[dist-epoch]

There is a mathematical structure in the harmonics of a piano note. There is a different one in the harmonics of a sax note. Yes, it's not 100% structure, there is noise too.

This is why the "timbre" of a piano is different from the timbre of a sax, and why the same C note might fit in a song if it's played by a piano, but not in the same song if it's played by a sax.

If you take a higher level view, the mathematical structure of the notes (the chords, ...) needs to fit the mathematical structure of the harmonics of the instruments (the timbres) to get a pleasing sound. Put another way, the timbres of some pairings of instruments might clash on a particular sequence of notes, so you need to search for a different instrument pairing which is harmonious on that sequence.

And then you have synthesizers, which are very mathematical in the way the timbre is generated (oscillators, filters, ...), but this allows you to exactly fine tune the timbre so it "fits"

[ChatGTP]

It seems a bit philosophical to me though, what came first, the music or the maths?

We can use mathematics to describe the theory behind music but the theory isn't the music itself, nor are the instruments, "mathematical".

When a great performer plays live, they're not generally perfectly reproducing what's written in the musical notation either.

Probably not a straightforward thing to argue about, as I said, I think it's mostly a matter of viewpoints.

[dist-epoch]

There was a paper which studied the neurological basis of music.

One of the function of the brain is to predict the future. Neurons and brain systems are rewarded for correct predictions and punished for incorrect ones.

The main theory of the paper was that music is pleasant because it's repetitive nature and simple patterns makes it easy for the brain to predict, thus there is a lot of reward to extract from it. They focused on modern trance music and ethnic african drumming, both which feature very long very repetitive sequences, which reward the brain so much that they can induce a trance like state.

Not any random combination of physical objects is a musical instrument (yes, avantgarde music exists, ...) The sax is built the way it is because it generates a sound spectrum with harmonic relations between the frequencies (integer ratios, ...). Those who originally built the sax didn't use math to design it, but it so happens that the result respects certain mathematical ratios, because those ratios are highly predictable and pleasant to the brain.