Dominant seventh sharp five chord: Difference between revisions

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A '''dominant seventh sharp five chord''', or '''augmented seventh chord''', is a [[tetrad]] comprising a root, [[major]] third, [[interval quality|augmented]] fifth, and [[minor]] seventh.

== In temperaments ==

In [[meantone]], on which traditional tonal harmony is built, the minor seventh tempers together [[9/5]]~[[16/9]], giving a chord with an [[intervallic odd limit]] of 125:

* (Meantone) 1/1 ‒ [[5/4]] ‒ [[25/16]] ‒ [[9/5]], with steps ~5/4, ~5/4, ~144/125.

In [[augmented (temperament)|augmented temperament]], which tempers out [[128/125]], a ~[[25/16]] augmented fifth is equated with an ~[[8/5]] minor sixth. The resulting chord is [[25-odd-limit]] due to the [[36/25]]~[[45/32]] interval between the ~[[9/5]] minor seventh and ~[[5/4]] major third.

* (Augmented) 1/1 ‒ [[5/4]] ‒ [[8/5]] ‒ [[9/5]], with steps ~5/4, ~5/4, ~9/8.

In [[archytas]] temperament, which tempers out [[64/63]], ~16/9 is equated with ~[[7/4]], giving another 25-odd chord:

* (Archytas) 1/1 ‒ [[5/4]] ‒ [[25/16]] ‒ [[7/4]], with steps ~5/4, ~5/4, ~28/25.

In [[marvel]] temperament, which tempers out [[225/224]], ~25/16 is equated with ~[[14/9]], and the ~[[64/45]] tritone between a ~5/4 major third a ~[[16/9]] minor seventh is equated with ~[[10/7]], yielding a [[9-odd-limit]] [[essentially tempered chord]]:

* (Marvel) 1/1 ‒ [[5/4]] ‒ [[14/9]] ‒ [[16/9]], with steps ~5/4, ~5/4, ~8/7.

[[12edo]] is a good tuning of augmented temperament and supports all of the above temperaments to varying degrees, and [[septimal meantone]] — which is well-represented by the historically prevalent [[quarter-comma meantone]] — supports marvel, so any of the above interpretations may be relevant for seventh sharp five chords found in contemporary and common-practice music.

== In just intonation ==

In the [[7-limit]]:

* [[16:20:25:28]] is a [[condissonant]] chord found in [[Euler-Fokker genus|genus]] 52{{dot}}7. In archytas temperament, it is [[tempered together]] with 144:180:225:256.

* [[36:45:56:64]] is a condissonant chord found in genus 32{{dot}}5{{dot}}7. In marvel temperament, it is tempered together with 144:180:225:256.

In the [[5-limit]]:

* [[20:25:32:36]], found on I+ ({{Frac|1|1}}) and VI+ ({{Frac|4|3}}) in the [[duodene]], is a simple [[preimage]] of the 25-odd-limit chord found in augmented temperament, substituting an 8/5 minor sixth in place of the augmented fifth.

* [[80:100:125:144]], found in genus 32{{dot}}53, is a 125-odd-limit preimage of the chord found in meantone.

* [[144:180:225:256]], found on ♭III+ ({{Frac|6|5}}) and ♭VII+ ({{Frac|9|5}}) in the [[duodene]], is a preimage of both the 5-limit meantone chord and the simpler 7-limit chords found in archytas and marvel temperaments.

* [[180:225:288:320]], found on V+ ({{Frac|3|2}}) and II+ ({{Frac|9|8}}) in the [[duodene]], is tempered together with 144:180:225:256 in augmented temperament, substituting an 8/5 minor sixth in place of the augmented fifth.

[[Category:Just intonation chords]]

[[Category:Dominant seventh sharp five chords| ]]

[[Category:Essentially tempered chords]]

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

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 {{Wikipedia|Augmented seventh chord}}
-{{stub}}
+A '''dominant seventh sharp five chord''', or '''augmented seventh chord''', is a [[tetrad]] comprising a root, [[major]] third, [[interval quality|augmented]] fifth, and [[minor]] seventh.
+== In temperaments ==
+In [[meantone]], on which traditional tonal harmony is built, the minor seventh tempers together [[9/5]]~[[16/9]], giving a chord with an [[intervallic odd limit]] of 125:
+* (Meantone) 1/1 ‒ [[5/4]] ‒ [[25/16]] ‒ [[9/5]], with steps ~5/4, ~5/4, ~144/125.
+In [[augmented (temperament)|augmented temperament]], which tempers out [[128/125]], a ~[[25/16]] augmented fifth is equated with an ~[[8/5]] minor sixth. The resulting chord is [[25-odd-limit]] due to the [[36/25]]~[[45/32]] interval between the ~[[9/5]] minor seventh and ~[[5/4]] major third.
+* (Augmented) 1/1 ‒ [[5/4]] ‒ [[8/5]] ‒ [[9/5]], with steps ~5/4, ~5/4, ~9/8.
+In [[archytas]] temperament, which tempers out [[64/63]], ~16/9 is equated with ~[[7/4]], giving another 25-odd chord:
+* (Archytas) 1/1 ‒ [[5/4]] ‒ [[25/16]] ‒ [[7/4]], with steps ~5/4, ~5/4, ~28/25.
+In [[marvel]] temperament, which tempers out [[225/224]], ~25/16 is equated with ~[[14/9]], and the ~[[64/45]] tritone between a ~5/4 major third a ~[[16/9]] minor seventh is equated with ~[[10/7]], yielding a [[9-odd-limit]] [[essentially tempered chord]]:
+* (Marvel) 1/1 ‒ [[5/4]] ‒ [[14/9]] ‒ [[16/9]], with steps ~5/4, ~5/4, ~8/7.
+[[12edo]] is a good tuning of augmented temperament and supports all of the above temperaments to varying degrees, and [[septimal meantone]] — which is well-represented by the historically prevalent [[quarter-comma meantone]] — supports marvel, so any of the above interpretations may be relevant for seventh sharp five chords found in contemporary and common-practice music.
+== In just intonation ==
+In the [[7-limit]]:
+* [[16:20:25:28]] is a [[condissonant]] chord found in [[Euler-Fokker genus|genus]] 5<sup>2</sup>{{dot}}7. In archytas temperament, it is [[tempered together]] with 144:180:225:256.
+* [[36:45:56:64]] is a condissonant chord found in genus 3<sup>2</sup>{{dot}}5{{dot}}7. In marvel temperament, it is tempered together with 144:180:225:256.
+In the [[5-limit]]:
+* [[20:25:32:36]], found on I+ ({{Frac|1|1}}) and VI+ ({{Frac|4|3}}) in the [[duodene]], is a simple [[preimage]] of the 25-odd-limit chord found in augmented temperament, substituting an 8/5 minor sixth in place of the augmented fifth.
+* [[80:100:125:144]], found in genus 3<sup>2</sup>{{dot}}5<sup>3</sup>, is a 125-odd-limit preimage of the chord found in meantone.
+* [[144:180:225:256]], found on ♭III+ ({{Frac|6|5}}) and ♭VII+ ({{Frac|9|5}}) in the [[duodene]], is a preimage of both the 5-limit meantone chord and the simpler 7-limit chords found in archytas and marvel temperaments.
+* [[180:225:288:320]], found on V+ ({{Frac|3|2}}) and II+ ({{Frac|9|8}}) in the [[duodene]], is tempered together with 144:180:225:256 in augmented temperament, substituting an 8/5 minor sixth in place of the augmented fifth.
 [[Category:Just intonation chords]]
 [[Category:Dominant seventh sharp five chords| ]]
+[[Category:Essentially tempered chords]]
+[[Category:9-odd-limit chords]]