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

Similarly, in [[marvel]] temperament, which tempers out [[225/224]] and tempers together 16/9~25/14 (but ''not'' 9/5), the ~64/45 tritone is tempered together with ~10/7, again resulting in a 9-odd-limit essentially tempered chord:

* (Marvel) 1/1 ‒ [[6/5]] ‒ [[10/7]] ‒ [[16/9]], 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 these sevenths (9/5~16/9~25/14), so any 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 another 7-odd-limit essentially tempered chord:

@@ 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
+Similarly, in [[marvel]] temperament, which tempers out [[225/224]] and tempers together 16/9~25/14 (but ''not'' 9/5), the ~64/45 tritone is tempered together with ~10/7, again resulting in a 9-odd-limit essentially tempered chord:
+* (Marvel) 1/1 ‒ [[6/5]] ‒ [[10/7]] ‒ [[16/9]], 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 these sevenths (9/5~16/9~25/14), so any 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 another 7-odd-limit essentially tempered chord: