Voice-Leading Transformation and Generative Theories of Tonal Structure

[0.1.1] As the theories of Heinrich Schenker were gaining currency in the United States in the mid-twentieth century, Milton Babbitt observed that Schenker's theory of voice-leading levels was "strikingly similar to transformational grammars in linguistics" (1965, 60). Babbitt's appraisal of Schenker's theory inspired the next generation of music theorists, who produced a multiplicity of generative approaches. Many of these were intended as formalizations of Schenker's method, such as Kassler (1967, 1975), Rahn (1979), and Smoliar (1980). Others were expressly Schenker-inspired, including Keiler (1977), Lerdahl and Jackendoff (1977, 1983), and Westergaard (1975). The generative principle remains a significant feature of how many theorists conceive of Schenkerian analysis, as exemplified by Brown (2005) and Neumeyer (2009).[0.1.2] Lerdahl and Jackendoff's Generative Theory of Tonal Music (1983) represents a high-water mark for the first wave of Schenker-inspired generative theories. The widely felt influence of their approach can be attributed in large part to two significant features: First, in observance of the standards of contemporary social science, Lerdahl and Jackendoff strive to be as precise and explicit as possible in the presentation of their theory. This has made their work especially valuable for the empirical study of tonal music,(1) and also led them to make distinctions that might have escaped a looser, more intuitive approach, many of which have shaped subsequent discourse in traditional music theory.(2) Second, Lerdahl and Jackendoff, like other authors cited above, present their work as a theory of tonality, unlike Schenker's own work, which, as Pastille (1985), Rothstein (1990), Snarrenberg (1997), Cook (2007), and others have pointed out, is about the interpretation of masterworks. This creative misprision of Schenker has a history as old as the Americanization of Schenker itself, and is what allowed Adele Katz (1945) and Felix Salzer (1952), for instance, as well as many others after them, to redirect the theory away from Schenker's narrow conservative music-historical agenda and draw larger implications from it. Temperley (2011) has advocated reformulating Schenkerian principles as a body of theoretical tenets that can, at least in principle, make empirically verifiable claims.(3)[0.1.3] This article is organized in four parts. Part 1 reviews generative theories that have been proposed, with special focus on the status of counterpoint, i.e. the structural separability of voices, within them. Parts 2 and 3 operationalize concepts of hierarchy and counterpoint taken from Schenker as an outline of how a generative system derived from his theory might look. Part 2 shows that the hierarchical syntax of Schenkerian symbols is different than the one assumed by most generative approaches. Part 3 discusses the coordination of hierarchy with counterpoint, unpacking a concept of voice-leading transformation that is fundamental to Schenker's approach. Voice-leading transformations cannot be accommodated by a purely generative system, in loose analogy to the transformational grammar of Chomskian linguistics (the parallel uses of the term "transformation" being, uncannily, fortuitous). Part 4 is an analytical application of the proposed generative system to the Menuetto of Beethoven's Piano Sonata op. 22, and shows that an explicitly worked out theory of voice-leading levels is needed to appreciate exceptional aspects of the work's tonal structure.[0.1.4] The purpose of this exercise is not only to further the development of systematic approaches to describing tonal structure, but also to inform the practice(s) of Schenkerian analysis and even the understanding of Schenker's theory itself. This effort should, therefore, be of interest to a wide spectrum of music theorists interested in tonal structure, not only those working from a mathematical or computational perspective. …