Teaching the Mathematics of Music Rachel Hall Saint Joseph’s University

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 Instrument Jim Pepper, Sophomore, History major, Music minor Predicted PitchPitchDesired Freq.Actual Freq.DifferencePredicted length Actual LengthDifference

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 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 inversion SR(S)R(S) I(S)I(S)RI(S) = IR(S) Canon #1Canon #2 theme

Canons 3 and 4 retrograde inversion SR(S)R(S) I(S)I(S)RI(S) = IR(S) Canon #3Canon #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 (handouts and other resource materials) MathandMusicLinks.htmlhttp:// MathandMusicLinks.html (over 30 links, grouped by topic) (my articles) me: