Pythagorean comma - Revision history

FloraC: This is not a negative interval! Diatonic degrees don't take positive or negative cuz they are ordinals. Pitch does. Also remove "tunings close to [Pythagorean tuning]" as that's vague

2026-06-10T06:27:50Z

This is not a negative interval! Diatonic degrees don't take positive or negative cuz they are ordinals. Pitch does. Also remove "tunings close to [Pythagorean tuning]" as that's vague

← Older revision		Revision as of 06:27, 10 June 2026
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	{{Wikipedia\| Pythagorean comma }}		{{Wikipedia\| Pythagorean comma }}

	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 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 - an augmented seventh - and 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.

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

	== Temperaments ==		== Temperaments ==

Eufalesio: Perhaps it is useful to keep this paragraph headed, not part of the intro

2026-06-09T18:24:51Z

Perhaps it is useful to keep this paragraph headed, not part of the intro

← Older revision		Revision as of 18:24, 9 June 2026
Line 13:		Line 13:
	{{Wikipedia\| Pythagorean comma }}		{{Wikipedia\| Pythagorean comma }}

	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 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.		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 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 - an augmented seventh - and an octave.

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

	== Temperaments ==		== Temperaments ==
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	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]].		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:		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]].		* [[Kalismic]], splitting it into 2 [[2835/2816\|fwiwismas]].
	* [[Landscape]], splitting it into 3 [[225/224\|marvel commas]].		* [[Landscape]], splitting it into 3 [[225/224\|marvel commas]].
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	== See also ==		== See also ==
	* [[Mercator's comma]], the difference between 53 perfect fifths and 31 octaves		* [[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]]		* [[Gallery of just intervals]]
	* [[Small comma]]		* [[Small comma]]

Eufalesio: Updated color notation

2026-05-01T14:50:21Z

Updated color notation

← Older revision		Revision as of 14:50, 1 May 2026
Line 8:		Line 8:
	\| Ratio = 531441/524288		\| Ratio = 531441/524288
	\| Name = Pythagorean comma, ditonic comma		\| Name = Pythagorean comma, ditonic comma
	\| Color name = ~~LLw-2~~, ~~Lalawa comma~~		\| Color name = LLwM, lalawama<br>pM, poma
	\| Comma = yes		\| Comma = yes
	}}		}}

Overthink: caps

2026-04-09T15:14:21Z

caps

← Older revision		Revision as of 15:14, 9 April 2026
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 [[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 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.		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 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]] or tunings close to it, this interval is an ''inverse'' diminished second. This is because adding ~~pythagorean~~ commas makes the interval go up in pitch, down the scale. This apparently counterintuitive notion is a result of just fifths naturally producing a [[TAMNAMS #Step ratio spectrum\|hard-of-basic]] [[5L 2s\|diatonic]] scale, which means that the [[chromatic semitone]] is wider, not narrower, than the [[diatonic semitone]].		In [[Pythagorean tuning]] or tunings close to it, this interval is an ''inverse'' diminished second. This is because adding Pythagorean commas makes the interval go up in pitch, down the scale. This apparently counterintuitive notion is a result of just fifths naturally producing a [[TAMNAMS #Step ratio spectrum\|hard-of-basic]] [[5L 2s\|diatonic]] scale, which means that the [[chromatic semitone]] is wider, not narrower, than the [[diatonic semitone]].

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

FloraC: Partial reversal. This is the diesis, not negative diesis, in pythagorean tuning. Misc. wording and linking improvements

2026-04-09T09:42:06Z

Partial reversal. This is the diesis, not negative diesis, in pythagorean tuning. Misc. wording and linking improvements

← Older revision		Revision as of 09:42, 9 April 2026
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	{{~~interwiki~~		{{Interwiki
			\| en = Pythagorean comma
	\| de = 531441/524288		\| de = 531441/524288
	~~\| en = Pythagorean comma~~
	\| es =		\| es =
	\| ja =		\| ja =
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	{{Wikipedia\| Pythagorean comma }}		{{Wikipedia\| Pythagorean comma }}

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

	== Temperaments ==		== 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.		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 the ~~simplest~~ [[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 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]].		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:		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]].		* [[Kalismic]], splitting it into 2 [[2835/2816\|fwiwismas]].
	* [[Landscape]], splitting it into 3 [[225/224\|marvel commas]].		* [[Landscape]], splitting it into 3 [[225/224\|marvel commas]].

Eufalesio: Added a little explanation for negative interval among notable temperaments that split this

2026-04-09T09:12:15Z

Added a little explanation for negative interval among notable temperaments that split this

← Older revision		Revision as of 09:12, 9 April 2026
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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.		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 ==		== 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.		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]].		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 ==		== See also ==

Videogamer1002: spelling

2026-02-08T21:14:32Z

spelling

← Older revision		Revision as of 21:14, 8 February 2026
Line 13:		Line 13:
	{{Wikipedia\| Pythagorean comma }}		{{Wikipedia\| Pythagorean comma }}

	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.		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]], this interval is an ''inverse'' diminished second, even though it has a positive size.

FloraC: - typo. Wording, style and links

2025-12-24T09:31:52Z

- typo. Wording, style and links

← Older revision		Revision as of 09:31, 24 December 2025
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 betweem six [[9/8]] major seconds and 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.		In [[pythagorean tuning]], this interval is an ''inverse'' diminished second, even though it has a positive size.

	== Temperaments ==		== 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 ~~a collection~~ 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 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 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 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]].

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

Overthink: /* See also */ explain what mercator's comma is

2025-12-24T04:02:44Z

See also: explain what mercator's comma is

← Older revision		Revision as of 04:02, 24 December 2025
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	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]].		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 ==		== See also ==
	* [[Mercator's comma]]		* [[Mercator's comma]], the difference between 53 perfect fifths and 31 octaves
	* [[Gallery of just intervals]]		* [[Gallery of just intervals]]
	* [[Small comma]]		* [[Small comma]]

Overthink: expanded description

2025-12-24T03:56:23Z

expanded description

← Older revision		Revision as of 03:56, 24 December 2025
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	{{Wikipedia\| Pythagorean comma }}		{{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 ==		== 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 ==		== See also ==
	* [[Mercator's comma]]		* [[Mercator's comma]]

← Older revision		Revision as of 15:14, 9 April 2026
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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 [[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 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.		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 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]] or tunings close to it, this interval is an ''inverse'' diminished second. This is because adding ~~pythagorean~~ commas makes the interval go up in pitch, down the scale. This apparently counterintuitive notion is a result of just fifths naturally producing a [[TAMNAMS #Step ratio spectrum\|hard-of-basic]] [[5L 2s\|diatonic]] scale, which means that the [[chromatic semitone]] is wider, not narrower, than the [[diatonic semitone]].		In [[Pythagorean tuning]] or tunings close to it, this interval is an ''inverse'' diminished second. This is because adding Pythagorean commas makes the interval go up in pitch, down the scale. This apparently counterintuitive notion is a result of just fifths naturally producing a [[TAMNAMS #Step ratio spectrum\|hard-of-basic]] [[5L 2s\|diatonic]] scale, which means that the [[chromatic semitone]] is wider, not narrower, than the [[diatonic semitone]].

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

← Older revision		Revision as of 09:12, 9 April 2026
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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.		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 ==		== 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.		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]].		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 ==		== See also ==