Pythagorean comma
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Ratio | 531441/524288 |
Factorization | 2^{-19} × 3^{12} |
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 (log_{2} nd) | 38.0196 |
Weil height (log_{2} 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}/2^{7} 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.