Mazzola, Guerino / Noll, Thomas / Lluis-Puebla, Emilio: Perspectives in Mathematical and Computational Music Theory

Figure 16:

Modulation has an opening telescope structure.

Super-local unfolding groups are structured by adding a control level above the existing wreath hierarchy, but such that it acts selectively on only part of the existing hierarchy. Such groups model situations, for example, in AutoCAD, where one freezes part of the existing structure and manipulates some unfrozen cross-hierarchy selection of elements; or conversely, situations, for example, in 3D Studio Max, where the cross-hierarchy selection is locked and manipulated over a sequence of steps.

Now consider sub-local unfoldings. Here, an extra fiber level is attached below only some part of the existing hierarchy. We shall soon see that this is the major basis of musical composition. However, in order to fully understand this, it is best to first examine an apparently different design process: mechanical CAD. This is design in mechanical engineering - forming the basis, for example, of the aerospace and automotive industries. It is generally accepted that mechanical CAD proceeds by a process called feature attachment. The remainder of the paper will do the following:

Give an algebraic theory of feature attachment in mechanical CAD.
Show that this is equivalent to musical composition.

To accomplish this, it is necessary to give more of the algebraic theory of object-oriented software, that we developed in Leyton (2001). First of all, within that theory, there is an analysis of class structure which says that each geometric class consists of an internal symmetry group, specified often in the invariants clauses of the software text for the class - and an external group consisting of command operations, such as deformations, specified in the feature clauses of the class text.

A principle claim of the theory is that the relation between the internal symmetry group and command structure, in the software text, is a wreath product, thus:

Gsym wO G(C)

where $G sym$ is the internal symmetry group and $G(C)$ is the group of command operations. This is no more than an algebraic formulation of the phenomenon of re-usability, within the specification text of the class. That is, according to our theory, re-usability is transfer and transfer is modelled by wreath products.