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Inversion by Minor sixth = Change in axis?

Okay, let me clear things up here. I have a short little motive in the first theme of my rondo that I think will lend itself to motivic development nicely. My second theme starts in Ab major. Just for context here, my first theme is in C minor. Here are the notes of the motive:

C, F, D, Eb, Ab, F, G, C

I have the melodic equivalent of a line cliche here. No problems with that. Here is the interval pattern:

(Ascending fourth, Descending third, Ascending second)x2, Ascending fourth

The rhythmic pattern is quarter, eighth, eighth, quarter, eighth, eighth, quarter, quarter. So something similar to this:



but with 2 extra quarter notes added.

The fact that both the intervals and the rhythm are in a pattern means that there are both rhythmic and melodic routes to developing the motive which means more development can be done without making the motive seem boring. And the fact that the motive is long means even more development is possible than if it were shorter.

Now here is where I get to inversion. I was thinking of inverting by an interval other than the unison so that my motive could not only go down but also be consonant with Ab major. Yes I know, I could have simply transposed the motive and then do your typical inversion by the unison. But I wanted to try something different.

First I wanted to see if the interval from C to Ab was symmetrical by scale steps alone. If it was, then, there would be 1 note that would stay the same. Turns out, it wasn't symmetrical by scale steps. Now this is where I had to bring in the half steps. I was wanting to figure out what note did stay the same assuming that the inversion is by half step numbers and not scale step numbers. Turns out, it is E natural. Now I also went outwards, this time by scale steps and there was a shared note, Bb, the only note in C natural minor that doesn't appear in my motive.

This made me think

Wait a minute. If I treat this as an inversion by minor sixth, then C becomes Ab, Db becomes G, D becomes Gb, Eb becomes F, E stays put, B becomes A, and Bb stays put. I end up in Db major this way which is at least closely related to Ab major. If I treat this as a change of axis followed by inversion by the octave, I get the scale symmetry back but all of a sudden I'm in Bb major when I'm trying to go to Ab major. That is not closely related to Ab major. The only key with a tonic of Bb that is closely related to Ab major is Bb minor.


So basically, I hit a dead end there with my non-octave, non-unison inversion. I was trying to invert it such that via inversion I would go from C minor to Ab major. And I figured that it would have to involve C going to Ab. But that lead me to 2 very different keys depending on whether I think chromatically or diatonically about the inversion. In either case, the E and Bb stay put and those notes are a tritone away so the fact that they stay put after inversion makes sense because the tritone is symmetrical across the octave.

But why is it that inversion by the minor sixth does not lead you to the key a minor sixth away but instead a minor second or minor seventh away? Does this mean that I would have to invert by the major 10th to get to Ab major from C minor via inversion alone? 

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