An Honors Seminar in
Music & Mathematics
Mathfest 2009
Alien Musical Scores; a drawing by Robert Mueller.
HONORS 151: Music & Math
Freshman level
Accepted for general education
requirement
Six students (two music majors)
Math backgrounds from algebra to calculus
All played an instrument (most often guitar)
Two could not read music!
Five broad themes:
• Sound
• Pitch
• Rhythm
• Melody
• Theory
My general approach:
Five reading assignments with reaction form
In-class discussion
Occasional lecture (mostly on math)
Five homework assignments
Two exams
Final project
Sound
Frequency, Harmonics, Acoustic spectrum
Consonance and dissonance
(Euclid, Galileo, Helmholtz)
Sine curves
Fourier’s Theorem
Reading: Kline, “The Sine of G Major,” Chapter
XIX from Mathematics in Western Culture
Pitch
Ratios for musical intervals
Pythagorean tuning and its flaws
Just intonation and its flaws
Equal temperament
(An aside: cents and logarithms)
Reading: Bibby, “Tuning and Temperament;
Closing the Spiral”, in Music and
Mathematics; from Pythagoras to Fractals
Boom Whackers
(part of Pitch)
Rhythm
Time signatures
Filling measures with notes (combinatorics)
Permutations:
cycle notation, composition, order, inverses
Readings:
Haack, The Mathematics of Steve Reich's
Clapping Music
Toussaint, The Euclidean Algorithm
Generates Traditional Musical Rhythms
Melody
Transposition, retrograde, inversion,
retrograde inversion
Translation, reflection, rotation
Composition, group properties
Twelve-tone music (Babbitt square)
Reading: Harkleroad, “How to Vary a Theme
Mathematically,” Chapter 4 from The Math
Behind the Music
Theory
Circle diagrams for intervals, chords, scales
“Maximally even” (mathematical definition)
The “Cardinality = Variety” property of diatonic
and pentatonic scales
Readings: Johnson, “Spatial Relations and
Musical Structures” and “Interval Patterns and
Musical Structures”, Chapters 1 and 2 from
Foundations of Diatonic Theory
Assessment:
Five homework assignments,
7-9 problems each
Two exams
Final project and presentation
Next time?
Better musical examples for Melody
Alternative representations for melodies
(Rachel Hall)
More with Boom Whackers
Practice Clapping Music
Create our own variations with permutations
Recruit more students!
Thanks for listening!