The automorphic action of the control group , on the fiber-group product , corresponds to the action of on the control set ..., , whose elements are the tonics and now appear as the indexes on the 12 fiber-group copies . This means that the fiber-group copies are moved up and down the control scale; i.e., there is modulation. In fact, in accord with the structure of a semi-direct product, carries out this action by conjugating the fiber-scale groups onto each other. The following observations are crucial:

0. The above discussion clearly illustrates our claim that modulation is a relative motion system. That is, the control group , which represents the modulation movement, corresponds to the absolute motion, with respect to which any fiber-group copy represents the relative motion within the ”frame” that has been moved by the absolute motion.
0. The above discussion also illustrates our claim that modulation is an object-oriented inheritance system. In accord with our algebraic theory of inheritance (section 10), the parent corresponds to the control group and the child corresponds to the fiber group. Thus, any movement by the parent is inherited by the child.

The wreath product gives the complete symmetry of the scale structure, as follows: The data set , in this example, is decomposable into the set of fiber-scale sets, rooted at the different tonics. It will be called the scale system. The wreath product acts on the scale system, in the following way: By inspection of the semi-direct product form (17) of the wreath product, an individual element from the wreath product is of the form

(18)

where and . The action of the group element (18) is then interpreted as follows: Each scale movement shifts the notes of its own scale , and then the remaining component performs a modulation across scales. Notice therefore that the group element (18) maps the scale system to itself, and that consequently the wreath product at (17) is a symmetry group of the scale system. The following profound point should be observed:

That is, in the symmetry structure we have developed, the fiber of the symmetry corresponds to a child object whose group of command operations consists of movements within a scale, and the control level of the symmetry corresponds to a parent object whose command group is modulation. The reader can see that successive modulation from the home key is given by an iterated wreath product:

Notice that this is structurally equivalent to the type of group we gave for the serial-link manipulator in expression (15); that is, the recursive substitution of a