Diminished seventh chord: Difference between revisions

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== In temperaments ==

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]]:

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]], ~~giving rise to~~ 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

(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]].)

(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 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 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:

* (Meantone) 1 – 6/5 – 36/25 – 128/75

@@ Line 5: / Line 5: @@
 == In temperaments ==
-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]]:
+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]], giving rise to 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
-(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]].)
+(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 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 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:
 * (Meantone) 1&thinsp;–&thinsp;6/5&thinsp;–&thinsp;36/25&thinsp;–&thinsp;128/75