Points of convergence -- music meets mathematics

TL;DR

This paper explores the mathematical phenomenon of phase-locking in oscillators, specifically through the Arnold circle maps, and discusses a successful interdisciplinary collaboration with a composer that influenced both science and art.

Contribution

It demonstrates the density of hyperbolic parameters in the Arnold family and details a pioneering collaboration between mathematicians and a composer, impacting artistic practice.

Findings

Hyperbolic parameters are dense in the Arnold family.

The collaboration led to new musical works inspired by mathematical concepts.

The project fostered a lasting influence on science and music integration.

Abstract

"Phase-locking" is a fundamental phenomenon in which coupled or periodically forced oscillators synchronise. The Arnold family of circle maps, which describes a forced oscillator, is the simplest mathematical model of phase-locking and has been studied intensively since its introduction in the 1960s. The family exhibits regions of parameter space where phase-locking phenomena can be observed. A long-standing question asked whether "hyperbolic" parameters~-- those whose behaviour is dominated by periodic attractors, and which are therefore stable under perturbation~-- are dense within the family. A positive answer was given in 2015 by van Strien and the author, which implies that, no matter how chaotic a map within the family may behave, there are always systems with stable behaviour nearby. This research was a focal point of a pioneering collaboration with composer Emily Howard, commencing with Howard's residency in Liverpool's mathematics department in 2015. The collaboration generated impacts on creativity, culture and society, including several musical works by Howard, and lasting influence on artistic practice through a first-of-its-kind centre for science and music. We describe the research and the collaboration, and reflect on the factors that contributed to the latter's success.