Tonality and Transformation (review)

In the midst of a burgeoning period of mathematic theories of music, Steven Rings’s Tonality and Transformation appears as an expansion of his eponymous dissertation (Yale University, 2006). Rings’s writing separates itself from its contemporaries in two ways: first, by cogently summarizing and glossing the formalism of David Lewin’s landmark publication, Generalized Musical Intervals and Transformations (New Haven: Yale University Press, 1987; reprinted and revised, Oxford: Oxford University Press, 2007), hereafter GMIT; and second, by implementing Lewinian Generalized Interval Systems (GISes) and transformational models to create a fresh analytical perspective on tonal (i.e., common-practice) music. In “assert[ing] that transformational theory may be used to illuminate certain specifically tonal aspects of tonal music” (p. 2), Rings simultaneously engages with the philosophical underpinnings of tonality in general and reexamines the role of traditional methodologies (e.g., Schenkerian theory) from a postmodern perspective in particular. More over, Rings’s comprehensible prose and his capacity to amalgamate recent and centuries-old theories of music contribute to this compelling and accessible work of scholarship.

The most gratifying portion for those unaccustomed to mathematical group theory resides in Rings’s lucid explanation of the seeds Lewin planted in GMIT. Chapter 1 (“Intervals, Transformations, and Tonal Analysis”) focuses on transmitting foundational Lewinian concepts (e.g., GIS) and the definitions associated with them (e.g., int(s, t) = i) in a manner that considerably mitigates Lewin’s intense formal language. Rings strategically uses tonal musical examples by J. S. Bach and Franz Schubert in order to demonstrate the concepts of GIS and transformation networks respectively. He also reinterprets Lewin’s frequent use of “intuitions” (throughout GMIT), which tend to carry inherent conceptual and cultural baggage that can lead to broad, global statements. Instead, Rings prefers “apperceptions,” which liberate the potential for new musical experiences, or adjust prior ones, through analysis (p. 18). Some readers may view this unorthodox look at tonal music through a transformational lens as fundamentally separate from the organic claims of Schenkerian theory. In Rings’s inclusive view, though, these differences are unproblematic and advantageous. The transformational approach seeks to illuminate “analytic” description of local tonal passages from a phenomenological perspective, while Schenkerian analyses provide a “synthetic” reading (p. 38). Rings’s embrace of both methodologies for different purposes “redirect[s] intellectual energy from polemic back to the business of generating insights into musical experience” (p. 40). Readers who follow this post-modern attitude will appreciate the interaction of Schenkerian and transformational methods at the end of chapter 3 (pp. 144–48) and in the Mozart and Brahms analyses (chaps. 5–7).

While the first chapter effectively presents Rings’s theory and methodology in miniature, the denser second (“A Tonal GIS”) and third (“Oriented Networks”) chapters formally define his unique transformational apparatuses. Although these two chapters are technically intense, Rings clarifies his concepts with an abundance of concrete music examples. Occasionally, he interjects succinct analytic vignettes that incorporate his analytical concepts in length-ier passages of music. Furthermore, at the risk of showing excessive connections between each object, Rings restricts most of his tonal GIS examples to instances of modulation and chromaticism.

Following Lewin’s original conception of GISes as closed interval systems applicable to a variety of specific musical contexts, Rings’s first original contribution, the “tonal GIS,” aims to model intervallic experiences within tonal music. The notions of tonal “qualia” and “chroma” lie at the heart of the tonal GIS, which Rings represents as ordered pairs of scale degrees and pitch classes (sd, pc) respectively. A two-dimensional grid of eighty-four cells, which charts the seven diatonic scale degrees and twelve pitch classes, visually renders a closed tonal GIS (p. 45). One advantage of this grid is its ability to distinguish between two en-harmonic intervals according to their contextual scale degrees. A similar grid (IVLS) represents the group structure of pitch-class and scale-degree intervals, which “allows us to model the familiar tonal-theoretic...