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David Clark Little Cornelis Springerstr. 14-2 1073 LJ AMSTERDAM
Abstact: In this paper the author shows where concepts and mathematical models derived from the developing field of Chaos Science can be applied to electroacoustic and instrumental composition. Examples of non-linear dynamics include Lorenz's model of fluid behaviour, Verhulst's model of population growth, Hénon's analysis of the multiple celestial body problem, Barry Martin's Algorithm which produces quasi-organic forms, and the 'Baker' mixing function. Besides broadening the numerical techniques available for electronic music generation, concepts such as fractal structure, feedback process and iterative function can be applied to 'ordinary' composition as well. For example, in designing melodic curve, defining meter, planning instrumentation, manipulating symbols, creating ornamentation and elaboration, etc. Some suggestions as to mapping are made, the critical boundary between science and art. Musical examples are used from the following works by the author: Harpsi-Kord for harpsichordist and tape, Fractal Piano for computer-guided pianola, Brain-Wave for recorder-players, The Five Seasons for 6 percussionists and tape, Modi-Fications for marimba & tape, and Hyperion's Tumble for tape.
300 years ago Newton formulated the laws of motion which laid the ground-work for a clockwork view of the universe. By the late 18th century the French astronomer Laplace optimistically stated that intelligent creatures could know any past or future state of the universe, if they only knew well enough its present state, what direction it was heading towards, and had powerful enough calculating methods. This deterministic world view has proved to need revision. Scientific and mathematical developments of the last 30 years have led to new insights into subjects, which because of their complexity, had previously been swept under the rug by the scientific establishment. Intractable problems in weather forecasting, the modelling of wildlife populations, the geometry of nature, the understanding of turbulent flow and bio-rhythms gave startling new results when revolutionary methods of analysis were applied. As a result, words such as "chaos", "order", "simple" and "complex" have been redefined; and a new concept formed: "fractal".
Ironically, it took the advent of the deterministic tool par excellence namely the computer, to cause many contemporary scientists to rethink the whole matter. With mathematical models they had been able to make accurate predictions of planetary motions and tides, for example. Everyone had thought that long-range weather prediction should also be possible; you just had to make much more calculations. In 1961 the meteorologist Edward Lorenz managed to model the Earth's weather on a computer; one could follow recurrent "rain storms", or "cyclones", etc. Only there was a problem: if he started the program with slightly different initial conditions of wind speed and temperature, the artificial weather would be the same as in a previous run only in the beginning. After a while, the "weather" would diverge from the previous run, and eventually end up completely different!
To appreciate what this means, one must remember that the computer model was using proven physical laws of gas and water behaviour; and the computer ran completely deterministically with no additional input after it was started. With Lorenz dawned the idea that long-range forecasting was impossible. Small errors in measurements would multiply, cascading upwards in the scale of turbulence: from a puff of wind to continent-sized spirals. Lorenz called it the "Butterfly Effect"- theoretically, a butterfly stirring its wings in Peking could start a storm over New York the next month!
Lorenz later developed a more general mathematical model of fluid behaviour. It describes the flow of heated fluid, called convection. For example, when a pan of water is heated, the hotter water at the bottom tends to rise, because it is less dense. At the top of the pan it comes into contact with air which cools it off somewhat. Then the cooled denser water sinks back to the bottom of the pan. This circulation of fluid is called a convection cell, and remains smooth and orderly as long as the heat under the pan is moderate. However, if the heat is high, the water moves to fast to cool off very much; the convection cell breaks up and flow is turbulent, as portions of the water compete with each other to get to the top. Lorenz took the Navier-Stokes equation which describes fluid flow, and simplified it to get an equation to model convection, using three variables in non-linear relationship. (A linear relationship is where a change in one variable is mirrored by a proportional change in another variable: its graph is a straight line. A graph of a non-linear relationship, on the other hand, might show breaks, reversals, bends, etc.)
xnew=x+d*a*(y-x) ynew=y+d*(x*(c-z)-y) znew=z+d*(x*y-b*z)a, b, c and d are constants with the values 10, 8/3, 28, and .003 respectively. A new value is calculated for each variable, dependent on its previous value and the other variables in various proportions. A loop is set up by plugging the new values from a calculation into the variables for the previous state (e.g. x=xnew). Then we can run the calculations all over again. The change in values of the variables with time can be traced out in what's called a phase diagram (See fig. 2.)
A point on the diagram represents the physical state of a system, actually in three dimensions. If a system heads toward a stable final state, its phase diagram would tend to localise to a point, called the attractor. For a periodic system, the phase diagram would tend to be a closed loop of some kind. Lorenz's model appears to be chaotic, with a kind of infinite complexity; it has a strange attractor! The trace of the model loops endlessly without repeating or crossing itself, flipping unpredictably from one side to the other. It does remain within bounds, however, and is not random; a pattern emerges resembling butterfly wings. Indeed, a new kind of order was discovered which was to reveal itself in analysis of many different natural phenomena; order within chaos. Lorenz's work started the revolution which was, like his "Butterfly Effect" to spread to many fields outside of meteorology.
At this point I'd like to describe pieces of mine which use some of the ideas just described. Harpsi-kord for tape and harpsichordist was composed in 1988. In this piece the central idea is order within chaos. Compositionally, it swings between the poles: regular/irregular, loud/soft, atonal/harmonic, the use of timbre from an ancient instrument or electronically generated. The middle ground is sought for by transformations sometimes possible only through new techniques: 'samples' of harpsichord sounds were adapted electronically. Sometimes techniques were turned on themselves; having sampled a tone-cluster, it was available on each note of a synthesizer. Clusters of clusters were made. Similarly, rhythmic or melodic structures were nested in several layers at times. For example, one samples not a single tone, but a melodic motive, and loops it. By holding several keys with the same sample, one generates a polymetric texture, because the same loop at higher pitch plays faster, hence is shorter. One can sample this whole texture, and repeat the process, achieving very soon the limits of human perception regarding detail! The harpsichordist relates to the tape in a quasi-improvisational manner. Although the timing and pitch material is exactly notated, he/she is given considerable freedom in performance. For example, only the pitches were notated in a square, with the rhythm and ordering "randomly" improvised. (See fig. 3.) In this way a "feedback loop" is created; the improviser must use his/her ears and think fast in order to create a proper "dialogue" with the tape.
The next two pieces, Shuffle and Fractal Piano 6 (both from 1988) were realised with the help of the "Vorsetzer". The Vorsetzer is a new form of pianola developed by the technicians of the Electronic Studio of the Sweelinck Conservatory, Amsterdam. It has 88 electromagnets mounted over the keys, which can be triggered with varied degrees of force by a computer. The obvious advantage of this system over the old method of punching out rolls of paper is the inherent flexibility and compactness of data storage with computers. In addition, the use of the computer offers new compositional possibilities.
To make Shuffle, an 88-note chromatic scale was produced and manipulated
by a computer. The scale accelerates smoothly from relative note values
of quarters in the lowest register to 32nds in the highest. It has a dynamic
curve of p Fractal Piano 6 is one of a series of studies
in which a computer program I developed was used in combination with the
Vorsetzer. The heart of this program is a mathematical
model of population growth, first derived in 1845 by P.F. Verhulst.
(It is often referred to as the logistic equation.) I'd like to describe
it in some detail because although simple, it contains profound implications.
The Malthusian Model describes the unbounded growth of a population
(of fruit flies, for example) with xnew=a*x. This formula tells us that
we can find the population of a new generation by multiplying the number
in the last generation with a productivity factor. Suppose the population
doubles each generation; then a=2, and starting with 2 parents, we'd get
the series: 4 children, 8 grandchildren, 16 great-grand children, etc.
It's easy to see that before many generations have been bred, we have a
gigantic number. By the 10th generation that is 1024 siblings!
In order to make a more realistic model, Verhulst considered that in
nature, the larger a population grows, the less productive it becomes,
perhaps because of lack of food or other overpopulation problems. So in
creating his (abstract) model, he says, let's set the upper limit of a
population at 1. (Think of it as 100% of the room available for growth).
Then the room left over by the environment for a new generation is 1-x.
This can be seen as a correction factor to unbounded growth. The Verhulst
Model for limited population growth then becomes: xnew=a*x*(1-x). The population
of a new generation is equal to the malthusian growth factor times the
old population, and scaled down by the amount of room available for growth.
In spite of its simplicity, it proves to be a fair model of what happens
in nature. If the productivity factor a is 2, then starting the formula
with a low seed value like 0.001, we see the population x rise and level
off at 0.5. This is what we might expect in nature with animals with a
healthy productivity. After an initial period of fast growth, the population
stabilises.
If we set the productivity factor a to higher values, strange things
happen. If a is 3.2, x grows rapidly at first, but then doesn't stabilise
to one value; rather it alternates between two values endlessly. (See Fig.
4a.)
It doesn't matter what the seed value was, x ends up alternating between
the same two values. If a is set a little bit larger than 3.4495, we find
the values for x orbiting between four values eventually. Carefully increasing
the value of a for still more trials, we find that the number of values
that x seems to land on keeps bifurcating (to 8, and 16) until there is
a value for a, 3.569946, just beyond which x fluctuates chaotically from
one value to the next. Sometimes it bounces back and forth between a couple
of values for a while, only to spin off again. (See fig. 4b.)
This type of chaotic behaviour is also observed in nature, for example
by an animal with a productivity so high that it overreaches the ability
of the environment to support it. The population crashes, only to build
up again. The interesting thing about the model is that it does show a
kind of regularity, with x-values jumping up and down, but it never repeats
itself exactly. This simple, deterministic mathematical formula can be
just as erratic as measurements of real populations in nature!
There are more mysteries lurking here. While searching for the
exact values of a where the behaviour of the model changed-where x values
would settle down eventually to one, two, four or eight values-the physicist
Mitchell Feigenbaum recently discovered a constant ratio between the a
values. Still more astonishing was the discovery that other quite different
mathematical formulas (still using an output-input loop to calculate a
new value from an old value), and also experimental data exploring the
onset of turbulent flow, also showed the doublings, and the same ratio
between them, 4.6692... In short, a new universal constant was discovered
by Feigenbaum, like the constant of gravity, the speed of light, or the
weight of an electron.
We're not yet finished with Verhulst's model. If a is increased to 3.83,
the chaotic behaviour eventually stops, and x circles eventually between
only three values. (See fig. 4c.)
Increasing a in small amounts for new trials results in period doubling
of the values where x eventually settles down to- 6,12; and again chaotic
behaviour sets in, up to a=4. (See fig. 4d.)
(We cannot set a to a number greater than 4, because that would produce
x values greater than 1, or exceeding our original definition of the maximum
population). With the help of a computer, a graph can be made of how Verhulst's
formula behaves for all settings of the a value. (See fig. 4e.)
We see the doublings of x at so called bifurcation points, followed
by chaotic regions, then windows, where x again has a low number of stable
values. We get a shock of recognition when we magnify the region where
x splits up again; the whole pattern reveals itself in miniature! (See
fig. 4f.)
Indeed, it seems that the pattern contains nested within itself, its
own replica! This kind of nested pattern is now called a "fractal".
The Polish-born mathematician Benoit Mandelbrot derived the term from the
Latin adjective fractus, meaning irregular or broken. Fractals are characterised
by intricately nested patterns within patterns, with self-similarity on
any scale. Fractals can be recognised in a wide range of natural phenomena
and shapes, such as trees and clouds. Analysis of Indonesian Gamelan music
reveals fractal structure. (See fig. 5).
The rhythmic punctuation fits a pattern based on the series 2,4,8,16,
etc.; and the nuclear melody is performed simultaneously on several instruments
at different speeds.
To compose Fractal Piano 6, values obtained from iteration of the Verhulst
formula where encoded using non-linear mapping (with a partially "shuffled"
scale, or with a selected or weighted element set) into pitches, lengths
and loudnesses. This MIDI-data was edited with the help of a commercial
sequencer program; stretched and squeezed time-wise, and layered in various
ways, using "fractal" structures. For example, the on/off pattern shown
in fig. 6 was used as a mask to create fragmented density in one part of
the piece. Say the upper register of the piano turns on and off at intervals
of 2 sec. When this register is "on", material is audible within this register.
When it is "off", it is silent. In a register just under the highest one,
the mask turns on and off every 4 sec.; in a register just under it, every
8 sec.; and so on. By applying such a mask over the (potentially endless)
chaotic material, I find a kind of musical tension is generated. Notice
that the whole mask pattern produces all possible on/off combinations for
the chosen number of registers. It is related to the I Ching, with its
64 possible combinations of 6 solid or broken lines.
Flocking animals co-ordinate in a remarkable and still incompletely
understood fashion. The reaction time of a group in danger, or in making
turns is considerably faster than the reaction time measured of isolated
individuals. In order to maintain the proper distances between neighbours
without collision, some sort of multi-sensory positive-negative feedback
mechanism is in operation. Neural physiology has revealed that the massively
interconnected neural network in the individual brain operates with feedback
processes. A neuron cell has a main body, an axon from which it receives
signals, and treelike extensions called dendrites which branch off in hundreds
to make contact with other cells. Connections between axons and dendrites
are effected across gaps, called synapses. Neurons send out impulses spontaneously
at a rate of about 10 per second. However, the rate of firing changes,
and depends on the sum total number and strength of the impulses it receives.
There are both excitatory and inhibitory synapses: signals from the former
tend to increase the firing rate of a cell, while the latter tend to reduce
the firing rate of a cell. The picture of ceaseless electrical activity;
signals amplifying, muting, modulating, crossing each other, and returning
in loops; all in incredibly complex and indecipherable wave-like patterns:
this picture gives us an idea how thought and memory are possible. Recent
investigation of the physiology of perception has led to the discovery
of chaos in the brain: complex behaviour which seems random, but has a
hidden order. Vast collections of neurons shift quickly from one complex
pattern to another, in response to the smallest of inputs (remember the
butterfly effect). An organism as a whole acts in its environment with
feedback mechanisms. The brain seeks information, and sends signals to
muscles to place sensory organs in position, and to sensitise parts of
the brain which will process signals. A burst of collective patterned activity
from all sensory organs is combined to form a gestalt. Then a fraction
of a second later, another search for information is demanded. It seems
that chaos in the brain is not pathological, as one might expect; but instead
is the basis of healthy functioning, indeed explains how the brain can
respond quickly and flexibly to an ever-changing outside world. Even what
we experience as a original idea (brain-storm) may be derived from a chaotic
neural firing pattern triggered in an ever-widening cascade from a small
initial impulse.
In my piece Brain-Wave for at least 3
recorders of any kind, (1989), I wanted to set up a self-regulating musical
situation. All musicians improvise on the same basic material, which is
arranged in four cycles, each with four events. (See fig. 7).
Performers should stand or sit spread out around the hall, possibly
on different levels. Each player should face an arbitrarily chosen direction.
Emphasis is placed on influences which performers take from their neighbours.
Players(s) in front of an individual give positive influence, and player(s)
in back give negative influence. Here is a table summarising these influences:
In 1989 I completed The Five Seasons
for 6 percussionists and tape. My inspiration source was an ancient Chinese
theory, in which the Seasons, Emotions, Colors, Elements and Directions
were grouped as follows:
The first part, Spring, begins with accelerandos of accelerandos.
First a pulse plan was worked out; the distance between pulses starts large,
with successive pulses scaled by a ratio such as 2/3rds down to small (fast)
intervals. Then an accelerating figure was fitted to each of the pulses.
(A similar slowly accelerating roll is found in Chinese opera and Korean
ceremonial music.) There are three layers, played by wood blocks, temple
blocks, and log drum. (See fig. 8).
Later on, a 16-note theme in quarter-notes is introduced in the bass
marimba. The melodic curve of this theme is a projection of a fractal graphic
design I made. Here a 4-note melodic motive is fitted or transposed into
a blown-up version of itself (See fig. 9).
Such nested patterns with scaled elements are characteristic of fractals,
as already described. There follows a metric canon; the theme enters in
eighth notes, then triplet eighths, and finally sixteenths. (See fig. 10).
One can consider the whole construction as a fractal of a fractal, since
the theme pattern (itself a fractal) occurs simultaneously at different
speeds and octaves. After another metric canon and a section with controlled
improvisation, this theme returns with a different treatment. It is split
up into 4-note fragments, and given a peculiar "doubling": not parallelism,
but an exaggeration of the melodic curve, using multiplication. Again,
such "scaling" is a common method of constructing geometrical fractals
(See fig. 11).
In the second part, Summer, a wiring scheme which includes two feedback
loops was used for electronic FM synthesis: the output of any generator
provides input control voltages for two other generators. In one loop,
when the voltage output of a Low-Frequency-Oscillator (LFO) is high, it
causes its neighbour LFO to oscillate at a higher rate. In the other loop,
inverted signals are sent out: in this case a high output voltage of a
LFO causes a lower rate of oscillation in a cross-connected LFO. The output
of all LFO's was used to control other electronic devices, to synthesize
a sound. Because of the interconnectedness, and the complex interaction
of positive and negative feedback loops, the results of such a circuit
can be unpredictable and chaotic.
The Summer is divided into four sections, each with a clearly
defined instrumentation. Each section and all instruments have similar
material rhythmically, generated from the Verhulst Model. There are different
scaling factors applied to the material for different instruments playing
together in an ensemble, controlling the relative densities of attack.
The third part, Aftersummer, uses an on/off masking scheme like that
used in Fractal Piano 6. Here, not registers on the piano, but six different
percussion timbres (all with a sharp decay) are "turned on or off". As
in Fractal Piano 6, we get all 64 possible combinations of the six elements,
and a kind of fractured crescendo. Rolls and repeated notes of various
tempi, but always in decrescendo, provide "thematic" self-similarity. (See
fig. 6 again.)
The fourth part, Fall, depicts musically the "Butterfly Effect",
previously described. In the last measures of Aftersummer, all six players
have finally come together. In the first measure of Fall, they play all
together again (this time on metal instruments); and then disperse. Sometimes
2 or 3 players synchronize for a while, but small deviations lead to larger
separations, and this part ends fragmented and scattered. Loosely spoken,
this part is an inversion of how Spring begins: Fall contains a ritardando
of ritardandos.
For the fifth and last part, Winter, I used a technique I call "nested
repeats" to create the metric structure. Difficult for a human, perhaps,
but a computer can easily carry out the following set of commands:
Modi-fications for large marimba and tape (1990)
makes use of what I call "transposing modes". These are constructed like
fractals, with an interval structure repeated indefinitely. For example,
take the interval cell [1,4,2] (a semitone=1). Starting with a low E and
repeating the cell, we get E,F,A,B,C,E,F#. Notice that because the elements
of the cell add up to 7, two cells don't complete an octave, but overreach
it. Indeed, we must repeat the cell 12 times before we get the same pitch-names.
Playing "scales" up and down through this mode, we get continuous transposition
through the cycle of fifths. The ordering of much of the material in this
piece was achieved with a computer program I worked out called "statistical
feedback". A weighted random choice between a string of elements; only
the order of preference among the elements is always changing, depending
on previous choices. What it does is make a "chaotisation" of serialism.
I have used this program on several different element sets in composing
the marimba part as well as the tape part of this piece. Pitch, note length
and dynamics; as well as larger structures: mode, section length, and electronic
timbre; all material and forms are subject to modification.
In 1993 I made Hyperion's Tumble for tape,
using computer algorithms. Observations of Hyperion, a small, irregularly-shaped
moon of Saturn, provided some of the first evidence that celestial motion
is not merely giant clockwork. With an eccentric orbit phase-locked in
3/4 ratio with Titan (Saturn's largest moon), Hyperion tumbles end-over-end
in sometimes periodic, sometimes chaotic fashion, subtly influenced by
gravitational forces.
Newton had solved the problem of 2 bodies interacting gravitationally:
depending on their energy and mass, they move in perfect curves: a circle,
ellipse, parabola or hyperbola. The problem of 3 bodies interacting gravitationally
has proved to be surprisingly difficult, and mathematician Henri Poincaré
has shown that in the long term, their motion can only be approximated,
and is in essence unpredictable. He invented a method to visualise the
complicated behaviour of such a system, now called a Poincaré map.
A 2-dimensional slice of a three-dimensional phase space will show either
one or a few points if the system is periodic, and a complicated figure
if it is chaotic. An object with a chaotic phase space might have a degenerate
or unstable orbit, causing it to crash into another body, or fly off into
infinite space. Close examination of these figures, called strange attractors,
proves that they are fractals. Curves are folded into themselves, with
infinite regress: increasing magnification shows evermore detail, but with
recurring proportional patterns (see fig. 12).
French astronomer Michel Hénon has also
demonstrated the theoretical possibility of chaos in the cosmos, when he
modelled stellar orbits in galaxies, with the computer. Depending on how
the model was set up, stellar orbits would show different behaviours: at
low energy levels, orbits were regular ellipses. Higher energy levels gave
more complicated orbits, which never exactly repeated themselves, and beyond
a certain energy level, the orbits became unstable and unpredictable. He
wrote a simple equation to explore the folding and remapping of an oval
onto itself, which produces an archetype of strange attractors (see fig.
13).
Blowing up a strand of the attractor reveals tiny strands within it,
spaced from each other in the same ratios as the parent strands.
Two computer programs I wrote based on chaos theory enabled me
to generate voltage fluctuations for synthesis. A
formula discovered by Barry Martin generates chaotic orbits, two-dimensional
plots of which resemble organic structures such as cells under a microscope
(see fig. 14).
Different initial values for the constants a,b, & c result in different
patterns and periodicities.
I call the second program the "Baker
function". A string of integers is folded into itself recursively,
mixing the integers completely. However there are strange periodicities
occurring, and eventually the original string mysteriously re-occurs. For
example, imagine picking up the integer-string in the first row, below,
by the middle. You have the '5' between your fingers, and two ends dangle
below. Read off the numbers, starting at your fingers and alternating between
the string ends as you move down, to get the second row. Repeat the process
to generate the other rows. This process is similar to the one a baker
uses to mix dough: flatten with a rolling pin, fold over a half, flatten
again, fold again, etc.
What holds our attention in listening to music? Music from Bach to Bartók,
Josquin to Xenakis, from Bali to Bolivia has some special kind of pattern
which hovers in a phase space between repetition and randomness, between
association and breaking-away, between order and chaos. Does a strange
attractor underlie a piece which gives us a feeling of anticipation &
resolution, of simplicity within complexity? Do its patterns show resemblance
to the fractal geometry identified in nature? Does the new definition of
chaos- that dynamics can be paradoxically both deterministic and unpredictable-
help us understand how a particular sequence of sounds gives us the feeling
both of inevitability and surprise? Perhaps composers as well as scientists
may do well to take a new look at Chaos.
June 2, 1995 David Clark Little (1952, USA). After receiving a
BS in chemistry, he studied harpsichord, finishing with Gustav Leonhardt,
and composition with Ton de Leeuw, in the Netherlands. He has been a finalist
and prize-winner in several composition competitions, including in the
USA, Germany, France and Greece; and has been given many grants, for example
to attend festivals and workshops in Germany, Holland, and the Soviet Union;
and has received many commissions for compositional work. Since 1988 he
has worked on compositional methods using the computer and based on the
new "chaos science" and "fractals". Scores of his music are available from
Donemus, Paulus Potterstr. 14, 1071 CZ Amsterdam, the Netherlands.
Positive influence
If: player(s) in front are playing, one may: start to play. play louder.
play more repeats of an event. attempt to imitate style of other player(s).
try to match the speed(s) of other player(s).
Negative influence
If: player(s) in back are playing, one may: stop playing. play softer.
stop, or go to next event. attempt to play in a style opposite to other
player(s). play fast when other player(s) are slow, and vice versa.The
object is not to synchronise exactly with the other players, but to correlate
what a player does with what the others do. Since there is no conductor,
each player must partly assume that function, being attentive to the ensemble
sound, and taking initiative to lead that sound were he/she thinks it should
go. The aim is to create an interactive situation such as is found in nature,
among flocks of birds, or in brain neurons, for example.
Spring Anger Green Wood East
Summer Joy Red Fire South
Aftersummer Sympathy Yellow Earth Center
Fall Grief White Metal West
Winter Fear Black Water North
This piece incorporates several techniques which were derived from Chaos
Science. My adaptation of the Verhulst Model was used to derive some of
the rhythmic material for the piece, as in Fractal Piano 6. Fractal structures
define the form of several sections. Some of the electronic sounds on the
tape were made using feedback loops for frequency modulation. The performers
are called on to improvise in one section. I'd like to go into more detail
with this piece to show how these techniques work.
j+[h+[f+[d+[b+[a]+c]+e]+g]+i]+k
=jhfdbaacbaacedbaacbaacegfdbaacbaacedbaacbaacegih
fdbaacbaacedbaacbaacegfdbaacbaacedbaacbaacegik.
Here, the [ ] signs indicate a simple repetition. [a]=aa, for example.
The nesting of the repeats makes that a gets printed 32 times, b and c
16 times, d and e 8 times, f and g 4 times, h and i 2 times; and j and
k get printed only once each. Whether you look at a small or large part
of the list above, it displays the self-similarity typical of a fractal.
For Winter, I desired a metrical structure with many changes but internally
consistent. First I decided what was going to happen in each measure, in
terms of instrumentation and so on, and then let the length in eighth-notes
of a measure be determined by substituting the numbers 2-12 for a-k in
the fractal pattern.
xnew=y+1-1.4*x^2
ynew=0.3*x
xnew=y-SQRT(ABS(b*x-c))*SIGN(x)
ynew=a-x
1 2 3 4 5 6 7 8 9
5 6 4 7 3 8 2 9 1
3 8 7 2 4 9 6 1 5
4 9 2 6 7 1 8 5 3
7 1 6 8 2 5 9 3 4
2 5 8 9 6 3 1 4 7
6 3 9 1 8 4 5 7 2
8 4 1 5 9 7 3 2 6
9 7 5 3 1 2 4 6 8
1 2 3 4 5 6 7 8 9
Irregular motoric sounds resulted from the transcription of the output
of these algorithms. A chaotic sound lies in a spectrum between a sinus-tone,
which is perfectly periodic, and white noise, which is perfectly aperiodic.
It can have a maximum of complexity, always producing more detail, or information,
within certain limits. In principle, computer chaos can be used to model
on any level, from musical structure to musical sound.
References:
Dewdney, A. K.: Wallpaper for the Mind; Computer Recreations, September 1986 Scientific American
Freeman, Walter J.: The Physiology of Perception, February 1991 Scientific American
Gleich, James: Chaos, Making a New Science; 1987 Viking
Hawking, Stephen W.: A Brief History of Time; 1988 Bantam Books
Hofstafter, Douglas R.: Godel, Escher, Bach; 1980 Vintage Books
Hofstafter, Douglas R.: Metamagical Themas (Mathematical Chaos and Strange Attractors); November 1981 Scientific American, and 1987 Penguin Books
Jones, Jesse: Fractals for the Macintosh; 1993 the Waite Group, Inc.
Lauwerier, Hans: Fractals; 1987 Aramith
Mandelbrot, Benoit: The Fractal Geometry Of Nature; 1983 W. H. Freeman and Company
Manzolli, Jonatas: FracWave Sound Synthesis
Peterson, Ivars: Newton's Clock, Chaos in the Solar System; 1993 W. H. Freeman and Company
Prigogine, Ilya: Order out of Chaos; 1985 Fontana Paperbacks
Russell, Peter: The Brain Book; 1979 Routledge
Stewert, Ian: Does God Play Dice? The Mathematics of Chaos; 1989 Basil Blackwell Ltd.
Watson, Lyall: The Dreams of Dragons; 1987 Morrow, New York
Biography David
Clark Little
