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{{Wikipedia}}
{{Wikipedia|Dominant seventh chord}}
A '''dominant seventh chord''' is a [[tetrad]] comprising a root, [[major]] third, fifth, and [[minor]] seventh.
A '''dominant seventh chord''' is a [[tetrad]] comprising a root, [[major]] third, fifth, and [[minor]] seventh.


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== In temperaments ==
== 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:
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.
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]]:
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).
[[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 just intonation ==
== In just intonation ==
In the [[3-limit]]:


In the [[7-limit]]:
* [[576:729:864:1024]] is found on the dominant scale degree (V or {{Frac|3|2}}) of the [[Pythagorean tuning|Pythagorean]] diatonic scale.
 
* [[4:5:6:7]], the ''harmonic seventh chord'', is a [[concord]] in the 7-limit, often used as a tuning target in {{w|Harmonic seventh chord #Barbershop seventh|barbershop music}}.
 
* [[28:35:42:50]] is a [[condissonant]] chord, and one of the possible interpretations of the dominant seventh in the starling, marvel, and septimal meantone temperaments.


In the [[5-limit]]:
In the [[5-limit]]:
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* [[128:160:192:225]], a 5-limit interpretation of an inversion of the {{w|Neapolitan chord|''Neapolitan''}} or {{w|Augmented sixth chord #German sixth|''German sixth chord''}}, is found rooted at the ♭II ({{Frac|16|15}}) and ♭VI ({{Frac|8|5}}) of the [[duodene]]. ([[225/128]] is often considered an augmented sixth rather than a minor seventh, but in septimal meantone and [[marvel]] temperament this chord is tuned identically to 4:5:6:7, and in [[12edo]] and its multiples it is tuned identically to 36:45:54:64 and 20:25:30:36.)
* [[128:160:192:225]], a 5-limit interpretation of an inversion of the {{w|Neapolitan chord|''Neapolitan''}} or {{w|Augmented sixth chord #German sixth|''German sixth chord''}}, is found rooted at the ♭II ({{Frac|16|15}}) and ♭VI ({{Frac|8|5}}) of the [[duodene]]. ([[225/128]] is often considered an augmented sixth rather than a minor seventh, but in septimal meantone and [[marvel]] temperament this chord is tuned identically to 4:5:6:7, and in [[12edo]] and its multiples it is tuned identically to 36:45:54:64 and 20:25:30:36.)


In the [[3-limit]]:
In the [[7-limit]]:
 
* [[4:5:6:7]], the ''harmonic seventh chord'', is a [[concord]] in the 7-limit, often used as a tuning target in {{w|Harmonic seventh chord #Barbershop seventh|barbershop music}}.
 
* [[70:90:105:126]] (1/1–9/7–3/2–9/5) is the ''subharmonic seventh chord'', a [[utonal]] [[9-odd-limit]] tetrad which is the inversion of [[6:7:9:10]], the subharmonic sixth chord.
 
* [[28:35:42:50]] is a [[condissonant]] chord, and one of the possible interpretations of the dominant seventh in the starling, marvel, and septimal meantone temperaments.


* [[576:729:864:1024]] is found on the dominant scale degree (V or {{Frac|3|2}}) of the [[Pythagorean tuning|Pythagorean]] diatonic scale.
* [[28:36:42:49]] is a septimal dominant seventh chord. A tempered version of this chord is found in the diatonic scale of [[superpyth]] temperament.


== See also ==
== See also ==
Retrieved from "https://en.xen.wiki/w/Dominant_seventh_chord"