Dominant seventh chord: Difference between revisions

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== In temperaments ==

In [[meantone]] (including [[12edo]]), on which traditional tonal harmony is built, the ~~dominant~~ seventh ~~chord has an~~ [[~~intervallic odd limit~~]] ~~of 25:~~

In [[meantone]] (including [[12edo]]), on which traditional tonal harmony is built, the interval of a minor seventh represents [[9/5]][[~]][[16/9]], and the tritone between ~5/4 and ~9/5 represents [[36/25]]~[[64/45]]~[[1024/729]], all [[tempered together]] into a single chord:

* (Meantone) 1/1 ‒ [[5/4]] ‒ [[3/2]] ‒ [[9/5]] ~~with steps 5/4~~, ~~6/5, 6/5.~~

~~Note the ~9~~/~~5 is simultaneously ~~~[[16/9]]~~, and the interval between the ~5/4 and ~9/5 is~~ [[36/25]]~[[64/45]].

* (Meantone) 1/1 ‒ [[5/4]] ‒ [[3/2]] ‒ [[9/5]], with steps 5/4, 6/5, 6/5.

In [[~~septimal meantone~~]] ~~(which is well-represented by the historically prevalent~~ [[~~quarter-comma meantone~~]]), ~~that ~~~[[36/25]]~[[64/45]] is tempered to ~[[10/7]], making the chord an [[essentially tempered chord]] in the [[9-odd-limit]]. (→ [[~~Didymic chords #Dominant seventh chord~~]])

This chord tempers together [[36:45:54:64]], [[20:25:30:36]], and [[108:135:160:192]], with a resulting [[intervallic odd limit]] of 25 due to the simplest interpretation of its tritone being ~36/25.

~~~~

In [[starling]] temperament, which tempers out [[126/125]] and tempers together 9/5~[[25/14]], the ~36/25 tritone is tempered together with ~[[10/7]], making the chord an [[essentially tempered chord]] in the [[9-odd-limit]]. However, note that in starling temperament the seventh of this chord does not also represent the ~16/9 seventh.

* (Starling) 1/1 ‒ [[5/4]] ‒ [[3/2]] ‒ [[9/5]]

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 ‒ [[5/4]] ‒ [[3/2]] ‒ [[16/9]]

[[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 dominant seventh chords found in common-practice music. (→ [[Didymic chords #Dominant seventh chord]])

== In just intonation ==

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 == In temperaments ==
-In [[meantone]] (including [[12edo]]), on which traditional tonal harmony is built, the dominant seventh chord has an [[intervallic odd limit]] of 25:
+In [[meantone]] (including [[12edo]]), on which traditional tonal harmony is built, the interval of a minor seventh represents [[9/5]][[~]][[16/9]], and the tritone between ~5/4 and ~9/5 represents [[36/25]]~[[64/45]]~[[1024/729]], all [[tempered together]] into a single chord:
-* (Meantone) 1/1 ‒ [[5/4]] ‒ [[3/2]] ‒ [[9/5]] with steps 5/4, 6/5, 6/5.
-Note the ~9/5 is simultaneously ~[[16/9]], and the interval between the ~5/4 and ~9/5 is [[36/25]]~[[64/45]].
+* (Meantone) 1/1 ‒ [[5/4]] ‒ [[3/2]] ‒ [[9/5]], with steps 5/4, 6/5, 6/5.
-In [[septimal meantone]] (which is well-represented by the historically prevalent [[quarter-comma meantone]]), that ~[[36/25]]~[[64/45]] is tempered to ~[[10/7]], making the chord an [[essentially tempered chord]] in the [[9-odd-limit]]. (→ [[Didymic chords #Dominant seventh chord]])
+This chord tempers together [[36:45:54:64]], [[20:25:30:36]], and [[108:135:160:192]], with a resulting [[intervallic odd limit]] of 25 due to the simplest interpretation of its tritone being ~36/25.
-<!-- TODO: There are two temperings happening in two different interpretations of this chord: starling takes ~36/25 to ~10/7, and marvel takes ~64/45 to ~10/7. Septimal meantone equates both of those preimages, but we should describe them separately. -->
+In [[starling]] temperament, which tempers out [[126/125]] and tempers together 9/5~[[25/14]], the ~36/25 tritone is tempered together with ~[[10/7]], making the chord an [[essentially tempered chord]] in the [[9-odd-limit]]. However, note that in starling temperament the seventh of this chord does not also represent the ~16/9 seventh.
+* (Starling) 1/1 ‒ [[5/4]] ‒ [[3/2]] ‒ [[9/5]]
+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 ‒ [[5/4]] ‒ [[3/2]] ‒ [[16/9]]
+[[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 dominant seventh chords found in common-practice music. (→ [[Didymic chords #Dominant seventh chord]])
 == In just intonation ==