Pythagorean comma

Interval information

Ratio	531441/524288
Factorization	2-19 × 312
Monzo	[-19 12⟩
Size in cents	23.46001¢
Names	Threerish 12-tone's comma, Pythagorean comma, ditonic comma
Color name	LLw-2, Lalawa comma
FJS name	[math]\displaystyle{ \text{d}{-2} }[/math]
Special properties	reduced, reduced harmonic
Tenney norm (log2 nd)	38.0196
Weil norm (log2 max(n, d))	38.0391
Wilson norm (sopfr(nd))	74
Comma size	small
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Pythagorean comma

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.

See also

• Gallery of just intervals
• Small comma