4.1 Iterative processes

An iterative process is the repeated application of a mathematical procedure where each step is applied to the output of the preceding step (Figure 4.1).

Figure 4.1: An iterative process whereby the output is fed back to the input.

Mathematically, an iterative process is defined as a rule that describes the action that is to be repeatedly applied to an initial value x ₀. 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 x ₀ is written as: O ^F( x ₀). For example, consider the following rule F: x _{n +1} = x _n + 2. This rule indicates that the next value of the orbit x _{n +1} is calculated by adding the two units to the previous value. If one specifies that the initial value of x ₀ is equal to zero, then the result of the iterated application of F onto x ₀ will be O ^F(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...