arXiv subjects

Drew Flieder

Publications and source records attributed to Drew Flieder.

3 recordsLinked to original sources

arXiv2026

Mathematical Principles of a Generalized Tonal Practice

We develop mathematical principles for a generalized tonal practice. We begin with an algebraic theory of modes and scales, formalized as groups and torsors over cyclic groups, respectively. We identify three musically salient classes of mode isomorphism---translations, rotations, and deformations. Upon this basis we organize successive layers of harmonic structure into a hierarchy of Grothendieck fibrations, built from scale types (generalizing the diatonic scale), modes (generalizing the diatonic modes), orbit covers (generalizing their covering by tertian triads), deformation families (generalizing chromatic alteration, such as modal mixture and secondary chords), and harmonic-function assignments (generalizing the assignment of tonic, subdominant, and dominant function). Equipping the resulting structure with a generative syntax---generalizing Rohrmeier's generative grammar of tonal harmony---yields a formal definition of a harmonic practice: a harmonic vocabulary together with an assignment of harmonic functions and a generative syntax accommodating hierarchical harmonic organization. We show that harmonic practices form a category, with morphisms constituting an inheritance and enrichment of structure from one harmonic practice to another. The theory originates in the author's own compositional practice, developed initially through the series of compositions Op. 26--31.

math.GM↗

arXiv2026

A Theory of Scales and Orbit Covers

This paper develops a formal theory of musical scales and their harmonic coverings and introduces orbit covers: coverings obtained by translating a fixed subset across a scale via a group action. Orbit covers generalize familiar constructions, such as the covering of the diatonic scale by tertian triads, and are motivated by the search for a generalized harmonic framework extending common-practice tonality. We model modes as group structures associated with pitch-class sets and scales as torsors, introducing scale covers and, in particular, orbit covers. To each orbit cover we associate a nerve complex encoding its intersection structure and associated topological invariants. We classify triadic orbit covers of heptatonic scales up to affine symmetry and nerve isomorphism. These results support a broader theory of harmonic organization with analytical and compositional applications.

math.GM↗

arXiv2025

Type Theory for the Working Mathematical Music Theorist

Many formal languages of contemporary mathematical music theory -- particularly those employing category theory -- are powerful but cumbersome: ideas that are conceptually simple frequently require expression through elaborate categorical constructions such as functor categories. This paper proposes a remedy in the form of a type-theoretic symbolic language that enables mathematical music theorists to build and reason about musical structures more intuitively, without relinquishing the rigor of their categorical foundations. Type theory provides a syntax in which elements, functions, and relations can be expressed in simple terms, while categorical semantics supplies their mathemusical interpretation. Within this system, reasoning itself becomes constructive: propositions and proofs are treated as objects, yielding a framework in which the formation of structures and the reasoning about them take place within the same mathematical language. The result is a concise and flexible formalism that restores conceptual transparency to mathemusical thought and supports new applications, illustrated here through the theory of voice-leading spaces.

math.CT↗