Mathematics and Music

Homework Assignment #8

Note: This assignment covers material important for the final exam, but there is no written work to be turned in.

Linear polyphony destroys itself by its very complexity; what one hears is in reality nothing but a mass of notes in various registers. The enormous complexity prevents the audience from following the intertwining of the lines and has as its macroscopic effect an irrational and fortuitous dispersion of sounds over the whole extent of the sonic spectrum. There is consequently a contradiction between the polyphonic linear system and the heard result, which is a surface or mass. This contradiction inherent in polyphony will disappear when the independence of sounds is total. In fact, when linear combinations and their polyphonic superpositions no longer operate, what will count will be the statistical mean of isolated states and of transformations of sonic components at a given moment. The macroscopic effect can then be controlled by the mean of the movements of elements which we select. The result is the introduction of the notion of probability, which implies, in this particular case, combinatory calculus. Here, in a few words, is the possible escape route from the "linear category" in musical thought.
Iannis Xenakis, "The Crisis of Serial Music" 1955

1. Read Chapter 8 of the course text, Composing with numbers: sets, rows and magic squares by Jonathan Cross. As you read, try to follow the musical (mathematical) analysis given by the author to further your understanding of the material. Some of the works were discussed in class.

2. Listen to the CD Math and Music 3: Musical Group Theory and Modern Music available on reserve in the music library. Liner notes for the CD are available here. You may be tested on some of this music so be sure to read the notes and listen carefully.