Sunday, May 10, 2009

music: complex shapes and memory stores?

Leyton | The book A Generative Theory of Shape (Michael Leyton, Springer-Verlag, 2001) develops new foundations to geometry in which shape is equivalent to memory storage. With respect to this, the argument is given that art-works are maximal memory stores. The present paper reviews some of the basic principles concerning our claim that, in particular, musical works are maximal memory stores. The argument is that maximizing memory storage explains the structure of musical works.

We first review the basic geometric theory of the book: A generative theory of shape is developed that has two properties regarded as fundamental to intelligence – maximization of transfer and maximization of recoverability. Aesthetic structuration is taken to be equivalent to intelligence. Thus aesthetics is brought into the very foundations of the new theory of geometry. A mathematical theory of transfer and recoverability is developed, using symmetry-breaking wreath products.

From this, it becomes possible to develop a theory of musical composition, as follows:

Musical works are complex shapes.

A theory of complex-shape generation is presented, in which any structure is described as unfolded from a maximally collapsed version of that structure, called an alignment kernel. This process is formalized by proposing a new class of groups called unfolding groups. The alignment kernel is a subgroup of that structure, consisting of symmetry ground-states which are themselves formalized by a new class of groups called iso-regular groups. In music, the iso-regular groups represent the anticipation hierarchies, for example the regular meters of the work. The process of musical composition is then described by an unfolding group, which "unfolds" the work, by successively breaking the iso-regular groups of the alignment kernel.