New Approaches to Tonal Theory

Two new and keenly anticipated books by young American theorists, Steven Rings’s Tonality and Transformation¹ and Dmitri Tymoczko’s A Geometry of Music: Harmony and Counterpoint in the Extended Common Practice,² re-examine the fundamental tenets of how we think about tonal music. Reading these books in close succession, what is most striking is that although they seem to depart from a similar premiss—the rethinking of tonal theory from a mathematical perspective—their emphases are decidedly different, revealing significant philosophical divergences in their approaches to theory and analysis.

Each author wrestles in his own way with the legacy of David Lewin, arguably the most influential theorist of the last thirty years. Lewin’s research, particularly his magnum opus Generalized Musical Intervals and Transformations (New Haven, 1987; repr. Oxford and New York, 2007), has led to a variety of mathematical formalizations and analytical tools, including ‘neo-Riemannian’ theory, which reframes chromatic progressions of triads as transformations in a mathematical space based on Hugo Riemann’s Schritte and Wechsel. Rings’s Tonality and Transformation is explicitly framed as continuation of Lewin’s project both from a technical and a methodological angle, particularly in the focus on analytical close reading and the use of a variety of transformations to model different ‘hearings’ of a passage. And though Tymoczko has been critical of several aspects of Lewin’s theories, his ‘geometry of music’ is unthinkable without the precursor of neo-Riemannian theory, which similarly emphasizes close voice-leading connections between chords.

The two authors draw on different branches of mathematics: Tymoczko on geometry, and Rings (like Lewin) on group theory, a branch of abstract algebra. One of the main differences between the approaches is that Rings’s groups of operations relate discrete points, while Tymoczko’s geometries describe a continuous space. Intriguingly, this same opposition was essential to Boethius’ classification of the mathematical arts of the quadrivium: while music and arithmetic dealt with discrete quantities (multitudes), geometry and astronomy dealt with continuous quantities (magnitudes). The apparently abstract and mathematical choice between geometry and group theory brings with it tendencies towards specific philosophical and methodological stances.

Rings notes that the choice of a transformational viewpoint encourages a pluralist, ‘prismatic’ approach, ‘in which phenomenologically rich local passages are refracted and explored from multiple perspectives’ (p. 38). Any single conception of a musical interval is insufficient to model the many different, contextually dependent ways it may be experienced; the variety of these experiences can begin to be expressed, however, by combining the many interval systems offered by group theory. Rings illustrates such ‘apperceptive multiplicity’ by demonstrating different ways of conceptualizing a single major tenth from the Prelude of Bach’s Cello Suite in G, BWV 1007: as a span in a diatonic scale, a traversal of four steps in a tonic arpeggio, a skip between overtones of a Rameauian fundamental bass, and so on. The idea that an interval is not a single thing but rather a multitude of intervallic experiences in different conceptual spaces focuses our attention on the heterogeneity of experience, not the immanent properties of ‘the music itself’. Rings avoids the temptation (as described by Henri Bergson) of flattening this heterogeneity into a single, homogenous space. This concept is one of Rings’s many inheritances from Lewin: ‘we do not really have one intuition of something called “musical space”. Instead, we intuit several or many musical spaces at once.’³

As his title implies, Tymoczko’s book is heavily vested in the geometrical modelling of musical structures. Tymoczko makes the argument that the essential property of musical spaces is their ability consistently and quantitatively to measure the voice-leading distance between any two chords. Such measurements do not engage the range of subjective experience that Rings seeks to model, but take a more immanent and objective approach to musical events: Tymoczko professes that he is ‘primarily concerned with what composers do, rather than what listeners hear’ (p. 8). Instead of the many co-existing, ‘prismatic’ intervallic spaces proposed by Rings, Tymoczko focuses on the single class of spaces that meets his criteria for consistency in measurement: metric spaces in various...