Since I’m coming up dry on actual math, I thought I’d give a few lessons on music theory from a mathematical viewpoint. The overarching argument I’m going to try to make is that students of music (and of music composition in particular) need to know some math and physics.
I think for the most part you can go all the way through a bachelor’s degree in music and never learn about these things. This is truly a shame because as I’ll point out when it comes up, these ideas are not some abstract theoretical nonsense, but are extremely important to fully understand if you write (and sometimes play) music.
The first thing to get out of the way is something called the overtone series. Pretty much everyone learns this, so I’ll go through it quickly. Here’s how I like to think of it. If you sing or play a note, then there are a series of notes above and below it that are related to it.
If you take a wind instrument (for our purposes just think of it as a tube of metal), then depending on the length and size there is some “fundamental tone” that you can produce on it. Think of this as the lowest note you can play. For the sake of argument suppose the frequency of the wavelength is 64 Hz (this corresponds to a low C).
It turns out the next note that is playable on the tube/instrument will be at 128 Hz just by simple physics of waves considerations (recall that when you solve the eigenvalue problem for the wave equation you get some discrete set of eigenvalues which only allows certain solutions which are in bijection with the natural numbers). This is a C one octave higher. The next note playable will be at 192 Hz, a G. The frequencies continue: 256, 320, 384, …
Now we could have started at any fundamental frequency, so it is the ratios that matter and not the starting number (again, just solve the wave equation if you don’t believe me). So let’s normalize and see if we’ve ever seen this pattern before. Let’s call the first pitch above the fundamental 1 Hz. The next one is . The next one is and so on.
This is the well-known partial sums of the harmonic series! Of course this is no accident. The ancients knew about the overtone series and that’s why this series got that name. Like I mentioned, everything I’ve said so far is quite standard and well-known. If this was too sparse to follow I suggest you glance at the numerous internet sources explaining this in more depth before moving on.
One thing that is often glossed over, but will be crucial in later posts is that the overtone series is a physical reality that exists in any note that is produced in a natural way. If you sing a note, all the notes in its overtone series are sitting inside it. If you pluck a string, then the overtone series is in that sounding note. The only way you could get rid of the overtones is to produce a “pure” tone with the overtones stripped out using a computer.
This means that the overtone series is a physical part of the nature of how sound works. If you don’t believe this, then it would be well worth your time to listen to the following clip. He is only singing a single note the whole time, but he draws out the overtones in that tone: