Jocelyn Neal
Eastman School of Music
The nature and intrinsic structure of the fifty hexachordal set classes has been investigated throughout the theoretical, analytical, and compositional literature. Many interrelationships are fully defined and explained mathematically; others are inferred. Yet these set classes contain implied geometric structures and interesting partition graphs that can illustrate visually those relationships already understood as well as those yet to be explored. Of particular interest are the trichord partitions of the hexachords, which capitalize both on the manageable size of the trichords and the symmetric nature of the partition.
This study examines the nature of the collection of all 50 hexachords according to their M- and Z- relations and patterns of their trichord partitioning. It includes an investigation of selected abstract (set-class) two-partition graphs, then the literal (pitch-class) realizations of some of those graphs. This approach is an extension of previous work by Robert Morris, James Boros, and Wayne Slawson.
The trichord two-partition graph of a hexachord shows the network of all its possible trichordal combinations. For that specific hexachord, the symmetric properties of literal two partition graphs define a mathematical group of operations, related to the internal nature of the hexachord. The properties of invariance and symmetry of the embedded trichords for the hexachord govern the structure of the graph. Applications of graph theory to trichord partitions of hexachords reveals intrinsic structures in the hexachords that provide rich compositional and analytical resources.