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Olivier Messiaen's permutations symétriques in theory and practice Sawatzky, Grant Michael


This study begins by looking at the three compositional techniques that exemplify what Olivier Messiaen called the “charm of impossibility.” Messiaen sought this aesthetic desideratum by way of techniques that in some way impose a limit on the generation of new musical material, while also implying some kind of symmetric structure. In the first chapter, using precise, formal language, I describe each technique as the application of a particular function to the elements of a particular domain. This approach exposes the similarities (as well as certain differences) between these techniques. My explanation of the more well-known techniques—modes of limited transposition, and non-retrogradable rhythms—is somewhat atypical, in that I define them in terms of functions, objects, and cycles/orbits; but this approach is purposeful, because it fosters a clearer understanding of the often misunderstood permutations symétriques. The second chapter focuses exclusively on symmetric permutations (SPs), surveying some existing explanations of SPs before considering the ways that their applications manifest symmetry. Further, it goes into considerable mathematical detail in order to relate Messiaen’s SP orbits (consisting of m different orderings of n elements) to the larger context of the symmetric group Sn. The third and final chapter turns to real musical examples, starting with serial procedures that are precursors to the SP technique proper, and then examining ways in which Messiaen used SP to manipulate pitch-class and/or rhythmic series in his compositions between 1950 and 1992. Finally, the conclusion outlines what it is that a rigorous theoretical approach to Messiaen’s music might contribute to the existing Messiaen literature—namely, that successful analysis of this kind can better inform some of the more speculative, philosophical lines of inquiry into Messiaen’s music, which often allude to the mathematical nature of his techniques.

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