24, 128, approach infinity -- I know there are microtonal systems, but if you want to understand them, choose the nearest multiple-of-twelve (or really, choose whatever TET contains the closest approximations to the exact notes) TET and look where their intervals land with respect to the symmetries (factors) within that TET, and where those symmetries are broken by a note moving by a perceptual 'slide' (one note 'moves' into the other position) rather than sounding like a new distinct voice.

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bonzini an hour ago | parent [-]

More like, only if you're playing Schoenberg it's about modulo 12.

Almost all sheet music that you play uses 5 or 7 unequal subdivisions of the octave, and occasionally the rest. Sure, sometimes you'll have one or two of the rest on a regular basis (blues, harmonic/melodic minor), but still most music won't use all twelve.

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drdaeman 43 minutes ago | parent [-]

Isn't modulo-twelve (if we ignore microtonal) the core underlying structure, and then you pick scale (like 2 2 1 2 2 2 1) and a key as a mapping, and then use degree as an 1-based index into the scale?

I don't really know a thing about music, so I could be wrong here - but that mental model at least made some sense to me when I tried to map all the concepts I've heard about into some sort of a mental system.

	▲	bonzini 16 minutes ago \| parent [-]
		The point is that to a player the mapping is more important than the core 12-note structure, because highly chromatic (12-note) passages happen but they are the exception. At least in western music you pick the seven basic notes on the scale, and that gives you the sharps and flats you place at the beginning of each line. That choice, from the point of view of notation, is even more important than the exact scale. And the choice is constrained to 15 sets of sharps and flats because it implicitly tells the player (veeeery roughly) to expect some patterns that probably will occur. It doesn't always work great, for example if you want to write the Simpsons theme the set of notes in the scale doesn't fit any of those 15 sets. But it uses an unfamiliar scale in any case, so there's not much to gain and as a player you just deal with it. See around 32:30 in https://youtu.be/Eq3bUFgEcb4?t=1950