French sixth chord: Difference between revisions
describe the chord as essentially-tempered in septimal meantone and marvel |
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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 | == 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, | 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 | * (Meantone) 1 – 5/4 – 25/18 – 125/72 | ||
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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: | ||
* (Meantone) 1 – 5/4 – 7/5 – 7/4 | * ([[Didymic chords#Septimal meantone chords|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]]. | 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]]. | ||
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* [[128:160:180:225]], one of the possible 5-limit interpretations of the meantone chord, is found rooted at ♭II ({{Frac|16|15}}) and ♭VI ({{Frac|8|5}}) in the [[duodene]]. | * [[128:160:180:225]], one of the possible 5-limit interpretations of the meantone chord, is found rooted at ♭II ({{Frac|16|15}}) and ♭VI ({{Frac|8|5}}) in the [[duodene]]. | ||
[[Category:French sixth chords| ]] <!-- main article --> | |||
[[Category:Just intonation chords]] | [[Category:Just intonation chords]] | ||
[[Category: | [[Category:Essentially tempered chords]] | ||
[[Category:9-odd-limit chords]] | |||
Revision as of 03:19, 22 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:
- (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.