French sixth chord: Difference between revisions
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→In temperaments: clarify conventional / septimal meantone distinction; refer to starling instead of marvel for the general essentially-tempered chord (marvel doesn't get that close!) |
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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: | 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 chords|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 | or [[20:25:28:35]] (step sizes 5/4, 28/25, and 5/4). Since the [[28/25]] interval in this chord also represents [[10/9]], in septimal meantone (and, more generally, in [[starling]] 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 05:28, 24 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 10/9, in septimal meantone (and, more generally, in starling 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.