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

The Bodily Incorporation of Mechanical Devices: Ethical and Religious Issues (Part I)

A substantial portion of the developed world's population is increasingly dependent on machines to make their way in the everyday world. For certain privileged groups, computers, cell phones, PDAs, Blackberries, and IPODs, all permitting the faster processing of information, are commonplace. In these populations, even exercise can be automated as persons try to achieve good physical fitness by riding stationary bikes, running on treadmills, and working out on cross-trainers that send information about performance and heart rate.This essay is drawn from collaborative research conducted under the auspices of the “Altering Nature: How Religious Traditions Assess the New Biotechnologies” project at Rice University from 2002 to 2004 through the financial support of The Ford Foundation (Grant #1010-1601). The paper is submitted with the permission and acknowledgement of the project coordinators and The Ford Foundation. The authors also acknowledge the research and editorial assistance of Dr. Siobhan Baggot and Ms. Sarah Gehrke