Diminished seventh chord

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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, 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. 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 a 125/108 augmented wholetone in place of one of the minor thirds in the stack.
• 225:270:320:384 is a 75-odd chord found on vii^o7 (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 (5⁄4) in the duodene.

In higher limits:

• 10:12:14:17 is a 17-limit interpretation of the chord associated with François-Joseph Fétis.