French sixth chord: Difference between revisions

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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 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 [[9/8]] 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]] 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]].

* [[72:90:100:125]], another possible interpretation of the meantone chord, is found in [[Euler-Fokker genus|genus]] 3²·5³.

== See also ==

* [[Dominant seventh flat five chord]], a chord that is enharmonically equivalent to a French sixth in 12edo.

[[Category:French sixth chords| ]]

[[Category:Just intonation chords]]

[[Category:Essentially tempered chords]]

[[Category:9-odd-limit chords]]

@@ Line 2: / Line 2: @@
 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&thinsp;–&thinsp;5/4&thinsp;–&thinsp;25/18&thinsp;–&thinsp;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 chords|Septimal meantone]]) 1&thinsp;–&thinsp;5/4&thinsp;–&thinsp;7/5&thinsp;–&thinsp;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]].
+<!-- 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. -->
 == 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]] 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]].
+* [[72:90:100:125]], another possible interpretation of the meantone chord, is found in [[Euler-Fokker genus|genus]] 3²&#xb7;5³.
+== See also ==
+* [[Dominant seventh flat five chord]], a chord that is enharmonically equivalent to a French sixth in 12edo.
+[[Category:French sixth chords| ]] <!-- main article -->
 [[Category:Just intonation chords]]
+[[Category:Essentially tempered chords]]
+[[Category:9-odd-limit chords]]