SUNY Fredonia

**Department of Mathematical
Sciences**

**MATH 307 –
Math and Music**

**Catalog Description:**

Frequency: B

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

**Rationale:**

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.

**Textbooks:**

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.

**Bibliography:**

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*, 2^{nd} ed., W. W. Norton and Company, Inc., New
York, 1977.

11.
Thomas
D. Rossing, *The Science of Sound*, 2^{nd}
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*,
2^{nd} 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*,

http://www.smt.ucsb.edu/mto/issues/mto.98.4.4/mto98.4.4.scholtz.html

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.

**Course Schedule: **

**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 20^{th} 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).