Scale Perception
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L. V. Avdeev and P. B. Ivanov

A Mathematical Model of Scale Perception

Journal of Moscow Physical Society, v. 3, pp. 331–353 (1993).
Printed in the UK


A hierarchical approach to scale formation in human perception is applied to musical scales. The model provides an adequate mathematical description of the already known scales and reveals some other new possibilities, in particular, a universal 19-tone musical system.

A formula for the information difference between two probability distributions is employed to construct a numerical estimate of a contradiction between two compound tones. The discordance function obtained in this way possesses a number of minima which correspond to the degrees of 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 historical development from simple to ever more complex scales is thus traced. 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. The analysis of the discordance functions indicates the ways of releasing the tensions and shows the musical consequences of any melodic move.


1. Introduction and general outline
2. The informational measure of a change in the distribution
3. The discordance value for a pair of partials
4. The discordance function for compound tones
5. A criterion of timbre stationarity
6. The discordance Fourier spectrum. Optimal timbres
7. Formation of scale zones
8. Timbres and scales
9. Scale hierarchy and musical context

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Also see: [Musical Scales] [Scales in the Visual Arts] [Musical Scale Hierarchy]