Barberpole tempo illusions
Daniele Ghisi · Journal of Mathematics and Music · 2021
“Barberpole” tempo illusions are a family of auditory illusions based on the synchronization of faded rhythmic streams playing at different rates, often manufacturing experiences of seemingly eternal acceleration or deceleration. The forefather of all such illusions, based on layers whose rates are powers of two apart (“octaves”), was studied by Jean-Claude Risset in the late seventies and is now known as Risset rhythm. This article provides a mathematical framework for barberpole tempo illusions, generalizing Risset rhythms for arbitrary numbers of subdivisions, non-integer proportions, arbitrary rate modulation, and increasingly accelerating tempi. Furthermore, this article describes a new illusion of eternal rallentando/accelerando based on the full harmonic spectrum of rates. This construction shows that Risset rhythms are related to barberpole variable-rate polyrhythms. A notable application of the study of divisional structures that barberpole illusions underpin is the construction of bistable auditory figures (accelerating or decelerating depending on the stream being focused).