The Distance Geometry of Music

We demonstrate relationships between the classic Euclidean algorithm and many other fields of study, particularly in the context of music and distance geometry. Specifically, we show how the structure of the Euclidean algorithm defines a family of rhythms which encompass over forty timelines (\emph{ostinatos}) from traditional world music. We prove that these \emph{Euclidean rhythms} have the mathematical property that their onset patterns are distributed as evenly as possible: they maximize the sum of the Euclidean distances between all pairs of onsets, viewing onsets as points on a circle. Indeed, Euclidean rhythms are the unique rhythms that maximize this notion of \emph{evenness}. We also show that essentially all Euclidean rhythms are \emph{deep}: each distinct distance between onsets occurs with a unique multiplicity, and these multiplicies form an interval $1,2,...,k-1$. Finally, we characterize all deep rhythms, showing that they form a subclass of generated rhythms, which in turn proves a useful property called shelling. All of our results for musical rhythms apply equally well to musical scales. In addition, many of the problems we explore are interesting in their own right as distance geometry problems on the circle; some of the same problems were explored by Erd\H{o}s in the plane.Comment: This is the full version of the paper: "The distance geometry of deep rhythms and scales." 17th Canadian Conference on Computational Geometry (CCCG '05), University of Windsor, Canada, 200

Marriages of Mathematics and Physics: A Challenge for Biology

The human attempts to access, measure and organize physical phenomena have led to a manifold construction of mathematical and physical spaces. We will survey the evolution of geometries from Euclid to the Algebraic Geometry of the 20th century. The role of Persian/Arabic Algebra in this transition and its Western symbolic development is emphasized. In this relation, we will also discuss changes in the ontological attitudes toward mathematics and its applications. Historically, the encounter of geometric and algebraic perspectives enriched the mathematical practices and their foundations. Yet, the collapse of Euclidean certitudes, of over 2300 years, and the crisis in the mathematical analysis of the 19th century, led to the exclusion of “geometric judgments” from the foundations of Mathematics. After the success and the limits of the logico-formal analysis, it is necessary to broaden our foundational tools and re-examine the interactions with natural sciences. In particular, the way the geometric and algebraic approaches organize knowledge is analyzed as a cross-disciplinary and cross-cultural issue and will be examined in Mathematical Physics and Biology. We finally discuss how the current notions of mathematical (phase) “space” should be revisited for the purposes of life sciences

Rhythmic maximal evenness: rhythm in voice-leading space

Maximal evenness was first introduced in the music theory domain by John Clough and Jack Douthett. Later, the concept was explored by others such as Dmitri Tymoczko and Richard Cohn. Although maximal evenness was first explored with respect to pitch-classes, the concept can be understood in the rhythmic domain. An explanation of voice-leading space can be found here to create a conceptual foundation before departing to the implications of maximal evenness on rhythm. This thesis will then explore the concept further by exploring music from Steve Reich and György Ligeti to demonstrate the applicability and deeper understanding of the concept

Science, Art and Geometrical Imagination

From the geocentric, closed world model of Antiquity to the wraparound universe models of relativistic cosmology, the parallel history of space representations in science and art illustrates the fundamental role of geometric imagination in innovative findings. Through the analysis of works of various artists and scientists like Plato, Durer, Kepler, Escher, Grisey or the present author, it is shown how the process of creation in science and in the arts rests on aesthetical principles such as symmetry, regular polyhedra, laws of harmonic proportion, tessellations, group theory, etc., as well as beauty, conciseness and emotional approach of the world.Comment: 22 pages, 28 figures, invited talk at the IAU Symposium 260 "The Role of Astronomy in Society and Culture", UNESCO, 19-23 January 2009, Paris, Proceedings to be publishe

Preferences in Musical Rhythms and Implementation of Analytical Results to Generate Rhythms

Rhythm is at the heart of all music. It is the variation of the duration of sound over time. A rhythm has two components: one is the striking of an instrument – called the onset – and the other is silence. Historically, musical forms and works were preferred and became popular by their rhythmic properties. Therefore, to study rhythm is to study the underpinnings of all of music. In this thesis, we explore basic rhythmic preferences in traditional music and, using this as a point of reference, methods are implemented to generate similar types of rhythms. Finally, a software platform to facilitate such an analysis is developed – it is the first of its kind available to our best knowledge as this research field has only recently emerged

Preferences in Musical Rhythms and Implementation of Analytical Results to Generate Rhythms

Rhythm is at the heart of all music. It is the variation of the duration of sound over time. A rhythm has two components: one is the striking of an instrument – called the onset – and the other is silence. Historically, musical forms and works were preferred and became popular by their rhythmic properties. Therefore, to study rhythm is to study the underpinnings of all of music. In this thesis, we explore basic rhythmic preferences in traditional music and, using this as a point of reference, methods are implemented to generate similar types of rhythms. Finally, a software platform to facilitate such an analysis is developed – it is the first of its kind available to our best knowledge as this research field has only recently emerged