If we I want to analyze a a piece by Beethoven e.g. with an Italian 6th chord (iv aug6) the notes are the first inversion of F#,Ab,C => Ab,C,F#.
So what chord symbol is correct? Is it F#b3b5/Ab ... or what I am missing?
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The last isn't really augmented, but I included to complete the list of unusual chord symbols.
What you might be missing is that when we look at the function of the augmented 6th chords and how they are analyzed for the common practice period, we consider there to be an augmented 6th interval contained in the chord, which is enharmonic to a minor seventh interval. If you want to spell the chord for easy reading for performance, it's better to base the chord name on the dominant seventh chord of the lowered 6th scale degree, rather than an altered first inversion of the raised fourth scale degree.
As with all chord spellings, once you go enharmonic, it can get very flexible. For example, the C major Fr6 could also be spelled D7b5/Ab instead of Ab7#5.
I consider jazz symbols as based on a hypothetical dominant 13th chord, in major, with all the extensions and modifications from that point of reference.
So, the root is given as the specific pitch
F#, then add notation to mod from a dominant seventh chord: drop the seventh, the major third needs to be lowered by two half steps (
𝄫3), the perfect fifth lowered a half step (
♭5), the bass is given as the specific pitch (
/A♭), so that would be
F#𝄫3♭5/A♭. Or, you could add
min for one of the half steps on the lowered third, like
F#m♭3♭5/A♭. Or, maybe base it on a diminished triad
I'm not very confident in those as a jazz/pop chord labels, because the classical augmented sixth chord is considered the same as the jazz tritone substitution which would simply call the chord
A♭7 with maybe the notation "omit 5th" to specify the Italian augmented sixth voicing.
The usual RNA symbol is
It+6 regardless of inversion, but it seems that
#iv+6 should be clear for the first inversion form.
I found an online textbook that shows both the jazz tritone label and RNA label used alternately: Hutchinson, Music Theory for the 21st-Century Classroom
That author seems to prefer labeling it