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this section, the theory will be illustrated using the visual domain, because, we will argue, in the next section, that there are deep abstract relationships between the visual domain and musical composition. This particularly concerns a profound correspondence between the iso-regular groups of the visual and musical domains. Consider the main problem for establishing a generative theory of complex structure. According to section 14, recoverability is possible only if the generative operations are symmetry-breaking. But this means that, as one proceeds forward in the generative sequence, the symmetry group of the structure quickly reduces to nothing. This means that there is a loss of algebriac information, which means a loss of generativity. This problem will now be solved, using the theory of symmetry-breaking of section 15. What we will do here is develop a symmetry group for a complex environment. This will be a powerful structure because it will contain all the required information for usability, navigation, manipulation, etc. The theory will become fundamental to the theory of musical composition presented in the next section. It is necessary to solve the fundamental problem of concatenation. Consider figure 15. Each of the two objects individually has a high-degree of symmetry. However, the combined structure, shown, looses much of this symmetry; i.e., causes a severe reduction in symmetry group. We want to develop a group theory that encodes exactly what the eye can see. In particular, in the combined situation, one can still see the individual objects. Therefore, we want to develop a symmetry group of the concatenated structure in which the symmetry groups of the individual objects are preserved, and yet there is the extra information of concatenation.
The solution to be proposed is this: The generative history starts out with the two independent objects, and therefore the symmetry of this starting situation is given thus:  which is the direct product of the groups of the two independent objects. The reader should carefully notice the following: The direct product symbol here should not be regarded as representing a direct product between fibers, as previously. It will be within a single fiber. Now, by the maximization of transfer, the starting group, i.e., this direct product group, must be transferred onto subsequent states in the generative history, |
