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

11 Theory of Relative Motion

We shall soon argue the following:

Musical works are relative motion systems.

This will allow us to give a detailed algebraic theory of music, because relative motion is an inheritance phenomenon, and thus it will be possible to use the algebraic theory of inheritance from the previous section.

Relative motion is a powerful organizing force in cognitive representation. For example, it is a classic Gestalt result that the visual system organizes motion into hierarchies of relative motion; furthermore, basic decomposition theorems in classical and quantum mechanics allow momentum to be organized into hierarchies of relative momenta - a basic tool for problem-solving in physics.

Computer animators know that relative motion is an inheritance phenomenon. Thus, using our algebraic theory of inheritance, it is now possible to give an algebraic theory of relative motion:

Algebraic Theory of Relative Motion. A relative motion system corresponds to a wreath product in which the relative motion is given by the fiber group and the absolute motion, to which it is judged, is given by the control group:

relative motion wO absolute motion.

This theory will be used to explain both melodic and rhythmic organization in music. However, it is instructive to first look at a visual example. figure 11 shows a wheel moving along the ground. If one follows a single point on the wheel, it makes a complex curve called a cycloid. However, the human eye does not organize the movement in this way. Instead it decomposes the motion into a relative motion hierarchy, in which the point is seen as executing circular motion around the wheel-center, and the wheel is seen as moving as whole in a straight line. Our algebraic theory explains this as follows: The visual system organizes the motion into a wreath product in which the fiber is the rotation group $SO(2)$ , and the control is the translation group $R$ :

SO(2) wO R.

Fundamental Rule of Relative Motion. Decompose the motion into two symmetry groups, such that one group transfers the other.

This gives a wreath product where the transferring symmetry group is the control group and the transferred symmetry group is the fiber group. As shown in Leyton (2001), the above theory explains relative motion in human perception, classical and quantum mechanics, robotics, and computer animation.