Introduction

John Clough's contributions have tremendous potential for the teaching of music theory at all levels. He used a rigorous style more inviting to professional music theorists than to students, but his work can be relevant and interesting even to beginners. Beginning students need some help, however—not because the work of Clough and his collaborators lacks clarity, but because the students lack the necessary background.

This chapter introduces several of Clough's ideas to beginning students in music theory. Some of my prior work suggests methods and materials for introducing certain aspects of diatonic theory to beginning music students. My textbook and accompanying Instructor Resources focus primarily on three essential features of the diatonic collection—cardinality equals variety (CV), structure implies multiplicity (SM), and maximal evenness (ME)—identified in two articles, one by Clough and Gerald Myerson and another by Clough and Jack Douthett. Building on this prior work, here I suggest pedagogical approaches for concepts developed in two more recent essays by John Clough, as discussed below.

Clough, Nora Engebretsen, and Jonathan Kochavi examine earlier scholars’ work in diatonic theory to reveal important interrelationships. Apparently disparate ways of identifying and understanding scales—and more generally sets (or collections, the term I prefer)—turn out to be related if one looks at interval cycles and other associated constructs—including CV, SM, and ME.

Clough's chapter in this book, “Flip-Flop Circles and their Groups,” also uses interval cycles, and ultimately the diatonic collection, to explore neo-Riemannian theory. Although Clough's two essays seem to deal with quite different topics, they are related in both their cyclic approaches and their diatonic goals. My aim here is to shape Clough's musical and mathematical approaches into a form suitable for beginning students, without trivializing these approaches. Although I am assuming the role of an interpreter here, most students will need another interpreter, their teacher, to guide them through the activities I suggest and to discuss the various outcomes. Although the ultimate test of this chapter's ideas is whether they succeed with students, teachers are my immediate audience, especially teachers of introductory music theory or music fundamentals.

Features and Interval Cycles

The principal aim of Clough, Engebretsen, and Kochavi is to develop a taxonomy of scales (specifically) or pitch-class sets (generally)—or, more familiarly, collections—based on certain features they possess as related to interval cycles.