81/80 - Revision history

TallKite at 03:55, 29 April 2026

2026-04-29T03:55:36Z

← Older revision		Revision as of 03:55, 29 April 2026
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	{{Infobox Interval		{{Infobox Interval
	\| Name = syntonic comma, Didymus' comma, meantone comma, Ptolemaic comma		\| Name = syntonic comma, Didymus' comma, meantone comma, Ptolemaic comma
	\| Color name = g1, ~~Gu comma~~, <br/> ~~gu unison~~		\| Color name = g1, gu unison,<br/>gM, guma
	\| Comma = yes		\| Comma = yes
	\| Sound = audacity pluck 81 80.wav		\| Sound = audacity pluck 81 80.wav

Overthink: more improvements

2025-12-24T03:16:06Z

more improvements

← Older revision		Revision as of 03:16, 24 December 2025
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	{{Wikipedia\|Syntonic comma}}		{{Wikipedia\|Syntonic comma}}

	The '''syntonic comma''', also known as the '''Didymus' comma''', the '''meantone comma''' or the '''Ptolemaic comma''', with a frequency ratio '''81/80''', is the difference between many [[3-limit]] and [[5-limit]] ratios in [[just intonation]]. Adding or subtracting this comma to/from any complex 3-limit [[ratio]] (such as [[32/27]] or [[81/64]]) creates a 5-limit ratio with a much lower odd-limit (such as [[6/5]] or [[5/4]]). Thus potentially dissonant 3-limit harmonies can often be sweetened via a commatic adjustment. For example, the [[64:81:96]] ~~chord~~ is quite dissonant, but flattening the third by 81/80 ~~gives~~ the much more consonant [[4:5:6]]. However, adding/subtracting this comma to/from the [[4/3\|perfect fourth]], [[3/2\|fifth]], or [[2/1\|octave]] creates a wolf interval of [[odd limit]] 27 or higher, such as the [[40/27]] wolf fifth. Any attempt to tune a fixed-pitch instrument (e.g. guitar or piano) to ~~5-limit just intonation~~ will create ~~such wolves~~; thus, for those who wish to avoid such wolves in composition, [[tempering out]] 81/80 is desirable. This leads to [[meantone]] temperament, which equates the complex pythagorean intervals with the simpler 5-limit ones~~, and~~ also equates [[9/8]] ~~and~~ [[10/9]], giving a tuning for the [[tone\|whole tone]] which is intermediate between ~~10/9 and 9/8~~; hence the name "meantone".		The '''syntonic comma''', also known as the '''Didymus' comma''', the '''meantone comma''' or the '''Ptolemaic comma''', with a frequency ratio '''81/80''', is the difference between many [[3-limit]] and [[5-limit]] ratios in [[just intonation]]. Adding or subtracting this comma to/from any complex 3-limit [[ratio]] (such as [[32/27]] or [[81/64]]) creates a 5-limit ratio with a much lower odd-limit (such as [[6/5]] or [[5/4]]). Thus potentially dissonant 3-limit harmonies can often be sweetened via a commatic adjustment. For example, the pythagorean major triad, [[64:81:96]], is quite dissonant, but flattening the 81/64 major third by 81/80 leads to the much more consonant [[4:5:6]] chord, with a 5/4 major third in place of 81/64. However, adding/subtracting this comma to/from the [[4/3\|perfect fourth]], [[3/2\|fifth]], or [[2/1\|octave]] creates a wolf interval of [[odd limit]] 27 or higher, such as the [[40/27]] wolf fifth. Any attempt to tune a fixed-pitch instrument (e.g. guitar or piano) to make intervals and chords pure in one key will create wolf intervals in others; thus, for those who wish to avoid such wolves in composition, [[tempering out]] 81/80 is desirable. This leads to [[meantone]] temperament, which equates the complex pythagorean intervals with the simpler 5-limit ones. This also equates [[10/9]] with [[9/8]], giving a tuning for the [[tone\|whole tone]] which is intermediate between them; hence the name "meantone".

	81/80 is the smallest [[superparticular]] interval which belongs to the [[5-limit]], and in fact 81/80 is a [[square superparticular]], being the difference between [[10/9]] and [[9/8]], the product of which is the just major third, [[5/4]].		81/80 is the smallest [[superparticular]] interval which belongs to the [[5-limit]], and in fact 81/80 is a [[square superparticular]], being the difference between [[10/9]] and [[9/8]], the product of which is the just major third, [[5/4]].

Overthink: oops

2025-12-24T03:07:50Z

oops

← Older revision		Revision as of 03:07, 24 December 2025
Line 11:		Line 11:
	{{Wikipedia\|Syntonic comma}}		{{Wikipedia\|Syntonic comma}}

	The '''syntonic comma''', also known as the '''Didymus' comma''', the '''meantone comma''' or the '''Ptolemaic comma''', with a frequency ratio '''81/80''', is the difference between many [[3-limit]] and [[5-limit]] ratios in [[just intonation]]. Adding or subtracting this comma to/from any complex 3-limit [[ratio]] (such as [[32/27]] or [[81/64]]) creates a 5-limit ratio with a much lower odd-limit (such as [[6/5]] or [[5/4]]). Thus potentially dissonant 3-limit harmonies can often be sweetened via a commatic adjustment. For example, the [[64:81:96]] chord is quite dissonant, but flattening the third by [[81/80]] gives the much more consonant [[4:5:6]]. However, adding/subtracting this comma to/from the [[4/3\|perfect fourth]], [[3/2\|fifth]], or [[2/1\|octave]] creates a wolf interval of [[odd limit]] 27 or higher, such as the [[40/27]] wolf fifth. Any attempt to tune a fixed-pitch instrument (e.g. guitar or piano) to 5-limit just intonation will create such wolves; thus, for those who wish to avoid such wolves in composition, [[tempering out]] 81/80 is desirable. This leads to [[meantone]] temperament, which equates the complex pythagorean intervals with the simpler 5-limit ones, and also equates [[9/8]] and [[10/9]], giving a tuning for the [[tone\|whole tone]] which is intermediate between 10/9 and 9/8; hence the name "meantone".		The '''syntonic comma''', also known as the '''Didymus' comma''', the '''meantone comma''' or the '''Ptolemaic comma''', with a frequency ratio '''81/80''', is the difference between many [[3-limit]] and [[5-limit]] ratios in [[just intonation]]. Adding or subtracting this comma to/from any complex 3-limit [[ratio]] (such as [[32/27]] or [[81/64]]) creates a 5-limit ratio with a much lower odd-limit (such as [[6/5]] or [[5/4]]). Thus potentially dissonant 3-limit harmonies can often be sweetened via a commatic adjustment. For example, the [[64:81:96]] chord is quite dissonant, but flattening the third by 81/80 gives the much more consonant [[4:5:6]]. However, adding/subtracting this comma to/from the [[4/3\|perfect fourth]], [[3/2\|fifth]], or [[2/1\|octave]] creates a wolf interval of [[odd limit]] 27 or higher, such as the [[40/27]] wolf fifth. Any attempt to tune a fixed-pitch instrument (e.g. guitar or piano) to 5-limit just intonation will create such wolves; thus, for those who wish to avoid such wolves in composition, [[tempering out]] 81/80 is desirable. This leads to [[meantone]] temperament, which equates the complex pythagorean intervals with the simpler 5-limit ones, and also equates [[9/8]] and [[10/9]], giving a tuning for the [[tone\|whole tone]] which is intermediate between 10/9 and 9/8; hence the name "meantone".

	81/80 is the smallest [[superparticular]] interval which belongs to the [[5-limit]], and in fact 81/80 is a [[square superparticular]], being the difference between [[10/9]] and [[9/8]], the product of which is the just major third, [[5/4]].		81/80 is the smallest [[superparticular]] interval which belongs to the [[5-limit]], and in fact 81/80 is a [[square superparticular]], being the difference between [[10/9]] and [[9/8]], the product of which is the just major third, [[5/4]].

Overthink: more understandable

2025-12-24T03:07:05Z

more understandable

← Older revision		Revision as of 03:07, 24 December 2025
Line 11:		Line 11:
	{{Wikipedia\|Syntonic comma}}		{{Wikipedia\|Syntonic comma}}

	The '''syntonic comma''', also known as the '''Didymus' comma''', the '''meantone comma''' or the '''Ptolemaic comma''', with a frequency ratio '''81/80''', is ~~helpful for comparing~~ [[3-limit]] and [[5-limit]] [[just intonation]]. Adding or subtracting this comma to/from any 3-limit [[ratio]] ~~with an~~ [[~~odd limit~~]] ~~of 27~~ or ~~higher~~ creates a 5-limit ratio with a much lower odd-limit. Thus potentially dissonant 3-limit harmonies can often be sweetened via a commatic adjustment. However, adding/subtracting this comma to/from ~~any~~ 3~~-limit ratio of odd limit~~ 3 ~~or less (the 4th~~, ~~5th~~ or ~~8ve),~~ creates a wolf interval of odd limit 27 or higher. Any attempt to tune a fixed-pitch instrument (e.g. guitar or piano) to 5-limit just intonation will create such wolves, thus, for those who ~~have no interest or desire~~ to ~~utilize~~ such wolves in composition, [[tempering out]] 81/80 is desirable. This ~~gives~~ a tuning for the [[tone\|whole tone]] which is intermediate between 10/9 and 9/8~~, and leads to [[meantone\|meantone temperament]],~~ hence the name meantone ~~comma~~.		The '''syntonic comma''', also known as the '''Didymus' comma''', the '''meantone comma''' or the '''Ptolemaic comma''', with a frequency ratio '''81/80''', is the difference between many [[3-limit]] and [[5-limit]] ratios in [[just intonation]]. Adding or subtracting this comma to/from any complex 3-limit [[ratio]] (such as [[32/27]] or [[81/64]]) creates a 5-limit ratio with a much lower odd-limit (such as [[6/5]] or [[5/4]]). Thus potentially dissonant 3-limit harmonies can often be sweetened via a commatic adjustment. For example, the [[64:81:96]] chord is quite dissonant, but flattening the third by [[81/80]] gives the much more consonant [[4:5:6]]. However, adding/subtracting this comma to/from the [[4/3\|perfect fourth]], [[3/2\|fifth]], or [[2/1\|octave]] creates a wolf interval of [[odd limit]] 27 or higher, such as the [[40/27]] wolf fifth. Any attempt to tune a fixed-pitch instrument (e.g. guitar or piano) to 5-limit just intonation will create such wolves; thus, for those who wish to avoid such wolves in composition, [[tempering out]] 81/80 is desirable. This leads to [[meantone]] temperament, which equates the complex pythagorean intervals with the simpler 5-limit ones, and also equates [[9/8]] and [[10/9]], giving a tuning for the [[tone\|whole tone]] which is intermediate between 10/9 and 9/8; hence the name "meantone".

	81/80 is the smallest [[superparticular]] interval which belongs to the [[5-limit]], and in fact 81/80 is a [[square superparticular]], being the difference between [[10/9]] and [[9/8]], the product of which is the just major third, [[5/4]].		81/80 is the smallest [[superparticular]] interval which belongs to the [[5-limit]], and in fact 81/80 is a [[square superparticular]], being the difference between [[10/9]] and [[9/8]], the product of which is the just major third, [[5/4]].

Overthink: swap sections; made superparticular relations table collapsed by default

2025-12-01T02:15:57Z

swap sections; made superparticular relations table collapsed by default

← Older revision		Revision as of 02:15, 1 December 2025
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	* The amount by which [[27/25]] exceeds [[16/15]].		* The amount by which [[27/25]] exceeds [[16/15]].
	* The amount by which [[16/15]] exceeds [[256/243]].		* The amount by which [[16/15]] exceeds [[256/243]].

	~~== Approximation ==~~
	If one wants to treat the syntonic comma as a musical interval in its own right as opposed to tempering it out, one can easily use it in melodies as either an {{w\|appoggiatura}}, an {{w\|acciaccatura}}, or a quick passing tone. It is also very easy to exploit in [[comma pump]] modulations, as among the [[Meantone comma pump examples\|known examples]] of this kind of thing are familiar-sounding chord progressions. Furthermore, not tempering out 81/80 both allows wolf intervals like [[40/27]] and [[27/20]] to be deliberately exploited as dissonances to be resolved, and it also allows one to contrast intervals like 5/4 and [[81/64]]. The [[barium]] temperament exploits the comma by setting it equal to exactly 1/56th of the octave, thus tempering out the [[barium comma]] ({{monzo\| -225 224 -56 }}).

	== 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 '''5 comma''', or '''5C''' for short, because the simplest interval it notates is 5/1 (equiv. 5/4), as for example in C–E{{nbhsp}}{{sagittal \| \! }}. The upward version is called '''1/5C''' or '''5C 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 '''5 comma''', or '''5C''' for short, because the simplest interval it notates is 5/1 (equiv. 5/4), as for example in C–E{{nbhsp}}{{sagittal \| \! }}. The upward version is called '''1/5C''' or '''5C up''' and is represented by {{sagittal\| /\| }}.

			== Approximation ==
			If one wants to treat the syntonic comma as a musical interval in its own right as opposed to tempering it out, one can easily use it in melodies as either an {{w\|appoggiatura}}, an {{w\|acciaccatura}}, or a quick passing tone. It is also very easy to exploit in [[comma pump]] modulations, as among the [[Meantone comma pump examples\|known examples]] of this kind of thing are familiar-sounding chord progressions. Furthermore, not tempering out 81/80 both allows wolf intervals like [[40/27]] and [[27/20]] to be deliberately exploited as dissonances to be resolved, and it also allows one to contrast intervals like 5/4 and [[81/64]]. The [[barium]] temperament exploits the comma by setting it equal to exactly 1/56th of the octave, thus tempering out the [[barium comma]] ({{monzo\| -225 224 -56 }}).

	== Relations to other superparticular ratios ==		== Relations to other superparticular ratios ==
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	Names in brackets refer to 7-limit [[meantone family\|meantone extensions]], or 11-limit rank-3 temperaments from the [[didymus rank-3 family]] that temper out the respective ratios as commas. (Cases where the meantone comma is expressed as a difference, rather than a product, usually correspond to [[exotemperament]]s.)		Names in brackets refer to 7-limit [[meantone family\|meantone extensions]], or 11-limit rank-3 temperaments from the [[didymus rank-3 family]] that temper out the respective ratios as commas. (Cases where the meantone comma is expressed as a difference, rather than a product, usually correspond to [[exotemperament]]s.)

	{\| class="wikitable"		{\| class="wikitable mw-collapsible mw-collapsed"
			\|+ Relations between 81/80 and other superparticular ratios
	\|-		\|-
	! Limit		! Limit
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	* [[40/27]] – its [[fifth complement]]		* [[40/27]] – its [[fifth complement]]
	* [[1ed81/80]] – its equal multiplication		* [[1ed81/80]] – its equal multiplication
	* [[Syntonoschisma]], the difference by which a stack of seven 81/80s falls short of [[12/11]]
	* [[Mercator's comma]]
	* [[Pythagorean comma]]		* [[Pythagorean comma]]
			* [[64/63]] – the septimal comma or Archytas' comma
	* [[Small comma]]		* [[Small comma]]
	* [[List of superparticular intervals]]		* [[List of superparticular intervals]]

FloraC: Sound examples are right below

2025-11-30T11:27:59Z

Sound examples are right below

← Older revision		Revision as of 11:27, 30 November 2025
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	== Comma pumps ==		== Comma pumps ==
	The familiar ~~vi-ii-V-I~~ progression requires that 81/80 be tempered out in order for the root in the vi chord to be the same as the root in the final I chord. If 81/80 is not tempered out, the new root will be 81/80 lower than the original root.		The familiar vi–ii–V–I progression requires that 81/80 be tempered out in order for the root in the vi chord to be the same as the root in the final I chord. If 81/80 is not tempered out, the new root will be 81/80 lower than the original root.
	~~{{Todo\|inline=1\|add sound example}}~~

	~~== Use in recorded music ==~~		A passage ([https://youtu.be/DO7yTiM-YJk?si=e4wVU4IlbITCAaNG&t=325 listen]) from [[Ben Johnston]]'s 9th string quartet, near the end of movement 1, makes a sudden and prominent use of the 81/80 comma, which demonstrates how a simple progression with held common tones can quickly lead to severe interference [[Beat\|beating]], rupturing the diatonic collection routinely associated with the [[5-limit]] and exposing "C major" as anything but simple.
	[https://youtu.be/DO7yTiM-YJk?si=e4wVU4IlbITCAaNG&t=325 ~~This passage~~] from [[Ben Johnston]]'s 9th string quartet, near the end of movement 1, makes a sudden and prominent use of the 81/80 comma, which demonstrates how a simple progression with held common tones can quickly lead to severe interference [[Beat\|beating]], rupturing the diatonic collection routinely associated with the [[5-limit]] and exposing "C major" as anything but simple.

	[[Monroe Golden]]'s ''Incongruity'' uses just-intonation chord progressions that exploit this comma<ref>[http://untwelve.org/interviews/golden UnTwelve's interview to Monroe Golden]</ref>.		[[Monroe Golden]]'s ''Incongruity'' uses just-intonation chord progressions that exploit this comma<ref>[http://untwelve.org/interviews/golden UnTwelve's interview to Monroe Golden]</ref>.

Overthink: removed irrelevant fourth power section

2025-11-30T05:56:22Z

removed irrelevant fourth power section

← Older revision		Revision as of 05:56, 30 November 2025
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	The '''syntonic comma''', also known as the '''Didymus' comma''', the '''meantone comma''' or the '''Ptolemaic comma''', with a frequency ratio '''81/80''', is helpful for comparing [[3-limit]] and [[5-limit]] [[just intonation]]. Adding or subtracting this comma to/from any 3-limit [[ratio]] with an [[odd limit]] of 27 or higher creates a 5-limit ratio with a much lower odd-limit. Thus potentially dissonant 3-limit harmonies can often be sweetened via a commatic adjustment. However, adding/subtracting this comma to/from any 3-limit ratio of odd limit 3 or less (the 4th, 5th or 8ve), creates a wolf interval of odd limit 27 or higher. Any attempt to tune a fixed-pitch instrument (e.g. guitar or piano) to 5-limit just intonation will create such wolves, thus, for those who have no interest or desire to utilize such wolves in composition, [[tempering out]] 81/80 is desirable. This gives a tuning for the [[tone\|whole tone]] which is intermediate between 10/9 and 9/8, and leads to [[meantone\|meantone temperament]], hence the name meantone comma.		The '''syntonic comma''', also known as the '''Didymus' comma''', the '''meantone comma''' or the '''Ptolemaic comma''', with a frequency ratio '''81/80''', is helpful for comparing [[3-limit]] and [[5-limit]] [[just intonation]]. Adding or subtracting this comma to/from any 3-limit [[ratio]] with an [[odd limit]] of 27 or higher creates a 5-limit ratio with a much lower odd-limit. Thus potentially dissonant 3-limit harmonies can often be sweetened via a commatic adjustment. However, adding/subtracting this comma to/from any 3-limit ratio of odd limit 3 or less (the 4th, 5th or 8ve), creates a wolf interval of odd limit 27 or higher. Any attempt to tune a fixed-pitch instrument (e.g. guitar or piano) to 5-limit just intonation will create such wolves, thus, for those who have no interest or desire to utilize such wolves in composition, [[tempering out]] 81/80 is desirable. This gives a tuning for the [[tone\|whole tone]] which is intermediate between 10/9 and 9/8, and leads to [[meantone\|meantone temperament]], hence the name meantone comma.

	81/80 is the smallest [[superparticular]] interval which belongs to the [[5-limit~~]]. Like [[16/15~~]], ~~[[625/624]], [[2401/2400]]~~ and [[4096/4095]], it has a fourth power as a numerator. Fourth powers are squares, and any superparticular comma with a square numerator is the ratio between two wider successive superparticular intervals, because {{nowrap\|''n''<sup>2</sup>/(''n''<sup>2</sup> − 1) {{=}} ''n''/(''n'' − 1) ÷ (''n'' + 1)/''n''}} (which is to say 81/80 is a [[square superparticular]]~~). 81/80 is in fact~~ the difference between [[10/9]] and [[9/8]], the product of which is the just major third, [[5/4]]~~. That the numerator is a fourth power entails that the wider of these two intervals itself has a square numerator; 9/8 is the interval between the successive superparticulars 4/3 and 3/2~~.		81/80 is the smallest [[superparticular]] interval which belongs to the [[5-limit]], and in fact 81/80 is a [[square superparticular]], being the difference between [[10/9]] and [[9/8]], the product of which is the just major third, [[5/4]].

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

Overthink: + comma pumps section

2025-11-30T05:53:14Z

+ comma pumps section

← Older revision		Revision as of 05:53, 30 November 2025
Line 13:		Line 13:
	The '''syntonic comma''', also known as the '''Didymus' comma''', the '''meantone comma''' or the '''Ptolemaic comma''', with a frequency ratio '''81/80''', is helpful for comparing [[3-limit]] and [[5-limit]] [[just intonation]]. Adding or subtracting this comma to/from any 3-limit [[ratio]] with an [[odd limit]] of 27 or higher creates a 5-limit ratio with a much lower odd-limit. Thus potentially dissonant 3-limit harmonies can often be sweetened via a commatic adjustment. However, adding/subtracting this comma to/from any 3-limit ratio of odd limit 3 or less (the 4th, 5th or 8ve), creates a wolf interval of odd limit 27 or higher. Any attempt to tune a fixed-pitch instrument (e.g. guitar or piano) to 5-limit just intonation will create such wolves, thus, for those who have no interest or desire to utilize such wolves in composition, [[tempering out]] 81/80 is desirable. This gives a tuning for the [[tone\|whole tone]] which is intermediate between 10/9 and 9/8, and leads to [[meantone\|meantone temperament]], hence the name meantone comma.		The '''syntonic comma''', also known as the '''Didymus' comma''', the '''meantone comma''' or the '''Ptolemaic comma''', with a frequency ratio '''81/80''', is helpful for comparing [[3-limit]] and [[5-limit]] [[just intonation]]. Adding or subtracting this comma to/from any 3-limit [[ratio]] with an [[odd limit]] of 27 or higher creates a 5-limit ratio with a much lower odd-limit. Thus potentially dissonant 3-limit harmonies can often be sweetened via a commatic adjustment. However, adding/subtracting this comma to/from any 3-limit ratio of odd limit 3 or less (the 4th, 5th or 8ve), creates a wolf interval of odd limit 27 or higher. Any attempt to tune a fixed-pitch instrument (e.g. guitar or piano) to 5-limit just intonation will create such wolves, thus, for those who have no interest or desire to utilize such wolves in composition, [[tempering out]] 81/80 is desirable. This gives a tuning for the [[tone\|whole tone]] which is intermediate between 10/9 and 9/8, and leads to [[meantone\|meantone temperament]], hence the name meantone comma.

	81/80 is the smallest [[superparticular]] interval which belongs to the [[5-limit]]. Like [[16/15]], [[625/624]], [[2401/2400]] and [[4096/4095]] it has a fourth power as a numerator. Fourth powers are squares, and any superparticular comma with a square numerator is the ratio between two wider successive superparticular intervals, because {{nowrap\|''n''<sup>2</sup>/(''n''<sup>2</sup> − 1) {{=}} ''n''/(''n'' − 1) ÷ (''n'' + 1)/''n''}} (which is to say 81/80 is a [[square superparticular]]). 81/80 is in fact the difference between [[10/9]] and [[9/8]], the product of which is the just major third, [[5/4]]. That the numerator is a fourth power entails that the wider of these two intervals itself has a square numerator; 9/8 is the interval between the successive superparticulars 4/3 and 3/2.		81/80 is the smallest [[superparticular]] interval which belongs to the [[5-limit]]. Like [[16/15]], [[625/624]], [[2401/2400]] and [[4096/4095]], it has a fourth power as a numerator. Fourth powers are squares, and any superparticular comma with a square numerator is the ratio between two wider successive superparticular intervals, because {{nowrap\|''n''<sup>2</sup>/(''n''<sup>2</sup> − 1) {{=}} ''n''/(''n'' − 1) ÷ (''n'' + 1)/''n''}} (which is to say 81/80 is a [[square superparticular]]). 81/80 is in fact the difference between [[10/9]] and [[9/8]], the product of which is the just major third, [[5/4]]. That the numerator is a fourth power entails that the wider of these two intervals itself has a square numerator; 9/8 is the interval between the successive superparticulars 4/3 and 3/2.

			== Comma pumps ==
			The familiar vi-ii-V-I progression requires that 81/80 be tempered out in order for the root in the vi chord to be the same as the root in the final I chord. If 81/80 is not tempered out, the new root will be 81/80 lower than the original root.
			{{Todo\|inline=1\|add sound example}}

	== Use in recorded music ==		== Use in recorded music ==

Lucius Chiaraviglio: /* Approximation */ Link to the barium comma

2025-11-24T00:01:15Z

Approximation: Link to the barium comma

← Older revision		Revision as of 00:01, 24 November 2025
Line 36:		Line 36:

	== Approximation ==		== Approximation ==
	If one wants to treat the syntonic comma as a musical interval in its own right as opposed to tempering it out, one can easily use it in melodies as either an {{w\|appoggiatura}}, an {{w\|acciaccatura}}, or a quick passing tone. It is also very easy to exploit in [[comma pump]] modulations, as among the [[Meantone comma pump examples\|known examples]] of this kind of thing are familiar-sounding chord progressions. Furthermore, not tempering out 81/80 both allows wolf intervals like [[40/27]] and [[27/20]] to be deliberately exploited as dissonances to be resolved, and it also allows one to contrast intervals like 5/4 and [[81/64]]. The [[barium]] temperament exploits the comma by setting it equal to exactly 1/56th of the octave.		If one wants to treat the syntonic comma as a musical interval in its own right as opposed to tempering it out, one can easily use it in melodies as either an {{w\|appoggiatura}}, an {{w\|acciaccatura}}, or a quick passing tone. It is also very easy to exploit in [[comma pump]] modulations, as among the [[Meantone comma pump examples\|known examples]] of this kind of thing are familiar-sounding chord progressions. Furthermore, not tempering out 81/80 both allows wolf intervals like [[40/27]] and [[27/20]] to be deliberately exploited as dissonances to be resolved, and it also allows one to contrast intervals like 5/4 and [[81/64]]. The [[barium]] temperament exploits the comma by setting it equal to exactly 1/56th of the octave, thus tempering out the [[barium comma]] ({{monzo\| -225 224 -56 }}).

	== Notation ==		== Notation ==

Lucius Chiaraviglio: /* Relations to other 5-limit intervals */ Add the amount by which 648/625 exceeds 128/125.

2025-11-23T22:53:51Z

Relations to other 5-limit intervals: Add the amount by which 648/625 exceeds 128/125.

← Older revision		Revision as of 22:53, 23 November 2025
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	* The amount by which [[25/24]] exceeds [[250/243]].		* The amount by which [[25/24]] exceeds [[250/243]].
	* The amount by which [[135/128]] exceeds [[25/24]].		* The amount by which [[135/128]] exceeds [[25/24]].
			* The amount by which [[648/625]] exceeds [[128/125]].
	* The amount by which [[128/125]] exceeds [[2048/2025]].		* The amount by which [[128/125]] exceeds [[2048/2025]].
	* The amount by which [[27/25]] exceeds [[16/15]].		* The amount by which [[27/25]] exceeds [[16/15]].