Half-diminished seventh chord: Difference between revisions
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add Meantone and Archytas interpretations |
correct Archytas diminished fifth — it's based on 64/45, not 36/25. |
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* (Meantone) 1/1 ‒ [[6/5]] ‒ [[36/25]] ‒ [[9/5]] | * (Meantone) 1/1 ‒ [[6/5]] ‒ [[36/25]] ‒ [[9/5]] | ||
In [[archytas]] temperament, which [[tempering out|tempers out]] [[64/63]], a ~[[16/9]] minor seventh is equated with ~[[7/4]], a ~ | In [[archytas]] temperament, which [[tempering out|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, 6 | * (Archytas) 1/1 ‒ 6/5 ‒ 7/5 ‒ 7/4, with steps 6/5, 7/6, 5/4 | ||
== In just intonation == | == In just intonation == | ||
Revision as of 02:30, 10 September 2024
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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