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    I find music theory fascinating from a philosophical perspective because:

    • our perception of pitch, and of pleasing combinations of pitches, is mathematical. Our ears/brains experience pitches logarithmically and like small-integer ratios.
    • So music theory is basically simple applied math — integer ratios, modular arithmetic, logs base 2.
    • We expect that mathematical structures will fit together perfectly, because usually they do. When you prove a=b, then a is precisely equal to b, no slop, no rounding error.
    • But when you do the very fundamental exercise described in this article, which Pythagoras was probably not the first to try, you end up with a beautiful structure of 12 notes … but it doesn’t quite fit. By all rights it ought to be a perfect 12-pointed star with all its lovely symmetries, but the damn thing doesn’t close. And yet we use this star, this circle of fifths, as a foundational structure of music in nearly every human culture.
    • This problem has been bothering musicians, and causing real problems, for about 2,000 years. You couldn’t transpose pieces to a different key or play them on certain instruments. Composers couldn’t use certain intervals or harmonies because they sound like shit. In fact no matter what you do you can’t get all harmonies to sound right. Composers were literally engaging in flame wars and nearly coming to blows in 18th-century Europe over this.
    • Our current Western tuning is a kind of hack that’s very symmetrical — 12 equal steps of the 12th root of 2 — but which makes all intervals except octaves slightly wrong. The wrongness is small enough to ignore, and it turns out if you grow up with it the true integer-ratio intervals sound weird and wrong.

    Anyway. And it’s all based on this odd coincidence that (3/2)^12 is almost equal to 2^7. If that weren’t the case, music would be indescribably different.

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      Jacob Collier does a great quick demo of how different the integer-ratio vs 12th-root-2 notes can sound: https://www.youtube.com/watch?v=XwRSS7jeo5s

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        Rhythm is also fascinating! It is so syntactical

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          I don’t think the circle of fifths is used outside western music. Can you explain why you say “nearly every human culture”?

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            12 tone equal temperament (which the circle of fifths arises from) has been around for a couple thousand years and has influenced a plurality of non-westerners. Especially after the internet, our music tastes have started to converge across the world. There are of course non-western cultures that don’t use 12 tones and thus don’t have the same circle of fifths, but if you listen to the radio in Eastern Europe, South America, East Asia, etc (big first-world cultural hubs), you’ll find plenty of pop songs structured very similarly to American pop songs. Bad Bunny and Higher Brothers are some examples of converging music tastes imo.

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              Sorry, I was being a bit lazy there. What’s universal is the use of simple pitch ratios as musical intervals. Every culture that has any sort of music has discovered & used pentatonic scales (or so I’ve read.)

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            There are interesting bigger circles of fifths with e.g. 19 notes and 31 notes in which C# != Db (and actually C# is lower than Db). Wikipedia equal temperament page has more info.

            Also there’s some great music using various other equal divisions of the octave (EDO), some cool examples being Sevish’s Gleam in 22EDO and Brendan Byrnes’ Sunspots in 27EDO (lots of frets on that guitar!).

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              I find microtonality intellectually interesting, but I haven’t been able to adapt my ears to the point where it sounds good. The pieces I’ve listened to all seem kind of unpleasantly out-of-tune. Which I know is just because it’s unfamiliar, but it hasn’t incentivized me to listen further.

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                Indeed some microtonal stuff can sound pretty crunchy :p

                One microtonal thing that immediately sounded lovely to me is Ben Johnston’s version of Amazing Grace (uses just intonation).