13 Musical Modulation

With the above concepts, it is now possible to understand deeply modulation in music. The first thing to observe is that modulation, rather than being simply a translation system, is actually a relative motion system. For example, when one talks about a musical piece has having modulated to the dominant, one means that movement is now judged as within the dominant key, yet the dominant key is judged, as a whole, relative to the tonic key. Notice that the same movement could be judged as within the tonic key. However, its position is instead judged through a hierarchy of relative motion.

The fact that modulation is a relative motion system allows us to see that it necessarily involves an inheritance hierarchy. For example, movement within the tonic key is the transform belonging the parent, and movement within the dominant key is the transform belonging to the child. The latter inherits the former in the hierarchical manner described above. Thus, let the symbol be the group of movements within a scale. Then, the ability to move the scale to any position within the scale is given by the following wreath product:

The control group represents the action of modulation, and the fiber group represents the key to which one modulates. This will be now be explained using the details of section 6.

For the purposes of illustration, we will assume that the group of scale movements is given by acting along the semitone scale. Thus both the control group and fiber group will have the structure of , and, to distinguish between these two roles, we will let the control group be denoted by , and the fiber group by . Furthermore, both the control set and fiber set will be the set of twelve semitones, and, again, to distinguish between these two roles, the control set will be denoted by and the fiber set by . Members of these two sets will be indicated by and respectively.

Now, for each of the 12 members of the control semitone set , make a copy of the fiber action. Thus there will now be 12 copies of the fiber set, indexed in the control semitone scale . Each of these copies will itself be the semitone scale (as fiber) rooted at a different tonic , within the control semitone scale. That is, we now understand the control semitone set to be the set of available tonics for modulation.

Corresponding to the 12 fiber-set copies, there will be 12 copies of the fiber group, also indexed in the control set . Each fiber-group copy (copy of the group of scale movements) will act on its own ”personal” copy of the fiber-set; i.e., its own personal scale. One can now define the regular wreath product:

(16)

where this group is the semi-direct product:

(17)