Diminished seventh chord: Difference between revisions

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In the [[5-limit]]:

* [[125:150:180:216]] is the 125-odd chord produced by stacking three [[6/5]] minor thirds. Its [[rotation|rotations]], [[90:108:125:150]], [[75:90:108:125]], and [[108:125:150:180]], represent chords that would be enharmonically equivalent to itself in dimipent temperament, substituting a [[~~125~~/~~108~~]] augmented ~~wholetone in place~~ of ~~one~~ of the ~~minor thirds in the stack~~.

* [[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]], [[90:108:125:150]], [[75:90:108:125]], and [[108:125:150:180]], represent chords that would be enharmonically equivalent to itself in dimipent temperament, substituting the augmented second for one of the minor thirds.

* [[225:270:320:384]] is ~~a 75-odd~~ chord found on viio7 ({{Frac|15|8}}) ~~in the duodene~~, 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.

* [[75:90:~~108~~:~~128~~]] is a ~~75-odd~~ chord found on ~~iii~~o7 ({{Frac|5|4}}) ~~in the [[duodene]]~~.

* 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:

** [[75:90:108:128]] (6/5, 6/5, 32/27) is an [[otonal]] chord found on iiio7 ({{Frac|5|4}}).

** [[225:270:320:384]] (6/5, 32/27, 6/5) is an [[ambitonal]] chord found on viio7 ({{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.

** [[675:800:960:1152]] (32/27, 6/5, 6/5) is a [[utonal]] chord found on ♯ivo7 ({{Frac|45|32}}).

In higher limits:

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 In the [[5-limit]]:
-* [[125:150:180:216]] is the 125-odd chord produced by stacking three [[6/5]] minor thirds. Its [[rotation|rotations]], [[90:108:125:150]], [[75:90:108:125]], and [[108:125:150:180]], represent chords that would be enharmonically equivalent to itself in dimipent temperament, substituting a [[125/108]] augmented wholetone in place of one of the minor thirds in the stack.
+* [[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]], [[90:108:125:150]], [[75:90:108:125]], and [[108:125:150:180]], represent chords that would be enharmonically equivalent to itself in dimipent temperament, substituting the augmented second for one of the minor thirds.
-* [[225:270:320:384]] is a 75-odd chord found on vii<sup>o7</sup> ({{Frac|15|8}}) in the duodene, 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.
-* [[75:90:108:128]] is a 75-odd chord found on iii<sup>o7</sup> ({{Frac|5|4}}) in the [[duodene]].
+* 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:
+** [[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.
+** [[675:800:960:1152]] (32/27, 6/5, 6/5) is a [[utonal]] chord found on ♯iv<sup>o7</sup> ({{Frac|45|32}}).
 In higher limits: