Symbolic structures in music theory and composition, binary keyboards, and the Thue–Morse shift

We address the broad idea of using mathematical models to inform music theory and composition by implementing them directly in the process of music creation. The mathematical model we will use is the Thue–Morse dynamical system. We briefly survey previously published works that were similarly motivated, and then discuss a new piece of music we composed, inspired by this symbolic dynamical system. In the course of our analysis, we also present an alternative proof of the well-know fact that the Thue–Morse subshift has topological entropy zero that motivated us to think of the map between binary sequences and musical scales as “binary keyboards.” Indeed, the binary representation allows to study musical scales through mathematical properties of the sequences that define them; here we present the set of standard Thue–Morse scales and compare them with other well-known musical scales.