"Roots" of Augmented Sixth Chords
Simon Sechter, in 'Die Grundsätze der musicalischen Komposition', explains the chord of the French Sixth A♭—C—D—F♯ in the key of C as being a chromatically altered version of a seventh chord on the second degree of the scale, and therefore gives the root as 'D'. The German Sixth A♭—C—E♭—F♯ is explained as a chromatically altered ninth chord on the same root, but with the root omitted. (In Sechter's theory, the diminished seventh chord F♯—A—C—E♭ is invariably described in the same way, i.e. a ninth chord on D with the root omitted, hence its equivalence to the augmented sixth.)
The tendency of the interval of the augmented sixth to resolve outwards is therefore explained by the fact that the A♭, being a dissonant note, a diminished fifth above the root (D), and flatted, must fall, whilst the F♯ - being chromatically raised - must rise.
On the contrary, Tchaikovsky considers the augmented sixth chords, rather than being built on the minor sixth degree (A♭ in C), as being altered dominants. In his Guide to the practical study of harmony considers the augmented sixth chords to be inversions of the diminished triad and of dominant and diminished seventh chords with the second degree chromatically lowered, and accordingly resolving into the tonic. He notes that, "some theorists insist upon resolution not into the tonic but into the dominant triad, and regard them as being erected not on the altered 2-nd degree, but on the altered 6-th degree in major and on the natural 6-th degree in minor", yet calls this view, "fallacious", insisting that a, "chord of the augmented sixth on the 6-th degree is nothing else than a modulatory degression into the key of the dominant". This would make the chord of the augmented sixth a member of a large group of chords with an altered second degree (which includes the Neapolitan chord). For an exhaustive discussion of the possibility of augmented sixth chords resolving to tonic (or other scale degrees), see Daniel Harrison's article, "A Supplement to the Theory of Augmented-Sixth Chords."
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