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

coordinates of another side. Notice that, given an individual point $(t,rh)$ on a side, its four transfer-equivalent copies (on each of the four sides), are now given by the diagonal embedding of the fiber group into the fiber-group product, thus: $t- --> (t,t,t,t)$ .

Figure 5:

The coordinates of four points.

Figure 6:

The control-nested structure of those coordinates.

Now, deformed shapes are handled in our system by adding extra layers of transfer. For example, to obtain a parallelogram, one adds the general linear group $GL(2, R)$ onto the two-level group of the square thus:

R wO Z4Ow GL(2,R).

(11)

Notice that the operation used to add $GL(2, R)$ on to the lower structure $ROwZ 4$ is, once again, the wreath-product which means that $GL(2,R)$ acts by transferring $R wOZ 4$ , as follows: Since the fiber group $ROwZ 4$ represents the structure of the square, this means that $GL(2,R)$ transfers the structure of the square onto the parallelogram. In particular, it transfers the generative coordinates of the square onto the parallelogram. For example, $GL(2,R)$ transfers the four points on the square in figure 5 onto the corresponding four points on the parallelogram, as shown in figure 7.

More deeply still, the fiber group $R wOZ 4$ in expression (11) is itself a transfer structure, as seen in figure 6, where rotation transferred the translation process from the top side onto the right side. This transfer structure is itself transferred, by $GL(2,R)$ , onto the parallelogram, as shown in figure 8. That is, we have transfer of transfer. This recursive transfer is encoded by the successive $wO$ operations in expression (11).

Figure 7:

The transferred coordinates from a square.

Figure 8:

The transfer of transfer.

What has been illustrated here is our principle of the maximization of transfer: The parallelogram is given a generative description, all the way up from a point, that maximizes transfer. That is, the point is transferred by translations to create a side, the side is transferred by rotations to create a square, and the square is transferred by the general linear group to create a parallelogram. Everything is re-used. This is the basis of our theory of aesthetics. For example this is the basis of a symphonic movement by Beethoven. We are giving here a simple first illustration of the mathematical principles involved.