5-34 and its hexachords

 The five-note pitch-class set for the dominant ninth chord is Forte-label 5-34 (0, 2, 4, 6, 9). Here it is with its inverse (the set is symmetrical and so both forms have the same notes, just with D at the bottom or at the top). As a "science experiment" (to quote Dr. Emmett Brown), I've written out the hexachords that can be made by adding one note to 5-34. In each case, I've marked the new note.

The first pair (with either C# or Eb) further increases the already strong whole-tone character of the dominant ninth chord when written as a scale. The second pair (with C or E) does the opposite: bring out the diatonic-scale qualities. I'm not sure what to say about the third pair (with A# or F#), but the last one (with the Ab axis of symmetry) almost succeeds in undermining the whole-tone quality with the fully diminished (as F-Ab-B-D).

Here are all eight again, this time written out as chords, followed by "shortened" versions that might be familiar (or at least usable) as voiced harmonies. The second system gives an alternate voicing for each of the seven hexachords followed by a plausible tonic-chord voicing.

Note that hexachord 1 is the dominant with a sharp 11. Hexachord 2 is the dominant ninth with both diatonic and raised 5th. In the second system I've simplified it to a dominant ninth with (just) raised fifth. Hexachord 3 is the generally unusable diatonic dominant 11th, but it has several -1 and -2 voicings, two of which are shown. Hexachords 5 & 6 split a note: for hexachord 5 it's the ninth, A-A#, OR the third, B-Bb; for hexachord 6 it's the 7th, F-F#. The last hexachord combines qualities of major and minor ninth chords. (The last two hexachords might perhaps be understood as splitting the root--for 6, G-Gb; for 7, G-G#--but I've never been able to absorb that idea.)