German sixth chord: Difference between revisions

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== In meantone ==

In [[~~meantone~~]], the ~~augmented~~ sixth represents both [[125/72]] and [[225/~~128~~]]. In the historically-prevalent [[quarter-comma meantone]] it is tuned only a few cents shy of a just [[7/4]]~~, and in the closely related [[septimal meantone]] it explicitly represents [[7/4]] as well~~, so the meantone German sixth chord can be considered to approximate ~~both~~ [[4:5:6:7]] and ~~[[128~~:~~160:192:225]]~~.

In 5-limit meantone, the traditional foundation of tonal harmony, the augmented sixth represents both [[225/128]] and [[125/72]]. It is treated as a rare and special dissonance, especially when used on the sixth degree. The 5-limit meantone German sixth chord,

* (Meantone) 1 – 5/4 – 3/2 – 125/72

simultaneously represents both [[72:90:108:125]] (with steps 5/4, 6/5, and [[125/108]]) and [[128:160:192:225]] (with steps 5/4, 6/5, and [[75/64]]).

However, in the historically prevalent [[quarter-comma meantone]], the augmented sixth is tuned only a few cents shy of a just [[7/4]], so the meantone German sixth chord can be considered to approximate the ''harmonic seventh chord'' as in [[septimal meantone]]:

* (Septimal meantone) 1 – 5/4 – 3/2 – 7/4

representing [[4:5:6:7]] (with steps 5/4, 6/5, and 7/6).

== In just intonation ==

~~Because the German sixth chord is often built on the sixth degree of a minor scale,~~ the [[128:160:192:225]] found on the ♭VI of the [[duodene]] ~~is a good candidate for its interpretation in the~~ [[~~5-limit~~]].

In the [[5-limit]]:

* [[128:160:192:225]], one of the possible 5-limit interpretations of the meantone chord, is found rooted at the ♭II ({{Frac|16|15}}) and ♭VI ({{Frac|8|5}}) of the [[duodene]].

[[Category:Just intonation chords]]

[[Category:German sixth chords| ]]

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 == In meantone ==
-In [[meantone]], the augmented sixth represents both [[125/72]] and [[225/128]]. In the historically-prevalent [[quarter-comma meantone]] it is tuned only a few cents shy of a just [[7/4]], and in the closely related [[septimal meantone]] it explicitly represents [[7/4]] as well, so the meantone German sixth chord can be considered to approximate both [[4:5:6:7]] and [[128:160:192:225]].
+In 5-limit meantone, the traditional foundation of tonal harmony, the augmented sixth represents both [[225/128]] and [[125/72]]. It is treated as a rare and special dissonance, especially when used on the sixth degree. The 5-limit meantone German sixth chord,
+* (Meantone) 1&thinsp;–&thinsp;5/4&thinsp;–&thinsp;3/2&thinsp;–&thinsp;125/72
+simultaneously represents both [[72:90:108:125]] (with steps 5/4, 6/5, and [[125/108]]) and [[128:160:192:225]] (with steps 5/4, 6/5, and [[75/64]]).
+However, in the historically prevalent [[quarter-comma meantone]], the augmented sixth is tuned only a few cents shy of a just [[7/4]], so the meantone German sixth chord can be considered to approximate the ''harmonic seventh chord'' as in [[septimal meantone]]:
+* (Septimal meantone) 1&thinsp;–&thinsp;5/4&thinsp;–&thinsp;3/2&thinsp;–&thinsp;7/4
+representing [[4:5:6:7]] (with steps 5/4, 6/5, and 7/6).
+<!-- Note: The close approximation of 7/4 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 German sixth. -->
 == In just intonation ==
-Because the German sixth chord is often built on the sixth degree of a minor scale, the [[128:160:192:225]] found on the ♭VI of the [[duodene]] is a good candidate for its interpretation in the [[5-limit]].
+In the [[5-limit]]:
+* [[128:160:192:225]], one of the possible 5-limit interpretations of the meantone chord, is found rooted at the ♭II ({{Frac|16|15}}) and ♭VI ({{Frac|8|5}}) of the [[duodene]].
+[[Category:Just intonation chords]]
+[[Category:German sixth chords| ]] <!-- main article -->