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Models and Music Theory: Reconsidering David Lewin’s Graph of the 3-2 Cohn Cycle

[1] In a recent issue of this journal, Jacob Reed and Matthew Bain (2007) review David Lewin's (1996, 1998) graphic network of the set-class (sc) 3-2 Cohn cycle and his analysis of J. S. Bach's F-sharp Minor Fugue, WTC I. Exploring different modeling options, Reed and Bain reformulate Lewin's "Graph" into a Tonnetz, per Cohn 1997. After a brief consideration of the Tonnetz, however, Reed and Bain reject this model and Lewin's graph as inappropriate candidates for visualizing the transformations of the instantiations of sc 3-2 in the exposition of Bach's fugue. Since these two-dimensional models exist in pitch-class (pc) space they are unequipped to solve "the problems of register, pitch-specific structure and order" (Reed and Bain 2007, [1.6]) in the unfolding fugal process. Finally, Reed and Bain present an animated three-dimensional tetrahelix as a solution to this modeling problem. Inasmuch as their tetrahelix is crafted specifically for the purpose of animating Bach's fugue, Reed and Bain's final result successfully integrates our imagination of the music progressing through pitch-space and time with a visual representation of that imagined motion.[2] The authors' comparison of Lewin's graph and their Tonnetz raises some questions concerning the relationship between music theory models and their subjects, the theoretical concepts that they illustrate. We may say generally that music theory models render observable abstract theoretical concepts that would otherwise remain in the imagination. They accomplish this by depicting abstract theoretical relations with an enduring and visible iconic design.(1) Lewin's graph and the Tonnetz are examples of iconic models; both use visual imagery to depict concretely an abstract theoretical Cohn cycle. Such iconic models pervade music theory literature, especially that of neo-Riemannian and transformational theory.[3] Reed and Bain argue that the Tonnetz representation of the 3-2 Cohn cycle, even though ultimately inadequate for their purposes, is a more satisfactory and economical model of the theory than Lewin's graph.(2) Here, I would like to consider-albeit indirectly-the larger question of the relationship between music theory models and their subjects by using as a case study Reed and Bain's arguments in favor of their Tonnetz. I suggest alternatively that Lewin's original graph is more successful than the Tonnetz in depicting the theoretical concept of a Cohn cycle as Lewin 1996 describes.(3) I shall do this by examining and answering each of Reed and Bain's criticisms of Lewin's graph and by considering the perceived solutions that the Tonnetz offers with a particular focus on how the iconic components of each model relate to the theory of Cohn cycles. Unduly privileging a personal taste for visual aesthetics over a more rigorous modeling of the theory, Reed and Bain's Tonnetz relates to its subject less clearly and directly than Lewin's graph. I begin with brief analyses of the two models.Lewin's Graph[4] Lewin's two graphic presentations of the 3-2 Cohn cycle are reproduced below. Figure 1 is the original model that appears in "Cohn Functions" (Lewin 1996); Figure 2 is an expanded version of the same model that Lewin uses for his analysis in "Notes on the Opening of the FMinor Fugue from WTCI" (Lewin 1998). The only difference between the two versions other than their scope is Lewin's legend of the connections between the pcsets underneath the graph in Figure 1.(4)Figure 1. Lewin 1996Figure 2. Lewin 1998Lewin's graph is a network of trichordal nodes interconnected by vertical and diagonal axes.(5) Each node represents a particular unordered pc instantiation of sc 3-2; each axis corresponds to one of the three possible Cohn flips of the Cohn function derived from the successive-interval array (sia) of sc 3-2, (129).(6) Therefore, every node is connected to three others by one of the three possible Cohn flips of this Cohn function. …