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Re: Pitching in

Dear Greg,

Thanks for your words of support.

>Has anybody of recent ilk tried to defend the mel scale?

Not that I know of, off hand. But let me briefly return to the
original question:

>Sometimes they say pitch is proportional to the logarithm of
frequency. What is true? What is not?

and add some words of explanation to my previous letter, in which I
claimed that:

> For complex tones, it is generally safe to assume that pitch is
> proportional to the logarithm of frequency ... over a wider
> range of frequencies than for pure tones.


The pitch of pure tones (as reflected by critical-band rate --
Zwicker & Terhardt, 1980 or ERB rate -- Moore & Glasberg, 1983) is
roughly proportional to frequency for frequencies greater than about
500 Hz. The (main virtual) pitch of a harmonic complex tone is
determined by certain "dominant" harmonics (Plomp, 1967). Ritsma
(1967) found that the pitch of a harmonic complex tone with all low
harmonics and a fundamental frequency in the range 100 to 400 Hz
depends only (or primarily) on the frequencies of the 3rd, 4th, and
5th harmonics. Assuming that the pitch of harmonics WITHIN a complex
tone is proportional to log frequency down to about 500 Hz, it
follows that the pitch of typical complex tones is roughly
proportional to frequency down to about 100 Hz (NB: ballpark estimate

The above argument seems to be consistent with musical experience. It
is no surprise to musicians that "psychoacoustical complex-tone
pitch" (if I may call it that) is roughly proportional to musical
pitch in semitones above about G2 (that's the bottom line of the bass
clef). Below that, melodies start to feel a bit cramped.

And while on the subject of music -- It is difficult to scale pitch
(and especially complex-tone pitch) in the way that Stevens did, due
to the confounding effect of musical experience (subjects are
reminded of musical intervals). That's why I prefer to rely on
round-about arguments such as those of the preceding paragraph. The
aim of the exercise is to give musicians (and others) a sensible
answer to a reasonable question, even if the reasoning is not
absolutely watertight.

Does that make sense? I'd be interested in feedback.

(NB: The following is probably of pedagogical rather than research

I think that communication about the perception of complex tones
could be rendered more transparent by more careful use of the word
"pitch". We often refer loosely to tone sensations as "pitches" or
even "timbres". But in a strict psychoacoustical definition, pitch is
not a "thing in itself" (kein Ding an sich, right?) but an ATTRIBUTE
of a thing, namely, of a tone or tone sensation. Take for example the
following quite innocent (and true) sentence from Punita's letter:

> Depending on the relationship between the components, a single

> unified pitch or multiple candidate pitches may be evoked.

Perhaps the following version would be more clear to "non-pitchers"?

Depending on the relationship between the components, either a single
tone sensation (whose pitch may be ambiguous) or multiple tone
sensations may be evoked.

The reason I'm mentioning this is (as well as the above material on
pitch scaling) that I happen to have a vested interest in applying
research in complex-tone perception to music theory. So I want to
make things as clear as possible to a music-theoretic audience.
Again, suggestions are welcome.

Richard Parncutt

P.S. I was wrong: The refs to Henning (1966) and Wier et al. (1977)
in my last epistle concerned pure, not complex tones.


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.

Ritsma (1967). Frequencies dominant in the perception of the pitch of
complex tones. JASA, 42, 191-198.

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,