Why Do Augmented-Sixth Chords Converge on the Dominant?

Augmented-sixth chords are usually labelled Italian, French or German, which can draw attention towards the names rather than the sound. What gives them direction is the movement of two voices: the lowered sixth scale degree falls by a semitone while the raised fourth rises by a semitone, and both arrive on the dominant.

In C minor, those notes are A-flat and F-sharp. Set them as an augmented sixth and add the tonic C, and you have the basic material of an Italian augmented-sixth chord. A-flat then falls to G while F-sharp rises to G. Two widely separated notes contract towards the same dominant pitch. The chord sounds unfamiliar yet purposeful because its chromatic colour does not erase tonal direction; it concentrates the pressure before a cadence. French and German forms add different inner notes, but retain this contrary-motion framework.

At the piano, treating it merely as a special chord that must be played heavily turns its tension into a static effect. Hear A-flat and F-sharp instead as unfinished lines, each only a semitone from its destination. The arrival on the dominant should not become complete relaxation either, because the dominant still awaits the tonic. The augmented sixth matters because it reveals that harmonic identity depends first on where voices come from and where they are going.

https://musictheory.pugetsound.edu/mt21c/AugmentedSixthIntroduction.html
https://musictheory.pugetsound.edu/mt21c/TypesOfAugmentedSixthChords.html