Quasimusic: tilings and metre

Abstract

In this paper, I try to explain how, by using concepts and ideas from the mathematical theory of tilings, we can approach metre in music through a geometric and algebraic point of view, being pinned down by a subgroup of $\mathbb{R}$ with the hierarchical structure, leading to an abstract approach to rhythm, tempo and time signatures. I will also describe an algorithmic approach to write down sound using this structure which gives a way in which music can be written in an irrational metre.

Keywords:

• Tilings
• music
• rhythm

Acknowledgments

The audiofiles made were used using Ken Schutte's midi script for Matlab and Reaper. I am grateful to Noah Giansiracusa for providing valuable feedback on an early draft of this paper. I am also grateful to Dmitri Tymoczko for pointing out earlier works of which I was not aware.

Disclosure statement

No potential conflict of interest was reported by the authors.

Correction Statement

This article has been republished with minor changes. These changes do not impact the academic content of the article.

Notes

1 Since I am completely ignorant of the structure of other musical traditions, I am narrowing my claim here in acknowledgement of this ignorance and not claiming that these properties are unique to the Western musical tradition. I would be delighted to learn whether or not the ideas here can also be adapted to non-Western musical systems.

2 It has also been claimed, and strongly disputed, that the intricate artwork in Islamic architecture over centuries and over a large geographic area has underlying aperiodic structure (Bonner, 2017; Cromwell, 2015).

3 Sturmian sequences are non-periodic sequences with the lowest possible complexity.

4 This is not the chair tiling, that is, it is not the substitution rule which is called “the chair tiling” in the tiling literature, but a variation. The canonical chair tiling has an expansion factor of 2 whereas the variation here has an expansion factor of 4.

5 The curious reader may want to verify the following: after n steps applying the inflation and substitution rule, we obtain patches which are tiled by roughly ${\lambda }^{2n}$ tiles.

6 Note that if a letter is substituted into a word which begins with the same letter, the words obtained by applying the substitution rule stabilize.

7 Some people follow the convention that would make this matrix the transpose of the substitution matrix.

8 A companion matrix $M_p$ for a polynomial $p\left(x\right)$ is the matrix A such that its characteristic polynomial is $p\left(x\right)$. One can check that $x^3-x-1$ is the characteristic polynomial os S above.

Additional information

Funding

This work was supported by Simons Foundation.