Row Properties

Not all tone rows are alike in structure. Analysts examine several special properties that shape a row's compositional potential. Combinatoriality describes the relationship between two row forms whose hexachords together contain all twelve pitch classes. If the first hexachord of P0 and the first hexachord of I_n together exhaust all twelve pcs (and the second hexachords do the same), the pair is semi-combinatorial. A row is all-combinatorial if its first hexachord can be combined with a transposition of its own Retrograde, Inversion, and Retrograde-Inversion as well. Schoenberg frequently used combinatorial rows to create a continuous twelve-tone "aggregate" across two simultaneous voices.

Invariance refers to pitch classes or subsets that remain fixed — or that retain the same ordering — under a particular transformation. For example, if T6 maps a row onto itself, it is said to have transpositional symmetry at T6. Symmetrical interval patterns in the row are the source of such invariances. A derived row is constructed by applying transformations to a small generating cell (typically a three- or four-note segment) to produce the entire twelve-note series; Webern was especially fond of rows derived from trichords, giving his music extreme motivic unity.

These properties are not merely theoretical curiosities — they directly influence compositional choices. A combinatorial row permits the superposition of two simultaneous row forms without either duplicating a pitch class within a given harmonic moment, maintaining the twelve-tone "aggregate" across both voices. Invariance allows a composer to return to a recognizable pitch-class subset under transformation, creating a sense of motivic return without literal repetition. Understanding a row's properties is therefore essential to understanding a twelve-tone work's structure.