Addition and Withdrawal of the Dot

Definition: A hybrid transformation combining proportional scaling with non-proportional modification, achieved by adding dots to (or removing dots from) note values during augmentation or diminution.

Messiaen's Treatment: Messiaen identifies augmentation by dot addition and diminution by dot withdrawal as particularly interesting procedures. Example 21 demonstrates augmentation by dotting: a simple rhythm in part A receives dots on all notes in part B, creating an augmentation that multiplies values by 1.5 (the effect of dotting) rather than by 2 (simple doubling). This represents a middle ground between identity and classical augmentation. The cross marking indicates the added value component.

Example 22 shows the inverse: diminution by dot withdrawal. A dotted rhythm loses its dots, reducing each value to two-thirds of its original duration (since a dotted value equals 1.5× the undotted value, removing the dot creates a 2/3 reduction).

This technique demonstrates Messiaen's systematic thinking: if dots can be added to create added values (Chapter III), and if rhythms can be proportionally scaled (classical augmentation/diminution), then dots can be added or removed as part of the scaling process, creating proportional transformations that are not integer multiples. The 1.5× and 0.667× ratios create intermediate degrees of augmentation and diminution unavailable through simple doubling or halving.

Modern Context: Contemporary theory would recognize these as rational (but non-integer) augmentation/diminution ratios. The transformation factor of 3:2 (dotted to undotted) or 2:3 (undotted to dotted) creates proportional relationships analogous to the sesquialtera proportion in Renaissance mensural theory.

This anticipates later explorations of:

• Irrational rhythms: Non-integer proportional relationships (triplets, quintuplets, septuplets against regular divisions)
• Tempo relationships: Complex ratios between simultaneous tempi or between sections
• Proportional notation: Scores specifying durations through spatial relationships rather than conventional note values

The dot addition/withdrawal technique also connects to Messiaen's added value concept—the dot functions simultaneously as a proportional multiplier (1.5×) and as an added small value (half the original note's duration). This dual nature makes dotting a bridge between proportional and additive transformation logics.

Examples: Example 21 (augmentation by dot addition), Example 22 (diminution by dot withdrawal).