Conceptualizing Music through Mathematics and the Generalized Interval System

Anyone who has listened to a piece of music and tried to anticipate the next few measures understands that music is full of recognizable patterns. For centuries, theorists have attempted to analyze these patterns in order to better appreciate musical works. Only in the last century, however, has music theory been significantly extended into the world of mathematics. One of the primary figures behind this development, David Lewin connected music theory with abstract algebra by creating the Generalized Interval System, or GIS. The GIS forms an excellent basis for analysis because it incorporates the notions of pitch classes and pitch-class intervals which are so fundamental to music theory. At the same time, the GIS provides mathematic structure by rendering the musical notions of transposition and inversion as mathematical group actions on the musical space associated with a GIS. The following sections will formally define the Generalized Interval System, and then provide examples of both the musical and mathematical importance of this construct.