12th-octave temperaments

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12edo is the predominating tuning system in the world today, so some of the 12th octave temperaments have been known for a very long time.

By far the most notable 12th-octave temperament is compton or aristoxenean, associated with the Pythagorean comma, which sets 3/2 to 7\12. Given that the circle of 12 fifths closing was the basis of music in Europe and independently China, it can be argued that the 12th-octave temperaments are the oldest form of musical tradition in history.

Furthermore, 12th-octave temperaments like compton usually temper other intervals to subsets of 12edo, like 9/8 to 6edo whole tone scale, and 5/4 to 3edo.

Another temperament which was hugely known is atomic, stemming from a comma described as the Kirnberger's atom. 3/2 flattened by 32805/32768, the schisma, is extremely close to 7\12.

For temperament data, see:

Magnesium (5-limit)

Subgroup: 2.3.5

Comma list: [-157 -24 84

Mapping[12 2 23], 0 7 2]]

mapping generators: 78125/73728 = 1\12, [71 11 -38 = 243.137

Supporting ETs: 84, 360, 528, 612, 696, 864, 1920, ...