This is an updated copy of a question I asked on Physics Stack Exchange not too long ago. Since I work primarily in mathematics, I thought it would be a good idea to ask it here as well (especially after being inspired by the popularity of this question).
As part of some work I've been doing in signal processing and Fourier analysis , I recently ran into a couple of questions regarding "octave equivalence," the psycho-acoustical sensation that frequencies differing from each other by a power of 2 are somehow "the same." I'd like to incorporate this into a construction I'm doing in my work, although I'd prefer it to not be ad hoc or even mostly biological; that is, I'd like some kind of mathematical statement from which I could "read off" the phenomenon of octave equivalence in the same way that one can "read off" the phenomenon of beats from (if I recall correctly), the double angle formula.
There doesn't seem to be too much literature on this subject, but here is what I've found so far. Apparently, the cochlea in our ear does something like a windowed Fourier transform. After extracting the frequency content from an auditory signal, the brain infers the fundamental from the partials available. This is why when actually hearing frequencies of, say, 400, 500, ..., 900 Hz, we may perceive a fundamental of 100 Hz. So, according to this logic, the phenomenon of octave equivalence arises because the frequencies $f$ and $2f$ share so many partials within the given window, in fact, more than with any other frequency $nf$ for $n>2$.
Is this sufficient? I'm not sure it is. For example, I was playing around on Maple with pure sine waves, and it seems as if one's ability to perceive pitch affinities, as they're called, is significantly dulled when complex signals are not used. I'm not sure why this is the case.
Also, to anticipate people's statements, I do know that these matters have a cultural component as well. However, from what I've read, octave equivalence has been detected in rats and human infants, making it seem at least somewhat universal. What is done with this phenomenon, of course, is completely subject to the particularities of one's place int the world.
If someone could point me to anything in the mathematical physics literature on this topic, I'd be very grateful.