been fitted to the data obtained in a frequency range from 31.5 to 5000 Hz. CONCLUSIONS me discrepancies between pitch functions obtained by various methods suggest that pitch scaling is particularly susceptible to prwedurd effects. me reason for that seems to be that the sensation of relative pitch includes two separate sub-dimensions: the continuous scale of pitch distance and the categorical scale of musical intervals. me fixation of the latter scale influences listeners’ judgments of pitch distance. me effect of the musical-interval scale can be minimized by constructing the pitch scale from direct estimates of pitch magnitude (pitch value), rather than from judgments of pitch distance. me present data are generally in agreement with the hypothesis that pure-tone pitch differences represent eqd cochlear distances (8, 9). me change of the pitch function steepness reported here represents the transition of the cochlear frequency mapping from linear to logarithmic (2, 9). However, further research is nded to explain why at frequencies above 5~ Hz the relation of pitch to cochlear distance does not follow the linear function. REFERENCES 1. Beck, J, and Shaw, W.A. “Magnitude estimations of pitch,” J. Acoust. Sot. Am. 34, 92-98 (1962). 2. Greenwood, D. D. “Critical bandwidth and the frequency coordinates of the basilar membrane,” J. Acoust. Sot, Am. 33, 1344-1356 (1961). 3. Greenwood, D.D. “me mel scale’s disqualifying bias and a consistency of pitchdifference equisections in 1956 with equal cochlear distances and equal frequency ratios,” Hearing Res. 103, 199–224 (1997). 4. Harris, J,D. “Scaling of pitch intervals,” J. Acousr. Sot. Am. 32, 1575-1581 (1960). 5. Rakowski, A. “Musicti and psychological scales of pitch,” Proceedings of the 13th Annual Meeting of the International Socie~ for Psychophysics, Poznafi, Poland, 91-96. 6. Stevens, S.S. and Galanter, E.H. “Ratio scales and category scales for a dozen perceptual continua,” J. Exp. Psycho/. 54, 377-411 (1957). 7. Stevens, S. S., Volkman, J. and Newman, E,B. “A scale for the measurement of the psychological magnitude pitch,” J. Acoust. Sot. Am. 8, 185-190 (1937). 8. Stevens, S. S., Volkman, J. ‘The relation of pitch to frequency: A revised scale,” Am. J. Psychol. 53, 329-353 (1940). 9. Zwicker, E., Feldtkeller, R. Das Ohr als Nachrichtenempfdnger, Hirzel, Stuttgart (1957). 10. Zwislocki, J.J. and Goodman, D.A. “Absolute scaling and sensory magnitudes: A validation. Percept, Psych ophys. 28, 28-38 (1990). 162.
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