Dominant seventh sharp five chord: Difference between revisions
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* (Meantone) 1/1 ‒ [[5/4]] ‒ [[25/16]] ‒ [[9/5]], with steps ~5/4, ~5/4, ~144/125. | * (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. | 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. | * (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 | 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, ~ | * (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]]: | 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]]: | ||
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== In just intonation == | == 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:Just intonation chords]] | ||
[[Category:Dominant seventh sharp five chords| ]] | [[Category:Dominant seventh sharp five chords| ]] | ||
[[Category:Essentially tempered chords]] | |||
[[Category:9-odd-limit chords]] | |||
Latest revision as of 03:47, 17 September 2024
A dominant seventh sharp five chord, or augmented seventh chord, is a tetrad comprising a root, major third, 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:
In 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.
In archytas temperament, which tempers out 64/63, ~16/9 is equated with ~7/4, giving another 25-odd chord:
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:
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 genus 52 ⋅ 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 ⋅ 5 ⋅ 7. In marvel temperament, it is tempered together with 144:180:225:256.
In the 5-limit:
- 20:25:32:36, found on I+ (1⁄1) and VI+ (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 ⋅ 53, is a 125-odd-limit preimage of the chord found in meantone.
- 144:180:225:256, found on ♭III+ (6⁄5) and ♭VII+ (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+ (3⁄2) and II+ (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.