This paper proposes a refinement of our understanding of pitch-class inversion in atonal and twelve-tone music. Part I of the essay establishes the theoretical foundation. It reviews the index number approach formulated by Milton Babbitt, examines characteristics of even and odd index numbers, and outlines a partitional approach to pitch-class inversion. Part II explores analytical implications of the partitional model and outlines a methodology for the analysis of note-against-note and free inversional settings. The analyses use the set-class inventories for even and odd index numbers to reduce polyphonic surfaces to note-against-note backgrounds and to evaluate the realizations of inversional designs. Part III generalizes the partitional model to enumerate and classify the distinct background structures for two-, three-, and four-voice inversional settings.

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