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On pitch and periodicity (was "correction to post")
At 5:55 PM +0300 8/2/11, ita katz wrote:
The periodicity is determined by the least-common-multiple of the
periodicities of the present harmonics, so if (for example) a sound
is composed of sines of frequencies 200Hz, 300Hz, and 400Hz, the
periods are 5msec, 3 1/3msec, and 2.5msec, so the
least-common-multiple is 10msec (2 periods of 5msec, 3 periods of
3.33msec, and 4 periods of 2.5msec), which is of course the
periodicity of the sum of the sines, or in other words 100Hz.
(actually it is the same as the greatest-common-divisor of the
Ita, that explanation is sort of OK, but as written implies that the
auditory system has the ability to do number-theory operations on
periods (or frequencies), and depends on there being harmonics
present and separately measureable.
It would be much more robust to say that "The pitch is determined
based on an approximately common periodicity of outputs of the
cochlea," which I believe is consistent with your intent.
Why is this better? First, it doesn't say the periodicity is
determined; what is determined is the pitch (even that is a bit of
stretch, but let's go with it). Second, it doesn't depend on whether
the signal is periodic, that is, whether harmonics exist. Third, it
doesn't depend on being able to isolate and separately characterize
components, harmonic or otherwise. Fourth, it doesn't need
"multiples" (or divisors), but relies on the property of periodicity
that a signal with a given period is also periodic at multiples of
that period, so it only needs to look for "common"
periodicities--which doesn't require any arithmetic, just simple
neural circuits. Fifth, it admits approximation, so that things like
"the strike note of a chime" and noise-based pitch can be
accommodated. Sixth, it recognizes that the cochlea has a role in
pitch perception. It's still not complete or perfect, but I think
presents a better picture of how it actually works, in a form that
can be realistically modeled.
Is this "tortured use of existing signal processing techniques" as
Randy puts it? I don't think so. Is it "a unique way to do
frequency analysis and to meet the dictum in biology that 'form
follows function'"? Sure, why not? But why call it "frequency
analysis"? How about "a unique way to do sound analysis" (if by
"unique" we mean common to many animals)?
I do have some sympathy for Randy's concern that we are far from a
complete understanding, and that hearing aids are not as good as they
would be if we understood better, but yes, he sounds way too harsh in
overblowing it so. I'm wondering what's behind that, and whether
it's just confusion about all the confusing literature on pitch
perception, which I agree is a complicated mess -- or is the problem,
indicated by Randy's previous posts, just that he doesn't understand
basic linear systems and signal processing, and that's why it all