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, as in septimal meantone and starling, 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
|
This page is a stub. You can help the Xenharmonic Wiki by expanding it. |