The augmented sixth chord is not actually a triad. Instead, it is a collection of tones that resolve very smoothly in a specific way. These chords include the scale degrees Fi and Le (sharp fourth and sharp sixth scale degrees of the key you are currently in) which both resolve by half step to Sol. Because of this, they almost always resolve to a root position five chord. The interval between low Le and Fi above it is an augmented sixth.
These chords also all share another common tone: Do (the tonic). This tone often resolves down by step to Ti (seventh scale degree), becoming the third of the V chord. “Do” in an augmented sixth chord can also remain on Do and become the root of a I 6/4 chord.
Italian Augmented Sixth Chords
In an Italian augmented sixth chord, the root of the key is always doubled. Therefore, there are only three unique notes in an italian augmented sixth chord: Le, Fi, and Do. These are also the only tones required for any type of augmented sixth chord.
French Augmented Sixth Chords
French augmented sixth chords include Le, Fi, and Do, but they also include Re, or the second scale degree. This tone often stays on Re to become the fifth of the V chord. If the french augmented sixth chord resolves to a I 6/4, the Re can resolve up to the third, or down to the root of the I 6/4 chord.
German Augmented Sixth Chords
German augmented sixth chords include Le, Fi, and Do, but they also include Me. German augmented sixth chords prefer to resolve to a I 6/4 chord. When they do this, Me often resolves up to the third of the I 6/4 chord.
The roman numeral notation is always either It+6, Fr+6, or Ger+6. There are no inversions to these chords, because they aren’t actually triads. The leadsheet notation for these chords are the same as the roman numerals.
The augmented sixth chord is almost always predominant, because they almost always resolve to V, or I6/4, both of which are always dominant.
Augmented sixth chords
Think of the components of these chords using scale degrees!
Italian is the “basic” flavor: it just contains all the notes that all the +6 chords have. In my experience, I actually had more trouble IDing It+6 chords than the other two because I psyched myself out looking for a fourth tone that obviously was not there. So, be aware that It+6s are going ot have a different shape than Ger+6 and Fr+6.
Speaking of shape, Ger+6s will usually appear in the same shape as a V6/5, while Fr+6 will usually appear in the same shape as a V4/3. The default shape for a It+6 is like a V6. (V(7) is used as a filler for simplicity: remember that these all have predominant function.ex: Ab C Eb F# can be renamed as Ab C Eb Gb to make a Ab7 chord instead of Aug6 chord)
The “inversion” doesn’t affect how we label these. No matter which note is in the bass, it’s always written as the default (x)+6. This is because the chord is not made up of stacked thirds, so our inversion labeling system doesn’t translate properly.
Common fix for Ger+6 resolution: have it go to a I6/4! Also be aware that composers may enharmonically write any of the tones for voice leading purposes (again, the primary use of these chords is just cool-sounding voice leading)
Augmented sixth chords resolving to a non-dominant harmony
This was a lot of info, so here’s the most important things to remember: