Examples

Musical gestures can be produced on the beat or event scope but they can also be part of melodic phrases, as articulations of combinations of musical notes. Consequently a parameter for expression must be chosen along, keeping in mind that changes developed, are either perceived as mutations in duration or spectra. A mathematical expression or a set of rules or probabilities give values that change always as function of time. Examples of functions which provide a variety of numbers for musical parameter manipulation in Physical Models are:

The random function

$$R(n)$$ (1)

which returns an integer greater than zero and less than or equal to the given value n. This function is the basis for creating Random Walks or different ways ordering and sorting of a musical event.

The Henon Map:[7, Weeks, 1999]

$$X(n+1)=1 + AX(n)^2 + BY(n)$$ (2)

$$Y(n+1)= X(n)$$ (3)

is an example of a set of rules for generating periodic, quasi periodic and chaotic signal which can be mapped to a set of N pitches in a sequence of notes. Careful choice of values A and B manages the behavior of the signal

The Restrepo Map: [6, Restrepo, 1997]

$$X(n+1)= \frac{X(n)*X(n-1)}{X(n-1)}$$ (4)

is a result of Teager's filter and gives a periodic predictable region plus a chaotic region. Values different to 0 and 1 and on the intervals

$$X(0)=0.01, x(1)=1.99 ; x(0)=1.99, x(1)=1.99$$ (5)

can be mapped to rhythmic and drumming patterns in addition to a sequence of pitches.

The Henon Map and the Restrepo map are two eamples of recursive functions that can be mapped to midinote values using the following normalizing function suggested by Craig Sapp [3, Sapp, 2000]:

$$INT((x+1)/2 * 127+1/2)$$ (6)