German sixth chord: Difference between revisions

(15 intermediate revisions by 2 users not shown)

Line 1:

A '''German sixth chord''' is a [[tetrad]] comprising a root, major third, fifth, and augmented sixth~~. In [[meantone]] (including [[12edo]])~~, on ~~which traditional tonal harmony is built,~~ the ~~German~~ sixth ~~chord is [[Essentially tempered chord|essentially just]], tuned identical to~~ the ~~[[harmonic seventh chord]]:~~

A '''German sixth chord''' is a [[tetrad]] comprising a root, major third, perfect fifth, and augmented sixth, often built on the sixth degree of the minor scale.

* 1 – 5/4 – 3/2 – 7/4 with steps 5/4, 6/5, 7/6.

~~Alternatively~~, a 5-limit ~~interpretation~~ is ~~given~~ as [[128:160:192:225]].

== In meantone ==

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 ==

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| ]]

@@ Line 1: / Line 1: @@
 {{Wikipedia|Augmented sixth chord #German sixth}}
-A '''German sixth chord''' is a [[tetrad]] comprising a root, major third, fifth, and augmented sixth. In [[meantone]] (including [[12edo]]), on which traditional tonal harmony is built, the German sixth chord is [[Essentially tempered chord|essentially just]], tuned identical to the [[harmonic seventh chord]]:
+A '''German sixth chord''' is a [[tetrad]] comprising a root, major third, perfect fifth, and augmented sixth, often built on the sixth degree of the minor scale.
-* 1&thinsp;–&thinsp;5/4&thinsp;–&thinsp;3/2&thinsp;–&thinsp;7/4 with steps 5/4, 6/5, 7/6.
-Alternatively, a 5-limit interpretation is given as [[128:160:192:225]].
+== In meantone ==
+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 ==
+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 -->