Half-diminished seventh chord: Difference between revisions

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* (Meantone) 1/1 ‒ [[6/5]] ‒ [[36/25]] ‒ [[9/5]], with steps 6/5, 6/5, 5/4

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]]:

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 [[9-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:

[[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

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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]]32{{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]] 32{{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.

* [[35:42:50:63]] is a [[condissonant]] chord found in genus 32{{dot}}52{{dot}}7, and tempered together with 25:30:36:45 in starling temperament.

@@ Line 9: / Line 9: @@
 * (Meantone) 1/1 ‒ [[6/5]] ‒ [[36/25]] ‒ [[9/5]], with steps 6/5, 6/5, 5/4
-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]]:
+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 [[9-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:
+[[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
@@ Line 24: / Line 26: @@
 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.
 * [[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.