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 ==

In [[~~Diminished (temperament)|diminished temperament~~]]~~, which~~ is ~~named for this chord and~~ [[tempering out|~~tempers~~ out]] ~~both~~ [[~~648/625~~]] ~~and~~ [[36/35]], the ~[[36/25]] diminished fifth is ~~equivalent to~~ ~[[7/5]], and the ~[[216/125]] diminished seventh is ~~equivalent to~~ a ~[[5/3]] major sixth, giving ~~rise to~~ a [[7-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]], 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 between ~6/5 and ~5/3 tempers to ~10/7.)

(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 ~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:

~~In the 5-limit, the simplest interpretation as a stack of ~~~6/5 ~~is a [[~~25~~-odd-limit]] [[essentially tempered chord]] of the [[dimipent]] temperament, another temperament named after this chord.~~

* (Meantone) 1 – 6/5 – 36/25 – 128/75

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

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

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

~~== In meantone ==~~

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

Before the ubiquity of [[12edo]], a diminished seventh chord did not imply an equal tuning of all four thirds (technically three minor thirds and an augmented second). It can be viewed as a 9-odd-limit essentially tempered chord of [[starling]]. See [[starling chords]]. This is still compatible with 12edo and is perhaps as authentic as the 25-odd-~~limit interpretation above.~~

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

== In just intonation ==

* [[10:12:14:17]]

In the [[7-limit]]:

* [[125:150:180:216]] <!-~~- stacked~~ minor thirds -->

* [[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.

* [[225:270:320:384]] <~~!-- Wikipedia~~: ~~“incomplete~~ dominant ninth”, and ~~found in~~ the ~~duodene~~. -->

* [[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. 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.

** [[675:800:960:1152]] (32/27, 6/5, 6/5) is a [[utonal]] chord found on ♯ivo7 ({{Frac|45|32}}).

~~{{stub}}~~

In higher limits:

* [[10:12:14:17]] is a 17-limit interpretation of the chord associated with François-Joseph Fétis.

[[Category:Just intonation chords]]

@@ Line 1: / Line 1: @@
 {{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 [[Diminished (temperament)|diminished temperament]], which is named for this chord and [[tempering out|tempers out]] both [[648/625]] and [[36/35]], the ~[[36/25]] diminished fifth is equivalent to ~[[7/5]], and the ~[[216/125]] diminished seventh is equivalent to a ~[[5/3]] major sixth, giving rise to a [[7-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]], 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 between ~6/5 and ~5/3 tempers to ~10/7.)
+(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 ~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:
-In the 5-limit, the simplest interpretation as a stack of ~6/5 is a [[25-odd-limit]] [[essentially tempered chord]] of the [[dimipent]] temperament, another temperament named after this chord.
+* (Meantone) 1&thinsp;–&thinsp;6/5&thinsp;–&thinsp;36/25&thinsp;–&thinsp;128/75
-* (Dimipent) 1&thinsp;–&thinsp;6/5&thinsp;–&thinsp;25/18&thinsp;–&thinsp;5/3
+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&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
-== In meantone ==
+edo: 0-320-640-880 cents, 1/1-6/5-16/11-5/3
-Before the ubiquity of [[12edo]], a diminished seventh chord did not imply an equal tuning of all four thirds (technically three minor thirds and an augmented second). It can be viewed as a 9-odd-limit essentially tempered chord of [[starling]]. See [[starling chords]]. This is still compatible with 12edo and is perhaps as authentic as the 25-odd-limit interpretation above.
+edo: 0-282-565-918 cents, 1/1-33/28-11/8-56/33
-<!-- This needs more detail. What is the actual starling chord in question, and how does this relate in any way to (especially 5-limit) meantone? -->
 == In just intonation ==
-* [[10:12:14:17]] <!-- Wikipedia: François-Joseph Fétis -->
+In the [[7-limit]]:
-* [[125:150:180:216]] <!-- stacked minor thirds -->
+* [[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.
-* [[225:270:320:384]] <!-- Wikipedia: “incomplete dominant ninth”, and found in the duodene. -->
+* [[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. 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.
+** [[675:800:960:1152]] (32/27, 6/5, 6/5) is a [[utonal]] chord found on ♯iv<sup>o7</sup> ({{Frac|45|32}}).
-{{stub}}
+In higher limits:
+* [[10:12:14:17]] is a 17-limit interpretation of the chord associated with François-Joseph Fétis.
 [[Category:Just intonation chords]]