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.

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 Gerard 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.

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.

These proceedings comprise 26 refereed research papers that were presented at theSecondInternationalConferenceonMathematicsandComputationinMusic (MCM 2009), which met in conjunction with the John Clough Memorial C- ference during June 19-22, 2009, at Yale University in New Haven, Connecticut, USA. The International Conference on Mathematics and Computation in Music (MCM) is the ?agship conference of the Society for Mathematics and Com- tation in Music. The inaugural conference of the society took place in 2007 in Berlin.Thestudyofmathematicsandmusicdatesbacktothetimeoftheancient Greeks.The riseof computing andthe digitalagehas addedcomputationto this august tradition. MCM aims to provide a dedicated platform for the commu- cation and exchange of ideas amongst researchers in mathematics, informatics, music theory, composition, musicology, and related disciplines. TheJohnCloughMemorialConferencehonorsamathematicalmusictheorist whose research modeled the virtues of collaborative work across the disciplines, and who generously fostered a cooperative attitude with and among younger researchers. The quadrennial conferences that Clough ?rst organized at Bu?alo in 1993 positioned neo-Riemannian theory on the map of musical scholarship. TheJohnCloughMemorialConferencecarriesthespiritofthosesessionsbeyond his passing in 2003, while embracing the entire domain of mathematical music theory. High-quality contributions - including research papers, invited sessions or panels, tutorials, andexhibits - were solicitedin allareasrelatedto the mission of the societ