French sixth chord: Difference between revisions
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describe the chord as essentially-tempered in septimal meantone and marvel |
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* (Meantone) 1 – 5/4 – 25/18 – 125/72 | * (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]]. | represents a range of JI interpretations including both [[72:90:100:125]] (step sizes 5/4, 10/9, and 5/4) and [[128:160:180:225]] (step sizes 5/4, 9/8, and 5/4). | ||
However, in the historically prevalent [[quarter-comma meantone]], the augmented sixth 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: | However, in the historically prevalent [[quarter-comma meantone]], the augmented sixth 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 also approximate the [[septimal meantone]] chord: | ||
* (Meantone) 1 – 5/4 – 7/5 – 7/4 | * (Meantone) 1 – 5/4 – 7/5 – 7/4 | ||
or [[20:25:28:35]] (step sizes 5/4, 28/25, and 5/4). Since the [[28/25]] interval in this chord also represents [[9/8]], in septimal meantone (and, more generally, in [[marvel]] temperaments) this chord is an [[essentially tempered chord]] in the [[9-odd-limit]]. | |||
<!-- 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. --> | <!-- 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. --> | ||
Revision as of 04:56, 21 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 (step sizes 5/4, 10/9, and 5/4) and 128:160:180:225 (step sizes 5/4, 9/8, and 5/4).
However, in the historically prevalent quarter-comma meantone, the augmented sixth 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 also approximate the septimal meantone chord:
- (Meantone) 1 – 5/4 – 7/5 – 7/4
or 20:25:28:35 (step sizes 5/4, 28/25, and 5/4). Since the 28/25 interval in this chord also represents 9/8, in septimal meantone (and, more generally, in marvel temperaments) this chord is an essentially tempered chord in the 9-odd-limit.
In just intonation
In the 5-limit:
- 128:160:180:225, one of the possible 5-limit interpretations of the meantone chord, is found rooted at ♭II (16⁄15) and ♭VI (8⁄5) in the duodene.