Review of Dmitri Tymoczko, A Geometry of Music: Harmony and Counterpoint in the Extended Common Practice (Oxford University Press, 2011)

[1] Near the beginning of his book Style and Music, Leonard Meyer repeats a parable told by the American philosopher John Searle, describing what might happen when an intelligent observer who knows nothing of American football watches a game for the first time.(2) To paraphrase somewhat, the observer watches a quarter or two of the game and eventually declares, "I have figured out football. Football consists of a statistically regular alternation of circular clustering, linear clustering, and random interpenetration."[2] The observer's description of football is not wrong: the players on the field do arrange themselves in statistically predictable ways as described. Clearly, though, this explanation misses the point of football in important ways. The statistical phenomena it details are nothing more than incidental byproducts of the things that football is really about: the objectives, the rules, and the strategies of the game, about all of which our observer remains blissfully clueless. In Searle's terms, the observer has noted some superficial brute facts, but has failed to account for the deeper institutional facts, which create constraints that shape the brute facts in essential ways.[3] In my more cynical moments, I find myself suspecting (and this is Meyer's point as well) that the understanding of music displayed in much of what is said and written on the subject is ultimately not much better than this observer's understanding of football. We make many observations about music, and while these observations are usually correct in some sense, they often seem to miss important points: we observe brute facts while remaining oblivious to more fundamental principles that underlie, motivate, and explain those facts.[4] Dmitri Tymoczko evidently harbors such suspicions as well, perhaps more often and more strongly than I do. While there are some brute facts in Tymoczko's new book A Geometry of Music (henceforth AGOM), the book is set apart by its aspirations to unearth deeper institutional facts that others have overlooked. Tymoczko is unafraid to question conventional wisdom, spurn the beaten path, and seek new viewpoints on familiar landmarks, and AGOM does not lack for bold claims with far-reaching consequences. It promises nothing less than a rational foundation for tonality; a set of common principles that unify the music of not just one or two centuries but broad swaths of music history spanning the better part of a millennium; new perspectives on (among other things) scales, modulation, functional harmony, chromaticism, and the place of jazz in music history; and a powerful, overarching geometric model that ties it all together.[5] The reader who suspects that a book of 450 pages cannot possibly deliver on such a sweeping array of promises may be surprised by the remarkable extent to which it succeeds. The geometric model that takes pride of place in the title is a monumental achievement in its own right, and is brought to bear revealingly on fundamental aspects of harmony, voice leading, consonance, tonality, and style, aptly illustrated through hundreds of figures and musical examples in a wide range of styles. The upshot of it all is a book that occasionally vexes but more often illuminates, inspires, and exhilarates: a treasury of big ideas, by any reasonable standard a major treatise for our field.[6] The essential features of the geometric model are developed in the third of AGOM's ten chapters, "A Geometry of Chords." The chord spaces described here have variously been dubbed "CQT spaces" (after the groundbreaking article by Callender, Quinn, and Tymoczko, published in Science in 2008, expanding upon Tymoczko 2006), "OPTIC spaces" (after the five kinds of symmetries the theory admits, defined by equivalence under octave shifts, permutation of notes within a collection, transposition, inversion, and cardinality change), and "orbifold spaces" (after the mathematical name for the topological structures formed by these spaces, which in general are non-Euclidean and include singularities of various kinds). …