ABSTRACT
Mathematically, an iterative process is defined as a rule that describes the action that is to be repeatedly applied to an initial value x0. The outcome of an iterative process constitutes a set, technically referred to as the orbit of the process; the values of this set are referred to as the points of the orbit. Thus, the orbit O that arises from the iterated application of a rule F to an initial value x0 is written as: OF(x0). For example, consider the following rule F: xn+1 = xn + 2. This rule indicates that the next value of the orbit xn+1 is calculated by adding the two units to the previous value. If one specifies that the initial value of x0 is equal to zero, then the result of the iterated application of F onto x0 will be OF(0) = {0, 2, 4, 6, ...}. This is certainly a very simple orbit, but iterative processes have the potential to produce fascinating orbits, some of which can be used to generate interesting
musical sequences. Essentially, an iterative process may produce three classes of orbits:
1 orbits whose points tend towards a stable fixed value 2 orbits whose points tend to oscillate between specific elements 3 orbits whose points fall into chaos
As an example of the first class of orbits, consider the rule xn+1 = (xn/2). If x0 = 1, then the result of the iteration will be OF(1) = {1, 0.5, 0.25, 0.125, ...}. In this case the orbit will invariably tend towards zero, no matter what the initial value is.
