A SAT Encoding to Compute Aperiodic Tiling Rhythmic Canons

Abstract: In Mathematical Music theory, the Aperiodic Tiling Complements Problem consists in finding all the possible aperiodic complements of a given rhythm $A$. The complexity of this problem depends on the size of the period $n$ of the canon and on the cardinality of the given rhythm $A$. The current state-of-the-art algorithms can solve instances with $n$ smaller than $180$. In this paper we propose an ILP formulation and a SAT Encoding to solve this mathemusical problem, and we use the Maplesat solver to enumerate all the aperiodic complements. We validate our SAT Encoding using several different periods and rhythms and we compute for the first time the complete list of aperiodic tiling complements of standard Vuza rhythms for canons of period $n={180,420,900}$.