irrational tempo ratio

A tempo relationship between two voices that cannot be expressed as a ratio of simple whole numbers — for example, e/π (approximately 0.865), √2/2, or the cube root of 13/16. Such ratios cannot be performed by human musicians to any practical accuracy and require the mechanical precision of a player piano.

Nancarrow's use of irrational ratios means the voices in his late canons never converge on any predictable structural point — the music never 'locks in' to a common pulse. This creates a continuous sense of drift and independence between layers, fundamentally different from the closure-oriented convergence points of his rational-ratio canons.