Dominant seventh chord: Difference between revisions

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== In temperaments ==

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.

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

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.~~

[[Septimal meantone]], which is well-represented by the historically prevalent [[quarter-comma meantone]], tempers the tritone to ~[[10/7]], making the chord an [[essentially tempered chord]] in the [[9-odd-limit]]. In fact, tempering out the starling comma [[126/125]] alone is enough to convert it to a 9-odd-limit essentially tempered chord:

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

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

[[~~Septimal meantone~~]]~~, which is well-represented by~~ the ~~historically prevalent~~ [[~~quarter-comma meantone~~]], ~~tempers~~ together all three of the sevenths (9/5~16/9~25/14), so either of the above interpretations may be relevant for dominant seventh chords found in common-practice music. (→ [[Didymic chords #Dominant seventh chord]])

However, in [[starling]] the seventh of this chord represents 9/5~[[25/14]], but not 16/9. Septimal meantone tempering is necessary to temper together all three of the sevenths (9/5~16/9~25/14), so either of the above interpretations may be relevant for dominant seventh chords found in common-practice music. (→ [[Didymic chords #Dominant seventh chord]])

In [[archytas]] temperament, which tempers out [[64/63]], ~16/9 is equated with ~[[7/4]] rather than 25/14, resulting in an [[Dyadic chord#Essentially tempered dyadic chord|essentially just]] [[7-odd-limit]] chord that tempers together [[4:5:6:7]] and [[36:45:54:64]]:

* (Archytas) 1/1 - [[5/4]] - [[3/2]] - [[7/4]]

* (Archytas) 1/1 – [[5/4]] – [[3/2]] – [[7/4]]

[[Dominant (temperament)|Dominant temperament]] combines archytas with meantone, tempering out both 81/80 and 64/63, and as a result also tempers out [[36/35]], equating 4:5:6:7 with all of the 5-limit dominant seventh chords of meantone. Since [[12edo]] is a good tuning of Dominant temperament, this simpler septimal interpretation may also be relevant for dominant seventh chords in music originally composed for 12edo — particularly in performance styles that use more flexible intonation (such as Barbershop).

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 == In temperaments ==
 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.
+* (Meantone) 1/1&thinsp;‒&thinsp;[[5/4]]&thinsp;‒&thinsp;[[3/2]]&thinsp;‒&thinsp;[[9/5]], with steps 5/4, 6/5, 6/5.
 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.
+[[Septimal meantone]], which is well-represented by the historically prevalent [[quarter-comma meantone]], tempers the tritone to ~[[10/7]], making the chord an [[essentially tempered chord]] in the [[9-odd-limit]]. In fact, tempering out the starling comma [[126/125]] alone is enough to convert it to a 9-odd-limit essentially tempered chord:
-* (Starling) 1/1 ‒ [[5/4]] ‒ [[3/2]] ‒ [[9/5]]
+* (Starling) 1/1&thinsp;‒&thinsp;[[5/4]]&thinsp;‒&thinsp;[[3/2]]&thinsp;‒&thinsp;[[9/5]]
-[[Septimal meantone]], which is well-represented by the historically prevalent [[quarter-comma meantone]], tempers together all three of the sevenths (9/5~16/9~25/14), so either of the above interpretations may be relevant for dominant seventh chords found in common-practice music. (→ [[Didymic chords #Dominant seventh chord]])
+However, in [[starling]] the seventh of this chord represents 9/5~[[25/14]], but not 16/9. Septimal meantone tempering is necessary to temper together all three of the sevenths (9/5~16/9~25/14), so either of the above interpretations may be relevant for dominant seventh chords found in common-practice music. (→ [[Didymic chords #Dominant seventh chord]])
 In [[archytas]] temperament, which tempers out [[64/63]], ~16/9 is equated with ~[[7/4]] rather than 25/14, resulting in an [[Dyadic chord#Essentially tempered dyadic chord|essentially just]] [[7-odd-limit]] chord that tempers together [[4:5:6:7]] and [[36:45:54:64]]:
-* (Archytas) 1/1 - [[5/4]] - [[3/2]] - [[7/4]]
+* (Archytas) 1/1&thinsp;–&thinsp;[[5/4]]&thinsp;–&thinsp;[[3/2]]&thinsp;–&thinsp;[[7/4]]
 [[Dominant (temperament)|Dominant temperament]] combines archytas with meantone, tempering out both 81/80 and 64/63, and as a result also tempers out [[36/35]], equating 4:5:6:7 with all of the 5-limit dominant seventh chords of meantone. Since [[12edo]] is a good tuning of Dominant temperament, this simpler septimal interpretation may also be relevant for dominant seventh chords in music originally composed for 12edo — particularly in performance styles that use more flexible intonation (such as Barbershop).