> The problem is that the twelve-tone system was designed for a very specific "use case"
This isn't really correct though. Twelve is used because it has certain properties of divisibility that make life easier for the musician.
When the wavelengths of two notes form simple mathematical ratios, they have a stability that sounds nice, and this makes it easier to construct intervals and chords.
Wouldn't choosing a non-twelve division be more of a limit?
Twelve is used because it has certain properties of divisibility that make life easier for the musician.
Not true. It's used because (in short) when you try to construct a scale from octaves and fifths, finding a certain number of octaves that are close to another number of fifths (which comes to finding continued fraction approximants to 117/200, i.e. log 3/log 2 - 1) there are very few contenders, and only 12 notes per octave has neither too few nor too many for our tastes. It's based on the coincidence that 7 octaves is very nearly 12 fifths, i.e. 2^7~(3/2)^12.
Also, the idea that symmetry sounds nice is intuitively appealing, but nothing sounds better to us, more pleasing, than the (highly irregular) major scale, or scarier, more ominous than diminished and augmented chords - musically, a square and equilateral triangle. Or more disorienting than a whole tone scale - a hexagon.
> Also, the idea that symmetry sounds nice is intuitively appealing, but nothing sounds better to us, more pleasing, than the (highly irregular) major scale, or scarier, more ominous than diminished and augmented chords - musically, a square and equilateral triangle. Or more disorienting than a whole tone scale - a hexagon.
Yes, the perceived consonance has less to do with the simplicity/symmetry of the concept used to construct the scale and more to do with the 'simplicity' of the denominators of the ratios that make up the intervals. The smaller (simpler) the denominators involved, the more pleasing/consonant the sound will be (provided that they're integer ratios). Frequency ratios like 3/2 and 4/3 are among the simplest possible and are both present in the major scale but absent in the latter two examples.
Hmm that's not quite what I was saying - symmetry in scales in actually unpleasant, asymmetry pleasant. It seems that with these symmetrical scales/chords, the ear suffers vertigo, cognitive dissonance, because it can't hear a tonal centre. With e.g. a major scale, the asymmetry makes it obvious what key you're in.
I see what you mean. Still I would argue that both tritones and minor thirds only appear to possess the symmetrical property as a result of temperament. It doesn't exist 'in nature' as such.
The frequency ratios that they represent, 7/5 and 6/5, respectively, don't lend themselves to the same type of symmetry. If you have a tritone, 7/5, you would need a 10/7 ratio to make the octave, which would be a different interval (although close).
With a minor third (6/5), if you were to stack the intervals starting from let's say 500 Hz, you would go to 600, 720, 864, and 1036.8, and you'd be off 36.8 Hz from the octave.
This isn't really correct though. Twelve is used because it has certain properties of divisibility that make life easier for the musician.
When the wavelengths of two notes form simple mathematical ratios, they have a stability that sounds nice, and this makes it easier to construct intervals and chords.
Wouldn't choosing a non-twelve division be more of a limit?
<disclaimer, I'm merely an amateur musician>