[thomasqbrady]

This piece is a good example of circular reasoning, isn’t it? The question “Why are there 12 notes in Western scales?” Is answered first by presuming that 4ths and 5ths sound pleasant (to whom? a Westerner?), the “4th” and “5th” being intervals ON a Western scale, which the author then reverse-engineers back to the 12-note scale which they assumed from the start. There are other scales you could start from, in which 4ths and 5ths aren’t so special…

[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.

[jollybean]

No, 4ths and 5ths are resonant. They sound better to a lot of people. They are completely distinguishable from arbitrary intervals right near there.

Most of the notes on the Western scale fall into this orientation, i.e. there is reasoning for it. It's not just arbitrary.

[bryanlarsen]

The why is because the 5th is the primary non-octave overtone of a vibrating string, and the 4th is an inverted 5th.

[herghost]

Thank you! After reading the article I was left unsatisfied, but couldn't put my finger on it. I was somewhere around thinking that we hadn't yet established why 4th and 5th intervals were particularly special and so I couldn't see why the conclusion worked.

You nailed it.

[pdonis]

> I was somewhere around thinking that we hadn't yet established why 4th and 5th intervals were particularly special

Other responses to the GP have explained that. The article itself also mentions the reason (small integer ratios).

[anon291]

4ths and fifths are used in most standard systems of music.

Actually it's just fifths because fourths are just fifths.

And fifths sound good because of physics.

For example the pentatonic scale used in a lot of eastern music is subsumed by the western scale.

[mcphage]

That's not really circular, though. It does start from the assumption that 4ths and 5ths sound pleasant, but uses that to build possible scales, some of which are more compatible with 4ths and 5ths than others.