A Mathematical Model of the Diatonic System

Abstract

In this article a mathematical model of the is presented, that is, the notion diatonic system is defined in mathematical terms. We believe the proposed model to be of considerable descriptive power, and potentially at least, of considerable explanatory power as well. That is to say, the model not only successfully captures the intuitive sense of diatonic but also opens possibilities towards answering deeper questions regarding the system, such as the source of its remarkable persistency through centuries, if not millennia, of human musical history (Western musical history, at any rate). In Agmon (1986) it is argued at length that the model represents a cognitive theory of diatonicism.' While this point is crucial from a broader theoretical perspective, we shall not pursue it again here except for pointing out that the empirical domain which the model describes is not a perceptual domain of pitches, but a more abstract domain of mental representation. In particular, to a considerable (and possibly surprising) extent, the model is independent of the question of how it may or may not be implemented in terms of pitches, so that the whole issue of intonation (including, for example, the status of 12-tone equal temperament) is for the