The attached article (pdf) outlines the underlying pitch-class structure of an original technique of twelve-tone musical composition which I frequently employ. I call this structure the system of pitch-symmetric tetrachord partitions (PSTP). I use this system because I consider that it affords systematically a sense of harmonic progression while retaining the essential atonality of twelve-tone music.
A particular technique of twelve-tone composition which I frequently employ makes extensive use of inversionally symmetric four-element pitch-class sets. The attached article (pdf) enumerates these, then goes on to generalize the concept to k-element sets in the context of n-degree scales for arbitrary k and n. It also enumerates the counterpart six-element sets in a twelve-degree scale, exhibiting the relationship between inversional pitch symmetry as defined here and combinatoriality in the usual serial-composition sense.
The ‘tropes’ (hexachordal partitions) of J.M. Hauer afford a compact enumeration of the six-element sets in a twelve-degree scale. The attached article (pdf) comprises an illustrated listing of these in the order used for “six-note collections” by G. Perle (Serial Composition and Atonality, 1991). This brings together hexachord types with the same combinatorial character. Appendices cross-index the hexachords by Forte number and Hauer’s original ordering of the tropes.