German sixth chord: Difference between revisions
Per discussion, list the 5-limit chord separately. |
clarify meantone vs. septimal meantone in chord summaries |
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== In meantone == | == 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). | |||
<!-- 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 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 --> | |||
Latest revision as of 05:31, 24 August 2024
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.
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 (16⁄15) and ♭VI (8⁄5) of the duodene.