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
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 7-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 essentially-tempered chord of diminished temperament, found in genus 32 ⋅ 52 ⋅ 7.
- 35:42:50:60 is a preimage of the essentially-tempered chord of 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 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. 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 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.