Pairwise Well-Formed Scales and a Bestiary of Animals on the Hexagonal Lattice

Abstract

Some pitch-class collections may be represented as subsets of a two-dimensional lattice or generalised Tonnetz. Whereas a well-formed scale of cardinality n is formed as a simple interval chain, and thus defined unambiguously by the size of its generating interval, there are a great number of inequivalent ways of forming connected n-subsets of the two-dimensional lattice defined by a given pair of basis intervals. Only very few of these connected subsets or lattice animals ever turn out to correspond to collections that possess the pairwise well-formed property. Pwwf scales are found to correspond to members of a small family of lattice animals that is independent of the generators at the basis of the lattice. Finally a method is shown for constructing a pair of generators that will yield any given heptatonic pwwf scale; the method is easily extended to other cardinalities.