Pythagorean comma

The Pythagorean or ditonic comma is the interval with the ratio 531441/524288 (= [-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).

Interval information

Ratio	531441/524288
Factorization	2-19 × 312
Monzo	[-19 12⟩
Size in cents	23.46001¢
Names	Pythagorean comma, ditonic 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
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Temperament

Tempering out this comma leads to the Pythagorean family of temperaments. 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 also

• Gallery of just intervals
• Small comma
• Wikipedia: Pythagorean comma