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

The automorphic action of the control group $Z 4$ , on the fiber-group product $R ×R × R × R c1 c2 c3 c4$ corresponds to the action of $Z 4$ on the control set ${c ,c,c ,c} 1 2 3 4$ , whose elements now appear as the indexes on the four fiber-group copies $R ci$ . This means that the fiber-group copies are rotated around the square. In fact, in accord with the structure of a semi-direct product, $Z 4$ carries out this action by conjugating the fiber-group copies onto each other.

The data set $F × C$ , in this example, is the Gestalt completion of the square - i.e., given by the four infinite lines containing the four finite sides, as indicated in figure 4. We can think of this as four infinite wires overlapping each other.

The wreath product $R wOZ 4$ acts on the data set (the Gestalt completion), in the following way: By inspection of the semi-direct product form (9) of the wreath product, an individual element from the wreath product is of this form:

< ( T , T , T , T ) |r > c1 c2 c3 c4 h

(10)

where $Tci (- Rci$ and $rh (- Z4$ . The action of the group element (10) is then interpreted as follows: Each translation $Tci$ moves its own infinite wire along itself by the amount indicated by that translation, and then the remaining component $rh$ rotates the four wires by the amount $h$ . Notice therefore that the group element (10) maps the Gestalt completion of the square to itself, and that consequently the wreath product at (9) is a symmetry group of the Gestalt completion.

In our generative theory, the Gestalt completion is cut down to its visible portion, the finite square, by placing what we call an occupancy group, $Z2$ (a cyclic group of order 2), at each point along the infinite line containing a side. The group switches between two states, ”occupied” and ”non-occupied,” and is wreath sub-appended below the above group thus

Z2Ow R wO Z4.

Notice the power of this wreath product is that it is a regular one. That is, going from left to right, there is one copy of $Z2$ for each element in the group $R$ above it, and there is one copy of $R$ for each element of the group $Z4$ above that. For ease of exposition, the occupancy level will be ignored in the present paper.

Now observe that the group at (9) gives generative coordinates to the square, as follows. Since the wreath product is regular, we can identify the members $ci$ of the control set with the members $rh$ of the control group. Thus, any fiber-group copy can be labelled $Rrh$ , and its elements can be labelled $trh$ . Therefore, any point on the square can be described by a pair of coordinates:

(t,rh) = trh (- Rrh.

The first coordinate gives the generative (translational) distance along a side, and the second coordinate gives the generative (rotational) distance of a side from the first generated side. Therefore a point is given a complete generative description from the origin. (This relies on the fact that the fiber-action is transitive.) Figure 5 illustrates this by giving the coordinates of four of the points.

The crucial thing to observe is that the coordinates maximize transfer. figure 6 illustrates this by showing that the coordinates on one side are a transfer of the