Pythagorean comma: Difference between revisions

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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 between six [[9/8]] major seconds and an octave.

In [[pythagorean tuning]], this interval is an ''~~inverse~~'' ~~diminished second~~, ~~even though it has~~ a ~~positive size~~.

In [[pythagorean tuning]] or tunings close to it, this interval is a ''negative'' [[Diesis (scale theory)|diesis]]. This is because adding adding pythagorean commas makes the interval go up in pitch, down the scale.

This counterintuitive notion is a result of just fifths naturally producing a ''[[hard]]'' [[5L 2s|diatonic]] [[MOS scale|MOS]], which means that the [[chroma]] (chromatic semitone) is ''wider'', not narrower, than the small step (diatonic semitone).

== Temperaments ==

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 the family 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 fairly 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 sharper than the 12edo fifth of 700{{c}}, such as [[41edo]] and [[53edo]], map the pythagorean comma to a positive small 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 fairly 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 the simplest [[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 (scale theory)|diesis]]). The majority of these edos support [[meantone]], which equates the pythagorean major third [[81/64]] to the 5-limit major third [[5/4]].

Since it is reached by 12 fifths, a highly composite number, there are many temperaments that split this comma whilst keeping fifths unsplit. Notably:

* [[Kalismic]], splitting it into 2 [[2835/2816|fwiwismas]].

* [[Landscape]], splitting it into 3 [[225/224|marvel commas]].

* [[Nexus]], splitting it into 3 [[243/242|rastmas]].

* [[Atomic]], splitting it into 12 schismas.

== See also ==

@@ Line 15: / Line 15: @@
 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 between six [[9/8]] major seconds and an octave.
-In [[pythagorean tuning]], this interval is an ''inverse'' diminished second, even though it has a positive size.
+In [[pythagorean tuning]] or tunings close to it, this interval is a ''negative'' [[Diesis (scale theory)|diesis]]. This is because adding adding pythagorean commas makes the interval go up in pitch, down the scale.
+This counterintuitive notion is a result of just fifths naturally producing a ''[[hard]]'' [[5L 2s|diatonic]] [[MOS scale|MOS]], which means that the [[chroma]] (chromatic semitone) is ''wider'', not narrower, than the small step (diatonic semitone).
 == Temperaments ==
 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 the family 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 fairly 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 sharper than the 12edo fifth of 700{{c}}, such as [[41edo]] and [[53edo]], map the pythagorean comma to a positive small 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 fairly 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 the simplest [[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 (scale theory)|diesis]]). The majority of these edos support [[meantone]], which equates the pythagorean major third [[81/64]] to the 5-limit major third [[5/4]].
+Since it is reached by 12 fifths, a highly composite number, there are many temperaments that split this comma whilst keeping fifths unsplit. Notably:
+* [[Kalismic]], splitting it into 2 [[2835/2816|fwiwismas]].
+* [[Landscape]], splitting it into 3 [[225/224|marvel commas]].
+* [[Nexus]], splitting it into 3 [[243/242|rastmas]].
+* [[Atomic]], splitting it into 12 schismas.
 == See also ==