Diminished seventh chord: Difference between revisions
Emphasize that the temperament representations are specifically dealing with stacks of ~6/5 minor thirds. (I haven't looked at non-meantone temperaments that also handle 32/27 in an interesting way.) |
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{{Wikipedia}} | {{Wikipedia}} | ||
A '''diminished seventh chord''' is a [[tetrad]] comprising a root, [[minor]] third, [[interval quality|diminished]] fifth, and diminished seventh, conventionally formed by stacking three minor thirds. | |||
== In temperaments == | == In temperaments == | ||
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* (Meantone) 1 – 6/5 – 36/25 – 128/75 | * (Meantone) 1 – 6/5 – 36/25 – 128/75 | ||
However, if [[126/125]] is tempered out instead or in addition, as in [[starling]] and [[septimal meantone]], the chord becomes | However, if [[126/125]] is tempered out instead or in addition, as in [[starling]] and [[septimal meantone]], the chord becomes a [[7-odd-limit]] [[essentially tempered chord]]: | ||
* (Starling) 1 – 6/5 – 10/7 – 12/7 | * (Starling) 1 – 6/5 – 10/7 – 12/7 | ||
Since [[12edo]] is a good tuning of dimipent and supports both diminished temperament and septimal meantone, and the historically prevalent [[quarter-comma meantone]] is a good tuning of septimal meantone (although it was historically usually analyzed as a 5-limit temperament), any of the above interpretations may be relevant for diminished chords found in common-practice and contemporary music. | Since [[12edo]] is a good tuning of dimipent and supports both diminished temperament and septimal meantone, and the historically prevalent [[quarter-comma meantone]] is a good tuning of septimal meantone (although it was historically usually analyzed as a 5-limit temperament), any of the above interpretations may be relevant for diminished chords found in common-practice and contemporary music. | ||
== In equal temperaments == | |||
4edo (and multiples thereof): 0-300-600-900 cents | |||
13edo: 0-277-554-923 cents, 1/1-7/6-11/8-12/7 | |||
15edo: 0-320-640-880 cents, 1/1-6/5-16/11-5/3 | |||
17edo: 0-282-565-918 cents, 1/1-33/28-11/8-56/33 | |||
== In just intonation == | == In just intonation == | ||
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In the [[7-limit]]: | In the [[7-limit]]: | ||
* [[15:18:21:25]] is a [[preimage]] of the essentially-tempered chord of diminished temperament, found in [[Euler-Fokker genus|genus]] 3<sup>2</sup>{{dot}}5<sup>2</sup>{{dot}}7. | * [[15:18:21:25]] is a [[preimage]] of the essentially-tempered chord of diminished temperament, found in [[Euler-Fokker genus|genus]] 3<sup>2</sup>{{dot}}5<sup>2</sup>{{dot}}7. | ||
* [[35:42:50:60]] is a preimage of the essentially-tempered chord of starling temperament, also found in genus 3 | * [[35:42:50:60]] is a preimage of the essentially-tempered chord of starling temperament, also found in genus 3{{dot}}5<sup>2</sup>{{dot}}7. | ||
* [[25:30:35:42]] is the result of using step pattern 6/5, 7/6, 6/5. | |||
* [[30:35:42:49]] is the result of using step pattern 7/6, 6/5, 7/6. | |||
In the [[5-limit]]: | In the [[5-limit]]: | ||