Pythagorean comma: Difference between revisions

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~~The '''~~Pythagorean~~''' or '''ditonic~~ comma~~''' (about 23.460[[Cent~~|~~¢]]) is the interval~~ 531441/524288 ~~or {{Monzo~~| ~~-19 12~~ }} ~~in [[Monzos~~|~~Monzo]] notation. 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 [[2187/2048|apotome]] and the [[256/243~~|Pythagorean ~~minor second]]~~, ~~([[2187/2048]])/([[256/243]]). 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]].~~

{{Interwiki

| en = Pythagorean comma

| de = 531441/524288

| es =

| ja =

}}

{{Infobox Interval

| Ratio = 531441/524288

| Name = Pythagorean comma, ditonic comma

| Color name = LLwM, lalawama pM, poma

| Comma = yes

}}

: '' ~~See~~ also [[~~Gallery of Just Intervals~~]], [[~~comma~~]], [~~http:~~//en.~~wikipedia.org~~/~~wiki/Pythagorean_comma Pythagorean comma - Wikipedia~~]''

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.

[[~~Category:Comma~~]]

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]].

[[~~Category:Pythagorean~~]]

== Temperaments ==

[[de:~~531441~~/~~524288~~]]

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]]

[[Category:Compton]]

[[Category:Commas named after polymaths]]

[[Category:Commas with unknown etymology]]

@@ Line 1: / Line 1: @@
-The '''Pythagorean''' or '''ditonic comma''' (about 23.460[[Cent|&cent;]]) is the interval 531441/524288 or {{Monzo| -19 12 }} in [[Monzos|Monzo]] notation. 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 [[2187/2048|apotome]] and the [[256/243|Pythagorean minor second]], ([[2187/2048]])/([[256/243]]). 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]].
+{{Interwiki
+| en = Pythagorean comma
+| de = 531441/524288
+| es =
+| ja =
+}}
+{{Infobox Interval
+| Ratio = 531441/524288
+| Name = Pythagorean comma, ditonic comma
+| Color name = LLwM, lalawama<br>pM, poma
+| Comma = yes
+}}
+{{Wikipedia| Pythagorean comma }}
-: '' See also [[Gallery of Just Intervals]], [[comma]], [http://en.wikipedia.org/wiki/Pythagorean_comma Pythagorean comma - Wikipedia]''
+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.
-[[Category:Comma]]
+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]].
-[[Category:Pythagorean]]
-<!-- interwiki -->
+== Temperaments ==
-[[de:531441/524288]]
+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]]
+[[Category:Compton]]
+[[Category:Commas named after polymaths]]
+[[Category:Commas with unknown etymology]]