Overview
• Sophomore-level course for math majors (non-proof)• Calc II and some musical experience required• Topics
– Rhythm, meter, and combinatorics in Ancient India– Acoustics, the wave equation, and Fourier series– Frequency, pitch, and intervals– Tuning theory and modular arithmetic– Scales, chords, and baby group theory– Symmetry in music
Semester project
Each student completed a major project that explored one aspect of the course in depth.
• Topics included – the mathematics of a spectrogram; – symmetry groups, functions and Bach; – Bessel functions and talking drums; – change ringing; – building an instrument; and – lesson plans for secondary school.
• Students made two short progress reports, a 15-minute final presentation, and wrote a paper about the mathematics of their topic. They were required to schedule consultations throughout the semester.
Logarithms and music: A secondary school math lesson
Christina Coangelo, Senior, 5 yr Math Ed program
Major Math Content Covered• Functions
– Linear, Exponential, Logarithmic, Sine/Cosine, Bounded, Damping
– Graphing & Manipulations
• Ratios
Building a PVC InstrumentJim Pepper, Sophomore, History major, Music minor
Predicted Pitch Pitch Desired Freq. Actual Freq. Difference Predicted lengthActual Length Difference
48 48.25 130.81 132.715498 1.905498 47.59574391 48.25 0.654256
49 49.1 138.59 139.394167 0.804167 45.35126555 46.25 0.898734
50 50.1 146.83 147.682975 0.852975 42.84798887 43.23 0.382011
51 51 155.56 155.563492 0.003492 40.71539404 41 0.284606
52 52.05 164.81 165.290467 0.480467 38.3635197 37.75 -0.61352
53 53.05 174.61 175.11915 0.50915 36.25243506 36 -0.25244
54 54 185 184.997211 -0.00279 34.35675658 33.75 -0.60676
55 55 196 195.997718 -0.00228 32.47055427 32 -0.47055
56 56 207.55 207.652349 0.102349 30.69021636 31.5 0.809784
57 57.3 220 223.845532 3.845532 28.52431467 28 -0.52431
58 58.1 233.08 234.43211 1.35211 27.27007116 26.25 -1.02007
59 58.8 246.94 244.105284 -2.83472 26.21915885 25.25 -0.96916
60 59.85 261.63 259.368544 -2.26146 24.72035563 25 0.279644
Frequency Difference
-4
-3
-2
-1
0
1
2
3
4
5
1 2 3 4 5 6 7 8 9 10 11 12 13
Series1
QuickTime™ and a decompressor
are needed to see this picture.
The Mathematics of Change RingingEmily Burks, Freshman, Math major
Resources
Assigned texts• David Benson, Music: A Mathematical Offering• Dan Levitin, This is Your Brain on Music
Other resources• Fauvel, Flood, and Wilson, eds., Mathematics and
music• Trudi Hammel Garland, Math and music: harmonious
connections (for future teachers)• Lots of web resources• YouTube!
Symmetry and group theory
Steve Reich’s Clapping Music
Performed by jugglers
http://www.youtube.com/watch?v=dXhBti625_s
J.S. Bach’s 14 Canons on the Goldberg Ground
Timothy Smith’s site:
http://bach.nau.edu/BWV988/bAddendum.html
Exercises (choose one)Clapping music• Describe the structure.• Write your own clapping music.• Why did Reich use this particular pattern?Groups in music• Read “Variations on a Theme.” Describe the
structure of the group of translations, inversions, delay, and retrograde.
Bach’s 14 Canons on the Goldberg Ground• How are canons #1-4 related to the solgetto and to
each other?• Write your own canon, using the template on the
back or your own template.Write your own exercise