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

In our theory of music, the concept of anticipation hierarchies will be crucial. We shall argue that, for deep mathematical reasons, such hierarchies have a role corresponding to the 3D shape primitives of mechanical CAD. Therefore it will be useful here to illustrate our theory of geometry with three-dimensional shape. For example, consider the structure of a cylinder. The standard group-theoretic description of a cylinder is

SO(2) × R

(12)

where $SO(2)$ , the group of planar rotations around a fixed point, gives the rotational symmetry of the cross-section, and $R$ gives the translational symmetry along the axis. Notice that in (12) the operation linking the two groups is the direct product operation $×$ .

For us, the problem with this expression is that it does not give a generative description of the cylinder. In computer vision and graphics, cylinders are described generatively as the sweeping of the circular cross-section along the axis, as shown in figure 9. To our knowledge, the group of this sweeping structure has never been given. We propose that the appropriate group is:

SO(2) wO R.

(13)

Notice that it uses the wreath-product operation rather than the direct product $×$ , and therefore the group has a very different structure from that in expression (12). The operation means that this new group has a fiber-control structure, in which $SO(2)$ is the fiber group and $R$ is the control group. This is exactly what is seen in the sweeping structure shown in figure 9. The cross-section is generated first as a fiber, and then its position is controlled by translation.

Figure 9:

The sweep structure of a cylinder.

We conclude this section by stating, more precisely, the principle of the maximization of transfer. It says two things: Given a data set: (1) Generate the set by maximizing re-use of the parts of the generative sequence; i.e., maximize the height of the wreath product. (2) However, make the height non-spurious; i.e., do not introduce levels where there are no detectable distinguishabilities in the data set. This second condition relates to the theory of recoverability (section 14),