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I have a vague question which I have trouble googling an answer to.

Let $X$ be a circle. I want to think of $L^2(X)$ as embedded into the space of periodic functions on the real line, with period 1, and as such each element of $L^2(X)$ gives rise to a "sound" (to define what sound precisely we have to fix units on x and y axes for time and pressure respectively).

Now, characters form a basis of $L^2(X)$, and the corresponding sounds are perceived as pure by humans. Obviously there are other bases of $L^2(X)$, for example all elements of a basis might correspond to some square waves.

But the corresponding sounds are not perceived as as pure. More extremely, in triangle waves I can distinctly hear several different tones, although it might be just my poor laptop speakers.

Thus human auditory system seems to have a preference for one specific orthogonal basis. I've had a look at the wikipedia's description of the human auditory system but it's quite complicated.

Question: Is it possible to pinpoint where does this preference for characters emerge in the human auditory system?

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    The human auditory system is analog, not digital. No Fourier or other orthogonal basis needed. – 2017-01-26
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    Sound is detected by vibration of the stereocilia, so the question comes down to why mechanical vibrations are sinusoidal and not square waves. – 2017-01-26
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    You might find the answer here: https://homepages.abdn.ac.uk/mth192/pages/html/maths-music.html – 2017-01-26
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    Thanks for the link Qiaochu, in fact this exact question is asked in that book (first mention is on p.X of the introduction, paragraph starting with "We begin the first chapter with...) Having said that I'm not entirely satisfied by the answer, but I think it's because of my lack intuition about biology and physics. – 2017-01-27
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    Btw, a paragraph in the Benson's book (p.14): The fact that these are the solutions of this differential equation is the explanation of why the sine wave, and not some other periodically oscillating wave, is the basis for harmonic analysis of periodic waves. For this is the differential equation governing the movement of any particular point on the basilar membrane in the cochlea, and hence governing the human perception of sound." In particular I don't understand the part starting with "and hence" – 2017-01-27
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    Ok, after reading wikipedia, the relevant bit of Benson's book and the article referenced by user275313, I'm fairly satisfied with my understanding. (Side remark about the article referenced by user275313: it would be so much easier if the original Helmholtz's hypothesis was true, because then it would be easy to "see in macroscale", using pretty much any instrument with more than 1 string, why sine waves are distinguished; whereas it's probably damn hard to build a macroscale model of cochlea...) Thanks to everyone who participated! – 2017-01-27

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It is my understanding that different frequency components resonate in different parts of the cochlea, and the activity is picked up by the hairs in the different locations and passed along to the brain. Basically, it's performing a spectral decomposition. But my understanding could be wrong.

Update (from Wikipedia): The basilar membrane of the inner ear spreads out different frequencies: high frequencies produce a large vibration at the end near the middle ear (the "base"), and low frequencies a large vibration at the distant end (the "apex"). Thus the ear performs a sort of frequency analysis, roughly similar to a Fourier transform. However, the nerve pulses delivered to the brain contain both rate-versus-place and fine temporal structure information, so the similarity is not strong.

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I don't know the mechanism, but it takes place in the cochlea. Check out this image from this article.