Here is a reference table with the transformations I have devised to date for use with the proto-backgrounds.
These are presented here as three groups of three: the first group results in stepwise changes, diatonic or chromatic (L, N, WEDGE); the second group adds a third note to a triad interval (DIVision, ADDINVersion, TRiadFLIP); and the third group manipulates a triad interval INVersion, EXPandUp, and TRiadTransposition). All of these assume inverses (L-1, N-1, etc.) but note that EXPU-1 contracts an interval -- it does not EXP downward (to make the point as clearly as possible, a fourth transformation, EXPandDown, is added to the third group: see the staff insert).
Caveats: (a) This is certainly not a complete list of what might be done in "triad space," much less in diatonic space; (b) these can only be regarded as informal -- I have not attempted formal definitions, here or elsewhere; and (c) to both the previous points, someone has without doubt done such work and, once I find it, this post will be updated accordingly.
