Moreno Andreatta

Diagrams, gestures and formulae in music

Abstract This paper shows an interplay of music and mathematics which strongly differs from the usual scheme reducing mathematics to a toolbox of formal models for music. Using the topos of directed graphs as a common base category, we develop a comprising framework for mathematical music theory, which ramifies into an algebraic and a topological branch. Whereas the algebraic component comprises the universe of formulae, transformations, and functional constraints as they are described by functorial diagrammatic limits, the topological branch covers the continuous aspects of the creative dynamics of musical gestures and their multilayered articulation. These two branches unfold in a surprisingly parallel manner, although the concrete structures (homotopy versus representation theory) are fairly heterogeneous. However, the unity of the underlying musical substance suggests that these two apparently divergent strategies should find a common point of unification, an idea that we describe in terms of a conjectural diamond of categories which suggests a number of unification points. In particular, the passage from the topological to the algebraic branch is achieved by the idea of the gestoid, an ‘algebraic’ category associated with the fundamental groupoid of a gesture.

Computation and Visualization of Musical Structures in Chord-Based Simplicial Complexes

We represent chord collections by simplicial complexes. A temporal organization of the chords corresponds to a path in the complex. A set of n-note chords equivalent up to transposition and inversion is represented by a complex related by its 1-skeleton to a generalized Tonnetz. Complexes are computed with MGS, a spatial computing language, and analyzed and visualized in Hexachord, a computer-aided music analysis environment. We introduce the notion of compliance, a measure of the ability of a chord-based simplicial complex to represent a musical object compactly. Some examples illustrate the use of this notion to characterize musical pieces and styles.

Z-relation and homometry in musical distributions

This paper defines homometry in the rather general case of locally-compact topological groups, and proposes new cases of its musical use. For several decades, homometry has raised interest in computational musicology and especially set-theoretical methods, and in an independent way and with different vocabulary in crystallography and other scientific areas. The link between these two approaches was only made recently, suggesting new interesting musical applications and opening new theoretical problems. We present some old and new results on homometry, and give perspective on future research assisted by computational methods. We assume from the readers basic knowledge of groups, topological groups, group algebras, group actions, Lebesgue integration, convolution products, and Fourier transform.

Towards a Categorical Theory of Creativity for Music, Discourse, and Cognition

This article presents a first attempt at establishing a category-theoretical model of creative processes. The model, which is applied to musical creativity, discourse theory, and cognition, suggests the relevance of the notion of “colimit” as a unifying construction in the three domains as well as the central role played by the Yoneda Lemma in the categorical formalization of creative processes.

Discrete phase retrieval in musical structures

This paper describes phase-retrieval approaches in music by focusing on the particular case of the cyclic groups (beltway problem). After presenting some old and new results on phase retrieval, we introduce the extended phase retrieval for a generalized musical Z-relation. This concept is accompanied by mathematical definitions and motivations from computer-aided composition. We assume from the reader basic knowledge of groups, topological groups, group algebras, group actions, Lebesgue integration, convolution products, and Fourier transform.

A Categorical Generalization of Klumpenhouwer Networks

This article proposes a functorial framework for generalizing some constructions of transformational theory. We focus on Klumpenhouwer Networks for which we propose a categorical generalization via the concept of set-valued poly-K-nets (henceforth PK-nets). After explaining why K-nets are special cases of these category-based transformational networks, we provide several examples of the musical relevance of PK-nets as well as morphisms between them. We also show how to construct new PK-nets by using some topos-theoretical constructions.

Using Formal Concept Analysisto Represent Chroma Systems

The article discusses the application of Formal Concept Analysis to the algebraic enumeration, classification and representation of musical structures. It focuses on the music-theoretical notion of the Tone System and its equivalent classes obtained either via an action of a given finite group on the collection of subsets of it or via an identification of Forte’s corresponding interval vector and Lewin’s interval function. The use of concept lattices, applied to a simple case such as the division of the octave into five equal parts and the associated Chroma System, clearly shows that these approaches are conceptually different. The same result is obtained for a given subsystem of the traditional Tone System, as we will show by analysing the case of the pentatonic system. This opens a window towards generic tone systems that can be used as starting point for the structural analysis of other finite chroma systems.

Representation of musical structures and processes in simplicial chord spaces

In this article, we present a set of musical transformations based on the representations of chord spaces derived from the Tonnetz. These chord spaces are formalized as simplicial complexes. A musical composition is represented in such a space by a trajectory. Spatial transformations are applied on these trajectories and induce a transformation of the original composition. These concepts are implemented in two applications, the software HexaChord and the Max object bach.tonnetz, dedicated to music analysis and composition, respectively.

Mathematics and Computation in Music

This book constitutes the refereed proceedings of the Third International Conference on Mathematics and Computation in Music, MCM 2011, held in Paris, France, in June 2011. The 24 revised full papers presented and the 12 short papers were carefully reviewed and selected from 62 submissions. The MCM conference is the flagship conference of the Society for Mathematics and Computation in Music. This year’s conference aimed to provide a multi-disciplinary platform dedicated to the communication and exchange of ideas amongst researchers involved in mathematics, computer science, music theory, composition, musicology, or other related disciplines. Areas covered were formalization and geometrical representation of musical structures and processes; mathematical models for music improvisation and gestures theory; set-theoretical and transformational approaches; computational analysis and cognitive musicology as well as more general discussions on history, philosophy and epistemology of music and mathematics.

An Introduction on Formal and Computational Models in Popular Music Analysis and Generation

This article provides a first introduction to some formal and computational models applied in the analysis and generation of popular music (including rock, jazz, and chanson). It summarizes the main philosophy underlying the project entitled “Modeles formels dans et pour la musique pop, le jazz et la chanson”, which constitutes one of the research axes of the GDR ESARS (Esthetique, Art & Science). Initially conceived as an extension of the MISA project carried on by the Music Representation Team at IRCAM, this research axis aims at bringing together researchers from different horizons, from the traditional MIR community of Music Information Retrieval to the most sophisticated approaches in mathematical music theory and computational musicology. It also includes an epistemological and critical evaluation of the relations between music and mathematics, together with some programmatic reflections on the possible cognitive and perceptual implications of this research.