Introduction

Until now, diatonic systems and neo-Riemannian transformations have generally been considered separately, and group and graph theoretic approaches have dominated the neo-Riemannian and transformational theory literature. In this paper, an alternative approach will be explored; techniques similar to those used in the study of dynamical systems in science will be adopted to study neo- Riemannian theory and its connection to diatonic theory.

Dynamical systems are probably best known today for the fractals they sometimes generate (e.g., Koch's snowflake, the Dragon curve, Mandelbrot's set, the Julia set, etc.), but fractals are only part of this field of study. As Strogatz puts it in his text on nonlinear dynamics and chaos, dynamics ”… is the subject that deals with change, with systems that evolve in time. Whether the system in question settles down to equilibrium, keeps repeating in cycles, or does something more complicated, it is dynamics we use to analyze the behavior.”

Dynamical systems related to the topics that will be discussed here will utilize point-symmetric concentric circles that rotate through time and stroboscopic portraits, which record events at particular time intervals. These dynamical systems can be thought of as sequence generators which, with the appropriate choice of control parameters, produce periodic orbits (cycles) of scales and chords well known in both diatonic and neo-Riemannian theory.

Douthett and Steinbach's Relation Definition will also be used:

Let X and Y be pcsets. Then X and Y are Pm,n-related if there exists a set {xk}m+n+1k=0 and a bijection τ : X →Y such that X \ Y (the set of pcs in X that are not in Y ) ={xk}m+n+1k=0, τ(x ) = x if x ∊X ∩Y , and

The requirement that τ be a bijection implies that X and Y have the same cardinality.