Fractal Sound

2013 Fractal Sound

Robert Braileanu University of West London

Module Title: Module Code: Level: Course:

Experimental Sound MU60011E 6 BA Music Technology Specialist

Task 2  – Portfolio Submission

Project Document

Fractal Sound Student Full Name Student Number Contact Address Mobile Number Email Address Word Count*

*

Robert Braileanu 21137205 Flat 9, Emanuel Court, Emanuel Avenue, Acton Town, London, UK 0044 7596 489782 [email protected] / [email protected] 5994

The word count excludes all quotes derived from external sources of1any nature including (but not limited to) books, articles, websites, videos, module study guide, university documents and other lecture support documents.

Contents

1.

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

2.

Brief.......................... .......................... ......................... ........................... ......................... ................ 5

3.

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

4.

Research ................................................. ....................... .......................... .......................... ......................... ........................... ........... 6 4.1.

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.

5.

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

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

4.4.1.

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

4.4.2.

Terminology ................................................. ........................ ......................... ........................... ......................... .............. 13

4.4.3.

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

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 ............................................................................................................. Journey ............................................................................................................. 16

9.

8.1.

Introduction ................................................. ....................... .......................... .......................... ......................... ....................... 16

8.2.

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

8.3.

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

Track 2 – Snowflake  – Snowflake Dance ............................................................................................................ 17 9.1.

Introduction ................................................. ....................... .......................... .......................... ......................... ....................... 17

9.2.

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

9.3.

Production Process .......................... ......................... .......................... ......................... .......... 17 2

10.

Track 3 – Scaled  – 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 – Structure – Sketches  Sketches ................................................................................................... 24

3.

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

4.

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

Audio CD content: Order 1 2 3

Content Track 1 – personal  – personal composition Track 2 – personal  – personal composition Track 3 – personal  – personal composition

Data CD content  – located in Research Folder: Order Content 1 Michael Hogg - Slow Deep Mandelbrot Zoom 2 John Cage – Cage – Atlas Eclipticalis 3 Lejaren Hiller- Illiac Suite for String Quartet - Part 1 4 Iannis Xenakis-ST/10=1,080262 5 Fractal Sound Generator 3

Type Video Audio Audio Audio MaxMSP application

1.

Executive Summary

‘Fractal Sound 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 – audio – described  described in detail in section 3; it is part artistic venture and part scientific fact, 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. sound 1) 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 under standing all other historical, technical, logistical and ethical considerations together with the implications they have with reg ards 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  – these can be found in the separate research folder by means of its content list. 1

Furthermore, red coloured superscript indexes (e.g. ) 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 – approach – the  the entire project needs to be accounted for – for – therefore  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 p roject.

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 1 1Mandelbrot  

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. 2

A software application named ‘Fractal Sound Generator’5 was written in 2Max MSP 7  – visual  – 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 – Polyphonic  – where  where the instrument acts as an additive synthesizer  – each  – 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 – Monophonic – a  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. 5

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 1 2 3

Type of Fractal Cantor Set Koch Snowflake Mandelbrot Set

Detailed description of track structure Section 8.2 Section 9.2 Section 10.2

Further information on the type of fractal (research folder) Section 5.1 Section 5.2 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 – research – fractal  fractal geometry & sound - will be discussed, followed by satellite themes towards the end of this sec tion.

4.1.

Project Title

4.1.1. Fractal From an etymological standpoint the term ‘fractal’ derives from the French word ‘fractale’ – ‘broken’ or 1

‘uneven’, as mentioned by Mandelbrot (1977) . 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 3

Oxford dictionary . 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 4

air or another medium . This scientific connotation describes sound as a physical phenomenon, measured in 4Hertz, disregarding any formally acknowledged musical system such as tonality. Secondly, 5

as mentioned in the Oxford Dictionary (2013) , ‘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  – 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.’ 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  – mathematicians Pierre 6Fatou2

6

7

 and Gaston 7Julia 3 , 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 7

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 fractals 5 exist. An in-depth description of these can be found in the research folder in section 5.

4.2.2. Mandelbrot set: characteristi characteristics cs One fundamental aspect of fractals is their property of self-similarity – self-similarity  – each  each individual part is similar to 9

the entire element.  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, n ebulae, 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 10

temperature and weather conditions also play an important role . 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

8

which nature operates. The principles behind waves as a force are fundamentally different from those 9

making up the structure of the coastline on an atomic level . Conversely, self-similarity in fractals can be seen most easily in the Sierpinski Gasket 6 – a  – 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 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.

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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  – 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 12

measured . This has given us an insight into the inner-workings of diverse fields, some of which are included below – below – for  for a more detailed account please see research folder (section 4):

9

Astronomy Galaxies Rings of Saturn

Biology / Chemistry Bacteria Cultures Chemical Reactions Human Anatomy Molecules Plants Population Growth

4.3.

Creative Fractal Art Fractal Music 8CGI Special Effects

Other Clouds Coastlines and Borderlines Data Compression Diffusion Economy Weather

Sound

4.3.1. Brief Introduction Building on the definitions of the word ‘sound’ sound’, as discussed in the project title sub-section, the term can inherit multiple meanings depending on the angle from which it is r egarded. 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 – front  – in  in a sense, sound can be viewed as a means of transmitting data; audio data which is then decoded by a receiver: the human 4

auditory system for example.

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 i s 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 16

(1996)

mentions: ‘The ‘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 15

10

th

century’s ‘canonic’ music –

by which a singer is given a single melody and a set of rules to derive subsequent voices. The rule or set 16

th

16

of rules was called a ‘canon’ (Grout, 1996) 199 6) . In the 20  century, John Cage  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 de vised a way of generating music in an ‘Illiac High-Speed High-Speed Digital Computer’  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

17

Music’. A comprehensive history of algorithmic composition is included in the research folder 6. In addition, a list of songs located on the separate data CD  –   –  part of the research folder – folder  – can  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 m usic. Composer John Cage

Piece Atlas Eclipticalis2

Lejaren Hiller

Illiac Suite for String Quartet - Part 13 ST/10=1,0802624

Iannis Xenakis

Description A score paper was placed on top of an astronomical chart and notes were placed where stars were present  –   –  chance composition Composed by the ‘Illiac  ‘Illiac  High-speed High-speed Digital Computer’ and then transcribed into traditional music notation. 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 15

is a breakdown of some of the models used for such compositions  –   –  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 – systems  – proposes  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 ‘L-system’. ’. Assuming a set of rules: A=BAC ; B=ACB ; C=CBA 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.

11



Evolutionary Systems – Systems  – these  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  – 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 – Systems – a  a combination of the above – above  – which  which is perhaps the most used me thod 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 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

Terminology

fundamental to the object of the project, they provide an account for important aspects in the MaxMSP

Copyright

Fractal Sound

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.

Press Release Brochure

Note that both

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

diagram. Although these consist of a research component, they will be acknowledged separately – 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 2

Max MSP7  is a graphical-interface programming language designed by Cycling 74© . 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

12

2

MacOSX© . 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 ca n include a 21

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

are now considered a standard. These include : 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, se ction 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’ activity ’ and ‘the ‘the branch of knowledge that deals with moral 14

principles’. principles’.

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.

13

Another important area of ethics is health and safety. Although the work was produced entirely on a digital platform – platform  – within  within the computer – computer – these  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 19

as copyright. In the UK, copyright co pyright is defined in the ‘Copyright, Designs and Patents Act 1988’ , 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‘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  – UWL –  – and  and the writer  – myself,  – myself, according to the university’s univ ersity’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 sugge sted 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

14

provided more flexibility in terms of programming. Sketches of how to structure research were then devised – devised – these  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 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 – software – photos  photos of the early versions are present in appendix 3. In addition, an unforeseen issue was discovered – discovered  – the  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 – equation – a  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. 2

The Mandelbrot set equation (z=z +c), when used to generate images, works by assigning complex numbers of the form (a, bi) 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 this value, the 22

point on the graph is considered to ‘go to infinity’ and the iteration process is stopped.   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’, 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 1

Intuitively titled ‘Cantor ‘Cantor’s ’s Journey’ , track 1 (1’09’’)  (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 lie s 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 – fractal  – the  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,  – bass-heavy, mid-rangy etc. – etc.  – and  and panning has been used on some elements.

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9.

Track 2 – Snowflake Dance 9.1.

Introduction

2

Snowflake Dance (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 – 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 – snowflake  – this  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 – structures – these  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

17

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

10.

Track 3 – Scaled Roughness 10.1. Introduction 3

The name ‘Scaled Roughness’ (1’09’’) describes (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 – set – which  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.

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11.

Conclusion

In much the same way fractals are thought of as being a mathematical description of nearly all living 9

things , 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. 1 2 3 4

Term Mandelbrot set Max MSP Iteration Hertz (Hz)

5 6

IBM Complex number

7 8 9

Cartesian coordinate system CGI Chant

10

Stochastic

11

Syncretism

Description Mathematical formula named after mathematician Benoit Mandelbrot. Visual programming language. Software. The repetition of a process or utterance. Unit of frequency in the International System of Units. Named after Heinrich Rudolf Hertz. Acronym. International Business Machines Corporation Number of the form a + bi where a and b are real numbers and I defines an imaginary component. An X/Y system on which each point’s position can be defined by two numerical coordinates. Computer generated images (from French: chanter) Repeated rhythmic singing/speaking of 14 sounds/words ‘Having ‘Having a random probability distribution or pattern that may be analysed 14 statistically but may not be predicted precisely’ (Oxford Dictionaries, 2013) The fusion of differing systems of belief or disciplines.

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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 O xford dictionary (British & World English) . [ONLINE] Available at: http://www.oxforddictionaries.com/definition/english/fractal. http://www.oxforddictionaries.com/definition /english/fractal. [Accessed 14 October 13]. th Rumsey, F, 2009. Sound and Recording . 6 Edition. Focal Press.

Online Dictionary

Oxford Dictionaries. 2013. Sound: definition of sound in Oxfor d dictionary (British & World English).  [ONLINE] Available at: http://www.oxforddictionaries.com/definition/english/sound?q=sou http://www.oxforddictionaries.com/definition /english/sound?q=sound. nd. [Accessed 15 October 13]. University of Arkansas. 2011. Types of Fractals - Math2033. [ONLINE] Available at: http://math2033.uark.edu/wiki/index.php/Types_of_Fractals. http://math2033.uark.edu/wiki/index.php/T ypes_of_Fractals. [Accessed 07 October 13].

Online Dictionary

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. 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. 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://www03.ibm.com/ibm/history/ibm100/us/en/icons/fractal/. 03.ibm.com/ibm/history/ibm100/us/en/icon s/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. Michael Hogg. 2010. Michael Hoog - Software - FractalNet . [ONLINE] Available at: http://www.michael-hogg.co.uk/fractalnet.php. [Accessed 05 January 14].

Book

4 5

5

11

Book

Website

Website

12

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

Book

13

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

Book

20

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. https://ccrma.stanford.edu/~blackrse/algorithm.h tml. [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]. The National Archives. 2013. Copyright, Designs and Patents Act 1988 . [ONLINE] Available at: http://www.legislation.gov.uk/ukpga/1988/48/contents. http://www.legislation.gov.uk/ukpga/1988/48/contents. [Accessed 10 December 13].

Online Video

20

Chernin, Ei (1988). "The 'Harvard system': a mystery dispelled", British Medical Journal. October 22, 1988, pp. 1062 –1063.  –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. http://www.morgoth.org/projects/fractalmusic/mandel_mus.ht ml. [Accessed 10 December 13].

Website

19

21

Online Legal Document

14.

Appendix

1.

Press Release Brochure – Front  – Front

22

Press Release Brochure – Brochure – Back  Back

23

2.

Research Structure – Sketches

24

25

3.

Fractal Sound Generator – Early Versions

26

4.

Fractal Sound Generator – Final Version

27