Some added things to consider with Augmented Sixth chords:
Never double either tone of the +6 interval. In four-voice texture, only the Italian +6 would involve doubling, and you would double the third of the chord (tonic), the only other chord tone not part of the +6 interval.
You should learn the figured bass along with the national names. I can’t notate this clearly in the blog except through a graphic:
Augmented sixth chords will sometimes resolve to scale degrees other than ^5. The most common is tonic, but others are possible. In this way, +6 chords act as secondary dominants, and should be labeled as secondary chords. (It+6/I, for example)
The +6 interval could be inverted to a °3. This creates a type of substitution for V/V in the ii6 – V/V – V progression and other similar ones. The bass line steps up ^4, #^4, ^5. The resulting chromaticism in the upper voices is very striking, and much different from the secondary dominant of V.
The +6 chord can resolve deceptively. I won’t go into this too much at this point.
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