The paper advances a pedagogical program that models small cyclic systems before teaching the twelve-element chromatic system of atonal theory. The central properties, relations and protocols of atonal theory (complementation, inclusion, invariance, transpositional equivalence, set classification and labeling, maximal evenness) are introduced in the smallest cyclic system to which they apply. All cyclic systems of 2 to 9 elements have at least one familiar musical application, modeling beat-class (rhythmic) cycle, pentatonic and diatonic scales. By the time students have scaled up to a twelve-element universe, they are technically prepared to explore it, and to appreciate its special properties. Along the way, they have learned a model of meter, an otherwise under-theorized aspect of music pedagogy.
Cohn, Richard Lawrence (2016) Teaching Atonal and Beat-Class Theory, Modulo Small. In: MusMat: Brazilian Journal of Music and Mathematics, Vol. 1, No. 1, pp. 15-24. Available at https://openmusiclibrary.org/article/812815/.