9. Quick review: levels of abstraction

Our aim in using pitch-class set analysis is to explore the pitch structure of (usually) atonal music. Music analysis always invokes abstract concepts that help us to classify and make sense of our concrete musical experience. As you have discovered, pc set analysis deals in a few levels of abstraction, some of which we have been routinely using on tonal music -- perhaps without considering them consciously. We'll quickly review these levels.

1. The pitch material of music consists of pitches. These pitches are separated in pitch-space by pitch intervals: ordered (directed) ones between melodic pitches, unordered ones between harmonic pitches.
   
   
2. We classify the pitches we hear into pitch classes (pcs) using axioms of octave and enharmonic equivalence. There are 12 pitch classes.
   
   
3. We then consider groups or sets of pitch classes, abstracted from the musical segments we analyze, for example, sets [1,2,3,6,7], [2,3,4,7,8], and [10,11,2,3,4]. We usually use the normal form names for these sets. We are mostly interested in sets of between three and nine pcs.
   
   
4. Using axioms of transpositional and inversional set equivalence, we can classify sets into pc set classes. We usually name set classes by citing either their prime forms, for example, (01256), or their Forte set-class names, for example, 5-6.
   
   
5. With the growing abstraction of pitch-related concepts, concepts of interval also grow abstract. We measure the distance between pitch classes -- pitch-class intervals -- using modulo 12 arithmetic. As with tonal pc intervals, interval pairs which add together to make an octave are considered "inverses" of each other. We classify such pairs as equivalent, grouping them into interval classes (ics). Pc set classes are usually characterized by unique ic contents (conveniently written as ic vectors), though some pairs of classes ("Z-related" classes) happen to share their ic profile.

Each of these levels of abstraction and of classification tells us something about the relatedness of concrete pitches and groups of pitches to each other.

We can say that the process of analyzing an atonal work is a three-stage one, though in real analysis we likely interweave these stages:

1. segmenting the music
2. gathering and classifying the pc sets that we find in the segments, and
3. interpretating the set data we've gathered.

The last, interpretive, stage is, of course, the real point of any analysis. The first stage lays the crucial (if at times problematical) groundwork for the analysis. The second stage is the one involving the basic mechanics of pc set analysis, mechanics we have been practising. Happily, many of these sometimes tedious mechanics can be computerized, leaving us more time to spend on interpretation. This guide is accompanied, for instance, by an excellent Web-based pc-set calculator, written by David Walters. (See the list of Other Sources for some other on-line set calculators.) As with arithmetic and the hand-held calculator, however, we must learn the mechanics and their rationale before we can sense how the data we gather might be interpreted.

Page last modified 3 October 2001 / GRT