IE 144: Mathematics and Music: The Cosmic Harmony
Instructors: Bill Alves and Michael Orrison
Anthology of articles and reserve readings including selections from:
- Assayag, Feichtinger, and Rodrigues. Mathematics and Music: A Diderot Mathematical Forum. Berlin: Springer-Verlag, 2002.
- Benson, Dave. Mathematics and Music. Text available at http://www.math.uga.edu/~djb/html/math-music.html, 2001.
- Cope, David. Virtual music: Computer Synthesis of Musical Style. Cambridge: MIT Press, 2001.
- Taylor, Timothy. Strange Sounds: Music, Technology, and Culture. New York: Routledge, 2001.
- Xenakis, Iannis. Formalized Music. Bloomington: University of Indiana Press, 1971.
- Xenharmonikon [journal].
Prerequisites: Math 61, 63, 64 (HMC mathematics core) or equivalent.
There is a popular perception of a close connection between math and music, presumably noting a perceived logical system in the conventions described by music theory that is not as evident in, say, painting. While we will discuss this view and what it tells us about our preconceived notions of the nature of art and the creative process, as well as the definition of "mathematics," we will also point out that this perception is the echo of an ancient coupling of music and mathematics that was also extended to architecture and fine arts. Though the mathematics and art have changed, artists faced many of the same issues in recent times, though now, artists have exploited advances in mathematics and computing as they occur. Algorithmic composition, which students will explore, involves the generation of notes and other musical events purely from predefined mathematical algorithms. As we enter the twenty-first century, computing power and further research has allowed these algorithms to mimic historical styles or create new ones, which raise fascinating questions about the nature of creativity and its supposed ineffability. Mathematics applied with computing power has also made possible music synthesis, recording, and compression, creating new ethical issues and questions of this technology's effects on musical style and culture.
Short assignments (approximately 13, 5% each) due from class to class or sometimes weekly, consisting of responses to assigned readings, short creative projects, analysis of art, and open-ended mathematical problems relating to issues under discussion. Sometimes students will have a choice of problems or assignments which they will then present at the next class. When appropriate, problems or issues will be assigned to groups.
In courses such as this in which thoughtful reflection and analysis are so important, your communication of your insights is critical. Therefore not only the content but the style, structure, and mechanics of any prose writing required for assignments or projects will be considered as part of your evaluation.
Final project (25%): At the end of the semester, students will have a choice of producing a major research paper or a significant creative project with accompanying paper which will ciritically examine an issue that explores the integration of mathematics and music from a contemporary cultural perspective. These papers and projects will be presented during Presentation Days. Proposals for projects will be due towards the last quarter of the semester and interim versions may be required.
Class participation (10%): Much of the class will be in a seminar format, so participation is crucial and a significant part of the student's grade. A student's conscientious participation in group projects and other class activities will also be considered.
Part I: The Philosophical and Aesthetic Basis of Number
Part II: Algorithmic composition
- Evolving definitions of mathematics, number, and music; readings from Plato, Russell, others.
- The humanistic basis of mathematics in history; Idealism and the "mathematical" in music
- Pythagorean cosmic harmony; readings in Ptolemy, Aristoxenus
- Proportion in music, visual arts, and architecture
- Music and mathematical notation
Part III: Mathematics in musical practice
- Aesthetic basis of algorithmic composition; readings from Xenakis, others
- Artificial intelligence and the mathematical basis of musical style
- Guest lecture from Prof. Thom
- Nature of musical creativity; readings from Cope, Hofstaeder
Part IV: Music, technology, and culture
- Pythagorean and just intonation
- Experimental tunings: extended JI, recurrent pattern scales, combination product sets
- Experimental Temperaments: microtonal divisions, well-temperament, algorithmic derivation of scales
- Guest lecture from Ervin Wilson
- Tuning and non-Western cultures
- Mathematical and psychological bases of rhythm
- Influences of technology on music and vice-versa
- Mathematics of synthesis and its cultural effects
- Mathematics and social impact of signal compression
- Ethics and social effects of downloading and sampling
- Technology, economics, and electronica culture
- Mathematics and music as cultural reflections
Updated on November 20, 2003, by Bill Alves (alves @hmc.edu).