Half-diminished seventh chord: Difference between revisions

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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:

* (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 [[~~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]]:

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

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* [[5:6:7:9]]

* [[20:24:28:35]]

* [[35:42:50:63]]

In the [[5-limit]]:

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

[[Category:Just intonation chords]]

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

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

[[Category:Essentially tempered chords]]

@@ Line 7: / Line 7: @@
 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 [[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]]:
+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
@@ Line 22: / Line 26: @@
 * [[5:6:7:9]]
 * [[20:24:28:35]]
+* [[35:42:50:63]]
 In the [[5-limit]]:
@@ Line 31: / Line 36: @@
 [[Category:Half-diminished seventh chords| ]]
 [[Category:Just intonation chords]]
+[[Category:7-odd-limit chords]]
+[[Category:9-odd-limit chords]]
+[[Category:Essentially tempered chords]]
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