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 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 ratio between ~7/4 and ~6/5 is equated with a ~3/2 perfect fifth, yielding a 9-odd-limit essentially tempered chord:
- (Archytas) 1/1 ‒ 6/5 ‒ 7/5 ‒ 7/4, with steps 6/5, 7/6, 5/4
In just intonation
In the 7-limit:
In the 5-limit:
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