Diminished seventh chord

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Diminished seventh chord

A 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 stack of three ~6/5 minor thirds is tempered to leave another ~6/5 to close the octave. The ~36/25 diminished fifth is equated with its complement (~25/18), and the ~216/125 diminished seventh is equated with a ~5/3 major sixth, forming 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, and the stack of ~6/5 thirds becomes 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.)

In 5-limit meantone, which tempers out 81/80, a stack of three ~6/5 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 a 7-odd-limit essentially tempered chord:

• (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 equal temperaments

4edo (and multiples thereof): 0-300-600-900 cents

13edo: 0-277-554-923 cents, 1/1-7/6-11/8-12/7

15edo: 0-320-640-880 cents, 1/1-6/5-16/11-5/3

17edo: 0-282-565-918 cents, 1/1-33/28-11/8-56/33

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 3 ⋅ 5² ⋅ 7.
• 25:30:35:42 is the result of using step pattern 6/5, 7/6, 6/5.
• 30:35:42:49 is the result of using step pattern 7/6, 6/5, 7/6.

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 iii^o7 (5⁄4).
  • 225:270:320:384 (6/5, 32/27, 6/5) is an ambitonal chord found on vii^o7 (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^o7 (45⁄32).

In higher limits:

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