Steve Coleman and Dominant/Half-Diminished Seventh Symmetry

Steve Coleman shares an example of the first sixteen bars on a "Rhythm Changes."

Notice the prevalence of half-diminished seventh chords in melodic outline. In his discussion of Charlie Parker's solo on the bridge out of "Celebrity," Coleman points to the use of such tetrachords (he shares the beboppers's preference for calling them "minor chords with added sixth") as upper extensions on an underlying dominant seventh chord. For instance, when Bird outlines C half-diminished seventh (E-flat minor--added 6th), he is arpeggiating the upper chord tones (3-5-7-9) of A-flat dominant seventh, which itself is the tritone substitution for D dominant seventh, the first chord of the bridge of "Rhythm Changes." From a voice-leading perspective, B-flat and E-flat represent chromatic appoggiature to the chord tones A and D which arrive in the following bar. 

Coleman is also aware of course that the dominant seventh and half-diminished seventh tetrachords have the same interval content---both belong to set-class (0258). Dominant seventh and half-diminshed seventh chords can therefore be related by inversion; e.g., D dominant seventh and the C half-diminished seventh that Bird superimposes on it are symmetrical. If we imagine the twelve chromatic pitch-classes as lying on a clock-face, then inverting (geometrically "reflecting") D and A (2 9) over the tritone axis of C--F-sharp (0 6) yields B-flat and E-flat (t 3), respectively. In other words, D dominant seventh and C half-diminished seventh are inversionally symmetrical about their shared tritone axis, C--F-sharp/G-flat: