Related papers: Towards A Categorical Approach of Transformational…
Transformational music theory is a recent field in music theory which studies the possible transformations between musical objects, such as chords. In the framework of the theory initiated by David Lewin, the set of all transformations…
Chords in musical harmony can be viewed as objects having shapes (major/minor/etc.) attached to base sets (pitch class sets). The base set and the shape set are usually given the structure of a group, more particularly a cyclic group. In a…
Transformational musical theory has so far mainly focused on the study of groups acting on musical chords, one of the most famous example being the action of the dihedral group D24 on the set of major and minor chords. Comparatively less…
Transformational music theory, pioneered by the work of Lewin, shifts the music-theoretical and analytical focus from the "object-oriented" musical content to an operational musical process, in which transformations between musical elements…
Transformational music theory, pioneered by David Lewin, uses simply transitive group actions to analyze music. In this paper, we construct a simply transitive group action on a disjoint union of two sets, built from a simply transitive…
In the field of transformational music theory, which emphasizes the possible transformations between musical objects, Klumpenhouwer networks (K-Nets) constitute a useful framework with connections in both group theory and graph theory.…
In this work we provide a categorical formalization of several constructions found in transformational music theory. We first revisit David Lewin's original theoretical construction of Generalized Interval Systems (GIS) to show that it…
The paper deals with the question of homometry in the dihedral groups $D_{n}$ of order $2n$. These groups have the specificity to be non-commutative. It leads to a new approach as compared as the one used in the traditional framework of the…
We develop a categorical analogue of Clifford theory for strongly graded rings over graded fusion categories. We describe module categories over a fusion category graded by a group $G$ as induced from module categories over fusion…
The sequence of pitches which form a musical melody can be transposed or inverted. Since the 1970s, music theorists have modeled musical transposition and inversion in terms of an action of the dihedral group of order 24. More recently…
Ornaments aim at taming the multiplication of special-purpose datatype in dependently-typed theory. In its original form, the definition of ornaments is tied to a particular universe of datatypes. Being a type theoretic object,…
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…
Symbolic Music Generation relies on the contextual representation capabilities of the generative model, where the most prevalent approach is the Transformer-based model. The learning of musical context is also related to the structural…
Despite ample evidence that our concepts, our cognitive architecture, and mathematics itself are all deeply compositional, few models take advantage of this structure. We therefore propose a radically compositional approach to computational…
Polyphonic music generation is still a challenge direction due to its correct between generating melody and harmony. Most of the previous studies used RNN-based models. However, the RNN-based models are hard to establish the relationship…
We classify three-tone and four-tone chords based on subgroups of the symmetric group acting on chords contained within a twelve-tone scale. The actions are inversion, major-minor duality, and augmented-diminished duality. These actions…
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…
This paper introduces the concept of distorted monoidal categories, a generalization of monoidal and braided monoidal categories that supports non-reversible and direction-sensitive tensor structures. Unlike the classical setting, where the…
We develop aspects of music theory related to harmony, such as scales, chord formation and improvisation from a combinatorial perspective. The goal is to provide a foundation for this subject by deriving the basic structure from a few…
We generalise classical reconstruction results in algebra, using the language of monads, monoidal categories, module categories, as well as various notions of duality, such as closedness, Grothendieck--Verdier duality (also known as…