Season 7 Episode 9
Music and Maths go together really well; in this episode we're exploring intervals, tuning, and Fourier transforms, with special guest Maria.
Further Reading
Video recommendations
Maria has put together a YouTube playlist of recommended videos on tones, tuning, and tempraments.
We briefly mentioned the separate topic of polyrhythms too; I like Project JDM on YouTube for this.
One more recommendation; Jacob Collier is a singer/song-writer. He can drum polyrhythms on one hand and he’s famous for conducting the audience at his shows.
Perfect fifths
We saw on the stream that if you keep increasing the frequency in steps of 3/2, then you’ll never get to a power of 2. Something you can do along the way is to divide anything larger than 2 by 2 (dropping down an octave) to keep the numbers between 1 and 2 (question: how often do you have to divide by 2?). If you do that then you get the following sequence;
$$1,\quad \frac{3}{2},\quad \frac{9}{8},\quad \frac{27}{16},\quad \frac{81}{64},\quad \frac{243}{128},\quad \frac{729}{512},\quad \frac{2187}{2048},\quad \dots $$
We could keep going, but this sequence will never hit 1. It has just got rather close to 1 though! This is (almost) the Pythagorean Scale. The only difference is that Pythagoras sneakily pretends that the 2187/2048 term in the sequence really is exactly 1 (so that the sequence is periodic), and hides that change a little by changing the 729/512 to 4/3. That’s sort of like working backwards from the 1 at the end. Of course there’s still a problem there; now the ratio from 243/128 to 4/3 isn’t quite right. We can’t hide the fact that 2187 isn’t the same as 2048, we’re just moving the problem around a bit!
The Pythagorean Scale:
| Note | C | D | E | F | G | A | B | C |
|---|---|---|---|---|---|---|---|---|
| Ratio | 1 | 9/8 | 81/64 | 4/3 | 3/2 | 27/16 | 243/128 | 2/1 |
We saw a different solution on the stream; keep going with the sequence until you’ve taken 12 steps and got to $(3/2)^{12}/2^7$. If you pretend that’s 1 and shuffle things around a bit, you get a 12-tone scale.
A different fix is to modify the number 3/2 slightly, so that we really do end up at 1 after 7 or 12 or however many steps. Since this is a sort of geometric series, we end up taking the ratio to be something like $2^{7/12}$. This is called equal temperament, and it’s almost ubiquitous in modern music.
Because we're going up in fifths, and we've got back to where we started, this is sometimes presented in a circular diagram called the circle of fifths. There’s a diagram for the circle of fifths here. If you’re learning music theory, you might have seen the circle of fifths when you were learning about key signatures.
Størmer's theorem
Given a finite set of primes $P$, there are only finitely many pairs of consecutive numbers $(S,S+1)$ such that each has prime factors only in the set $P$.
See Wikipedia for a pretty good overview of how the calculations work. For a proof, I recommend Lehmer’s 1964 paper from the references on the Wikipedia page.
If you want to really get into this, then I suggest this blog post from Flying Colours Maths as a starting point to learn about continued fractions and the Pell equation (the start of the article refers to a previous post called a sock puzzle).
Discrete Fourier Transform
This is a tool for signal analysis. This webpage is a helpful starting point; An Introduction to the Discrete Fourier Transform.
The notation is probably a bit more unfamiliar than the algebra you’ve seen at A-level or equivalent, but hopefully the section leading up to equation 4 gives you an idea of what we’re trying to do.
Here’s a link to the vector calculator that was featured on the livestream; Jenn's Visual Pitch Class Vector Calculator.
I also like this tool from the same website which lets you apply this to a piece of music (either selected from a corpus or uploaded) to see how the components might change over the duration of the piece.
More!
Maria's given me some more links and reading suggestions for you;
- A reference for barbershop tuning on Wikipedia
- A book recommendation How Equal Temperament Ruined Harmony (And Why You Should Care) by Ross W. Duffin; bit controversial in some circles, but worth a read.
- A massive list of musical interval names from the Huygens-Fokker Foundation
- A page by Kyle Gann about Just Intonation
- Two papers on Javanese Gamelan tuning;
- A paper on the classical Indian just intonation tuning system
If you want to get in touch with us about any of the mathematics in the video or the further reading, feel free to email us on oomc [at] maths.ox.ac.uk.