21/20 - Revision history

FloraC: Don't need an extra section for that

2025-12-26T19:25:54Z

Don't need an extra section for that

← Older revision		Revision as of 19:25, 26 December 2025
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	'''21/20''' is a small semitone of about 85 cents. It may be found in [[7-limit]] [[just intonation]] as, for example, the difference between [[4/3]] and [[7/5]], [[8/7]] and [[6/5]], or [[5/3]] and [[7/4]].		'''21/20''' is a small semitone of about 85 cents. It may be found in [[7-limit]] [[just intonation]] as, for example, the difference between [[4/3]] and [[7/5]], [[8/7]] and [[6/5]], or [[5/3]] and [[7/4]].

			In what is known as an authentic cadence, there is a resolution from the V chord to the I chord. If the V is a [[4:5:6:7\|harmonic seventh chord]], its harmonic seventh ([[21/16]] above the tonic) resolves down to the major third of the I chord ([[5/4]]) by a step of 21/20.

	== Terminology ==		== Terminology ==
	21/20 is traditionally called a ''chroma'', perhaps for its proximity (and conflation in systems like septimal [[meantone]]) with the major chroma [[135/128]]. However, it is a ''diatonic semitone'' in both [[Helmholtz–Ellis notation]] and [[Functional Just System]], viewed as the Pythagorean minor second [[256/243]] altered by [[5120/5103]]. [[Marc Sabat]] has taken to call it the ''minor diatonic semitone'' in the same material where [[15/14]] is also named as the major chromatic semitone<ref>Marc Sabat. [https://masa.plainsound.org/pdfs/crystal-growth.pdf ''Three Crystal Growth Algorithms in 23-limit constrained Harmonic Space'']. Plainsound Music Edition, 2008.</ref>.		21/20 is traditionally called a ''chroma'', perhaps for its proximity (and conflation in systems like septimal [[meantone]]) with the major chroma [[135/128]]. However, it is a ''diatonic semitone'' in both [[Helmholtz–Ellis notation]] and [[Functional Just System]], viewed as the Pythagorean minor second [[256/243]] altered by [[5120/5103]]. [[Marc Sabat]] has taken to call it the ''minor diatonic semitone'' in the same material where [[15/14]] is also named as the major chromatic semitone<ref>Marc Sabat. [https://masa.plainsound.org/pdfs/crystal-growth.pdf ''Three Crystal Growth Algorithms in 23-limit constrained Harmonic Space'']. Plainsound Music Edition, 2008.</ref>.

	~~== In music ==~~
	In what is known as an authentic cadence, there is a resolution from the V chord to the I chord. If the V is a [[4:5:6:7\|harmonic seventh chord]], its harmonic seventh ([[21/16]] above the tonic) resolves down to the major third of the I chord ([[5/4]]) by a step of 21/20.
	== Approximation ==		== Approximation ==
	{{Interval edo approximation\|21/20}}		{{Interval edo approximation\|21/20}}

	== Interval chain ==		== Interval chain ==
	An [[interval chain]] of 21/~~20s~~ stacked on top of one another comes close to approximating some important [[JI]] intervals. The error between the approximation and the target JI interval may be tempered out in some [[regular temperaments]].		An [[interval chain]] of 21/20's stacked on top of one another comes close to approximating some important [[JI]] intervals. The error between the approximation and the target JI interval may be tempered out in some [[regular temperaments]].

	Some examples include:		Some examples include:
	* A stack of two 21/20 upwards is ~4¢ from [[11/10]]		* A stack of two 21/20's upwards is ~4{{c}} from [[11/10]];
	* A stack of seven 21/20 upwards is ~9¢ from [[7/5]]		* A stack of seven 21/20's upwards is ~9{{c}} from [[7/5]];
	* A stack of ten 21/20 upwards is ~4¢ from [[13/8]]		* A stack of ten 21/20's upwards is ~4{{c}} from [[13/8]];
	* A stack of twelve 21/20 upwards is ~4¢ from [[9/5]]		* A stack of twelve 21/20's upwards is ~4{{c}} from [[9/5]];
	and		and
	* A stack of six 21/20 downwards is ~~~10¢~~ from [[3/2]]		* A stack of six 21/20's downwards is ~10{{c}} from [[3/2]];
	* A stack of nine 21/20 downwards is ~5¢ from [[9/7]]		* A stack of nine 21/20's downwards is ~5{{c}} from [[9/7]];
	* A stack of eleven 21/20 downwards is ~4¢ from [[7/6]]		* A stack of eleven 21/20's downwards is ~4{{c}} from [[7/6]].

	When treated as a scale, this interval chain can be called the '''[[ambitonal sequence]] of 21/20''' ('''AS21/20''' or '''1ed21/20''').		When treated as a scale, this interval chain can be called the '''[[ambitonal sequence]] of 21/20''' ('''AS21/20''' or '''1ed21/20''').

	1ed21/20 is equal to approximately 14.~~2067 EDO~~, and as a result of tethering between compressed 14 and heavily stretched 15. It is quite [[xenharmonic]] in its sound. It is related to the [[nautilus]], [[sextilifourths]] and [[floral]] temperaments.		1ed21/20 is equal to approximately 14.2067edo, and as a result of tethering between compressed 14 and heavily stretched 15. It is quite [[xenharmonic]] in its sound. It is related to the [[nautilus]], [[sextilifourths]] and [[floral]] temperaments.

	1ed21/20 offers a possible approximation of the no-3s [[11-limit]], or alternatively of the 2.9.5.7.11.17 [[subgroup]].		1ed21/20 offers a possible approximation of the no-3s [[11-limit]], or alternatively of the 2.9.5.7.11.17 [[subgroup]].

	{{Harmonics in equal\|1\|21\|20\|intervals=integer\|columns=11}}		{{Harmonics in equal\|1\|21\|20\|intervals=integer\|columns=11}}
	{{Harmonics in equal\|1\|21\|20\|intervals=integer\|collapsed=1\|start=12\|columns=12}}		{{Harmonics in equal\|1\|21\|20\|intervals=integer\|collapsed=1\|start=12\|columns=12}}
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	[[Category:Septisemi]]		[[Category:Septisemi]]
	[[Category:Commas named after their interval size]]		[[Category:Commas named after their interval size]]
	{{~~todo~~\|improve synopsis}}		{{Todo\|improve synopsis}}

Squib: 21/20 as resolution from 21/16 to 5/4

2025-12-25T22:10:45Z

21/20 as resolution from 21/16 to 5/4

← Older revision		Revision as of 22:10, 25 December 2025
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	{{Wikipedia\|Septimal chromatic semitone}}		{{Wikipedia\|Septimal chromatic semitone}}

	'''21/20''' is a small semitone of about 85 cents. It may be found in [[7-limit]] [[just intonation]] as, for example, the difference between [[4/3]] and [[7/5]], [[8/7]] and [[6/5]], or [[5/3]] and [[7/4]].		'''21/20''' is a small semitone of about 85 cents. It may be found in [[7-limit]] [[just intonation]] as, for example, the difference between [[4/3]] and [[7/5]], [[8/7]] and [[6/5]], or [[5/3]] and [[7/4]].

	== Terminology ==		== Terminology ==
	21/20 is traditionally called a ''chroma'', perhaps for its proximity (and conflation in systems like septimal [[meantone]]) with the major chroma [[135/128]]. However, it is a ''diatonic semitone'' in both [[Helmholtz–Ellis notation]] and [[Functional Just System]], viewed as the Pythagorean minor second [[256/243]] altered by [[5120/5103]]. [[Marc Sabat]] has taken to call it the ''minor diatonic semitone'' in the same material where [[15/14]] is also named as the major chromatic semitone<ref>Marc Sabat. [https://masa.plainsound.org/pdfs/crystal-growth.pdf ''Three Crystal Growth Algorithms in 23-limit constrained Harmonic Space'']. Plainsound Music Edition, 2008.</ref>.		21/20 is traditionally called a ''chroma'', perhaps for its proximity (and conflation in systems like septimal [[meantone]]) with the major chroma [[135/128]]. However, it is a ''diatonic semitone'' in both [[Helmholtz–Ellis notation]] and [[Functional Just System]], viewed as the Pythagorean minor second [[256/243]] altered by [[5120/5103]]. [[Marc Sabat]] has taken to call it the ''minor diatonic semitone'' in the same material where [[15/14]] is also named as the major chromatic semitone<ref>Marc Sabat. [https://masa.plainsound.org/pdfs/crystal-growth.pdf ''Three Crystal Growth Algorithms in 23-limit constrained Harmonic Space'']. Plainsound Music Edition, 2008.</ref>.

			== In music ==
			In what is known as an authentic cadence, there is a resolution from the V chord to the I chord. If the V is a [[4:5:6:7\|harmonic seventh chord]], its harmonic seventh ([[21/16]] above the tonic) resolves down to the major third of the I chord ([[5/4]]) by a step of 21/20.
	== Approximation ==		== Approximation ==
	{{Interval edo approximation\|21/20}}		{{Interval edo approximation\|21/20}}

FloraC: Text replacement - " {{Interval_Edo_Approximation | " to "{{Interval edo approximation|"

2025-11-03T13:17:53Z

Text replacement - " {{Interval_Edo_Approximation | " to "{{Interval edo approximation|"

← Older revision		Revision as of 13:17, 3 November 2025
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	== Approximation ==		== Approximation ==
	{{~~Interval_Edo_Approximation~~ \| 21/20}}		{{Interval edo approximation\|21/20}}
	== Interval chain ==		== Interval chain ==
	An [[interval chain]] of 21/20s stacked on top of one another comes close to approximating some important [[JI]] intervals. The error between the approximation and the target JI interval may be tempered out in some [[regular temperaments]].		An [[interval chain]] of 21/20s stacked on top of one another comes close to approximating some important [[JI]] intervals. The error between the approximation and the target JI interval may be tempered out in some [[regular temperaments]].

Pailiaq at 06:37, 3 November 2025

2025-11-03T06:37:36Z

← Older revision		Revision as of 06:37, 3 November 2025
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	== Approximation ==		== Approximation ==
	{{Interval_Edo_Approximation \| 21/20}}		{{Interval_Edo_Approximation \| 21/20}}


	== Interval chain ==		== Interval chain ==
	An [[interval chain]] of 21/20s stacked on top of one another comes close to approximating some important [[JI]] intervals. The error between the approximation and the target JI interval may be tempered out in some [[regular temperaments]].		An [[interval chain]] of 21/20s stacked on top of one another comes close to approximating some important [[JI]] intervals. The error between the approximation and the target JI interval may be tempered out in some [[regular temperaments]].

Pailiaq at 06:37, 3 November 2025

2025-11-03T06:37:22Z

← Older revision		Revision as of 06:37, 3 November 2025
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	== Terminology ==		== Terminology ==
	21/20 is traditionally called a ''chroma'', perhaps for its proximity (and conflation in systems like septimal [[meantone]]) with the major chroma [[135/128]]. However, it is a ''diatonic semitone'' in both [[Helmholtz–Ellis notation]] and [[Functional Just System]], viewed as the Pythagorean minor second [[256/243]] altered by [[5120/5103]]. [[Marc Sabat]] has taken to call it the ''minor diatonic semitone'' in the same material where [[15/14]] is also named as the major chromatic semitone<ref>Marc Sabat. [https://masa.plainsound.org/pdfs/crystal-growth.pdf ''Three Crystal Growth Algorithms in 23-limit constrained Harmonic Space'']. Plainsound Music Edition, 2008.</ref>.		21/20 is traditionally called a ''chroma'', perhaps for its proximity (and conflation in systems like septimal [[meantone]]) with the major chroma [[135/128]]. However, it is a ''diatonic semitone'' in both [[Helmholtz–Ellis notation]] and [[Functional Just System]], viewed as the Pythagorean minor second [[256/243]] altered by [[5120/5103]]. [[Marc Sabat]] has taken to call it the ''minor diatonic semitone'' in the same material where [[15/14]] is also named as the major chromatic semitone<ref>Marc Sabat. [https://masa.plainsound.org/pdfs/crystal-growth.pdf ''Three Crystal Growth Algorithms in 23-limit constrained Harmonic Space'']. Plainsound Music Edition, 2008.</ref>.

			== Approximation ==
			{{Interval_Edo_Approximation \| 21/20}}


	== Interval chain ==		== Interval chain ==

BudjarnLambeth: /* Interval chain */

2025-09-21T22:53:12Z

Interval chain

← Older revision		Revision as of 22:53, 21 September 2025
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	When treated as a scale, this interval chain can be called the '''[[ambitonal sequence]] of 21/20''' ('''AS21/20''' or '''1ed21/20''').		When treated as a scale, this interval chain can be called the '''[[ambitonal sequence]] of 21/20''' ('''AS21/20''' or '''1ed21/20''').

	1ed21/20 is equal to approximately 14.2067 EDO, and as a result of tethering between compressed 14 and heavily stretched 15 it is quite xenharmonic in its sound. It is related to the [[nautilus]], [[sextilifourths]] and [[floral]] temperaments.		1ed21/20 is equal to approximately 14.2067 EDO, and as a result of tethering between compressed 14 and heavily stretched 15. It is quite [[xenharmonic]] in its sound. It is related to the [[nautilus]], [[sextilifourths]] and [[floral]] temperaments.

	1ed21/20 offers a possible approximation of the no-3s [[11-limit]], or alternatively of the 2.9.5.7.11.17 [[subgroup]].		1ed21/20 offers a possible approximation of the no-3s [[11-limit]], or alternatively of the 2.9.5.7.11.17 [[subgroup]].

BudjarnLambeth: /* Interval chain */ Since both fifths are 40c off, I can't see it being used as dual-fifth, only no-fifth makes sense

2025-09-21T22:52:37Z

Interval chain: Since both fifths are 40c off, I can't see it being used as dual-fifth, only no-fifth makes sense

← Older revision		Revision as of 22:52, 21 September 2025
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	1ed21/20 is equal to approximately 14.2067 EDO, and as a result of tethering between compressed 14 and heavily stretched 15 it is quite xenharmonic in its sound. It is related to the [[nautilus]], [[sextilifourths]] and [[floral]] temperaments.		1ed21/20 is equal to approximately 14.2067 EDO, and as a result of tethering between compressed 14 and heavily stretched 15 it is quite xenharmonic in its sound. It is related to the [[nautilus]], [[sextilifourths]] and [[floral]] temperaments.

	1ed21/20 offers a possible approximation of the no-3s [[11-limit]]. ~~It might alternatively be used as a~~ [[~~dual-fifth tuning~~]].		1ed21/20 offers a possible approximation of the no-3s [[11-limit]], or alternatively of the 2.9.5.7.11.17 [[subgroup]].
	{{Harmonics in equal\|1\|21\|20\|intervals=~~prime~~}}		{{Harmonics in equal\|1\|21\|20\|intervals=integer\|columns=11}}
			{{Harmonics in equal\|1\|21\|20\|intervals=integer\|collapsed=1\|start=12\|columns=12}}

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

BudjarnLambeth: Merge from 1ed21/20

2025-08-01T23:07:52Z

Merge from 1ed21/20

@@ Line 11: / Line 11: @@
 == Terminology ==
 /20 is traditionally called a ''chroma'', perhaps for its proximity (and conflation in systems like septimal [[meantone]]) with the major chroma [[135/128]]. However, it is a ''diatonic semitone'' in both [[Helmholtz–Ellis notation]] and [[Functional Just System]], viewed as the Pythagorean minor second [[256/243]] altered by [[5120/5103]]. [[Marc Sabat]] has taken to call it the ''minor diatonic semitone'' in the same material where [[15/14]] is also named as the major chromatic semitone<ref>Marc Sabat. [https://masa.plainsound.org/pdfs/crystal-growth.pdf ''Three Crystal Growth Algorithms in 23-limit constrained Harmonic Space'']. Plainsound Music Edition, 2008.</ref>.
 == See also ==
 * [[40/21]] – its [[octave complement]]
 * [[10/7]] – its [[fifth complement]]
 * [[List of superparticular intervals]]
 * [[Gallery of just intervals]]

FloraC: Improve linking and style

2025-01-08T11:49:26Z

Improve linking and style

← Older revision		Revision as of 11:49, 8 January 2025
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	== Terminology ==		== Terminology ==
	21/20 is traditionally called a ''chroma'', perhaps for its proximity (and conflation in systems like septimal [[meantone]]) with the major chroma [[135/128]]. However, it is a ''diatonic semitone'' in both [[~~Helmholtz-Ellis~~ notation]] and [[Functional Just System]], viewed as the Pythagorean minor second [[256/243]] altered by [[5120/5103]]. [[Marc Sabat]] has taken to call it the ''minor diatonic semitone'' in the same material where [[15/14]] is also named as the major chromatic semitone<ref>[https://~~marsbat~~.~~space~~/pdfs/crystal-growth.pdf ~~Marc Sabat (2008)~~ Three Crystal Growth Algorithms in 23-limit constrained Harmonic Space]</ref>.		21/20 is traditionally called a ''chroma'', perhaps for its proximity (and conflation in systems like septimal [[meantone]]) with the major chroma [[135/128]]. However, it is a ''diatonic semitone'' in both [[Helmholtz–Ellis notation]] and [[Functional Just System]], viewed as the Pythagorean minor second [[256/243]] altered by [[5120/5103]]. [[Marc Sabat]] has taken to call it the ''minor diatonic semitone'' in the same material where [[15/14]] is also named as the major chromatic semitone<ref>Marc Sabat. [https://masa.plainsound.org/pdfs/crystal-growth.pdf ''Three Crystal Growth Algorithms in 23-limit constrained Harmonic Space'']. Plainsound Music Edition, 2008.</ref>.

	~~<references/>~~

	== See also ==		== See also ==
	* [[40/21]] – its [[octave complement]]		* [[40/21]] – its [[octave complement]]
	* [[10/7]] – its [[fifth complement]]		* [[10/7]] – its [[fifth complement]]
	* [[1ed21/20]] - its ~~equal multiplication~~		* [[1ed21/20]] – its [[ambitonal sequence]]
	* [[List of superparticular intervals]]		* [[List of superparticular intervals]]
	* [[Gallery of just intervals]]		* [[Gallery of just intervals]]
	* [[Septisemi temperaments]], where it is tempered out		* [[Septisemi temperaments]], where it is tempered out

			== References ==
			<references/>

	[[Category:Second]]		[[Category:Second]]

BudjarnLambeth at 07:12, 3 November 2024

2024-11-03T07:12:51Z

← Older revision		Revision as of 07:12, 3 November 2024
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	[[Category:Chroma]]		[[Category:Chroma]]
	[[Category:Septisemi]]		[[Category:Septisemi]]
			[[Category:Commas named after their interval size]]
	{{todo\|improve synopsis}}		{{todo\|improve synopsis}}