Teaching the Mathematics of Music

Rachel HallSaint Joseph’s University

rhall@sju.edu

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