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

Claims

1. extended-common-practice
   In A Geometry of Music (2011), Tymoczko identifies five features characteristic of tonality in a broad sense, including harmonic consistency and smooth voice leading. He proposes that in various combinations they characterize an extended common practice stretching from eleventh-century polyphony to contemporary jazz. Of the five, centricity is the one that most sharply separates the eighteenth- and nineteenth-century common practice from the rest of this extended practice.

   "As early as page 4 of AGOM, he identifies five features characteristic of “tonality” taken broadly. (These should be relatively uncontroversial; harmonic consistency and smooth voice leading are two of them.) He revisits these features throughout the book, describing how their interactions are constrained by geometry, and postulating that in various manifestations and combinations they characterize an “extended common practice” that stretches from eleventh-century polyphony through contemporary jazz ... Significantly, of the five features of tonality, the one that most sharply discriminates the traditional “common practice” of the eighteenth and nineteenth centuries from the remainder of Tymoczko's “extended common practice” is centricity"

   Locator: paragraph: 12 and note 8 · Quote language: en
2. optic-spaces
   The chord spaces developed by Callender, Quinn and Tymoczko, also called OPTIC or orbifold spaces, situate chords in a continuous space according to equivalences under octave shift, permutation, transposition, inversion and cardinality change. A voice leading is a path through such a space, and the distance between two chords reflects how smoothly they can be connected. The model draws together set theory, scale theory, neo-Riemannian theory and earlier work on voice leading.

   "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” ... a voice leading is a path through an appropriately chosen OPTIC space, and the distance between any two chords in that space reflects the potential to connect them via smooth voice leading. As such, the OPTIC spaces use sophisticated techniques from contemporary mathematics to pull together many threads from musical scholarship of the last half-century, including set theory, scale theory, neo-Riemannian theory, and an assortment of work on voice leading and geometric representations."

   Locator: paragraph: 6 · Quote language: en