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Western music consists of 12 unique notes that are repeated across octaves. Is there any set number of unique notes in microtonal music? Or does the number of notes vary depending on the octave?

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There's no set number of notes outside a specific system — the term "microtonal" is very broad. From Wikipedia:

Microtonal music can refer to all music which contains intervals smaller than the conventional contemporary Western semitone. The term usually refers to music containing very small intervals but can include any tuning that differs from the western 12-tone equal temperament. Traditional Indian systems of 22 śruti; Indonesian gamelan music; Thai, Burmese, and African music, and music using just intonation, meantone temperament or other alternative tunings may be considered microtonal

It's quite possible that there are systems where the number of tones depends on the octave (I'm not personally aware of one), but that is not generally a feature of most systems. I would guess that this is because you would probably want to increase the number of tones as you ascended (since the absolute frequency difference between octaves is increasing and thus you have more "space" for division), but human perception of the distinction between notes also drops off as you go higher and counteracts the spacing.

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