CCRMA OVERVIEW - CCRMA - Stanford University

A chaotic signal is generated by a mathematical expression that contains some sort of randomness.
The periodicity of the values given by the expression depends upon probability distributions and rules
which also are dependent on past or present values. The behavior of these functions show symmetry or
tendency causing attraction to the value with the highest periodicity. A chaotic behavior is considered
as a quantified level of unpredictability [Schroeder, 1991] which needs to be normalized and mapped to
values meaningful and within the bounds for synthesis of Physical Models.
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] 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: [Weeks, 1999]
X[n+1]= (AX[n])squared + BY[n] y[n+l]= X[n]
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: [Restrepo, 1997]
X[n+1]= X[n]*(X[n]-l)/X[n-l])
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[l] = 1.99 ; x[0] = 1.99, x[l]=1.99 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 [Sapp, 2000]:
INT[(x+l)/2 * 127+1/2]
Results
Phrasing and melodic transformations are another class of music gesture which permit use of chaotic
expressions. In this, the relationship among a sequence of musical events, past, present and future also
determine expressiveness. A group or combination of notes describes not only a melody but how these
notes are tied together[1995, Jaffe, Smith]. There can be a function or rule which conducts how these
notes are glued together. Thus it can specify durations of musical events as well as space (silence)
between each event.
When a chaotic or Random function is applied to the frequency (pitch) of a sound, different shades of
noise are obtained. Nevertheless these values can also be applied to filter coefficients providing musical
effects. For example, In the case of the Physical Model of the Piano, they can handle detunning factor,
stiffness, hammer strike angle, etc. The advantage of using chaos is that the effect seems more natural
and reflects the state (energy or entropy) of a real musical instrument.
Conclusions
It is important to understand that a musical instrument is not a stable or balanced system in reality.
Its behavior and responsiveness depends on various factors and consequently a Physical Model should
not be framed as a squared box or a fixed parametric system. The degree of musical expression is
proportional to the number of degrees of freedom provided by the Physical Model. Nevertheless the
complexity of the model is inverse to its degrees of freedom. Therefore, the more flexible the model
the harder to understand and handle. More information and parameter descriptions and documentation
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