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These days I'm digging into the math of 'just intonation' - one of the most popular tuning systems in the late Middle Ages and early Renaissance. The idea here is that all frequency ratios should be products of the primes 2, 3 and 5. In this article I explain why it became more popular than the earlier Pythagorean system which only used the primes 2 and 3.

Basically, it seems that around 1300 English musicians started writing music that relied heavily on the frequency ratio

5/4 = 1.25

Pythagorean tuning approximates this by

81/64 = 1.265625

but that's sort of annoying. So, the English system spread into Europe - perhaps helped by the Hundred Years’ War (1337–1453), when northern France was sometimes occupied by the English.

The math of this tuning system goes a lot deeper than I explain here... but that's why this is only Part 1!

One problem is that people who know math jargon often don't know music jargon, and vice versa. So, if you want to use the concepts of "free commutative monoid" and "major triad on the fourth", you either have to explain both concepts, explain one and alienate one audience while boring the other... or explain neither and alienate almost everyone. There are a few people who write about music theory assuming you know both math and music jargon. They publish it in serious journals. But they are talking to very few people, I'm afraid! That's a pity, since the stuff is so cool. So I'm pulling my punches and trying to be really gentle.

https://johncarlosbaez.wordpress.com/2023/10/30/just-intonation-part-1/