Diminished seventh chord: Difference between revisions

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The '''diminished seventh chord''' is a [[tetrad]] ~~traditionally described as~~ comprising a root, [[minor]] third, [[interval quality|diminished]] fifth, and diminished seventh. In ~~contemporary~~ [[~~12edo~~]] ~~music~~, it is ~~a stack of four identical minor thirds of~~ [[~]][[6/5]] ~~closing at~~ the [[~~octave~~]], [[~~tempering out~~]] [[~~648/625~~]]. In the 5-limit, the simplest interpretation is a [[25-odd-limit]] [[essentially tempered chord]] of the [[dimipent]] temperament, ~~the~~ temperament named after this chord.

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.

* (Dimipent) 1 – 6/5 – 25/18 – 5/3~~, with steps 6/5, 6/5, 6/5, 6/5.~~

== 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]]:

* (Diminished) 1 – 6/5 – 7/5 – 5/3

(Note that the interval between ~6/5 and ~5/3 tempers to ~10/7.)

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.

* (Dimipent) 1 – 6/5 – 25/18 – 5/3

== In meantone ==

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.

== In just intonation ==

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

* [[125:150:180:216]]

* [[225:270:320:384]]

[[Category:Just intonation chords]]

[[Category:Diminished seventh chords| ]]

[[Category:Diminished]]

[[Category:Essentially tempered chords]]

[[Category:7-odd-limit chords]]

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 {{Wikipedia}}
-The '''diminished seventh chord''' is a [[tetrad]] traditionally described as comprising a root, [[minor]] third, [[interval quality|diminished]] fifth, and diminished seventh. In contemporary [[12edo]] music, it is a stack of four identical minor thirds of [[~]][[6/5]] closing at the [[octave]], [[tempering out]] [[648/625]]. In the 5-limit, the simplest interpretation is a [[25-odd-limit]] [[essentially tempered chord]] of the [[dimipent]] temperament, the temperament named after this chord.
+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.
-* (Dimipent) 1&thinsp;–&thinsp;6/5&thinsp;–&thinsp;25/18&thinsp;–&thinsp;5/3, with steps 6/5, 6/5, 6/5, 6/5.
+== 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]]:
+* (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.)
+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.
+* (Dimipent) 1&thinsp;–&thinsp;6/5&thinsp;–&thinsp;25/18&thinsp;–&thinsp;5/3
 == In meantone ==
-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.
+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.
+<!-- 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 -->
+* [[125:150:180:216]] <!-- stacked minor thirds -->
+* [[225:270:320:384]] <!-- Wikipedia: “incomplete dominant ninth”, and found in the duodene. -->
-{{Stub}}
+{{stub}}
 [[Category:Just intonation chords]]
 [[Category:Diminished seventh chords| ]] <!-- main article -->
+[[Category:Diminished]]
+[[Category:Essentially tempered chords]]
+[[Category:7-odd-limit chords]]