Abstract
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.
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Notes
- 1.
A transformation of a pitch-class set (in our case, a scale) consisting in raising or lowering one pitch-class (note) by a semitone, while sustaining the rest of them.
- 2.
The vector listing the number of times each of the 6 dyads (intervals from 1 to 6 semitones) is contained in a given set type or set class (in our case, scale type or scale class). It characterizes, to a great extent, the sonority of a set class. In [1], it was called interval vector.
- 3.
The term tetrachord also means “4-note chord”. However, throughout this paper, its right meaning will easily be determined by the context.
- 4.
The intervallic form is the sequence of intervals, in semitones, between every two adjacent pitch classes in a set type (in our case, a scale type), including the interval between the last and the first ones, or any of its circular shifts. If it starts from a scale tonic, then it matches the “intervallic structure” previously used in this section.
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Nuño, L. (2022). Parsimonious Graphs for Selected Heptatonic and Pentatonic Scales. In: Montiel, M., Agustín-Aquino, O.A., Gómez, F., Kastine, J., Lluis-Puebla, E., Milam, B. (eds) Mathematics and Computation in Music. MCM 2022. Lecture Notes in Computer Science(), vol 13267. Springer, Cham. https://doi.org/10.1007/978-3-031-07015-0_3
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