[Math] The mathematics of music – why sine waves

Question

Of course, the Fourier transform is an extremely elegant mathematical method of overwhelming simplicity, and this straight away puts sine waves (or complex exponentials) on a high pedestal.

But what if, instead, we started with solving the wave equation for a plucked string, which through d'Alembert's solution gives a nondifferentiable wave (inherently a more complicated object), and based music theory (and harmonics, etc) on that?

We could just as well pick some other basis of $L^2(\mathbb{R})$ instead of $\sin(nx)$, $\cos(nx)$, such as the above waves describing the motion of a plucked string at various frequencies. What effect would this have on the mathematics of harmony and resonance in music, if any? And what about completely different bases all together; does the standard $\sin(nx)$, $\cos(nx)$ basis have any musical importance, rather than just mathematical simplicity? Is the name "pure tone" really deserved? Could a spectrogram based on a decomposition into motions of plucked strings provide a different angle than just a Fourier transform decomposition?

Edit: A similar question arises when considering the vibrational modes of a circular membrane, see e.g. this Wikipedia article. This allows decomposition of any motion as a superposition of particular Bessel functions, which are no doubt chosen for mathematical simplicity.

Answer 1

It sounds like you really want to read Benson's Music: a Mathematical Offering (freely available at the link). I'm not completely sure what you're asking, but if it's anything like "why is it natural to think about music in terms of sine waves," this question is addressed right in the introduction:

...what's so special about sine waves anyway, that we consider them to be the "pure" sound of a given frequency? Could we take some other periodically varying wave and define it to be the pure sound of this frequency?

The answer to this has to do with the way the human ear works. First, the mathematical property of a pure sine wave that's relevant is that it is the general solution to the second order differential equation for simple harmonic motion. Any object that is subject to a returning force proportional to its displacement from a given location vibrates as a sine wave. The frequency is determined by the constant of proportionality. The basilar membrane inside the cochlea in the ear is elastic, so any given point can be described by this second order differential equation, with a constant of proportionality that depends on the location along the membrane.

The result is that the ear acts as a harmonic analyser [emphasis added]. If an incoming sound can be represented as a sum of certain sine waves, then the corresponding points on the basilar membrane will vibrate, and that will be translated into a stimulus sent to the brain.