Introduction

When we analyze tonal music, the basic principle of tonality -- that a single pitch class (pc) is the gravitational centre for a work's pitch structure -- generates most of the terms of reference for our analysis. For instance, in analyses of "common-practice" tonal music (much of the music written between c.1650 and c.1900), such concepts as scale steps, chord functions, and tonicization help to describe a system -- an evolving system, to be sure -- of pitch relations.

Some twentieth-century music, however, is not tonal. Not only does such music avoid the conventions of common-practice tonality; it avoids projecting any clear sense of a central pitch class. Analyzing this music is challenging. There appears to be no generalized system of non-tonal pitch structure; instead, each piece seems to create its own contextual structure. And it may be unclear just what the bases of such a contextual structure might be.

An approach that has gained favour with musicians intrigued by non-tonal music is pitch-class set analysis. Like other analytical methods, pc set analysis has a few essential features:

• basic axioms, assumptions upon which the analytical method is founded,
• abstract concepts used for classifying and interpreting musical events,
• common analytical operations with which to probe the music, and
• conventions for describing the relationships and patterns the analysis discovers.

Just what use we make of pc set analysis depends on our interests and skills. The music of Schoenberg, Berg, Webern, Stravinsky, Bartók, Debussy, Scriabin, and their contemporaries has formed the core repertory to which pc set analysis has been applied, but it has also been used to examine later and earlier music, including tonal music. Indeed, studying pc set analysis doesn't just give us a new toolbox for probing many pieces of music; it can give us a new way of thinking about pitch design in general.

Pc set analysis builds upon the observation that, in the absence of tonality's centralizing force, pitch structure often seems to be grounded in the intervallic relationships among pitches. At times these relationships also throw certain pitches or pitch classes into prominence. Some intervallic relationships may be obvious to the listener. Many others may not be, and pc set analysis has proved useful at uncovering and categorizing these relationships.

I.  Gathering Sets	This part introduces the primary axioms and concepts of pc set analysis, including the notions of pitch class, interval class, pc set, pc set class and interval vector.
II. Interpreting Sets	This part defines the kinds of relationships and patterns among sets and set classes for which analysts usually look. Uncovering these relationships and patterns is the aim of pc set analysis. Note: At present (September 2001), Part II is still under construction.

This guide includes some exercises, some of which you are guided through, with answers, to give you practice in dealing with analytical concepts and operations. In addition, the guide includes a pc set-class table and an annotated bibliography of basic printed and on-line resources in pc set analysis.

Acknowledgements

This guide in its present form has principally been developed over two summers, with the financial support of two grants and with the much appreciated assistance of two of my students. A grant from Mount Allison University's Innovative Teaching Fund supported the work of Jordan Fleming In the summer of 1999, and one from Mount Allison's Purdy Crawford Centre for Teaching allowed me to work with David Walters in the summer of 2001. I extend my thanks to the University and especially to Jordan and David. Their computer (and especially JavaScript) knowledge and their rapid assimilation of pc set analysis and of my aims made working with both of them a pleasure and an education. In addition to his many valuable insights throughout this guide, David is wholly responsible for the excellent JavaScript set calculator that accompanies it. I am solely responsible, of course, for any mistakes remaining in this guide.

Page last modified 3 October 2001 / GRT