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/
John Carlos Baez /
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2023-11-01 12:33:21
Author Public Key
npub17u6xav5rjq4d48fpcyy6j05rz2xelp7clnl8ptvpnval9tvmectqp8pd6mPublished at
2023-11-01 12:33:21Event JSON
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"content": "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.\n\nBasically, it seems that around 1300 English musicians started writing music that relied heavily on the frequency ratio \n\n5/4 = 1.25\n\nPythagorean tuning approximates this by \n\n81/64 = 1.265625 \n\nbut 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.\n\nThe math of this tuning system goes a lot deeper than I explain here... but that's why this is only Part 1! \n\nOne 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.\n\nhttps://johncarlosbaez.wordpress.com/2023/10/30/just-intonation-part-1/",
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