Symbolic dynamical scales: modes, orbitals, and transversals

TL;DR

This paper explores the mathematical structure of musical scales using symbolic dynamics, analyzing modes, orbitals, and transversals to understand their properties across different tuning systems.

Contribution

It introduces a novel framework linking symbolic dynamics with musical scales, providing explicit formulas for orbital and transversal dimensions in various tuning systems.

Findings

Derived generating functions for scale classes

Calculated orbital and transversal dimensions for all n

Applied results specifically to 12-TET tuning system

Abstract

We study classes of musical scales obtained from shift spaces in symbolic dynamics through the first symbol rule, which yields scales in any $n$-TET tuning system. The modes are thought as elements of orbit equivalence classes of cyclic shift actions on languages, and we study their orbitals and transversals. We present explicit formulations of the generating functions that allow us to deduce the orbital and transversal dimensions of classes of musical scales generated by vertex shifts, for all $n$, in particular for the 12-TET tuning system.