On the Proto-Theory of Musical Structure

Abstract

JUST AS music history is a species of history, so no doubt is music theory a species of theory. Hence music theory should have some structure in common with other theories. The question is, then, what is this structure? And of what species of theory is music theory? To specify a theory in logically acceptable terms is usually no easy matter. Every theory involves an underlying logic, and many alternative logics clamor here for attention. Even after underlying logic has been agreed upon to be of such and such a kind, we then must worry about (i) special domain of objects allowed as values for variables and (ii) basic classes (or domains or sorts) into which these objects may be grouped as well as basic relations whereby they may be related. Item (i) has to do with choice of an ontology for theory and (ii) with choice of suitable non-logical predicate or relational constants as primitive or undefined. Only after (i) and (ii) have been well settled is it of any interest to start search for axioms. In process of getting a theory, gaining of axioms comes at very end.1 Let us begin then at beginning with (i) and (ii), ontology and primitive predicates. This much might be called the proto-theory of music. We should first note that proto-theory need not involve complexities exhibited by many scientific or philosophical theories. For example, music theory need not, it seems, involve modal logic, or threevalued logic. In modal logic, it will be recalled, notions of necessity and possibility play fundamental roles, and in three-valued logic a value intermediary between truth-values truth and falsehood is admitted. It is very doubtful that such notions need play any role in music theory. Nor for that matter need music theory assume such complicated mathematical axioms as axiom of choice, or, say, continuum hypothesis. In fact it is not clear that music theory need even