Scope, Method, and Aim of Scale Theory

Researchers have, in the past several decades, used formal approaches to diatonic theory in an attempt to show why the features of certain pitch collections have had such appeal for composers. The results relate either to what musicians have discovered they can do with a given collection—through moves, routines, or processes within the collection, or through manipulation of the collection itself—or to how a given collection functions cognitively, based upon measures of symmetry versus asymmetry, simplicity versus complexity, or information versus redundancy.

Investigations of the first type use transformational theory and analysis. For example, harmonic triads, and the usual pentatonic and diatonic sets, all participate in maximally smooth cycles, the starting-point for neo-Riemannian analysis. Triads in an octatonic setting, or dominant and half-diminished seventh chords in non-maximally smooth settings, also suggest a neo-Riemannian approach; certain pitch-class sets in an atonal setting allow for analogous procedures using transformational analysis.

Investigations of the second type tend to be too general to apply to analysis, but help explain the popularity of certain systems—always within particular cultural contexts, be it understood. Such explanations are by no means the exclusive province of self-described diatonicists and other scale theorists. Carl Dahlhaus offers a diatonicist distinction between Guido's hexachord, on the one hand, and the usual pentatonic and diatonic, on the other: the latter are “systems,” the former “is not, in contrast to the heptatonic and pentatonic scales, a self-significant system of tones,” but is “a mere auxiliary construction.” What Dahlhaus means is that in the pentatonic and diatonic one always understands which intervals are steps and leaps, relative to the system: “minor thirds” in the pentatonic are always “steps,” while in the diatonic they are always “leaps,” as opposed to the situation in the hexachord, where “the listener would have to alternate between … the idea of the minor third as a ‘step’ and as a ‘leap’.” He labels this an “absurd consequence.” The vehemence of Dahlhaus's language here must be understood within the context of a particular historical/theoretical discussion, on the development of the tonal system. But the implicit assumption, that a simpler relationship between intervals and their description in terms of scale steps provides for smoother cognitive processing, has more general applications.