1 / 16

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

justus
Download Presentation

Teaching the Mathematics of Music

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  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

More Related