[thaumasiotes]

> A 12 TET chromatic is 2^(1/12), and a 12 TET fifth would be 2^(7/12). A perfect fifth is a 3:2 ratio. Those numbers are slightly different, and that’s enough to understand it.

Note this this is normally called the "Pythagorean comma".

https://en.wikipedia.org/wiki/Pythagorean_comma

[dahart]

Yes, thanks! And that article points at the “problem” setup in the article: the syntonic or diatonic comma, or the discrepancy between four stacked perfect fifths and two octaves plus a major third.

I like how the Pythagorean comma simplifies to 3^12 / 2^19 - pure powers of three vs pure powers of two. That makes it so obvious (to me anyway) why we can’t have perfect fifths.