century, but whose origins can be traced back to Euclid’s fundamental concern with congruence. Euclid’s congruence program was later generalized as Klein’s invariants program, and the latter, in turn, became the basis of 20th century geometry and physics - for example, Cartan’s moving-frame classification of curves and surfaces, Einstein’s special and general principles of relativity, Wigner’s method of classifying quantum-mechanical particles, etc.

Our book argues that the invariants program is fundamentally destructive of the needs of modern computation. The reason is that invariants are those aspects that are memoryless with respect to applied action: i.e., no action can be recovered from an object which it leaves invariant, i.e., from a geometric object in Klein’s definition. Since computing systems are required to increase memory, the invariants program defeats this fundamental purpose. A related way of saying this is that invariants defeat generativity, which is basic to computation; i.e., an object that is invariant under a generative operation will not alter under the action of the operation, and thus negate the purpose of the operation.

In contrast, the theory of geometry developed in our book is concerned with the needs of computation. For this, we elaborate a theory in which geometric objects store the effects of actions, i.e., act as memory stores for action. In fact, our basic claim is this:

This directly opposes the Klein program. As an example, consider the shape of the human body. There is very little that is congruent or invariant between the developed body and the original spherical egg from which it arose. Thus Euclid’s or Klein’s program has almost nothing to say about this situation. However, notice that, from the developed body, one can recover a considerable amount of the history of embryological development and subsequent growth, that the body underwent. Therefore, it is much more valuable to argue that the shape of the body is equivalent to the history that it underwent, instead of invariants that should have survived in the situation. In fact, observe that it is the absence of invariants that allows the history to be recovered. Thus we argue that geometry should not be the study of invariants, or equivalently, memoryless-ness; but rather it should be the study of memory storage. The book shows that this new theory of geometry allows one to develop powerful analyses of perception, robotics, and design.

This paper presents the theory of music that arises from the book. The purpose of the paper is to show that music is a particular example of the theory of geometry as memory storage.

2 Two Basic Requirements

Let us begin by looking at the general theory of geometry, developed in the book. The basic argument is that one substantially increases the power of geometry by establishing a generative theory of shape founded on the following two criteria which we regard as fundamental to intelligent and insightful behavior: