Pythagorean comma: Difference between revisions
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{{Wikipedia| Pythagorean comma }} | |||
The '''Pythagorean''' or '''ditonic comma''' is the interval with the ratio '''531441/524288''' ( | The '''Pythagorean''' or '''ditonic comma''' is the interval with the ratio '''531441/524288''' ([[monzo]]: {{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)<sup>12</sup>/2<sup>7</sup> 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 leads to the [[ | 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 == | == See also == | ||
* [[Gallery of just intervals]] | * [[Gallery of just intervals]] | ||
* [[Small comma]] | * [[Small comma]] | ||
[[Category:Compton]] | |||
Revision as of 04:22, 22 December 2022
| Interval information |
ditonic comma
reduced harmonic
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.