Relation of Modes of Limited Transpositions and Nonretrogradable Rhythms
Definition: A fundamental structural parallel between the symmetrical properties of the modes (operating in the vertical/transpositional dimension) and nonretrogradable rhythms (operating in the horizontal/temporal dimension). Both embody the principle of symmetry as a compositional foundation, creating what Messiaen describes as a "theological rainbow" of unified musical language.
Messiaen's Treatment: Messiaen returns to reflections from Chapter V, article 3, establishing the parallel explicitly: modes of limited transpositions realize in the vertical direction (transposition) what nonretrogradable rhythms realize in the horizontal direction (retrogradation). These modes cannot be transposed beyond a certain number without falling again into the same notes, enharmonically speaking; likewise, these rhythms cannot be read in retrograde without one finding again exactly the same order of values as in the right sense. These modes cannot be transposed because they are without polytonality; these rhythms frame a central value common to each group.
The analogy extends further: modes exist in the modal atmosphere of several keys simultaneously and contain in themselves small transpositions; rhythms contain in themselves small retrogradations. These modes are divisible into symmetrical groups; these rhythms also exhibit this difference—the symmetry of rhythmic groups is a retrograde symmetry. Finally, the last note of each group of these modes is always common with the first of the following group, and the groups of these rhythms frame a central value common to each group. The analogy is now complete.
Messiaen quotes from Chapter V, article 3: "Let us think of the hearer of our modal and rhythmic music; he will not have time at the concert to inspect the nontranspositions and the nonretrogradations, and, at that moment, these questions will not interest him further; to be charmed will be his only desire. And that is precisely what will happen; in spite of himself he will submit to the strange charm of impossibilities: a certain effect of tonal ubiquity in the nontranspositions, a certain unity of movement (where beginning and end are confused because identical) in the nonretrogradation, all things which will lead him progressively to that sort of theological rainbow which the musical language, of which we seek edification and theory, attempts to be."
Messiaen concludes by asking rhetorically whether, having arrived at this place in the treatise, it is not useful to repeat these lines.
Modern Context: This theoretical connection represents one of Messiaen's most profound insights—the recognition that symmetry can operate as a unifying principle across different musical parameters. Contemporary theory has explored similar cross-domain symmetries, particularly in the work of composers like Bartók, Webern, and later serialists who sought to systematize relationships between pitch and rhythm. However, Messiaen's formulation remains distinctive in its theological interpretation: symmetry becomes not merely a compositional technique but a manifestation of divine order, a "theological rainbow" bridging temporal and transpositional dimensions.
The parallel between limited transposability and nonretrogradability can be understood through group-theoretic concepts: both represent structures exhibiting non-trivial automorphisms—the modes map onto themselves under certain transpositions, while the rhythms map onto themselves under retrograde inversion. This mathematical elegance appealed to later composers working with serial and algorithmic techniques, though Messiaen characteristically emphasizes perceptual and spiritual rather than purely formal dimensions.
The "charm of impossibilities" concept—that listeners will be unconsciously affected by structural properties they cannot analytically perceive—anticipates later theoretical work on subliminal structural perception and embodied musical cognition. Whether audiences genuinely perceive symmetry at unconscious levels remains an empirical question, but Messiaen's confidence in this principle shaped his compositional aesthetic profoundly. The theological dimension—symmetry as manifestation of eternal, unchanging divine truth—connects to long traditions in sacred architecture, visual art, and mystical mathematics where symmetry signifies perfection and transcendence.
The explicit linkage of these two chapters reveals the systematic unity of Messiaen's treatise—what might appear as separate rhythmic and harmonic techniques are revealed as manifestations of a single underlying structural principle. This holistic thinking distinguishes Messiaen from composers who treated rhythm and pitch as independent parameters, suggesting instead that both dimensions should be governed by parallel organizational principles to achieve aesthetic and theological unity.
Examples: Conceptual discussion drawing on examples from both Chapter V (rhythms) and Chapter XVI (modes)