Nonlinear Dynamics, the Missing Fundamental, and Harmony

We review the historical and current theories of musical pitch perception, and their relationship to the intriguing phenomenon of residue pitch. We discuss the nonlinear dynamics of forced oscillators, and the role played by the Fibonacci numbers and the golden mean in the organization of frequency locking in oscillators. We show how a model of the perception of musical pitch may be constructed from the dynamics of oscillators with three interacting frequencies. We then present a mathematical construction, based on the golden mean, that generates meaningful musical scales of different numbers of notes. We demonstrate that these numbers coincide with the number of notes that an equal-tempered scale must have in order to optimize its approximation to the currently used harmonic musical intervals. Scales with particular harmonic properties and with more notes than the twelve-note scale now used in Western music can be generated. These scales may be rooted in objective phenomena taking place in the nonlinearities of our perceptual and nervous systems. We conclude with a discussion of how residue pitch perception may be the basis of musical harmony.