Relationship to Other Chapters

Chapter V represents a culmination and pivot point in the treatise:

• Chapter I (Charm of Impossibilities): This chapter fulfills Chapter I's preview of the first impossibility—rhythms that cannot be retrograded—and establishes the fundamental analogy with modes of limited transposition (the second impossibility).

• Chapter II (Rāgavardhana): The nonretrogradable fragment discovered in rāgavardhana (Example 5 of Chapter II) provides the seed from which this entire chapter develops.

• Chapters III–IV (Added Values, Augmentation/Diminution): Examples 28–29 demonstrate that nonretrogradable structures can incorporate added values and proportional transformations—the techniques combine rather than exclude each other.

• Chapter VI (Polyrhythm and Rhythmic Pedals): The superposition of a rhythm upon its retrograde (previewed here) receives development in Chapter VI as a species of polyrhythmic texture.

• Chapters XVI–XIX (Modes of Limited Transposition, etc.): The analogy established here receives its pitch-domain realization in the harmonic chapters, where modes of limited transposition are developed systematically.

The chapter also marks a transition from primarily technical exposition to more explicit aesthetic and theological reflection. Messiaen's discussion of "tonal ubiquity," "unity of movement," and "theological rainbow" reveals the spiritual motivations underlying his technical innovations.