I almost totally agree with you. My intention wasn't to imply that the auditory system has some arithmetic-related capabilities --- I was just mentioning the simple arithmetic principles.
In other words (and relating to a different phenomenon), although the trajectory of a tennis ball flying through air can be modeled by some complex partial differential equations, I would not state that a person catching a flying tennis ball have some part of its brain actually integrating these equations.
The only question I wanted to ask, is why there are many references to the missing fundamental as being an auditory illusion, while the periodicity is not an illusion at all. I agree that saying that the auditory system percepts periodicity is oversimplifying and not 100% correct.
Thanks for your comments.
On Tue, Aug 2, 2011 at 8:49 PM, Richard F. Lyon <DickLyon@xxxxxxx>
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 frequencies).
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 seems "tortured"?