Maximal Evenness as Conceptual Apparatus for a Course on Post-Tonal Theory and Analysis

Adam Ricci

Maximal Evenness as Conceptual Apparatus for a Course on Post-Tonal Theory and Analysis by Adam RicciIn "Music Theory's New Pedagogability," Richard Cohn observes that as recently as 25 years ago "the boundary between research and teaching at the introductory levels seemed inevitable and unbreachable...[and that]...we are at a somewhat different juncture now. A number of central concepts have emerged...that can be taught at the introductory level" (Cohn 1998, [3]). Among the central concepts that he mentions is Clough and Douthett's concept of maximal evenness. A maximally even set "is a set whose elements are distributed as evenly as possible around the chromatic circle" (Clough and Douthett 1991, 96). Each maximally even set is labeled ME(c,d), where c is the cardinality of the universe - in the case of the chromatic scale, 12 - nd d is the cardinality of the set contained within it. Timothy Johnson's textbook Foundations of Diatonic Theory, as if in answer to Cohn's call, puts maximal evenness front and center; indeed, Johnson himself acknowledges the connection in his Preface. As to the intended audience, the "material in this text was originally designed for use as a supplement in traditional Theory I courses, equally appropriate for courses in the fundamentals of music...and for stand-alone courses [in]...mathematics and music" (Johnson 2003, vii). Maximal evenness can also profitably be used as a central concept for a course on twentieth-century music theory and analysis. In such a course, typically the final one in the undergraduate curriculum, students possess substantial music theory knowledge, and therefore maximal evenness can serve synthetic ends, as a means of consolidating (and expanding) what students already know. Such synthetic ends are appropriate in a course that often constitutes the conclusion of a student's formal training in music theory.This paper will outline such a course, based on one that I teach at my own institution. In this course, instead of introducing theoretical concepts using musical examples, a typical strategy in required music theory courses, the focus is more on music theory - on exploring the underlying structure of the objects students have been studying and tasks students have been carrying out for the previous two years. What I offer here is a new way to organize the final course in the undergraduate theory curriculum, a way that suggests a different ordering and combination of topics - and manner of talking about topics - than that offered by current approaches. Central to the course is the concept of maximal evenness, but woven in are closely related ideas from neo-Riemannian theory and elementary combinatorial theory.