A new paper appeared on the arXiv this week applying some simple topological ideas to the analysis of musical modes. This contains some simple ideas about fundamental groups and so on and I was thinking it might make a project for a musically-oriented student in my MASS course. There have been several papers of this sort recently: the earliest that I’m aware of is Rachel Hall’s article in Science five years ago. (Dr Hall is a professor at Saint Joseph’s University in Philadelphia and before that was a student in the Geometric Functional Analysis group at Penn State.)
A mode is a seven-note scale. The familiar major scale of Western music (C,D,E,F,G,A,B) is a mode (Ionian mode), but one can obtain different modes by starting from a different note (e.g. Aeolian mode, A,B,C,D,E,F,G) or by using a different starting scale, or both. The authors analyze a mode into two chords: the seventh chord made by notes I, III, V, and VII of the scale, and the triad made by notes II, IV and VI. For instance, Ionian mode is a major seventh (CEGB) plus a minor triad (DFA); Aeolian mode is a minor seventh (ACEG) plus a diminished triad (BDF). Spaces of chords are natural examples of configuration spaces (selections of \(n\) distinct points from the space of notes, with order irrelevant) so this allows the representation of a mode as a subspace of a product of two configuration spaces.
I’m not really competent to judge the music theory here (as a guitarist, I’m usually sticking with two or three of the innumerable possible scales that the authors identify), but it is a fascinating point (this is also what I got out of Rachel Hall’s paper) that music is full of natural examples of configuration spaces, including knots and Möbius bands and other topological exotica.
Bergomi, Mattia G., and Alessandro Portaluri. Modes in Modern Music from a Topological Viewpoint. ArXiv e-print, September 3, 2013. http://arxiv.org/abs/1309.0687.
Budney, Ryan, and William Sethares. Topology of Musical Data. ArXiv e-print, July 4, 2013. http://arxiv.org/abs/1307.1201.
Callender, Clifton, Ian Quinn, and Dmitri Tymoczko. “Generalized Voice-Leading Spaces.” Science 320, no. 5874 (April 18, 2008): 346–348. doi:10.1126/science.1153021.
Hall, Rachel Wells. “Geometrical Music Theory.” Science 320, no. 5874 (April 18, 2008): 328–329. doi:10.1126/science.1155463.