Some Thoughts on Maximally-Smooth Voice Leading among Pcsets and Set Classes

André Brégégère, William Paterson University & Queensborough Community College, CUNY

The study of voice-leading among pitch sets and pitch-class sets has been a prominent topic of music theory in the past twenty years. A variety of approaches, focused on harmonic similarity, transformational networks, or parsimonious voice-leading, have more recently been subsumed under a geometrical model based on a mapping of pitch or pc sets onto dimensional coordinates using the semitone as a metric. These developments have led to some early attempts to establish a general typology of voice-leading sets (vlset) and voice-leading classes (vlclass), as a higher-level analog to the pc-set and set-class typologies.

My paper examines vlsets and vlclasses from a purposely narrowed perspective, limited to instances of Maximally-Smooth (MS) voice leading -- i.e., wherein motion between pcs is limited to one semitone. I show that there exist, for each cardinality, only a relatively limited number of MS-vlsets, and an even smaller number of MS-vlclasses. Focusing initially on pcsets and set-classes of cardinality two and three (including multisets), I examine the properties of MS- vlsets and vlclasses, corresponding to various types of relations (T/I, K-net isographies, constant sums), and explore the various geometrical features of the resulting voice-leading spaces. I then extend these observations to other cardinalities, and conclude with suggestions for a unified, systematic typology of MS-vlsets and vlclasses for all cardinalities.