joseph schillinger embraced not only contrapuntal inverse, retrograde, and retrograde-inverse—operations of matrix multiplication in euclidean vector space—but also their rhythmic counterparts as well. thus he could describe a variation of theme using the same pitches in same order, but employing its original time values in retrograde order. he saw the scope of this multiplicatory universe beyond simple reflection, to include transposition and rotation (possibly with projection back to source), as well as dilation which had formerly been limited in use to the time dimension (via augmentation and diminution) (). thus he could describe another variation of theme, or even of a basic scale, by multiplying the halfstep counts between each successive pair of notes by some factor, possibly normalizing to the octave via modulo-12 operation ().