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Suppose we are in the key of C:
Ab7 would be the "augmented-six" chord.
But doesn't any 7th chord have the same intervals as the bVI7?
Why does bVI7 only have the particular name of "augmented six"?
Is it because bVI resolve to the V (because its only half step above)?
An augmented sixth chord most often functions as a predominant. When it does so, it is enharmonic with (or at least very similar to) the ♭VI7 chord. In the key of C:
We call these chords augmented sixths and not ♭VI7 chords because of how they are spelled: the interval from A♭ to F♯ is not a minor seventh, but an augmented sixth. And because of that spelling, these pitches resolve differently: in the augmented sixth interval the F♯ will move up to G, but in a ♭VI7 the G♭ will want to move down.
It's important to note that not all augmented sixth chords function as predominants. And in at least one particular instance, this non-predominant augmented sixth chords is not enharmonic to ♭VI7:
G B D F. If we lower that chordal fifth (D to D♭) and put it in second inversion, we have what we call a "French V43"; the augmented sixth interval is from D♭ up to B. You'll note that this "French V43" is the same as a French augmented sixth in F! I discuss this chord a bit more in Are there any pre-20th century examples of an augmented 6th chord with a leading tone? and What chord does C, E, G♭ , and B♭ make?Lastly, an augmented sixth chord can also function to just expand tonic; this is often done by means of what we call a common-tone augmented sixth chord. These are relatively rare, but normally these result from the chordal fifth of tonic moving in contrary motion by half step. In C, we begin with C E♭ G; if the G then splits and moves outward by half step to A♭ F♯, we're then with an augmented sixth that is spelled "correctly" but doesn't function as a predominant. I discuss this chord in Analyzing Tschaikowsky's "Wintermorgen 2"
The Augmented Sixth is characterized by the outward resolution of an augmented sixth (hence the name) to an octave. This intervallic resolution dates back to the Medieval Era. The resolution to the octave consists of two half-step moves. The particular forms (termed Italian, French, and German, for no particular reason) combine the double half-step resolution with a tritone resolution (or two in the case of the French Sixth). This combination is enharmonic with some (perhaps altered) dominant seventh resolutions but the voice move differently from that of a dominant chord.
There are several possible derivations but composers seem to treat the Augmented Sixths as objects in themselves rather than altered chords. The most common location is on the flattened sixth step which leads to V chord (perhaps through a I64 to avoid consecutive fifths and octaves if necessary), however there are others. An augmented sixth on the flattened second step does occur throughout the 1650-1900 time frame; it resolves to the tonic. There are secondary Augmented Sixths which resolve to a secondary dominant. The German Sixth is the dominant of the Neapolitan Sixth so these chords can occur consecutively in either order. The Aug6-V movement is so strong that it can signal a modulation without playing the new tonic immediately.
The French Sixth has two tritone resolutions in addition to the resolution of the augmented sixth; it is enharmonic to a dominant seventh with a lowered fifth. The German Sixth is enharmonic to a dominant seventh. Both chords can be resolved as either. In some jazz pieces, the tritone resolution is treated as more important than the augment sixt resolution leading to the idea of tritone substitutes. The difference is in the resolution; chords are identified by their resolution (the sound of the music) rather than by their labels or spelling. Some classical and romantic composers approach the augmented sixth as an augmented sixth and leave as a dominant seventh (of a chord a tritone away from the other resolution); the other way around is also used. These are useful for making smooth connections between chords which don't share any tones.
In Classical and Romantic era music, the Augmented Sixths are used as pre-dominants. It seems that one can never have enough pre-dominants to choose from so that the dominant resolution can be delayed. In the cadential pattern ii-V-I or ii0-V-i (or II-V-I or ii65-V7-I, or any of these), an Augmented Sixth can be placed after the ii chord to extend the cadence. One can combine these with a Neapolitan Sixth to extend things further (and through a I64 in just before the V for that matter.) Sousa's marches tend to have a German Sixth somewhere.
The use of tritone substitution is different and I don't know use it enough to comment.
I think you are misunderstanding what the root of the augmented sixth chord actually is.
Enharmonically the German augmented sixth chord seems to be a root position dominant seventh chord. By this perspective you are thinking the root is Ab and then there is a minor seventh above it at Gb. Your full spelling is then Ab C Eb Gb which is root position chord.
The problem with that is where is the augmented sixth?
In actuality the augmented sixth is from Ab to F#! This gives us the correct spelling of Ab C Eb F# which is a first inversion seventh chord with one very important additional characteristic_: the root is chromatically altered!
Rather than the chord being rooted on Ab it is rooted on F#. F being the subdominant degree in the key of C, it's actually a chromatically altered subdominant chord.
If you write that as a Roman numeral analysis symbol it should be #iv6/5.
While that augmented sixth chord as the sonority of a dominant seventh chord it doesn't resolve like a dominant seventh.
With an actual dominant seventh in the key of C the root of the V7 (G7) chord will be G and it will be held as a common tone when resolving to i.
In the case of the augmented sixth chord the alter fourth - the F# - is not held and has a strong pull to resolve up to the dominant scale degree G. That particular movement makes clear the subdominant function of augmented sixth chords. It resolves to the dominant.
...the same interval as the bVI7
No. In C minor the chord would be Ab C Eb G with a G natural which makes a major seventh chord. That different should help underscore that the German augmented sixth chord is not a type of bVI chord.
You may be interested to know there is another augmented sixth chord spelled Ab C D F#. Called the French augmented sixth chord.
The root of that chord is D which makes is a chromatically altered supertonic seventh chord. You could use the Roman numeral symbol ii#6/4/3 for it.
In the jazz world both of these chords can appear in enharmonic respellings as tritone substitutions.
The German augmented sixth as a tritone substitution could look like bII7 as in Dm7 Db7 C6.
The French augmented sixth as a tritone substitution could look like bII7b5 as in Dm7 Db7b5 C6.
In that jazz context notice that the chords are substituting the dominant chord V and that would be the rationale for why they are enharmonically respelled and don't fulfill the subdominant function of the alternate spellings of the German or French augmented sixth chords.
The important take away point is: enharmonic spellings should reflect how the chord is functioning!
In a piece in C major, we are very likely to come across D7, E7, F7, G7, A7...and more... (Yes, a chromatic chord or a secondary dominant doesn't have to mean there's a change of key), and they're all 'dominant 7th shape' chords. But only G7 is THE dominant 7th of C.
An Augmented 6th chord is classical harmony's nervous first attempt the idea of a 'b5 substitution'! There's a special name for the case where it's rooted on the b6 and moves to the dominant - 'Augmented 6th'. You can of course use a tritone substitution for any dominant 7th shape chord that CONTAINS a tritone. But the label 'augmented 6th' is generally reserved for that particular one.
