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
- 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. Three of the works were discussed in class and
can be heard on CD #4.
- Listen to CD #4 Composing with Numbers: Bells, Rows
and Magic Squares distributed in class on April 20th.
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.