The augmented sixth chord inherits the subdominant function of the IV degree chord. The raised note makes it more like the secondary dominant of the V degree. We can see this very clearly in the French sixth chord. We can analyze this chord as a secondary dominant of the V degree (B - D# - F# - A) with a lowered fifth (F natural in this case):

The German sixth is enharmonically equivalent to a dominant seventh chord. Here is the German sixth chord of A minor and the same chord transformed into the dominant seventh of Bb major:

We can also change a dominant chord into an augmented sixth. Here we see the C major dominant chord transformed into the German sixth (G6) of B minor:

We can use these procedures to modulate to remote keys. Here we show the German sixth of A minor used as the dominant chord of Bb major:

In this example, the dominant chord of C major is used as German sixth of B major:

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José Rodríguez Alvira.