An Honors Seminar in Music & MathematicsMathfest 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
Rhythm Time signatures Filling measures with notes (combinatorics) Permutations: cycle notation, composition, order, inverses Readings: Haack, The Mathematics of Steve Reich'sClapping MusicToussaint, 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!