Tonal Scales and Minimal Simple Pitch Class Cycles

Abstract

Abstract. Numerous studies have explored the special mathematical properties of the diatonic set. However, much less attention has been paid to the sets associated with the other scales that play an important rôle in Western tonal music, such as the harmonic minor scale and ascending melodic minor scale. This paper focuses on the special properties of the class, T , of sets associated with the major and minor scales (including the harmonic major scale). It is observed that T is the set of pitch class sets associated with the shortest simple pitch class cycles in which every interval between consecutive pitch classes is either a major or a minor third, and at least one of each type of third appears in the cycle. Employing Rothenberg's definition of stability and propriety, T is also the union of the three most stable inversional equivalence classes of proper 7-note sets. Following Clough and Douthett's concept of maximal evenness, a method of measuring the evenness of a set is proposed and it is shown that T is also the union of the three most even 7-note inversional equivalence classes.

GeneralizedTonnetzeandZeitnetze, and the topology of music concepts

“…For example, we could treat the thirds as generators of the diatonic, constructing a scalar Tonnetz with a tertian logic. As Meredith (2011) observes, this captures what are typically taught as the standard tonal scales, including the harmonic minors and majors (4-4-3-4-3-3-3) as well as the melodic minor/acoustic (4-4-3-3-4-3-3), but not the whole-tone-plus-one. In addition, this Tonnetz includes a more exotic type of collection including a diminished second (i.e., enharmonic variant of a single pitch class), 4-4-4-3-3-3-3, e.g., C-D -E -F -G-A-B.…”