In the late afternoon of October 24, 1999, about one hundred people were gathered in a large rehearsal room of the Rotterdam Conservatory. They were listening to a discussion between representatives of nine European countries about the teaching of music theory and music analysis. It was the third day of the Fourth European Music Analysis Conference.¹ Most participants in the conference (which included a number of music theorists from Canada and the United States) had been looking forward to this session: meetings about the various analytical traditions and pedagogical practices in Europe were rare, and a broad survey of teaching...

Pitch-class set theory is a musical application of mathematical set theory. As with the latter, it defines a set as a collection of things, and it refers to these things as the elements of the set. Sets are usually denoted by capital letters, likeA,B,C, etc. If not specified, the elements are denoted by lower-case letters:a,b,c, etc. In pitch-class set theory, too, a set is defined by its elements only, not by a particular arrangement of these elements or by their quantities. When two setsAandBconsist of the same elements, they are...

Each combination of tones can be identified on the basis of its PC set. However, for such a combination to be considered of structural interest—that is, worth identifying at all—it is a necessary (though not a sufficient) condition that it bear a relation to other combinations. A relation between two combinations of tones can sometimes be conceived of as a transformation. PC set theory has defined several operations that transform one PC set into another, the most important of which are transposition (T), inversion (I), and multiplication (M). No doubt, transposition and inversion are backed by the longest...

The operations discussed in chapter 3 are generalized representations of compositional techniques whereby PC sets are derived from each other. However, not all relations between PC sets are based on derivation. This chapter and the following ones will deal with other relations, seen within a historical framework. In this chapter I will discuss the evolution of the concept of PC set equivalence.

Like the term “operation,” the term “relation” evokes the world of mathematics. In mathematics, more specifically in algebra, a relation is commonly conceived as an open sentence, designatedP, connecting the elements of two collections,SandT....

The discussion of “similarity” has been one of the major threads running through PC set theory. It revolves around the question of whether PC sets, when they are not connected by a more or less obvious operation, or do not share a more or less obvious property, can still be “related.” The aim has been to model such relatedness, and thus render it tangible. There is no apparent link with the familiar geometrical concept of similarity, which refers to the equal proportions of different-sized objects. Nor is there a link with the set-theoretic relation of the same name, which involves...

Much of what we call “pitch-class set theory” today originally served as theintroductionto a theory. An inventory of classes of musical objects, and of relations between these classes, its function was to prepare the ground for a model of large-scale pitch organization in music of the twentieth century: thepitch-class set complex. This model was the main focus of Allen Forte’s seminal article “A Theory of Set-Complexes for Music” from 1964 and of his bookThe Structure of Atonal Musicfrom 1973.

Set-complex theory deals with the analysis of entire compositions, or movements and sections of compositions. More...

Music theory, it was argued in chapter 1, can be a discipline at the service of music historians, providing them with the conceptual apparatus to analyze and assess music from the past. But music theory is also apartof music history, presenting itself for analysis and assessment. This double identity can be viewed as a special case of a problem fundamental to historiography: the problem of a historian’s own historicity, which renders an objective truth outside history inherently impossible. Historians trained in modern hermeneutics know about this. They know that their reading of a source is at best a...

From a narrativistic point of view, history is a staging of the past.¹ This book has placed PC set theory, an evolving theoretical discourse on music, in the fore-ground, without adding much scenery. It has only made casual reference to its social, cultural, and technological setting. Now that we are about to bring two crucial elements of that setting onstage—the computer and the university—we should realize that their mere presence does not provide an explanation for their influence on music theory. They have exerted that influence because music theorists reacted to them, probably for a complex of reasons....