Half-diminished seventh chord
A half-diminished seventh chord is a tetrad comprising a root, a minor third, a diminished fifth, and a minor seventh.
In temperaments
In meantone (including 12edo), on which traditional tonal harmony is built, the interval of a diminished fifth tempers together 36/25~64/45~1024/729, and the minor seventh tempers together 9/5~16/9, giving a chord with an intervallic odd limit of 25:
In starling temperament, which tempers out 126/125, the ~36/25 diminished fifth is equated with ~10/7, reducing the chord to the 7-odd-limit as an essentially tempered chord:
In archytas temperament, which tempers out 64/63, a ~16/9 minor seventh is equated with ~7/4, a ~64/45 diminished fifth is equated with ~7/5, and the ~35/24 ratio between ~7/4 and ~6/5 is equated with a ~3/2 perfect fifth, yielding another 7-odd-limit essentially tempered chord:
- (Archytas) 1/1 ‒ 6/5 ‒ 7/5 ‒ 7/4, with steps 6/5, 7/6, 5/4
Alternately, if we take the ~9/5 minor seventh instead, archytas gives a 9-odd-limit essentially just chord, tempering together 5:6:7:9 with 45:54:64:81:
- (Archytas) 1/1 ‒ 6/5 ‒ 7/5 ‒ 9/5, with steps 6/5, 7/6, 9/7
In just intonation
In the 7-limit:
In the 5-limit:
- 25:30:36:45
- 45:54:64:80
- 45:54:64:81, another preimage of the meantone half-diminished seventh chord, is found in genus 34 ⋅ 5, and tempered together with 5:6:7:9 in archytas temperament.
- 135:160:192:240
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