Hyperscales and the Generalized Tetrachord

Abstract

We begin with the concept of the Western diatonic scale as the union of two tetrachords-an idea whose historical importance needs no elaboration here. In sections 2 and 3, we first generalize the conjunction of diatonic tetrachords to the arbitrary set composed of two hypertetrachords and then explore the interaction of such sets with ambiguity (especially the ambiguous tritone), interval content, and maximal evenness. As a result, we are able to strengthen the definition of hyperdiatonic given in a previous paper (Clough and Douthett 1991). In the final two sections of this paper, we first develop necessary mathematical tools (consecutive integer sequences, multiplicity sequences) and then extend the work of a previous paper (Clough et al. 1993) that deals with the similarities between a class of ancient Indian heptatonic scales (the gramas) and the Western diatonic scale. We formalize the notion of iterated maximally-even sets given there, develop sets of axioms based on the similarities between the gramas and the diatonic, and explore classes of pcsets (we call them hyperscales) that satisfy the axioms. This approach, we feel, yields additional insight to the affinities between the gramas and the diatonic.