Furthermore, imposing the condition of maximization of transfer demands that the structure be actually of the form .

10 Algebraic Theory of Inheritance

Our theory of music is inherently object-oriented, in the sense of object-oriented programming. Indeed we argue that object-oriented inheritance is a fundamental part of musical perception and composition. A central component of our generative theory of shape is a mathematical theory of inheritance, which will now be described, and which becomes essential to every part of the music theory.

The term inheritance, in object-oriented programming, refers to the passing of properties from a parent to a child. The child incorporates these parent properties, but also adds its own. This kind of structure covers two types of situation. The first is class inheritance, which is a static software concept, and the second is a type of dynamic linking created at run-time. The book gives an algebraic theory of both types of inheritance, in the geometrical domain; i.e., related to shape. Notice that, since our theory of shape is generative, spatial movement and deformation are understood as part of the specification of shape. Thus the command operations in shape classes are understood as part of the specification of higher order shapes (e.g., configurations). Notice that, in shape classes, the command operations - which include spatial movements and deformations - form groups.

In this paper, we will have time to deal with only the dynamic type of inheritance created at run-time. This is fundamental to all computer-aided design, assembly, robotics, animation, etc. A typical example is a child object inheriting the transform of a parent object, and adding its own. It is instructive for the music theorist to consider the following example in architectural CAD: Here a door is defined as a child of a wall, and moves with the wall if the designer decides to change the position of the wall. However, the door can also open and close with respect to its attached position in the wall. This means that the door inherits the movement of the wall, but adds its personal movement with respect to the latter. Clearly, examples of this type are profuse in music. For instance, we shall later show that modulation has exactly this structure. Now for the basic statement of our algebraic theory of inheritance:

Algebraic Theory of Inheritance. Inheritance arises from a wreath product:

This can be illustrated by returning to the door/wall example. Let us suppose that the command group of motions for the wall is the Euclidean group on the plane (i.e., the base-line of the wall can be translated and/or rotated within the plane of the floor plan - which is a typical operation in architectural CAD). Let