As part of an exploration of people's knowledge about elementary physics, Michael McCloskey and his associates designed a set of exper- iments that presented subjects with simple non-quantitative problems involving the behavior of moving objects. Here are the instructions for one such problem: The diagram [Example 1] shows a thin curved metal tube. In the diagram you are looking down on the tube. In other words, the tube is lying flat. A metal ball is put into the end of the tube indicated by

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... e arrow and is shot out of the other end of the tube at high speed. (McCloskey 1983, 300) After reading the instructions subjects were asked to draw the path the ball would follow after it emerged from the tube, ignoring air resistance and any spin the ball might have. Slightly over half of their subjects, who were college undergraduates whose background ranged from never having taken a course in physics in either high school or college to having completed at least one college physics course, gave the solution shown in Example 2a. According to Newton's first law, in the absence of a net applied force, an object in motion will travel in a straight line. Thus, from the view of classical physics the correct answer is that after the ball 193 Example 1. McCloskey, Diagram of spiral tube a b Examples 2a and 2b. McCloskey, Solutions to the spiral tube problem leaves the tube it will travel in a straight line in the direction of its instantaneous velocity at the moment it exits the tube, as shown in Example 2b. In order to explain the remarkable consistency exhibited by his subjects' incorrect responses to this and similar problems, McCloskey proposed that these responses were based on what he called a naive impetus theory. This theory rests on two fundamental assertions about motion. The first assertion is that the act of setting an object in motion imparts to the object an internal force or "impetus" that serves to maintain the motion (ibid., 306). Thus the curvilinear impetus imparted to the metal ball as it travels through the spiral tube shapes its path once it exits the tube. The second assertion is that a moving object's impetus gradually dissipates (either spontaneously or as a result of external influences), and as a consequence the object gradually slows down and comes to a stop. As a reflection of this assertion, subjects tended to show the path of the ball straightening out the farther it got from the end of the tube. McCloskey noted that this theory was highly similar to pre-Newtonian physical theories popular in the fourteenth through sixteenth centuries, vestiges of which can be seen in the work of such pioneers as Galileo Galilei (ibid., 317). McCloskey's work vividly demonstrates that theories-or what I shall later formalize as conceptual models'-are central to how we structure 194 our understanding of the world. An extensive body of recent work by cognitive scientists has suggested that cognitive structures of this sort provide essential guides to inference and reason, and are in general basic to thought. In this essay, I want to explore the role conceptual models play in theories of music. Basic to my approach is the assumption that theorizing about music is an activity specialized only in its domain, not in the cognitive processes it involves. Conceptual models are accordingly an integral part of theories of music. Our analyses of music are conditioned and constrained by the conceptual models we employ, just as our accounts of the behavior of physical objects are shaped by our theories of motion. The effect different conceptual models have on accounts of music can be seen in three analyses of the opening eight measures of the rhythm theorist's hobby horse, the theme from the first movement of Mozart's A major piano sonata, K. 331 (Example 3). Edward Cone gave his analysis of this theme in his Musical Form and Musical Performance of 1968. The analysis comes just after the introduction of a striking metaphor for the melodic and harmonic shape of a musical phrase: Cone likens the shape of a phrase to a game of catch. If I throw a ball and you catch it, the completed action must consist of three parts: the throw, the transit, and the catch. There are, so to speak, two fixed points: the initiation of the energy and the goal toward which it is directed; the time and distance between them are spanned by the moving ball. In the same way, the typical musical phrase consists of an initial downbeat (/), a period of motion (-), and a point of arrival marked by a cadential downbeat (\). (Cone 1968, 26-27) Cone uses this notion of the shape of a musical phrase in his treatment of the rhythm of Mozart's theme; his analysis of the opening period is given in Example 4. He rejects a simple alternation of strong and weak measures, an alternation typical of many rhythmic analyses, on the basis of harmony: the first and fifth measures of the period should be strong because of the firm statement of tonic; the fourth and eighth measures should also be strong because of the emphasis provided by the cadences. A consideration of the motivic structure of the period refines this analysis. The first phrase consists of two individual sequential measures followed by a two-measure unit. The shape of the two-measure unit duplicates the shape of the phrase in miniature: it consists of two half-measure units followed by a full measure. As shown in Example 4, the internal dynamic of the two-measure unit is the same as that of the complete phrase. The second phrase begins with the same sequential measures as the first, but in m. 7 the compression that prepares the closing cadence brings the rise from A to C# into prominence. Cone's hearing of this rise as a third member of the sequence prompts him to assign it the same 195 Andante grazioso (IMVir&:n t rr t J cr ; c* rM O?r X 'l _ * rr~ . I' f P Example 3. Mozart K. 331/I/theme, mm. 1-8 rhythmic symbol as the second member of the sequence, heard in m. 6. Voice-leading concerns and the forward energy of the sforzando cause Cone to group the final eighth of m. 7 with the cadential material of m. 8; this constitutes the closing downbeat. Although Leonard Meyer's analysis of Mozart's theme appeared after Cone's (first in a Bloch lecture of 1971, then in his Explaining Music of 1973), his methodology is essentially the same as that which he had developed with Grosvenor Cooper over a decade earlier. Cooper and Meyer understood rhythmic structure to be perceived as "an organic process in which smaller rhythmic motives, while possessing a shape and structure of their own, also function as integral parts of a larger rhythmic organization" (Cooper and Meyer 1960, 2). Rhythmic relationships are analyzed as patterns of beats, in which a stable accent and one or more weak beats are grouped together in different ways; these low-level, foreground patterns combine with one another in various ways to form more extended rhythmic groupings. In this way the musical surface gives rise to a hierarchy of rhythmic groupings (Meyer 1973, 27-28). Meyer's analysis of the opening period of Mozart's theme is given in Example 5. The first level of the analysis starts at the half-measure level of the music, important for Meyer because the dotted-eighth/sixteenth/ eighth patterns of the first two measures of each phrase do much to determine the grouping and accentual pattern of the complete bar. In mm. 3 and 4 Meyer's analysis on level 2 corresponds closely with Cone's reading: although the symbols are slightly different (Meyer's indication for a retrospectively weak accent replacing Cone's symbol for an initial downbeat), on this level as well as the third and fourth the analyses for the most part agree. The main difference between the two is that Cone starts his analysis at what for Meyer is the third level of a rhythmic hierarchy. In his 1978 article "The Theory and Analysis of Tonal Rhythm" Robert Morgan noted that, although both Cone's and Meyer's work reflected the