Half-diminished seventh chord: Difference between revisions

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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 [[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]], 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 [[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

[[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_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.

In the [[5-limit]]:

* [[25:30:36:45]] is found on iiiø7 ({{Frac|5|4}}) and viø7 ({{Frac|5|3}}) in the [[duodene]].

* [[45:54:64:80]] found on viiø7 ({{Frac|15|8}}) in Ptolemy's intense diatonic scale ([[Zarlino]]), perhaps the most common 5-limit diatonic. It is also found on Vø7 ({{Frac|3|2}}) in the [[duodene]].

* [[45:54:64:81]], another [[preimage]] of the meantone half-diminished seventh chord, is found in [[Euler-Fokker genus|genus]] 34{{dot}}5, and tempered together with 5:6:7:9 in archytas temperament.

* [[135:160:192:240]] is found on iiø7 ({{Frac|9|8}}) and ♯ivø7 ({{Frac|45|32}}) in the [[duodene]].

[[Category:Half-diminished seventh chords| ]]

[[Category:Just intonation chords]]

[[Category:7-odd-limit chords]]

~~{{stub}}~~

[[Category:9-odd-limit chords]]

[[Category:Essentially tempered chords]]

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 {{Wikipedia}}
+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 [[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]], 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 [[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
+[[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_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.
+In the [[5-limit]]:
+* [[25:30:36:45]] is found on iii<sup>ø7</sup> ({{Frac|5|4}}) and vi<sup>ø7</sup> ({{Frac|5|3}}) in the [[duodene]].
+* [[45:54:64:80]] found on vii<sup>ø7</sup> ({{Frac|15|8}}) in Ptolemy's intense diatonic scale ([[Zarlino]]), perhaps the most common 5-limit diatonic. It is also found on V<sup>ø7</sup> ({{Frac|3|2}}) in the [[duodene]].
+* [[45:54:64:81]], another [[preimage]] of the meantone half-diminished seventh chord, is found in [[Euler-Fokker genus|genus]] 3<sup>4</sup>{{dot}}5, and tempered together with 5:6:7:9 in archytas temperament.
+* [[135:160:192:240]] is found on ii<sup>ø7</sup> ({{Frac|9|8}}) and ♯iv<sup>ø7</sup> ({{Frac|45|32}}) in the [[duodene]].
 [[Category:Half-diminished seventh chords| ]]
 [[Category:Just intonation chords]]
+[[Category:7-odd-limit chords]]
-{{stub}}
+[[Category:9-odd-limit chords]]
+[[Category:Essentially tempered chords]]