Diminished seventh chord - Revision history

FloraC: - arbitrary mentions of certain edos

2026-01-27T19:53:16Z

- arbitrary mentions of certain edos

← Older revision		Revision as of 19:53, 27 January 2026
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	Since [[12edo]] is a good tuning of diminished and supports septimal meantone, and the historically prevalent [[quarter-comma meantone]] is a good tuning of septimal meantone (although it was historically usually analyzed as a 5-limit temperament), any of the above interpretations may be relevant for diminished chords found in common-practice and contemporary music.		Since [[12edo]] is a good tuning of diminished and supports septimal meantone, and the historically prevalent [[quarter-comma meantone]] is a good tuning of septimal meantone (although it was historically usually analyzed as a 5-limit temperament), any of the above interpretations may be relevant for diminished chords found in common-practice and contemporary music.

	~~== In equal temperaments ==~~
	~~4edo (and multiples thereof): 0-300-600-900 cents~~

	~~13edo: 0-277-554-923 cents, 1/1-7/6-11/8-12/7~~

	~~15edo: 0-320-640-880 cents, 1/1-6/5-16/11-5/3~~

	~~17edo: 0-282-565-918 cents, 1/1-33/28-11/8-56/33~~

	== In just intonation ==		== In just intonation ==

FloraC at 19:52, 27 January 2026

2026-01-27T19:52:40Z

← Older revision		Revision as of 19:52, 27 January 2026
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	== In temperaments ==		== In temperaments ==
			If [[648/625]] is [[tempering out\|tempered out]], as in the [[diminished (temperament)\|diminished]] temperament (named for this chord), a stack of three [[~]][[6/5]] minor thirds is tempered to leave another ~6/5 to close the octave. The ~[[36/25]] diminished fifth is equated with its [[complement]] (~[[25/18]]), and the ~[[216/125]] diminished seventh is equated with a ~[[5/3]] major sixth, forming a [[25-odd-limit]] [[essentially tempered chord]]:
	If [[648/625]] is [[tempering out\|tempered out]], as in the [[diminished (temperament)]] temperament (named for this chord), a stack of three [[~]][[6/5]] minor thirds is tempered to leave another ~6/5 to close the octave. The ~[[36/25]] diminished fifth is equated with its [[complement]] (~[[25/18]]), and the ~[[216/125]] diminished seventh is equated with a ~[[5/3]] major sixth, forming a [[25-odd-limit]] [[essentially tempered chord]]:

	* (Diminished) 1 – 6/5 – 25/18 – 5/3		* (Diminished) 1 – 6/5 – 25/18 – 5/3

FloraC: Classical diminished is now canon

2025-09-11T12:57:04Z

Classical diminished is now canon

← Older revision		Revision as of 12:57, 11 September 2025
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	== In temperaments ==		== In temperaments ==

	If [[648/625]] is [[tempering out\|tempered out]], as in the [[~~dimipent~~]] temperament (~~loosely~~ named for this chord), a stack of three [[~]][[6/5]] minor thirds is tempered to leave another ~6/5 to close the octave. The ~[[36/25]] diminished fifth is equated with its [[complement]] (~[[25/18]]), and the ~[[216/125]] diminished seventh is equated with a ~[[5/3]] major sixth, forming a [[25-odd-limit]] [[essentially tempered chord]]:		If [[648/625]] is [[tempering out\|tempered out]], as in the [[diminished (temperament)]] temperament (named for this chord), a stack of three [[~]][[6/5]] minor thirds is tempered to leave another ~6/5 to close the octave. The ~[[36/25]] diminished fifth is equated with its [[complement]] (~[[25/18]]), and the ~[[216/125]] diminished seventh is equated with a ~[[5/3]] major sixth, forming a [[25-odd-limit]] [[essentially tempered chord]]:

	* (~~Dimipent~~) 1 – 6/5 – 25/18 – 5/3		* (Diminished) 1 – 6/5 – 25/18 – 5/3

	If [[36/35]] is also tempered out, giving ~~[[Diminished (temperament)\|~~diminished temperament~~]] (also named for this chord)~~, the ~36/25 diminished fifth is equated with ~[[7/5]], and the stack of ~6/5 thirds becomes a [[7-odd-limit]] [[essentially tempered chord]]:		If [[36/35]] is also tempered out, giving septimal diminished temperament, the ~36/25 diminished fifth is equated with ~[[7/5]], and the stack of ~6/5 thirds becomes a [[7-odd-limit]] [[essentially tempered chord]]:

	* (Diminished) 1 – 6/5 – 7/5 – 5/3		* (Diminished) 1 – 6/5 – 7/5 – 5/3
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	* (Starling) 1 – 6/5 – 10/7 – 12/7		* (Starling) 1 – 6/5 – 10/7 – 12/7

	Since [[12edo]] is a good tuning of ~~dimipent~~ and supports ~~both diminished temperament and~~ septimal meantone, and the historically prevalent [[quarter-comma meantone]] is a good tuning of septimal meantone (although it was historically usually analyzed as a 5-limit temperament), any of the above interpretations may be relevant for diminished chords found in common-practice and contemporary music.		Since [[12edo]] is a good tuning of diminished and supports septimal meantone, and the historically prevalent [[quarter-comma meantone]] is a good tuning of septimal meantone (although it was historically usually analyzed as a 5-limit temperament), any of the above interpretations may be relevant for diminished chords found in common-practice and contemporary music.

	== In equal temperaments ==		== In equal temperaments ==
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	In the [[5-limit]]:		In the [[5-limit]]:
	* [[125:150:180:216]] is the 125-odd chord produced by stacking three [[6/5]] minor thirds, leaving a [[125/108]] augmented second to close the octave. Its [[rotation~~\|rotations~~]] represent chords that would be enharmonically equivalent to itself in ~~dimipent~~ temperament, substituting the augmented second for one of the minor thirds.		* [[125:150:180:216]] is the 125-odd-limit chord produced by stacking three [[6/5]] minor thirds, leaving a [[125/108]] augmented second to close the octave. Its [[rotation]]s represent chords that would be enharmonically equivalent to itself in diminished temperament, substituting the augmented second for one of the minor thirds.

	* In the [[duodene]], there are three unique diminished seventh chords, each combining one [[32/27]] and two 6/5 minor thirds and leaving a [[75/64]] augmented second to close the octave. In just intonation the sequence of the intervals uniquely identifies the root and [[rotation]] of the chord, whereas they would all be [[tempered together]] in meantone.		* In the [[duodene]], there are three unique diminished seventh chords, each combining one [[32/27]] and two 6/5 minor thirds and leaving a [[75/64]] augmented second to close the octave. In just intonation the sequence of the intervals uniquely identifies the root and [[rotation]] of the chord, whereas they would all be [[tempered together]] in meantone.

MisterShafXen at 13:09, 20 April 2025

2025-04-20T13:09:15Z

← Older revision		Revision as of 13:09, 20 April 2025
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	Since [[12edo]] is a good tuning of dimipent and supports both diminished temperament and septimal meantone, and the historically prevalent [[quarter-comma meantone]] is a good tuning of septimal meantone (although it was historically usually analyzed as a 5-limit temperament), any of the above interpretations may be relevant for diminished chords found in common-practice and contemporary music.		Since [[12edo]] is a good tuning of dimipent and supports both diminished temperament and septimal meantone, and the historically prevalent [[quarter-comma meantone]] is a good tuning of septimal meantone (although it was historically usually analyzed as a 5-limit temperament), any of the above interpretations may be relevant for diminished chords found in common-practice and contemporary music.

			== In equal temperaments ==
			4edo (and multiples thereof): 0-300-600-900 cents

			13edo: 0-277-554-923 cents, 1/1-7/6-11/8-12/7

			15edo: 0-320-640-880 cents, 1/1-6/5-16/11-5/3

			17edo: 0-282-565-918 cents, 1/1-33/28-11/8-56/33

	== In just intonation ==		== In just intonation ==

BudjarnLambeth: /* In just intonation */

2024-12-04T06:21:02Z

In just intonation

← Older revision		Revision as of 06:21, 4 December 2024
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	* [[35:42:50:60]] is a preimage of the essentially-tempered chord of starling temperament, also found in genus 3{{dot}}5<sup>2</sup>{{dot}}7.		* [[35:42:50:60]] is a preimage of the essentially-tempered chord of starling temperament, also found in genus 3{{dot}}5<sup>2</sup>{{dot}}7.
	* [[25:30:35:42]] is the result of using step pattern 6/5, 7/6, 6/5.		* [[25:30:35:42]] is the result of using step pattern 6/5, 7/6, 6/5.
	* [[30:35:42:29]] is the result of using step pattern 7/6, 6/5, 7/6.		* [[30:35:42:49]] is the result of using step pattern 7/6, 6/5, 7/6.

	In the [[5-limit]]:		In the [[5-limit]]:

BudjarnLambeth: /* In just intonation */

2024-12-04T06:20:51Z

In just intonation

← Older revision		Revision as of 06:20, 4 December 2024
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	* [[35:42:50:60]] is a preimage of the essentially-tempered chord of starling temperament, also found in genus 3{{dot}}5<sup>2</sup>{{dot}}7.		* [[35:42:50:60]] is a preimage of the essentially-tempered chord of starling temperament, also found in genus 3{{dot}}5<sup>2</sup>{{dot}}7.
	* [[25:30:35:42]] is the result of using step pattern 6/5, 7/6, 6/5.		* [[25:30:35:42]] is the result of using step pattern 6/5, 7/6, 6/5.
			* [[30:35:42:29]] is the result of using step pattern 7/6, 6/5, 7/6.

	In the [[5-limit]]:		In the [[5-limit]]:

BudjarnLambeth: /* In just intonation */ add link to orphaned page

2024-12-04T06:18:06Z

In just intonation: add link to orphaned page

← Older revision		Revision as of 06:18, 4 December 2024
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	* [[15:18:21:25]] is a [[preimage]] of the essentially-tempered chord of diminished temperament, found in [[Euler-Fokker genus\|genus]] 3<sup>2</sup>{{dot}}5<sup>2</sup>{{dot}}7.		* [[15:18:21:25]] is a [[preimage]] of the essentially-tempered chord of diminished temperament, found in [[Euler-Fokker genus\|genus]] 3<sup>2</sup>{{dot}}5<sup>2</sup>{{dot}}7.
	* [[35:42:50:60]] is a preimage of the essentially-tempered chord of starling temperament, also found in genus 3{{dot}}5<sup>2</sup>{{dot}}7.		* [[35:42:50:60]] is a preimage of the essentially-tempered chord of starling temperament, also found in genus 3{{dot}}5<sup>2</sup>{{dot}}7.
			* [[25:30:35:42]] is the result of using step pattern 6/5, 7/6, 6/5.

	In the [[5-limit]]:		In the [[5-limit]]:

Bcmills: The → A, since there are multiple diminished seventh chords

2024-09-07T21:57:45Z

The → A, since there are multiple diminished seventh chords

← Older revision		Revision as of 21:57, 7 September 2024
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	{{Wikipedia}}		{{Wikipedia}}

	~~The~~ '''diminished seventh chord''' is a [[tetrad]] comprising a root, [[minor]] third, [[interval quality\|diminished]] fifth, and diminished seventh, conventionally formed by stacking three minor thirds.		A '''diminished seventh chord''' is a [[tetrad]] comprising a root, [[minor]] third, [[interval quality\|diminished]] fifth, and diminished seventh, conventionally formed by stacking three minor thirds.

	== In temperaments ==		== In temperaments ==

Bcmills: phrasing consistency

2024-09-07T00:42:32Z

phrasing consistency

← Older revision		Revision as of 00:42, 7 September 2024
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	* (Meantone) 1 – 6/5 – 36/25 – 128/75		* (Meantone) 1 – 6/5 – 36/25 – 128/75

	However, if [[126/125]] is tempered out instead or in addition, as in [[starling]] and [[septimal meantone]], the chord becomes ~~an [[essentially tempered chord]] in the~~ [[7-odd-limit]]:		However, if [[126/125]] is tempered out instead or in addition, as in [[starling]] and [[septimal meantone]], the chord becomes a [[7-odd-limit]] [[essentially tempered chord]]:

	* (Starling) 1 – 6/5 – 10/7 – 12/7		* (Starling) 1 – 6/5 – 10/7 – 12/7

Bcmills: /* In just intonation */ correct genus of 35:42:50:60

2024-09-05T03:50:28Z

In just intonation: correct genus of 35:42:50:60

← Older revision		Revision as of 03:50, 5 September 2024
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	In the [[7-limit]]:		In the [[7-limit]]:
	* [[15:18:21:25]] is a [[preimage]] of the essentially-tempered chord of diminished temperament, found in [[Euler-Fokker genus\|genus]] 3<sup>2</sup>{{dot}}5<sup>2</sup>{{dot}}7.		* [[15:18:21:25]] is a [[preimage]] of the essentially-tempered chord of diminished temperament, found in [[Euler-Fokker genus\|genus]] 3<sup>2</sup>{{dot}}5<sup>2</sup>{{dot}}7.
	* [[35:42:50:60]] is a preimage of the essentially-tempered chord of starling temperament, also found in genus 3~~<sup>2</sup>~~{{dot}}5<sup>2</sup>{{dot}}7.		* [[35:42:50:60]] is a preimage of the essentially-tempered chord of starling temperament, also found in genus 3{{dot}}5<sup>2</sup>{{dot}}7.

	In the [[5-limit]]:		In the [[5-limit]]: