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

Course Goals

• Use the medium of musical analysis to• Explore mathematical concepts such as Fourier series and

tilings that are not covered in other math courses • Introduce topics such as group theory and combinatorics

covered in more detail in upper-level math courses

• Discuss the role of creativity in mathematics and the ways in which mathematics has inspired musicians

• Use mathematics to create music• Have fun!

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 and a 15-minute final presentation and wrote a paper about the mathematics of their topic. They were required to schedule consultations throughout the semester. The best projects involved about 40 hours of work.

Logarithms and music: A secondary school math lesson

Christina Coangelo, Senior, 5 yr M. Ed. program

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

The Mathematics of Change RingingEmily Burks, Freshman, Math major

Symmetry and group theoryexercises

Sources:

J.S. Bach’s 14 Canons on the Goldberg Ground

Timothy Smith’s site:

http://bach.nau.edu/BWV988/bAddendum.html

Steve Reich’s Clapping Music

Performed by jugglers

http://www.youtube.com/watch?v=dXhBti625_s

Bach’s 14 Canons on the Goldberg Ground

• How are canons 1-4 related to the solgetto and to each other?

• How many “different” canons have the same harmonic progression?

• Write your own canons.

Bach composed canons 1-4 using transformations of this theme.

Canons 1 and 2

retrograde retrogradeinversion

inversion

S R(S)

I(S) RI(S) = IR(S)

Canon #1 Canon #2

theme

Canons 3 and 4

retrograde retrogradeinversion

inversion

S R(S)

I(S) RI(S) = IR(S)

Canon #3 Canon #4

The template

• How many other “interesting” canons can you write using this template?

• (What makes a canon interesting?)• Define a notion of “equivalence” for canons.

Steve Reich’s Clapping Music

• Describe the structure.• Why did Reich use this particular pattern?• Write your own clapping music.

Performer 1

Performer 2

Challenges

• Students’ musical backgrounds varied widely. I changed the course quite a bit to accommodate this.

• Two students did not meet the math prerequisite. They had the option to register for a 100-level independent study, but chose to stay in the 200-level course. One earned an A.

For next time…

• Spend more time on symmetry and less on tuning• Add more labs• More frequent homework assignments

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)• My own stuff• Lots of web resources• YouTube!

Learn more

• http://www.sju.edu/~rhall/Mathofmusic (handouts and other resource materials)

• http://www.sju.edu/~rhall/Mathofmusic/-MathandMusicLinks.html (over 30 links, grouped by topic)

• http://www.sju.edu/~rhall/research.htm(my articles)

• Email me: rhall@sju.edu