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 the order relations in two sets. The most likely source of the concept is information science, especially insofar as it deals with uncertainty. Similarity relations are defi ned there as “fuzzy” relations. They involve a degree of imprecision, and thus better approximate everyday human thought and speech than “crisp” equivalence relations (Klir and Folger 1988, 83–85.).

In PC set theory, “similarity” is an umbrella term covering concepts with different origins, ranging from geometry to statistics. In this chapter, these concepts will be analyzed with respect to their underlying questions, intuitions, and assumptions. None of them has become the “most common” concept of PC set similarity; nor do they all refer to the same properties. PC sets can be similar in different ways; generally speaking, however, similarity is a matter of degree. In that respect, similarity relations are different from equivalence relations. PC sets are either in an equivalence relation or not, but they can be more or less similar. The study of PC similarity has been motivated by the prospect of a subtle and fl exible analytical framework.

The terms “equivalent” and “similar” may both be appropriate to a single pair of PC sets. In other words, they are not mutually exclusive; they refl ect different ways of thinking about relations. However, similarity relations involve a much greater number of PC sets: any pair of PC sets will be more or less similar on some scale. This inclusiveness has been another reason to search for reliable measures of PC set similarity.

Concept and Referent

What does it mean when two PC sets are called “similar”? How do these PC sets resemble one another? And in what regard can PC set similarity be meaningful? A PC set can be realized in a multitude of ways, causing different musical sensations.