A Geometric Model of Musical Harmony

This thesis develops a mathematical framework for modeling harmonic structure and voice leading through orbifold geometry. An $n$-note sonority is represented as a point of $\mathbb{R}^n$ in logarithmic pitch coordinates. Musical equivalences such as octave shifts ($O$), voice permutations ($P$), transpositions ($T$), and inversions ($I$) act on $\mathbb{R}^n$, and each subgroup $G \le \langle O, P, T, I \rangle$ determines a different level of musical analysis. The corresponding quotient $\mathbb{R}^n / G$ is an orbifold whose singular strata arise where the action is not free, corresponding to symmetric pitch configurations. The topology of several such quotients is examined, with explicit homeomorphisms and fiber bundle descriptions obtained in musically relevant cases, including Coxeter-type orbifolds. We describe their local models, compute fundamental groups, and analyze how isotropy affects homotopy within these spaces. A new categorical formalism, the \emph{Voice-Leading Category}, is introduced. Its objects are classes of $n$-note sonorities, and its morphisms are endpoint-fixed homotopy classes of paths in orbifolds. This construction extends the Poincaré groupoid to the orbifold setting, providing a rigorous geometric interpretation of harmonic identity and transformation.