Interesting Little Problem

This past weekend I worked on an interesting little musical problem.
Perhaps some of you would also find it interesting to ponder.

I'm writing an 88CET melodic-inversion utility program. As with the
transposition program I wrote earlier, it has a "pseudodiatonic" option.
That is the 88CET analogy to saying that, if you start with a melody in,
say D major, you end up with a melody also in D major. It adjusts the
pitches to bring them back into the D major scale.

I was having problems figuring out what exactly that table of
adjustments would be in the case of inversions. In the case of
transpositions, it was pretty simple: For example, if you start out in D
major and transpose it up a major second up chromatically, you end up with
a melody in E major. That means that your adjustment table must indicate
that G#s and D#s should be adjusted downward by a half step.

So I have a pretty simple function to create an adjustment table for
transpositions. But what about the adjustment table for melodic
inversions? For example, if your melody is in D major and you invert it
about D, a supertonic (E) will chromatically invert to a subtonic (C), and
would have to be adjusted up a half step to a leading tone (C#).

I thought about it a bit and found out that I can use the function that
makes adjustment tables for transpositions also for making adjustment
tables for inversions. The puzzle is "what transposition factor produces
the same adjustment table as inversion about a given diatonic scale
degree?"

(Actually, I did this for my 2122121 88CET mode rather than the 2212221
major mode, but it works out analogously.)

I'll describe the solution in a later posting.