2013

Robert Braileanu

University of West London

Fractal Sound

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Task 2 – Portfolio Submission

Project Document

Fractal Sound

Student Full Name Robert Braileanu

Student Number 21137205

Contact Address Flat 9, Emanuel Court, Emanuel Avenue, Acton Town, London, UK

Mobile Number 0044 7596 489782

Email Address 21137205@student.uwl.ac.uk / robert.braileanu@gmail.com

Word Count* 5994

Module Title: Experimental Sound Module Code: MU60011E Level: 6 Course: BA Music Technology Specialist

* The word count excludes all quotes derived from external sources of any nature including (but not limited to) books, articles, websites, videos, module

study guide, university documents and other lecture support documents.

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Contents

1. Executive Summary ......................................................................................................................... 4

2. Brief................................................................................................................................................. 5

3. Experimental Audio Focus................................................................................................................ 5

4. Research .......................................................................................................................................... 6

4.1. Project Title .............................................................................................................................. 6

4.1.1. Fractal ............................................................................................................................. 6

4.1.2. Sound .............................................................................................................................. 7

4.2. Fractal Geometry ..................................................................................................................... 7

4.2.1. Brief Introduction. The Mandelbrot Set ............................................................................ 7

4.2.2. Mandelbrot set: characteristics ........................................................................................ 8

4.2.3. Mandelbrot set: applications4 ........................................................................................... 9

4.3. Sound..................................................................................................................................... 10

4.3.1. Brief Introduction ........................................................................................................... 10

4.3.2. Algorithmic composition ................................................................................................. 10

4.4. Satellite Themes ..................................................................................................................... 12

4.4.1. Max MSP ........................................................................................................................ 12

4.4.2. Terminology ................................................................................................................... 13

4.4.3. Press Release Brochure .................................................................................................. 13

5. Ethical Issues ................................................................................................................................. 13

5.1. Professional ethics ................................................................................................................. 13

5.2. Inter-personal ethics .............................................................................................................. 13

6. Copyright ....................................................................................................................................... 14

7. Project Development ..................................................................................................................... 14

8. Track 1 – Cantor’s Journey ............................................................................................................. 16

8.1. Introduction ........................................................................................................................... 16

8.2. Experimental Audio Focus ...................................................................................................... 16

8.3. Production Process ................................................................................................................ 16

9. Track 2 – Snowflake Dance ............................................................................................................ 17

9.1. Introduction ........................................................................................................................... 17

9.2. Experimental Audio Focus ...................................................................................................... 17

9.3. Production Process ................................................................................................................ 17

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10. Track 3 – Scaled Roughness ........................................................................................................ 18

10.1. Introduction ....................................................................................................................... 18

10.2. Experimental Audio Nature ................................................................................................ 18

10.3. Production Process ............................................................................................................. 18

11. Conclusion ................................................................................................................................. 19

12. Glossary of Terms ...................................................................................................................... 19

13. References ................................................................................................................................. 20

14. Appendix ................................................................................................................................... 22

1. Press Release Brochure .............................................................................................................. 22

2. Research Structure – Sketches ................................................................................................... 24

3. Fractal Sound Generator – Early Versions ................................................................................... 26

4. Fractal Sound Generator – Final Version .................................................................................... 27

Audio CD content:

Order Content

1 Track 1 – personal composition

2 Track 2 – personal composition

3 Track 3 – personal composition

Data CD content – located in Research Folder:

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. Executive Summary

‘Fractal Sound’ proposes the idea of interaction between sound and mathematics. This document

presents a theoretical concept based on a combination of original ideas and research. It is aimed to

support the body of work comprised of three audio tracks developed as an experimental approach to

audio. These are original compositions which employ fractals as their fundamental building blocks.

Furthermore, fractal principles form the very fabric of this project, ranging from audio content to

structure and all adjacent media featured as part of this project.

This project presents an experimental approach to working with audio – described in detail in section 3;

it is part artistic venture and part scientific fact, merging the notions of ‘sound’ and ‘fractals’ into a single

entity, in much the same way history has seen 11syncretism in arts and science.

The project also serves as a case study for the implementation of fractal geometry principles as the basis

for music composition, in an attempt to gain knowledge on the subject and to raise interest for more

research to be conducted.

Research supporting the ideas presented in this document aims to provide a context for all aspects of

the work. The material is organized into core and satellite themes, in a fractal manner where each point

mentioned opens new doors for ideas to form and thus creating a ‘fractal web’ of information. Research

is discussed in depth in section 4; additionally, more information can be found in the research folder by

following the subscript indexes in this document (e.g. sound1) and the table of contents in the separate

research folder. The two core themes discussed here are sound and fractal geometry; these will span

out into different sub-areas covering the object of this document. Separately, a number of satellite

themes are be dealt with; these include terminology, technical considerations, ethics, copyright issues

and press release conventions. Despite being described as satellite themes, these play an essential role

in fully engaging with the project and help towards understanding all other historical, technical, logistical

and ethical considerations together with the implications they have with regards to the work at hand.

Alongside this document – the main written account – the presentation package also contains a

Redbook standard audio CD with the audio clips, a separate data CD containing extra material, an

additional data CD containing only the softcopy of the press release brochure as a single PDF file, the

printed version of the press release brochure, as well as a separate research folder. The data is

organised in such a way that it is made easily accessible: superscript indexes (e.g.1) are used to signify

external references found under section 13; references to more in-depth areas are marked with a

subscript index (e.g.1) – these can be found in the separate research folder by means of its content list.

Furthermore, red coloured superscript indexes (e.g.1) depict the original audio pieces on the main audio

CD, while red coloured subscript indexes (e.g.1) make reference to external material located on the data

CD in the research folder. Lastly, a blue coloured subscript index number preceding a word makes

references to the glossary of terms located at the end of this document in section 12 (e.g. 2CGI).

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2. Brief

The requirements for this project are to create a body of work – a portfolio of audio material – of

experimental nature, encompassing multiple skills derived from the course such as programming,

recording, editing etc. The nature of the task is to treat audio in an experimental manner and to support

all original ideas with relevant research, clearly organized and structured in the written work and

research folder. One important aspect of this task is its holistic approach – the entire project needs to be

accounted for – therefore areas such as ethics, copyright, resources etc. must be addressed. In addition,

the work needs to be structurally cohesive with the concept chosen. Furthermore, a press release

brochure needs to be produced for promotion of the project.

3. Experimental Audio Focus The nature of this project is experimental in the sense that audio will be created with regards to the

ideas mentioned above. The concept is to employ generative fractal algorithms such as the

1Mandelbrot1 set and to map the resulting numbers onto the frequency and time domain. It will

therefore portray a sonic representation of a fractal, by combining multiple variations of polyphonic and

monophonic renderings to create cohesive pieces.

A software application named ‘Fractal Sound Generator’5 was written in 2Max MSP27 – visual interface

programming language – to generate a Mandelbrot set of a finite number of 12 3iterations due to

complexity of the calculations and hardware considerations. The numbers generated are scaled into the

human hearing range of approximately 20Hz to 20kHz and can then be altered via a range of controls,

which include an intuitive range setting and oscillator blend control. In addition, one of two basic modes

of operations can be selected:

Polyphonic – where the instrument acts as an additive synthesizer – each iteration generates a

specific frequency and all of these are added together to create a sonic texture. In polyphonic

mode, the instrument automatically sets the amplitude of each iteration by scaling the result to

control individual levels. The resulting audio can be considered both a chord in western music or

as a single sound composed of multiple harmonics;

Monophonic – a random number generator selects only one iteration at a time within the given

range of the set’s domain (where the modulus of the calculation result is not greater than the

number 2) to play its specific frequency. Timing is controlled by the fractal values themselves

and therefore, it can be argued that a primitive fractal rhythmical pattern is created.

Timbre has been specifically left to the user’s control by means of the oscillator blend component. This

decision has been made after testing the software and reviewing visual renders of fractals which, if left

purely at the algorithms’ mercy, will most likely produce a less pleasing result than those controlled by

the user.

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The program allows for audio files to be created via the record and save buttons. These files are then

sequenced to produce the compositions available on the audio CD. Images of the final versions can be

seen in appendix 4.

Structurally, the three compositions submitted with this project are also based around the idea of

fractals. The structure of each track is an audio translation of a specific type of fractal or a property of

fractals. The table shows the type of fractals used for each track and also provides a reference for

further information.

Track Number

Type of Fractal Detailed description of track structure

Further information on the type of fractal (research folder)

1 Cantor Set Section 8.2 Section 5.1

2 Koch Snowflake Section 9.2 Section 5.2

3 Mandelbrot Set Section 10.2 Section 4.2.1 (main written work)

To sum up, this approach tackles the four fundamental aspects of sound: pitch, amplitude, time and

timbre by relying mostly on fractals and thus, creating an audible translation of fractals. One can

therefore see this method as an experimental way of dealing with sound.

4. Research As mentioned in the executive summary, the idea of combining fractals with sound has risen from

intellectual curiosity. For this project there are two main research areas based on the core themes

expressed in the title: fractals and sound. Research will present a brief history of both realms which will

introduce any reader into the subject matter. It will also provide an etymological account into what can

be understood by the two terms which will link the audio material to the ideas presented in the

document. The research will show how the two areas converge and provide the basis on which this

entire document has been written.

Provided below is a description of the title, after which the main areas of research – fractal geometry &

sound - will be discussed, followed by satellite themes towards the end of this section.

4.1. Project Title

4.1.1. Fractal

From an etymological standpoint the term ‘fractal’ derives from the French word ‘fractale’ – ‘broken’ or

‘uneven’, as mentioned by Mandelbrot (1977)1. Used as a noun, a fractal describes ‘a curve or

geometrical figure, each part of which has the same statistical character as the whole’ according to the

Oxford dictionary3. However, in this title it is used in the form of an adjective to depict the fractal nature

of sound from a micro and macro perspective. On the one hand, the micro-perspective refers to the

inner-workings of a sound, the distribution of harmonics and synthesis methods while on the other

hand, the macro-perspective takes into account how sounds interact to create music and the laws that

govern this process.

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4.1.2. Sound

In this context, ‘sound’ inherits multiple meanings. Firstly, it refers to vibrations travelling through the

air or another medium4. This scientific connotation describes sound as a physical phenomenon,

measured in 4Hertz, disregarding any formally acknowledged musical system such as tonality. Secondly,

as mentioned in the Oxford Dictionary (2013)5, ‘sound’ can describe ‘a distinctive quality of the music of

a particular composer, performer or particular instrument’ – for example, the sound of violins or the

sound of Mozart. In this case however, the term will quote the sound of fractals – using fractals as the

premises for abstract composition and piece structure.

4.2. Fractal Geometry

4.2.1. Brief Introduction. The Mandelbrot Set

‘Geometry. Its principles are taught to young students across the world. The Pythagorean theorem;

Surface area and volume; Pi; This classical, or Euclidean, geometry is perfectly suited for the world that

humans have created. But if one considers the structures that are present in nature, that which are

beyond the realm of smooth human construction, many of these rules disappear. Clouds are not perfect

spheres, mountains are not symmetric cones, and lightning does not travel in a straight line. Nature is

rough, and until very recently this roughness was impossible to measure. The discovery of fractal

geometry has made it possible to mathematically explore the kinds of rough irregularities that exist in

nature.’ 8, 18

This is perhaps the most important idea promoted by Benoit Mandelbrot, one of the founders of fractal

geometry. While working at 5IBM, in 1975 he coined the term ‘fractal’ to describe a geometry

characterized by roughness and not by straight lines and perfect circles. Although the company who

employed him considered his discovery a breakthrough for dealing with noise in telephone signal

transmission, Mandelbrot realised the implications it can have in a vast range of areas from cartography

to image compression. His ideas were based on the work of his predecessors – mathematicians Pierre

6Fatou26 and Gaston 7Julia3

7, who proposed a simple formula to map values on the complex plane:

z = z² + c

The equation uses a variable ‘z’ and a

constant ‘c’ to define 6complex numbers on a

7Cartesian coordinate system. At the time of

its first incarnation, the technology available

prohibited Fatou and Julia to tap on to its

true potential and the issue was considered

impractical by the mathematical community

of that time. However, at IBM Mandelbrot

was given access to some of the most

advanced computers available and therefore,

he was able to iterate the equation

thousands and thousands of times, giving

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birth to some of the most stunning images of fractals known today – the above image represents a

visual fractal based on the Mandelbrot formula iterated 2 million times. As Mandelbrot realised, the key

to harnessing the formula’s potential was the immense number of iterations, made possible by

advances in the field of computers.

Apart from the Mandelbrot set, different variations of fractals5 exist. An in-depth description of these

can be found in the research folder in section 5.

4.2.2. Mandelbrot set: characteristics

One fundamental aspect of fractals is their property of self-similarity – each individual part is similar to

the entire element.9 Self-similarity is a constant throughout nature; it can be seen in outlines of maps,

coastlines, edges of mountains and canyons, tree branches and leaf structures, magnified snowflakes,

river networks, the nervous system, sutures between skull plates, lung structure, clouds in the

atmosphere, plasma loops on the surface of the sun, nebulae, etc.4

Therefore, one can see that self-similarity has implications in many aspects of our universe and that

studying this property can be done by understanding the principles behind fractal geometry. However,

these so-called natural fractals are different from mathematical models in the sense that nature makes

use of multiple forces combined in various processes. For example, coastlines are formed by the forces

generated by waves, cliff erosion, rivers flowing into the ocean, accumulation of sediments etc. while

temperature and weather conditions also play an important role10. Comparing this to an established

generative function such as the Mandelbrot set could make the latter seem rather primitive. Another

fundamental difference between our man-made algorithms and nature’s fractals is the limited scale in

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which nature operates. The principles behind waves as a force are fundamentally different from those

making up the structure of the coastline on an atomic level9.

Conversely, self-similarity in fractals can be seen most easily in the Sierpinski Gasket6 – a triangle-shaped

fractal which appears the same regardless of the magnification/reduction factor applied. The generation

method produces three new triangles ½ the height and width of the original. Theoretically, the process

can be repeated an infinite number of times.

In doing so, one can deduce another property of

fractals – scale ambiguity. While Euclidian

geometrical shapes have finite perimeter and area,

a fractal object can have a finite perimeter - as in

the case of the Mandelbrot fractal – but the area

can be considered infinite because, theoretically, a

fractal of infinite complexity can be generated. As

a general observation, it is these properties that

make fractals resemble shapes found in nature.

A visual representation of both self-similarity and

scale ambiguity can be seen in Michael Hogg’s

render1 of an M-set iterated 90456 billion times. ‘Slow deep Mandelbrot zoom’1, featured on the data

CD located in the separate research folder, took 12 days 1 hour and 17 minutes to render using a

commercially available computer.11

Following the discovery of fractal geometry along with some of its characteristics, a number of practical

applications have been derived. These are discussed in more detail below.

4.2.3. Mandelbrot set: applications4

Perhaps the most important application of fractals in general is data analysis. By analysing fractal

characteristics of large quantities of data, a pattern can be deduced, from which the data’s fractal

dimension can be extracted. Fractal dimension is a ratio – describing complexity – of how details in a

fractal pattern – be it a theoretical model, a tree, a cloud etc. – change with the scale at which it is

measured12. This has given us an insight into the inner-workings of diverse fields, some of which are

included below – for a more detailed account please see research folder (section 4):

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Astronomy Biology / Chemistry Creative Other

Galaxies Bacteria Cultures Fractal Art Clouds

Rings of Saturn Chemical Reactions Fractal Music Coastlines and Borderlines

Human Anatomy 8CGI Data Compression

Molecules Special Effects Diffusion

Plants Economy

Population Growth Weather

4.3. Sound

4.3.1. Brief Introduction

Building on the definitions of the word ‘sound’, as discussed in the project title sub-section, the term can

inherit multiple meanings depending on the angle from which it is regarded.

From a physical standpoint, sound is a universal phenomenon transmitted through a medium such as

air. Vibrations from a source travel through the medium acting as a wave front – in a sense, sound can

be viewed as a means of transmitting data; audio data which is then decoded by a receiver: the human

auditory system for example.4

From a psychological standpoint, sound is a powerful entity with qualities far beyond the physical realm.

Tapping into the very fabric of human nature, sound can alter emotions; it can influence one’s state of

mind. History has shown how sound was used both as a healing and destructive mechanism.13

Sound represents the primordial requirement for music, and while there is no exact definition for

‘music’, understanding its ‘language’ is an innate quality of humans. Throughout history, mankind has

been fascinated by sound and music, and has questioned, studied and advanced the knowledge based

around the two, spanning out similar to the branches of a tree – an evolution model which one can

consider to be of fractal nature.

From the most basic 9chants to the development of mechanical instruments, the history of music shows

the curiosity to develop new means of creating sound and music as a constant.

4.3.2. Algorithmic composition

Algorithmic composition refers to the use of algorithms as a generative engine for music. What also

started out as a component of curiosity quickly spanned out over vast distances, being explored in

different parts of the world by artists and scientists alongside.15

The idea of a formal set of instructions to create music stretches back to the ancient Greeks, as Grout

(1996)16 mentions: ‘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 inseparable 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’. Another

historical example of early algorithmic composition can be found in the 15th century’s ‘canonic’ music –

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by which a singer is given a single melody and a set of rules to derive subsequent voices. The rule or set

of rules was called a ‘canon’ (Grout, 1996)16. In the 20th century, John Cage16 experimented with the use

of randomness in his compositions, while the end of World War II brought ‘twelve-tone serialism’ as a

form of music composition.6

In the digital realm, Lejaren Hiller and Leonard Isaacson devised a way of generating music in an ‘Illiac

High-Speed Digital Computer’ at the University of Illinois in 1955-1956. The result then had to be

transcribed into traditional music notation to be played by a string quartet.17 Five years later, Iannis

Xenakis pioneered the use of chance and probability to create music; this is referred to as 10‘Stochastic

Music’.17 A comprehensive history of algorithmic composition is included in the research folder6. In

addition, a list of songs located on the separate data CD – part of the research folder – can be found

below. These are pieces by some of the composers mentioned in this section – the pioneers of

algorithmic music and in terms, the pioneers of fractal music.

Composer Piece Description

John Cage Atlas Eclipticalis2 A score paper was placed on top of an astronomical chart and notes were placed where stars were present – chance composition

Lejaren Hiller Illiac Suite for String Quartet - Part 13

Composed by the ‘Illiac High-speed Digital Computer’ and then transcribed into traditional music notation.

Iannis Xenakis ST/10=1,0802624 Composed by Xenakis’ own computer program, it follows stochastic laws to define pitch, timing, duration and timbre (arco, pizzicato etc.)

Algorithmic composition includes different ways of generating music including the use of fractals. Below

is a breakdown of some of the models used for such compositions15 – it is worth noting that a clear

distinction between the different models cannot be accurately produced as some components are

common in more than one category:

Mathematical models – rely on equations and stochastic processes mapped to different

parameters in varying degrees. For example, one can assign a value to a frequency or it can

round of its number to match a note on an instrument.

Knowledge-based systems – proposes the analysis of a specific style of music (by the user), by

studying its characteristic and then replicating them into a model, with the hope of creating

similar compositions to those of the original choice of study.

Grammars – this category provides a formal language upon which music is created. One

example is a type of fractal called an ‘L-system’. Assuming a set of rules: A=BAC ; B=ACB ; C=CBA

and an initial axiom : A C B A, one can generate music by assigning a parameter to these symbols

(e.g. pitch) and then following the set of rules to replace each initial symbol with its definition.

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Fractal Sound

Terminology

Copyright

Ethics Press

Release Brochure

MaxMSP

Evolutionary Systems – these are models which replicate biological functions such as mutations

or processes of natural selection into a model which is then used to control the properties of

sounds.

Learning Systems – programs that have the capacity to collect data from material provided by

the programmer/user and then devising and constantly improving an algorithm to generate

music.

Hybrid Systems – a combination of the above – which is perhaps the most used method today.

The method used to generate sounds and the pieces on the audio CD can be considered a hybrid

system. The ‘Simple Fractal Generator’ software employs stochastic components in the form of random

generators to produce an initial value for the software to operate with. It also contains a mathematical

model in the form of a Mandelbrot set equation, while different types of fractals are used to generate

melodies and structures.

4.4. Satellite Themes

Having briefly discussed the core themes in sections

4.2 and 4.3, adjacent satellite themes will be

mentioned below. Although these might not seem

fundamental to the object of the project, they

provide an account for important aspects in the

development stages of the work at hand and also

provide a context in terms of the legal framework

and ethics involved. The diagram on the left hand

side depicts the five main areas of research

surrounding the core theme. Note that both

‘copyright’ and ‘ethics’ are included in the adjacent

diagram. Although these consist of a research

component, they will be acknowledged separately –

having their own individual headings – as they

portray an overview of the work at hand from a legal and ethical point of view.

4.4.1. Max MSP

Max MSP7 is a graphical-interface programming language designed by Cycling 74©2. Its main uses

include the development of audio, video and multimedia applications which can be run internally or

exported into a format understood by most major operating systems – Microsoft Windows© ; Apple

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MacOSX©2. The software has provided an ideal environment for developing an application which can

generate audio from a fractal formula. A more detailed description can be found in the research folder.

4.4.2. Terminology

Due to the nature of this project, specialist terminology is used. A requirement for this project is for it to

be accessible to a diverse range of people, including those who are not familiar with the subject being

discussed. Therefore a glossary of terms which provides a basic understanding for some of the terms

used has been made available; it can be found in section 12.

4.4.3. Press Release Brochure

A press release is the first point of contact with the media, in the professional realm. It is intended to

promote and to ‘sell’ an idea or product. It can be in the form of a formal letter or it can include a

graphical design.21 Certain conventions have been perpetuated throughout the years, conventions which

are now considered a standard. These include21: a genuine headline; a striking design; concise content;

applicable to desired audience; mentions partners; provide contact details. A press release brochure is

included at the end of this document, in appendix 1, section 14.

5. Ethical Issues The term ‘ethics’ is defined by the Oxford Dictionary as ‘moral principles that govern a person’s

behaviour or the conducting of an activity’ and ‘the branch of knowledge that deals with moral

principles’.14 From the first definition two distinct areas with their own implications can be derived:

professional ethics and inter-personal ethics.

5.1. Professional ethics These are to do with the content of the work and the implications it has on the professional

individuals/bodies/institutions/establishments to which it is referring to and to those with which the

work is directly and/or indirectly related to such as Benoit Mandelbrot, IBM©, Cycling 74©, Microsoft©,

Apple©, The University of West London etc. Considering ethics, one must make sure the work will not

produce any moral and physical damage to their image or bring them into disrepute. In addition, should

the work have made reference to, provided illegal materials or promoted bad practises, these would

have constituted a serious offence both from an ethical and legal point of view (e.g. pornographic

images in the press release brochure). I hereby declare that no damage was made to any of the

individuals, bodies, institutions and/or establishments mentioned or related to this project.

5.2. Inter-personal ethics Referring to the relationship with the people involved in the development of the project, inter-personal

ethics is related to the well-being of others and creating a healthy and pleasant environment for the

work to be produced in. As the work did not require the involvement of others, this aspect does not to

apply this project.

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Another important area of ethics is health and safety. Although the work was produced entirely on a

digital platform – within the computer – these issues can still appear. One example would be to plug the

computer into a faulty power outlet which could potentially be harmful.

6. Copyright The right for intellectual property along with its multiplication and distribution is commonly referred to

as copyright. In the UK, copyright is defined in the ‘Copyright, Designs and Patents Act 1988’19, as

amended. As described in the document, the works eligible for protection are of: literary, dramatic,

artistic or musical nature; the typographical arrangement of a published edition, a sound recording, a

film, or a broadcast.19

The issue is raised on how copyright affects scientific research, as it falls in neither of these categories.

However, the act provides a ‘fair dealing’ provision which allows for a ‘reasonable proportion’ of the

work to be copied for ‘non-commercial research or private study’, but no specific method is described.19

In terms of data collected for research purposes, a fact isn’t protected, but despite this, a collection of

data can make use of database rights. Considering the above, from an ethical standpoint, external

material can be sourced provided acknowledgement of its source and/or ownership is made available.

Formal models such the Harvard referencing system exist as a convention to provide a framework for

the use of non-original material in personal work. This is a complex mechanism which applies to a

variety of sources including text and multimedia formats.20

For this project, copyright applies to all external material used for informative purposes, directly or

indirectly integrated within this document. The Harvard system was used to acknowledge ownership

and sourcing; a list of all materials referenced using this system can be found in section 13.

The same principles mentioned above applies for the distribution of this document, however, copyright

is jointly owned by the institution – UWL – and the writer – myself, according to the university’s rules

and regulations.

7. Project Development Initially, the project started out from the idea of fractals to create music. However, because of the

nature of the project, sound was chosen over music. The idea was to create a piece of software that

could generate sounds in the form of soundscapes and melodies using a multitude of fractal formulas.

Having discussed the concept with the lecturer, and considering the limited time frame, it was suggested

that approaching only certain elements would be more appropriate: using only one type of fractal and

relating it to sound, reflected by frequencies, rather than imposing musical notation, scales etc.

Therefore, only one type of fractal had to be used and a way of understanding how exactly it worked

had to be studied. Having reviewed L-systems, Julia sets and Mandelbrot sets, the latter was chosen as it

15

provided more flexibility in terms of programming. Sketches of how to structure research were then

devised – these can be found in appendix 2.

A high proportion of the allocated 200 hours was used to create the ‘Fractal Sound Generator’ software

(appendix 3) for which countless problems had to be overcome. The application was developed in Max

MSP 5 on a Microsoft Windows system, using only Max’s internal objects. Some of the early incarnations

of the program were unable to produce a reasonable number of iterations due to improper design of

data structures within the software – photos of the early versions are present in appendix 3. In addition,

an unforeseen issue was discovered – the programming language has a peculiar upper limit in terms of

the numbers it can calculate: after reaching values higher than 7 million, the results would appear as

‘infinite’, making it impossible to transform into a frequency value. This was overcome by scaling down

the initial numbers to an interval of 0-1, allowing for the software to calculate up-to 12 iterations. Due

to the architecture of Max MSP, it does not permit creating a finite loop by making the result of a

calculation act as a variable within its generative equation – a fundamental requirement for the iterative

formulas of fractals. Therefore, only a small number of iterations are possible and while in theory it

proves that the concept works, the resulting audio might seem rather primitive. One must understand

that the complex images of visual fractals are the result of millions of iterations. If the number of

iterations for generating audio was similar, one can assume the resulting sounds would have been of

higher standard than the ones achieved by ‘fractal sound generator’. Alternative programming

languages which support iterative equations such as CMusic and Pure Data are available, however, due

to time constraints of the module, learning and devising a piece of software in one of these languages

would have been unrealistic.

The Mandelbrot set equation (z=z2+c), when used to generate images, works by assigning complex

numbers of the form (a, bi) to ‘z’ and ‘c’. This is because the values need to be plotted on a Cartesian

coordinate system with an x and y axis. However, for audio purposes, the approach has been simplified.

The formula uses regular numbers as these are transformed into frequencies, amplitudes etc. For

integrity purposes, the formula attains the rule used in the production of visual images: each iteration

result modulus must not be greater than the number ‘2’. In the visual realm, if it exceeds this value, the

point on the graph is considered to ‘go to infinity’ and the iteration process is stopped.22 The same

applies to ‘Fractal sound generator’: if a number exceeds this value on the first iteration, a new set of

values is automatically generated and the process starts over again. If, however, the result is within the

‘fractal limit’, it is scaled to an audible frequency and used to control amplitude and the result is passed

onto the next iteration, repeating the cycle.

For reasons mentioned above, ‘Fractal sound generator’ should be considered an integral part of this

project. The final version has been made available on the separate data CD located in the research

folder.

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8. Track 1 – Cantor’s Journey

8.1. Introduction Intuitively titled ‘Cantor’s Journey’1, track 1 (1’09’’) aims to introduce the listener into the fractal

universe. It is an audio exploration of essentialist world, dominated by shapes of vast complexity yet

somehow echo a sense of simplicity and calmness.

8.2. Experimental Audio Focus The experimental nature of this track lies in its production process. Firstly, all the material used has been

created via the ‘fractal sound generator’. As mentioned above, the software uses a Mandelbrot set to

map out frequency, duration and amplitude. Secondly, as can be seen in the photo below, the piece is

structured in the form of another type of fractal – the cantor set (more details on the Cantor set can be

found in the research folder). Therefore, the track is a sonic representation of the fractal realm both in

terms of source material and structural composition.

8.3. Production Process Having generated the 7 basic audio components, these have been sequenced using Steinberg Cubase 5.

A number of virtual processing units have been used to add texture to some of the elements and to

create a sense of depth and space in the mix.

The polyphonic fractal audio elements

are accompanyed by a simple fractal

melody, also generated by the

software, which has been pitchshifted

to match the harmonic content of the

piece. Although not a part of the

cantor set, the melody has been added

for aesthetical reasons, adding interest

to the overal piece.

A creative delay has been used on the

shortest clips of the track, adding a

rythmical element to the piece. The

unit uses a random generator to clock

each delay hit and therefore, the track

still retains its integrity with the aparent chaotic and random behaviour of fractals.

Textures have been imposed on almost every element by means of distortion and modulation. Each

individual track has been equed according to its nature – bass-heavy, mid-rangy etc. – and panning has

been used on some elements.

17

9. Track 2 – Snowflake Dance

9.1. Introduction Snowflake Dance2 (1’22’’) aims to evoke positive emotions while still portraying the same sense of

calmness and familiarity as track 1. As it is structurally based around ‘Koch Snowflake’ fractals, the goal

was to create soft textures in a composition which portrays ‘self-similarity’ – a characteristic of fractals.

From an artistic point of view, the slow movement of elements aims to replicate snow fall in the form of

delicate particles floating in a fractal universe.

9.2. Experimental Audio Focus Consisting of an audio translation of two distinct type of fractals, the M-set and Koch Snowflake, the

piece is experimental by means of its content and structure. Similarly to track 1, all audio used was

created with ‘Fractal sound generator’ by iterating a Mandelbrot set in both polyphonic and

monophonic modes. Structurally, the piece replicates a Koch snowflake – this fractal can be created by

splicing a 1 unit line into 3 equal sections and replacing the middle one with two sides of an equilateral

triangle and then repeating the process. The shape generated will now have four thirds of its original

length. Considering each audio track within the piece a ‘1 unit line’, each track was split into 3 equal

sections and the middle section was removed. The following audio track features two clips of equal

length, also equal to each section of the track above, thus replicating the generative pattern of the Koch

snowflake.

9.3. Production Process Having generated numerous clips using the fractal generator, a selection of those clips which had similar

harmonic content was made – clips with chord structures or melodies that would work well from a

aesthetic point of view were chosen. Based on the fractal complexity of each clip, the simplest were set

as the base (1 unit line) of the Koch snowflake. More complex clips were layered above and below the

two bases, as the idea was to create two such structures – these can be seen in the adjacent image.

These subsections were

then spliced according to

the Koch fractal model. A

combination of distortion

filters and modulation was

used to give each

component track a distinct

texture while reverb and

delays were used to create

a sense of spaciousness and

to add interest to the mix.

In addition, as can be seen

in the picture, an extra

segment of the pad sound

18

(the grey block) was added at the end of the piece for aesthetic reasons.

10. Track 3 – Scaled Roughness

10.1. Introduction The name ‘Scaled Roughness’3 (1’09’’) describes the underlying principles on which the piece is build. In

mathematics fractals are usually describe high degrees of roughness and complexity. However, scaling

down the numbers derived from the M-set, one can create slow evolving textures. Although in the

theoretical domain this might seem impossible, this composition stands out as a contradiction to the

theory, promoting the idea by which everything in the universe can be brought down to its essence. It

aims to portray a universal sound, echoing a primordial formula of simple nature – such as the

Mandelbrot set – which has the power to create everything surrounding us.

10.2. Experimental Audio Nature The experimental nature of this track lies in its ‘building blocks’ and structural model. All sounds used

were generated by the iterations of multiple M-sets and the structure is based on the complexity

property of fractals – each iteration increases the resulting fractal’s scale and complexity degree. The

composition emulates this characteristic by introducing new elements after each completion of a

melodic cycle.

10.3. Production Process Similarly to the previous track, the

production process of track 3 starts

by manually choosing each clip by

means of their harmonic nature.

Following this, the clips are then

treated for removal of unwanted

artefacts and they are given a

structure. In this case, the song

builds upon the concept of infinite

complexity – a fundamental

property of fractals. Each individual

element is then texturized using

various tools such as distortion,

filters, modulation and pitch shifting, after which they are all allocated a space in the mix in terms of

frequency spectrum, spatial distribution and depth perception. Finally, automation controls the majority

of parameters from volume to specific functions of the processing units. In addition, a master buss

treatment is applied consisting of equalisation and compression.

19

11. Conclusion In much the same way fractals are thought of as being a mathematical description of nearly all living

things9, it is a personal belief that fractals could act as a key to advances in the realm of sound and music

technology. This project stands as an exemplification of this belief, providing an experimental approach

to dealing with audio by using fractals to generate and control sound.

From a critical standpoint, the workload seems to surpass the allocated 200-hour limit, considering the

complexity with which it has been produced. However, a more simplistic approach would have been

unsuitable for the object of the project. In terms of the audio, although it might appear rudimental, it

shows the feasibility of the concept. Provided that sufficient time and resources are allocated, I believe

the underlying ideas can be developed into a complex project containing audio of similar complexity and

of a high level of quality. Considering the project as a whole, the original software, audio material,

research documentation, press release brochure and all adjacent media, I believe the brief requirements

have been met. Comparing the results to the initial ideas, I it can be implied the project matches my

initial goals. From an academic standpoint, appropriate vocabulary has been used, writing conventions

have been respected and sourcing of external material has been referenced accordingly. In addition,

ethical and copyright issues have been addressed, to some extent.

It can therefore be concluded that, from a personal perspective, the work at hand seems to be of

satisfactory quality and it portrays an adequate level of commitment and engagement with the module.

12. Glossary of Terms

No. Term Description

1 Mandelbrot set Mathematical formula named after mathematician Benoit Mandelbrot.

2 Max MSP Visual programming language. Software.

3 Iteration The repetition of a process or utterance.

4 Hertz (Hz) Unit of frequency in the International System of Units. Named after Heinrich Rudolf Hertz.

5 IBM Acronym. International Business Machines Corporation

6 Complex number Number of the form a + bi where a and b are real numbers and I defines an imaginary component.

7 Cartesian coordinate system

An X/Y system on which each point’s position can be defined by two numerical coordinates.

8 CGI Computer generated images

9 Chant (from French: chanter) Repeated rhythmic singing/speaking of sounds/words14

10 Stochastic ‘Having a random probability distribution or pattern that may be analysed statistically but may not be predicted precisely’ (Oxford Dictionaries, 2013)14

11 Syncretism The fusion of differing systems of belief or disciplines.

20

13. References

No. Reference Source

1 Mandelbrot, B, 1962. Fractals: Form, Chance and Dimension. W.H.Freeman & Co Ltd. Book

2 Cycling '74. 2013. Max is powerful software. [ONLINE] Available at: http://cycling74.com/products/max/. [Accessed 03 December 13].

Website

3 Oxford Dictionaries. 2013. Fractal: Definition of fractal in Oxford dictionary (British & World English). [ONLINE] Available at: http://www.oxforddictionaries.com/definition/english/fractal. [Accessed 14 October 13].

Online Dictionary

4 Rumsey, F, 2009. Sound and Recording. 6th Edition. Focal Press. Book

5 Oxford Dictionaries. 2013. Sound: definition of sound in Oxford dictionary (British & World English). [ONLINE] Available at: http://www.oxforddictionaries.com/definition/english/sound?q=sound. [Accessed 15 October 13].

Online Dictionary

5 University of Arkansas. 2011. Types of Fractals - Math2033. [ONLINE] Available at: http://math2033.uark.edu/wiki/index.php/Types_of_Fractals. [Accessed 07 October 13].

Website

6 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].

University Article

7 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].

University Article

8 IBM. 2013. IBM 100 - Fractal Geometry. [ONLINE] Available at: http://www-03.ibm.com/ibm/history/ibm100/us/en/icons/fractal/. [Accessed 05 December 13].

Website

9 Yale University. 2013. Fractal Geometry. [ONLINE] Available at: http://classes.yale.edu/fractals/. [Accessed 05 December 13].

Online Course

10 Yumiko Kura, 2001. Pilot Analysis of Ecosystems: Coastal Ecosystems (Pilot Analysis of Global Ecosystems). Edition. World Resources Inst.

Book

11 Michael Hogg. 2010. Michael Hoog - Software - FractalNet. [ONLINE] Available at: http://www.michael-hogg.co.uk/fractalnet.php. [Accessed 05 January 14].

Website

12 Kenneth Falconer. 2003. Fractal Geometry: Mathematical Foundations and Applications. 2nd Edition. Wiley.

Book

13 Siu-Lan Tan, 2010. Psychology of Music: From Sound to Significance. 1 Edition. Psychology Press.

Book

21

14 Oxford University Press. 2013. Oxford Dictionaries. [ONLINE] Available at: http://www.oxforddictionaries.com/. [Accessed 08 January 14].

Online Dictionary

15 Jacob, B, L, 1996. Algorithmic composition as a model of creativity. Organised Sound, Volume 1, Issue 3, pp 157-165.

University Article

16 Grout, Donald Jay and Claude V. Palisca (1996), A History of Western Music. 5th ed. W. W. Norton & Company: New York

Book

17 John A. Maurer. 1999. The History of Algorithmic Composition. [ONLINE] Available at: https://ccrma.stanford.edu/~blackrse/algorithm.html. [Accessed 08 January 14].

Website

18 TED Talks. (2010). Benoit Mandelbrot: Fractals and the art of roughness. [Online Video]. 21 July. Available from: http://www.ted.com/talks/benoit_mandelbrot_fractals_the_art_of_roughness.html. [Accessed: 16 October 2013].

Online Video

19 The National Archives. 2013. Copyright, Designs and Patents Act 1988. [ONLINE] Available at: http://www.legislation.gov.uk/ukpga/1988/48/contents. [Accessed 10 December 13].

Online Legal Document

20 Chernin, Ei (1988). "The 'Harvard system': a mystery dispelled", British Medical Journal. October 22, 1988, pp. 1062–1063.

Journal

21 WikiHow. 2013. How to write a press release. [ONLINE] Available at: http://www.wikihow.com/Write-a-Press-Release. [Accessed 10 December 13].

Website

22 John Price. 2002. Mandelbrot Music. [ONLINE] Available at: http://www.morgoth.org/projects/fractalmusic/mandel_mus.html. [Accessed 10 December 13].

Website

22

14. Appendix

1. Press Release Brochure – Front

23

Press Release Brochure – Back

24

2. Research Structure – Sketches

25

26

3. Fractal Sound Generator – Early Versions

27

4. Fractal Sound Generator – Final Version