Half-diminished seventh chord: Difference between revisions
add Archytas essentially-just chord |
add starling essentially-tempered chord; fix odd limit of archytas chord |
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In meantone (including [[12edo]]), on which traditional tonal harmony is built, the interval of a diminished fifth [[tempering together|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 meantone (including [[12edo]]), on which traditional tonal harmony is built, the interval of a diminished fifth [[tempering together|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]] | * (Meantone) 1/1 ‒ [[6/5]] ‒ [[36/25]] ‒ [[9/5]], with steps 6/5, 6/5, 5/4 | ||
In [[ | In [[starling]] temperament, which [[tempering out|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]]: | ||
* (Starling) 1/1 ‒ [[6/5]] ‒ [[10/7]] ‒ [[9/5]], with steps 6/5, 6/5, 5/4 | |||
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 | * (Archytas) 1/1 ‒ 6/5 ‒ 7/5 ‒ 7/4, with steps 6/5, 7/6, 5/4 | ||
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* [[5:6:7:9]] | * [[5:6:7:9]] | ||
* [[20:24:28:35]] | * [[20:24:28:35]] | ||
* [[35:42:50:63]] | |||
In the [[5-limit]]: | In the [[5-limit]]: | ||
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[[Category:Half-diminished seventh chords| ]] | [[Category:Half-diminished seventh chords| ]] | ||
[[Category:Just intonation chords]] | [[Category:Just intonation chords]] | ||
[[Category:7-odd-limit chords]] | |||
[[Category:9-odd-limit chords]] | |||
[[Category:Essentially tempered chords]] | |||
{{stub}} | {{stub}} | ||
Revision as of 03:24, 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 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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