than had been assumed by the researchers who first proposed the prototype theor y of category structure.34 In most Type 1 categor ies, some attr ibutes are more impor-tant to categor y structure than other s. As a result, some of the statistically most prevalent features of members of a category may be disregarded in for mulating the
34. Armstrong et al., “What Some Concepts Might Not Be”; Gleitman et al., “On Doubting the Concept ‘Concept’ ”; Barsalou, “The Instability of Graded Str ucture”; Lawrence W. Barsalou, “Intracon-cept Similar ity and Its Implications for Intercon“Intracon-cept Similar ity,” in Similarity and Analogical Reasoning, ed.
S.Vosniadou and A. Ortony (Cambr idge: Cambridge University Press, 1989), 76 – 121.
cate g ori zat i on 45
figure 1.3 Diagram of the categor y structure for the categor y motive forms from the opening of Beethoven’s Fifth Symphony, mm. 1– 37
prototype. For example, attributes such as natural environment and means of locomo-tion have a strong influence on how we categor ize whales, leading us to disregard such taxonomic attr ibutes as mode of respiration and means of reproduction. In conse-quence, a whale seems closer to a prototypical fish than to a prototypical mammal, despite its proper Linnean categor ization.35 Research has also shown that judgments of typicality change over time, even when no new individuals are categor ized. It appears that, rather than constituting a stable cognitive entity, the prototype for a category is actually fluid and variable and it changes with the circumstances of categorization.
An alter native view of category structure suggests that categor ies are organized around conceptual models. In brief, conceptual models consist of concepts in specified relationships; each model per tains to a specific domain of knowledge. Such models are central to reasoning and inference, and they will be discussed in g reater detail in chapter 3; in what follows, the emphasis is on the role conceptual models play in categorization, rather than on the str ucture of such models. With regard to
catego-35. For a discussion of some of the factors involved in this particular categor ization, see John Dupré,
“Are Whales Fish?” in Folkbiology, ed. Douglas L. Medin and Scott Atran (Cambr idge, Mass.: MIT Press, 1999), 461– 76.
rization, the primary function of the conceptual model is to supply a guide for rea-soning about accepted and potential member s of the categor y. This is accomplished through a simplified representation of category structure that incor porates knowl-edge about what values are most typical for a select g roup of attributes for the g iven category. These attr ibutes are selected according to the goals of categor ization, which are themselves infor med by more global conceptual models applicable to a broad range of categorization tasks.36 The conceptual model for a g iven category thus reflects infor mation of the sort summar ized in frame diag rams but conditioned by knowledge about the overall goals of categor ization.
As a means of developing a preliminary understanding of the concepts and rela-tionships constituent of a conceptual model for a categor y, let us return once more to the opening of Beethoven’s symphony and consider a conceptual model for the category motive forms from the opening of Beethoven’s Fifth Symphony. The model, dia-grammed in figure 1.4, consists of four cor related conceptual elements, character ized as the things necessar y for a melodic fragment to count as an instance of the motive:
• the rhythmic patter n Ω≈ç « h
• a statement by the full orchestra
• a fortissimo dynamic
• an intervallic patter n consisting of three repeated notes followed by a descending diatonic third37
These elements g ive priority to select attr ibutes of the motive (rhythmic patter n, orchestration, dynamics, and intervallic profile) and assign def ault values to these attributes — that is, they define what counts as a “typical” member of the categor y.
This network of attributes and values then provides a guide for categor izing the various versions of the motive that occur in the opening measures. Accordingly, ver-sions of the motive such as those of mm. 1– 2 and mm. 22 – 24 (which instantiate all aspects of the conceptual model) would be judged most typical of the category;
motive forms such as those of mm. 6 – 7 and mm. 29 – 30 (which instantiate only two aspects of the conceptual model) would be less typical; and motive forms such as those of mm. 7 – 8 and mm. 35 – 36 (which instantiate only the rhythmic aspect of the conceptual model) would be least typical of the category.
What, then, of the evidence provided by figure 1.3? That is, what can one make of the lack of correlation between the conceptual model and the individual cate-gory members, since most of these would be reckoned to be less typical members according to the conceptual model for the categor y? The answer to this question requires delving a bit deeper into the way conceptual models both reflect and shape our understanding of music.
36. Barsalou, “Deriving Categor ies to Achieve Goals”; Lawrence W. Barsalou, Wenchi Yeh, Barbara J. Luka, Karen L. Olseth, Kelly S. Mix, and Ling-Ling Wu, “Concepts and Meaning,” in Chicago Lin-guistics Society 29: Papers from the Parasession on the Correspondence of Conceptual, Semantic, and Grammatical Representations, ed. Kathar ine Beals, Gina Cooke, David Kathman, Sotaro Kita, Karl-Er ik McCullough, and David Testen (Chicago: University of Chicago, Chicago Linguistics Society, 1993), 23 – 61.
37. I should emphasize that the conceptual models I descr ibe here are by no means the simplest ele-ments of conceptual str ucture. Each conceptual cluster of the present model, for instance, could be rep-resented by a still smaller conceptual model cor relating the concepts that make up the cluster.
cate g ori zat i on 47
u, u, -3 intervallic pattern stated at fortissimo dynamic
stated with full orchestra q q q h rhythmic pattern
motive forms from the opening of Beethovens Fifth Symphony
figure 1.4 Conceptual model for the categor y motive forms from the opening of Beethoven’s Fifth Symphony
The conceptual model shown in figure 1.4 gives undeniable priority to the statements of the motive that open the movement. The preference given the first statements of musical mater ials we encounter is not absolute — there are, of course, works in which the pr incipal mater ials emerge only g radually — but it does reflect the fact that we must process musical mater ials in time and that it is often the case that significance accr ues to the first events in any psychological process. Note that Beethoven departs from his opening mater ials only gradually; in particular, diver-gences from the inter vallic patter n of the motive (the aspect of the typicality shown by fig. 1.3 that best matches the proposed conceptual model) occur somewhat later in the opening measures and cor relate with the compositional strateg ies of devel-opment and transition.38 The initial statements of the motive thus represent anchor points for the process of elaboration that Beethoven undertakes: the variations evi-dent in figure 1.3 are variations on the musical mater ials summar ized in the con-ceptual model of figure 1.4.
The conceptual model of figure 1.4 is also infor med by our ideas about musical themes: that is, such themes should be strongly and clearly stated at pr ivileged points within a movement (such as the beg inning of the movement or after a change of key or tempo), after which they may be modified or embellished in confor mance with compositional strateg ies. Our local model for the motive of Beethoven’s sym-phony is thus shaped by a global model of what constitutes an appropriate musical theme. Such global models are developed through abstraction from any number of local models, and, perhaps more important, they are taken from the broader base of knowledge that constitutes culture.39 For instance, in the case of Beethoven’s sym-phony, the relevant global model for a musical theme reflects the influence of nine-teenth-centur y German and Austrian musical culture and, in particular, the efforts
38. Relationships between processes of categor ization and compositional strategy are developed more fully in chap. 4.
39. This perspective on culture is explored more fully in chaps. 3 and 5.
of the nineteenth-centur y critic and theor ist A. B. Marx and others to come to terms with Beethoven’s compositional style.40
Our global model of what counts as a theme (which applies to any number of works by any number of composers) is thus in rappor t with the local models through which we organize our under standing of specific musical works. Without a global model, it can be difficult to decide what aspects of a phenomenon are relevant for categorization. Without local models, there is no way to anchor global models to specific phenomena.
It should be emphasized that the conceptual model sketched here and dia-grammed in figure 1.4 represents limited features of a network of interrelated propositions that might be used to guide reasoning about how motive forms in the opening of Beethoven’s symphony relate to one another. The model does not rep-resent, in a simple way, what someone “has in mind” when confronting this music.
The simplicity of the model (which reflects in par t the global models relative to which it is framed) contr ibutes to its efficiency. It is specific enough to allow for clear distinctions between the motive of the opening measures and other thematic mater ials in the symphony (such as the new theme that enters in m. 63). It is gen-eral enough, however, to accommodate the wide range of different for ms the open-ing motive takes. The model also reflects our intuition that there is just one main form of the opening motive, despite evidence to the contrar y. Intuitions such as this contribute to the plausibility of Type 2 categor ies, with their cr isp boundar ies and clear definitions. Features of the conceptual model for a g iven categor y will often be transfor med into the necessar y and sufficient conditions character istic of Type 2 categor ies: take a Type 1 category and specify limits for it by regarding the con-stituent elements of the conceptual model relative to which it is str uctured as nec-essar y and sufficient conditions for categor y membership, and you create a Type 2 category. In descr iptions of music, this process yields “textbook” definitions of the sort that often lead to considerable debate among music theor ists. Although such debates appear to be about music, they are in fact about how to define the cate-gories through which we organize our under standing of music.