Department of Mathematical Sciences
MATH 307 – Math and Music
Explores how mathematical ideas have been used to understand and create music, and how musical ideas have influenced math and science. Topics include the history of tuning and alternative tuning, the Music of the Spheres doctrine, historical theories of consonance, contributions to music theory by mathematicians, mathematical analysis of sound, philosophical and cognitive connections between math and music, and math in music composition and instrument construction. This course is not intended for math majors. An ability to read music is recommended. Junior or senior standing is required.
Prerequisite Courses: None Credits: 3
This course is taken primarily by students to satisfy a general education requirement (Part IIIA of the GCP, or the Upper Level Requirement in the CCC). Most math courses taken by non-math majors (e.g. calculus and statistics) aim at the mastery of skills necessary for subsequent coursework, allowing little time for discussion of the history, aesthetics, and role of math in civilization. Thus many students complete their college education with little appreciation for the depth, scope, utility, and beauty of mathematics. The department currently offers two general education courses designed to cultivate that appreciation, namely MATH 117 – Why Math? at the lower division level and MATH 307 at the upper division. MATH 117 may be used to help satisfy the Sciences Requirement in the CCC, and is generally taken in the freshman or sophomore years. MATH 307 satisfies the Upper Level Requirement (formerly Part III), and as such provides an interdisciplinary, reading, and writing-intensive “capstone” experience. It is taken by juniors and seniors and draws on the skills and knowledge acquired during their first two years of college. Furthermore, the material is attractive and useful to the large number of music majors enrolled at Fredonia, including students who might otherwise avoid mathematics courses.
Students in this course are not expected to have especially strong mathematical or scientific backgrounds; most calculations are elementary, and more advanced material is taught as needed. Nevertheless, students are exposed to ideas from number theory (e.g. prime factorization and modular arithmetic), trigonometry and calculus, group theory, geometry, probability, and mathematical modeling. Concepts are applied immediately to the problems that motivated them, such as the desire to divide the octave into smaller musical intervals or to describe symmetry and pattern in composition. In this way, liberal arts students gain meaningful experience in a variety of mathematical fields.
The course is built around the underlying question, “What sounds good and why?” It explores mathematical approaches to answering this question from antiquity to the present, including critical discussion of the basic idea that mathematics and science are appropriate tools for addressing such a question of emotional response in the arts. In fact, critical analysis of assumptions, bias, and methodology is an integral part of students’ activities both in and out of class, especially as students read and comment on historical texts. The development of these advanced scholarship skills is an important element in any course satisfying the Upper Level Requirement. Students learn that mathematics has utility across the scope of human inquiry, as well as the risks of inappropriate or naive applications of math.
1. Math and Music: Harmonious Connections, by Trudi Hammel Garland and Charity Vaughan Kahn, Dale Seymour Publications, 1995.
2. Emblems of Mind: The Inner Life of Music and Mathematics, by Edward Rothstein, Avon Books, 1995.
3. Temperament: The Idea that Solved Music’s Greatest Riddle, by Stuart Isacoff, Alfred A. Knopf, 2001.
4. MATH 307 Reader, a compilation of articles, excerpts, and handouts, available at the Connections Bookstore.
Objectives: By the end of the course, students should develop the following:
· Content knowledge (an understanding of the applications of math to music and the use of musical ideas in math and science);
· Structural knowledge (an understanding of relationships between ideas from different areas of scholarship and from different periods in history);
· Historical context (an understanding of the sequence of events in the common history of math and music, and how they relate to developments in civilization);
· Critical thinking, especially in the analysis of primary and secondary sources and the interpretation of historical theory;
· Technical reading and writing skills;
· Appreciation for the beauty of mathematics and its applicability to the humanities.
Instructional Methods and Activities:
Methods and activities for instruction include lecture/discussion; demonstration (including use of manipulatives, musical instruments, and scientific apparatus); cooperative groups; long-term projects and writing assignments; student demonstrations or presentations; guided discovery. Refer to individual syllabus for details.
Evaluation and Grade Assignment:
Students will be evaluated using grades received on homework, projects, quizzes, and exams. Refer to individual syllabus for details.
1. Trudi Hammel Garland and Charity Vaughan Kahn, Math and Music: Harmonious Connections, Dale Seymour Publications, 1995.
2. Edward Rothstein, Emblems of Mind: The Inner Life of Music and Mathematics, Avon Books, 1995.
3. Stuart Isacoff, Temperament: The Idea that Solved Music’s Greatest Riddle, Alfred A. Knopf, 2001.
4. H.F. Cohen, Quantifying Music: The Science of Music at the First Stage of the Scientific Revolution, 1580-1650, D. Reidel Publishing Co., 1984.
5. Erich Neuwirth, Musical Temperaments, text and CD, Springer-Verlag/Wien, New York, 1997.
6. Philip Wheelwright, The Presocratics, Odyssey Press, ITT Bobbs-Merrill Educational Publishing Company, Inc., Indianapolis, 1985.
7. A.E. Taylor, Aristotle on his Predecessors, The Open Court Publishing Company, London, 1949.
8. Nan Cooke Carpenter, Music in the Medieval and Renaissance Universities, University of Oklahoma Press, Norman, 1958.
9. Raymond J. Seeger, Galileo Galilei, His Life and His Works, Pergamon Press, Oxford, 1966.
10. John Backus, The Acoustical Foundations of Music, 2nd ed., W. W. Norton and Company, Inc., New York, 1977.
11. Thomas D. Rossing, The Science of Sound, 2nd ed., Addison-Wesley Publishing Company, 1990; accompanying CD.
12. Calvin M. Bower (translator), Boethius’ The Principles of Music, an Introduction, Translation, and Commentary, Ph.D. dissertation, School of Music, George Peabody College for Teachers, 1966.
13. Edward A. Lippman, Musical Thought in Ancient Greece, Columbia University Press, New York, 1964.
14. Richard Hope (translator), Aristotle’s Metaphysics, Columbia University Press, New York, 1952.
15. Oliver Strunk, Source Readings in Music History, from Classical Antiquity through the Romantic Era, W.W. Norton and Company, Inc., New York, 1950.
16. J. A. Philip, Pythagoras and Early Pythagoreanism, University of Toronto Press, 1966.
17. Philip Merlan, From Platonism to Neoplatonism, 2nd ed., Martinus Nijhoff, The Hague, 1960.
18. Sir James Jeans, Science and Music, Dover Publications, Inc., New York, 1968; originally published 1937.
19. Iannis Xenakis, Formalized Music: Thought and Mathematics in Music, Pendragon Press, Hillsdale, NY, 1992.
20. Hermann Helmholtz, On the Sensations of Tone, Dover Publications, Inc., New York, 1954; originally published 1885.
21. Ian Stewart, Another Fine Math You’ve Got Me Into, W. H. Freeman and Co., New York, 1992.
22. Juan G. Roederer, The Physics and Psychophysics of Music, Springer Verlag, New York, 1995.
23. Ernest G. McClain, The Myth of Invariance: The Origin of the Gods, Mathematics and Music from the Rg Veda to Plato, Nicholas-Hays, Inc., York Beach, Maine, 1976.
24. Dave Benson, Mathematics and Music, http://www.math.uga.edu/~djb/math-music.html (text can be viewed and downloaded in its entirety), 2002.
25. Margo Schulter, Pythagorean Tuning, http://www.medieval.org/emfaq/harmony/pyth5.html.
26. Mark Shepard, Simple Flutes: How to Play or Make a Flute of Bamboo, Wood, Clay, Metal, Plastic, or Anything Else, Simple Productions, Los Angeles, 2001; available as an e-book at www.markshep.com/flute.
27. Robert J. Zatorre and Carol L. Krumhansl, “Mental Models and Musical Minds”, Science, 13 December 2002, Vol. 298.
28. Mike May, “Did Mozart Use the Golden Section?” American Scientist, March-April 1996; available at http://www.sigmaxi.org/amsci/Issues/Sciobs96/Sciobs96-03MM.html.
29. Gioseffo Zarlino, The Art of Counterpoint (Part 3 of Le Istitutioni Harmoniche), 1558, translated by Guy A. Marco and Claude V. Palisca, Da Capo Press, New York, 1983.
30. Ptolemy, Harmonics, translated and commentary by Jon Solomon, Koninklijke Brill NV, Leiden, The Netherlands, 2002.
31. Patrice Bailhache, “Music translated into Mathematics: Leonhard Euler”, translation from French by Joe Monzo; and “La Musique, une pratique cachée de l’arithmétique?”, an article on the musical writings of Leibniz; both available at http://bailhache.humana.univ-nantes.fr/thmusique.html.
32. Ernest G. McClain, “Chinese Cyclic Tunings in Late Antiquity”, Ethnomusicology, V. 23, n2, 1979.
33. Rachel W. Hall and Kresimir Josic, “The Mathematics of Musical Instruments”, http://www.sju.edu/~rhall/newton/, 2000.
34. Erich Neuwirth, “Designing a Pleasing Sound Mathematically”, Mathematics Magazine, Vol. 74, No. 2, April 2001.
35. Kenneth P. Scholtz, “Algorithmis for Mapping Diatonic Keyboard Tunings and Temperaments”, Music Theory Online,
36. Kyle Gann, “An Introduction to Historical Tunings”, and “Just Intonation Explained”, both available at http://home.earthlink.net/~kgann.html.
37. André Barbera, “The Consonant Eleventh and the Expansion of the Musical Tetractys: A Study of Ancient Pythagoreanism”, Journal of Music Theory, V. 28, n2, 1984.
Week 1: basic music concepts and vocabulary; musical notation; harmonic ratios
Week 2: summary of tuning and history; mathematically defined scales; vocabulary and arithmetic of intervals; Pythagorean scale; Pythagorean school
Week 3: music in ancient Greece; Music of the Spheres; empiricism vs. theory; Neopythagoreans; Ptolemy; early Chinese music theory
Week 4: Medieval music theory and quadrivium; justifications of scales
Week 5: polyphony, consonant thirds and Just Intonation; Theories of Consonance: Zarlino and senario; Kepler
Week 6: continue Theories of Consonance: versions of Coincidence Theory
Week 7: tonal drift and modulation; temperaments: Meantone, Irregular, and Equal; logarithms and cents
Week 8: modern acoustical theory; trig, calculus, and physics; sound analysis and synthesis
Week 9: begin modern Theory of Consonance: Helmholtz, ear anatomy, psychoacoustics
Week 10: finish modern Theory of Consonance; alternative scales and temperaments in 20th Century
Week 11: mathematical “tour of orchestra”; instrument design; field trip?
Week 12: begin pattern in composition: symmetry, geometry, permutations; campanology
Week 13: continue pattern: algorithmic music, 12 Tone music, probability in composition/performance, composition by computer
Week 14: philosophical connections (Rothstein)
Week 15: cognitive connections; semester summary
Policies: Refer to individual syllabus (details vary with semester).