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Teaching the Mathematics of Music

Teaching the Mathematics of Music. Rachel Hall Saint 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

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Teaching the Mathematics of Music

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  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. 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!

  4. 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.

  5. Logarithms and music: A secondary school math lessonChristina Coangelo, Senior, 5 yr M. Ed. program Math Content Covered • Functions • Linear, Exponential, Logarithmic, Sine/Cosine, Bounded, Damping • Graphing & Manipulations • Ratios

  6. Building a PVC InstrumentJim Pepper, Sophomore, History major, Music minor

  7. The Mathematics of Change RingingEmily Burks, Freshman, Math major

  8. Symmetry and group theoryexercises Sources: J.S. Bach’s 14Canons 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

  9. 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.

  10. Canons 1 and 2 I(S) RI(S) = IR(S) S R(S) theme retrograde inversion retrograde inversion Canon #1 Canon #2

  11. Canons 3 and 4 I(S) RI(S) = IR(S) S R(S) retrograde inversion retrograde inversion Canon #3 Canon #4

  12. 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.

  13. Performer 1 Performer 2 Steve Reich’s Clapping Music • Describe the structure. • Why did Reich use this particular pattern? • Write your own clapping music.

  14. 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

  15. 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!

  16. 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

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