No: Having 12 notes is a neat but accidental outcome of the musical scale.
On the derivation of the scale:
Two pure tones go good together when they have "ratios" of frequency. So, 440Hz and 660Hz would interfere in a pleasing way. This is the same way that it's "nice" when tiles on a floor match your gait in a way you can follow a pattern, or when two blinkers sync up.
So, it's nice when tones are fractions of one another like 3/2, 4/3, 5/3, etc. A ratio like pi/2 would sound weird, and very close frequencies (like 440Hz and 440.5Hz) would interfere to make a beat of 0.5Hz. (I'm sure we can all agree that a ratio of 4/3 is a much nicer fraction 440/441. In practice, this doesn't matter much, because we rarely use pure tones. This is why equal temperament scales don't sound abysmal.)
A natural ratio is just "2/1". This division gives you your octave. One octave up from 440 is 880. One octave down from 440 is 220.
This plays nicely into the second problem: Human perception is logarithmic in many things, tone included. For a musical scale to be perceived to have equal differences in pitch between notes, it needs to be roughly evenly spaced on the logarithmic scale.
So, we need to select a set of frequencies in [440Hz, 880Hz) (where 440Hz is arbitrary) that (1) arenice fractions of one another, but are also (2) evenly spaced on the logarithmic scale.
By nice mathematical luck, the 12-tone chromatic scale fulfills that!
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On the questionable qualities of "12":
I don't think the nice divisibility of the number 12 matters here, in a scale where only 7 of the notes (with the most pleasing ratio) get the title of "major". Notation is written on a musical staff where the other 5 notes are folded away.
Furthermore, if you swap out the base of 2 for a base of 3 and re-derive the scale, you get other scales. One is the Bohlen-Pierce scale, which has 13 notes. I derived a similar scale one time, and I remember finding a 7 or 11 note scale. I forget which exactly, but the point being that these are prime numbers.
So, I'm wondering, how would 12 being divisible matter for music? I don't compose much, so I mean this question genuinely.
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On the main point:
> I'm not sure that the linked article, or the current top comment (Circle of Fifths) meaningfully extends beyond this "useful parts" hypothesis.
I ask this in good faith: Did you read the linked article before commenting?
If you didn't, then :\
If you did, then I'm curious:
The scale derivation I described here is described in the article, with nice visualizations. If you did read the article, how do you now have this take? How could a 10-tone scale fit into this, and how would having a less nicely-divisible number matter?
Let me start by making the same point, but starting from a different place.
When you see pi, you look for a circle. When you see an 8 or 12 or 24 or 60, you see a few small primes multiplied together, and you might look for things that like being flexibly divided into clean subgroups.
Now I spent a couple of hours trying to extend my original argument, using the internet.
12tet (12 tone equal temperament) gives you good fourths and fifths (which as you say sound great) and TET. What you don't get is being able to correctly voice say a barbershop seventh, or persian music, or dissonant death metal, or microtonal pop.
While n-TET for any n is possible, Wikipedia lists 5, 7, 12, 15, 17, 19, 22, 23, 24, 26, 27, 29, 31, 34, 41, 46, 53, 72, and 96tet as being in use, some much more popular than others. Arabic music shifted from 17tet to 24tet, with the exception of some holdouts that find 24tet too 'commercial'.
You don't need bang-on fourths and fifths to be happy in n-tet; given n-tet, you figure out what intervals to use, what kind of chords you want to build, and what kind of scales you are willing to learn to use those chords.
So, fourths and fifths are good in 12tet, and you covered that. Is there anything else to say about 12tet and its popularity, particularly about there being 12 notes and its prime factorization into 2s and 3s?
From stackexchange: You can create a circle of any number of steps mutually prime to the total number of pitches in the octave.
And what it looks like is that you get a Circle of Fifths that has some particularly nice properties in 12tet due to the math with 2/3/5, and that makes scales and chord building particularly easy, making 12tet easy to work with, and that further popularizes 12tet. I just waved my hands there pretty hard; I did not do anywhere near enough due diligence on sources. But, yes, the number of notes does end up being important. Probably.
So I think I was minorly correct in being suspicious of '12' and how divisible it was into small primes, but majorly wrong in most other aspects.
As an aside, I did read your article, and just had a different take on what was going on; I was sure that the prime factorization structure of '12' would explain more than it does.
On the derivation of the scale:
Two pure tones go good together when they have "ratios" of frequency. So, 440Hz and 660Hz would interfere in a pleasing way. This is the same way that it's "nice" when tiles on a floor match your gait in a way you can follow a pattern, or when two blinkers sync up.
So, it's nice when tones are fractions of one another like 3/2, 4/3, 5/3, etc. A ratio like pi/2 would sound weird, and very close frequencies (like 440Hz and 440.5Hz) would interfere to make a beat of 0.5Hz. (I'm sure we can all agree that a ratio of 4/3 is a much nicer fraction 440/441. In practice, this doesn't matter much, because we rarely use pure tones. This is why equal temperament scales don't sound abysmal.)
A natural ratio is just "2/1". This division gives you your octave. One octave up from 440 is 880. One octave down from 440 is 220.
This plays nicely into the second problem: Human perception is logarithmic in many things, tone included. For a musical scale to be perceived to have equal differences in pitch between notes, it needs to be roughly evenly spaced on the logarithmic scale.
So, we need to select a set of frequencies in [440Hz, 880Hz) (where 440Hz is arbitrary) that (1) arenice fractions of one another, but are also (2) evenly spaced on the logarithmic scale.
By nice mathematical luck, the 12-tone chromatic scale fulfills that!
---
On the questionable qualities of "12":
I don't think the nice divisibility of the number 12 matters here, in a scale where only 7 of the notes (with the most pleasing ratio) get the title of "major". Notation is written on a musical staff where the other 5 notes are folded away.
Furthermore, if you swap out the base of 2 for a base of 3 and re-derive the scale, you get other scales. One is the Bohlen-Pierce scale, which has 13 notes. I derived a similar scale one time, and I remember finding a 7 or 11 note scale. I forget which exactly, but the point being that these are prime numbers.
So, I'm wondering, how would 12 being divisible matter for music? I don't compose much, so I mean this question genuinely.
---
On the main point:
> I'm not sure that the linked article, or the current top comment (Circle of Fifths) meaningfully extends beyond this "useful parts" hypothesis.
I ask this in good faith: Did you read the linked article before commenting?
If you didn't, then :\
If you did, then I'm curious:
The scale derivation I described here is described in the article, with nice visualizations. If you did read the article, how do you now have this take? How could a 10-tone scale fit into this, and how would having a less nicely-divisible number matter?