Parsimonious Graphs for Selected Heptatonic and Pentatonic Scales

Abstract

[EN] Tonal music is based on major, melodic and harmonic minor scales. In some cases, the harmonic major scale is also used. In this paper, four additional heptatonic scale types, derived from them, are considered. The harmonic characteristics of these eight scale types are analyzed by the trichord- and tetrachord-type vectors, which list, respectively, the number of times each trichord and tetrachord type is contained in a set type. Then, a novel parsimonious graph is provided, called 7-Cyclops, which relate those scales by single-semitonal transformations. On the other hand, their complements are eight pentatonic scales, whose harmonic characteristics are also analyzed and the corresponding parsimonious graph, called 5-Cyclops, is given. These graphs highlight the cycles of fifths and fourths, which are the only possible circumferences linking the same scale types in these graphs. Other parsimonious transformations, like moving one note by a whole tone, are easily found in these graphs, too. The acoustical relationship between those heptatonic and pentatonic scale types is analyzed by the pentachord-type vector, which lists the number of times each pentachord type is contained in a set type. With the inclusion of a musical example, all this information is intended both for theorists and composers.