Pythagorean comma

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Interval information
Ratio 531441/524288
Factorization 2-19 × 312
Monzo [-19 12
Size in cents 23.46001¢
Names Pythagorean comma,
ditonic comma
Color name LLw-2, Lalawa comma
FJS name [math]\text{d}{-2}[/math]
Special properties reduced
Tenney height (log2 nd) 38.0196
Weil height (log2 max(n, d)) 38.0391
Wilson height (sopfr (nd)) 74
Harmonic entropy
(Shannon, [math]\sqrt{nd}[/math])
~4.12762 bits
Comma size small
open this interval in xen-calc
English Wikipedia has an article on:

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). In addition, it also equates six 9/8 major seconds with an octave.


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