The augmented 6th chord originated as a result of chromatic voice-leading between the first inversion subdominant and the dominant.
Eventually, the vertical chromatic note combination was treated as an independent chord which has three versions.
The combination shown in the above examples, of a major third and augmented sixth interval above the bass note, is known as the Italian 6th.
When the intervals above the bass note consist of a major third, augmented fourth and augmented sixth, it is known as a French 6th.
And, when the intervals above the bass note consist of a major third, perfect fifth and augmented sixth, it is known as a German 6th.
Note: all augmented 6th chords contain a major third and augmented sixth interval. The interval of the augmented sixth must resolve outwards to the root of the following chord. Typically, the augmented 6th is followed by the dominant, as is shown in the initial examples above, however, it may also move onto a cadential six-four (this is a common move for a German 6th chord as it breaks the consecutive fifths between the root and fifth of the German 6th and the root and fifth of the dominant).
Notice, in a major key, when preceding a cadential six-four with a German 6th chord, the accidental creating the perfect fifth above the German 6th’s root must be cancelled for the cadential six-four. In the above example, the B flat of the German 6th must become a B natural in the cadential six-four otherwise the chord would be G minor, not G major.
In all of the examples shown so far, the augmented 6th is found a minor sixth above the root of the key, which is its usual location. In a major key this requires the inclusion of an accidental
whereas the minor key already contains a minor sixth above the root
The augmented 6th may also occasionally be found on other degrees of a major or minor scale, such as the minor second above the root
in this location it typically resolves to the tonic
or to the second inversion subdominant.
Once again, the fifth above the German 6th’s root, in the above example, A flat, must be restored to an A natural once the German 6th resolves to the subdominant.
You may have noticed that the German 6th sounds like a dominant seventh chord, as does the Italian 6th, although with the Italian 6th the dominant seventh would be without a fifth. When shown side by side, you can see that the Dominant seventh and German 6th contain the same notes, but the top notes are written enharmonically: in this case F and E sharp.
Aurally, it is the chords which follow these chords which indicate whether we are hearing a dominant seventh or a German 6th: the dominant seventh will typically move to the tonic or submediant, or to another dominant seventh, whereas the German 6th, on the minor sixth degree of the scale, will move to the dominant. The resolution of the notes within these chords is also different. In the German 6th chord the outer notes move outwards, while the seventh of the dominant seventh resolves downwards.
The duality of these chords has been used by composers for effecting modulation through the use of a pivot chord. For example, the German 6th in one key may be left as a dominant seventh in another, or vice versa.