in Proceedings of the 12th Annual BRIDGES Conference: Mathematics,
Music, Art, Architecture, Culture
Banff, Canada, July 2009, to appear
The musical plane is different than the Euclidean plane: it has two
different and incomparable dimensions, pitch-space and time, rather than
two identical dimensions. Symmetry and transformations in music have been
studied both in musical and geometric terms, but not when taking this
difference into account. In this paper we show exactly which
transformations apply to musical space and how they can be arranged into
repeating patterns (frieze patterns and variations of the wallpaper
groups). Frieze patterns are created intuitively by composers, sometimes
with timbral color patterns or in sequence, and many examples are shown.
Thinking about symmetry in the musical plane is useful not just for
analysis, but as inspiration for composers.
One of my current composition projects is a seven-movement piano trio with one
movement inspired by each of the seven possible geometric frieze patterns.
Here are piano sketches of the first three movements: