The Line of Fifths

Abstract

The concept of pitch-class depends, as is well-known, on the assumption of octave equivalence: any two notes one or more octaves apart are members of the same category. But the pitch-class idea also entails another kind of equivalence, that is, enharmonic equivalence: any two notes of the same pitch belong to the same category. The assumptions of octave and enharmonic equivalence have proved to be extremely useful to music theory. Flrst proposed with serial music in mind, the pitch-class system has more recently opened up a new approach to the study of tonal music as well.l FIowever, we should remember that enharmonic equivalence is not the usual assumption in traditional tonal theory and music notation. There, the same pitch-class may be represented in different ways: as Ab or G, for example. Retaining the assumption of octave equivalence, we might call such categories 'tonal pitch-classes' (TPCs) as opposed to the twelve 'neutral pitch-classes' (NPCs) of atenal set theory. In this article, I shall explore the tonal pitch-class system and the role of tonal pitch-class distinctions in musical cognition. In emphasising the differentness of Ab and G, I am not in any way denying that they are, in another important sense, the same any more than distinguishing between C3 and C4 denies the essential similarity between them. It seems clear that both the neutral pitch-class and tonal pitch-class systems play important roles in tonal music. However, the tonal pitch-class system has not received the theoretical attention it deserves. I begin by proposing a very simple spatial representation of tonal pitchclasses. I then present a model of how TPC labels are inferred from neutral ones: that is, the principles for choosing one spelling of a pitch over another. I also discuss the issue of ambiguity: cases where a single event seems to be functioning as two different TPCs. In the second half of the article, I explore some of the ways in which spelling choices are consequential for other aspects of music cognition; I also consider their relevance to the problem of keyH1nding. In closing, I discuss the relevance of the TPC system to highly chromatic and post-tonal music. I should H1rst mention two issues that I do not intend to diwuss in this article. The Elrst is intonation. The premise of this study is that inferring the spelling of notes is an important part of tonal cognition; but I will argue that