I made up a little musical example which I hope will help clarify the often confusing augmented sixth chords and their function. I believe that all harmony has a contrapuntal origin arising from basic voice-leading procedures. In the case of augmented sixth chords the voice-leading that generates these chords is chromatic, as would be the case for all of the "altered chords" that we encounter in tonal classical music as well as jazz harmony. Please open the pdf file below to see the examples and let me know what you think.