Dmitri TYMOCZKO, Root motion, function, scale-degree: a grammar for elementary tonal harmony

The paper considers three theories of tonal harmony: root-motion theories, scale-degree theories and function theories, in order to determine whether they can produce a simple grammar of elementary tonal harmony, i. e. a simple set of principles that generates all and only the standard tonal chord progressions. The study is based on the analysis of 30 of Bach's chorale harmonizations in the major mode.
   Theories of root motion involve a principle of scale-degree symmetry, which asserts that all diatonic harmonies participate equally in the set of allowable root motions, and a principle of root-motion asymmetry, which claims that certain types of root motions are preferable to others. Neither is entirely verified. Theories of scale degrees, examined here in the form of a first-order Markov model, do a suprisingly good job of approximating the progressions of elementary diatonic harmony, much better than the pure root-motion perspective. The function view (envisaged here without consideration of its possible psychological implications) merely adds that some chords behave in similar enough ways to justify grouping them together in categories.
   The scale-degree theory therefore yields the best grammar of tonal harmony. The root-motion theory is too restrictive and the function theory overly permissive.