harmony-counterpoint · tymoczko2006-geometry-chords

Western music lies at the intersection of two seemingly independent disciplines: harmony and counterpoint. Harmony delimits the acceptable chords (simultaneously occurring notes) and chord sequences. Counterpoint (or voice leading) is the technique of connecting the individual notes in a series of chords so as to form simultaneous melodies. Chords are usually connected so that these lines (or voices) move independently (not all in the same direction by the same amount), efficiently (by short distances), and without voice crossings (along nonintersecting paths) (Fig. 1, A to C). These features facilitate musical performance, engage explicit aesthetic norms (1, 2), and enable listeners to distinguish multiple simultaneous melodies (3).