Augmented sixth chord
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An augmented sixth chord is a chord which has the interval of an augmented sixth between its highest and lowest notes and also a major third above the lowest note. For example, an augmented sixth chord with an A flat in the bass consists of the notes (from the bottom up) A flat, C, F sharp.
Here are the standard "sixth" chords: (Note that spellings are done from the bottom note.)
Italian = major third + augmented sixth from bottom note. (Can also be spelled using scale degrees b6 + 1 + #4 if spelled note to note.)
French = major third + augmented fourth + augmented sixth from bottom note. (Or b6 + 1 + 2 + #4 note to note)
German and English are spelled with the same pitches:
M3 + P5 + A6 (Or b6 + 1 + b3 + #4)
And named thus:
German if it resolves to the dominant.
English if it resolves immediately to the tonic and is enharmonically equivalent to a tritone substitution dominant seventh and itself a substitute for the dominant of the dominant (V/V).
The etymology of all these names is unclear. All augmented sixth chords have a flattened sub-mediant (sixth degree of the scale (A flat in C major, for example) and a raised fourth (F# in C).
Walter Piston (1987) also defines the Swiss sixth, which is simply a German augmented sixth chord spelled with a raised second scale degree instead of a lowered third scale degree (i.e., D sharp instead of E flat in C major). This type of chord would generally resolve to I64 (the major second-inversion tonic triad), and the spelling of the raised second degree of the scale is used to emphasize that part's resolution to the major third degree of the scale.
Augmented sixth chords usually have the flattened sixth as the bass note. When they do, they tend to resolve to the dominant and are an example of tonal chromaticism.
These chords are unique in tertial harmony in that they contain intervals other than major and minor thirds. Because of this, there is no definite root in the chord; the lowered pitch which forms the bottom note of the augmented sixth interval is treated as a "sounding root." Some theorists, however, consider these chords to be chromatically modified V/V chords.