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But the whole point of 12-TET is it is a close approximation of those just intervals while still allowing multiple keys to be played without retuning the instrument. If the close mathematical approximation didn't exist, there would be no reason to tune your instruments with a logarithmic recurrence relation.

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> allowing multiple keys to be played without retuning the instrument

That's another simplification, real pianos are not tuned to 12TET, but use stretch tuning (which can be over 30cents off in the lower range).

I'd rather argue that people like what they're used to and so "people like pure ratios" is seemingly only true if "pure ratios" are not really pure. And that's ignoring a lot of the music that doesn't have roots in Europe.

FWIW it doesn't get better than 12-TET until you get to (like) 53-TET.

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