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Mathematics and music: New perspectives on the connections between these ancient arts

The link between music and mathematics is ancient, but we are understanding it better every year

World Scientific

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From Pythagoras to Leibniz to Diderot, philosophers have long drawn profound connections between mathematics and music. In the liberal arts education of medieval universities, music was one part of the quadrivium of mathematical and scientific subjects, alongside arithmetic, geometry, and astronomy, while in ancient China, music and mathematics were two of the Six Arts, the standard curriculum.

Today, the connections between mathematics and music have renewed relevance: at the core of the digital music revolution are mathematical and computational techniques that have driven scientific advances.

Mathemusical Conversations celebrates the understanding of music through mathematics, and the appreciation of mathematics through music. This volume collects the writings of world experts and leading scholars of mathematics and music, all related to the intersection between these two universal human activities. Many of the contributions 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.

The chapters are grouped around five themes:

  • Mathemusical Engagement: perspectives on compelling ways to combine music
  • and mathematics in composition, music games, and performance;
  • Mathemusical Creativity: perspectives on the different types of creativity that exist, and how creativity can emerge from inter-agent and interdisciplinary interaction;
  • Shaping Performance: perspectives on how performers are shaped by musicians, by mathematical relationships within a piece, or by interacting with semi-autonomous accompaniment systems;
  • Educating the Mathemusical: perspectives on how we can bring mathematics
  • into music education and vice versa, and on how interdisciplinary research can be nurtured;
  • Geometries: perspectives on geometrical models of pitch and time, their history, and novel tools for visualizing and utilizing them.

The rise of computing devices and the ensuing digitization of music have led to new ways to represent, create and perform music; to study and understand it; and to make scientific discoveries about how and why music is made. Almost every subfield of mathematics--including number theory, algebraic geometry, topology, geometric analysis, probability, statistics and mathematical physics--as well as every modeling technique--including optimization, stochastic systems, game theory, and network analysis--have contributed to these advances in digital music research. Every step of the music process, from composition (including improvisation, a form of real-time composition) to performance, from conceptualization to perception and cognition, can be modeled and studied using mathematical and computational means.

Many mathematicians and scientists are also fine musicians: the young geometer Donald Coxeter composed music and wrote on the parallels between composition and mathematical proof; Albert Einstein was a talented and enthusiastic violinist; and at the Massachusetts Institute of Technology, 60% of incoming freshmen declare advanced proficiency in music (usually performance), and 25% declare proficiency in other performing arts. Conversely, many prominent musical figures have acknowledged their passion for mathematics, including the popular American classical composer Paul Schoenfield.

The 19 contributors of Mathemusical Conversations are all world-leading scholars or experts on music or mathematics. They hail from universities, conservatories and engineering schools from across the globe--over a dozen prestigious institutions in Australia, Canada, China, France, Singapore, the United Kingdom and the United States.

Interested individuals and institutions can purchase the book from World Scientific (quote WSMTH1220 to receive a 20% discount), Amazon (bit.ly/Mathemusical-Amazon), and Amazon-UK (bit.ly/Mathemusical-AmazonUK).

More at bit.ly/ec-MathemusicalConversations2.

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About the Editors

Elaine Chew is a Professor of Digital Media at Queen Mary University of London, where she is affiliated with the Centre for Digital Music and serves as its Director of Music Initiatives and co-Lead of the Cognition, Creativity, and Expression research theme. Prior to her move to the UK in Fall 2011, she was a faculty member at the University of Southern California, where she founded and directed research at the Music Computation and Cognition Laboratory, first as an Assistant Professor, then as a tenured Associate Professor. She received PhD and SM degrees in Operations Research at MIT, a BAS in Mathematical and Computational Sciences (honors) and in Music (distinction) at Stanford, and FTCL and LTCL diplomas in Piano Performance from Trinity College, London.

An operations researcher and pianist by training, her research goal is to explain and demystify the phenomenon of music and its performance--what it is that musicians do, how they do it, and why--through formal scientific methods. Her research centers on the design of mathematical and computational tools to model, analyze, and visualize music structures. Her research has been recognized by the National Science Foundation Faculty Early Career Development Award and the Presidential Early Career Award in Science and Engineering, and a fellowship cluster on Analytical Listening through Interactive Visualization at the Radcliffe Institute for Advanced Study. Chew also disseminates her work through boundary-crossing lecture-recitals designed to help audiences think more deeply about the music and performances that they hear. As a pianist, she has performed as chamber musician and soloist at numerous venues worldwide and frequently collaborates with composers to commission and present eclectic post-tonal music. At MIT, she was selected by Pulitzer-prize winning composer John Harbison to accompany Yo-Yo Ma in an open rehearsal of his new Cello Concerto; she won a MISTI grant to conduct a field study on contemporary Chinese piano music in Beijing; and, was awarded the prestigious Laya and Jerome Wiesner Student Art award for her "sustained, ubiquitous, unfailingly enlivening contribution" to musical life at the Institute. Her performances can be heard on public radio, the WGBH-EBU exchange online program Art of the States, and on Albany and Neuma Records.

Gérard Assayag is head of IRCAM's Research Lab "Sciences and Technologies of Music and Sound" and of its Music Representation Team, which he founded in 1992. His research interests are centered on music representation issues, including programming languages, machine learning, constraint and visual programming, computational musicology, music modeling, and computer-assisted composition. He has designed with his collaborators OpenMusic and OMax, two music research environments which have received international acclaim, and are used in many contexts all over the world for computer assisted composition, analysis and improvisation.

Assayag is a founding member of AFIM (Association Francaise d'Informatique Musicale) and SMCM (Society for Mathematics and Computation in Music). He serves in the Editorial Boards of SMCM's Journal of Mathematics and Music, and of the Journal of New Music Research (JNMR). He has organized or co-organized the "Forum Diderot, Mathematique et Musique" for the European Mathematical Society in 1999 as well as several international computer music conferences, including the Sound and Music Computing 2004 conference, the 3rd Mathematics and Computation in Music conference in 2011 and the Improtech Paris--New York 2012 conference on Improvisation and new technologies.

Jordan B. L. Smith is a postdoctoral research scientist at the National Institute for Advanced Industrial Science and Technology (AIST) in Japan, working in the Media Interaction Group. He works with Prof. Masataka Goto on a variety of projects, from automated music analysis to music-based puzzles. While advised by Prof. Elaine Chew, he obtained his PhD at Queen Mary University of London in Electrical Engineering and Computer Science, and a Master's of Science in Operations Research at the University of Southern California. Previously, he earned a Master's in Music Technology at McGill University and a Bachelor's in Music and Physics at Harvard College.

Smith has applied his diverse background to multi-disciplinary research about a crucial musical phenomenon: musical structure, or how a piece of music may be composed of repeated and varied sections. He has studied how structure is perceived by listeners and improvisers, how it is annotated by music theorists, and how to estimate structure algorithmically. His doctoral and master's research was supported by national research grants from Canada (SSHRC) and Québec (FQRSC).

About World Scientific Publishing Co.

World Scientific Publishing is a leading independent publisher of books and journals for the scholarly, research, professional and educational communities. The company publishes about 600 books annually and about 130 journals in various fields. World Scientific collaborates with prestigious organizations like the Nobel Foundation, US National Academies Press, as well as its subsidiary, the Imperial College Press, amongst others, to bring high quality academic and professional content to researchers and academics worldwide. To find out more about World Scientific, please visit http://www.worldscientific.com. For more information, contact Amanda Yun at heyun@wspc.com.

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