The tritone - the dissonant sound halfway between any sound and the one that vibrates twice as fast - is sometimes called "diabolus in musica". Yes: THE DEVIL IN MUSIC! 😈
Believe it or not, this devil also afflicts modern data storage. But first: there's a lot of misinformation about the tritone. Was it really banned in the Middle Ages? Adam Neely's great video here clears it up. It goes into a lot of detail. But not enough detail for me!
As I studied just intonation in Renaissance music - where frequency ratios should be simple fractions built by multiplying and dividing the numbers 2, 3, and 5 - I realized that the tritone is DEVILISHLY DIFFICULT for this system!
You see, the tritone vibrates with a frequency of √2 ≈ 1.414 times that of the sound it's sitting over. This number is not only irrational - making Pythagoras turn in his grave - there are also four competing ways to approximate it in just intonation:
25/18 ≈ 1.38888
45/32 ≈ 1.40625
64/45 ≈ 1.42222
36/25 = 1.44
And the ratios of these frequencies vexed early music theorists so much they all have their own individual names!
For example,
(36/25)/(45/32) = 128/125 = 2⁷/3³
is called the 'lesser diesis'. This number shows up automatically when you try to approximate powers of 5 by powers of 2. But notice:
128/125 = 1024/1000
also shows up when you try to approximate powers of 10 by powers of 2. For example, when we talk about a kilobyte, we usually don't mean 1000 bytes - we usually mean 1024. So we're a bit off! And our error is the "lesser diesis" - a number discovered in the Renaissance, or even earlier, by musicians fighting the devil in music.
(1/2)
@johncarlosbaez of course approximating powers of 10 with powers of 2 is the same as approximating powers of 5 with powers of 2, simply because 10=5*2 and the twos cancel out
@mapto - right. By the way, the other important glitch in just intonation is 81/80, which is about approximating powers of 3 by powers of 2 and 5.