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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 (“\(\left.\text{It}^{+6}\right.\)”), the French augmented sixth chord (“\(\left.\text{Fr}^{+6}\right.\)”), and the German augmented sixth chord (“\(\left.\text{Ger}^{+6}\right.\)”). 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 ♭\(\hat{6} \) and ♯\(\hat{4} \). To these two scale degrees, the \(\left.\text{It}^{+6}\right.\) adds \(\hat{1} \). The three notes of the \(\left.\text{It}^{+6}\right.\) (♭\(\hat{6} \), ♯\(\hat{4} \), and \(\hat{1} \)) form the foundation of the \(\left.\text{Fr}^{+6}\right.\) and \(\left.\text{Ger}^{+6}\right.\). The \(\left.\text{Fr}^{+6}\right.\) adds \(\hat{2} \) to the Italian augmented sixth chord’s ♭\(\hat{6} \), ♯\(\hat{4} \), and \(\hat{1} \), and the \(\left.\text{Ger}^{+6}\right.\) adds ♭\(\hat{3} \) to the Italian’s ♭\(\hat{6} \), ♯\(\hat{4} \), and \(\hat{1} \), 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 \(\left.\text{}^{+6}\right.\) or \(\left.\text{EnGer}^{+6}\right.\)—respells the ♭\(\hat{3} \) as a ♯\(\hat{2} \) because the \(\left.\text{EnGer}^{+6}\right.\) resolves only to major \(\left.\text{I}^{6}_{4}\right.\). The \(\left.\text{EnGer}^{+6}\right.\) does not occur in minor.
The 1964 Harvard Dictionary of Music states these chords are “rather pointlessly…distinguished as ‘Italian,’ ‘German,’ and ‘French’ sixth…”