Diminished seventh chord: Difference between revisions

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== 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 ~[[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]]:

* (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]]:

* (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]], and the interval of ~[[25/21]] between ~7/5 and ~5/3 tempers to ~[[12/7]].)

In 5-limit [[meantone]], 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 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

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, as in [[septimal meantone]] and [[starling]], the chord becomes an [[essentially tempered chord]] in the [[9-odd-limit]]:

~~== In meantone ==~~

* (Starling) 1 – 6/5 – 10/7 – 12/7

~~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.

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.

<!-~~- 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 ==

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 == 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 ~[[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]]:
+* (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]]:
 * (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]], and the interval of ~[[25/21]] between ~7/5 and ~5/3 tempers to ~[[12/7]].)
+In 5-limit [[meantone]], 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 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
+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, as in [[septimal meantone]] and [[starling]], the chord becomes an [[essentially tempered chord]] in the [[9-odd-limit]]:
-== In meantone ==
+* (Starling) 1&thinsp;–&thinsp;6/5&thinsp;–&thinsp;10/7&thinsp;–&thinsp;12/7
-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.
+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.
-<!-- 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 ==