For some reason it’s taken me decades of playing guitar to become good enough at tuning and also sensitive enough to really feel the fact that I can’t tune the guitar. Recently I finally grokked the simple reason that 12 TET cannot be perfect, and it doesn’t take a long article to see it. I was kind-of aware of the major third problem, but I naively thought fifths were still perfect.

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. Another way of thinking about it is that if you were to complete the cycle of fifths purely by stacking fifths, you should end up on the note you started with but many octaves higher. But you should be able to see that starting on C1 and going by octaves will produce a number that is purely powers of 2, whereas stacking fifths will necessarily involve powers of both 2 and 3, so they can never be equal, I can stack fifths and never land on my original note’s octaves.