Diminished seventh chord: Difference between revisions

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~~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 ==

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

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]]:

* (Dimipent) 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]], ~~giving rise to~~ a [[7-odd-limit]] [[essentially tempered chord]]:

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]]:

* (Diminished) 1 – 6/5 – 7/5 – 5/3

(Note that the interval of ~[[25/18]] between ~6/5 and ~5/3 tempers to ~[[10~~/7]], and the interval of ~[[25/21]] between ~7/5 and ~5/3 tempers to ~[[12~~/7]].)

(Note that the interval of ~[[25/18]] between ~6/5 and ~5/3 tempers to ~[[10/7]].)

In 5-limit [[meantone]], which tempers out [[81/80]], a stack of three minor thirds tempers to ~[[128/75]], leaving a ~[[75/64]] augmented second to close the octave. The resulting chord has an [[intervallic odd limit]] of 75:

In 5-limit [[meantone]], which tempers out [[81/80]], a stack of three ~6/5 minor thirds tempers to ~[[128/75]], leaving a ~[[75/64]] augmented second to close the octave. The resulting chord has an [[intervallic odd limit]] of 75:

* (Meantone) 1 – 6/5 – 36/25 – 128/75

~~Note that ~[[36/25]] also tempers to ~[[64/45]], so this chord also represents a tempered [[225:270:320:384]].~~

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]]:

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~~ [[~~9-odd-limit~~]]:

* (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.

== 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 the [[7-limit]]:

* [[15:18:21:25]] is a [[preimage]] of the chord ~~found in~~ diminished temperament, found in [[Euler-Fokker genus|genus]] 32{{dot}}52{{dot}}7.

* [[15:18:21:25]] is a [[preimage]] of the essentially-tempered chord of diminished temperament, found in [[Euler-Fokker genus|genus]] 32{{dot}}52{{dot}}7.

* [[35:42:50:60]] is a preimage of the chord ~~found in~~ starling temperament, also found in genus 3~~2~~{{dot}}52{{dot}}7.

* [[35:42:50:60]] is a preimage of the essentially-tempered chord of starling temperament, also found in genus 3{{dot}}52{{dot}}7.

* [[25:30:35:42]] is the result of using step pattern 6/5, 7/6, 6/5.

* [[30:35:42:49]] is the result of using step pattern 7/6, 6/5, 7/6.

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.

* 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. ~~The~~ sequence of the intervals uniquely identifies the root and [[rotation]] of the chord:

* 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.

** [[75:90:108:128]] (6/5, 6/5, 32/27) is an [[otonal]] chord found on iiio7 ({{Frac|5|4}}).

** [[225:270:320:384]] (6/5, 32/27, 6/5) is an [[ambitonal]] chord found on viio7 ({{Frac|15|8}}), and is closely related to the [[36:45:54:64]] dominant seventh chord, stacking on an additional minor third (to make a “dominant ninth”) and dropping the root.

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 {{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 ==
-If [[648/625]] is [[tempering out|tempered out]], as in the [[dimipent]] temperament (loosely named for this chord), a ~[[36/25]] diminished fifth is equated with its [[complement]] (~[[25/18]]), a ~[[216/125]] diminished seventh is equated with a ~[[5/3]] major sixth, and the resulting stack of three ~[[6/5]] minor thirds is a [[25-odd-limit]] [[essentially tempered chord]]:
+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]]:
 * (Dimipent) 1&thinsp;–&thinsp;6/5&thinsp;–&thinsp;25/18&thinsp;–&thinsp;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]], giving rise to a [[7-odd-limit]] [[essentially tempered chord]]:
+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]]:
 * (Diminished) 1&thinsp;–&thinsp;6/5&thinsp;–&thinsp;7/5&thinsp;–&thinsp;5/3
-(Note that the interval of ~[[25/18]] between ~6/5 and ~5/3 tempers to ~[[10/7]], and the interval of ~[[25/21]] between ~7/5 and ~5/3 tempers to ~[[12/7]].)
+(Note that the interval of ~[[25/18]] between ~6/5 and ~5/3 tempers to ~[[10/7]].)
-In 5-limit [[meantone]], which tempers out [[81/80]], a stack of three minor thirds tempers to ~[[128/75]], leaving a ~[[75/64]] augmented second to close the octave. The resulting chord has an [[intervallic odd limit]] of 75:
+In 5-limit [[meantone]], which tempers out [[81/80]], a stack of three ~6/5 minor thirds tempers to ~[[128/75]], leaving a ~[[75/64]] augmented second to close the octave. The resulting chord has an [[intervallic odd limit]] of 75:
 * (Meantone) 1&thinsp;–&thinsp;6/5&thinsp;–&thinsp;36/25&thinsp;–&thinsp;128/75
-Note that ~[[36/25]] also tempers to ~[[64/45]], so this chord also represents a tempered [[225:270:320:384]].
+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]]:
-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 [[9-odd-limit]]:
 * (Starling) 1&thinsp;–&thinsp;6/5&thinsp;–&thinsp;10/7&thinsp;–&thinsp;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.
+== In equal temperaments ==
+edo (and multiples thereof): 0-300-600-900 cents
+edo: 0-277-554-923 cents, 1/1-7/6-11/8-12/7
+edo: 0-320-640-880 cents, 1/1-6/5-16/11-5/3
+edo: 0-282-565-918 cents, 1/1-33/28-11/8-56/33
 == In just intonation ==
 In the [[7-limit]]:
-* [[15:18:21:25]] is a [[preimage]] of the chord found in 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 chord found in 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.
+* [[25:30:35:42]] is the result of using step pattern 6/5, 7/6, 6/5.
+* [[30:35:42:49]] is the result of using step pattern 7/6, 6/5, 7/6.
 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.
-* 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. The sequence of the intervals uniquely identifies the root and [[rotation]] of the chord:
+* 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.
 ** [[75:90:108:128]] (6/5, 6/5, 32/27) is an [[otonal]] chord found on iii<sup>o7</sup> ({{Frac|5|4}}).
 ** [[225:270:320:384]] (6/5, 32/27, 6/5) is an [[ambitonal]] chord found on vii<sup>o7</sup> ({{Frac|15|8}}), and is closely related to the [[36:45:54:64]] dominant seventh chord, stacking on an additional minor third (to make a “dominant ninth”) and dropping the root.