Journal of Mathematics and Music 

About: Journal of Mathematics and Music is an academic journal published by Taylor & Francis. The journal publishes majorly in the area(s): Music theory & Computer science. It has an ISSN identifier of 1745-9737. Over the lifetime, 243 publications have been published receiving 1847 citations. The journal is also known as: Journal of mathematics and music.

Towards a generative syntax of tonal harmony

Abstract: This paper aims to propose a hierarchical, generative account of diatonic harmonic progressions and suggest a set of phrase-structure grammar rules. It argues that the structure of harmonic progressions exceeds the simplicity of the Markovian transition tables and proposes a set of rules to account for harmonic progressions with respect to key structure, functional and scale degree features as well as modulations. Harmonic structure is argued to be at least one subsystem in which Western tonal music exhibits recursion and hierarchical organization that may provide a link to overarching linguistic generative grammar on a structural and potentially cognitive level.

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 conjec...

David Lewin and maximally even sets

Abstract: David Lewin originated an impressive number of new ideas in musical formalized analysis. This paper formally proves and expands one of the numerous innovative ideas published by Ian Quinn in his dissertation, to the import that Lewin might have invented the much later notion of Maximally Even Sets with but a small extension of his very first published idea, where he made use of Discrete Fourier Transform (DFT) to investigate the intervallic differences between two pc-sets. Many aspects of Maximally Even Sets (ME sets) and, more generally, of generated scales, appear obvious from this original starting point, which deserves, in our opinion, to become standard. In order to vindicate this opinion, we develop a complete classification of ME sets starting from this new definition. As a pleasant by-product we mention a neat proof of the hexachord theorem, which might have been the motivation for Lewin's use of DFT in pc-sets in the first place. The nice inclusion property between a ME set and its compl...

Minimum description length modelling of musical structure

Abstract: This article presents a method of inductive inference whose aim is to build formal quantitative models of musical structure. The models are constructed by statistical extraction of significant patterns from a musical corpus. The minimum description length (MDL) principle is used to select the best model from among the members of a non-parametric model family characterized by an unbounded parameter set. The chosen model achieves optimal compromise between goodness-of-fit and model complexity, thereby avoiding the over-fitting normally associated with such a family of models. The MDL method is illustrated through its application to the Hidden Markov Model (HMM) framework. We derive an original mathematical expression for the MDL complexity of HMMs that employ a finite alphabet of symbols; these models are particularly suited to the symbolic modelling of musical structure. As an illustration, we use the proposed HMM complexity expression to construct a model for a common melodic formula in Greek church chant...

An algebra for periodic rhythms and scales

Abstract: This paper shows how scale vectors (which can represent either pitch or rhythmic patterns) can be written as a linear combination of columns of scale matrices, thus decomposing the scale into musically relevant intervals. When the scales or rhythms have different cardinalities, they can be compared using a canonical form closely related to Lyndon words. The eigenvalues of the scale matrix are equal to the Fourier coefficients, which leads to a number of relationships between the scale vectors and the decompositions. Overcomplete dictionaries of frame elements can be used for more convincing representations by finding sparse decompositions, a technique that can also be applied to tiling problems. Scale matrices are related to familiar theoretical properties such as the interval function, Z-relation or homometry, all of which can be efficiently studied within this framework. In many cases, the determinant of the scale matrix is key: singular scale matrices correspond to Lewin's special cases, regular matric...