Music From Chaos:
Nonlinear Dynamic Systems as Generators of^{1}

Rick Bidlack

In recent decades, a great deal of
attention from the scientific and mathematical
communities
has been focused on the exploration of a class of mathematical objects known
as
nonlinear dynamical systems. Although the symbolic specification of many of
these systems
is simple to the point of apparent triviality, their behavior can often be
extremely rich
and complex, even unpredictable.
This
unpredictability, arising as it
does from processes
which are completely deterministic, has a certain flavor to it which is
quite distinct from the uncorrelated
randomness of white noise or other probabilistic distributions. The generic
term for this brand of deterministic
unpredictability is *chaos, *
and the body of
theory which describes
it is often called *chaos theory. *
(The terms 'nonlinear dynamical system,' `chaotic
dynamical system,' or simply 'chaotic system,' are considered in the present
context to be synonymous.) Nonlinear dynamical systems are interesting to
scientists because they exhibit structural
characteristics which are shared by forms found in nature. Chaotic systems are thought to underlie or model important aspects of
the motion (change in location) and/or evolution (change in form) over time of
phenomena in the real world. The complexity of behavior exhibited by certain chaotic mathematical objects - specified
as sets of equations -rivals that of real-world phenomena in many
instances.

The meshing of a number of events has raised the subject of chaos to its
currently
topical level within the scientific community. Improved methods for the
analysis and theoretical description of
dynamical systems were developed around the turn of the century by Poincaré and others. The advent of cheap
and plentiful computers has made practical the computation and analysis of numerical solutions to nonlinear
equations. Moreover, computers have made graphical representations of
the behaviors of these systems possible, revealing extraordinary structures
which would not immediately be apparent from the raw numerical data. Today,
increasingly well-defined behavioral and structural parallels between chaotic
systems and real-world phenomena are being found and described. The range of phenomena which are presently being studied in
terms of underlying chaotic structure is diverse, and includes: fluid
flow, aerodynamics and turbulence^{2}, orbital deviations and^{3}, the physiological structures (organs) in living things''^{4}, the
dynamics of superconductivity,^{5} structural mechanics^{6}, celestial mechanics^{7},
ecology and population dynamics'^{8}, and many others. The interest in chaotic
systems has generated a certain sense of excitement among the general public as
well, as manifested by the popularity of books such as James Glieck's
non-fiction best-seller *Chaos: Making a New Science*^{9}*, *
or Benoit Mandelbrot's
treatise *The
Fractal Geometry of Nature*^{10}* *
Striking
illustrations of intricate fractal objects, the "remnants"
of nonlinear dynamical
processes"^{11}, have been provided by Peitgen, Richter, Saupe
and others.^{12}

The
exploration of chaos is forcing a change in the definition of this word. Chaos is *
not
*
formless anarchism, it is a highly structured phenomenon
pervasive throughout the natural world. Its variations are
rich and complex, and
its properties are intriguing. Its
study has invigorated
large segments of the mathematical and scientific communities; perhaps it has
the potential to inspire artistic activities
as well.

Nonlinearity is a concept which is intuitively understood and
applied in a wide range of everyday
situations. The popular adage about the straw that breaks the camel's back is based on this understanding. In this
quintessentially nonlinear situation, a small *quantitative
*change to the "input" (the addition of
one straw) is seen to cause a dramatic *qualitative *change
in the "output" (the demise of the camel). Another example, used by
Glieck^{13}, is the traditional verse:

For want of a nail the shoe was lost.

For want of a shoe the horse was lost.

For want of a horse the rider was lost.

For want of a rider the battle was
lost.

For want of a battle the kingdom was lost.

This is a
perfect example of what is more formally known as *sensitive dependence on initial conditions, *
the idea that
a seemingly insignificant variance at the outset of a chain of events may
nonetheless precipitate effects down the line that build up to irrevocably and
drastically
alter
the final outcome. One might also think of the performance of the economy (seemingly
immune to all theory and attempts at regulation), the occurrence of earthquakes
(sudden jolts resulting from presumably
constant tectonic pressures), or even the outbreak of war (an abrupt shift of international tension from the
political to the military arena) as familiar examples of nonlinear
dynamics at work.

Nonlinear dynamical systems are mathematical objects which behave in
similarly unpredictable ways. To understand how these expressions
can be manipulated to produce musical
results, certain basic concepts of dynamics and nonlinear equations must
be understood. Among the terms to be introduced in this section and in
the following chapters are: phase space,
orbit, bifurcation, attraction,
dissipative and conservative systems, transients,
sensitive dependence on initial conditions, Poincaró section, iterated
map and continuous flow. An operational
definition of chaos, if not a rigorously mathematical one, will
hopefully follow from a familiarity with these concepts.

Dynamical systems convey in their
notation an implicit recognition of the passage of time. The length of time that passes is
rather arbitrary; what is more important is the quality of the time, or how it is
fundamentally conceived. If the mathematical expression of a dynamical system implies the
passage of time in terms of discrete steps of a uniform
size, then the system is a *difference *
equation,
of the form

*x _{t+1 }= f(xt)*

This
is akin to a simple feedback system, in which the current output of the equation
becomes the
input for the next iteration. Systems expressed as difference equations are
also called *iterated
maps, *
because the effect of the equation is to map one value, x_{t},
to another value,
x_{t+1},_{ }for each iteration of the equation. If, on the
other hand, time is considered to be a continuous quantity, advancing smoothly, then
the system is expressed as a *differential *
equation
in the form

*x = dx1dt = f(x),*

where *x *
is a measure of the rate of
change of the dependent variable x with respect to time.
These
systems are called *continuous flows, *
and
what they produce are smooth, unbroken
curves. In a digital computer,
however, continuous time cannot exist - it is a theoretical concept that must be approximated in simulation.
This is accomplished by reformulating the differential equations of the system into difference equations, a
process called integration. Several
methods have been developed for integrating differential equations, but only
the simplest of these, Euler's
method, is used in this study. Essentially, this procedure calculates a value (x)„ from the function *f(x _{n}), *
then
computes a new

*x*_{n+1} =
x_{n }+_{ }
x_{n }
∆*t,*

where Delta t is typically a very small value, 0.01 or less. The effect
of this procedure is to
move forward along the curve by very
small, discrete steps (of unequal length, but equally
spaced in time.)

In mathematical terms, nonlinear equations are simply those which are
not linear.
Linear *algebraic
*equations
are those in which the degree of the variables is unity, e.g., none of the
variables of the equation are raised to any power other than one. An example is
the linear equation *y = mx b, *
which defines a straight line of slope *m *
and y-intercept *b. *
On the other hand,the
second-degree equation 9 = *mx b, *
which defines a parabola,
is not linear. This distinction applies in *differential *
equations
as well, although in
this case the only variables for which
the degree is significant are the *dependent *
variables,
the ones for
which a derivative term occurs in the equation. For example, the equation

*x ^{2} (dy/dx)
+ y(x^{3} - *2) = 2x

is linear in *y, *
even
though *x *
is both squared and cubed, because the
dependent variable *y *
is always of degree one. The
equation

*A(dy ^{2,}/dx^{2}) + B(dy/x) + *Cy

is nonlinear in *y *
because the
dependent variable *y *
is raised to the
third power.

While the degree of an equation is critical in the distinction between linear and nonlinear dynamical systems, it is not the only factor in determining whether or not a given equation is nonlinear. The presence within an equation of a trigonometric function, or, in a differential equation, the multiplication of two or more of the dependent variables with one another, are also sufficient causes to make the equation nonlinear.

One of the simplest nonlinear equations known to exhibit chaotic behavior is
the *
logistic *
difference equation, also known as the quadratic or parabolic equation. This
equation has been applied in studies of the population dynamics of a species
from generation
to generation^{14}. It specifies the population *
P at *
successive time increments 0, 1, 2, ... ,n
as a function
of its prior value:

P_{n+1} = KP_{n}(1
- P_{n}),

where K is a parameter whose value normally remains constant once an iteration
sequence
is initiated. The equation is nonlinear by virtue of the quantity P^{2}
(remember that *
K P *
(1 - *
P) = K
P - K P ^{2}). *
Despite its
apparent simplicity, the dynamics of this equation are sufficiently complex to
have warranted attention from many authors in the mathematical community; see
for example Thompson and Stewart

The equation may be iterated indefinitely, to produce an arbitrarily long
sequence of numbers ^{
P}o,
•"
Rn•
This sequence is called the *
orbit *
or *
trajectory. *
The term
orbit does not necessarily imply a smooth, continuous curve. While the orbit
of the earth
around the sun may be quite smooth, orbits of iterated maps such as the
logistic function
can be quite
erratic. The equation becomes interesting when the orbits resulting from the
adoption of different values for the constant *
K *
are compared. Table 1 lists the first thirty
iterates of eight different orbits, each following from a different value for
*
K. *
Although the
initial value
*P _{0}
*.5
in all cases, each orbit is quite distinct from the others.

Three distinctly different modes of behavior are demonstrated by the orbits
shown in Table 1. First, for K-values of 0.5, 1.0, 1.5, 2.0,2.5 and 4.0, the
orbit is drawn to
a single point, or fixed point, where it remains. (For K *
=*1.0,
the orbit is eventually drawn to
0.) This point is called an *
attractor. *
The rate at which the sequence of values converges on
the
single-point attractor varies from immediate *
(K
*2.0) to
extremely gradual *
(K = 1.0).
*
Secondly, for *
K.
*3.5, the
orbit is attracted to a *
limit cycle,
*
where it oscillates indefinitely. In this case, the limit cycle is a periodic
oscillation among four points. This is a periodic
attractor.
Third, for *K.
*3.0, the orbit describes a damped oscillation wherein
the high and low values converge asymptotically. The two values will converge to a single
point only when the limit of resolution of the program which is
computing the orbit is reached - in other words,
when the computer is no longer able to make a distinction between the two
values. Depending on what this
resolution is, this may not occur until after tens of thousands of
iterations have been calculated.

Table 2 shows the first eighty
iterations of eight more orbits of the logistic
equation, each from an initial point P_{0} = .5, for values of
*K *
starting at 3.55 and increasing in
steps of 0.01125. These orbits illustrate
modes of behavior which are again quite distinct from those shown in
Table 1. Within the span of eighty iterations listed, only two of the orbits
attain an unambiguously periodic state: the
first, for *
K *3.55, and
the last, for *
K
*3.62875.
In both of these cases, the P-values comprising each cycle do not repeat
*exactly *until
after the sixtieth
iteration. The initial portion of the orbit, during which the eventual,
settled behavior of the
equation under the specified conditions (the value of the parameter *
K *and the
initial value of *
P)
*is approached, is called
the *
transient *
segment of the orbit. In the case of these
two
periodic orbits, the span of latent periodicity prior to the sixtieth
iteration (approximately) comprises the
transient portion. For the orbits shown in Table 1 which are attracted to a
single point, the transient state consists of those points through which
the orbit passes before it settles
onto the fixed point. There is not necessarily a firm dividing line between
the transient portion of an orbit and
its settled state. Some orbits ease into their settled behaviors very
gradually, while others show no transients at all.

Four other orbits in Table 2
appear to be headed toward a regime of periodicity,
but none of them reaches this
state within the span of eighty iterations shown here. For *
K *= 3.56125,
the orbit tends toward a period 8 oscillation. For *
K = *3.57250 the
tendency is toward a
period 24 oscillation, for *
K =
*3.58375 it could be a
period 12 (although it gets
"noisier" rather than cleaner
as it progresses), and for *
K *3.60625 there
is a clear period 10
oscillation. In practice, it is sometimes difficult to determine whether or
not a given orbit is
being attracted to a periodic state without following the orbit for a much
longer span than has
been illustrated in Table 2. These orbits may still be in a transient state,
headed toward an ultimately periodic condition; on
the other hand, it is possible that one or more of them has
already settled into a long-term state of "noisy periodicity" within the
span shown here. Adding to the
ambiguity is the fact that a determination of a given orbit's periodicity is
contingent on the numerical resolution used in the representation of
that orbit. Long transient orbits appear to
become shorter when fewer significant figures after the decimal point are
used. On the other hand, the
seemingly exact periodicities finally attained in the first and
eighth orbits may simply be artifacts of there being only five
significant figures after the decimal point.

**Table 1:**
Eight orbits of the logistic equation, in numerical form for different values
of
*
the parameter
K. *
All orbits begin
at the same point P

The remaining two orbits in
Table 2, for *K = *3.595 and *K = *3.61750, tend
neither toward a fixed point,
nor to a limit cycle. These are *chaotic *orbits, in which the
sequence of values is quite
erratic, and apparently random. The fact that these orbits are
the result of a completely
deterministic procedure (the equation itself) is one distinction
between true randomness and chaos.

Taken together, Tables 1 and 2 demonstrate a remarkably
rich (and potentially bewildering) range of behavior for an equation as
apparently trivial as the logistic equation.
Graphical methods of representation may be
employed to clarify this behavior and relate it
within a global context. One method is the
construction of a *bifurcation diagram, *in which a
large number of partial orbits are computed
and plotted against a range of values of the constant parameter. Figure 1
shows two versions of a bifurcation diagram for the logistic equation
over the range 0 < *K < *4. In the first version, the first 150 points of
each orbit are plotted, while in the second
version iterations 150 through 300 are plotted. Thus, the first version
illustrates the transient behavior of the system, while the second version
shows the settled behavior (this assumes that most transients have died down
by the 150th iteration). Figure 2 shows two
successive magnifications of the diagram, the first in the range 3 < *K <
*4, the second in the range 3.55
< *K < *3.62875. Several important features of the behavior of the
logistic equation emerge from an examination of these diagrams. When the value
of *K *is less than 3.0, the orbit is attracted to a single point (for *
K < *1, this point is 0). At *K = *
3.0, a *bifurcation *of the attractor
is evident. At *K *apx. 3.46, a further bifurcation is seen,
forming a period 4 oscillation (compare
Table 1, for the orbit at *K = 3.5). *Yet another
bifurcation occurs just before *K= *3.55, and again between 3.56125
and 3.5725. The period of the
oscillation is thus doubled at each bifurcation node, in a sequence called the
*period-doubling cascade. *Eventually, the nature of the attractor
undergoes a change from a condition of
periodicity to one of chaos. For the logistic equation, the chaotic *regime
*begins at around *K *3.567
and extends to the maximum value of *K *4.0 (orbits at higher values
of *K *are unstable). Nonetheless, periodic windows are found
interspersed throughout the chaotic regime; note, for example, the large
period 3 window around *K *3.84. Figure 2(b)
provides a new perspective on the eight orbits listed in Table 2. The
period 24 window at *K = *3.5725 is barely discernible, while the period
12 orbit at *K *3.58375 is seen to lie right
on the edge of a periodic window. Chaotic
orbits for *K *3.595 and *K *3.6175 lie squarely within
chaotic regimes.

**Figure 1:** Bifurcation diagrams of the
logistic equation, over the range 0 < *
K < *
4, with

*PO
*.5;
(a) showing transient behavior (first 150 iterations),
(b) showing settled behavior (next 150 iterations).

**Figure 2:**
Magnification sequence of bifurcation diagrams of the logistic equation; (a) in
the range 3 *< K < *
4, (b) in the range 3.55 < *K < *
3.62875. Iterations 150 through 300 are plotted.

Thus far, all of the numerical orbits listed, as well as the
four bifurcation diagrams plotted, have
been initiated from P_{0}
= 0.5. In fact, any initial value of *
Po *
between 0 and 1 could have been used to
produce basically the same results. For those values of *
K *
which lead to
periodic orbits, only the initial transient portions of the orbits would be
different. For those values of *
K *
which lead to chaotic orbits, the exact sequence of points
along the orbit would be quite different
for different values of *
Po, *
but the range of values over
which the orbit
wanders would remain the same. Thus, the two bifurcation diagrams in
Figure 2, which do not plot the transient
portion of the orbits, would look essentially the same
no matter what the initial value of *
P _{o}.
*
This is the significance of the term

Another intriguing aspect of
chaotic orbits is their *
sensitive
dependence
on initial
conditions.
*Two orbits which begin at
__almost __
the same point will eventually diverge from one another. An
example of this is shown in Table 3. Here two orbits are followed
through twenty-four iterations. For both
orbits, *
K *
4.0, while the initial value
*P _{o}_{
}*
differs by
only 1/100,000.
Nonetheless, this difference causes a complete divergence of the two orbits
within the span of fourteen iterations. This effect
can be demonstrated with any chaotic
dynamical system, no matter how small the initial difference. Incidentally, a
comparison of these two orbits with the last orbit listed in Table 1
(for

**
Table 3:
**Two
orbits of
the logistic equation, demonstrating sensitive dependence on initial
conditions.
*K.
*
4.0 for
both orbits, while P_{0 }differs only by 1/100,000. The orbits remain
fairly similar through
fourteen iterations.

The phenomenon of sensitive dependence
on initial conditions has
wide-ranging ramifications for anyone
working with chaotic systems. Since real numbers can
only be represented with finite precision in a computer, rounding errors in
the calculation of any orbit are inevitable. These errors eventually
accumulate, causing the calculated orbit to
diverge exponentially from the theoretical orbit. Slight differences in the
implementation of an algorithm to compute an orbit (for example, the order in
which mathematical operations
are carried
out), the precision employed, or the use of a different computer, are
sufficient to cause a divergence of
trajectories. One may rely on the *
global *
behavior of a chaotic system
to remain roughly the same from machine to machine, but not the precise
sequence of a
given orbit.

This study approaches the issue of the generation of musical materials by means of nonlinear dynamical systems through an examination of two distinct systems: the Hénon map and the Lorenz system. These systems, only slightly more complex than the logistic equation just discussed, were chosen not only because they are well-known and widely studied, but because they are exemplary of certain fundamental classifications of nonlinear dynamical systems in general.

The Hénon map is an iterated map, an object in which the
passage of time is represented in discrete
steps of equal size, as would be appropriate in the periodic
measurements of things such as populations,
water levels, or stock prices. The phase *
space, *or the area in which the dynamics of
the system take place, is two-dimensional. In the
case of the Hénon map, the significant portion of the phase space may be
represented on the Cartesian plane,
between approximately -1 .5 and 1 .5 on the x-axis, and between
approximately -0.4 and 0.4 on the y-axis.

The Lorenz system is a
continuous flow, a phenomenon in which the passage
of time is - theoretically - represented in an unbroken manner, as would be
appropriate to the calculation of celestial
orbits or to the representation of wind speeds over a continuous span
of time. The stable phase space of the
Lorenz attractor is three-dimensional, roughly
symmetrical around the origin along both
*
x *
and *
y *
axes, and lying
entirely in the positive region along the z-axis.

The Hénon and Lorenz systems
are *
dissipative, *
that is, they are
representatives of a class of phenomena in which the total
energy of the system is dissipated over
time. (The logistic equation is also a member of this class.) In analytic
terms, this means that the phase space of the system shrinks over time. The
dissipation of energy occurs during
the initial transient portion of the orbit. Orbits of dissipative systems are
drawn to an attractor, which may be either a point (a single-point
attractor), a set of points (a limit cycle
or periodic attractor), or a complexly folded region of space (a chaotic
attractor or strange attractor). The great majority of earth-bound dynamical systems
are dissipative, due to energy loss through friction.

Other systems not discussed
here are *
conservative. *
They are representative
of phenomena in which friction
does not play a role, those in which energy is conserved.
These systems are also called
*
Hamiltonian *
systems or
*
area preserving *
systems, because
their phase spaces do not shrink with time. Thus, they
cannot properly be said to have an
attractor, although they do exhibit periodic and chaotic behaviors in a manner
parallel to that of the dissipative
systems. Such systems occur within the realm of celestial mechanics, and
on earth, within the electromagnetic storage rings of particle
accelerators.

The Experimental Apparatus

It is desirable, in an initial exploration of a potential compositional technique,that highly subjective issues be removed from the picture, or at least minimized as much as possible. The question at hand is not: "How can I make music with this dynamical system?"Rather, it is: "How much music inherently resides in this system?" Once that question is answered, the details of the extraction of the music become the idiosyncratic, personal domain of the composer, and lie outside the domain of this study.

Although many different mappings of chaotic orbits into musical space can be imagined, only one such mapping will be employed for each of the two systems examined in the following chapters. To simplify the presentation and evaluation of the musical product of the chaotic systems in question, the orbits are projected into a uniform pitch space encompassing a four-octave range centered around middle C, in which dynamic levels are allowed to vary between the extremes of loud and soft. Except for the first example from the Lorenz system, the temporal axis (rhythm) is considered non-dimensional and constant.

A graphical notation system has been devised to provide a visual reference for the score examples. The coordinate system of the scores is similar to that of traditional musical notation, in that pitch is read on the vertical axis and time is read horizontally. The biggest difference is that the pitch axis of the graphical scores is continuous, rather than discrete, and the distance of a semitone is everywhere the same, rather than variable in size as is the case with traditional notation. This allows an accurate notation of pitch without the use of accidentals, and is as readily adaptable to the notation of infinitely variable pitch as to the equal-tempered scale. For the sake of reference, each score is inscribed with the lines of the traditional grand staff. Dynamic levels in the graphic scores are indicated by the shading of the notes; a very dark note would be quite loud, while a lightly-shaded one is softer. Figure 3 illustrates the notation of a one-octave C major scale, beginning on middle C. The velocity of each note is conveyed by its shading - the darker a note, the louder it is.

It is desirable, in an initial exploration of a potential compositional technique, that highly subjective issues be removed from the picture, or at least minimized as much as possible. The question at hand is not: "How can I make music with this dynamical system?" Rather, it is: "How much music inherently resides in this system?" Once that question is answered, the details of the extraction of the music become the idiosyncratic, personal domain of the composer, and lie outside the domain of this study.

Although many different mappings of chaotic orbits into musical space can be imagined, only one such mapping will be employed for each of the two systems examined in the following chapters. To simplify the presentation and evaluation of the musical product of the chaotic systems in question, the orbits are projected into a uniform pitch space encompassing a four-octave range centered around middle C, in which dynamic levels are allowed to vary between the extremes of loud and soft. Except for the first example from the Lorenz system, the temporal axis (rhythm) is considered non-dimensional and constant.

A graphical notation system has been devised to provide a visual reference for the score examples. The coordinate system of the scores is similar to that of traditional musical notation, in that pitch is read on the vertical axis and time is read horizontally. The biggest difference is that the pitch axis of the graphical scores is continuous, rather than discrete, and the distance of a semitone is everywhere the same, rather than variable in size as is the case with traditional notation. This allows an accurate notation of pitch without the use of accidentals, and is as readily adaptable to the notation of infinitely variable pitch as to the equal-tempered scale. For the sake of reference, each score is inscribed with the lines of the traditional grand staff. Dynamic levels in the graphic scores are indicated by the shading of the notes; a very dark note would be quite loud, while a lightly-shaded one is softer. Figure 3 illustrates the notation of a one-octave C major scale, beginning on middle C. The velocity of each note is conveyed by its shading - the darker a note, the louder it is.

**Figure
3:** Example of the graphical scoring system. A one-octave C major scale is
notated. The scale
starts softly and increases in volume. Note that the tick marks on each "bar
line" indicate the positions
of C's, and that the horizontal dashed lines indicate positions where
ledger lines would normally be
written.
Time is measured in five-second intervals.

The Hénon Map - Technical Description

The Hénon system is a dissipative,
two-dimensional iterated map which was first introduced in 1976 by Michel
Hénon of the Observatory in Nice^{19}. The Hénon map has
no explicit counterpart in the real world, but is an abstract system
formulated for the express
purpose of studying chaotic systems in general. At
the time the map was introduced, most
of the known chaotic dynamical
systems were defined as continuous flows in three or more dimensions. The
computation of numerical solutions to these flows involves the integration
of differential equations, a time-consuming and computationally expensive
process. The
Hénon map, on the other hand, exhibits all of the
interesting features of a higher-dimensional
chaotic flow (sensitive dependence on initial conditions, interspersed regimes
of chaos and
periodicity, complex topological structure, a strange attractor, etc.), but is
much easier and
faster to
compute since it is specified in terms of difference equations. In addition,
the inaccuracies which arise inevitably in
the process of integration may be avoided altogether, rendering the
Hénon system more susceptible to mathematical analysis than a
higher-dimensional flow.

The Hénon map is expressed by the equations
where the succession of points (x_{o},
y_{o}),
(x_{1}, y_{i}),(x_{n},_{ }y_{n})*
*
specifies the orbit on the plane, and
*
A *
and *
B *
are constant parameters. The equations are nonlinear by
virtue of the squared variable
*
x. *
In his initial discussion of the system, Hénon assigned the values 1.4 and 0.3
to *
A *
and *
B, *
respectively; these have since become
the "classical" values used to produce the familiar form of the attractor; see
Figure 4. A program to generate orbits of the Hénon map
in numerical
form may be found in Appendix A.2. Hénon's paper, as well as information
presented by Thompson and Stewart^{20},
and Fluelle^{21},
provides the substance of the Summary of the behavior of the equations which
follow.

Table 4 presents in numerical form the first twenty iterates of the
orbit
for the parameter values A *
=1.4 *
and *
B = 0.3, *
from an initial point (0,0). Like the logistic map,
the orbit of the Hénon system wanders from point to point along the attractor
in an erratic
manner.

Table 4:
Orbit of the Hénon map in
numerical form through the first twenty iterates. *A=1.4, B=0.3, *
the initial
point is (0,0).

Technically speaking, the attractor itself can never be represented. The points plotted in the three phase portraits of Figure 4 lie on the attractor, but they do not comprise it. (A phase portrait or phase plot is simply a plot of the points along an orbit in the phase space of the system.) Iterates 0 through 3 can still be individually distinguished in Figure 4(c) (the zeroeth iterate lies exactly in the center of the plot), and clearly lie far from the attractor; these points, along with the next two or three, constitute the initial transient portion of the orbit before it settles onto the attractor. There is no specific number of iterates after which one can say that the transient portion has ended and the attractor has been reached. For example, although the fourth iterate can not be individually distinguished in Figure 4(c), at a higher resolution it could be, and would also be seen not to lie on the attractor.

**Figure 4:** The
Hénon map, (a) after 100 iterations, (b) after
1000 iterations, The Hénon Map after

10,000 iterations. The initial point is
(0,0), the parameters are *
A=1.4, B=0.3. *
The area of the inset

square is
magnified in Figure 5.

The attractor itself consists of an apparently infinite number of finely spaced filaments, along which the points of a given trajectory are scattered. Figure 5 shows a magnified view of the area within the small inset rectangle in Figure 4(c). The structure of the spacing of the filaments at different levels of magnification is indicative of the fractal Cantor set structure underlying the Hénon attractor. Note that, due to the increasingly small area graphed in the magnification sequence of Figure 5, many more points of the orbit must be computed in order to fill the graphed area to an adequate level.

**Figure 5**: Magnification sequence of the Hénon map. Initial conditions are as for Figure 4. Note that

many more iterations of the map must be computed in order to produce a
sufficient number of points

to fall
within the graphed areas; (a) after 100,000 iterations, (b) after 500,000
iterations.

**Figure 6: **
Basin of attraction of the
Hénon map with the
attractor superimposed, for parameters
*
A=1.4,B=0.3. *
Any orbit starting in the shaded areas is repelled from
the basin of attraction and tends
to
infinity.

The *
basin of attraction *
is that set of points, any one of which taken as an
initial
point *
(x _{0},
y_{0}) *
of an orbit, produces an orbit which is stable and is drawn to the
attractor. Figure 6 shows the basin of attraction as the unshaded area, with
the attractor itself superimposed. Points which lie outside the basin of
attraction produce orbits which are repelled from this area and escape to
infinity.

Like the
logistic equation, the form of the Hénon attractor varies as the
parameter values *
A *
and *
B *
change. The bifurcation diagrams shown
in Figures 7, 8, and 9
exhibit a period-doubling cascade quite similar in form to
that of the logistic equation, followed by
a regime of chaos interspersed with periodic windows of varying widths. These
diagrams were made by holding the
parameter *
B *
constant at 0.3 and sweeping *
A *
across the
range of values
indicated for each diagram. Phase plots corresponding to some of the
values indicated in the diagrams with
vertical dotted lines are shown in Figure 10. Note the correspondence, for
example, between the four and six-band chaotic regimes located at
*
A
=1.076 *
and *
A=1.078
*in
both the bifurcation diagrams and in the two plots of Figures 10(b) and (c).
Figures 10(c) and (d) illustrate co-existing attractors for the *
same
*value of
*A
*
emerging from different initial points of the orbit. Other marked positions in
the bifurcation
diagrams correspond to orbits chosen for the musical
examples following in the next section.

Figure 7a: (a) Bifurcation diagram of the Hénon map, initial transient behavior (the first 150 iterations)

**Figure 7b:** settled, stable
behavior (the next 150 iterations). Each plot sweeps parameter *
A
*from 0
to 1.4, holding *
B *
constant at 0.3. The initial point of all orbits is the origin, (0,0).

**Figure 8:**
Magnification of a section of the bifurcation diagram from *
A.1 *
to *
A.1.4. *
A period 16
bifurcation is discernible just before the onset of the chaotic regime at
approximately *
A=1.055.
*
Note
the
large period 7 window around *
A.1.25, *
as well as
other, less extensive regimes of periodicity interspersed through the chaotic
regime. This diagram shows the settled behavior of the system.

**Figure 9:**
Further magnification of the Hénon bifurcation diagram, focusing on the
transition to turbulence. The two
illustrations plot (a) transient behavior (the first 150 iterations) b) settled
behavior (the
next 150 iterations). At this level of resolution, an additional period 32
bifurcation can be barely discerned just above *
A=1.054. *
Note as well the abrupt shift in the structure of the attractor,
from four
bands of chaos to six, which occurs between *
A.1.076 *
and *
A.1.078.*

**Figure 10: **Phase portraits of the Hénon map. Parameter
*
B=0.3 *
for all orbits. Initial point is the origin
(0,0) for all orbits except (d); (a) period 16 orbit at *
A=1.054, *
(b) chaotic attractor at *
A=1.076, *
(c) chaotic
attractor at *
A=1.078, *
(d) co-existing chaotic attractor at *
A=1.078, *
from initial point (.1, 0), (e) first 150
iterations of apparently chaotic orbit at *
A=1.3, *
however, (f) period 7 orbit emerges from chaos after
approximately 120 iterations at *
A=1.3.*

**Figure 11:** Bifurcation diagram of the Hénon attractor plotting
*
B *
against *
x. *
Parameter *
A *
is constant
at 1.1
for all trials. The first 150 points of each orbit are plotted.

For the
sake of comparison, a bifurcation diagram sweeping the parameter *
B *and
holding *
A *
constant is
shown in
Figure 11. Bifurcation diagrams plotting changes in *
y
may *
be produced as well, and are similar in appearance to
those plotted against *
x, *
except for the change in scale. The behavior of both variables is globally
similar: if *
x *
is periodic, *
y *
is also
periodic, if *
x *
varies
chaotically,
so does
*y.*

Further
discussion on the Hénon map may be found in Shaw, Eckmann and Ruelle^{23}t,
Ruelle^{24}, and Froehling, et
al..25.

Musical Examples

Musical mappings of the Hénon
system exploring a range of characteristic
behaviors were made
over 9 different values of the parameter *
A.
*These
values, 0.2, 0.5,
1.0, 1.046, 1.054, 1.076,
1.078, 1.3, and 1.4, may all be located among the phase plots and
bifurcation diagrams in the
preceding section. The initial point (0,0) is used for all examples
(Example 10 excepted), and
the parameter *B
*is held constant at 0.3
throughout.

There are a
large number - infinite, actually - of possible mappings of the phase
space to
the musical domain. Each of the variables *
x
*and
*
y *
could be
mapped to any musical
parameter;
in addition, a variety of orientations are possible. For example, values along
each axis could be mapped
*inversely *
to a given parameter, such that an
increase in the value along the axis
would cause a decrease in the value of the musical parameter. The phase
space could be quantized, interpreted
logarithmically, or multiplied. Clearly a comprehensive exploration of
possible mapping configuration would take several lifetimes.

For simplicity and the sake of space, only one mapping configuration is used in the examples included in Appendix A. This configuration maps values along the y-axis to pitch, and values along the x-axis to dynamic level. In all of the examples, the extent of the Cartesian phase is taken to be -1.43 to 1.43 on the x-axis, and -0.43 to 0.43 on the y-axis. These values were discovered empirically to encompass the totality of the phase space of the Hénon map under the conditions employed in the musical examples.

The first examples are straightforward and require little explanation or comment. Example 1 begins with an oscillation between two values which, during a brief transient phase, is quickly damped to a single point, a sequence of repeated notes. In Example 2, the transient phase consists of a widening oscillation which stabilizes on a period-2 attractor. In Example 3, the transient portion of the sequence appears shorter at first glance than it does in Examples 1 and 2, but in fact it is just about as long, requiring 16 iterations before it stabilizes. In Example 4, the transient portion is both longer and more irregular than in the previous examples. The period-8 orbit is finally stabilized after twelve seconds. Example 5 illustrates a period-16 oscillation following a short transient phase.

Example 6 is taken from a regime in which the phase space is divided into four distinct bands of chaos. Two broad layers are easily perceived, each of which is in turn subdivided into two bands. In the lowest band of the lowest layer, extending from approximately B--D# in the octave below middle C, a downward-tending three-note pattern is apparent as the primary constituent, which is interspersed with a longer pattern of either four or five notes. The second band, extending from approximately the F# below middle C to the D above, is similar to the first band, but with a contrary sense of motion. In the upper layer, the third band, from the C an octave above middle C to the F# above that, is similar to the lowest band, but with a slightly wider ambitus. The fourth and highest band is also like the first and third, but compressed in ambitus and offset in phase slightly.

Although it appears quite periodic, six distinct chaotic bands mark the texture of Example 7. The effect is of greater regularity or periodicity than in Example 6, because of the reduced range in which the values of each band may vary. Examples 6 and 7 are illustrations of what might be called "noisy periodicity." Example 8 illustrates an interesting sequence in which a long transient phase eventually settles into a period 7 orbit.

Example 9 is generated from the full-blown chaotic attractor plotted in Figure 4. Note that the louder notes are clustered in the middle of the range, corresponding to middle portion of the boomerang-shaped attractor. A clearly audible complex hocketing effect may be discerned from sonic realizations of this texture as the ear forms independent melodic lines from notes of similar velocity levels arrayed within consistent registral dispositions.

Example 10 is an example of sensitive dependence on initial conditions as applied to the generation of harmony. Three voices are generated synchronously, which proceed from initial points which are very close together: (0.000000, 0), (0.000001, 0), and (0.000002, 0). The three voices/orbits remain virtually identical through at least twenty iterations before the effect of their different initial positions precipitates a rapid divergence of the orbits from one another. Note that the first voice in this example (the one which begins exactly at the origin) is identical with the sequence in Example 9, with the reduced tempo taken into account.

The Lorenz System - Technical Description

The earliest explicit recognition that stable regimes of chaotic behavior
could arise
spontaneously
from the dynamics of a set of nonlinear equations is generally attributed to
the mathematician and meteorologist Edward
Lorenz in a landmark 1963 paper entitled
"Deterministic nonperiodic flow"^{26}. From a conventional model
of the dynamics of fluid convection, Lorenz evolved a simplified version expressed as a system of
three partial differential equations,

x - σ(y-x)

y = Rx- y -yx

z = xy = Bz

where, R and *
B
*
are positive constant parameters. The system becomes unstable if any of
these parameters are negative. The equations are nonlinear by virtue of the
multiplication of
the variable
*x *
with each of
the other two variables of the system. Solutions to the equations
may be described as a continuous trajectory
or a flow embedded in a three-dimensional
Euclidian space. The equations specify a
dissipative system whose basin of attraction
encompasses all of real space. The clear exhibition of extreme sensitivity
to initial conditions that Lorenz was able to demonstrate in the
behavior of this system had profoundly negative
implications for the feasibility of long-term weather prediction. Since,
by definition, an accurate measurement of the instantaneous state of
the atmosphere is impossible (because of the finite
resolution of measuring devices), then the long-term predictions of any
computer simulation used to model atmospheric conditions forward from the
present state will eventually diverge exponentially from the actual conditions.
This situation has come to be known as the
"butterfly effect," because of the implication that a phenomenon as seemingly
insignificant as the flutter of a
butterfly's wings in
Japan may exert an enormous influence on the weather
in California
a week later.

Lorenz adopted the parameter values 6 =
*
10, R = 28 *and *
B = 8/3 *
in his
numerical study of the equations. These
values reveal a chaotic attractor bounded within a
region of space whose extent is a function of the three constant
parameters. The orbit along
the
attractor describes an alternating sequence of spiral revolutions around two
*
stationary
points, *P1 and P2, located
at *(1√(B (R - 1)),
√(B(R - 1)), (R - 1)) *
and (-*√(*(B(R - 1)}, *
-i√(B(R -1)),
(R - 1)); *see
Figures 12 and 13. A typical orbit spirals out from one of the stationary
points a variable
number of times (as few as one, or as many as 50 or more revolutions) until
it is attracted towards
the neighborhood of the other stationary point, and begins the cycle
again.

Like the logistic equation and
the Hénon map, the Lorenz system exhibits
regimes of
single-point attraction, periodicity, and chaos. The creation of bifurcation
diagrams for the Lorenz system, however, is extraordinarily more expensive and
time-consuming to compute due to the fact
that the equations must be integrated over extended spans of time in
order for the underlying order to become apparent. In his extensive survey of
the system, Sparrow^{27} provides several hand-drawn sketches of the
locations of some of the bifurcation nodes in the total phase space. It is
from Sparrow's study that the following summary of the global behavior of the
system is condensed.

A third stationary point (in addition to P1 and P2) exists
at the origin (0, 0, 0), and is stable for
all parameter values -meaning that any orbit which begins there, stays
there. The origin is also globally attracting for
*
0 < R < 1 *meaning that all
orbits tend toward the origin and remain there if *
R < 1, *
no matter where they
are initiated.

The z-axis is invariant for all parameter values. Any orbit which starts on
the
z-axis (the line in the phase space along which both *
x *
and *
y
*
are 0) will remain on that axis
and tend
toward the origin.

**Figure 12:** Phase portrait of the Lorenz
system projected onto the (x,z) plane along the y-axis. The
attractor lies within a region of real space measured in the positive
direction along the z-axis. The
crosshairs at the centers of the two lobes mark the locations of the two
stationary points P1 and P2.
Constant parameters are *
a = 10, R = 28, B = 8/3. *
The initial point is (.1, .1, .1), located at the bottom
of the
graph right above the "X".

**Figure 13:** Phase portraits of the Lorenz
system projected onto (a) the (x,y) plane, and (b) the (y,z)
plane. Crosshairs mark the locations of the stationary points. Parameters
values and initial point are
the same
as for Figure 12. Both graphs plot the same orbit.

The critical
value *
R - 24.74
*marks a
reversal in the character of the two
stationary points P1 and P2. At the supercritical value adopted by Lorenz,
*
R 28,
*
these two
points are
non-stable or repelling, whereas at subcritical values of *
R, *
they are stable
or attracting. In other words, for subcritical values of *
R, *
all orbits are attracted to one of the stationary points P1
or P2, or to the origin. Orbits are not necessarily chaotic for all
super-critical values of *
R, *
however. Regimes of periodicity are interspersed within regimes
of chaos across a wide range of values. Figure 14 (a) illustrates a periodic
orbit at *
R = 150.
*A
change in the value of parameter *
B *
to 0.25,
however, while leaving *
R *
at
150, produces another chaotic orbit; see Figure 14(b).

**Figure 14: **Two orbits of the Lorenz attractor at
*
R = 150,
sigma
= 10; *
(a) periodic
orbit for *B=8/3,
*shown
without the initial transient; (b) chaotic orbit for *
B=1/4, *
including the
transient portion. Both
portraits are projections onto the (x,z)
plane along the y-axis, and the initial point is also (.1, .1, .1) for

both. Crosshairs mark the locations of the stationary points P1 and P2.

An alternate
perspective on the structure of continuous dynamical systems is
afforded by the *surface of section *
obtained by plotting the intersection of the orbit with a
plane which slices through its phase space. The resulting two-dimensional map
plots the
position of
the trajectory as it "punctures" the plane of the section. A method of
analysis of dynamical systems based on this
procedure was developed by Poincaré toward the end of the nineteenth
century. Also called *
Poincaré sections *
or *
return maps, *
these
two-dimensional reductions of the dynamics
of a higher-dimensional flow bear a strong resemblance to iterated maps
such as the Hénon system. For example, a Poincaré section of a periodic orbit
would show simply as a few points on the
plane, similar to phase plots of periodic regimes of the Hénon map. For a
chaotic orbit, a scattering of points across the plane is produced.
Figures 15 and 16 illustrate surfaces of
section taken across planes which pass through one
or both of the stationary points P1 and P2. These sections correspond to
the parameter values and initial
conditions plotted in Figures 12 and 14(b). Figure 16, especially, illustrates
the reduction of phase-space as the
orbit passes through its transient phase and into the region of the
attractor in the center of the plot.

Despite the
data reduction represented in a Poincaré section, its computation requires as
much time as the computation of the entire trajectory does, because it is
still necessary to compute all of the points
along the orbit between punctures of the sectional plane in order to
know where the next puncture will be. The derivation of a set of difference
equations (an iterated map) from a given set
of differential equations (a continuous flow), in a manner which would
allow a direct computation of the Poincaré section of the flow, is
unfortunately neither trivial nor obvious. There is some speculation that a
higher-dimensional flow exists for every true iterated map (of which the map
might be considered the surface of section
of the hypothetical flow), but the existence of such has not been demonstrated
to date. Nonetheless, Shaw^{28}
demonstrates morphological similarities between the Hénon
attractor and
the hypothetical section of a certain generic flow.

Further
discussion on the Lorenz equations may be found in Thompson and Stewart ^{
29},
Lichtenberg and Lieberman^{30},Helleman^{31},Ruelle^{32},
and Farmer et al.^{33}.

**Figure
****15: **Surfaces of section through the
Lorenz attractor, for the parameter values *sigma = 10, R = 28, *and *B
= 8/3;(a) *on the (x,y) plane at *z=(R-1)=27, *(b) on the (x,z) plane
at *y = 8.485281. *
Crosshairs mark the locations of the stationary points.

**Figure 16:** Surfaces of section through
the Lorenz attractor, parameter values *sigma = 10, R = 150,
*and *B = 1/4; (a) *on the (x,y) plane at

*z=(R-1)=149, *(b) on
the (x,z) plane at *y = 6.103278. *Crosshairs
mark the
locations of the stationary points.

Musical Examples

Musical mappings of orbits of the Lorenz equations were made with the two
sets of parameters
explored in the preceding technical section, namely *
sigma = 10, *
R = *
28, *
B *
=8/3 *
(Set A), and
*
sigma=10, *
R = *
150, *
B *
1/4 *
(Set B).
Because of the different region of phase space occupied by the orbit under
each of these two sets of parameters values, the
ranges along each of the three
axes of the coordinate space which map to minimum and
maximum values of the musical domain space are different as
well. These relationships are summarized in
Table 5. The first example (of Appendix B) explores the musical potential of
the Lorenz equations as a continuous
flow, while examples 2 through 5 map surfaces of
section into the musical domain. Except for
the second voice in the first example, the initial point of each
sequence is (.1, .1, .1).

Example 1 charts the courses
of two orbits which begin at almost the same
location, specifically (.1, .1, .1) and (.1001, .1001, .1001). The two
orbits remain virtually
identical for at least 100 seconds before the difference in their initial
positions causes them
to diverge. This example lasts
just over five minutes, a span barely long enough for the initial transients
to subside and for behavior more typical of the settled orbit to manifest
itself. Pitch varies according to
*
z, *
velocity according to
*
x, *
and the
inter-note onset time according to *
y. *
A given mapping
configuration has the effect of aligning parameters into consistent
relationships: for example,
around middle C notes are all at a medium velocity and speed, while the lowest
notes are also the loudest and fastest, etc. The low, fast dips correspond
to single revolutions of the orbit around the
stationary point located in the (-x, *
-y, z)
*quadrant. On the
grand staff, this point is located at approximately the F below middle C. The
"humps"
in the upper register
correspond to revolutions around the other stationary point, in the
*
(x, y,
z) *
quadrant. This point is located at approximately G above middle C.
Although the orbit makes only single and double
revolutions around the stationary points at the beginning, this behavior is
only transitory. A sequence of five revolutions around the lower stationary
point begins after 150 seconds (note the dips located approximately at 152,
157, 162, 168 and 176 seconds), and a
similar sequence around the upper stationary point begins after 180
seconds. The outward spiral of the orbit as
it is repelled from a stationary point is reflected in the increasing ambitus
of the oscillations in each of these sequences. The last revolution
before a transition to the opposite stationary point is always the greatest in
extent.

Example 2 is made from a
Poincaré section taken on the plane at *
z = R-1 =
27 *
(see Figure
15(a)), which includes both of the stationary points P1 and P2. In the graphical score, these two points are located at the
vertices of the sideways V-shaped figures
visible throughout the examples. These figures, opening to the right as time
moves forward, are evidence of the
outward spiral of the orbit around each point. The orbital information which is represented over a span of five minutes in Example 1 occupies
only the first ten seconds of Example 2. These examples thus present in
a condensed format a picture of the
behavior of the Lorenz system over a much longer span of time. As with the
continuous flow in Example 1, the
mapping aligns musical parameters in a consistent manner: all of the notes in the lower part of the range (around the lower stationary point) are
softer than those in the upper part of the range. Example 3 is a section taken
at *y =.413(R-1)) = 8.485282. *
Although this example is made from precisely the same
orbit as that of example 2, the pattern recorded is entirely different
due to change in the geometry of the sectional plane with respect to the course of the flow.

Example 4 is
made from a section at *
z = 149 (= R - 1) *
using the
parameters of Set B; see Figure 16(a)).
The drastic reduction in the range of the musical texture as the
initial transient dies out in the first
several seconds is a demonstration of the
loss
of energy
common to all
dissipative systems. The nature of the attractor visible in the center of the
pitch range is also clearly quite
different from that of examples 2 and 3, even considering the
change of scale. Example 5 is made from a
section at *
y *
14B(R-1)} = 6.103278 (see Figure
16(b)).
Interestingly, irregularities in the texture occur here at locations where the
corresponding texture in Example 5 is
fairly smooth. Like Examples 2 and 3, these sections
demonstrate the variation in textural contour which can be attained by
judicious placement of the sectioning plane.

**Table 5:** Parameters and coordinate space mappings

Musical Implications

In a discussion concerning historical changes in the structure of musical forms which have been brought about by changes in the nature of local structures, Boulez begins with a quotation from Claude Levi-Strauss:

Form and content are of the same nature and amenable to
the same analysis. Content
derives its reality from its structure, and what is
called *form *is the "structuring" of local
structures, which are called
content.^{34}

Boulez goes on to say:

We have seen
that the generation of networks of possibilities, which are the raw
material for *Popérateur -to *use a
significant term of Mallarmé' - has from the outset
tended increasingly to produce a material
that is constantly evolving. [W]e can only work towards connections
that are constantly evolving, and in the same way this morphology will be
matched by a correspondingly non-fixed syntax.^{35}

Although that was written well before the subject of chaos burst onto the international scene, Boulez' thoughts dovetail very nicely with the quality of material that one can expect to generate as a matter of course by means of nonlinear dynamical systems such as those discussed in this work. The subject matter of chaotic systems is, indeed, the structuring of local structures, in connections that are constantly evolving.

Nonetheless, there is a quality to the music of chaos which is distinct and
not to be confused with
products of the human mind -although there is also an undeniably
natural feel to much of the material produced by the
methods explored in this work. Still, the
quality of these materials is closer to that of figuration than to
*bel canto
*aria. If music may
be thought of as the
art of projecting, in an interesting way, a moving line or a group of
moving lines through a multi-dimensional grid measured in
terms of frequency, volume, time and timbre (which is itself
multi-dimensional), then nonlinear dynamical systems furnish a powerful and
novel means for the specification and control of these lines through such a
grid space. The process of understanding
these systems is the process of learning how they
behave under a variety of conditions. Their
global behaviors are learnable and therefore predictable from the point
of view of being useful in the generation of musical raw materials. Orbits
which are known to produce desirable effects may be calculated on demand. One
may treat these systems as musical commodities, natural resources to be
exploited at will. In fact, the term "natural resource" is doubly meaningful
because of its dual associations: textures
generated from these systems are "natural" in that they undeniably share
qualities of real-world phenomena to
which we are able to respond. Textures generated from these
systems are also "resources" in that they may
be exploited (or not) like any other musical resource.

There remains much room for
experimentation with more complex mapping
schemes not explored in this study. For example, the succession of (x,y)
coordinate pairs in orbits of the Hénon system need not be
consistently mapped to an invariant set of musical
parameters throughout the course of a
sequence, but may alternate between two
complementary mappings; e.g.,
(x_{0},y_{0}) to (frequency,
amplitude), (x_{1},y_{1}) to (frequency,
amplitude), (x_{2},y_{2})*
*back to (frequency,
amplitude), and so on. This produces a new musical
texture quite different from those presented in the earlier examples generated
from the Hénon map, but which still
contains essential qualities of that system. Orbital mappings may be
quantized on the computer to any degree, in any dimension: pitch, time,
volume, timbre. Additionally, the output of one system may be coupled to the
input of another system, and chaotic
sequences may be used to drive production grammars in place of white or
colored noises which are often used.
Nor need the utility of these systems be confined to the
generation of note textures. The Lorenz
attractor, for example, might be quite interesting as a
three-dimensional spatialization path, or in a waveform synthesis algorithm.

A number of interesting qualities are exhibited in chaotically produced textures. The paradoxical condition of a "consistent variety" of motivic material generated is foremost among them. The materials carry with them an inherent suggestion that they be employed in sequences of (at least) intermediate duration, because the character of the material is made manifest only after a number of iterations. While the overall behavior of a given chaotic orbit is, in a strict sense, not predictable, in an operational sense, it is. It is clear, for example, that the settled orbit of the Hénon map will not suddenly visit a point far from the attractor, or that a periodic oscillation will suddenly come to an end and turn chaotic. Nonetheless, these systems are productive of sequences of events in which sudden, unexpected changes may occur - albeit within the limits of their global repertoire of behavioral possibilities. This is a type of bahavior not seen in purely random

sequences, which maintain
an even consistency throughout, incapable of evoking surprise. Spectra of
chaotic systems
do, however,
resemble 1/f noise in that longer-term correlations are present and obvious (see
Farmer, et al.^{36}). Nonetheless,
the character is noticeably different. Chaos is generated
by *
deterministic *
procedures (there is therefore a presence of some measure of "volition,"
even if it is
only exercised by mindless equations), whereas white noise and 1 /f-noise are
*
probabilistic *
outcomes.

These qualities notwithstanding, the *
long-term *
behavior of all of these systems
is static. The equations contain no inherent specifications of drive or
impetuousness. There
is no
overarching teleological force guiding the sequence of notes to some grand
conclusion. There is no phrase structure. What
is missing in these materials is *
directionality, *
the sense
that the structure evolves according to the dictates of a human will. All of
these qualities
must be added
by the musician "by hand." Moments of localized directionality, in which a
serendipitous turns of events catches the attention, must be isolated and
elaborated.

Nonlinear
dynamical systems have the potential for creating the theoretical foundation of
a new and idiosyncratic idiom of computer music, of an intrinsic fluency wholly
independent of external derivations from tape
and instrumental music. Chaotic systems are a means by which computer music may
move beyond the limited possibilities of sequencing (a technique
appropriated from analogue electronic music) or the role of sound processor to
which it is often consigned. Dynamical
systems are versatile generators of
"meta-sequences" - sequences of motivic material made more flexible by virtue of
their specification as *
rules
*of natural
behavior rather than as fixed sets of notes. Chaos affords composers working
with computers the opportunity to address musical issues at a level of
organization which has remained largely uninvestigated in the medium to date:
the generation of melodic and harmonic materials at the note level.

Appendix A - Score Examples

**Example 7:** chaos in six
bands (A - 1.078)

**Example 8:** period-7
oscillation afater and extended transient (A- 1.3)

**Example 9:** chaos (A - 1.4)

**Example 10:**chaos,
illustrating sensitive dependence on initial conditions

**Example 11:** two voices
divergin due to sensitive dependence on initial conditions

**Example 12:** section at z=
27

**Example 13:** section at y -
8.4852828

**Example 14:** section at z -
149

**Example 15:** section at y -
6.103278

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