The S-canon and the multi-S-canon: an introduction

The technique of rhythmic tiling can be used to embed canons within a monophonic texture by ensuring that no two parts play on the same beat. This paper introduces the S-canon, a type of canon based on Sidon sets, in which the parts do overlap rhythmically but always do so in unison. We also introduce the multi-S-canon, which involves sets of S-canons that tile with each other in terms of rhythm and pitch. Multi-S-canons are defined and constructed in terms of Sidon Zn-tilings. We demonstrate that Sidon Zn-tilings have redundant representations and define a standard form that addresses these redundancies. Finally, we give an algorithm for finding the Sidon Zn-tilings that cover Zn using as few Sidon sets as possible, and we look at some of the results. There is an Online Supplement to this paper, which can be accessed at http://dx.doi.org/10.1080/17459737.2016.1168492). It includes audio files to accompany a few of the musical examples, as well as a spreadsheet listing minimal Sidon Zn-tilings for 12≤n≤30.