Relation of Nonretrogradable Rhythms and Modes of Limited Transpositions

Definition: The fundamental analogy between temporal (horizontal) and pitch (vertical) symmetries in Messiaen's system, establishing that nonretrogradable rhythms and modes of limited transposition are parallel manifestations of the same mathematical principle operating in different musical dimensions.

Messiaen's Treatment: This section articulates the core theoretical insight of Messiaen's treatise. He recalls Chapter I's promise to examine the relationship between these two impossibilities and provides the systematic explanation:

Modes of limited transposition cannot be transposed beyond a certain number of positions without returning to the same pitch-class content because they lack polytonality—they exist in the modal atmosphere of several keys simultaneously and contain within themselves small transpositions. These modes cannot be transposed because they are, in a sense, already in multiple keys at once through their symmetrical construction.

Nonretrogradable rhythms cannot be retrograded because they contain within themselves small retrogradations—the internal mirroring means that retrograding the whole produces the same pattern, since local retrograde relationships are already built into the structure.

The analogy operates precisely:

• Modes of limited transposition: Symmetry in the vertical direction (pitch-class space) → transpositional limitation
• Nonretrogradable rhythms: Symmetry in the horizontal direction (temporal succession) → retrograde limitation

Both represent structures where internal symmetry collapses what would normally be distinct transformations into identity. The modes are "divisible into symmetrical groups" (their pitch-class collections can be partitioned into symmetrical subsets), and "the symmetry of the rhythmic groups is a retrograde symmetry" (the temporal structure mirrors itself around a central axis).

Messiaen emphasizes the completeness of the analogy: both techniques create impossibilities through symmetry, both involve internal repetition of transformational relationships that prevent external application of those same transformations, and both frame a central value common to each group (the last note of each modal group connects to the first of the following group; the central durational value connects the two mirrored rhythmic groups).

Theological and Aesthetic Implications: Messiaen then articulates his aesthetic philosophy. The modal and rhythmic music employing these impossibilities will not interest the listener at a concert who lacks time for theoretical inspection and reflection. However, the listener will submit to the "strange charm of impossibilities" manifesting as:

• Tonal ubiquity in the nontransposition: The sense that the mode exists everywhere harmonically, not localized to a single key
• Unity of movement in the nonretrogradation: The temporal unity of beginning and ending being confused because they are identical, suggesting timelessness

These effects lead progressively toward a "theological rainbow"—Messiaen explicitly connects mathematical impossibility to spiritual transcendence. The music seeks edification and theory, implying that understanding the structural impossibilities deepens appreciation of their aesthetic and theological significance.

Modern Context: This analogy represents Messiaen's most sophisticated theoretical contribution. Contemporary music theory recognizes several important aspects:

1. Parametric isomorphism: Messiaen identifies a structural parallel between operations in different musical domains (pitch vs. time). This anticipates transformation theory's recognition that similar mathematical structures can govern multiple musical parameters.

2. Symmetry as compositional principle: Messiaen elevates symmetry from an occasional feature to a systematic organizing principle. This prefigures later composers' exploitation of symmetry (Bartók's axis system, spectral music's harmonic-timbral symmetries, minimalism's palindromic processes).

3. Limitation as aesthetic resource: Rather than viewing limitations (inability to transpose, inability to retrograde) as constraints to be overcome, Messiaen embraces them as sources of distinctive aesthetic qualities. This represents an inversion of traditional compositional values that prize maximal transformational freedom.

4. Cross-domain unity: The analogy suggests that Messiaen conceives his musical language as a unified system rather than a collection of independent techniques—pitch and rhythm governed by parallel principles.

5. Mathematical aesthetics: Messiaen's claim that mathematical properties (symmetry, invariance) produce specific perceptual and spiritual effects represents a Pythagorean worldview—mathematical relationships as sources of beauty and meaning.

Contemporary theorists might analyze these structures through:

• Neo-Riemannian theory: Symmetrical divisions of the octave (which generate modes of limited transposition)
• Transformation theory: Structures invariant under specific transformations
• Cognitive music theory: Perceptual effects of symmetrical vs. asymmetrical patterns
• Spectral analysis: Harmonic vs. inharmonic spectra (symmetrical vs. asymmetrical pitch structures)

The theological dimension remains distinctive. While many composers employ mathematical structures, few articulate explicit connections between mathematical properties and spiritual transcendence as directly as Messiaen. His position reflects his Catholic faith and his conviction that musical structures can embody and communicate theological truths.

Examples: The analogy is conceptual; specific musical examples of modes of limited transposition appear in Chapters XVI–XIX, while examples of nonretrogradable rhythms appear throughout this chapter.