Symmetry and Transformations in the Musical Plane

Vi Hart
Proceedings of Bridges 2009: Mathematics, Music, Art, Architecture, Culture (2009)
Pages 169–176 Regular Papers


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 isometric 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.