A dynamical systems approach to musical tuning

Abstract

At the end of the 1990s Jaroslav Stark was supervising my PhD studies, on the subject of strange nonchaotic attractors in quasiperiodically forced systems [R.J. Sturman, Strange nonchaotic attractors in quasiperiodically forced systems, Ph.D. thesis, University College London, 2001]. Like several others at the time [A.S. Pikovsky and U. Feudel, Characterizing strange nonchaotic attractors, Chaos Interdis. J. Nonlinear Sci. 5(1) (1995), p. 253; A. Witt, U. Feudel, and A. Pikovsky, Birth of strange nonchaotic attractors due to interior crisis, Physica D: Nonlinear Phenomena 109(1–2) (1997), pp. 180–190], we approximated irrational rotations on the circle by rational approximations of increasing accuracy [R. Sturman, Scaling of intermittent behaviour of a strange nonchaotic attractor, Phys. Lett. A 259(5) (1999), pp. 355–365]. During one conversation about circle maps, Jaroslav asked me about the connection between musical tuning and rational and irrational rotations; he had correctly recognized the link between this dichotomy and that which separates equal temperament from Pythagorean tuning. He was delighted when I described to him the natural extension to nonlinear circle maps which forms the basis of this article. Recently the ideas for such nonlinear tunings have been used as the basis for original compositions.