1. Teaching the Mathematics of Music Rachel Hall Saint Joseph’s University rhall@sju.edu

2. 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

3. 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.

4. 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

5. Building a PVC Instrument Jim Pepper, Sophomore, History major, Music minor Predicted PitchPitchDesired Freq.Actual Freq.DifferencePredicted length Actual LengthDifference 4848.25130.81132.7154981.90549847.5957439148.250.654256 4949.1138.59139.3941670.80416745.3512655546.250.898734 5050.1146.83147.6829750.85297542.8479888743.230.382011 51 155.56155.5634920.00349240.71539404410.284606 5252.05164.81165.2904670.48046738.363519737.75-0.61352 5353.05174.61175.119150.5091536.2524350636-0.25244 54 185184.997211-0.0027934.3567565833.75-0.60676 55 196195.997718-0.0022832.4705542732-0.47055 56 207.55207.6523490.10234930.6902163631.50.809784 5757.3220223.8455323.84553228.5243146728-0.52431 5858.1233.08234.432111.3521127.2700711626.25-1.02007 5958.8246.94244.105284-2.8347226.2191588525.25-0.96916 6059.85261.63259.368544-2.2614624.72035563250.279644

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

7. 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!

8. 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

9. 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