French sixth chord

English Wikipedia has an article on:

Augmented sixth chord #French sixth

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) 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, 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.
• 72:90:100:125, another possible interpretation of the meantone chord, is found in genus 3²·5³.

See also

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