Introduction

Vuza canons are aperiodic rhythmic tilings of cyclic groups, named after Dan Tudor Vuza. They create musical canons where neither the rhythm nor entry points repeat within the cycle.

Mathematical Definition

A Vuza canon $(A, B)$ tiles $\mathbb{Z}_n$ perfectly (every position covered exactly once) where both $A$ and $B$ are aperiodic.

Why Are They Rare?

Vuza canons only exist for special values called Vuza orders (72, 108, 120, 144...). Even within these values, valid canons are extremely sparse.

Current Research

Research combining algebraic construction, constraint programming, and computational enumeration has catalogued over 19,000 Vuza canons.

Construction Methods

Vuza canons can be systematically generated using algebraic methods developed by Jedrzejewski.

The Coven-Meyerowitz Constraints

Number-theoretic constraints discovered by Coven and Meyerowitz can efficiently filter out invalid candidates during search.

Sparsity and Search Difficulty

Valid Vuza canons are extremely rare and isolated in the search space, making local search methods ineffective.

Musical Interpretation

Set $A$ is a rhythmic pattern and set $B$ defines entry points for multiple voices that tile the cycle perfectly.

Musical Properties and Perception

The aperiodic structure creates unpredictable textures with constant rhythmic density and no sense of repetition or return.

Interactive Demo

The visualization shows the circular tiling with colored dots for set $A$ positions and triangles for set $B$ entry points.

Open Questions

Key open problems include completeness of enumeration, growth rates across periods, and the double-orbit phenomenon observed at $n=144$.

References

Key papers include Vuza (1991-1993), Jedrzejewski (2013), Coven & Meyerowitz (1999), and Amiot (2016).

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