128_Teaching the Mathematics of Music

Teaching the Mathematics
of Music

Rachel Hall
Saint Joseph’s University



Overview
• Sophomore-level course for math majors (nonproof)
• Calc II and some musical experience required
• Topics
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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
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–
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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 Instrument

Jim Pepper, Sophomore, History major, Music minor
Predicted Pitch

Pitch

Desired Freq.

Actual Freq.

Difference

Actual
Length

Predicted 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

60

59.85

261.63

259.368544

5
-2.26146
4
3
2
1
0
-1
-2
-3
-4

Difference
26.21915885 Frequency 25.25

-0.96916

24.72035563

0.279644

25

Series1
1

2

3

4

5

6

7

8

9

10

11

12

13


The Mathematics of Change Ringing
Emily Burks, Freshman, Math major


Symmetry and group theory
exercises
Sources:

J.S. Bach’s 14 Canons on the Goldberg Ground
Timothy Smith’s site:
/>Steve Reich’s Clapping Music
Performed by jugglers
/>

Bach’s 14 Canons on the Goldberg Ground
Bach composed
canons 1-4 using
transformations of
this theme.

• 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.


Canons 1 and 2
I(S)

S

RI(S) = IR(S)

R(S)
theme

retrograde

Canon #1

inversion

retrograde
inversion

Canon #2


Canons 3 and 4
I(S)

S

RI(S) = IR(S)

R(S)
retrograde

Canon #3

inversion

retrograde
inversion

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
Performer 1
Performer 2

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


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
• />(handouts and other resource materials)

• />(over 30 links, grouped by topic)

• />(my articles)

• Email me: