Classical Augmentation and Diminution

Definition: Proportional scaling of all durational values in a rhythmic pattern by a constant multiplicative factor, preserving the pattern's internal proportional relationships while changing its absolute temporal span.

Messiaen's Treatment: Messiaen positions classical augmentation and diminution as one approach among many for rhythmic transformation. He references Bach's practice in canonic writing—thematic material systematically doubled (augmentation) or halved (diminution)—but treats this as a foundation to be extended rather than a complete method.

In Messiaen's taxonomy, classical augmentation appears as one option: addition of the values to themselves (2x multiplication). Example 20 demonstrates this clearly: rhythm A spans five sixteenth-notes, rhythm B (augmentation of A) spans five eighth-notes (each value doubled), rhythm C spans five quarter-notes (varied double augmentation, where values are quadrupled). The statement of the rhythm followed immediately by its augmentation or diminution creates a particular temporal effect—recognition of the same pattern at different time-scales.

Messiaen notes that augmentation can occur in more complex forms beyond simple doubling, and that diminution permits inverse operations (halving, but also reduction by other factors). The key characteristic of classical augmentation/diminution is proportional uniformity—all elements scale by the same factor, maintaining rhythmic Gestalt while altering tempo or time-scale.

Modern Context: Contemporary music theory recognizes proportional augmentation/diminution as rhythmic transformation preserving similarity under scaling. In mathematical terms, these operations constitute similarity transformations in the temporal domain—the rhythmic contour remains invariant (all proportional relationships preserved) while absolute magnitude changes.

This relates to:

• Metric modulation: Elliott Carter's technique of using augmentation/diminution ratios to pivot between tempi
• Temporal perception: Psychoacoustic research on rhythm recognition across different time-scales (rhythms remain recognizable when proportionally scaled within certain limits)
• Canonic technique: Renaissance and Baroque canons per augmentationem and per diminutionem represent historical precedents
• Fractal rhythms: Self-similar temporal structures at multiple scales (explored by composers like Ligeti and Ferneyhough)

Messiaen's innovation lies not in the technique itself—which has deep historical roots—but in systematizing it alongside non-proportional transformations (added values) and partial applications (inexact augmentations), creating a comprehensive toolkit of rhythmic transformation procedures.

Examples: Example 20 demonstrates simple, varied, and complex augmentation.