2013

Robert Braileanu

University of West London

Fractal Sound. Research Folder

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Contents

1. Introduction .................................................................................................................................... 3

2. Pierre Fatou ..................................................................................................................................... 4

Pierre Joseph Louis Fatou ........................................................................................................................ 4

Born: 28 February 1878 in Lorient, France Died: 10 August 1929 in Pornichet, France...................... 4

3. Gaston Julia ..................................................................................................................................... 6

Gaston Maurice Julia ............................................................................................................................... 6

Born: 3 February 1893 in Sidi bel Abbès, Algeria Died: 19 March 1978 in Paris, France ..................... 6

4. Natural Fractals3 ............................................................................................................................ 11

4.1. Atronomy ............................................................................................................................... 11

4.1.1. Galaxies .......................................................................................................................... 11

4.1.2. Rings of Saturn ............................................................................................................... 12

4.2. Bio/Chem ............................................................................................................................... 13

4.2.1. Bacteria Cultures ............................................................................................................ 13

4.2.2. Chemical Reactions ........................................................................................................ 13

4.2.3. Human Anatomy ............................................................................................................ 14

4.2.4. Molecules ....................................................................................................................... 15

4.2.5. Plants ............................................................................................................................. 18

4.3. Other ..................................................................................................................................... 19

4.3.1. Clouds ............................................................................................................................ 19

4.3.2. Coastlines & Borderlines ................................................................................................. 21

4.3.3. Data Compression .......................................................................................................... 22

4.3.4. Special Effects ................................................................................................................ 23

5. Other Fractals4 ............................................................................................................................... 24

A Simple Explanation Of Fractal Geometry ......................................................................................... 24

5.1. The Cantor Set ....................................................................................................................... 28

5.2. The Koch Curve ...................................................................................................................... 28

5.3. The Sierpinski Triangle ........................................................................................................... 30

6. Algorithmic Composition5 .............................................................................................................. 30

6.1. Introduction ........................................................................................................................... 30

6.2. Pre/Non-Computer Practices.................................................................................................. 31

6.3. Use Of The Computer ............................................................................................................. 33

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6.4. Closing ................................................................................................................................... 39

7. Max MSP ....................................................................................................................................... 40

7.1. Introduction ........................................................................................................................... 40

7.2. Language................................................................................................................................ 40

9. References ..................................................................................................................................... 42

Data CD content

Order Content Type

1 Michael Hogg - Slow Deep Mandelbrot Zoom

Video

2 John Cage – Atlas Eclipticalis Audio

3 Lejaren Hiller- Illiac Suite for String Quartet - Part 1

Audio

4 Iannis Xenakis-ST/10=1,080262 Audio

5 Fractal Sound Generator MaxMSP application

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1. Introduction

The purpose of this document is to provide additional information regarding matters discussed in the

main written document – Fractal Sound; it is an integral part of this project, aiding the reader in fully

engaging with the subject. The information provided here is in the form of website excerpts, which have

been referenced accordingly in section 9 – a superscript index identifies the reference number in the list

(eg.1). Furthermore, original ideas based on materials gathered as part of this project are included in this

document. This document has only been submitted as a physical copy, however, should a softcopy be

required, a digital version can be produced at the reader’s request. Contact details can be found in the

opening page of the main document.

This collection of research material has been produced for logistic reasons. A limited word count applies

to the main body of work and as such, this document includes additional information which could not be

included in the main written account. A notation system is in place towards the end of the executive

summary, linking the two documents together. It is recommended that the system is acknowledged and

used appropriately, as it provides a simple means of fully grasping the ideas being discussed.

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2. Pierre Fatou

Pierre Joseph Louis Fatou

Born: 28 February 1878 in Lorient, France

Died: 10 August 1929 in Pornichet, France

Pierre Fatou entered the École Normanle Supérieure in Paris in 1898 to study

mathematics. He graduated in 1901 and then decided that the chance of obtaining a

mathematics post was so low that he would apply for a position in the Paris

Observatory.

Having been appointed to the astronomy post, Fatou continued to work on

mathematics for his thesis. He submitted his thesis in 1906 which was on integration

theory and complex function theory. Fatou proved that if a function is Lebesgue

integrable, then radial limits for the corresponding Poisson integral exist almost

everywhere. This result led to generalisations by Privalov, Plessner and Marcel Riesz.

Although not giving a complete solution, Fatou's work also made a major contribution

to finding a solution to the related question of whether conformal mapping of Jordan

regions onto the open disc can be extended continuously to the boundary. In 1907

Fatou received his doctorate for this important work.

The book [2] presents a beautiful historical account of the global theory of iteration of

complex analytic functions. Fatou enters this history in a rather complicated way and

the book does an excellent job in explaining an interesting episode in the history of

mathematics.

In 1915, the Académie des Sciences in Paris gave the topic for its 1918 Grand Prix.

The prize would be awarded for a study of iteration from a global point of view. The

author of [2] suggests that mathematicians such as Appell, Émile Picard, and Koenigs

had put forward the idea to the Académie des Sciences because they were hoping for

developments of Montel's concept of normal families. Fatou wrote a long memoirs

which did indeed use Montel's idea of normal families to develop the fundamental

theory of iteration in 1917. Although we do not know for certain that he was intending

to enter for the Grand Prix, it seems almost certain that he undertook the work with

that in mind.

Given that the topic had been proposed for the prize, it is not surprising that another

mathematician would also work on the topic, and indeed Julia also produced a long

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memoir developing the theory in a similar way to Fatou. The two, however, chose

different ways to go forward. During the later half of 1917 Julia deposited his results

in sealed envelopes with the Académie des Sciences. Fatou, on the other hand,

published an announcement of his results in a note in the December 1917 part

of Comptes Rendus. It later became evident that they had discovered very similar

results.

Julia wrote a letter to Comptes Rendus concerning priority which was published on 31

December 1917. Julia had asked the Académie des Sciences to inspect his sealed

envelopes and Georges Humbert had been asked to carry out the task. In the same 31

December 1917 part of Comptes Rendus Georges Humbert has a letter reporting on

Julia's papers. Almost certainly as a result of these letters Fatou did not enter for the

Grand Prix and it was awarded to Julia. Fatou did not lose out completely, however,

and even though he had not entered for the prize, the Académie des Sciences gave him

an award for his outstanding paper on the topic.

Fatou was given the title of "astronomer" in 1928 and, as an astronomer, he also made

contributions to that topic. Using existance theorems for the solutions to differential

equations, Fatou was able to prove rigorously certian results on planetary orbits which

Gauss had suggested by only verified with an intuitive argument. He also studied the

motion of a planet in a resistant medium with the intention of explaining how twin

stars would form with the capture of one moving in the atmosphere of the other.

We have mentioned some of his important mathematical work above. We should also

mention his work on Taylor series where he examined the convergence and the

analytic extension of the series. Perhaps Fatou's most famous result is that a harmonic

function u > 0 in a ball has a nontangential limit almost everywhere on the boundary.1

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3. Gaston Julia

Gaston Maurice Julia

Born: 3 February 1893 in Sidi bel Abbès, Algeria

Died: 19 March 1978 in Paris, France

Gaston Julia's parents were Delorès Delavent and Joseph Julia. Two generations

before, the family had left the Spanish Pyrenees to become established in Algeria after

the French colonised the area. Joseph Julia, who was a mechanic, was working in

Sidi-bel-Abbès when his son was born. Gaston became interested in mathematics and

music when he was young. He entered school when he was five years old, and was

taught by Sister Théoduline. She gave young Gaston certain principles which he

followed throughout his life, in particular to always aim at being top in everything he

did. She also encouraged Gaston's mother to provide financial support to allow her

son to have a good schooling, something that was very difficult to achieve given that

the family were very poor. Gaston studied with the Frères des Écoles Chrétiennes

(Brothers of the Christian Schools) from the age of seven. His outstanding abilities

were quickly spotted, and his teachers encouraged Gaston's parents to try to get a

scholarship to allow him to study at high school.

In 1901, when Gaston was eight, the family moved to Oran, a city on the

Mediterranean coast in northwest Algeria 70 km north of Sidi-bel-Abbès. There

Gaston's father earned his living repairing agricultural machinery. Gaston entered the

Lycée in Oran, and his parents wanted him to begin his studies in grade 5. However,

the teachers pointed out that pupils in that grade had already studied German for one

year while Gaston had no knowledge of the language. However, Gaston requested that

they give him a month in the class to prove that he could catch up. Learning on his

own from books, he soon caught up and was allowed to remain in this class. By the

end of one year he was the best pupil in German as well as in every other subject that

he studied. He graduated with distinction in the baccalaureate examinations in

science, modern languages, philosophy and mathematics.

Julia won a scholarship which allowed him to go to Paris and spend the year 1910-11

at the Lycée Janson-de-Sailly where he took classes in higher mathematics. Despite

his outstanding abilities, Julia did not find life easy. First, he was still young and had

left the familiar country in which he was brought up for the very different life in

France. Second, he contracted typhoid fever before he had even begun his studies and

was taken to hospital. It was November of 1910 before he was well enough to embark

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on a course which normally took two years but which he had to complete in the

remaining eight months. Despite these difficulties he was still able to reach a higher

standard than any other student. Somehow, he was also able to continue his interest in

music, playing on a violin he mother had given him, and it was during this time that

he fell in love with the music of Bach, Schubert, and Schumann. Throughout his life

these continued to be his favourite composers. He sat the entrance examinations for

the École Normale Supériore and the École Polytechnique and was placed first in both

entrance examinations. He could choose either university but decided to enter the

École Normale on the grounds that it was the stronger of the two establishments for

mathematics.

Entering the École Normale Supériore in 1911, Julia had just completed the

examinations for his first degree in mathematics when political events in Europe

interrupted his studies. Matters came to a head in July 1914 with various declarations

of war, and on 3 August Germany declared war on France. Events had been moving

quickly and Julia received his call up papers one day later. He trained with the

57th Infantry Regiment at Libourne and was soon made a corporal, then a sub-

lieutenant. He saw action on the western front with the 144th Infantry Regiment when

sent to the Chemin des Dames ridge. Kaiser Wilhelm II of Germany had his birthday

on 27th January and the German troops wished to mark the occasion with successes.

Accordingly, on 25 January they launched a strong attack on the French lines where

Julia and his men had just arrived. The following is a report of what happened to Julia

that day:-

January 25, 1915, showed complete contempt for danger. Under an extremely violent

bombardment, he succeeded despite his youth (22 years)to give a real example to his

men. Struck by a bullet in the middle of his face causing a terrible injury, he could no

longer speak but wrote on a ticket that he would not be evacuated. He only went to the

ambulance when the attack had been driven back. It was the first time this officer had

come under fire.

Many on both sides were wounded in the action called the 'attack of the Creute farm'

in which the Germans captured the remaining allied positions on the plateau. Julia's

injury was an extremely painful one and many unsuccessful operations were carried

out in an attempt to repair the damage. Eventually, in 1918, he resigned himself to the

loss of his nose and he had to wear a leather strap across his face for the rest of his

life. Between these painful operations he had carried on his mathematical researches

often in his hospital bed. He undertook research at the Collège de France, beginning in

1916, and in 1917 he submitted his doctoral dissertation Étude sur les formes binaires

non quadratiques à indéterminées réelles ou complexes, ou à indéterminées

conjuguées. The examiners of his thesis were Émile Picard, Henri Lebesgue and

Pierre Humbert, with Picard as president of the examining committee.

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In 1918 Julia married Marianne Chausson, one of the nurses who had looked after him

while he was in hospital. Marianne was the daughter of the romantic composer Ernest

Chausson, who had died in 1899 in a freak accident on his bicycle. Gaston and

Marianne Julia had six children: Jérôme, Christophe, Jean-Baptiste, Marc, Daniel, and

Sylvestre.

When only 25 years of age, Julia published his 199 page masterpiece Mémoire sur

l'iteration des fonctions rationelles which made him famous in the mathematics

centres of his day. The beautiful paper, published in Journal de Math. Pure et

Appl. 8 (1918), 47-245, concerned the iteration of a rational function f. Julia gave a

precise description of the set J(f) of those z inC for which the nth iterate f n(z) stays

bounded as n tends to infinity. He received the Grand Prix of the Academy of

Sciences for this remarkable piece of work.

In November 1919 he was invited to give the prestigious Peccot Foundation lectures

at the Collège de France and was appointed as Maître de Conférences at the École

Normale Supériore. At the same time he was appointed répétiteur in analysis at the

École Polytechnique, examiner at the École Navale, and professor at the Sorbonne.

This appointment to a professorship at the Sorbonne came without a specific chair, but

in 1925 he was named to the Chair of Applications of Analysis to Geometry at the

Sorbonne. In 1931 he was appointed to the Chair of Differential and Integral Calculus,

then in 1937 he was appointed to the Chair of Geometry and Algebra at the École

Polytechnique when Maurice d'Ocagne retired.

Seminars were organised in Berlin in 1925 to study Julia's work on iteration and

participants included Richard Brauer, Heinz Hopf and Kurt Reidemeister. H Cremer

produced an essay on his work which included the first visualisation of a Julia set.

Although he was famous in the 1920s, his work on iteration was essentially forgotten

until Benoit Mandelbrot brought it back to prominence in the 1970s through his

fundamental computer experiments. However, Julia was very active mathematically

over a wide range of different topics which is perhaps best summarised by looking

briefly at the six volumes of his collected works which were published between 1968

and 1970 edited by Jacques Dixmier and Michel Hervé. Of course the volumes were

published before Julia's death so he was able to write the Preface to the volumes

himself. In addition to the Preface, Volume 1 contains a list of Julia's 232 publications

from 1913 to 1965. These 232 publications consist of 157 research papers, 30 books,

and 45 articles on the history of science or miscellaneous topics.

Volume 1 contains works on iteration and its applications.

Volume 2, in three parts, consists of articles on (i) J points of functions of one

variable, (ii) Jpoints of functions of several variables, and (iii) Series of iterates.

Volume 3 contains four parts: (i) Functional equations and conformal mapping; (ii)

Conformal mapping; (iii) General lectures; and (iv) Isolated works in analysis on

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Implicit function defined by the vanishing of an active function, and on certain series.

Volume 4 is again in four parts: (i) Functional calculus and integral equations; (ii)

Quasianalyticity; (iii) Various techniques of analysis; and (iv) Works concerning

Hilbert space.

Volume 5 contains works on (i) Number theory; and (ii) Geometry, mechanics, and

electricity.

Volume 6 contains Julia's miscellaneous writings.

What about the 30 books? Let us mention Eléments de géométrie

infinitésimale (1927),Cours cinématique (1928), and Exercices d'Analyse (4 vols.)

(1928-38). Reviewing the first of the four volumes of Exercices d'Analyse, Einar Hille

writes:-

This book is a worthy descendent of a long line of French Exercices sur le calcul

infinitésimal. Such collections of problems are intended primarily for the students who

prepare themselves for the licence or the agrégation and contain problems of the type

set in these examinations. A thorough knowledge of the theory is expected as well as

skill in calculation and the training is directed towards developing both qualities in

the students. The present book contains a small number of carefully chosen problems,

each problem followed by one or more complete solutions. About two-thirds of the

first volume is devoted to the applications of analysis to geometry. An admirable

account of the theory of Fourier series (pp. 120-190) is eminently suitable as outside

reading for first year graduate students. This part of the book will probably be found

the most useful one to the general mathematical public outside of France.

The classic Principes Géométriques d'Analyse (1930) was reviewed by Virgil Snyder

who wrote:-

The present volume has for its purpose the development and explanation of those

geometric concepts which are employed in connection with rational, and particularly

linear, transformations of a complex variable z, and the consequent transformations

of uniform and of multiform functions of z.

Two years later Julia produced a second volume of Principes Géométriques

d'Analysewhich was reviewed by W Seidel:-

This book presents a continuation of the first volume of the author dealing with those

aspects of the modern theory of functions of a complex variable which are derivable

from simple geometrical principles. As the author himself points out in the preface to

the first volume, the most important of these principles is the conformal

correspondence between two regions of planar character or two Riemann surfaces

realized by an analytic function. The book serves the excellent purpose of unifying by

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means of geometric concepts various branches of the theory of functions which have

hitherto been scattered in the literature. The presentation throughout is lucid,

rigorous, and elegant.

Another classic text Introduction Mathématique aux Theories Quantiques also

appeared in two volumes, the first in 1936 and the second in 1938. Francis

Murnaghan, reviewing the first volume, wrote:-

This book is the sixteenth of the well known series, 'Cahiers Scientifiques,' and is the

first of a series which proposes to give the mathematical foundation of quantum

mechanics. In this first volume the essential difficulties of quantum mechanics (some

of which concern the fact that Hubert space is not finite dimensional) are merely

foreshadowed, the attention being directed in the main to vector analysis in a space of

finite dimensions. However, the treatment is sophisticated and designed, as far as

possible, to carry over to the infinite dimensional case.

The second volume was reviewed by Marshall Stone:-

The topics included in the book are presented from a purely mathematical point of

view in a clear and lively style. The applications to the theory of matrices and

equations, which are largely implicit, in certain of the more abstract treatments, are

elaborated here with a wealth of detail which renders them unusually accessible to

the student. The author's approach to the modern theory of operators is obviously a

cautious one, presumably because of his desire to keep the reader on ground which

shall appear as nearly familiar as possible at every stage.

Further books by Julia include L'espace hilbertien (1949) and Eléments

d'algèbre (1959).

Julia received many honours for his outstanding mathematical contributions. He was

elected to the Academy of Sciences on 5 March 1934, filling the place left vacant by

the death of Painlevé in the previous year. He was elected President of the Academy

in 1950. He was also elected to the Upsal Academy in Sweden, the Pontifical

Academy of Rome, and many other European Academies. He was also President of

the French Mathematical Society. In 1950 he was made an officer of the Légion

d'Honneur.2

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4. Natural Fractals3

4.1. Atronomy

4.1.1. Galaxies

Looking at the structure of our universe, you can find it to be very self-similar. It is composed of

gigantic superclusters, which are in turn composed of clusters. Every cluster is composed of

galaxies, which are in turn composed of star systems such as the solar system, which are further

composed of planets with moons revolving around them. Truly, every detail of the universe

shows the same clustering patterns. The cluster fractals, such as the Cantor Square below are

indeed useful in modeling the universe:

Cluster fractals are formed by repeatedly cutting out pieces of a polygon. The fractal above is

obviously not a good model, and making it more random helps a lot. The fractal dimension of

such fractals can be found very easily using

the similarity method. In the Cantor

Square, for example there are 4 smaller

squares, the sides of each of which are 1/3

of the entire picture. The fractal dimension

will thus be log 4 / log 3 = 1.26. This is

remarkably close to the fractal dimension of

the universe according to one of the

experiments, where it was found to be

about 1.23. The fact that this is a fraction is

yet another proof of the universe’s fractal

geometry.

Another way universe can be modeled

is by using IFS fractals that resemble

the galaxies, such as the one on the

left.

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4.1.2. Rings of Saturn

Saturn is perhaps most famous for the ring that it has around itself. Originally, it was

believed that the ring is a single one. After some time, a break in the middle was

discovered, and scientists considered it to have 2 rings. However, when Voyager I

approached Saturn, it discovered that the two ring were also broken in the middle, and

the 4 smaller rings were broken as well. Eventually, it identified a very large number

of breaks, which continuously broke even small rings into smaller pieces. The overall

structure is amazingly similar to... Cantor Set, which is formed by continuously

cutting out middles of the segments:

If you put circles through the points in the last picture above, you will get a simple

model of the rings of Saturn:

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4.2. Bio/Chem

4.2.1. Bacteria Cultures

Some of the most amazing applications of fractals can be found in such distant areas as the

shapes of bacteria cultures. A bacteria culture is all bacteria that originated from a single

ancestor and are living in the same place. When a culture is growing, it spreads outwards in

different directions from the place where the original organism was placed. Just like plants the

spreading bacteria can branch and form patterns which turn out to be fractal. The spreading of

bacteria can be modeled by fractals such as the diffusion fractals, because bacteria spread

similarly to nonliving materials.

If you are familiar with fractals, you will probably bet money on the fact that the above picture is

a fractal. You would be right, but we still need a real mathematical proof to be sure of that. The

way to do it is quite simple – just place the culture on a piece of graph paper and count the

number of occupied squares. This kind of data will let you calculate the fractal dimension of the

culture using the box-counting method. In an example experiment performed by Tohey

Matsuyama and Mitsugu Matsushita, the fractal dimension of a culture of Salmonella

anatum was found to be about 1.77. The fact that the dimension is a fraction is enough to prove

that the culture is a fractal.

4.2.2. Chemical Reactions

If you know some chemistry, you are probably familiar with the concept of forward and

backward reactions. Most reactions are accompanied by a backward reaction, in which the

products turn back into the reactants. At equilibrium, the rates of these reactions become equal

and the overall composition of the system does not change. However, the fact that is usually

missed here is that talking about the rates of reactions we are talking aboutaverage rates, since

the rates depend on the movement of particles, which involves a lot of chance. Sometimes,

however, the rates become different for a short interval of time and the composition of the

system changes. As you might guess, these changes would be very chaotic... Aha! In one of the

lessons, we have already established the connection of chaos and fractals. Maybe if we view

every three consecutive concentrations of a substance as coordinates of a point in space... we can

get something that is fractal in shape! Such fractal would be a strange attractor because we

know that this is the type of fractals based on changing numbers.

Indeed, fractal shapes were found after graphing many different systems,

even such common ones as hydrogen and oxygen reacting to make water.

One of the scientists who tried to study this mathematically was Otto

Rossler. He came up with three formulas that could model chemical

reactions. When these three formulas are used to create a strange

attractor, they create the famous 3-dimensional Rossler Attractor:

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4.2.3. Human Anatomy

If you are still not convinced that that fractals, being a math topic, are very important in real life,

your opinion might change after finding out that you yourself are made of fractals!

THE LUNGS

The first place where this is found is rather obvious to anyone who knows fractals — in the

pulmonary system, which you use to breathe. The pulmonary system is composed of tubes,

through which the air passes into microscopic sacks called alveoli. The main tube of the system

is trachea, which splits into two smaller tubes that lead to different lungs, called the bronchi. The

bronchi are in turn split into smaller tubes, which are even further split. This splitting continues

further and further until the smallest tubes, called the bronchioles which lead into the alveoli.

This description is similar to that of a typical fractal, especially a fractal canopie, which is

formed by splitting lines:

The endpoints of the pulmonary tubes, the alveoli, are extremely close to each other. The

property of endpoints being interconnected is another property of fractal canopies.

THE ALVEOLI

Another supporting evidence that your lungs are fractal comes from measurements of the

alveolar area, which was found to be 80 m2 with light microscopy and 140 m

2 at higher

magnification with electron microscopy. From the geometric method we know that the increase

in size with magnification is one of the properties of fractals!

THE BLOOD VESSELS

Similarly to bronchial tubes, splitting can also be found in blood vessels. Arteries, for example

start with the aorta, which splits into smaller blood vessels. The smaller ones split as well, and

the splitting continues until the capillaries, which, just like alveoli, are extremely close to each

other. Because of this, blood vessels can also be described by fractal canopies.

THE BRAIN

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The surface of the brain, where the highest level of thinking takes place contain a large number

of folds. Because of this, a human, who is the most intellectually advanced animal, has the most

folded surface of the brain as well. Geometrically, the increase in folding means the increase in .

Instead of 2, which is the dimension of a smooth surface, the surface of a brain has a dimension

greater than 2. In humans, it is obviously the highest, being as large as between 2.73 – 2.79.

Here’s another topic for science fiction: super-intelligent beings with a fractal brain of dimension

up to 3!

MEMBRANES

The surface folding similar to that of a brain was found in many other surfaces, such as the ones

inside the cell on mitochondria, which is used for obtaining energy and the endoplasmic

reticulum, which is used for transporting materials. The same kind of folding was found in the

nasal membrane, which allows sensing smells better by increasing the sensing surface. However,

in humans this membrane is less fractal than in other animals, which makes them less sensitive to

smells.

The fractal dimension of some anatomical structures are given below. Note that all dimensions

are greater than you would expect them to be, and most are fractions, which automatically

implies that the structures are fractal.

Anatomical Structure Fractal Dimension

Bronchial Tubes very close to 3

Arteries 2.7

Brain 2.73 – 2.79

Alveolar Membrane 2.17

Mitochondrial Membrane (outer) 2.09

Mitochondrial Membrane (inner) 2.53

Endoplasmic Reticulum 1.72

Fractals, in addition to the anatomical structures above can be found in the body on smaller

scales in various molecules.

4.2.4. Molecules

In addition to anatomical structures, fractals were found in living organisms on even smaller

scales — in molecules.

DNA

As you probably already know, DNA is a long sequence of nucleotides that code all

the genetic information about us. The nucleotides can be either adenine, guanine,

16

cytosine, or thymine (abbreviated A, G, C, and T). One of the fractal patterns that

were studied was in the sequence of nucleotides in what is called the DNA walk.

The DNA walk is a graphical representation of the DNA sequence in which you move

up if you hit C or T and down if you hit A or G. For example, for the sequence CATG

you will get the following picture:

Fractal patterns were found in many DNA walks. These pattern are remarkably

similar to Brownian motion. The fractal below is a model of a fractal DNA walk:

CHROMATIN

Chromatin is a fibrous material inside a cell’s nucleus that contains the genetic

material. As you can see below, chromatin tends to cluster:

17

We have seen on the example of galaxies, clusters are fractal in shape. Recently,

scientists have found ways to measure the fractal dimension of chromatin.

Interestingly, experiments performed a couple of years ago at the Mount Sinai

research center in New York showed that the fractal dimension of chromatin might be

somehow connected with cancer. Current experiments are attempting to detect breast

cancer by measuring the fractal dimension. Talk about useful applications!

PROTEINS AND POLYMERS

A polymer is a molecule that is composed of a series of "building blocks" (called monomers)

connected to one another in a chain. If you take a polymer, you will find that its monomers are

not connected in a straight line. Instead, the angles between the monomers can be different and

the entire molecule can twist into pretty complicated shapes. The same is true for proteins, which

are formed by amino acid bonding together in a chain. Twisting, as well as folding and breaking

often implies by itself that the shape is fractal. Proteins and many other polymers are, indeed,

fractal and various methods exist for finding their fractal dimension. For some interesting

proteins the results are shown below. Note that the dimensions are much higher than 1, which

you would expect from a linear chain. This is another proof that proteins are fractal.

Protein Fractal Dimension

Lysozyme (egg-white) 1.614

Hemoglobin (oxygen carrier in the blood) 1.583

Myoglobin (muscle protein) 1.728

SPECTRUM

If you hold a substance above in the flame, the flame will turn some color that is characteristic of

that substance. If you then let the light from that flame pass through a spectroscope, the light will

break into several colors of the rainbow. Shortly after the discovery of fractals, Harter found

spectra of some molecules that remarkably resembled the Cantor Set. The picture below if a

simulation of a spectrum that is perfectly fractal:

18

4.2.5. Plants

Most plants show some form of branching. This happens when the main stem (of

trunk) splits into a number of branches. Each of those branches splits into smaller

branches, and this kind of splitting continues until the smallest branches. You have

probably noticed that a tree branch looks similar to the entire tree and a fern leaf looks

almost identical to the entire fern. This property, called self-similarity is one of the

most important properties of fractals. Because of numerous ways branching can be

achieved geometrically, there are several ways of creating models of plants as well.

One classic way of creating fractal plants is by means of l-systems. Lindenmayer,

who is the founder of l-systems, introduced them in a book called The Algorithmic

Beauty of Plants, where he first used them to create models of plants. Some of the

fractal plants he created became classic examples. Here are some of them in addition

to several other ones:

Another way of creating fractal plants is using fractal canopies or Pythagoras trees.

Fractal canopies are formed by splitting lines, which is very similar to branching.

Pythagoras trees, such as the one below do the same more realistically by using

squares and triangles instead of lines:

19

One of the properties of fractal canopies is the endpoints being interconnected. This is

especially interesting in its similarity to broccoli, where the branches’ endpoints form

an interconnected surface:

The final way of creating plant models is by using IFS fractals such as the Barnsley

Fern below, which resemble plant shapes:

4.3. Other

4.3.1. Clouds

Clouds look very irregular in shape. At some point in your life you probably did look at them

wondering how their diverse shapes are capable of resembling many common objects, animals,

and people. Yet, for the purpose of this website, the word "irregular" automatically triggers the

word "fractal." Yes, indeed, clouds are fractal in shape just like most other objects in nature. Let

us first look at experimental evidence that can prove this.

Usually, to prove that something is a fractal it is enough to find its fractal dimension. For

something like a cloud it is the best to do it using the geometric method. Obviously, it is not

done by measuring the actual cloud, but by measuring its 2D projection, which is the shade. We

20

can make several measurements of the cloud’s perimeter using different magnifications. This is

achieved by using different sized "yardsticks." If, let’s say, our yardstick is 1 kilometer long, the

magnification is higher and measurement would be more exact than the one where the yardstick

is 10 kilometers long. We also know that in fractals, more detail adds additional irregularities,

which adds to the measurement. If we graph log(magnification) against log(perimeter) we should

get a line with a positive slope since the perimeter of fractals increases with magnification.

Indeed, when graphing it for the clouds, we get something like this:

By adding 1 to the slope (see geometric method) we find the fractal dimension. According to

the findings of Lovejoy in 1981, the fractal dimension for most clouds is about 1.164.

Now, having proved that clouds are fractals, it would be good to try using fractals to generate

computer models of them. We know for sure that, since clouds are very irregular, we have to use

fractals that are random and have Brownian self-similarity. The best ones to use are plasma

fractals. To make plasma fractals look like clouds, we can use a color map which uses colors

similar to ones on a real cloud photograph. The pictures we can generate using this method are

something like this:

We can control how fragmented the clouds are by changing a parameter in plasma

fractals called roughness.

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4.3.2. Coastlines & Borderlines

Benoit Mandelbrot, the founder of fractals has first noticed the properties of fractals

on the coast of Britain. He realized that no matter how small a piece of the coast is, it

will still have its own bays, harbors, and capes. Basing himself on Richardson’s data,

he was able to prove that many coasts as well as borderlines are fractal.

Richardson searched many encyclopedias to find data about the lengths of certain

borderlines. He found enormous differences in data from different countries. For

example, Portugal claimed its border with Spain to be 1214 km, while Spain claimed

it to be 987 km. Portugal, as a smaller country would definitely measure its border

more accurately. Thus, we know that the increase of accuracy increased the

measurement... which is one of the properties of fractals! This happens because

fractals are figures with an infinite amount of detail, and measuring more accurately

adds more of these details, which adds to the overall size. Mandelbrot claimed that the

difference in the two measurements were due to the fact that Spain used a "yardstick"

that was bigger than Portugal’s. If, for example, Spain measured the border with a 2

kilometer yardstick, its measurement would be less exact than Portugal, which used a

1 kilometer yardstick. If we graph log(total length) against log(length of yardstick),

we get lines with negative slopes since the total length decreases with the increase of

the size of the yardstick:

Using this graph, we can find fractal dimensions of the coasts and borders by using a

modified version of the geometric method. Since the magnification, used in the

geometric method is equal to 1 / size of yardstick, the identity log (1/x) = – log (x)

will tell you that the slopes of the line above are the negated slopes of the lines in

geometric method. A little bit of algebra will tell you that in order to get fractal

dimension, you need to subtract the slope from 1. Note that in the above diagram, the

lines of more irregular lines, such as the coast of Britain are more steep than the lines

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of more smooth lines, such as the coast of South Africa. This is due to the fact that the

more irregular a curve is, the higher its fractal dimension.

Simple models of coasts can be made with base-motif fractals that use polygons for

the bases. Such fractals are also called Koch Islands. Below are three examples of

the Koch Snowflake which uses a triangle, a Quadric Koch Island which uses a

square, and the Gosper Island which uses a hexagon:

These pictures, obviously, are terrible in modeling real-life coats since they are too

perfectly symmetrical and self-similar. The solution to this problem is to use fractals

with Brownian self-similarity, such as the plasma fractals. This gives the coasts

randomness, which makes them more realistic. Below is an example of a coastline

made using a plasma fractal:

4.3.3. Data Compression

In December 1992, Microsoft released a compact disk entitled the Encarta Encyclopedia. It

contains thousands of articles, 7000 photographs, 100 animations, and 800 color maps. All of

this is in less than 600 megabytes of data. How was it possible? The answer lies in the

mathematics of fractal data compression.

Consider the Mandelbrot Set. A color full-screen GIF image of it occupies about 35 kilobytes.

However, all you need to store it is the formula z = z^2 + c, which takes no more than 7 bytes.

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That’s a 99.98% compression – talk about efficiency! Well, maybe if it works for the Mandelbrot

Set... it could work for a flower diagram, a map of Africa, or a photo of Kennedy as well! The

goal is too find functions, each of which produces some part of the image. For a complex image

that is not a fractal, you might need hundreds of such functions. Yet, it would still take up less

space than hundreds of thousands of colored pixels. IFS are the functions usually used for

compressing data. The mathematical foundation of the image compression was established by

Michael Barnsley, who is the founder of IFS fractals as well.

4.3.4. Special Effects

Computer graphics has been one of the earliest applications of fractals. Indeed, fractals can

achieve realism, beauty, and require very small storage space because of easy compression.

Very beautiful fractal landscapes were published as far back as in Mandelbrot’s Fractal

Geometry of Nature. Although the first algorithms and ideas are owed to the discoverer of

fractals himself, the artistic field of using fractals was started by Richard Voss, who generated

the landscapes for Mandelbrot’s book. This sparked the imagination of many artists and

producers of science fiction movies. A little later, Loren Carpenter generated a computer movie

of a flight over a fractal landscape. He was immediately hired by Pixar, the computer graphics

division of Lucasfilms. Fractals were used in the movie Star Trek II: The Wrath of Khan, to

generate the landscape of the Genesis planet and also in Return of the Jedi to create the

geography of the moons of Endor and the Death Star outline. The success of fractal special

effects in these movies lead to making fractals very popular. Today, numerous software allows

anyone who only knows some information about computer graphics and fractals to create such

art. For example, we ourselves were able to generate all landscapes throughout this website, such

as the one below.

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5. Other Fractals4

A Simple Explanation Of Fractal Geometry While the classical Euclidean geometry works with objects which exist in integer dimensions, fractal

geometry deals with objects in non-integer dimensions. Euclidean geometry is a description lines,

ellipses, circles, etc. Fractal geometry, however, is described in algorithims -- a set of instructions on

how to create a fractal.

The world as we know it is made up of objects which exist in integer dimensions, single dimensional

points, one dimensional lines and curves, two dimension plane figures like circles and squares, and three

dimensional solid objects such as spheres and cubes. However, many things in nature are described

better with dimension being part of the way between two whole numbers. While a straight line has a

dimension of exactly one, a fractal curve will have a dimension between one and two, depending on

how much space it takes up as it curves and twists. The more a fractal fills up a plane, the closer it

approaches two dimensions. In the same manner of thinking, a wavy fractal scene will cover a

dimension somewhere between two and three. Hence, a fractal landscape which consists of a hill

covered with tiny bumps would be closer to two dimensions, while a landscape composed of a rough

surface with many average sized hills would be much closer to the third dimension.

A More Complete Explanation of Fractal Geometry and Fractal Dimensions

Fractal Dimensions can be demonstated by first defining a fractal set as Nn = C/rnD where Nn is the

number of fragments with the linear dimension defined as rn, C is some constant, and D defines the

fractal dimension. If this equation is rearranged with simple algebra, the outcome is D = [ ln(Nn + 1/Nn) ]

/ [ ln(rn/rn + 1) ]. Given a line of unit length, we can divide it in varying ways and do different things with

each segment. For the first example (figure 1a), if the segment is divided into two parts, making r1 =

1/2. One of the parts is kept and the other is disposed of, so N1 = 1. If we divide the remaining segment

into two parts and again only keep one of the fragments, then r2 = 1/4 and N2=1. If this process is

repeated (iterated), D turns out to be zero, which give the equivalent to the Euclidean point. Regardless

of the number of iterations, at order n, Nn=1. Hence, D will always be zero. This way of thinking makes

sense because if you were to take a line segment and continually divide it into two, keeping only one of

the pieces, the length of the line segment will approach zero as the order approaches infinity.

A Euclidean line which exists in the first dimension can be demonstrated just as easily. This example is

modeled in figure 1b. The line segement is again divided into two parts; however, we keep all the

fragments, sor1 = 1/2 and N1 = 2. Iterating again, we get r2 = 1/4 and N2 = 4. Hence, ln(2) / ln(2) =

1. This also makes sense because we never remove any part of line so it will always remain of unit

length.

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In the first two examples, the results were both Euclidean figures with dimensions of zero and one,

respectively. It is, however, just as easy to create a line segment with a fractal dimension between zero

and one. Infigure 1c we divide the line segment into three different parts and keep only the two end

pieces. After the first iteration, we get r1 = 1/3 and N1 = 2. When this process is repeated, we get r2 =

1/9 and N2 = 4.Therefore, D = ln(2) / ln(3) = 0.6309. To show how to generate line segements with a

varying fractal dimension, we start with a line segment of unit length and divide it into five distinct parts

(figure 1d). By keeping only the two end pieces and the center piece, we get r1 = 1/5 and N1 =

3. Iterating again, we get r2 = 1/25 and N2 = 9. In this example D = ln(3) / ln(5) = 0.6826. As this process

is iterated, the infinite set of points is called dust. This term will be explained later.

Figure 1. Demonstration of fractal dimensions with Euclidean line segments.

Fractal dimensions are not limited to being between zero and one. We can also apply the same method

to the Euclidean square to produce items with a fractal dimension between zero and two. For each of

the following examples, each square will be divided into nine squares of equal size, making r1 = 1/9. The

iterations continues n times. To demonstrate the Euclidean point (figure 2a), we keep only one square

with each iterations, making N1=Nn=1. In the next example (figure 2b), we keep only the top three

squares with each iteration, making N1 = 3 and N2 = 9. Through this process we discover a Euclidean

line with a dimension of one. The last Euclidean figure which can be derived from this example is the

plane (figure 2c). To accomplish this, we keep all the squares with each iteration.

To produce a figure with a fractal dimension, we will keep only the two pieces in the upper left and

lower right corner with each iteration (figure 2d), making N1 = 2 and N2 = 4. Hence, at the second

orderD = ln(2) / ln(3) = 0.6309. On the other hand, if we remove only the center piece with each

iteration, as in figure 2e, we N1 = 8 and N2 = 64. This example produces a fractal dimension of 1.8928.

Figure 2. Demonstration of fractal dimensions with Euclidean planes.

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Calculating Fractal Dimensions

Now you understand what fractal dimensions are and where they come from, but how are they

calculated? For certain objects which with you have delt all of your life, such as squares, lines, and

cubes, it is easy to assign a dimension. You intuitively feel that a square has two dimensions, a line has

one dimension, and a cube has three dimensions. You might feel this way because there are two

directions in which you can move on a square, one direction on a lines, and three directions in a cube,

but what about fractals? Sometimes you can move in a certain number of directions and sometimes you

can move in a different number of directions. This is what causes fractal dimensions to be non-integers.

To derive a formula which will work with all figures, lets first look at how to calculate the dimensions for

the figures which we already know. A line can be divided into n = n1 seperate pieces. Each of those

pieces is 1/n the size of the whole line and each piece, if magnified n times, would look exactly the same

as the original. Repeating the process for a square, we find that is can be divided into n2 pieces. The

same concept holds true for a cube, we need n3 pieces to reassemble a cube. Each of the pieces would

be 1/n the size of the whole figure. The exponent in each of these examples is the dimension. For

fractals, we need a generalized formula, which can be derived from what we already know. The steps

bellow assume you have a working knowledge of logarithims and basic algebra.

Note: ln denotes loge and may be refered to as the natural logarithim. Because of the way in which this

formula ends up, it is independant of the base used for the logarithims.

for a line: ln(number of divisions) = ln(n1)

for a square: ln(number of divisions) = ln(n2)

for a cube: ln(number of divisions) = ln(n3)

If you look back, the figure was divided into pieces that when zoomed in on n times, revealed to starting

figure. Because of this, we divide the ln(number of divisions) by the natural logarithim of the

magnification facator. The resulting formula gives the dimension, represented by D.

D=ln(number of divisions)/ln(magnification factor)

for a line: D = ln(n1)/ln(n) = 1

for a square: D = ln(n2)/ln(n) = 2

for a cube: D = ln(n3)/ln(n) = 3

Each of these examples was easy because the magnification factor was always n. But for fractals,

magnification factor will be a constant, which varies for each fractal. Because you are unfamiliar with

specific fractals, we can not examine specific cases now. Under the section Individual Fractals the

dimension of the individual fractals will be examined in more detail.

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What Are Fractals?

For the most part, when the word fractal is mentioned, you immediately think of the stunning pictures

you have seen that were called fractals. But just what exactly is a fractal? Basically, it is a rough

geometric figure that has two properties: First, most magnified images of fractals are essentially

indistinguishable from the unmagnified version. This property of invariance under a change of scale if

called self-similiarity. Second, fractals have fractal dimensions, as were described above. The word

fractal was invented by Benoit Mandelbrot, "I coined fractal from the Latin adjective fractus. The

corresponding Latin verb frangere means to break to create irregular fragments. It is therefore sensible

and how appropriate for our needs! - that, in addition to fragmented, fractus should also mean irregular,

both meanings being preserved in fragment."

Graphical Representation Of Fractals

Graphically, fractals are images created out of the process of a mathematical exploration of the space in

which they are plotted. For this page, a computer screen will represent the space which is being

explored. Each point in the area is tested in some way, usually an equation iterating for a given period of

time. The equations used to test each point in the testing region are often extremely simple. Each

particular point in the testing region is used as a starting point to test a given equation in a finite period

of time. If the equation escapes, or becomes very large, within the period of time, it is colored white. If if

doesn't escape, or stays within a given range throughout the time period, it is colored black. Hence, a

fractal image is a graphical representation of the points which diverge, or go out of control, and the

points which converge, or stay inside the set. To make fractal images more elablorate and interesting,

color is added to them. Rather than simply plotting a white point if it escapes, the point is assigned a

color relative to how quickly it escaped. The images produced are very elaborate and possess non-

Euclidean geometry. Fractals can also be produced by following a set of instructions such as remove the

center third of a line segment. A more complete explanation of how to generate fractal images, specific

to individual fractals, follows.

Bellow are several sections, each dealing with an individual fractal. Of course not all of the fractals in the

world are listed bellow, but only ones which are well known or show and important point which

everyone should know. With each fractal, there is a picture, followed by some information about it. For

many of the fractals, there is also a link to a C/C++ or BASIC program which will generate a picture of the

fractal. For more working soure code, visit the Appendix Of Source Code. Even if you are not interested

in these specific fractals, it is strongly enouraged that you read through each one because many topics

other than the specific fractal are reviewed. For example, strange attractors and several applications of

fractals to real-life situations are discussed.

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5.1. The Cantor Set

Figure 3. The Cantor set

The Cantor set is a good example of an elementary fractal. The object first used to demostrate fractal

dimensions, figure 1c, is actually the Cantor set. The process of generating this fractal is very simple. The

set is generated by the iteration of a single operation on a line of unit lenght. With each iteration, the

middle third from each lines segment of the previous set is simply removed. As the number of iterations

increases, the number of seperate line segments tends to infinity while the length of each segment

approaches zero. Under magnification, its structure is essentially indiguishable from the whole, making

it self-similiar.

To calculate the dimension of the Cantor set, we first realize that its magnification factor is three, or the

fractal is self-similiar if magnified three times. Then we notice that the line segments decompose into

two smaller units. Using the formula given in the section entitled Calculating Fractal

Dimensions, we get:

D = ln(2) / ln(3)

D = 0.6931 / 1.0986 D = 0.6309

The Cantor set has a dimension of 0.6309.

5.2. The Koch Curve

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Figure 4. The Koch Curve

So far, all of the examples in this document have delt with removing pieces from various geometric

figures. Fractals, and fractal dimensions can also be defined by adding onto geometric figures. The Koch

curve was named after Helge von Koch in 1904. The generation of this fractal is simple. We begin with a

straight line of unit length and divide it into three equally sized parts. The middle section is replaced

with and equilateral triangle and its base is removed. After one iterations, the length is increased by

four-thirds. As this process is repeated, the length of the figure tends to infinity as the length of the side

of each new triangle goes to zero. Assuming this could be iterated an infinite number of times, the result

would be a figure which is infinitely wiggly, having no straight lines whatsoever.

To calculate the dimension of the Koch Curve, we look at the image of the fractal and realize that it has a

magnification factor of three and with each iteration, it is divided into four smaller pieces. Knowing this,

we get

D = ln(4) / ln(3)

D = 1.3863 / 1.0986

D = 1.2619

The Koch Curve has a dimension of 1.2619.

The Koch Snowflake

As would be expected, the Koch Snowflake is generated in very much the same way as the Koch Curve.

The only variation is that, rather than using a line of unit length as the intial figure, an equilater triangle

is used. It is iterated in the same way as the Koch Curve. The length of the resulting figure tends to

infinity as the length of the side of each new triangle goes to zero. Iterated an infinite number of times,

the Koch Snowflake, like the Koch Curve, has absolutely no straight lines in it. This fractal, if magnified

three times in any area, also displays the property of self-similiarity.

As mentioned above, the magnification factor of this fractal is three, and as with the Koch Curve, the

number of divisions in each magnification is four. With this we get:

D = ln(4) / ln(3)

D = 1.3863 / 1.0986

D = 1.2619

The Koch Snowflake has a dimension of 1.2619.

.

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5.3. The Sierpinski Triangle

Figure 6. The Sierpinski Triangle

Unlinke the Koch Snowflake, which is generated with infinite additions, the Sierpinski triangle is created

by infinite removals. Each triangle is divided into four smaller, upside down triangles. The center of the

four triangles is removed. As this process is iterated an infinite number of times, the total area of the set

tends to infinity as the size of each new triangle goes to zero.

After closer examinition of the process used to generate the Sierpinski Triangle and the image produced

by this process, we realize that the magnification factor is two. With each magnification, there are three

divisions of the triangle. With this data, we get:

D = ln(3) / ln(2)

D = 1.0986 / 0.6931

D = 1.5850

The Sierpinski Triangle has a dimension of 1.5850.

6. Algorithmic Composition5

6.1. Introduction

"Since I have always preferred making plans to executing them, I have gravitated

towards situations and systems that, once set into operation, could create music with

little or no intervention on my part. That is to say, I tend towards the roles of planner

and programmer, and then become an audience to the results" -Brian Eno (Alpern,

1995).

Algorithmic composition, sometimes also referred to as "automated composition,"

basically refers to "the process of using some formal process to make music with

minimal human intervention" (Alpern, 1995). Such "formal processes," as we will see,

have been familiar to music since ancient times. The title itself, however, is relatively

new—the term "algorithm" having been adopted from the fields of computer science

and information science around the halfway mark of the 20th century (Burns, 1997).

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Computers have given composers new opportunities to automate the compositional

process. Furthermore, as we will explore, several different methods of doing so have

developed in the last forty years or so.

To begin with the title itself, Webster's dictionary defines an "algorithm" simply as "a

predetermined set of instructions for solving a specific problem in a limited number of

steps." The "problem" composers are faced with, of course, is creating music; the

"instructions" for creating this music according to the definition are "predetermined,"

suggesting that intervention on the part of the human composer is superceded once the

compositional process itself is set into motion, as hinted at as well in the above Brian

Eno quote. Thus, "automated composition" also suitably describes this kind of music,

since "automation" refers to "anything that can move or act of itself."

6.2. Pre/Non-Computer Practices

ancient Greeks, canon, Mozart, John Cage, serialism

The idea of utilizing formal instructions and processes to create music dates back in

musical history as far back as the ancient Greeks. Pythagoras believed in a direct

relation between the laws of nature and the harmony of sounds as expressed in music:

"The word music had a much wider meaning to the

Greeks than it has to us. In the teachings of

Pythagoras and his followers, music was

inseperable from numbers, which were thought to

be the key to the whole spiritual and physical

universe. So the system of musical sounds and

rhythms, being ordered by numbers exemplified

the harmony of the cosmos and corresponded to

it" (Grout, 1996; italics added).

Thus, theoretical applications of numbers (i.e. "data," in a sense) and various

mathematical properties derived from nature were the formalisms, or "algorithms,"

upon which the ancient Greek musicians had constructed their musical

systems. Ptolemy and Plato, also, were two others who wrote about this practice.

Ptolemy, the "most systematic of the ancient theorists of music," was also a leading

astronomer of his time; he believed that mathematical laws "underlie the systems both

of musical intervals and of the heavenly bodies," and that certain modes and even

certain notes "correspond with particular planets, their distances from each other, and

32

their movements" (Grout, 1996). This idea was also given poetic form by Plato in the

myth of the "music of the spheres," the unheard music "produced by the revolutions of

the planets" (Grout, 1996), and the notion was later invoked by writers on music

throughout the Middle Ages, including Shakespeare and Milton (Grout, 1996).

These ancient Greek "formalisms," however, are rooted mostly in theory, and their

strict application to musical performance itself is probably questionable since Greek

music was almost entirely improvised (Grout, 1996). Thus, while Greek mathematical

conjectures certainly created the musical system of intervals and modes with which

the musician operated and probably also guided and influenced his/her performance

practice in some ways, the musician was by no means entirely removed from the

decision-making process. Ancient Greek music was not "algorithmic composition" in

any pure sense, therefore, but it is undoubtably important historically in music for its

tendency towards formal extra-human processes.

An extra layer of abstraction would later be achieved with the birth of "canonic"

composition in the late 15th century:

"The prevailing method was to write out a single

voice part and to give instructions to the singers to

derive the additional voices from it. The instruction

or rule by which these further parts were derived

was called a canon, which means 'rule' or 'law.' For

example, the second voice might be instructed to

sing the same melody starting a certain number of

beats or measures after the original; the second

voice might be an inversion of the first or it might

be a retrograde [etc.]" (Grout, 1996).

These "rules" of imitation and manipulation are indeed the "algorithm" by which

performers unfolded the music. In this case, then, as opposed to the previous one of

the ancient Greeks, we can see a clear removal of the composer from a large portion

of the compositional process: the composer himself only invents a kernel of music—a

single melody or section—from which an entire composition is automatically

constructed.

Mozart, too, used automated composition techniques in his Musikalisches

Wurfelspiel ("Dice Music"), a musical game which "involved assembling a number of

small musical fragments, and combining them by chance, piecing together a new

piece from randomly chosen parts" (Alpern, 1995). This very simple form of

33

"algorithmic" composition leaves creative decisions in the hands of chance, letting the

role of a dice to decide what notes are to be used.

There are more modern examples, as well, of algorithmic composition without the use

of the computer. John Cage, for example, like Mozart, utilized randomness in many

of his compositions, such as inReunion, performed by playing chess on a photo-

receptor equipped chessboard: "The players' moves trigger sounds, and thus the piece

is different each time it is perfomed" (Alpern, 1995). Cage also delegated the

compositional process to natural phenomena, as in his Atlas Eclipticalis (1961), which

was composed by laying score paper on top of astronomical charts and placing notes

simply where the stars occurred, again delegating the compositional process to

indeterminacy (Schwartz, 1993).

The twelve-tone method and serialism, furthermore, were movements of the post-

World War II era that tried to completely control all parameters of music and to

objectify and abstract the compositional process as much as possible. Decisions over

everything from notes to rhythms to dynamic markings were often subject to pre-

composed "series" and "matrices" of values, which, in effect, "automated" many of

these parameters by determining the order in which each must occur in a piece. These

series and matrices were, then, the "algorithms" that superceded the human creative

process. Serialism can thus be labeled "algorithmic" or "automated" composition in a

rather pure sense, especially when it strives to integrate as many musical parameters

as possible. Olivier Messiaen's 1949 piano etude, Mode de valeurs et d'inensites, for

example, had a thirty-six pitch series, each pitch of which was given specific

rhythmic, dynamic, registral, and attack characterstics with which to be used in the

composition (Kostka, 1995).

* * *

6.3. Use Of The Computer

2 early pioneers: Lejaren Hiller, Iannis Xenakis 3 general approaches: stochastic, rule-based, artificial intelligence (AI)

Computers introduced incredible new capacities available for algorithmic composition

purposes. Ada Lovelace, inventor of the "calculating engine," the precursor of

computers, had this to say about the possibilities of automated composition (Alpern,

1995) in the 19th century:

"Supposing, for instance, that the fundamental

relations of pitched sound in the signs of harmony

and of musical composition were susceptible of

34

such expression and adaptations, the engine might

compose elaborate and scientific pieces of music of

any degree of complexity or extent" (Alpern, 1995).

And so it happened, as Lovelace had predicted, that the computer (or modern

"calculating engine") brought scientists and composers together to construct such

"elaborate" pieces of music out of new algorithmic programming methods.

The earliest instance of computer generated composition is that of Lejaren

Hiller and Leonard Isaacson at the University of Illinois in 1955-56. Using

the Illiac high-speed digital computer, they succeeded in programming basic material

and stylistic parameters which resulted in the Illiac Suite (1957). The score of the

piece was composed by the computer and then transposed into traditional musical

notation for performance by a string quartet. What Hiller and Isaacson had done in

the Illiac Suite was to (a.) generate certain "raw materials" with the computer, (b.)

modify these musical materials according to various functions, and then (c.) select the

best results from these modifcations according to various rules (Alpern, 1995). This

"generator/modifier/selector" paradigm was also later applied to MUSICOMP, one of

the first computer systems for automated composition, written in the late 1950s and

early 1960s by Hiller and Robert Baker, which realized Computer Cantata: "Since

[MUSICOMP] was written as a library ofsubroutines, it made the process of writing

composition programs much easier, as the programmer/composer could use the

routines within a larger program that suited his or her own style" (Alpern, 1995; italics

added). This idea of building small, well-defined compositional functions—i.e.

"subroutines"—and assembling them together would prove efficient and allow the

system a degree of flexibility and generality (Alpern, 1995), which has made this

approach a popular one, as we will see, in many algorithmic composition systems

even into the present day.

Another pioneering use of the computer in algorithmic compostion is that of Iannis

Xenakis, who created a program that would produce data for his "stochastic"

compositions, which he had written about in great detail in his book Formalized

Music (1963). Xenakis used the computer's high-speed computations to calculate

various probability theories to aid in compositions like Atrées (1962) and Morsima-

Amorsima (1962). The program would "deduce" a score from a "list of note densities

and probabilistic weights supplied by the programmer, leaving specific decisions to a

random number generator" (Alpern, 1995). "Stochastic" is a term from mathematics

which designates such a process, "in which a sequence of values is drawn from a

corresponding sequence of jointly distributed random variables"

(Webster'sdictionary). As in the previous example of the Illiac Suite, these scores

were performed by live performers on traditional instruments.

35

With Xenakis, it should be noted, however, "the computer has not actually produced

the resultant sound; it has only aided the composer by virtue of its high-speed

computations" (Cope, 1984): in essence, what the computer was outputing was not the

composition itself but material with which Xenakis could compose. In contrast, the

work of Hiller and Isaacson attempted to simulate the compositional process itself

entirely, completely delegating creative decisions to the computer.

Already in these first two examples—Xenakis and Hiller—we find two different

methodologies that exist in computer-generated algorithmic composition: (1.)

"stochastic" vs. (2.) "rule-based" systems. As we will see, there is also a third

category, (3.) which we can label AI, or artificial intelligence systems.

Stochastic approaches, already somewhat touched upon, are the simplest. These

involve randomness and can be as simple as generating a random series of notes, as

seen already in the case of Mozart'sDice Music and in the works of John Cage, though

a great amount of conceptual complexity can also be introduced to the computations

through the computer with statistical theory and Markov chains. Basically, many of

the creative decisions in the stochastic method are merely left to chance, essentially

the same as drawing notes out of a hat. Another example of non-computer-oriented

"stochastic" composition can be found in Karlheinz Stockhausen's Klaveirstucke

XI in that the sequence of various fragments of music are to be performed by a pianist

in random sequence. A different slant to usages of unexpectedness is that of

applying chaos theory to algorithmic composition (Burns, 1997). These applications

employ various nonlinear dynamics equations that have been deduced from nature and

other chaotic structures such as fractals to relay different musical information:

"In recent years [the '70s and '80s], the behaviour

of systems of nonlinear dynamical equations when

iterated has generated interest into their uses as

note generation algorithms. The systems are

described as systems of mathematical equations,

and, as noted by Bidlack and Leach, display

behaviours found in a large number of systems in

nature, such as the weather, the mixing of fluids,

the phenomenon of turbulence, population cycles,

the beating of the human heart, and the lengths of

time between water droplets dripping from a leaky

faucet" (Alpern, 1995).

This is a large and mathematically complex field of algorithmic composition, and the

author refers the interested reader to my website on the topic as well as to the article

36

by Jeremy Leach ("Nature, Music, and Algorithmic Composition." Computer Music

Journal, 1995) as good starting points for more in-depth investigation.

A second approach to algorithmic composition using the computer is that of "rule-

based" systems and formal grammars: "An elementary example of a rule-based

process would center around a series of tests, or rules, through which the program

progresses. These steps are usually constructed in such a way that the product of the

steps leads to the next new step" (Burns, 1997). Non-computer parallels to rule-based

algorithmic composition that have been previously mentioned include the 15th-century

canon of the Renaissance period as well as the post-WWII twelve-tone method and

integral serialism. Rather than delegating decisions to chance as in the stochastic

methods just described, rule-based systems pre-compose a "constitution," so to say, or

a "grammer," by which the compositional process must behave once set into motion—

"grammar" being a term borrowed from linguistic theory which designates the formal

system of principles or rules by which the possible sentences of a language are

generated (Burns, 1997). Like Hiller's MUSICOMP, these efforts usually take the

form of a computer program or a unified system of subroutines, and often also involve

databases of various rules either collected from compositional techniques of the past

or newly invented. One example of using a "rule-based" method of algorithmic

composition is that of William Shottstaedt's automatic species counterpoint program

that writes music based on rules from Johann Joseph Fux' Gradus ad Parnassum, a

counterpoint instruction book from the early 18th-century aimed at guiding young

composers to recreate the strictly controlled polyphonic style of Palestrina (1525-

1594) (Grout, 1996):

"The program is built around almost 75 rules, such

as 'Parallel fifths are not allowed' and 'Avoid

tritones near the cadence in lydian mode.'

Schottstaedt assigned a series of 'penalties' for

breaking the rules. These penalties are weighted

based on the fact that Fux indicated that there

were some rules that could never be broken, but

others did not have to be adhered to as

vehemently. As penalties accumulate, the program

abandons its current branch of rules and backtracks

to find a new solution" (Burns, 1997).

Another example is that of Kemal Ebcioglu's automated system called CHORAL

which generates four-part chorales in the style of J. S. Bach according to over 350

rules (Burns, 1997).

37

One last unique approach that I found to algorithmic composition using the computer

is that of aritifical intelligence (AI) systems. These systems are like rule-based

systems in that they are programs, or systems of programs, based on some pre-defined

grammar; however, AI systems have the further capacity of defining their own

grammar—or, in essence, a capacity to "learn." An example of this is David Cope's

system called Experiments in Musical Intelligence (EMI). Like the previous

example of Shottstaedt and of Ebcioglu's CHORAL, EMI is based on a large database

of style descriptions, or rules, of different compositional strategies. However, EMI

also has the capactiy to create its own grammar and database of rules, which the

computer itself deduces based on several scores from a specific composer's work that

are input to it. EMI has been used to automatically compose music that evokes already

somewhat successfully the styles of Bach, Mozart, Bartók, Brahms, Joplin, and many

others.

Another interesting branch of AI techniques is that of "genetic programming," a very

recent technique in the field of computer science for "automatic programming" of

computers (Alpern, 1995). Rather than basing its grammar on scores input to the

computer as in EMI, genetic programming generates its own musical materials as

well as form its own grammar. The composer must also program a "critic" function,

therefore, which then listens to the numerous automatically produced outputs at

various stages of the processing to decide which are "fit" or suitable for final output

(the composer having final say, then, as to which of these to discard and which to

save). Below is a more in-depth description of the different processes involved in

genetic programming methods:

"[Genetic programming] is a method which actually

uses a process of artificially-created natural

selection to evolve simple computer programs. In

order to perfrom this process, one uses a small set

of functions and terminals, or constants, to

describe the domain one wishes an evolved

program to operate in. For example, if the human

programmer wishes to evolve a program which can

generate or modify music, one would give it

functions which manipulate music, doing things

such as transposition, note generations, stretching

or shrinking of time values, etc. Once the functions

have been decided on, the genetic programming

system will create a population of programs which

have been randomly generated from the provided

function set. Then, a fitness measure is determined

38

for each program. This is a number describing how

well the program performs in the given problem

domain. Since the initial programs are randomly

generated, their performance will be very poor—

however, a few programs are likely to do slightly

better than the rest. These will be selected in pairs,

proportionate to their fitness measure, and then a

new population of programs will be created from

these individuals, and the whole process will be

repeated, until a solution is reached (in the form of

a program which satisfies the critic), or a set of

number of iterations has passed. Operations which

may be performed in generating this new

population include reproduction (passing an

individual program on into the next generation

unchanged), crossover (swapping pieces of code

between two 'parent' programs in order to create

two unique 'children'), mutation, permutation, and

others" (Alpern, 1995).

The composer, thus, provides the system with a library of functions, or subroutines, as

we have already seen in the case of Hiller's MUSICOMP and other systems, which

can do various things to the generated musical materials: however, in this case, the

composer does not define the way in which these functions will be used—the

composer merely defines for the computer what is desirable in an output (i.e. designs

a "critic") and the computer in turn tries to automatically achieve these results using

the provided subroutines. This form of "algorithmic composition," thus, (using AI or

genetic programming) can be seen as an extreme case, abstracting itself even from its

own "algorithm" since the output it produces as well as the formal process by which it

performs is automatically constructed.

Besides the three various methods of algorithmic composition using the computer that

I have described—stochastic, rule-based, and AI—further distinction also occurs in

the type of musical output different algorithmic composition systems produce. Some

systems specify score information only (i.e. pitch, duration, and dynamic material) to

be realized by whatever acoustic or electronic instruments, as seen already in the early

cases of Hiller and Xenakis and which is also true in the case of Cope's EMI

compositions (the MIDI scores of which are fed into a Disclavier or other MIDI sound

device for output) and most others mentioned in this paper. Other systems, however,

do not create scores and focus instead on electronic sound synthesis or manipulation

of recorded sounds (i.e. musique concréte), or on a combination of these activities.

39

Sound synthesis algorithms, furthermore, "have been used in a variety of ways, from

the calculation of complex waveforms (building sounds), to the evolution of timbre

development over time" (Burns, 1997). A last approach is to combine both score and

electronic sound synthesis in the system's output, controlling both structural content

and its own timbral realization.

6.4. Closing

As for new developments in the field today, automatic listening programs seem to be

a new trend and focus: not only does the computer automatically compose, it is also

being designed to listen and respondto music being performed around it, a field of

music that is labelled "live electronics":

"Another tendency is to use the computer as an

accompanist who listens to what is being played

and responds appropriately in real-time. Here, the

human input is used to generate rules on which the

machine will base its output. This is seen in such

programs as Cypher (Rowe, 1993) and IBL-Smart

(Widmer, 1994)" (Jacob, 1996).

Another slant on "automatic listening" is that of Jonathan Berger and Dan Gang

(Berger, 2004) who have created computational models of perception and cognition of

music using AI approaches that have given new insights into the creative properties

inherent in listening and, furthermore, to the process of creativity itself . These new

techniques could also potentially improve algorithmic composition, it would seem,

since the "critic" functions that we have seen in examples of genetic programming

could gain much improvement from their insights into how humans listen to music:

the computer could, then, better judge itself as to the quality of its output.

Aesthetically speaking, the more recent and complicated brands of algorithmic

composition that utilize the computer are still in their infancy and much improvement

is, perhaps, left to be desired. As Cope himself remarks, for example, in regard to

Hiller's early experiments with the Illiac, many "directions in 'computer control' have

not proven to be great artistic successes" (Cope, 1984). These various new directions

with the computer (i.e. stochastic, rule-based, and AI) have been, nevertheless,

extraordinarily important for they have "opened the door to new vistas in the

expansion of the computer's development as a unique instrument with significant

potential" (Cope, 1984). They have also broadened our conception of music and how

40

it can be realized, as well as given us rare opportunities to test different compositional

theories, listen to them in action, and also then try to improve upon them. Thus, not

only has the composer been able to do new things with the computer through

algorithmic means, s/he has also been able to investigate him/herself more closely and

to gain new insights not only into his/her own compositional processes but into the

techniques and strategies of composers throughout history. These experiments are

thus intellectually stimulating and important in their own right for these reasons, and

time will tell whether they cannot also produce many "great artistic successes."

7. Max MSP

7.1. Introduction Max is a visual programming language for music and multimedia developed and maintained by San

Francisco-based software company Cycling '74. During its 20-year history, it has been used by

composers, performers, software designers, researchers, and artists for creating recordings,

performances, and installations.

The Max program itself is modular, with most routines existing in the form of shared libraries. An API

allows third-party development of new routines (called "external objects"). As a result, Max has a large

user base of programmers not affiliated with Cycling '74 who enhance the software with commercial

and non-commercial extensions to the program. Because of its extensible design and graphical interface

(which represents the program structure and the GUI as presented to the user simultaneously), Max has

been described as the lingua franca for developing interactive music performance software.6

7.2. Language Max is named after the late Max Mathews, and can be considered a descendant of MUSIC, though its

graphical nature disguises that fact. As with most MUSIC-N languages, Max/MSP/Jitter distinguishes

between two levels of time: that of an "event" scheduler, and that of the DSP (this corresponds to the

distinction between k-rate and a-rate processes in Csound, and control rate vs. audio rate in

SuperCollider).

The basic language of Max and its sibling programs is that of a data-flow system: Max programs (called

"patches") are made by arranging and connecting building-blocks of "objects" within a "patcher", or

visual canvas. These objects act as self-contained programs (in reality, they are dynamically-linked

libraries), each of which may receive input (through one or more visual "inlets"), generate output

(through visual "outlets"), or both. Objects pass messages from their outlets to the inlets of connected

objects.

Max supports six basic atomic data types that can be transmitted as messages from object to object: int,

float, list, symbol, bang, and signal (for MSP audio connections). A number of more complex data

41

structures exist within the program for handling numeric arrays (table data), hash tables (coll data), and

XML information (pattr data). An MSP data structure (buffer~) can hold digital audio information within

program memory. In addition, the Jitter package adds a scalable, multi-dimensional data structure for

handling large sets of numbers for storing video and other datasets (matrix data).

Max is typically learned through acquiring a vocabulary of objects and how they function within a

patcher; for example, the metro object functions as a simple metronome, and the random object

generates random integers. Most objects are non-graphical, consisting only of an object's name and a

number of arguments/attributes (in essence class properties) typed into an object box. Other objects

are graphical, including sliders, number boxes, dials, table editors, pull-down menus, buttons, and other

objects for running the program interactively. Max/MSP/Jitter comes with about 600 of these objects as

the standard package; extensions to the program can be written by third-party developers as Max

patchers (e.g. by encapsulating some of the functionality of a patcher into a sub-program that is itself a

Max patch), or as objects written in C, C++, Java, or JavaScript.

The order of execution for messages traversing through the graph of objects is defined by the visual

organization of the objects in the patcher itself. As a result of this organizing principle, Max is unusual in

that the program logic and the interface as presented to the user are typically related, though newer

versions of Max provide a number of technologies for more standard GUI design.

Max documents (called patchers) can be bundled into stand-alone applications and distributed free or

sold commercially. In addition, Max can be used to author audio plugin software for major audio

production systems.

With the increased integration of laptop computers into live music performance (in electronic music and

elsewhere), Max/MSP and Max/Jitter have received quite a bit of attention as a development

environment available to those serious about laptop music/video performance.7

42

9. References

No. Reference Source

1 J J O'Connor, E F Robertson. 2000. Fatou Biography. [ONLINE] Available at: http://www-history.mcs.st-and.ac.uk/Biographies/Fatou.html. [Accessed 05 December 13].

Online Article

2 J J O'Connor, E F Robertson. 2008. Julia Biography. [ONLINE] Available at: http://www-history.mcs.st-and.ac.uk/Biographies/Julia.html. [Accessed 05 December 13].

Online Article

3 Oracle Education Foundation. 1999. Fractal Applications. [ONLINE] Available at: http://library.thinkquest.org/26242/full/ap/ap.html. [Accessed 12 December 13].

Website

4 Oracle Education Foundation. 1999. Fractals and Fractal Geometry. [ONLINE] Available at: http://library.thinkquest.org/3493/frames/fractal.html. [Accessed 12 December 13].

Website

5 John A. Maurer. 1999. The History of Algorithmic Composition. [ONLINE] Available at: https://ccrma.stanford.edu/~blackrse/algorithm.html. [Accessed 08 January 14].

Website

6 Place, T. and Lossius, T.: Jamoma: A modular standard for structuring patches in Max. In Proc. of the International Computer Music Conference 2006, pages 143–146, New Orleans, US, 2006.

Online Article

7 Cycling 74. 2014. Max 5 Help and Documentation. [ONLINE] Available at: http://cycling74.com/docs/max5/vignettes/intro/docintro.html. [Accessed 14 December 13].

Online User Manual