Diminished seventh chord
The diminished seventh chord is a tetrad comprising a root, minor third, diminished fifth, and diminished seventh, conventionally formed by stacking three minor thirds.
In temperaments
If 648/625 is 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 (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 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, 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:
- (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:
- (Starling) 1 – 6/5 – 10/7 – 12/7
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.
In just intonation
In the 7-limit:
- 15:18:21:25 is a preimage of the chord found in diminished temperament, found in genus 32 ⋅ 52 ⋅ 7.
- 35:42:50:60 is a preimage of the chord found in starling temperament, also found in genus 32 ⋅ 52 ⋅ 7.
In the 5-limit:
- 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 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.
- 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 (5⁄4).
- 225:270:320:384 (6/5, 32/27, 6/5) is an ambitonal chord found on viio7 (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 (45⁄32).
In higher limits:
- 10:12:14:17 is a 17-limit interpretation of the chord associated with François-Joseph Fétis.