P. B. Ivanov
MUSICAL SCALE HIERARCHY
Proceedings of the International Congress of Humanities and Social Sciences Research, Paris 24-28 July 2012
(G. Tchibozo, ed.),
Strasbourg (France): Analytrics, 2012, pp. 131–139
L. Avdeev’s theory of musical scale hierarchy formation is discussed. The model provides an adequate mathematical description of the already known scales and reveals some other new possibilities. A measure of the information difference between two probability distributions is employed to construct a numerical estimate of a contradiction between two compound tones. The discordance function thus obtained allows to determine a musical scale. A dissonance function is introduced, which reveals the scale as a set of zones. Stationarity under nonlinear transformations and maximum regularity provide the numerical criteria for selecting the preferable scales. The formant structure of the internal timbre pattern characterizes the stability of local hierarchical structures. Harmonic, modal, and chromatic types of scale lability are described. Musical scale as a movable hierarchy of zone structures unfolds itself in various ways, forming the musical context.
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