Two Musical Paths to the Farey Series and Devil s Staircase

At the 2007 Helmholtz Workshop in Berlin, two seemingly disparate papers were presented. One of these, by Julyan Cartwright, Diego González, and Oreste Piro, dealt with a nonlinear dynamical model for pitch perception based on frequency ratios and forced oscillators, while the other, by Jack Douthett and Richard Krantz, focused on musical scales, maximally even (ME) sets, and their relationship to the one-dimensional antiferromagnetic Ising model. Both these approaches lead to a fractal structure involving Farey series known as a devil's staircase. Why is this? What is the connection between them? The ME sets approach is related to the Ising model of statistical physics; on the other hand, the forced oscillator model relates to the circle map of dynamical systems. Thus we find ourselves facing a deeper question: what are the links between these two paradigms, the Ising model and the circle map, that are fundamental to statistical physics on the one hand and to dynamical systems on the other? Here we present the two halves of the work side by side, so that