Tertian names for augmented sixth chords

Here are the rank-2 intervallic names for the three famous augmented sixth chords of classical music:

     [P1, M3, A6] # Italian augmented sixth chord

[P1, M3, A4, A6] # French augmented sixth chord 

[P1, M3, P5, A6] # German augmented sixth chord

They're traditionally rooted a major third below the tonic, where they have a subdominant (or predominant) function. So for example, in they key of C major these would be spelled in pitch classes as

    [Ab, C, F#] # Italian

    [Ab, C, D, F#] # French

    [Ab, C, Eb, F#] # German

When we spell chords intervallically in the tertian way, we interpret 6th intervals as 13th intervals (i.e. the augmented sixth interval, A6, is octave-displaced to give an augmented thirteenth interval, A13). The tertian names for these chords in C major are then:

    Ab.maj#13(no 5) # Italian

    Ab.maj#11#13(no 5) # French

    Ab.maj#13 # German

The French one can also be called D.7b5 with its fifth in the bass.

If we allow 12-TET enharmonic spellings, the Italian augmented 6 chord in C major has many names including: [C.aug#11b13(no 3), C.aug#11(no 3), C.dim#11b13(no 3), C.dimb13(no 3), F#.dim(add9)#11(no 3), F#.dim(add9)(no 3), Ab.7(no 5)]

If we allow 12-TET enharmonic spellings, the Italian augmented 6 chord in C major has names including [C.aug(add9)#11b13(no 3), C.aug(add9)#11(no 3), C.dim(add9)#11b13(no 3), C.dim(add9)b13(no 3), D.7b5, D.7b5#11, D.7#11(no 5), D.maj#11#13(no 5), F#.aug(add9)#11b13(no 3), F#.aug(add9)#11(no 3), F#.dim(add9)#11b13(no 3), F#.dim(add9)b13(no 3), Ab.7b5, Ab.7b5#11, Ab.7#11(no 5)]

And the German augmented 6th chord has names including [C.m#5#11, C.m#5#11b13, C.m#5#9#11, C.m#5#9#11b13, C.dimb13, C.dim#11b13, C.dim#9b13, C.dim#9#11b13, C.aug#9#11b13(no 3), C.aug#9#11(no 3), C.dim#9#11b13(no 3), C.dim#9b13(no 3), C.m#11b13(no 5), C.m#9#11b13(no 5), Eb.m#9(add11)(add13)(no 5), Eb.m(add11)(add13)(no 5), F#.dim(add9)#11(add13)(no 3), F#.dim(add9)(add13)(no 3), F#.dim13#11(no 3), F#.dim9#11(no 3), F#.dim9(add13)(no 3), F#.dim9(no 3), Ab.7].

The shortest of those are:

    Ab.7(no 5) # enharmonically Italian

    D.7b5 : # French, exactly but respelled

    Ab.7: # enharmonically German

and these enharmonic respellings just rely on the fact that the intervals A6 and m7 are separated by d2, which is tempered out in 12-TET.

You might wonder if these are superior chords to use. Does the traditional German augmented sixth chord, Ab.maj#13, really sound better than Ab.7 when used subdominantly in C major? I don't know. We'll have to listen to both later on and come to some conclusions.

I will say that Ab.7 is also used extensively as a pre-dominant chord: it's the secondary dominant of Db.7, which is the "tritone substitution" for G.7. Although the notion of tritone substitution also relies enharmonic equivalence, so this isn't as pure a relationship as it sounds at first. Let's compare G.7 and Db.7:

    G.7: [G, B, D, F]

    Db.7: [Db, F, Ab, Cb]

The F is retained exactly from one chord to another. The B of G.7 and the Cb of Db.7 are tuned the same in 12-TET and other tuning systems that temper out the diminished second. And that's enough connection for jazz theorists.

So an enharmonic spelling (Ab.7) of the German augmented sixth chord (Ab.maj#13) has an exact relationship with the tritone substitution (Db.7) of G.7, but tritone substitution is itself an enharmonic relationship, and so it's not obvious that this version of the German chord has a clearer / stronger / purer functional relationship to chords in C major than does the original.

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