Mazzola, Guerino / Noll, Thomas / Lluis-Puebla, Emilio: Perspectives in Mathematical and Computational Music Theory

Tiling Problems in Music Theory

Harald Fripertinger

Institut für Mathematik, Karl-Franzens-Universität Graz

harald.fripertinger@uni-graz.at

Abstract

In Mathematical Music Theory we often come across various constructions on $Zn$ , the set of residues modulo $n$ for $n > 2$ . Different objects constructed on $Zn$ are considered to be equivalent if there exists a symmetry motivated by music which transforms one object into the other one. Usually we are dealing with cyclic, dihedral, or affine symmetry groups on $Zn$ . Here we will compare partitions of $Zn$ , sometimes also called mosaics, and rhythmic tiling canons on $Zn$ . Especially we present a new method for the construction of regular complementary canons of maximal category.

1 Introduction

In the present paper we compare two tiling problems of

Zn = {0,1,...,n - 1},

the set of integer residues modulo $n$ for $n > 2$ . We discuss how to partition the set $Zn$ in essentially different ways, and we describe a special class of canons which also partition $Zn$ . When speaking about partitioning a set $X$ , in our case the set $X = Zn$ , we assume that there exist an integer $k > 1$ and nonempty subsets $P1,...,Pk$ of $X$ such that $X = P1 U ... U Pk$ , and the intersection $Pi /~\ Pj$ is the empty set for all $i /= j$ . Two partitions are called essentially different if there is no symmetry operation of $Zn$ which transforms one partition into the other one. Of course this notion heavily depends on what is assumed to be a symmetry of $Zn$ . The set of units in $Zn$ will be indicated by