A hypercube-graph model for n-tone rows and relations

Abstract

We investigate the representation of n-tone rows as paths on an n-dimensional hypercube graph with vertices labeled in the power set of the aggregate. These paths run from the vertex labeled by the null set to the one labeled by the full set, passing through vertices whose labels gradually accumulate members of the aggregate. Row relations are then given as hypercube symmetries. Such a model is more sensitive to the musical process of chromatic completion than those that deal more exclusively with n-tone rows and their relations as permutations of an underlying set. Our results lead to a graph-theoretical representation of the duality inherent in the pitch-class/order-number isomorphism of serial theory. © 2013 Springer-Verlag.