French sixth chord: Difference between revisions
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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 [[marvel]], and [[10/9]] in septimal meantone and [[starling]], this chord is an [[essentially tempered chord]] in those temperaments in the [[9-odd-limit]]. | 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 [[marvel]], and [[10/9]] in septimal meantone and [[starling]], this chord is an [[essentially tempered chord]] in those temperaments in the [[9-odd-limit]]. | ||
<!-- Note: The close approximation of the 7-limit chord in meantone 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 in meantone 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 18:13, 25 August 2024
A French sixth chord is a tetrad comprising a root, major third, augmented fourth, and augmented sixth.
In temperaments
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:
- (Septimal 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 marvel, and 10/9 in septimal meantone and starling, this chord is an essentially tempered chord in those temperaments in the 9-odd-limit.
In just intonation
In the 7-limit:
- 20:25:28:35 is a condissonant chord corresponding to an interpretation of the French sixth found in starling and marvel temperaments.
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.
- 72:90:100:125, another possible interpretation of the meantone chord, is found in genus 3²·5³.