Listening to similarity relations.

WHAT FOLLOWS is an extended argument for these two claims:

1. The ways in which we are accustomed to talking about similarity relations are not as productive as they seem to be, and there are better ways to do it.

2. Comparison of various similarity relations from such a different point of view shows that they are more related to each other, and to a lot of other theory, than they appear to be in traditional modes of discourse.

By the term similarity relations, I refer to a well-known class of functions that take as input pairs of pc set classes and return numbers--e.g., ASIM (Morris 1980) or IcVSIM (Isaacson 1990)--and not to structurally different entities that go by the same name, e.g., Forte's [R.sub.p] relation, although that relation will come up in the course of discussion. Some of the problems inherent in the term will be the subject of section 1 below.

I will argue for a new way of thinking and speaking about similarity relations, one that involves "listening to similarity relations," by which I intend less to anthropomorphize these most theoretical of entities than to refer to the fact that there is information hidden in the results of this diverse group of functions, information that conventional music-theory wisdom holds to be either nonexistent or irretrievable. My motive for finding a way to listen to similarity relations is a belief that I will call the Natural Kinds Hypothesis: Similarity relations serve as a model classification scheme for pc set classes that (a) corresponds to the intuitions of music scholars, (b) can be shown experimentally to correspond to certain judgments of people who are not music scholars, and (c) can be modeled with tools already at hand in pc set-class theory, which means that similarity relations tell us nothing that we do not already know--not in the intuitive sense, but in terms of the inherent properties of the twelve- tone equal-tempered universe of pitch classes, of which we have a formidable mathematical model already.

The Natural Kinds Hypothesis, in other words, means to suggest that certain relationships of similarity among pc set classes exist a priori, owing their nature to the structural properties of the twelve-pc system itself. Constructs like ASIM or ATMEMB or REL point out these relationships, model them, describe them, or what have you, but they do not create them. By focusing our scholarly efforts on the constructs (or, worse, on the efforts behind the constructs!) we get distracted from the arguably more interesting set-class relationships. The evidence for the Natural Kinds Hypothesis, as I've hinted, consists of a set of observations about the ways in which these various constructs agree with one another despite the differences we're used to hearing about.

My argument depends on a distinction between two modes of thinking and talking about similarity relations, a distinction roughly parallel to the one in the bromide about missing the forest for the trees. (Here the trees are the numbers that the similarity relations produce, and the forest is the set of interrelations among those numbers.) Were I to pursue this metaphor, I would characterize the received view as tree-oriented and the view I advocate as forest-oriented. In the tree-oriented view, I would argue, the forest is missed, and in the forest-oriented view the trees are missed; subsequently I would argue that when it comes to similarity relations, the forest provides more useful information than any tree, and that would complete the argument. But since my metaphorically forest-oriented view, as will emerge in section 3 below, is best represented by what are, in a somewhat more literal sense, trees, the potential for conceptual disarray is obvious, and I will let the metaphor drop for the moment.