15. Similarity relations, complexes, and genera.

We are used to the fact that in tonal music, motives and harmonies sometimes appear in variant guises, for instance, with one or two pitches or intervals changed. Generally we still regard these variants as variants, not as wholly unrelated musical segments. With its emphasis on pc and ic exactitude for classifying sets as equivalent, pc set analysis might be unable to address our perception of similar but non-equivalent sets. And, given the abstract nature of set classes, determining similarity among classes is not as intuitive as perceiving it among motives.

Some theorists have devised means to discuss the similarity of sets and classes according to shared pc and ic content. The conventions of naming pcs and ics with numbers has helped these theorists to formalize and quantify measurements of similarity in a mathematical way. In The Structure of Atonal Music, for instance, Allen Forte recognizes similarity relations of a few degrees between sets of equal size, based on their sharing all but one pc and on their having maximally or minimally similar ic contents (Forte 1973, 46-60). Other theorists have proposed their own criteria for determining similarity between sets, as well as ways to quantify degrees of similarity numerically.

The desire to express relatedness among sets in a work--and set classes in the abstract--has also led to theorizing--especially, again, by Allen Forte--about set complexes and set genera, super-classifications to which set classes can belong. In both, a single class or a small group of classes may be held to serve as a "nexus" or to have "generated" a whole web of classes, principally through inclusion relations (but also through complement and Z relations). Forte's theory of set complexes occupies the second part of The Structure of Atonal Music, pp. 93-177. His theory of set genera appeared in "Pitch-Class Set Genera and the Origin of Modern Harmonic Species," Journal of Music Theory 32 (1988): 187-271.

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