An Honors Seminar in Music & Mathematics
Mathfest 2009
William E. Fenton
Bellarmine University
Louisville, KY, USA 40205
wfenton@bellarmine.edu
In the spring 2008 semester, I taught a course entitled “Music & Math” for my university's
Honors program. Aimed at the freshman level, it sought to explore some of the ways
mathematics can be applied to the study of music. In a broad sense, the course content was
Sound, Pitch, Rhythm, Melody, and Theory, and how mathematics can be used for each of these
five topics.
TOPIC
Sound
(2 weeks)
Pitch
(2.5 weeks)
Rhythm
(3 weeks)
Melody
(2.5 weeks)
Theory
(2.5 weeks)
MUSIC
frequency, harmonics,
timbre,
acoustic spectrum
scales,
tuning systems
time signatures,
rhythm patterns
transposition, inversion,
retrograde
scales, keys
MATHEMATICS
sine curves,
Fourier’s Theorem
READINGS
Kline
ratios, logarithms,
irrational numbers
combinatorics,
permutations,
group properties
geometric transformations,
groups
sequences,
necklace diagrams
Bibby
Haack
Toussaint
Harkleroad
Johnson
I also brought in recordings and photocopies of musical excerpts. (Our classroom fortunately had
a piano and a stereo system.) On one fun day we analyzed “Boom Whackers”, which are plastic
tubes of various lengths that resonate on different pitches when struck.
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N. Bibby, “Tuning and Temperament; Closing the Spiral”, in Music and Mathematics; from
Pythagoras to Fractals, J. Fauvel, R. Flood, and R. Wilson, eds., Oxford University Press,
2006, pages 13-27.
Joel K. Haack, The Mathematics of Steve Reich's Clapping Music, Bridges Conference
Proceedings (1998), pp. 87–92.
L. Harkleroad, “How to Vary a Theme Mathematically,” Chapter 4 from The Math Behind
the Music, Cambridge University Press, 2006.
T. Johnson, “Spatial Relations and Musical Structures” and “Interval Patterns and Musical
Structures”, Chapters 1 and 2 from Foundations of Diatonic Theory, Key College Publishing,
2003.
M. Kline, “The Sine of G Major,” Chapter XIX from Mathematics in Western Culture,
Oxford University Press, 2008.
Godfried Toussaint, The Euclidean Algorithm Generates Traditional Musical Rhythms,
Rennaisance Banff Conference Proceedings (2005), pp. 47–56
Sample homework problems
1. List the frequencies of the first four overtones of a note having frequency 440 Hertz. Do the
same for a note of 8000 Hertz. Why is it harder to tell two instruments apart when they play a
note of 8000 Hertz than when they play a note of 440 Hertz?
2. The graph [not shown here] shows the acoustic spectrum of a trumpet. Use Maple software to
produce a graph of a trumpet playing a note with the fundamental frequency of 200 Hertz.
Then produce a graph of the note one octave higher.
3. In the C-major scale with Pythagorean tuning, verify that all whole steps have the same ratio.
Then show that this is not true in just tuning.
4. We saw that raising by pure fifths, those having a 3/2 ratio, will never reach an exact number
of octaves. What about raising by pure major thirds, which have a 5/4 ratio; will this ever
reach an exact number of octaves?
5. In how many ways could you fill a [4,4] measure with two quarter notes, three eighth notes,
and one eighth rest?
6. Steve Reich’s Clapping Music used a twelve beat pattern. He made variations of this pattern
by moving the first beat to the end of the pattern, over and over. Find a six-beat rhythm
pattern that has six different variations (using this same process to create the variations).
7. Write a short melody of at least eight notes that is the same in retrograde.
8. Simplify the following mixtures of operations: RT4IT7R, T1IRT2IRT3I.
9. Find a pattern of nine dots that is maximally even. Show that your answer is correct by
making a table of d-distances and c-distances. [diatonic and chromatic distances]
10. On the staff [not shown] is a series of three notes. Transpose the series diatonically so that a
series begins on each note of the C-major scale. Identify the interval pattern for each series.
Tell how many distinct interval patterns there are.
The Final Project
Since this course was part of the Honors Program at my university, students were expected to do
some writing. Thus I assigned a 6-8 page final paper:
“In this paper you are to explore a topic relating mathematics to music. You are expected
to seek out references and to learn about this topic. Summarize and synthesize what you
learn from your references.”
The students picked randomness in musical composition, the Fibonacci sequence in music, and
musical transformations. They also gave short presentations on their topic.
The most ambitious project was using permutations to write a piece of music. Michael (a
composition major) wanted to apply permutations to both rhythms and pitches to create a tonal
work with a clear harmonic progression. His simple melody followed the harmonic progression
C, C, F, G (one measure of each chord for a total of four measures), which then resolves back to
C. He collected the notes from each chord of his melody into a single cycle and then permuted
pitches within that chord. He used the same permutation on the lengths of the notes within his
melody. Thus one cycle permuted both notes and rhythms over a two measure span in C, the next
cycle permuted notes and rhythms over a single measure in F, and the final cycle permuted notes
and rhythms over a measure in G. His opus is similar to Clapping Music in that one piano plays
the same four measure pattern over and over, while a second piano plays the variations. (For his
presentation, Michael programmed his computer to play the two parts against each other. It was
impressive!)