128_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 (nonproof) • 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, 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 -2.26146 -1 -2 -3 -4 Difference 26.21915885 Frequency 25.25 -0.96916 24.72035563 0.279644 25 Series1 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: http://bach.nau.edu/BWV988/bAddendum.html Steve Reich’s Clapping Music Performed by jugglers http://www.youtube.com/watch?v=dXhBti625_s 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 and I(S) S RI(S) = IR(S) R(S) theme retrograde Canon #1 inversion retrograde inversion Canon #2 Canons and 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 Performer • 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 • 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 ... 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... 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: ... 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