Half-diminished seventh chord

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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:

• (Meantone) 1/1 ‒ 6/5 ‒ 36/25 ‒ 9/5

In archytas temperament, which tempers out 64/63, a ~16/9 minor seventh is equated with ~7/4, a ~36/25 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, 6/5, 5/4

In just intonation

In the 7-limit:

• 5:6:7:9
• 20:24:28:35

In the 5-limit:

• 25:30:36:45
• 45:54:64:80
• 135:160:192:240

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