|
COPYRIGHT 2001 University of Washington
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.
The distinction between the received view and the one I advocate is not a new one. Both Demske (1995) and Rahn (1989) have obliquely argued the utility of looking at the large picture painted by the individual numbers returned by similarity relations in response to individual pairs of pc set classes, but have given up such a goal as unattainable. Over the course of this paper, I will develop their arguments more fully, and then show that the goal can be reached quite easily.
1. SIMILARITY; RELATIONS, AND SIMILARITY RELATIONS
Notwithstanding the vagueness of the term similarity, which is not a primary concern of this essay, similarity relation is a problematic term. As many others have noted, most "similarity relations" are not relations at all, strictly speaking. This accounts for the fact that they have been called similarity measures and similarity indexes and similarity functions in the literature. The term similarity relations, however, has been used as often as it has not, and usually the terms are used interchangeably within articles despite the technical inaccuracy of the r-word. In this part of the essay I will argue first that they are not relations in standard modes of thinking, then that they are relations in the context of fuzzy set theory; some issues that fall out along the way will be useful later on in the essay, when the issue of transitivity arises.
My argument that they are relations, at least from the fuzzy point of view, is motivated by the fact that proponents of similarity relations want them to be relations, even though they aren't. That is, even though they are functions and not standard relations, they are used specifically to relate pc set classes, and not (in principle) for any other reason. In terms of their use, then, similarity relations have a relational character if not a classical relational structure.
1.1 RELATIONS AND FUNCTIONS IN GENERAL
Relations and functions are both kinds of mappings--ways of connecting elements of one set, the domain, to elements of another, the range. We will see shortly that functions are a special kind of relation.
Every relation can be characterized in two ways. The intension of a relation is a general rule for mapping (relating) elements of the domain to elements of the range. Such rules are normally set out in the form of necessary and sufficient conditions. The intension of the Z relation, for instance, is this:
Pc set class x is Z-related to pc set class y iff y has the same interval vector as x but no combination of transposition and inversion operations can turn x into y.
The extension of a relation is an enumeration of all the mappings made by the relation:
4-Z15 [0146] is Z-related to 4-Z29 [0137], and vice versa; 5-Z12 [01356] is Z-related to 5-Z36 [01247], and vice versa; 5-Z17 [01348] is Z-related to 5-Z37 [03458], and vice versa...
If two relations have the same domain, range, and intension, then they will have the same extension. The converse, however, is not necessarily true, since two general rules for a mapping may result in the same extension.
The extension of any relation R can be represented as a set of ordered pairs of the form (d, r,> where d is an element of the mapping's domain, and r is an element of the mapping's range. In a sense, the extension of a relation is a bare uninterpreted set of pairs, and the relation's intension gives meaning to the extension. The distinction between extension and intension is an important one, a point to which we will return in section 2 below.
If a relation { , ,..., } has the property that all of the d terms are distinct--that every element of the domain is mapped into the range at most once--then the relation is a function. Note that every function has an extension as well as an intension, since functions are relations.
1.2 RELATIONS AND FUNCTIONS IN PC SET THEORY
Pc set theory makes extensive use of relations, and these can be classified into three types. Type 1 relations include transformations, operations, the Z relation, the relation of pc set equivalence under [T.sub.n] and [T.sub.n]I, the literal and abstract inclusion relations, and the complement relation, among others. The distinctive structural features of Type 1 relations are as follows:
1. The domains and ranges of Type 1 relations are coextensive. The domain and range of the literal inclusion relation, for example, are both the set of pc sets, and the domain and range of the abstract complement relation are both the set of pc set classes.
2. The domains (and ranges) of Type 1 relations are sets of specifically pc set-theoretic entities-pc sets, pc set classes of various kinds, interval classes, and so forth.
3. Type 1 relations may be functions, or they may not. Any one of the [T.sub.n] relations is a function (also known as an operation), since they map each pc set uniquely into another. The [T.sub.n]/[T.sub.n]I equivalence relation, on the other hand, is not a function, since any given pc set can mapped into as many as 23 others, in addition to the trivial mapping of all pc sets into themselves under [T.sub.n]/[T.sub.n]I equivalence.
Type 2 relations include constructs like Lewin's embedding function (EMB), the degree-of-symmetry function (DSYM), and the cardinality function. They are structurally differentiated from Type 1 functions in a number of significant ways:
1. The domains and ranges of Type 2 relations are not coextensive.
2. The domains of Type 2 relations, like those of Type 1 relations, are sets (or sets of sets) of pc set-theoretic entities. The domain of the DSYM and cardinality functions is the set of pc set classes, as are the domains of many Type 1 functions. Type 2 domains can also be sets of sets; the domain of the EMB function, for instance, is the set of ordered pairs of pc sets (or of pc set classes).
3. The ranges of Type 2 relations, in contrast, are sets of numbers.
4. Type 2 relations are always functions; each pc set or pc set-class, or set thereof, is uniquely mapped to a single number.
The third type of pc set-theoretic relations includes those relations between one kind of pc set-theoretic entity and another. These include the relations that map pitches into pitch classes, pc sets into pc set classes, and rows into row classes. They also include mappings from pc sets into Klumpenhouwer networks and various other transformational mappings. These are the mappings between relatively concrete pc set-theoretic entities and relatively abstract ones. Type 3 relations share certain properties with both Type 1 and Type 2 relations:
1. As with Type 2 relations, the domains and ranges of Type 3 relations are not coextensive.
2. The domains of Type 3 relations, like those of both Type 1 and Type 2 relations, are sets (or sets of sets) of pc set-theoretic entities.
3. As with Type 1 relations, the ranges of Type 3 relations are also sets (or sets of sets) of pc set-theoretic entities.
4. Type 3 relations may be functions (as the relation from pc sets to pc set classes) or they may not (as the relation from pc sets to Klumpenhouwer networks).
Some relations of types 2 and 3 have as their domains not sets of pc set-theoretic entities, but sets of sets of them. Examples mentioned already include the pc-set-to-pc-set-class mapping and the EMB function. Relations of this kind do still have extensions made out of ordered pairs, but the first term in those ordered pairs will itself be a set. In the pc-set-to-pc-set-class mapping, those terms include
,
,
and others of that form. (Usually, of course, the first term is more sizeable.) The relation maps the sets of pc sets that constitute the left-hand terms in the ordered pairs into the pc set classes in the right-hand terms.
The EMB function's extension is structurally similar, but its meaning is a bit more opaque. The extension of EMB under [T.sub.n]/[T.sub.n]I equivalence includes ordered pairs like
, 1>
and
, 2>.
Here again, the left-hand terms of the ordered pairs are sets of pc sets (in this case, these sets are themselves ordered pairs), but the right hand terms are numbers. Thus EMB, from a technical point of view, is a relation from ordered pairs of pc sets to numbers.
1.3 RELATIONS, RELATIONSHIPS, AND PREDICATES
If the foregoing exegesis of EMB's relational character seems a bit at odds with our intuitions about what EME actually does, that is because it should. EMB is meant to describe a relationship between two pc sets, not a relationship between a pair of pc sets and a number.
The example of EMB brings out the difference between a relation and a relationship. A relation is a set-theoretic structure that abstractly describes or models a relationship among things in the world. (The question whether pc sets and their relationships have the status of real-world entities is left aside for present purposes.) A relation's intension gives meaning to its extension precisely by specifying the real-world relationship being modeled. In other words, relationships are purely intensional objects (pardon the pun), though they are capable of being modeled by one or more relations.
Relationships among pairs of things are a special case of predicates, terms describing properties that attach to entities or sets of entities. Phrases of the form "xis a P" assign the predicate IS--A--P to the entity x. In the case of one-place predicates the assignment is clear: "3-12 [048] is a pc set class" assigns the predicate IS--A--PC--SET--CLASS to the entity 3-12 [048]. Two-place predicates can be phrased in the form "y is R-related to z," in which case an assignment is made between the predicate IS--R--RELATED--TO and the ordered pair .
To see how two-place predicates generalize one-place predicates, and how they relate to relations, requires a bit more work. Without a change in meaning, we can rephrase "y is R-related to z" as " is an ordered pair whose first term is R-related to its second term." Not only do these mean the same thing, but they share the desirable property that the entities can be swapped out without affecting the predicate itself. Additionally, this way of phrasing it itself of the form "x is a P," where x, the entity, becomes the ordered pair , and where Is--A--P, the predicate, becomes IS--AN--ORDERED--PAIR--WHOSE--FIRST--TERM--IS--R--RELATED--TO--ITS--S ECOND--TERM. It is easy to see how this rewriting technique will work for predicates of three and more places as well. Although the technique is awkward, it suffices to know that it exists--the point of the exercise is to show that simple predicates, binary relationships, and more complex kinds of predicates can be put into a single conceptual framework.
A more meaningful way to distinguish among kinds of n-place predicates than the size of n is in terms of the way they are used in discourse. Specifically, I am concerned with the kinds of modifiers that can be applied to predicates. The philosopher Susan Haack (1980) identifies two categories of modifiers, degree modifiers and success modifiers, that are...
Read the full article for free courtesy of your local library.
|
Bookmark this article
Print this article
Link to this article
Email this article
Digg It!
Add to del.icio.us
RSS
