Music is comprised of sound waves. Waves can be modeled mathematically using a Fourier series. So, could we then write music in terms of mathematics by using a Fourier series to represent the sound waves? Would it be worthwhile to do so? Would it have any benefit over traditional music notation?

Music is composed of sound waves, but that's only part of the story. A Fourier analysis would run the risk of including too much information, overlooking the fact that there can be very different interpretations of the same piece.

It might be possible to fudge that point. (After all, compression schemes rely on Fourier analysis, but don't include every detail.) The real point is that to be useful, musical notation has to be something that musicians can efficiently read. To make the case that Fourier analysis would be useful for that purpose would be a pretty heavy lift, I suspect. Compare: it would be possible to give a Fourier analysis of a speech in a play, at least as performed by some actor. But I'd be willing to bet that any actor would find it a lot easier just to have the words written down. Musical notation, like written words and dance notation, is able to do its job precisely because it vastly, massively underspecifies the full detail of what any performance will actually be like. The point here isn't just about leaving room for interpretation. It's about what we can reasonably expect a human brain to process.

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