Re: scales of pitch

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A little tutorial on pitch...

"Frequency = pitch" is a convenient way of measuring or marking
pitch: The pitch of a sound may be measured by the frequency of a
pure tone whose pitch is judged to be the same as the pitch of that
sound  (e.g., Fletcher, 1934; Plomp, 1967, 1976; Terhardt, 1972,
1974; Terhardt et al, 1982). But neither the frequency scale nor its
logarithm is generally proportional to perceived pitch.


Stevens and Volkman (1940) attempted to create an approximate,
proportional scale for the pitch for pure tones of constant loudness.
The "mel" scale is related to critical-band rate (Zwicker & Terhardt,
1980) and ERB rate (Moore & Glasberg, 1983), which in turn are
related to distance along the basilar membrane (Greenwood, 1961,
1990).


The mel scale and its relatives listed above apply only to pure
tones, and not (or at least not directly) to music or speech. For
complex tones, it is generally safe to assume that pitch is
proportional to the logarithm of frequency. This is true over a wider
range of frequencies for complex tones than for pure tones. The range
over which the JND for the pitch of complex tones is approximately
constant is roughly 100 to 5000 Hz (e.g., Henning, 1966; Wier et al.,
1977).

For a nice overview see Moore (1977).


References

Fletcher, H. (l934). Loudness, pitch  and timbre of musical tones and
their relations to the intensity, the frequency and the overtone
structure. JASA, 6, 59-69.

Greenwood, D.D. (1961). Critical bandwidth and the frequency
coordinates of the basilar membrane. Journal of the Acoustical
Society of America, 33, 1344-1356.


Greenwood, D. (1990). A cochlear frequency-position function for
several species -- 29 years later. JASA 87, 2592-2605.


Henning, G.B. (1966). Frequency discrimination of random-amplitude
tones. JASA, 39, 336-339.

Moore, B.C.J., & Glasberg, B.R. 1983. Suggested Formulae for
Calculating Auditory-Filter Bandwidths and Excitation Patterns.
Journal of the Acoustical Society of America 74: 750-753.

Plomp, R. (1967). Pitch of complex tones. Journal of the Acoustical
Society of America, 41, 1526-1533.

Plomp, R. (1976). Aspects of tone sensation: A psychophysical study.
London: Academic.

Terhardt, E. (1972). Zur Tonhoehenwahrnehmung von Klaengen. Acustica,
26, 173-199.

Terhardt, E. (1974). Pitch, consonance, and harmony. Journal of the
Acoustical Society of America, 55, 1061-1069.

Terhardt, E., Stoll, G., & Seewann, M. (1982a). Pitch of complex
tonal signals according to virtual pitch theory: Tests, examples and
predictions. Journal of the Acoustical Society of America, 71,
671-678.

Terhardt, E., Stoll, G., & Seewann, M. (1982b). Algorithm for
extraction of pitch and pitch salience from complex tonal signals.
Journal of the Acoustical Society of America, 71, 679-688.

Moore, B.C.J. (1977). An Introduction to the Psychology of Hearing,
3rd edition. Academic, London. Third edition 1989.

Stevens, S.S. and J. Volkman. 1940. "The Relation of Pitch to
Frequency." American Journal of Psychology 53: 329-353.

Wier, C.C., Jesteadt, W., & Green, D.M. (1977). Frequency
discrimination as a function of frequency and sensation level. JASA,
61, 178-184.

Zwicker, E. and E. Terhardt. 1980. "Analytical Expressions for
Critical-Band Rate and Critical Bandwidth as a Function of
Frequency." Journal of the Acoustical Society of America, 68,
1523-1525