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    Contemporary pieces at the intersection of music and mathematics

    Dear Collected Wisdom!

    This year I’ve started teaching a non-major, first-year course called “Music, Mathematics, and Computation.” My plan for the course was a mixture of mathematical music theory (e.g., tuning and temperament, the Golden Section, etc.) and computational music analysis (key finding, similarity measures, etc.). When I first offered the course in the fall, what I found to be the most successful was to talk about post-1965 pieces with some mathematical angle: I’d lecture on the relevant mathematics (translating them for students who may not even have calculus), we’d listen to the piece together, and we’d talk about the experience of listening. We did things like ratios of tempo in Nancarrow, temperament in Young’s The Well-Tuned Piano and Partch’s Barstow, fractals in Ligeti’s Disordre, fractals in Norgard's infinity series, and algorithmic composition in Pärt’s Cantus, John Luther Adams’s For Lou Harrison, and the postcard pieces by James Tenney.

    Since this material was the most successful, I’d like to add more of it (probably double it) and ditch things like key-finding, which I'm starting to think only people who read this board care about. Can anyone think of other concepts/pieces I might add to the syllabus?

    Thanks so much!

    -Mitch Ohriner

    University of Denver

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    • 9 Comments sorted by Votes Date Added
    • Mitch,

      Jon Kochavi has a great demonstration of the serial design of Nono's Il Canto Sospeso--you might ask him about that. I do things with polyrhythm in Ligeti and Carter in my classes--something you might consider, and Xenakis is also an obvious choice.


      --Jason Yust


    • Hi, Mitch,

      You might have a look at Xenakis's Nomos Alpha (for solo cello).  It uses a compositional process that incorporates the group of rotations of a cube--something that first-year students can grasp fairly intuitively.  Formalized Music contains a nicely succinct description.

      Best wishes, Bob Peck

    • You might add to the syllabus a piece that incorprates chance/randomness. This is appropriate in the sense that "random", "pseudorandom" and related are central concepts in the theory of computation (and some other mathematical areas).

      I'm not sure what specific music to recommend. Of course, the first composer who comes to mind is Cage; but how successful are his works that incorporate chance ...

      Isaac Malitz, Ph.D.




    • Mitch, my recent article in MusMat, the new Brazilian journal, lays out some of the basics of non-majors course that Andrew Jones and I taught at Yale in 2014, and I taught solo at University of Sydney in 2015. 


      We use small cyclic universes to model metric cycles, and scale up to larger ones. Things start getting interesting, both musically and mathematically,  when you come to 6 beats in the cycle, and you could just start there, although there's a bit of a "throw 'em in the lake" aspect to that. Better though than the "throw 'em in the ocean" of mod 12.

      The possibilities for 6, 7, 8, and 9 -beat universes all fall directly from the properties of the cardinal of their respective universes. (And 7, of course, leads you into the rich body of diatonic set theory.)  This helps students cultivate an appreciation for the unique properties and musical potentials of  a 12-element universe, once you scale up to it. 

      We have extensive course materials that we would consider sharing.  I also recommend using them in conjunction with  Andrew Milne's "loop-generating" interactive graphics program. 


      There is also a special issue of the Journal of Mathematics and Music that gives some other recent pedagogical approaches. There is quite a bit going on this area right now!


      --Rick Cohn

    • Hi Mitch!

      I recommend exploring processes in (post-)minimalist pieces, like phasing in early Steve Reich and the signature transformations in Michael Torke's "Yellow Pages" from Telephone Book.  There are often very simple and elegant ways of modeling these processes and understanding them has an immediate effect on the listening experience.

      --Robert T. Kelley

    • Mitch! You just woke me up! So glad to learn you are doing this & especially that your initial experience teaching it has made you want to expand.

      I agree with Bob & Jason re Xenakis as well as Nono, Ligeti & Carter. There are two other IX sources that I would add to Formalized Music. One is The Instrumental Music of Iannis Xenakis: Theory, Practice, Self-Borrowing by Benoit Gibson. The other is Arts/Sciences: Alloys which is Xenakis' thesis defence before Messiaen & others on the committee at Sorbonne - it has a spirited discussion/debate about the intersection of music, mathematics, and architecture. Also, Appendix II is a reprint of his 'Sieve Theory' & Appendix I is a fascinating parallel history of significant events in music & maths: 'Correspondences between Certain Developments in Music and Mathematics'.

      The mention of Messiaen brings up permutations as a theme running through a huge amount of post-1945 music (and fits with Bob's suggestion about rotations of a cube & Nomos Alpha). Right now I'm trying to finish a paper on Messiaen's two Ile de feu piano pieces. No.2 especially is saturated with permutation examples (what OM called 'interversion') & a still unsolved analytical puzzle as well as a bunch of stuff for set theory. It would be interesting to do Idf2 along side Babbit's Three Compositions for Piano to compare the two distinct approaches in early integral/total serialism (permutations again) developed in France vs US.

      I also agree with Rick's suggestions. It's taken too long for the community to come around to the deep math relationships between pitch and rhythmic structures. I suspect most still aren't there. Much of the work I know of in this area has been done by Godfried Toussaint & his colleagues (but even he took a while to identify maximally even rhythms, preferring 'Euclidean rhythms' - a difference without much distinction & seems to play down max evenness as an important descriptor of rhythmic structures still. Nevertheless, his book, The Geometry of Musical Rhythm is an excellent compendium of the math basis relating rhythms worldwide. His web site also has many other essays pub & unpub on the topic.

      And for hands-on investigation & dicovery of the still unexplored possibilities in an undifferentiated 'pitch-rhythm matrix', Jeremiah Goyette offers a set of outstanding tools in his online Music Calculators. This would be a great teaching/learning aid for all things music set theory.

    • To follow up on Isaac's suggestion, there is an article by Popoff in MCM2015 about the Cage number pieces that might be useful. These would actually be a lot of fun to play with, to imagine all the different possible realizations and what aspecs of the design make them possible.


      --Jason Yust


    • It might be instructive or at least amusing to collect a few musical works that have utilized mathematical techniques and were not successful.   :)

      Taking a step back: I am a credentialed mathematician [mathematical logic] and also a musician. I have long been fascinated by the question of whether mathematics actually plays an important role in top-quality musical composition and analysis, or it just seems to.

      Isaac Malitz, Ph.D.




    • Thanks for these excellent suggestions everyone! I think I'll be able to incorporate many of them. Something I could have made more clear: since it's a non-major course, most of the students don't read music and there isn't time to teach them. So I'm finding that ideas related to rhythm and form (and Toussaint is a great direction here) gain more traction than ideas related to pitch. I'll let you now how it goes!