By Elaine Chew.

From the Preface: Blending ideas from operations research, music psychology, music theory, and cognitive science, this book aims to tell a coherent story of how tonality pervades our experience, and hence our models, of music. The story is told through the developmental stages of the Spiral Array model for tonality, a geometric model designed to incorporate and represent principles of tonal cognition, thereby lending itself to practical applications of tonal recognition, segmentation, and visualization. Mathematically speaking, the coils that make up the Spiral Array model are in effect helices, a spiral referring to a curve emanating from a central point. The use of “spiral” here is inspired by spiral staircases, intertwined spiral staircases: nested double helices within an outer spiral. The book serves as a compilation of knowledge about the Spiral Array model and its applications, and is written for a broad audience, ranging from the layperson interested in music, mathematics, and computing to the music scientist-engineer interested in computational approaches to music representation and analysis, from the music-mathematical and computational sciences student interested in learning about tonality from a formal modeling standpoint to the computer musician interested in applying these technologies in interactive composition and performance. Some chapters assume no musical or technical knowledge, and some are more musically or computationally involved.

The Mathematical Theory of Tone Systems patterns a unified theory defining the tone system in functional terms based on the principles and forms of uncertainty theory. This title uses geometrical nets and other measures to study all classes of used and theoretical tone systems, from Pythagorean tuning to superparticular pentatonics. Hundreds of examples of past and prevalent tone systems are featured. Topics include Fuzziness and Sonance, Wavelets and Nonspecificity, Pitch Granulation and Ambiguity, Equal Temperaments, Mean Tone Systems. Well Tempered Systems, Ptolemy Systems, and more. Appendices include extended lists of tone systems and a catalogue of historical organs with subsemitones.

Tonal music, from a historical perspective, is far from homogenous; yet an enduring feature is a background "diatonic" system of exactly seven notes orderable cyclically by fifth. What is the source of the durability of the diatonic system, the octave of which is representable in terms of two particular integers, namely 12 and 7? And how is this durability consistent with the equally remarkable variety of musical styles — or languages — that the history of Western tonal music has taught us exist? This book is an attempt to answer these questions. Using mathematical tools to describe and explain the Western musical system as a highly sophisticated communication system, this theoretical, historical, and cognitive study is unprecedented in scope and depth. The author engages in intense dialogue with 1000 years of music-theoretical thinking, offering answers to some of the most enduring questions concerning Western tonality. The book is divided into two main parts, both governed by the communicative premise. Part I studies proto-tonality, the background system of notes prior to the selection of a privileged note known as "final." After some preliminaries that concern consonance and chromaticism, Part II begins with the notion "mode." A mode is "dyadic" or "triadic," depending on its "nucleus." Further, a "key" is a special type of "semi-key" which is a special type of mode. Different combinations of these categories account for tonal variety. Ninth-century music, for example, is a tonal language of dyadic modes, while seventeenth-century music is a language of triadic semi-keys. While portions of the book are characterized by abstraction and formal rigor, more suitable for expert readers, it will also be of value to anyone intrigued by the tonal phenomenon at large, including music theorists, musicologists, and music-cognition researchers. The content is supported by a general index, a list of definitions, a list of notation used, and two appendices providing the basic mathematical background.

This volume highlights the papers presented at the National Academy of Engineering's 2010 U.S. Frontiers of Engineering Symposium. Every year, the symposium brings together 100 outstanding young leaders in engineering to share their cutting-edge research and technical work. The 2010 symposium was held September 23 - 25, and hosted by IBM at the IBM Learning Center in Armonk, New York. Speakers were asked to prepare extended summaries of their presentations, which are reprinted here. The intent of this book is to convey the excitement of this unique meeting and to highlight cutting-edge developments in engineering research and technical work.

With contributions by numerous experts

Mathemusical Conversations celebrates the understanding of music through mathematics, and the appreciation of mathematics through music. This volume is a compilation of the invited talks given at the Mathemusical Conversations workshop that took place in Singapore from 13-15 February 2015, organized by Elaine Chew in partnership with Gérard Assayag for the scientific program and with Bernard Lanskey for the artistic program. The contributors are world experts and leading scholars, writing on the intersection of music and mathematics. They also focus on performance and composition, two topics which are foundational both to the understanding of human creativity and to the creation of tomorrow's music technologies. This book is essential reading for researchers in both music and mathematics. It will also appeal more broadly to scholars, students, musicians, and anyone interested in new perspectives on the intimate relationship between these two universal human activities.

Ornette Coleman, Psychoanalysis, Discourse develops tools from psychoanalysis for the analysis of Ornette Coleman's discourse. In this psychoanalytic, philosophical and musical meditation on what it means to follow, A. L. James presents an approach to the analysis of discourse that is a kind of listening for listening – an attempt to discern in and between the lines of Coleman's speech the implication of new ways to listen, new ways to experience Coleman’s music as movement and space – as Movements in Harmolodic Space. Each chapter of this book is oriented with respect to fragments from Coleman’s discourse, dealing with a piece, or collection of pieces, from Coleman’s work, with particular attention to the implication of relations and relationality. Insofar as Coleman’s discourse about his work also contains allusions to fields beyond music, it develops tools that draw elements and structures from these fields together, finding in their relation echoes and parallels. Ornette Coleman, Psychoanalysis, Discourse will be of great interest to psychoanalysts, musicians, and musicologists. It will be relevant for academics and scholars of psychoanalytic and Lacanian studies, music, and cultural studies.