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Since I was a teenager I've been very strongly classified as a "math person" and pretty strongly classified as a "music person". And that's not terribly uncommon. A lot of people that like math (including physicist and other scientists) are also said to have played an instrument. Classic example, Einstein and his violin. For a more contemporary example, 3Blue1Brown is clearly into music.

So, then, it's not surprising that I got asked a question very often: "is music mathematical?". In fact, it was often not a question, just a statement. I remember back then I didn't really like that. My answer would typically be something along the lines of "well, you can describe music in a sort of mathematical way the same way can explain everything mathematically!". But, I was thinking about this recently and I'm not sure I totally agree with my past self.

Everything is mathematical, but music might be particularly mathematical.

So, clearly, there's a part of music that is not meaningfully mathematical. There is a lot of "taste" in music. Connecting human experiences, going after a feeling with the harmony and whatnot. While, pedantically, you could say that all of this is ultimately mathematical because you can explain it and yada yada I think that in practice, it just ain't. Let's talk about those that are people claim as mathematical, and asses if they really are.

Counting, tuplets and filling in bars

I don't think this is mathematical. Time signatures have numbers and in a bar you have to make sure everything fits, so you need to do some adding and dividing and maybe even adding fractions 😱😱

But doing calculations is not doing math. This is the kind of reason why I dismissed people when they said music is mathematical, because they often referred to this.

Powers of two

You might be surprised to hear that I do think this is mathematical, since I just complained on the previous section.

I'm sure that for a lot of people 2, 4, 8, 16 are the things you divide a bar into or the number of bars in a phrase, not powers of two. Of course, they are just powers of two but the structure arises from a different place.

Here's the insight. This structure arives because there is a notion of multiplying by two. Nobody multiplied anything by two consciously. But the abstract "multiply by two" idea happened. It seems to me that music has a particularly high number of different math operations, that are encoded in a way that is not people think of math (multiplying numbers), that have very visible results.

Note distance

What are closer, C and D or C and G? Physically, in the piano, in terms of pitch, of frequency and most obvious metrics D is closer C than G. But a lot of musicians might say that G is closer to C. This example is more extreme if you compare C# and G to C. G is clearly closer. But why? Well G is closer in the circle of fiths, and the circle of fiths kidna represents how close notes are ~harmonically~. Kinda. Not really. But it shows musical relationships better very often. Keys closer in the circles of fiths share more chords and have similar signatures. If you, for example, modulate from C to G, most notes are going to be the same, while from C to C# a lot of notes are going to be close but not quite identical. Technically, if you compare note to note a melody or a chord progression in C and G they will be further apart than C and C# but the note pool is more similar.

So what we have is that notes can be thought of as existing in at least two distinct spaces, and often the second one is much nicer, even though it is less intuitive on a first glance. This is a very common math thing. Changing your perspective, changing bases (in a concrete and abstract sense). Different measures for spaces. That's neat.

Also, scales in music have a modulus and, unlike the classic example of a clock, it is modulo 2 in log scale!! That's neat and kind of mathematical.

The harmonic series

The harmonic series is probably the quintessential "math" thing from music. Here's the thing, I don't think the harmonic series itself is that mathematical? Like, yes, it's clearly math; but it's as math-y as tennis in the sense that when you hit a ball it moves according to math equations.

However, the fact that everything in music is made from ratios does give sometimes a deeper musical understanding. Probably the best example I can think of is the modulation in Celine Dion's All By Myself. The Adam Neely explains a lot of thing beautifully but I think that what hits the hardest is that the median modulation literally inverts how we're thinking of the note and Celine goes from being the ornament on top to the fucking foundation of the entire instrumental!! I'm getting goosebumps just thinking about it in the train lol.

But does math explain that? I don't really think so, no.

Music theory analysis

This is pretty mathematical in the end, I think. Especially things relating keys, interpreting the same chord as different functions in relation to the key. There is of course that human taste I talked about earlier, but there can be these math elements too. Again, the seeing if a different interpretation/expression of the same value can lead to insight is very fundamental to a lot of interesting derivations in math. The difference is that in music you can't really reach "an answer".

Effect chains

This is more music production, but effect chains are really functions on sound waves! That is very mathy. You can observe things like the fact that functions are not necessarily commutative.

Conclusion

I guess I did come up with a few examples, but I don't think they're that persuasive... I mostly like the powers of two and the note distances, and those were in fact the inspiration for me writing this post. I thought I would think of more, but again it wasn't very interesting. I can't blame my past self too much for being jaded with the question, but I think there is some merit to calling music mathematical!

Really, the thing that is more math-y than anything else is the fact that music is abstract. It is hard to describe sounds because sounds are just sounds that sound like the thing they sound like. Math is even worse, in that nothing exists and any connection to the real world is a coincidence and unprovable. I am being slightly dramatic, yes, but it is kind of true. And probably the number one reason I enjoy music and math are both for their abstractness. And it is in that sense that music is mathematical.

Otherwise, not really.