Mathemusical Conversations: Mathematics And Computation In Music Performance And Composition

Mathematics and Computation in Music Performance and Composition

Jordan B L Smith, Elaine Chew

• Publisher: WSPC
• Year: 2016
• ISBN:
  9789813140110
• Topic: Matemáticas
• Subtopic: Matemática aplicada

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.

Contents:

• Foreword by Series Edited bys
• Foreword by Workshop Organizers
• Mathemusical Engagement:
  • Without Our Consent (Paul Schoenfield)
  • Approaches to Musical Expression in Harmonix Video Games (Eran Egozy)
  • Motion and Gravitation in the Musical Spheres (Elaine Chew)
• Mathemusical Creativity:
  • Improvising in Creative Symbolic Interaction (Gérard Assayag)
  • Music, Creativity, and Computers (Margaret A Boden)
  • Tiling Canons as a Key to Approaching Open Mathematical Conjectures? (Moreno Andreatta)
• Shaping Performance:
  • Musical Motives in Performance: A Study of Absolute Timing Patterns (Neta Spiro, Nicolas Gold and John Rink)
  • Playing with Variables: Anticipating One Particular Performance of Bach's Goldberg Variations (Bernard Lanskey and Stephen Emmerson)
  • The Informatics Philharmonic in the Indiana University Summer String Academy (Christopher Raphael)
• Educating the Mathemusical:
  • Mathematical Thought and Empirical Approaches in Higher Education in Music (Jian Yang)
  • Action and Symbol: An Essential Tension (Jeanne Bamberger)
  • Educating the Mathemusical: Balancing the Equation (Don McLean)
• Geometries:
  • Graph-theoretic and Geometric Models of Music (Richard Cohn)
  • In Quest of Musical Vectors (Dmitri Tymoczko)
  • A Topological Approach of Musical Relationships (Jean-Louis Giavitto and Antoine Spicher)
• List of Contributors

Readership: Advanced secondary school students; post-secondary school students; and scientists, mathematicians, musicians and members of the public interested in the mathematical music sciences.

Part I
Mathemusical Engagement

If music is the opiate of the masses and mathematics the language of the gods, it is no wonder that together the two make for a potent and enthralling cocktail. This section consists of three chapters that each present unique perspectives on compelling and widely engaging ways to combine the two: in composition, in music games, and in music making.

Without Our Consent. Paul Schoenfield, Professor of Composition at the University of Michigan and a noted composer of eclectic classical music that incorporates folk and jazz elements, writes on aesthetic associations between mathematics and music. Words used to describe his music (in extremely positive reviews) have included “Bad Culture”, “Really Annoying Music” and, “One doesn’t know whether to laugh or gape in awe at a mind so deranged.”

Schoenfield himself does not use mathematics to compose. He demonstrates the futility of elementary mappings between numbers and music by the pareidolia of the Fibonacci sequence imagined in any Beethoven Sonata. Focusing on deeper connections, composing is likened to proving theorems, the two being kindred experiences that share a profound aesthetic.

In mathematics, beauty is associated with simplicity and elegance. If a theorem’s first mathematical proof is not particularly elegant, subsequent generations work to simplify and beautify the proof. Music no longer exhibits a period-specific unifying style and frequently lapses into the past. Schoenfield argues that we have thus reached the age of re-proving in music. Re-proving in mathematics may be an accepted and normal activity, but composing successfully in a style of the past often does not receive the same kind of recognition.

So why compose? And why prove theorems? While ego may once upon a time raise its head as a motivator to compose or perform, for both music and mathematics, the lasting motivator is the heady blend of the intellect and aesthetic.

Approaches to Musical Expression in Harmonix Video Games. Eran Egozy, co-founder of Harmonix Music Systems and Professor of the Practice in Music and Theater Arts at the Massachusetts Institute of Technology, reflects on the feasibility of combining the often-contradictory objectives of winning in a game with the expressive goals of music.

The chapter centers on several Harmonix products: The Axe (a joystick-based improvisation engine), Amplitude (an arcade-style multitrack rhythm game), Karaoke Revolution (a singing game with microphone input), Guitar Hero and Rock Band (multi-instrument rock band simulation), and Fantasia: Music Evolved (Kinect-based musical exploration based on the classic Disney film). The evolving design decisions and products are evaluated according to player engagement and musical expressiveness.

Due to the inherent conflict between the scoring required in a game motivation loop and the nature of musical aesthetics, Harmonix has avoided creating game scenarios with aesthetic evaluation. In order to push the envelope with respect to engagement and expressiveness, Egozy advocates for a greater emphasis on intuitive interfaces and interaction design, and for finding the happy medium between the strict evaluation of game mechanics and the freedom of non-evaluation environments.

Motion and Gravitation in the Musical Spheres. Elaine Chew, Professor of Digital Media at the Centre for Digital Music at Queen Mary University of London, considers the parallels between concepts of gravitational pull and physical motion in music and physics, in the quest to represent and explain the craft of making music. Physical phenomena such as motion and gravitation have been successfully modeled using mathematics, which portends similar advances in music.

Chew explores concretely ideas of motion and gravitation in two music domains: tonality and time. Concepts of center of gravity are extended from physics to operations research to the mathematical modeling of tonality. Centers of gravity model the establishing of key, and movements away and toward this tonal center. The evolution of tonal context, expectation, and tension, the primary determinants of musical experience in tonal music, is shown visually in a geometric model for tonality. In music, time and space are often conflated so that the progression of time in music is conceptualized as a journey through space. Timing deviations are likened to wrinkles in time, and often ascribed to and used to simulate an imaginary terrain. Tipping points in the manipulation of timing in performance demonstrate the ideas of gravitation and motion in this concocted terrain. Examples are drawn from Schoenfield’s Trio and Café Music.

Common threads run through Schoenfield’s apology, where apology is used in the Platonic sense of the word, to Egozy’s practical concerns of designing durable and engaging music game experiences, to Chew’s theoretical pursuit to make plain the craft of music. The three chapters form an elegant cycle. Schoenfield’s and Egozy’s chapters share a common belief in the unquantifiability of musical aesthetics and ideas of motivation and reward. Egozy’s and Chew’s chapters are connected by a common search for intuitive and concrete handles on abstract musical concepts. Both Chew and Schoenfield seek deeper connections between music and mathematics beyond surface mappings of numbers and pitches or durations. Together, they span possibilities for mathemusical engagement afforded by composition, gaming, and performance.

Elaine Chew
Queen Mary University of London

Without Our Consent

Paul Schoenfield

In his book, 5000 B.C., Raymond Smullyan relates an anecdote about German-born mathematician Felix Klein, who, during a dinner party discussion on the relationship between math and music, exclaimed, “But I don’t see the connection at all; after all, math is beautiful.” [10, p. 35]

Klein was certainly aware of the claims of connection between applied mathematics and music going back to ancient times—in instrument building, harmonic theory, acoustical principles, to name but a few—yet he appeared (pretended to appear) oblivious to the deep aesthetic associations between music and mathematics exemplified, for example, by Galois Theory (which provided a bridge between field theory and group theory), Picard’s Theorem (regarding the ranges of analytic functions), and, of course, Euler’s identity, e^iπ + 1 = 0, frequently called the most “beautiful” equation in mathematics.

This association of mathematics with the language of aesthetics is frequently established in tones of reverence and awe by mathematicians employing adjectives like uncanny, mysterious, and sublime. Paul J. Nahin writes, for instance that the above equation is one of “exquisite beauty.” [7] And, Stanford University mathematics professor Keith Devlin has said, “like a Shakespearean sonnet that captures the very essence of love, or a painting that brings out the beauty of the human form that is far more than just skin deep, Euler’s equation reaches down into the very depths of existence.” [2]

Language employed to describe music is appropriately expropriated from descriptions of mathematics. It has been said many times that music is the language of the soul; to one sensitive to music, how can mathematics, with its infinite array of universal truths, not fail to produce the same shivers evoked by listening to a glorious passage of music, similar to having an eye on the ineffable, or a near transcendental experience? Consider, a, b and c being integers. This equation has never had and never will have a solution anywhere in the universe:

If many do not recognize an aesthetics of mathematics, it is perhaps owing to mathematical education that undermines a comprehension of deeper connections. When I was in school, we learned about each element of Euler’s identity separately, learning about π in sixth grade geometry and ‘i’ in 11th grade algebra. At that time ‘e’ wasn’t addressed until university calculus courses. Under these circumstances, who would have grasped the profound connection between these three universal constants?

J. J. Sylvester, the 19th century geometer, wrote, “May not Music be described as the Mathematics of sense, Mathematics, Music of reason? the soul of each the same!”

1. Mathematics as a Method of Composition

Composing is difficult, and if a mathematical process can help composers in their compositional journey, I am happy for them. In my experience, however, most of these processes are merely basic arithmetic, silly and arbitrary, producing bad mus...

Table of contents

• Cover page
• Title page
• Copyright
• Contents
• Foreword by Series Editors
• Foreword by Workshop Organizers
• Part I. Mathemusical Engagement
• Part II. Mathemusical Creativity
• Part III. Shaping Performance
• Part IV. Educating the Mathemusical
• Part V. Geometries
• List of Contributors