Generalizing the Orbifold Model for Voice Leading

We generalize orbifold models for chords and voice leading to incorporate loudness, allowing for the modeling of resting voices, which are used frequently by composers and arrangers across genres. In our generalized setting (strictly speaking, that of orbispaces rather than an orbifolds), passages with resting voices, passages with two or more voices in unison, and fully harmonized passages occupy distinct subspaces that interact in mathematically precise and musically interesting ways. In particular, our setting includes previous orbifold models by way of constant-loudness subspaces, and provides a way to model voice leading between chords of different cardinalities. We model voice leading in this general setting by morphisms in the orbispace path groupoid, a category for which we give a formal definition. We demonstrate how to visualize such morphisms as singular braids, and explore how our approach relates to (and is consistent with) selected previous work.