Algorithmic Composition Paradigms of Automated Music Generation
Algorithmic composition – composing by means of formalizable methods – has a century old tradition not only in occidental music history. This is the first book to provide a detailed overview of prominent procedures of algorithmic composition in a pragmati
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8ZWL]K\4QIJQTQ\a" 3, duplications of accumulation points are increasingly produced, until finally with values for r between 3.57 and 4, chaotic behavior begins.
Fig. 6.2 Feigenbaum diagram of the logistic equation.
The best-known example of complex system behavior goes back to Lorenz who in 1963 developed a system of three coupled non-linear differential equations as a simplified model for atmospheric flow. Small changes performed on the values of the variables lead to completely different results, i.e. temporal developments of the system. This high sensitivity of these so-called deterministic chaotic systems in regard to smallest modifications in the initial conditions, are illustrated by Lorenz with the “butterfly effect”: One flap of the wings of a butterfly causes a minimal turbulence which, however, in the course of the deterministic chaotic development of the system, may lead to completely unforeseeable meteorological consequences also in very distant places. 5
Named after the American physicist Mitchell Jay Feigenbaum (born 1948).
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6 Chaos and Self-Similarity
Fig. 6.3 Edward Lorenz. With kind permission of Edward Lorenz.
6.2 Strange Attractors When observing the long-term behavior of dynamical systems, the states of the system approach particular possible solutions. In other words, the phase space of the system evolves to a comparatively small region, which is indicated by the attractor. Geometrically, simple attractors may be fixed points, such as in a pendulum, for example, which evolves towards its resting state in the lowest point of the track. Another form would be the limit cycle in which the solution space is a sequence of values that are run through periodically. These simple attractors have in common that they have an integer dimension (see below) in the phase space. The structure of so-called strange attractors reflects the behavior of chaotic systems – they cannot be described with a closed geometrical form and therefore, since they have a non-integer dimension, are fractals (see below). Well-known examples of strange attractors as a representation for the limiting values of non-linear equation systems are the H´enon attractor, the R¨ossler attractor and the Lorenz attractor (figure 6.4), whose form resembles a butterfly.
6.3 Fractals
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Fig. 6.4 Lorenz attractor.
6.3 Fractals Fractals are geometric shapes that show a high degree of self-similarity (also: scale invariance), meaning that particular graphic patterns reoccur in identical or very similar shapes on several different orders of magnitude. Fractal structures can be found in processes such as crystallization, the shape of coastlines or also in numerous manifestations of plant growth, e.g. in the
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