Diminished seventh chord: Difference between revisions

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* (Meantone) 1 – 6/5 – 36/25 – 128/75

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

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* [[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.

* 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. ~~The~~ sequence of the intervals uniquely identifies the root and [[rotation]] of the chord:

* 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.

** [[75:90:108:128]] (6/5, 6/5, 32/27) is an [[otonal]] chord found on iii<sup>o7</sup> ({{Frac|5|4}}).

** [[225:270:320:384]] (6/5, 32/27, 6/5) is an [[ambitonal]] chord found on vii<sup>o7</sup> ({{Frac|15|8}}), and is closely related to the [[36:45:54:64]] dominant seventh chord, stacking on an additional minor third (to make a “dominant ninth”) and dropping the root.

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 * (Meantone) 1&thinsp;–&thinsp;6/5&thinsp;–&thinsp;36/25&thinsp;–&thinsp;128/75
-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 instead or in addition, as in [[starling]] and [[septimal meantone]], the chord becomes an [[essentially tempered chord]] in the [[9-odd-limit]]:
@@ Line 36: / Line 34: @@
 * [[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.
-* 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. The sequence of the intervals uniquely identifies the root and [[rotation]] of the chord:
+* 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.
 ** [[75:90:108:128]] (6/5, 6/5, 32/27) is an [[otonal]] chord found on iii<sup>o7</sup> ({{Frac|5|4}}).
 ** [[225:270:320:384]] (6/5, 32/27, 6/5) is an [[ambitonal]] chord found on vii<sup>o7</sup> ({{Frac|15|8}}), and is closely related to the [[36:45:54:64]] dominant seventh chord, stacking on an additional minor third (to make a “dominant ninth”) and dropping the root.