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The Charm of Impossibilities

Definition: The aesthetic principle that musical pleasure and voluptuousness derive from mathematical and structural constraints that create perceptual paradoxes—systems that possess internal symmetries preventing certain transformations.

Messiaen's Treatment: Messiaen identifies two primary "impossibilities" that generate this charm:

  1. Transpositional limitation: Modal systems that, due to internal symmetry, cannot be transposed beyond a limited number of positions before returning to the same pitch-class content
  2. Retrograde equivalence: Rhythmic patterns that, due to symmetrical construction, sound identical when reversed in time

These impossibilities share an underlying principle: symmetrical organization creates equivalence classes under transformation, collapsing what would normally be distinct musical objects into identity. Messiaen finds this quality both mathematically elegant and sensuously appealing—"voluptuous" is his term. He explicitly connects this to religious expression, suggesting that these structured impossibilities can convey contemplative and exalted states.

Modern Context: Contemporary music theory recognizes these structures through the concepts of:

  • Limited transposition: Modes exhibiting pitch-class set symmetry, formalized in pitch-class set theory as sets with high degrees of symmetrical partitioning. Messiaen's modes of limited transposition are specific instances of symmetrical collections (the whole-tone scale is Mode 1, the octatonic scale is Mode 2).

  • Palindromic or nonretrogradable rhythm: Structures exhibiting retrograde invariance, analyzed in metric theory and formal analysis as temporal symmetries. These relate to mathematical concepts of temporal reflection and invariance under time-reversal operations.

The aesthetic claim—that symmetry produces charm—represents Messiaen's distinctive contribution. While symmetry has long been recognized as structurally significant, Messiaen elevates it to a guiding compositional principle explicitly linked to sensual and spiritual experience.

Examples: Chapter V develops nonretrogradable rhythms; Chapter XVI develops modes of limited transposition.