64/63 - Revision history

FloraC: No disambiguation pages

2026-05-15T11:54:57Z

No disambiguation pages

← Older revision		Revision as of 11:54, 15 May 2026
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	== See also ==		== See also ==
	* [[Septimal comma]] (disambiguation page)
	* [[Gallery of just intervals]]		* [[Gallery of just intervals]]
	* [[List of superparticular intervals]]		* [[List of superparticular intervals]]

FloraC: + redirect notice

2026-05-15T11:53:21Z

+ redirect notice

← Older revision		Revision as of 11:53, 15 May 2026
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	{{~~interwiki~~		{{Interwiki
			\| en = 64/63
	\| de = 64/63		\| de = 64/63
	~~\| en = 64/63~~
	}}		}}
			: ''"Septimal comma" redirects here. For non-idiomatic usages, see [[Septimal]] and [[Comma]].''
	{{Infobox Interval		{{Infobox Interval
	\| Name = septimal comma, Archytas' comma		\| Name = septimal comma, Archytas' comma

TallKite: updated the comma color name

2026-04-29T04:00:48Z

updated the comma color name

← Older revision		Revision as of 04:00, 29 April 2026
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	{{Infobox Interval		{{Infobox Interval
	\| Name = septimal comma, Archytas' comma		\| Name = septimal comma, Archytas' comma
	\| Color name = r1, ru unison, <br/>~~Ru comma~~		\| Color name = r1, ru unison,<br/>rM, ruma
	\| Sound = Ji-64-63-csound-foscil-220hz.mp3		\| Sound = Ji-64-63-csound-foscil-220hz.mp3
	\| Comma = yes		\| Comma = yes

Overthink: /* Temperaments */ split long sentence

2025-12-03T06:19:45Z

Temperaments: split long sentence

← Older revision		Revision as of 06:19, 3 December 2025
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	== Temperaments ==		== Temperaments ==
	[[Tempering out]] this comma leads to [[superpyth]] temperament (sometimes called ''archy'' in the 2.3.7-subgroup), which equates 9/8 and 8/7, and also equates [[7/4]] with [[16/9]]~~, so~~ that the just dominant seventh chord, [[36:45:54:64\|1–5/4–3/2–16/9]], and the harmonic seventh chord, [[4:5:6:7\|1–5/4–3/2–7/4]], are equated to the same chord. Equal temperaments tempering out 64/63 include {{EDOs\| 12, 15, 17, 22, 27, 37, 49 and 59 }}.		[[Tempering out]] this comma leads to [[superpyth]] temperament (sometimes called ''archy'' in the 2.3.7-subgroup), which equates 9/8 and 8/7, and also equates [[7/4]] with [[16/9]]. This means that the just dominant seventh chord, [[36:45:54:64\|1–5/4–3/2–16/9]], and the harmonic seventh chord, [[4:5:6:7\|1–5/4–3/2–7/4]], are equated to the same chord. Equal temperaments tempering out 64/63 include {{EDOs\| 12, 15, 17, 22, 27, 37, 49 and 59 }}.

	Archytas' comma is similar to Didymus' or the syntonic comma, 81/80, in that when it is tempered out it makes a stack of four fifths [[octave reduction\|octave reduced]] equal a relatively consonant major third. In the case of 81/80, the major third is [[5/4]], while with Archytas' comma, the major third is [[9/7]].		Archytas' comma is similar to Didymus' or the syntonic comma, 81/80, in that when it is tempered out it makes a stack of four fifths [[octave reduction\|octave reduced]] equal a relatively consonant major third. In the case of 81/80, the major third is [[5/4]], while with Archytas' comma, the major third is [[9/7]].

Overthink: /* Temperaments */ link to superpyth

2025-12-01T23:00:03Z

Temperaments: link to superpyth

← Older revision		Revision as of 23:00, 1 December 2025
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	== Temperaments ==		== Temperaments ==
	[[Tempering out]] this comma equates 9/8 and 8/7, and also equates [[7/4]] with [[16/9]], so that the just dominant seventh chord, [[36:45:54:64\|1–5/4–3/2–16/9]], and the harmonic seventh chord, [[4:5:6:7\|1–5/4–3/2–7/4]], are equated to the same chord. Equal temperaments tempering out 64/63 include {{EDOs\| 12, 15, 17, 22, 27, 37, 49 and 59 }}.		[[Tempering out]] this comma leads to [[superpyth]] temperament (sometimes called ''archy'' in the 2.3.7-subgroup), which equates 9/8 and 8/7, and also equates [[7/4]] with [[16/9]], so that the just dominant seventh chord, [[36:45:54:64\|1–5/4–3/2–16/9]], and the harmonic seventh chord, [[4:5:6:7\|1–5/4–3/2–7/4]], are equated to the same chord. Equal temperaments tempering out 64/63 include {{EDOs\| 12, 15, 17, 22, 27, 37, 49 and 59 }}.

	Archytas' comma is similar to Didymus' or the syntonic comma, 81/80, in that when it is tempered out it makes a stack of four fifths [[octave reduction\|octave reduced]] equal a relatively consonant major third. In the case of 81/80, the major third is [[5/4]], while with Archytas' comma, the major third is [[9/7]].		Archytas' comma is similar to Didymus' or the syntonic comma, 81/80, in that when it is tempered out it makes a stack of four fifths [[octave reduction\|octave reduced]] equal a relatively consonant major third. In the case of 81/80, the major third is [[5/4]], while with Archytas' comma, the major third is [[9/7]].

FloraC: /* Temperaments */ this isn't where you discuss the temp in detail

2025-12-01T10:34:41Z

Temperaments: this isn't where you discuss the temp in detail

← Older revision		Revision as of 10:34, 1 December 2025
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	== Temperaments ==		== Temperaments ==
	[[Tempering out]] this comma equates 9/8 and 8/7, and also equates [[7/4]] with [[16/9]]. Equal temperaments tempering out 64/63 include {{EDOs\| 12, 15, 17, 22, 27, 37, 49 and 59 }}.		[[Tempering out]] this comma equates 9/8 and 8/7, and also equates [[7/4]] with [[16/9]], so that the just dominant seventh chord, [[36:45:54:64\|1–5/4–3/2–16/9]], and the harmonic seventh chord, [[4:5:6:7\|1–5/4–3/2–7/4]], are equated to the same chord. Equal temperaments tempering out 64/63 include {{EDOs\| 12, 15, 17, 22, 27, 37, 49 and 59 }}.

	Archytas' comma is similar to Didymus' comma, 81/80, in that when it is tempered out it makes a stack of four fifths [[octave reduction\|octave reduced]] equal a relatively consonant major third. In the case of 81/80, the major third is [[5/4]], while with Archytas' comma, the major third is [[9/7]].		Archytas' comma is similar to Didymus' or the syntonic comma, 81/80, in that when it is tempered out it makes a stack of four fifths [[octave reduction\|octave reduced]] equal a relatively consonant major third. In the case of 81/80, the major third is [[5/4]], while with Archytas' comma, the major third is [[9/7]].

	If one is using 9/7 major thirds, this also implies that the major third is split into two equal steps that represent both [[9/8]] and [[8/7]]: if a stack of four fifths (octave-reduced) reaches the interval 9/7, and a stack of two fifths reaches 9/8, then the difference must be (9/7)/(9/8) = 8/7. The 8/7 and 9/8 intervals are equated, however, as a result of the generation process.		If one is using 9/7 major thirds, this also implies that the major third is split into two equal steps that represent both [[9/8]] and [[8/7]]: if a stack of four fifths (octave-reduced) reaches the interval 9/7, and a stack of two fifths reaches 9/8, then the difference must be (9/7)/(9/8) = 8/7. The 8/7 and 9/8 intervals are equated, however, as a result of the generation process.

	The archy temperament contains a version of the [[5L 2s\|diatonic]] scale where the major third represents 9/7, and the minor third represents [[7/6]]. Tertian harmony can thus be used, with the major and minor triads representing [[~]][[14:18:21]] and ~[[6:7:9]] respectively, rather than the ~[[4:5:6]] and ~[[10:12:15]] triads in meantone. Since the fifth is sharp in archy rather than flat in [[meantone]], the large steps (major seconds) are wider, being around 212-222[[cent\|{{c}}]] depending on the tuning. The small steps (minor seconds) are thus narrower, being around 44-71{{c}}. This has the effect of large and small steps being more distinct compared to meantone diatonic, as well as stronger leading tones due to narrower small steps, though one may want to use milder tunings like [[17edo]] to avoid the small step being too narrow.		See [[Archytas family]] for the family of rank-3 temperaments where it is tempered out. See [[Archytas clan]] for the clan of rank-2 temperaments where it is tempered out.

	In the full [[7-limit]], the just dominant seventh chord, [[36:45:54:64\|1–5/4–3/2–16/9]], and the harmonic seventh chord, [[4:5:6:7\|1–5/4–3/2–7/4]], are equated to the same chord. This simplifies harmonic logic in temperaments like [[pajara]].

	See [[Archytas family]] for the family of rank-3 temperaments where it is tempered out. See [[Archytas clan]] for the clan of rank-2 temperaments where it is tempered out.

	== Comma pumps ==		== Comma pumps ==

Overthink: expanded temperaments section, + comma pumps section, swap notation and approximation sections

2025-12-01T02:07:50Z

expanded temperaments section, + comma pumps section, swap notation and approximation sections

← Older revision		Revision as of 02:07, 1 December 2025
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	== Temperaments ==		== Temperaments ==
	[[Tempering out]] this comma equates 9/8 and 8/7, and also equates [[7/4]] with [[16/9]]~~, so that the just dominant seventh chord, [[36:45:54:64\|1–5/4–3/2–16/9]], and the harmonic seventh chord, [[4:5:6:7\|1–5/4–3/2–7/4]], are equated to the same chord~~. Equal temperaments tempering out 64/63 include {{EDOs\| 12, 15, 22, 27, 37, 49 and 59 }}.		[[Tempering out]] this comma equates 9/8 and 8/7, and also equates [[7/4]] with [[16/9]]. Equal temperaments tempering out 64/63 include {{EDOs\| 12, 15, 17, 22, 27, 37, 49 and 59 }}.

	Archytas' comma is similar to Didymus' ~~or the syntonic~~ comma, 81/80, in that when it is tempered out it makes a stack of four fifths [[octave reduction\|octave reduced]] equal a relatively consonant major third. In the case of 81/80, the major third is [[5/4]], while with Archytas' comma, the major third is [[9/7]].		Archytas' comma is similar to Didymus' comma, 81/80, in that when it is tempered out it makes a stack of four fifths [[octave reduction\|octave reduced]] equal a relatively consonant major third. In the case of 81/80, the major third is [[5/4]], while with Archytas' comma, the major third is [[9/7]].

	If ~~you are~~ using 9/7 major thirds, this also implies that the major third is split into two equal steps that represent both [[9/8]] and [[8/7]]: if a stack of four fifths ~~gets you to~~ (octave-~~equivalent~~) 9/7, and a stack of two fifths ~~gets you to~~ 9/8, then the difference must be (9/7)/(9/8) = 8/7. The 8/7 and 9/8 intervals are ~~equal~~, however, as a result of the generation process.		If one is using 9/7 major thirds, this also implies that the major third is split into two equal steps that represent both [[9/8]] and [[8/7]]: if a stack of four fifths (octave-reduced) reaches the interval 9/7, and a stack of two fifths reaches 9/8, then the difference must be (9/7)/(9/8) = 8/7. The 8/7 and 9/8 intervals are equated, however, as a result of the generation process.

	~~See~~ [[~~Archytas family~~]] ~~for~~ the ~~family of rank~~-~~3 temperaments where it is tempered out; see~~ [[~~Archytas clan~~]] ~~for~~ the ~~clan~~ of ~~rank-2 temperaments where it is tempered out~~.		The archy temperament contains a version of the [[5L 2s\|diatonic]] scale where the major third represents 9/7, and the minor third represents [[7/6]]. Tertian harmony can thus be used, with the major and minor triads representing [[~]][[14:18:21]] and ~[[6:7:9]] respectively, rather than the ~[[4:5:6]] and ~[[10:12:15]] triads in meantone. Since the fifth is sharp in archy rather than flat in [[meantone]], the large steps (major seconds) are wider, being around 212-222[[cent\|{{c}}]] depending on the tuning. The small steps (minor seconds) are thus narrower, being around 44-71{{c}}. This has the effect of large and small steps being more distinct compared to meantone diatonic, as well as stronger leading tones due to narrower small steps, though one may want to use milder tunings like [[17edo]] to avoid the small step being too narrow.

	~~== Approximation ==~~		In the full [[7-limit]], the just dominant seventh chord, [[36:45:54:64\|1–5/4–3/2–16/9]], and the harmonic seventh chord, [[4:5:6:7\|1–5/4–3/2–7/4]], are equated to the same chord. This simplifies harmonic logic in temperaments like [[pajara]].
	If one wants to treat Archytas' comma as a musical interval in its own right as opposed to tempering it out, you will find that it acts as a sort of chroma – specifically, it functions as a septimal equivalent of [[~~55/54~~]], ~~from which it differs by a~~ [[~~385~~/~~384\|keenanisma~~]], ~~or of~~ [[56/55]], ~~from which it differs by a~~ [[~~441/440\|werckisma~~]]. ~~In addition, its incredible proximity to 1/44th of the octave – to the point where the~~ [[~~septimal ruthenia\|44-64/63 comma~~]] is tempered out ~~in edos as large as tens of thousands – enables the tuning of~~ [[~~ruthenium~~]] ~~temperament. As a result,~~ the ~~major second~~ of ~~[[22edo]]~~ is ~~a good approximation to [[17/15]]~~, ~~due to it being~~ the ~~[[mediant]] of [[~~9~~/8]]~~ and ~~[[8/7]]~~, so that the ~~~7:8:9 chord~~ is ~~much more accurately a 17/15–17/15 chord~~, ~~with~~ the ~~outer interval as 9~~/~~7, by tempering out [[2025/2023]]~~.
			See [[Archytas family]] for the family of rank-3 temperaments where it is tempered out. See [[Archytas clan]] for the clan of rank-2 temperaments where it is tempered out.

			== Comma pumps ==
			The septimal version of the common vi–ii–V–I progression, which uses the 6:7:9 subminor and 14:18:21 supermajor triads, requires that 64/63 be tempered out in order to avoid shifting the root. If 64/63 is not tempered out and intervals are kept pure, the root in the final I chord will be 64/63 higher than the root in the vi chord.
			{{todo\|add sound example}}

	== Notation ==		== Notation ==
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	=== Sagittal notation ===		=== Sagittal notation ===
	In the [[Sagittal]] system, the downward version of this comma (possibly tempered) is represented by the sagittal {{sagittal \| !) }} and is called the '''7 comma''', or '''7C''' for short, because the simplest interval it notates is 7/1 (equiv. 7/4), as for example in G–F{{nbhsp}}{{sagittal \| !) }}. The upward version is called '''1/7C''' or '''7C up''' and is represented by {{sagittal\| \|) }}.		In the [[Sagittal]] system, the downward version of this comma (possibly tempered) is represented by the sagittal {{sagittal \| !) }} and is called the '''7 comma''', or '''7C''' for short, because the simplest interval it notates is 7/1 (equiv. 7/4), as for example in G–F{{nbhsp}}{{sagittal \| !) }}. The upward version is called '''1/7C''' or '''7C up''' and is represented by {{sagittal\| \|) }}.

			== Approximation ==
			If one wants to treat Archytas' comma as a musical interval in its own right as opposed to tempering it out, you will find that it acts as a sort of chroma – specifically, it functions as a septimal equivalent of [[55/54]], from which it differs by a [[385/384\|keenanisma]], or of [[56/55]], from which it differs by a [[441/440\|werckisma]]. In addition, its incredible proximity to 1/44th of the octave – to the point where the [[septimal ruthenia\|44-64/63 comma]] is tempered out in edos as large as tens of thousands – enables the tuning of [[ruthenium]] temperament. As a result, the major second of [[22edo]] is a good approximation to [[17/15]], due to it being the [[mediant]] of [[9/8]] and [[8/7]], so that the ~7:8:9 chord is much more accurately a 17/15–17/15 chord, with the outer interval as 9/7, by tempering out [[2025/2023]].

	== See also ==		== See also ==

FloraC: Cleanup and amend on problematic statements

2025-11-30T11:20:21Z

Cleanup and amend on problematic statements

← Older revision		Revision as of 11:20, 30 November 2025
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	}}		}}
	{{Wikipedia\|Septimal comma}}		{{Wikipedia\|Septimal comma}}
	'''64/63''', the '''septimal comma''' (also '''Archytas' comma''', or more simply and systematically the '''archytas comma''' or '''archy comma'''), is a [[small comma\|small]] [[7-limit]] [[superparticular]] [[comma]] which separates [[9/8]] and [[8/7]] and has the eighth square number as a numerator. It can be considered the [[2.3.7 subgroup]] equivalent of the [[~~81/80~~]] ~~comma~~, and seperates complex pythagorean intervals from simpler 7-limit ones. For example, it is the difference between [[32/27]] and [[7/6]], and the difference between [[81/64]] and [[9/7]]. Since its numerator is a power of 2, it is a [[Mersenne comma]].		'''64/63''', the '''septimal comma''' (also '''Archytas' comma''', or more simply and systematically the '''archytas comma''' or '''archy comma'''), is a [[small comma\|small]] [[7-limit]] [[superparticular]] [[comma]] which separates [[9/8]] and [[8/7]] and has the eighth square number as a numerator. It can be considered the [[2.3.7 subgroup\|2.3.7-]][[subgroup]] equivalent of the [[syntonic comma]], and seperates complex pythagorean intervals from simpler 7-limit ones. For example, it is the difference between [[32/27]] and [[7/6]], and the difference between [[81/64]] and [[9/7]]. Since its numerator is a power of 2, it is a [[Mersenne comma]].

	== Temperaments ==		== Temperaments ==
	Tempering out this comma equates 9/8 and 8/7, and also equates [[7/4]] with [[16/9]], so that the just dominant seventh chord, ~~1-5~~/~~4-3~~/~~2-16~~/9, and the ~~otonal tetrad~~, 1–5/4–3/2–7/4, are equated to the same chord. Equal ~~divisions of the octave~~ tempering out 64/63 include {{EDOs\| 12, 15, 22, 27, 37, 49 and 59 }}.		[[Tempering out]] this comma equates 9/8 and 8/7, and also equates [[7/4]] with [[16/9]], so that the just dominant seventh chord, [[36:45:54:64\|1–5/4–3/2–16/9]], and the harmonic seventh chord, [[4:5:6:7\|1–5/4–3/2–7/4]], are equated to the same chord. Equal temperaments tempering out 64/63 include {{EDOs\| 12, 15, 22, 27, 37, 49 and 59 }}.

	~~The~~ Archytas' comma is similar to ~~the~~ Didymus or syntonic comma, [[81/80]], in that when it is tempered out it makes a stack of four fifths equal a major third ~~(octave equivalent)~~. In the case of 81/80~~, however~~, the major third is [[5/4]], while with ~~the~~ Archytas comma, the major third is [[9/7]]. ~~(Note that [[porcupine]], which tempers out 64/63, uses a minor tone as a generator and generally is considered to have 5/4 major thirds, so it doesn't depend on this equivalency.)~~		Archytas' comma is similar to Didymus' or the syntonic comma, 81/80, in that when it is tempered out it makes a stack of four fifths [[octave reduction\|octave reduced]] equal a relatively consonant major third. In the case of 81/80, the major third is [[5/4]], while with Archytas' comma, the major third is [[9/7]].

	If you are using 9/7 major thirds, this also implies that the major third is split into two equal steps that represent both [[9/8]] and [[8/7]]: If a stack of four fifths gets you to (octave-equivalent) 9/7, and a stack of two fifths gets you to 9/8, then the difference must be (9/7)/(9/8) = 8/7. The 8/7 and 9/8 intervals are equal, however, as a result of the generation process.		If you are using 9/7 major thirds, this also implies that the major third is split into two equal steps that represent both [[9/8]] and [[8/7]]: if a stack of four fifths gets you to (octave-equivalent) 9/7, and a stack of two fifths gets you to 9/8, then the difference must be (9/7)/(9/8) = 8/7. The 8/7 and 9/8 intervals are equal, however, as a result of the generation process.

	See [[Archytas family]] for the family of rank-3 temperaments where it is tempered out; see [[Archytas clan]] for the clan of rank-2 temperaments where it is tempered out.		See [[Archytas family]] for the family of rank-3 temperaments where it is tempered out; see [[Archytas clan]] for the clan of rank-2 temperaments where it is tempered out.

	== Approximation ==		== Approximation ==
	If one wants to treat Archytas' comma as a musical interval in its own right as opposed to tempering it out, you will find that it acts as a sort of chroma – specifically, it functions as a septimal equivalent of [[55/54]], from which it differs by a [[385/384\|keenanisma]], or of [[56/55]], from which it differs by a [[441/440\|~~Werckisma~~]]. In addition, its incredible proximity to 1/44th of the octave – to the point where the [[~~Septimal~~ ruthenia\|44-64/63 comma]] is tempered out in edos as large as tens of thousands – enables the tuning of [[ruthenium]] temperament. As a result, the major second of [[22edo]] is a good approximation to [[17/15]], due to it being the [[mediant]] of [[9/8]] and [[8/7]], so that the ~7:8:9 chord is much more accurately a 17/15–17/15 chord, with the outer interval as 9/7, by tempering out [[2025/2023]].		If one wants to treat Archytas' comma as a musical interval in its own right as opposed to tempering it out, you will find that it acts as a sort of chroma – specifically, it functions as a septimal equivalent of [[55/54]], from which it differs by a [[385/384\|keenanisma]], or of [[56/55]], from which it differs by a [[441/440\|werckisma]]. In addition, its incredible proximity to 1/44th of the octave – to the point where the [[septimal ruthenia\|44-64/63 comma]] is tempered out in edos as large as tens of thousands – enables the tuning of [[ruthenium]] temperament. As a result, the major second of [[22edo]] is a good approximation to [[17/15]], due to it being the [[mediant]] of [[9/8]] and [[8/7]], so that the ~7:8:9 chord is much more accurately a 17/15–17/15 chord, with the outer interval as 9/7, by tempering out [[2025/2023]].

	== Notation ==		== Notation ==

Overthink: Expanded intro

2025-11-30T05:58:41Z

Expanded intro

← Older revision		Revision as of 05:58, 30 November 2025
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	}}		}}
	{{Wikipedia\|Septimal comma}}		{{Wikipedia\|Septimal comma}}
	'''64/63''', the '''septimal comma''' (also '''Archytas' comma''', or more simply and systematically the '''archytas comma''' or '''archy comma'''), is a [[small comma\|small]] [[7-limit]] [[superparticular]] [[comma]] which separates [[9/8]] and [[8/7]] and has the eighth square number as a numerator. It is a [[Mersenne comma]].		'''64/63''', the '''septimal comma''' (also '''Archytas' comma''', or more simply and systematically the '''archytas comma''' or '''archy comma'''), is a [[small comma\|small]] [[7-limit]] [[superparticular]] [[comma]] which separates [[9/8]] and [[8/7]] and has the eighth square number as a numerator. It can be considered the [[2.3.7 subgroup]] equivalent of the [[81/80]] comma, and seperates complex pythagorean intervals from simpler 7-limit ones. For example, it is the difference between [[32/27]] and [[7/6]], and the difference between [[81/64]] and [[9/7]]. Since its numerator is a power of 2, it is a [[Mersenne comma]].

	== Temperaments ==		== Temperaments ==

Jerdle at 01:37, 25 October 2025

2025-10-25T01:37:14Z

← Older revision		Revision as of 01:37, 25 October 2025
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	}}		}}
	{{Wikipedia\|Septimal comma}}		{{Wikipedia\|Septimal comma}}
	'''64/63''', the '''septimal comma''' (also '''Archytas' comma''' (more simply and systematically the '''archytas comma''' or '''archy comma''')), is a [[small comma\|small]] [[7-limit]] [[superparticular]] [[comma]] which separates [[9/8]] and [[8/7]] and has the eighth square number as a numerator. It is a [[Mersenne comma]].		'''64/63''', the '''septimal comma''' (also '''Archytas' comma''', or more simply and systematically the '''archytas comma''' or '''archy comma'''), is a [[small comma\|small]] [[7-limit]] [[superparticular]] [[comma]] which separates [[9/8]] and [[8/7]] and has the eighth square number as a numerator. It is a [[Mersenne comma]].

	== Temperaments ==		== Temperaments ==