Almost all the music you hear is played in the 12-tone equal-tempered scale, or 12-TET for short. It has 12 notes, each vibrating 2^(1/12) times faster than the note below it.
Why do we like this scale so much? What makes the number 12 so special? That's what I talk about that in the article below. But I only tackle one small piece of this multi-faceted puzzle.
One reason 12-TET is great is that
2^(7/12) ≈ 1.498
This is really close to 3/2. So if we start at any note on a piano and climb up 7 notes, like this:
C C# D D# E F F# G
we get a note that vibrates almost 3/2 times as fast. And that sounds good!
In my article I systematically look for other numbers that work well like this. It turns out that 5-TET and 7-TET are the best we can do before 12-TET.
There's a lot of gamelan music that uses a 5-note scale called slendro that's close to 5-TET. And I've read that a lot of ancient Chinese music uses 7-TET - but I haven't been able to find out any details. 😢
After 12-TET the next winners are 29-TET, 41-TET and 53-TET. The last is especially good:
2^(31/53) ≈ 1.49994
This was discovered way back around 50 BC by the Chinese music theorist Jing Fang! It was rediscovered in the 1600s by the mathematician Nicholas Mercator - not the guy with the map, the guy who invented the term 'natural logarithm'. Then in 1694 William Holder pointed out that 53-TET also gives an excellent approximation to 4/3, another fraction of great importance in music:
2^(22/53) ≈ 1.33339
What comes after 53-TET? Read my article!
While thinking about stuff this I ran into an interesting math problem I don't know the answer to....