Half-diminished seventh chord: Difference between revisions
Copypaste information from dominant seventh chord page. As this is the inverse of the dominant seventh chord, all that applies to the dominant seventh chord applies here too |
→In just intonation: added 1/1 – 7/6 – 7/5 – 7/4 |
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* (Starling) 1/1 ‒ [[6/5]] ‒ [[10/7]] ‒ [[9/5]], with steps 6/5, 6/5, 5/4 | * (Starling) 1/1 ‒ [[6/5]] ‒ [[10/7]] ‒ [[9/5]], with steps 6/5, 6/5, 5/4 | ||
[[Septimal meantone]], which is well-represented by the historically prevalent [[quarter-comma meantone]], tempers together all three of the tritones (10/7~36/25~64/45) and sevenths (9/5~16/9~25/14), so either of the above interpretations may be relevant for half-diminished seventh chords found in common-practice music. (→ [[Didymic chords #Dominant seventh 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 a [[7-odd-limit]] 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 | |||
* (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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In the [[7-limit]]: | In the [[7-limit]]: | ||
* [[5:6:7:9]] is a highly [[concordant]] 7-limit interpretation of the half-diminished seventh, found in the [[Ringer_scale|Ringer 5]] scale and [[Euler-Fokker genus|genus]] 3<sup>2</sup>{{dot}}5{{dot}}7, and tempered together with 45:54:64:81 in archytas temperament. | * [[5:6:7:9]] is a highly [[concordant]] 7-limit interpretation of the half-diminished seventh, found in the [[Ringer_scale|Ringer 5]] scale and [[Euler-Fokker genus|genus]] 3<sup>2</sup>{{dot}}5{{dot}}7, and tempered together with 45:54:64:81 in archytas temperament. | ||
* [[60:70:84:105]] (1/1 - 7/6 - 7/5 - 7/4) is a subharmonic chord. | |||
* [[20:24:28:35]] is a [[condissonant]] chord found in genus 3{{dot}}5{{dot}}7, and tempered together with 45:54:64:80 in archytas temperament. | * [[20:24:28:35]] is a [[condissonant]] chord found in genus 3{{dot}}5{{dot}}7, and tempered together with 45:54:64:80 in archytas temperament. | ||
* [[35:42:50:63]] is a [[condissonant]] chord found in genus 3<sup>2</sup>{{dot}}5<sup>2</sup>{{dot}}7, and tempered together with 25:30:36:45 in starling temperament. | * [[35:42:50:63]] is a [[condissonant]] chord found in genus 3<sup>2</sup>{{dot}}5<sup>2</sup>{{dot}}7, and tempered together with 25:30:36:45 in starling temperament. | ||
Latest revision as of 23:42, 27 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 9-odd-limit as an essentially tempered chord:
Septimal meantone, which is well-represented by the historically prevalent quarter-comma meantone, tempers together all three of the tritones (10/7~36/25~64/45) and sevenths (9/5~16/9~25/14), so either of the above interpretations may be relevant for half-diminished seventh chords found in common-practice music. (→ Didymic chords #Dominant seventh 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 a 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:
- 5:6:7:9 is a highly concordant 7-limit interpretation of the half-diminished seventh, found in the Ringer 5 scale and genus 32 ⋅ 5 ⋅ 7, and tempered together with 45:54:64:81 in archytas temperament.
- 60:70:84:105 (1/1 - 7/6 - 7/5 - 7/4) is a subharmonic chord.
- 20:24:28:35 is a condissonant chord found in genus 3 ⋅ 5 ⋅ 7, and tempered together with 45:54:64:80 in archytas temperament.
- 35:42:50:63 is a condissonant chord found in genus 32 ⋅ 52 ⋅ 7, and tempered together with 25:30:36:45 in starling temperament.
In the 5-limit:
- 25:30:36:45 is found on iiiø7 (5⁄4) and viø7 (5⁄3) in the duodene.
- 45:54:64:80 found on viiø7 (15⁄8) in Ptolemy's intense diatonic scale (Zarlino), perhaps the most common 5-limit diatonic. It is also found on Vø7 (3⁄2) in the duodene.
- 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 is found on iiø7 (9⁄8) and ♯ivø7 (45⁄32) in the duodene.