Pythagorean comma
Jump to navigation
Jump to search
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,
reduced harmonic
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
| Interval information |
ditonic comma
reduced harmonic
(Shannon, [math]\sqrt{nd}[/math])
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.
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.