The augmented sixth, like most chromatic chords, usually leads to V. While the diminished seventh and secondary dominant are diatonic chords taken out of context, the augmented sixths does not have a direct diatonic origin.
The outer voices of examples a) and b) both move from a major sixth to an octave in contrary motion. The first is a last inversion secondary dominant in C major, while the second is a phrygian cadence in C minor. The voice-leading of example c) includes both the Ab from the phrygian and the F# from the secondary dominant, resolving the interval of an augmented sixth by semitone in contrary motion to an octave. This is not to say that the augmented sixth is based on these progressions, merely that it is a similar, but intensified, progression to the dominant. More important than possible harmonic origins is the voice-leading of the augmented sixth rising expanding to an octave. The three types of augmented sixth all share this characteristic but each have slightly different inner voices. The effect in all three cases is of a strong tension (the dissonant augmented sixth) resolving onto the dominant (chord V).
Italian augmented sixth
There is no good reason why the three different types of augmented sixth chord are named after European countries, but so many writers use them that there is no alternative but to adopt them.
The Italian augmented sixth is the most basic form - most similar to the phrygian cadence but with a raised sixth. It consists of the augmented sixth itself along with a major third above the root, and is the most common variant, particularly in earlier music (listen out for the Italian sixth near the beginning of Beethoven's famous Fifth Symphony).
French augmented sixth
This slightly more colourful sounding chord is more closely related to the secondary dominant than the Phrygian cadence. It is similar to progression b) in the first example but with the A lowered to Ab.
The German augmented sixth is enharmonically the same as a dominant seventh chord (it sounds the same but is 'spelt' differently). It has two possible which are theoretically dependent on the way the chord is spelt, but in practice composers do not bother with such niceties. Resolving the German sixth onto a cadential 6/4 makes it possible to avoid the parallel fifths that are the result of directly resolving onto the dominant (seen in the second example below).
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