[diffeomorphism]

No, not at all.

> 4th and 5th being intervals ON a Western scale.

That is where you went wrong. They are not. They naturally arise as small integer frequency multiples/fractions on any string instrument (wave lengths 1, 1/2, 1/3,...). They are hence quite obvious/loud (and humans recognize patterns as pleasant for whatever reason). Once you have 4ths and 5ths you repeat to get the Western system (handwaving away that this does not actually close, but "rounds" to make it fit into twelve. That is a whole other subject).

This is a pretty good example of inductive reasoning. We want a system that for any note also includes its first few harmonics, show that this implies....

[thomasqbrady]

I guess I wasn’t clear. I’m not saying the 4th and 5th notes don’t have a special sound to anyone. I’m saying that 1) just exactly how important their resonance is to you is influenced by your culture. It’s not that atonal musicians didn’t notice the resonance. They weren’t drawn to it as much. 2) The logic given was “if you want your scale to include the 4th and 5th, then 12 notes is inevitable.” The “if you want your scale to include the 4th and 5th” is the a built in assumption that you want something like the western scale, so it doesn’t seem that impressive to me that they then arrive at the 12-tone scale.