COMPLEXITY OF HIERARCHICALLY ORGANIZED SYSTEMS AND THE STRUCTURE OF MUSICAL EXPERIENCES

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Abstract
Many systems can be described much more simply with a hierarchical organization than without: if an identifiable subsystem occurs several times, it is easier to name it and then refer to its instances by name, rather than to repeat their description. This means appropriate measures of hierarchical complexity will have smaller values than corresponding linear measures. A key point for music, as well as many other applications, is to use a measure of complexity—that is, of information—in the general style of Kolmogorov, rather than the more restrictive type of statistical measure in the style of Shannon; this permits a more successful account of musical form and regularity. However, we find it more convenient to use general systems (here hierarchically structured) as a basis, than to use Turing machines as Kolmogorov did, or some other automata “Hearing” a piece of music is a cognitive as well as a physiological process, and what is.“heard.” depends on the conceptual systems which the listener brings to bear on his incoming perceptual stream. This paper presents the view that an understanding of a piece of music is a hierarchical analysis of it into simpler components, eventually into.“already understood.” basic subsystems. It follows that some aesthetic properties of the piece should be reflected in complexity measures of the analysis. The sequential character of music suggests a.“conditional.” complexity (of what is heard now given what preceded) may be useful in considering frustration and fulfillment of expectations, for example. Our approach is able to cope with indeterminacy in systems, in the forms of: multiple possibilities, any one of which is acceptable (non-determinism); robust vague instructions, interpreted appropriately at execution time (fuzziness); as well as (the more traditional) randomness.

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