Now let us turn to cloning in object-oriented programming. It is important to notice that, when one clones an object, one is producing a copy with the same instance values. This means that one is essentially creating a copy that is aligned with the original, as can be seen in such programs such as 3D Studio Max and Viz. This copy can then be manipulated via its command operations, which will then pull the clone out of alignment, i.e., break the symmetry of the object-clone pair.

We are now ready to turn to object creation and feature attachment. Feature attachment is the term used in mechanical design for the successive addition of structural units and components. It is, of course, the main process in any design. Our basic proposal is this:

Theory of Feature Attatchment When one creates objects and attaches them in the design structure, one is entering new instances into the alignment kernel, and positioning the command group for each new instance in the appropriate wreath position within the unfolding group corresponding to the inheritance hierarchy of the structure.

21 Theory of Musical Composition

With these concepts, it is now possible to give a theory of musical composition. Recall that our basic principle of aesthetics says:

This was seen in section 13 with respect to the structure of modulation which was modelled by an -fold wreath product, i.e., hierarchical transfer, of the scale structure; and it was seen also in section 19 with respect to musical meter which was modelled by an iso-regular group - again, a structure of hierarchical transfer. It will now be seen with respect to melodic form.

Significant progress has been made in understanding sequential organization by psychologists working on the generation of serial patterns. Herbert Simon, himself an outstanding musician, together with collegues, was the first to consider rule-systems for psychological sequence generation (Kotovsky and Simon, 1973). A further advance was made by Restle (1970b), who used hierarchies of rules. figure 17 shows an example typical of one of Restle’s hierarchies. Three generative rules are used in this hierarchy: = transpose by one unit upwards in the scale; = repeat; and = mirror about the scale center. Here the scale is assumed to consist of 12 notes. Each of the operators takes the entire subsequence that it dominates via its left node and maps it to the entire subsequence dominated by its right node. Notice that the tree is strictly nested, a term used by Greeno and Simon (1974), meaning that all the operators within a level are exactly the same. The condition of strict nesting is equivalent to the fact that the tree can be represented by a recursive formula. In the example shown, the formula is:

(19)

The symbol 1, in this formula, is the left most 1 in figure 17; and the formula generates the remainder of the sequence.