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

Abstract

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 ${\mathbb{Z}}_n$-tilings. We demonstrate that Sidon ${\mathbb{Z}}_n$-tilings have redundant representations and define a standard form that addresses these redundancies. Finally, we give an algorithm for finding the Sidon ${\mathbb{Z}}_n$-tilings that cover ${\mathbb{Z}}_n$ 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 ${\mathbb{Z}}_n$-tilings for $12\le n\le 30$.

Keywords:

• Sidon sequence/set
• Golomb ruler/set
• tiling
• tessellation
• music composition

2010 Mathematics Subject Classification:

• 43A46
• 00A65
• 05B45

Acknowledgements

We would like to thank the two anonymous reviewers and Editor Clifton Callender for their thoughtful consideration of this paper, as well as their many helpful suggestions.

Supplemental online material

Supplemental online material for this article can be accessed at http://dx.doi.org/10.1080/17459737.2016.1168492. This includes audio files to accompany Application 2.3, Example 3.3, and Example 3.4, as well as a spreadsheet listing minimal Sidon ${\mathbb{Z}}_n$-tilings for $12\le n\le 30$.