Emmanuel Amiot

An algebra for periodic rhythms and scales

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 Lewins special cases, regular matrices allow a simple method of recovering the argument of an interval function and elicit unique decompositions, large determinant values correspond to flat interval distributions.

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 Lewins use of DFT in pc-sets in the first place. The nice inclusion property between a ME set and its complement (up to translation) is also developed, as occurs in actual music.

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.

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.

New perspectives on rhythmic canons and the spectral conjecture

The musical notion of rhythmic canons has proved to be relevant to some non-trivial mathematical problems. After a survey of the main concepts of tiling rhythmic canons, we discuss recent developments that enable to make, or expect, definite progress on several open mathematical conjectures.

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.

A geometry of music: harmony and counterpoint in the extended common practice

This is the long-awaited magnum opus of one of the keenest and most imaginative researchers in the field nowadays, and as such, the publication of this book is a momentous event in itself. For most readers, and especially for its intended audience, it deserves unreserved praise. Its groundbreaking contents, clear pedagogical exposition of admittedly difficult and technical topics, numerous examples, wide field of application, additional resources (multimedia content and software), and immediate usability for many composers or students make this work a significant hallmark of music theory, and a reference volume for years to come, on a par with Basic Atonal Theory [1] or The Topos of Music [2], to name some of the few publications of this caliber. Reviewing it for JMM, however, compels us to cast a more critical eye on some of the author’s choices and strategies, which a few of the more mathematically inclined readers might find slightly irksome.

La Vérité du Beau dans la Musique

This book, written in French, stems from a series of lectures given in 2005 at the Ecole Normale Supérieure in Paris. According to the author, the French language was chosen because it is ‘la langue de la culture par excellence’. In present times, it certainly is a courageous editorial choice. The opus makes use of the richness and subtleties of the language, with an extensive vocabulary verging on the recherché.1 This might make the reading difficult for non-native readers, but it also does justice to the nuances of the author’s thought. As the course in Paris was purportedly for non-specialists, either in music or in mathematics, this book is far more readable than Mazzola’s magnum opus, The Topos of Music [1], which contains with due rigour all the mathematical tools and theorems of Mazzola’s ‘théorie mathématique de la musique’, together with many of the aesthetic ideas of the present book. More generally, most of the scientific results to be found in this book have already been published during the author’s career, from Geometrie der Töne [2] to The Topos of Music: Geometric Logic of Concepts, Theory, and Performance [1]. This might appear as a flaw to the happy few who have been able to follow Mazzola’s constructions through his tough, technical, previous works: they will know most of the notions of mathematical music theory that are presented here as illustrations of Mazzola’s idea about la vérité du beau, ‘the truth about beauty’ (its title notwithstanding, Mazzola’s theses are much bolder than Hanslick’s in his similarly titled essay [3]). But in accordance with the intended audience, the perspective is entirely different. Although it cannot hurt (not by a long shot) if the reader has some knowledge of mathematics, the math only exemplifies ideas of a philosophical nature, among which the most provocative and intriguing might be those about the relationship between mathematics and perception of beauty in music. Hence the book’s title is fully appropriate to its content; this is no ‘Topos of Music for Dummies’. On the other hand, it might give useful and valuable insights on the importance and power of Mazzola’s theory to those who cannot or will not delve into it [1].