French sixth chord: Difference between revisions
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add meantone details, parallel to German sixth chord |
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A '''French sixth chord''' is a [[tetrad]] comprising a root, major third, augmented fourth, and augmented sixth. | A '''French sixth chord''' is a [[tetrad]] comprising a root, major third, augmented fourth, and augmented sixth. | ||
== In meantone == | |||
In 5-limit meantone, the traditional foundation of tonal harmony, the augmented sixth represents both [[225/128]] and [[125/72]], and the augmented fourth represents all of [[45/32]], [[25/18]], and [[729/512]], so the 5-limit meantone French sixth chord, | |||
* (Meantone) 1 – 5/4 – 25/18 – 125/72 | |||
represents a range of JI interpretations including both [[72:90:100:125]] and [[128:160:180:225]]. | |||
However, in the historically prevalent [[quarter-comma meantone]], the augmented seventh is tuned only a few cents shy of a just [[7/4]], and the augmented fourth is only a few cents shy of [[7/5]], so the meantone French sixth chord can be considered to approximate the [[septimal meantone]] chord: | |||
* (Meantone) 1 – 5/4 – 7/5 – 7/4 | |||
representing [[20:25:28:35]]. | |||
<!-- Note: The close approximation of the 7-limit chord is specific to quarter-comma meantone. Many other meantones, such as 12edo and 19edo, do not approximate it nearly as closely, although they may nonetheless support the septimal meantone interpretation that unifies it with the French sixth. --> | |||
== In just intonation == | == In just intonation == | ||
Revision as of 04:13, 20 August 2024
A French sixth chord is a tetrad comprising a root, major third, augmented fourth, and augmented sixth.
In meantone
In 5-limit meantone, the traditional foundation of tonal harmony, the augmented sixth represents both 225/128 and 125/72, and the augmented fourth represents all of 45/32, 25/18, and 729/512, so the 5-limit meantone French sixth chord,
- (Meantone) 1 – 5/4 – 25/18 – 125/72
represents a range of JI interpretations including both 72:90:100:125 and 128:160:180:225.
However, in the historically prevalent quarter-comma meantone, the augmented seventh is tuned only a few cents shy of a just 7/4, and the augmented fourth is only a few cents shy of 7/5, so the meantone French sixth chord can be considered to approximate the septimal meantone chord:
- (Meantone) 1 – 5/4 – 7/5 – 7/4
representing 20:25:28:35.
In just intonation
In the 5-limit:
- 128:160:180:225 is found rooted at ♭II (16⁄15) and ♭VI (8⁄5) in the duodene.