Modeling Diatonic, Acoustic, Hexatonic, and Octatonic Harmonies and Progressions in Two- and Three-Dimensional Pitch Spaces; or Jazz Harmony after 1960

Abstract

The paper provides methods for modeling harmonies and harmonic progressions based on diatonic, acoustic, hexatonic, and octatonic collections. It examines these four spaces in relation to jazz harmony, particularly post-1960s jazz compositions in which harmonic function is suppressed or absent. It discusses the opening progression to Wayne Shorter’s composition “Vonetta” (from the 1967 Miles Davis Quintet recording Sorcerer ), and ultimately provides a three-dimensional Tonnetz that permits moves among ninth chords, showing them as trapezoidal moves in all four spaces. The article makes use of interactive graphics to illustrate its procedures.