ABSTRACT
Musical structures are typically generated in a hierarchical manner. Most compositions can be divided approximately into natural segments (e.g. movements of a sonata); these are again divided into smaller units (e.g. exposition, development, and coda of a sonata movement). These can again be divided into smaller parts (e.g. melodic phrases), and so on. Different parts even at the same hierarchical level need not be disjoint. For instance, different melodic lines may overlap. Moreover, different parts are usually closely related within and across levels. A general mathematical approach to understanding the vast variety of possibilities can be obtained, for instance, by considering a hierarchy of maps defined in terms of a manifold (see e.g. Mazzola 1990a). The concept of hierarchical relationships and similarities is also related to “self-similarity” and fractals as defined in Mandelbrot (1977) (see Chapter 3). To obtain more concrete results, hierarchical regression models have been developed in the last few years (Beran and Mazzola 1999a,b, 2000, 2001).
