Pythagorean comma: Difference between revisions

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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)12/27 ~~and it~~ also can be written as the ratio between the apotome and ~~the Pythagorean minor second~~, ([[2187/2048]])/([[256/243]]), and as the ratio between the Pythagorean augmented fourth and the Pythagorean diminished fifth, ([[729/512]])/([[1024/729]]). In addition, it also ~~equates~~ six [[9/8]] major seconds ~~with~~ an octave.

The '''Pythagorean comma''' or '''ditonic comma''' is the interval with the ratio '''531441/524288''' ([[monzo]]: {{monzo| -19 12 }}). It is the amount by which twelve fifths exceed seven octaves, or in other words (3/2)12/27. It also can be written as the ratio between the apotome and limma, ([[2187/2048]])/([[256/243]]), and as the ratio between the Pythagorean augmented fourth and the Pythagorean diminished fifth, ([[729/512]])/([[1024/729]]). In addition, it is also the difference betweem six [[9/8]] major seconds and an octave.

In pythagorean tuning, this interval serves as a ''descending'' diminished second, even though it has a positive size.

== 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.

If the pythagorean comma is [[tempering out|tempered out]], then the [[circle of fifths]] closes at 12 notes. This circle of fifths covers the entirety of [[12edo]], while larger multiples of 12edo such as [[24edo]] and [[72edo]] contain multiple such circles. If one takes this circle of fifths and adds an independent [[generator]] for prime [[5/1|5]], this leads to the 5-limit rank-2 [[compton]] temperament. See [[compton family]] for a collection of rank-2 temperaments where it is tempered out.

EDOs with a fifth sharper than the 12edo fifth of 700{{c}}, such as [[41edo]] and [[53edo]], map the pythagorean comma to a positive number of steps rather than tempering it out. The pythagorean comma is quite close to the [[81/80|syntonic comma]], only exceeding it by a [[schisma]]. It is also faily close to the [[64/63|septimal comma]], with the septimal comma exceeding the pythagorean comma by the [[garischisma]]. Tempering out both the schisma and the garischisma leads to [[garibaldi]] temperament, which is a relatively simple [[7-limit]] interpretation of the pythagorean chain of fifths.

EDOs with a fifth flatter than the 12edo fifth, such as [[19edo]] and [[31edo]], map the pythagorean comma negatively, and thus have a positive diminished second (also known as a diesis). The majority of these EDOs support [[meantone]], which equates the pythagorean major third [[81/64]] to the 5-limit major third [[5/4]].

== See also ==

* [[Mercator's comma]]

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 {{Wikipedia| Pythagorean comma }}
-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]]), and as the ratio between the Pythagorean augmented fourth and the Pythagorean diminished fifth, ([[729/512]])/([[1024/729]]). In addition, it also equates six [[9/8]] major seconds with an octave.
+The '''Pythagorean comma''' or '''ditonic comma''' is the interval with the ratio '''531441/524288''' ([[monzo]]: {{monzo| -19 12 }}). It is the amount by which twelve fifths exceed seven octaves, or in other words (3/2)<sup>12</sup>/2<sup>7</sup>. It also can be written as the ratio between the apotome and limma, ([[2187/2048]])/([[256/243]]), and as the ratio between the Pythagorean augmented fourth and the Pythagorean diminished fifth, ([[729/512]])/([[1024/729]]). In addition, it is also the difference betweem six [[9/8]] major seconds and an octave.
+In pythagorean tuning, this interval serves as a ''descending'' diminished second, even though it has a positive size.
 == 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.
+If the pythagorean comma is [[tempering out|tempered out]], then the [[circle of fifths]] closes at 12 notes. This circle of fifths covers the entirety of [[12edo]], while larger multiples of 12edo such as [[24edo]] and [[72edo]] contain multiple such circles. If one takes this circle of fifths and adds an independent [[generator]] for prime [[5/1|5]], this leads to the 5-limit rank-2 [[compton]] temperament. See [[compton family]] for a collection of rank-2 temperaments where it is tempered out.
+EDOs with a fifth sharper than the 12edo fifth of 700{{c}}, such as [[41edo]] and [[53edo]], map the pythagorean comma to a positive number of steps rather than tempering it out. The pythagorean comma is quite close to the [[81/80|syntonic comma]], only exceeding it by a [[schisma]]. It is also faily close to the [[64/63|septimal comma]], with the septimal comma exceeding the pythagorean comma by the [[garischisma]]. Tempering out both the schisma and the garischisma leads to [[garibaldi]] temperament, which is a relatively simple [[7-limit]] interpretation of the pythagorean chain of fifths.
+EDOs with a fifth flatter than the 12edo fifth, such as [[19edo]] and [[31edo]], map the pythagorean comma negatively, and thus have a positive diminished second (also known as a diesis). The majority of these EDOs support [[meantone]], which equates the pythagorean major third [[81/64]] to the 5-limit major third [[5/4]].
 == See also ==
 * [[Mercator's comma]]

Pythagorean comma: Difference between revisions

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