Pythagorean comma: Difference between revisions

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{{~~interwiki~~

{{Interwiki

| en = Pythagorean comma

| de = 531441/524288

~~| en = Pythagorean comma~~

| es =

| ja =

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| Ratio = 531441/524288

| Name = Pythagorean comma, ditonic comma

| Color name = ~~LLw-2~~, ~~Lalawa comma~~

| Color name = LLwM, lalawama pM, poma

| Comma = yes

}}

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]]). 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 [[3/2|fifths]] exceed seven [[2/1|octaves]], or in other words (3/2)12/27. It also can be written as the ratio between the Pythagorean 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 (i.e. an augmented seventh) and an octave.

In [[Pythagorean tuning]], this interval is an ''inverse'' diminished second. This is because adding Pythagorean commas makes the interval go up in pitch, down in [[5L 2s|diatonic]] [[degree]]s. This apparently counterintuitive notion is a result of just fifths naturally producing a [[TAMNAMS #Step ratio spectrum|hard-of-basic]] diatonic scale, which means that the [[chromatic semitone]] is wider, not narrower, than the [[diatonic semitone]].

== 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 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 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 one of the most intuitive [[7-limit]] interpretations 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, splitting octaves instead. 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 ==

* [[Mercator's comma]], the difference between 53 perfect fifths and 31 octaves

* [[41-comma]], the difference between 65 octaves and 41 perfect fifths

* [[Gallery of just intervals]]

* [[Small comma]]

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[[Category:Compton]]

[[Category:Commas named after polymaths]]

[[Category:Commas with unknown etymology]]

@@ Line 1: / Line 1: @@
-{{interwiki
+{{Interwiki
+| en = Pythagorean comma
 | de = 531441/524288
-| en = Pythagorean comma
 | es =
 | ja =
@@ Line 8: / Line 8: @@
 | Ratio = 531441/524288
 | Name = Pythagorean comma, ditonic comma
-| Color name = LLw-2, Lalawa comma
+| Color name = LLwM, lalawama<br>pM, poma
 | Comma = yes
 }}
 {{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]]). 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 [[3/2|fifths]] exceed seven [[2/1|octaves]], or in other words (3/2)<sup>12</sup>/2<sup>7</sup>. It also can be written as the ratio between the Pythagorean 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 (i.e. an augmented seventh) and an octave.
+In [[Pythagorean tuning]], this interval is an ''inverse'' diminished second. This is because adding Pythagorean commas makes the interval go up in pitch, down in [[5L 2s|diatonic]] [[degree]]s. This apparently counterintuitive notion is a result of just fifths naturally producing a [[TAMNAMS #Step ratio spectrum|hard-of-basic]] diatonic scale, which means that the [[chromatic semitone]] is wider, not narrower, than the [[diatonic semitone]].
 == 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 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 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 one of the most intuitive [[7-limit]] interpretations 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, splitting octaves instead. 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 ==
+* [[Mercator's comma]], the difference between 53 perfect fifths and 31 octaves
+* [[41-comma]], the difference between 65 octaves and 41 perfect fifths
 * [[Gallery of just intervals]]
 * [[Small comma]]
@@ Line 24: / Line 38: @@
 [[Category:Compton]]
 [[Category:Commas named after polymaths]]
+[[Category:Commas with unknown etymology]]