Diminished seventh chord: Difference between revisions

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== 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 [[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]]:~~

* (Diminished) 1 – 6/5 – 25/18 – 5/3

* (Dimipent) 1 – 6/5 – 25/18 – 5/3

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]]:

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

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* (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 just intonation ==

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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.

* [[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]]:

* [[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.

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 == 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 [[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]]:
+* (Diminished) 1&thinsp;–&thinsp;6/5&thinsp;–&thinsp;25/18&thinsp;–&thinsp;5/3
-* (Dimipent) 1&thinsp;–&thinsp;6/5&thinsp;–&thinsp;25/18&thinsp;–&thinsp;5/3
+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]]:
-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
@@ Line 23: / Line 22: @@
 * (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.
+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 just intonation ==
@@ Line 31: / Line 30: @@
 * [[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.
-* [[30:35:42:29]] 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]]:
-* [[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.