Pythagorean comma
| Interval information |
ditonic comma
reduced harmonic
(Shannon, [math]\sqrt{nd}[/math])
The Pythagorean or ditonic comma is the interval with the ratio 531441/524288 (monzo: [-19 12⟩). It could also be called the 12-comma as it is the amount by which twelve fifths exceed seven octaves, or in other words (3/2)12/27 and it also can be written as the ratio between the apotome and the Pythagorean minor second, (2187/2048)/(256/243), and as the ratio between the Pythagorean augmented fourth and the Pythagorean diminished fifth, (729/512)/(1024/729). In addition, it also equates six 9/8 major seconds with an octave.
Temperaments
Tempering out this comma in the 5-limit leads to the compton temperament. For edos up to 300, it is tempered out if and only if the edo is a multiple of 12, and hence for instance by 12edo, 24edo, 72edo and 84edo. See compton family for a number of rank-2 temperaments where it is tempered out.