Tonal Communication

Abstract

A heptad (Sect. 14.1) is a set of seven diatonic and first-order chromatic degrees conceived as a subset of a modal cluster (represented as a set of degrees). A heptad satisfies two properties. In a “generic” sense, it contains exactly one representative of each degree; moreover, the third degree of a triadic heptad is diatonic. Relative to a given mode, there exist exactly 32 dyadic heptads, and 16 triadic heptads. In particular, there are 16 Ionian (major) and 16 Aeolian (minor) triadic heptads. Two heptads are “type equivalent” if, in a specific sense, one can be represented as a cyclic permutation of the other. A “tonal” heptad (Sect. 14.2) is a member of a pair of triadic heptads, one major and the other minor, such that, among other properties, the heptads are not type equivalent. It is shown that a tonal heptad is either “natural major” or harmonic minor. Moreover, a modal communication system that employs “tonalities,” that is, keys, the scores of which are representable as tonal heptads, is a context-free system in the sense that no contextual cues are needed for the receiver to judge the image of the transmitted final as privileged.