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Section 21.2 Types of Augmented Sixth Chords

There are three general types of augmented sixth chords—the Italian augmented sixth chord (“It+6”), the French augmented sixth chord (“Fr+6”), and the German augmented sixth chord (“Ger+6”). These geographic labels have persisted throughout the years despite the fact that no reasoning has been found for these designations.  1 
All types of augmented sixth chords contain scale degrees ♭6^ and ♯4^. To these two scale degrees, the It+6 adds 1^. The three notes of the It+6 (♭6^, ♯4^, and 1^) form the foundation of the Fr+6 and Ger+6. The Fr+6 adds 2^ to the Italian augmented sixth chord’s ♭6^, ♯4^, and 1^, and the Ger+6 adds ♭3^ to the Italian’s ♭6^, ♯4^, and 1^, as is shown in the example below.
Figure 21.2.1. The Three Types of Augmented Sixth Chords in Major and Minor
The final chord on the first line—the Enharmonic German +6 or EnGer+6—respells the ♭3^ as a ♯2^ because the EnGer+6 resolves only to major I46. The EnGer+6 does not occur in minor.
The 1964 Harvard Dictionary of Music states these chords are “rather pointlessly…distinguished as ‘Italian,’ ‘German,’ and ‘French’ sixth…”