My understanding of melodic inversion (strict and diatonic) is taking a melody/motif, beginning on the same note and moving in the opposite direction. However, in the IB Music Revision Guide's analysis of Rachmaninoff's Rhapsody On A Theme Of Paganini, the first 4 note motif CBAB below is said to be an inversion of the 2nd 4 note motif ACBA. Can anyone tell me what we call this particular type of melodic inversion (that does not begin on the same first note) and what logic (if any) is used to create this particular type of melodic inversion, especially what note do we start on?
As others have said, this is neither inversion nor retrograde; and it's not even a type of retrograde inversion!
But if you're looking for a very famous example of inversion of the original A–C–B–A motive, check out the opening four notes (at the end of the second measure) of Variation 18.
Not differentiating specific interval quality, the example includes the following melodic intervals: descending step, ascending step, ascending third.
The only thing inverted is the single step. Descending in green, it's ascending inversion in red...
If the second figure was an inversion of the first it would look more like...
Descending step, descending step, ascending step
Fig 2 (just reverse the directions...)
Ascending step, ascending step, descending step