Why does the A?–C–F? augmented sixth chord resolve to G?
If I take the augmented sixth chord Ab-C-F#, why is its dominant G?
This augmented chord is in the F minor scale, isn't it?
But isn't G the dominant in C?
Maybe I missed something on how to understand the build up of augmented chords?
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There's a really simple reason why this sounds good and thus is found in music theory. D7 (which is the V to G maj) is spelled D F? A C, and A?7 (the chord you've cited) is spelled A? C E? G?. In both cases, we've spelled the chords using the 1st, 3rd, 5th, and 7th scale tones. The crucial fact here is that D7 and A?7 share the exact same notes as their third and seventh. For both D7 and A?7, the 3rd and 7th are C and F? / G?.
Why does sharing the 3rd and 7th make | A?7 | G maj | sound as good | D7 | G maj |? Besides the root, the 3rd and 7th are the most important tones for defining the quality of a chord. The presence of a flat vs. natural third and a flat vs. natural seventh distinguish major chords from minor chords from melodic minor chords from dominant seventh chords.* The 3rd and 7th are of crucial importance, and so any two chords which share the same 3rd and 7th are in a way interchangeable.
In fact, in jazz, taking a V–I progression like | D7 | G maj | and substituting in | A?7 | G maj | is called a tritone substitution and is probably the most common chord substitution in the entire genre. This is called a tritone substitution because we're replacing D7 with a chord that is exactly a tritone (six half steps) away. A?7 is in the key of D? maj, which is similarly a tritone away from the G maj tonal center.
In classical music, the traditional approach is for this chord (augmented sixth) to resolve to a dominant chord, not a major chord as I've shown above. So the classical music progression would be: | A?7 | G7 | C Maj | or | D7 | D?7 | G? Maj |. Additionally, classical musicians wouldn't call the first chord a dominant seventh chord--they would recognize the function it was serving and use that function to distinguish it as an 'augmented sixth' chord. This gives: | A? aug 6 | G7 | C Maj | or | D aug 6 | D?7 | G? Maj |. Jazz theory doesn't draw this distinction and calls the first chord a dominant 7th chord.
*See below for distinctions based on the 3rd and 7th:
major chord: natural 3rd, natural 7th
minor chord: flat 3rd, flat 7th
melodic minor chord: flat 3rd, natural 7th
dominant 7th chord: natural 3rd, flat 7th
An augmented-sixth chord typically precedes a dominant. These augmented-sixth chords are so named because of the augmented-sixth interval between two of its pitches (in this case, between A? and F?). These two pitches then move in contrary motion by semitone to reach scale-degree 5 of the key. Thus A? moves down a half step to G and F? moves up a half step to G. Since G is scale-degree 5 of C, we're looking at an augmented-sixth chord in C. This is why G is "the dominant" of your A?–C–F? augmented-sixth chord.
The collection A?–C–F? is not in the F-minor scale, since there is no F? in F minor.
You can also build an augmented-sixth chord backwards, by starting on the dominant. If we're in C, our scale-degree 5 is G. Now go up a half step from G and put that pitch (A?; this is scale-degree ?6) in the bass, then go down a half step from G and put that pitch (F?; this is scale-degree ?4) in an upper voice. Voila, there's your augmented-sixth interval, from which you can fill in your augmented-sixth chord.
Now, there are three types of augmented-sixth chords. All of them have that augmented-sixth interval between scale-degrees ?6 and ?4. They also all have scale-degree 1.
If you just have those three pitches, we call it an Italian augmented-sixth chord.
If you have those three pitches but add in scale-degree 2 (D in the key of C), we call it a French augmented-sixth chord.
If you have those three pitches but add in scale-degree ?3 (E? in the key of C), we call it a German augmented-sixth chord.
As so often when attempting to explain tonal harmony - a clue is 'hunt the tritone'. C and F# (however we spell them) are a tritone that want to resolve to B and G. This tritone interval occurs in D7 (which we all know resolves to G). But it's in the Aug6 chord as well.
The next stage is to realise that a tritone is symmetrical. It can resolve two ways. C and F# (Gb) can resolve to a Db chord - think of them as the engine of Ab7. (And that Aug6 chord looks a lot like Ab7, doesn't it!) You've nearly discovered 'tritone substitution'.
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