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 [[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

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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.

~~== 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 ==

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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 [[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&thinsp;–&thinsp;6/5&thinsp;–&thinsp;25/18&thinsp;–&thinsp;5/3
@@ Line 24: / Line 23: @@
 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 ==
-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 ==