The Progression Vector: Modelling Aspects of Post-Tonal Harmony
Paul Nauert · Journal of Music Theory · 2003
Post-tonal harmonic progressions can be conceived in terms of successions of pitch class sets (pcsets). The basic unit of harmonic progression, according to this conception, is the ordered pcset pair (X, Y), which represents harmony based on X followed by, or progressing to, harmony based on Y. These basic units can be chained together to form larger pcset progressions; linking the pairs (X, Y) and (Y, Z), for instance, yields a progression from X to Y to Z. The purpose of this article is to show how a construct called the progression vector, which characterizes pcset pairs intervallically, can assist in organizing pcset progressions. This construct has appeared before, in different guises, in work by Lewin (1959, 1977, 1987) and Morris (1987), as detailed in Part 1 of this article. The progression-vector applications developed in Part 2 are new, drawing on recent work by Block and Douthett (1995) and Scott and Isaacson (1998); this section lays out the main ideas of the article. Finally, Part 3 regards the progression vector, from another perspective, as the total potential voice leading between two pcsets, and considers what it is like to hear voice leading and other kinds of intervallic relationships between chords.