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1
USING STRANGE ATTRACTORS
TO MODEL SOUND
Submitted to
The University of London
for the Degree of
Doctor of Philosophy
Jonathan Mackenzie
King's College
April 1994
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Abstract
This thesis investigates the possibility of applying nonlinear dynamical systems
theory to the problem of modelling sound with a computer. The particular interest is in
the creative use of sound, where its representation, generation and manipulation are
important issues. A specific application, for example, is the modelling of
environmental sound for film sound-tracks.
Recently, there have been a number of major advances in the field of nonlinear
dynamical systems which include chaos theory and fractal geometry. It is argued that
these provide a rich source of ideas and techniques relevant to the issues of modelling
sound. One such idea is that complex behaviour may be generated from simple
systems. Such behaviour can often replicate a wide range of natural phenomena, or is
of interest in its own right because of its aesthetic appeal. This has been demonstratedoften through computer generated images and so an equivalent is sought in the audio
domain. This work is believed to be the first substantial attempt at this.
The investigation begins with a consideration of fractal and chaotic properties of
sound and with a comparison between established approaches to modelling and the
alternatives suggested by the new theory. Then, the inquiry concentrates on strange
attractors, which are the mathematical objects central to chaos theory, and on two
ways in which they may be used to model sound.
The first of these involves using static fractal functions to represent sound time
series. A technique is developed for synthesising complex abstract sounds from a
small number of parameters. A class of these sounds have the novel property that they
are simultaneously rhythms and timbres. It is believed these have potential for use in
computer music composition. Also considered is the problem of modelling a given
time series with a fractal function. An algorithm for doing this is taken from the
literature, shown to be of limited ability, and then improved. The results indicate that
data compression may be achieved for certain types of sound.
The second approach focuses on modelling the dynamics of a sound via the
embedded reconstruction of an attractor from a time series. Two models are presented,
one deterministic, the other stochastic. It is demonstrated that with the first of these,
certain sounds may be modelled such that their perceived qualities are preserved. For
some other signals, although the sound is not so well preserved, many statistical
aspects are. The second model is shown to provide a solution to the film sound-track
problem.
It is concluded that this investigation shows strange attractors to have considerable
potential as a basis for modelling sound and that there are many areas for continued
research.
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To
Valerie Duff
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Acknowledgements
I would very much like to thank my supervisor, Dr. Mark Sandler, for encouragingme to begin this research project, for finding the funding for it, and for everything he
has done towards making it such a stimulating and enjoyable experience. I am also
indebted to Solid State Logic for providing the sponsorship and to Chris Jenkins for
arranging it. I doubt whether I would have had the opportunity to pursue the project of
my choice otherwise.
I am enormously grateful to my colleagues at King's College who have always
been helpful, supportive and inspiring. These include Maaruf Ali, Julian Bean, Victor
Bocharov, Rob Bowman, Ian Clark, Chris Dunn, Jason Goldberg, Anthony Hare, RodHiorns, Simon Kershaw, Panos Kudumakis, Anthony Macgrath, Phillipa Parmiter,
Allan Paul, Marc Price, Mark Townsend, Mike Waters, and Jie Yu.
For sharing their knowledge and for always being helpful I would like to thank
Dr. Bill Chambers, Prof. Tony Davies, and Dr. Luke Hodgkin. I am also deeply
grateful to Peter King, Mustaq Mohammed and Talat Malik for their generous
technical support.
Finally, special thanks to Val, my family and friends for their support, enthusiasm,
patience and inspiration and for knowing never to ask "when are you going to finish?"
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Contents
Abstract ............................................................................................................... 2
Acknowledgements ....................................................................................................... 4
Contents ............................................................................................................... 5
List of Figures ............................................................................................................... 8
List of Tables ............................................................................................................. 14
List of Sound Examples .............................................................................................. 16
List of Acronyms......................................................................................................... 19
1. Introduction ........................................................................................20
2. Modelling Sound.................................................................................24
2.1. Sound and its Representation................................................................... 24
2.2. Music composition................................................................................... 25
2.3. The Roomtone Problem ........................................................................... 26
2.4. Digital Audio............................................................................................ 27
2.5. The Modelling Framework....................................................................... 28
2.6. Conventional Models ............................................................................... 29
2.6.1. Physical Modelling.................................................................... 292.6.2. Additive and Subtractive Synthesis........................................... 29
2.6.3. Frequency Modulation and Waveshaping................................. 32
2.7. Summary.................................................................................................. 33
3. Chaos Theory and Fractal Geometry ..............................................34
3.1. Introduction.............................................................................................. 34
3.2. The Significance of Chaos ....................................................................... 35
3.3. Dynamical Systems and State Space........................................................ 363.4. Stability .................................................................................................... 37
3.5. Attractors.................................................................................................. 39
3.6. Chaos........................................................................................................ 40
3.7. Visualisation............................................................................................. 42
3.8. Bifurcation................................................................................................ 44
3.9. Statistical Descriptions of Dynamics ....................................................... 47
3.10. Fractal Geometry.................................................................................... 48
3.11. Iterated Function Systems ...................................................................... 53
3.11.1. Contraction Mappings............................................................. 54
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3.11.2. The Random Iteration Algorithm............................................ 56
3.11.3. The Shift Dynamical System................................................... 58
3.11.4. The Collage Theorem.............................................................. 59
3.11.5. The Continuous Dependence of the Attractor on the IFS
Parameters.............................................................................. 60
3.12. Summary................................................................................................ 60
4. Applying Chaos and Fractals to the
Problem of Modelling Sound.............................................................62
4.1. The Reasons for Using Chaos Theory................................................. 62
4.2. Diagnosis of Chaotic Behaviour ......................................................... 64
4.2.1. Chaos and Woodwind Instruments ........................................... 65
4.2.2. Chaos and Gongs....................................................................... 66
4.2.3. Fractal Time Waveforms........................................................... 66
4.2.4. 1/f Noise.................................................................................... 67
4.3. Representing Sound Using Chaos and Fractals................................... 71
4.4. Summary ............................................................................................. 73
5. Fractal Interpolation Functions........................................................75
5.1. Theory ................................................................................................. 75
5.2. The Synthesis Algorithm..................................................................... 785.3. Experiments with the Synthesis Algorithm......................................... 80
5.4. Rhythm/Timbres ................................................................................. 85
5.5. Generating Time-Varying FIF Sounds................................................ 87
5.6. A Genetic Parameter Control Interface............................................... 90
5.6.1. Implementation ......................................................................... 91
5.6.2. Experiments .............................................................................. 95
5.7. Conclusions....................................................................................... 101
6. Modelling Sound with FIFs.............................................................103
6.1. Deriving Interpolation Points from Naturally Occurring Sound ........... Wa
6.2. Mazel's Time Series Models ............................................................. 107
6.3. Comparison with Requantisation ...................................................... 109
6.4. Mazel's Inverse Algorithm for the Self-Affine Model ...................... 114
6.4.1. Initial Results .......................................................................... 118
6.4.2. Error Weighting ...................................................................... 121
6.4.3. Interpolation Point Range Restriction..................................... 124
6.5. Conclusions....................................................................................... 128
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7. Chaotic Predictive Modelling..........................................................131
7.1. Chaotic Time Series .......................................................................... 131
7.2. Embedding ........................................................................................ 133
7.3. The Analysis/Synthesis Model.......................................................... 135
7.4. The Inverse Problem ......................................................................... 138
7.5. A Solution to the Inverse Problem................................................... 140
7.6. Experimental Technique ................................................................... 143
7.7. Experiments with a Lorenz Time Series ........................................... 148
7.8. Experiments with Sound Time Series............................................... 155
7.8.1. Air Noises................................................................................ 155
7.8.2. Gong Sounds ........................................................................... 162
7.8.3. Musical Tones ......................................................................... 164
7.9. Conclusions....................................................................................... 167
7.10. Further Work..................................................................................... 172
7.10.1. Using the Same Model with More Sounds ........................... 172
7.10.2. Optimising the Synthetic Mapping ....................................... 173
7.10.3. Stability Analysis .................................................................. 174
7.10.4. Connections with IFS............................................................ 174
7.10.5. Time Varying Sounds............................................................ 177
8. The Poetry Generation Algorithm..................................................178
8.1. Introduction ....................................................................................... 178
8.2. Description of the Algorithm............................................................ 179
8.3. Analysis of the PGA.......................................................................... 184
8.4. Implementation of the PGA for Sound ............................................. 187
8.5. Results............................................................................................... 191
8.6. Conclusions....................................................................................... 197
9. Summary and Conclusions..............................................................200
Appendix A. Previously Published Work ..........................................209
AES Preprint ................................................................................................. 210
ISCAS '94...................................................................................................... 221
References ............................................................................................225
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List of Figures
Figure 1.1 A synthetic cloud, fern and a Julia set [frac90]. ........................................ 20
Figure Error! Bookmark not defined..1 The analysis-synthesis scheme................. 25
Figure Error! Bookmark not defined..2 The sound modelling framework. ............ 28
Figure Error! Bookmark not defined..3 A schematic diagram for additive synthesis.
..................................................................................................................................... 30
Figure Error! Bookmark not defined..4 Karplus-Strong algorithm. Top, simplified
recursive linear filter and bottom, general delay-line view. ........................................ 31
Figure Error! Bookmark not defined..5 The basic units used within the FM (top)
and waveshaping (bottom) synthesis techniques......................................................... 32
Figure Error! Bookmark not defined..6 State space representation of a dynamical
system.......................................................................................................................... 37
Figure Error! Bookmark not defined..7 Illustration of the three regular attractor
types. ........................................................................................................................... 40
Figure Error! Bookmark not defined..8 Sequence of magnifications of the Lorenz
attractor showing its fractal, self-similar property. ..................................................... 42
Figure Error! Bookmark not defined..9 Two simulations of the Lorenz system for
similar initial conditions showing sensitive dependence on initial conditions. .......... 42
Figure Error! Bookmark not defined..10 Three phase portraits constructed from a
time series of observations of the Lorenz chaotic system. Delay values are: (a) 1, (b)
10, (c) 100. .................................................................................................................. 43
Figure Error! Bookmark not defined..11 The logistic mapping for 0 9. . .......... 45
Figure Error! Bookmark not defined..12 Bifurcation diagram for the logistic
mapping with corresponding time series plots............................................................ 46
Figure Error! Bookmark not defined..13 The exactly self-similar, triadic Koch
curve............................................................................................................................ 49
Figure Error! Bookmark not defined..14 General formula for similarity dimension
derived by inspection of standard Euclidean shapes. ................................................. 50
Figure Error! Bookmark not defined..15 Iterative construction of the triadic Koch
curve............................................................................................................................ 52
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Figure Error! Bookmark not defined..16 Area of closed Koch curve (dark grey) is
within area of circle (light grey) showing that it is finite. ........................................... 52
Figure Error! Bookmark not defined..17 Three affine contraction mappings on
X=R
2
and their single combination, W. ..................................................................... 55
Figure Error! Bookmark not defined..18 The repeated application of a contractive
mapping, W, to some initial set B, tending to the limit set, or attractor, A.................. 55
Figure Error! Bookmark not defined..19 Example of Random Itaration Algorithm
(RIA) in operation. The three images show the results of iterating the Markov process,
(a)~100, (b)~300, (c)~1000 times. .............................................................................. 57
Figure Error! Bookmark not defined..20 Examples of RIA attractors where the
mappings are weighted with different associated probabilities................................... 58
Figure Error! Bookmark not defined..21 Example of an IFS attractor partitioned
into three disjoint subsets according to the effect of the three individual contraction
mappings on the attractor............................................................................................ 59
Figure Error! Bookmark not defined..22 Bifurcation diagram showing a Hopf
bifurcation occurring at the threshold of oscillation in a wind instrument as the
blowing pressure is increased...................................................................................... 65
Figure Error! Bookmark not defined..23 Time series plots and spectral density
forms for 1/f noise compared with white noise and Brown noise............................... 69
Figure Error! Bookmark not defined..24 Power spectral densities of wind noise
(left) and an industrial roomtone (right) showing 1/f characteristic over the audible
range of frequencies. ................................................................................................... 70
Figure Error! Bookmark not defined..25 A demonstration of the property of
continuous dependence of IFS attractors on the parameters that define them. This also
illustrates the power of manipulation capable with chaotic models [frac90].............. 73
Figure Error! Bookmark not defined..26 An example of the effect of three shearmaps, w w w1 2 3, and on the area A and an illustration of one of the vertical scaling
factor, d1...................................................................................................................... 77
Figure Error! Bookmark not defined..27 The initial arbitrary set, B, and a sequence
of five iterations of the deterministic algorithm. ........................................................ 81
Figure Error! Bookmark not defined..28 FIF for equally spaced interpolation points
derived from a single cycle of a sinewave, but where the vertical scaling factors
increase for the mappings from left to right. ............................................................... 82
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Figure Error! Bookmark not defined..29 FIF where x values are spaced according to
a square law. Sequence of magnifications of windows is shown in (a)-(d). ............... 83
Figure Error! Bookmark not defined..30 Same interpolation points as Figure Error!
Bookmark not defined..29, but with 6 iterations showing the cumulative effect oferrors in the algorithm. The bottom plot is a magnification of the middle ~1000 points
of the top plot. ............................................................................................................. 84
Figure Error! Bookmark not defined..31 FIF generated from random x,y and d
values for the interpolation points. .............................................................................. 84
Figure Error! Bookmark not defined..32 (a) (left) FIF generated with random y
values, but evenly spaced x. All d= 0.9. (b) (right) FIF generated with random y, but
square law x values. All d= 0.9. ................................................................................. 85
Figure Error! Bookmark not defined..33 - see Table Error! Bookmark not
defined..1 .................................................................................................................... 86
Figure Error! Bookmark not defined..34 Development of two rhythm/timbres from
rhythmic design, top, through interpolation points, middle, to final waveform, bottom.
..................................................................................................................................... 87
Figure Error! Bookmark not defined..35 Control rule for time-varying FIF sound.
Left, pseudocode where jiji yx , is the ith interpolation point of the jth FIF and dij is
the vertical scaling factor for the ith map of the jth FIF. Right, graphical depiction ofthe effect on the interpolation points through time. .................................................... 88
Figure Error! Bookmark not defined..36 Left, time plot of the whole waveform
generated with the control rule shown in Figure Error! Bookmark not defined..35
with selected magnifications of individual FIFs to show how the sound develops
through time. Right, spectrogram of the first half of the sound showing how it
contains complex, time varying partials similar to those found in naturally occurring
musical sounds. ........................................................................................................... 89
Figure Error! Bookmark not defined..37 Pictorial representation of the FIF
parameter control used to generate the second example of a time-varying FIF sound.90
Figure Error! Bookmark not defined..38 Schematic diagram of the model for
biological evolution..................................................................................................... 92
Figure Error! Bookmark not defined..39 Schematic diagram of hardware used for
GEN program. ............................................................................................................. 92
Figure Error! Bookmark not defined..40 Example of mutation, (a), and
recombination, (b), of FIF parameters......................................................................... 94
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Figure Error! Bookmark not defined..41 A single screen-shot from the program
GEN............................................................................................................................. 96
Figure Error! Bookmark not defined..42 A sequence of populations generated with
the program GEN. In this case, the FIFs are produced from 6 interpolation points. Atthe start (waveform A - top left) all interpolation points and vertical scaling factors
are zeroed. At each stage, 7 mutations are produced and then a single survivor is
chosen by the operator (starred waveform), which reappears as waveform A in the
next generation............................................................................................................ 98
Figure Error! Bookmark not defined..43 Starting point (top left) and sequence of
starred waveforms from Figure Error! Bookmark not defined..42 shown in more
detail............................................................................................................................ 99
Figure Error! Bookmark not defined..44 Mutated varients of an FIF that is defined
by a relatively large number of parameters. It can be seen (and heard) that when this is
the case, low factor mutations are found not to be distinctive from one another...... 100
Figure Error! Bookmark not defined..45 Results of an experiment to extract
interpolation points by decimating a wind sound waveform and then constructing an
FIF with them............................................................................................................ 103
Figure Error! Bookmark not defined..46 Original wind sound waveform (top),
interpolation of peak points (bottom left), and reconstructed waveform (bottom right).
................................................................................................................................... 105
Figure Error! Bookmark not defined..47 Section of original wind sound (left) and
part of the composite FIF (right) constructed using groups of peak points............... 106
Figure Error! Bookmark not defined..48 Mapping of amplitudes in requantisation
process....................................................................................................................... 110
Figure Error! Bookmark not defined..49 Degradation against compression
performance of Mazel's inverse algorithms for a variety of data and model types
compared with the theoretically expected performance of requantisation................ 113
Figure Error! Bookmark not defined..50 First trial pair of interpolation points on
the original time series graph. ................................................................................... 115
Figure Error! Bookmark not defined..51 Mapping of whole time series to in
between the first pair of interpolation points. ........................................................... 115
Figure Error! Bookmark not defined..52 Maximum vertical extent of part of the
original time series between a pair of consecutive interpolation points and the
maximum vertical extent of the mapped original time series. The vertical scalingfactor is calculated so as to make these two extents equal........................................ 117
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Figure Error! Bookmark not defined..Error! Bookmark not defined. Error
weighting function parameterised by ..................................................................... 122
Figure Error! Bookmark not defined..53 Graph of the results shown in Table
Error! Bookmark not defined..9. ........................................................................... 123
Figure Error! Bookmark not defined..54 Comparison of performance between
requantisation and error-weighted version of Mazel's algorithm. The original is 1000
samples of wind noise which is processed as 10x100 sample sections. ................... 124
Figure Error! Bookmark not defined..12 Comparison of performance of the window
restricted inverse algorithm with that of requantisation. The original time series is
wind noise and processed as 10x100 sample sections. ............................................. 126
Figure Error! Bookmark not defined..13 Waveform plot of original wind noise
(left) and compressed FIF version (right) using the modified inverse algorithm. The
compression ratio in this case is 8.1:1, and the SNR is 22.6dB................................ 127
Figure Error! Bookmark not defined..14 Column chart showing the performance
figures given in Table Error! Bookmark not defined..11 for a variety of different
original sound time series. ........................................................................................ 128
Figure Error! Bookmark not defined..55 The proposed analysis/synthesis model
based upon the embedded attractor and measure representation of a sound time series.
................................................................................................................................... 136
Figure Error! Bookmark not defined..56 Left, an example recursive partition for
m=2 and right, the associated search tree.................................................................. 142
Figure Error! Bookmark not defined..57 Lorenz input, N=10,000, Q=256 and a
variety of embedding dimensions, m.................................................................. 149
Figure Error! Bookmark not defined..58 Lorenz input, N=10,000, m=7, and a
variety of number of domains, Q. ............................................................................. 150
Figure Error! Bookmark not defined..59 Lorenz input, Q=64, m=7 and a variety oforiginal time series lengths, N ....................................................................... 151
Figure Error! Bookmark not defined..60 Time series plots from original Lorenz
system (left) and the synthetic one shown as phase portrait Error! Bookmark not
defined..58(f) (right)................................................................................................. 152
Figure Error! Bookmark not defined..61 Estimates of amplitude probability
distributions for original, left, and synthetic, right, time series shown in Figure Error!
Bookmark not defined..60. ..................................................................................... 153
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Figure Error! Bookmark not defined..62 Time series plots and phase portraits for:
left, original fan rumble sound and right, best synthetic output, rc127..................... 157
Figure Error! Bookmark not defined..63 Time series plots and phase portraits for
some more outputs from the sound model using the fan rumble as input. Note thatonly about a third the length of the output appears in the phase portraits as it does in
the time series plots for the sake of clarity................................................................ 159
Figure Error! Bookmark not defined..64 Time series plots (first fifth of top plot
shown magnified as second plot), power spectra and phase portraits for original wind
noise, left, and synthetic version, right...................................................................... 161
Figure Error! Bookmark not defined..65 Time series plots, phase portraits and
amplitude histograms for original, left, and synthetic, right, lightly-struck gong sound.
Both amplitude histograms were computed with 10,000 samples and 100 bins....... 163
Figure Error! Bookmark not defined..66 Time series plots, phase portraits and
amplitude histograms for original, left, and synthetic, right, hard-strike gong sound.
Both amplitude histograms were computed with 10,000 samples and 100 bins....... 164
Figure Error! Bookmark not defined..67 Time series plots, power spectra and phase
portraits for original, left and synthetic, right, tuba tones. ........................................ 166
Figure Error! Bookmark not defined..68 Time series and phase portraits for
original, left, and synthetic, right, saxaphone tones. ................................................. 166
Figure Error! Bookmark not defined..69 Relative one-step prediction errors for the
best results found for each of the time series. .......................................................... 168
Figure Error! Bookmark not defined..70 Autocorrelation functions for original, left,
and synthetic, right, gently struck gong sound. The upper plot shows the function upto
8,000 delays, and the lower upto 100 delays. Both were calculated by convolving
10,000 samples of the time series with itself for different delays............................. 171
Figure Error! Bookmark not defined..71 The top line shows the interdependence of
the components of the RIA version of an IFS. The bottom line shows a suggested path
to obtain a solution to the inverse problem. .............................................................. 179
Figure Error! Bookmark not defined..72. Input to the algorithm treated as a circular
sequence. ................................................................................................................... 181
Figure Error! Bookmark not defined..73 Part of the state space, X, corresponding to
an example PGA showing some of the possible states and their associated transitions.
................................................................................................................................... 185
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Figure Error! Bookmark not defined..74 Crossfade envelopes applied to beginning
and end of original time series which are then added together to form modified time
series. This is then stored in the circular register so that there is no amplitude
discontinuity between its end and its beginning........................................................ 191
Figure Error! Bookmark not defined..75 Time domain plots of the original
roomtone showing 300 (left) and 3000 (right) samples. ........................................... 194
Figure Error! Bookmark not defined..76 Time domain plots of output time series
when (a) I=300, L=1, (b) I=3000, L=3, and (c) I=300, L=4...................................... 194
Figure Error! Bookmark not defined..77 Comparison between original (left) and
synthetic time series (right) showing: (a)&(b) time domain plots, (c)&(d) power
spectral densities calculated by averaging eleven 4096 point FFTs, and (e)&(f)
amplitude histograms calculated from 30,000 samples. ........................................... 195
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List of Tables
Table Error! Bookmark not defined..2 A summary of possible sound types. After
[ross82]........................................................................................................................ 25
Table Error! Bookmark not defined..3 (left) example set of interpolation points and
vertical scaling factors that define the FIF shown in Figure Error! Bookmark not
defined..27. ................................................................................................................ 80
Table Error! Bookmark not defined..4 (right) vertical scaling factors used in
generating Figure Error! Bookmark not defined..28............................................... 80
Table Error! Bookmark not defined..5 and Figure Error! Bookmark not
defined..78 Input data and waveform plot of the resulting FIF that is a rhythm/timbre...................................................................................................................................... 86
Table Error! Bookmark not defined..6 Summary of the results obtained by Mazel
for his four FIF based models/inverse algorithms..................................................... 109
Table Error! Bookmark not defined..7 Summary of results for reimplementation of
Mazel's algorithm for the self-affine model. Each original time series of length Ttot has
been processed as m=10 sections of length T=100. .................................................. 119
Table Error! Bookmark not defined..8 Running algorithm with wind noise as
original time series for a variety of section lengths T. .............................................. 120
Table Error! Bookmark not defined..9 Results of error weighting the inverse
algorithm for a range of weighting function gradients, . The original time series is
wind noise and is processed as 10x 100 sample sections.......................................... 122
Table Error! Bookmark not defined..10 Performance of modified FIF inverse
algorithm with a specified window restricting the range of the trial interpolation point.
................................................................................................................................... 126
Table Error! Bookmark not defined..11 Table of performance figures for window
restricted inverse algorithm using a variety of sound time series. Each original time
series is processed as 10x100 sample sections and the restriction window is set at l=15
and r=25 samples...................................................................................................... 127
Table Error! Bookmark not defined..12 Summary of results using fan rumble sound
as input to the dynamic model............................................................................... 156
Table Error! Bookmark not defined..13 Summary of analysis parameters for best
results using gong sounds.......................................................................................... 162
Table Error! Bookmark not defined..14 Analysis details for the musical tones. .. 165
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Table Error! Bookmark not defined..15 Example of the PGA acting on a short
paragraph of text for a variety of values of the seed length parameter. .................... 180
Table Error! Bookmark not defined..16 Example sequence of iterations of the PGA.
................................................................................................................................... 182
Table Error! Bookmark not defined..17 Simple example showing how the
preprocessing reorders the original input sequence. ................................................. 189
Table Error! Bookmark not defined..18 Summary of results obtained with PGA and
industrial roomtone as original time series. (Numbers in brackets are experiment
identification.) ........................................................................................................... 192
Table Error! Bookmark not defined..19 Summary of results for PGA used with
other roomtones having different qualities................................................................ 196
Table Error! Bookmark not defined..20 Summary of results obtained with PGA and
a variety of other background sounds........................................................................ 197
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List of Sound Examples
All sounds are created by playing 16-bit sound files at 48kHz or 44.1kHz sample-
rate unless otherwise stated. The sample-rate is indicated by the suffix of the sound
file name given in brackets after each description. For example, '.441' indicates an
original sound recording made with a sample-rate of 44.1kHz or a synthetic version
played-back at that rate. The suffix '.mbi' is used to indicates an abstract waveform
with no intrinsic sample-rate. These files are played at 48kHz.
Playback is via a Digital Audio Labs 'CardD Plus' system connected to an IBM
compatible P.C. This allows an AES/EBU compatible, serial digital audio data-stream
to be generated from the sound file. This is then passed to a Sony TCD-D10 digital
audio tape (DAT) recorder which is used as the digital-to-analogue device.
Chapter 5
1. FIF derived from 17 equally x-spaced interpolation points taken from a single
sinewave cycle, 5 iterations. (sine_5.mbi) .................................................................. 81
2. Same as Sound 1, but with increasing vertical scaling factors. (sine3.mbi) ......... 81
3. FIF derived from 129, square-law x-spaced interpolation points taken from a single
sinewave cycle, 3 iterations. (sine9_3.mbi) ................................................................ 82
4. Same waveform used in Sound 3, but played as a sequence where the speed of
playback is halved at each stage. (sine9_3.mbi) ......................................................... 83
5. FIF derived from randomised interpolation points and vertical scaling factors.
(rand4.mbi).................................................................................................................. 84
6. FIF derived from interpolation points whose y-values are randomised, but that are
regularly x-spaced. (rand2.mbi)................................................................................... 84
7. Same as Sound 6, but with square-law x-spacing. (rand3.mbi) .............................. 84
8. Original FIF rhythm/timbre. (fif1.mbi) ................................................................... 85
9. Same waveform used in Sound 8, but played as a sequence where the speed of
playback is halved at each stage. (fif1.mbi)............................................................... 85
10. First designed FIF rhythm/timbre. (rhy2_1_x.mbi) .............................................. 86
11. Second designed FIF rhythm/timbre. (rhy4_4.mbi).............................................. 86
12. Percussive sounding, time-varying FIF. (tv1.mbi)................................................ 89
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13. Second example of a time-varying FIF. (tv2.mbi) ................................................ 89
14. Audio output from the program GEN which accompanies Figure Error!
Bookmark not defined..79. Each of the 8 sounds is a member of a single evolved
population of FIFs. Played at 48kHz........................................................................... 95
15. Sounds to accompany Figure 5.17. Each of the 8 sounds is the chosen survivor of
a sequence of generations produced with GEN. Played at 48kHz. ............................. 96
16. Concatenated sequence of ~15 short, evolved FIFs. (mbi1log.mbi)..................... 97
17. Concatenated sequence of 4 related FIF rhythm/timbres. (goodone.mbi) ............ 97
18. Audio output from GEN which accompanies Figure 5.19. Each sound is the
member of one generation evolved from FIF parameters similar to those used in
Sound 3. It can be heard how there is little to distinguish the mutated offspring. Playedat 48kHz. ................................................................................................................... 100
Chapter 6
19. FIF whose interpolation points are the peak-points of a wind noise waveform.
(wp1.mbi).................................................................................................................. 105
20. As Sound 19, but using groups of peak-points. (wp2.mbi)................................. 106
Chapter 7
All the examples from Chapter 7 are presented as pairs of the original sound and
the synthetic version produced with the chaotic predictive model.
21. Original fan rumble air-noise. (fan_rmb5.48)..................................................... 157
22. Synthetic version of above. (rc127b.48) ............................................................. 157
23. Original wind noise. (wind6.48) ......................................................................... 160
24. Synthetic version of above. (rc162b.48) ............................................................. 160
25. Original lightly-struck gong sound. (gong4.48).................................................. 162
26. Synthetic version of above. (rc115b.48) ............................................................. 162
27. Original hard-strike gong sound. (gong6.48)...................................................... 162
28. Synthetic version of above. (rc148b.48) ............................................................. 162
29. Original tuba tone. (tuba2.48)............................................................................. 165
30. Synthetic version of above. (rc175x.48) ............................................................. 165
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31. Original saxophone tone. (sax9.48) .................................................................... 165
32. Synthetic version of above. (rc108x.48) ............................................................. 165
Chapter 8
33. Original industrial roomtone. (rmt4.441)............................................................ 192
34. Synthetic version of Sound 33 produced by PGA where I=300 and L=1.
(rmt4_111.441).......................................................................................................... 192
35. As 34, but I=3,000 and L=2. (rmt4_212.441) ..................................................... 192
36. As 34, but I=3,000 and L=3. (rmt4_213.441) ..................................................... 192
37. As 34, but I=3,000 and L=4. (rmt4_214.441) ..................................................... 192
38. As 34, but I=30,000 and L=2. (rmt4_312.441) ................................................... 192
39. As 34, but I=3,000 and L=5. (rmt4_315.441) ..................................................... 192
40. Original laboratory roomtone, played at 48kHz. (lab_rmt.48)............................ 196
41. Synthetic version of above produced with PGA, played at 48kHz.
(lab_313.48) .............................................................................................................. 196
42. Original 'rumble-like' industrial roomtone. (rmt11.441)..................................... 196
43. Synthetic version of above produced with PGA. (rt11_314.441) ....................... 196
44. Original industrial roomtone with drone. (rmt15.441)........................................ 196
45. Synthetic version of above produced with PGA. (rt15_314.441) ....................... 196
46. Original river sound. (river.48)........................................................................... 197
47. Synthetic version of above produced with PGA. (rive_313.48) ......................... 197
48. Original wind noise. (wind1.48) ......................................................................... 197
49. Synthetic version of above produced with PGA. (wind_313.48)........................ 197
50. Original audience applause sound. (applause.48) ............................................... 197
51. Synthetic version of above produced with PGA. (appl_312.48)......................... 197
52. Original rainforest ambience. (ecuador.48)......................................................... 197
53. Synthetic version of above produced with PGA. (ecua_314.48) ........................ 197
54. Original speech extract. (speech.48) ................................................................... 199
55. Synthetic version of above produced with PGA. (sp_pga.48) ............................ 199
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Summary of Acronyms
DAT Digital Audio Tape
DSP Digital Signal Processor
FFT Fast Fourier Transform
FIF Fractal Interpolation Function
FM Frequency Modulation
IFS Iterated Function System
jpdf joint probability density function
LPC Linear Predictive Coding
pdf probability density function
PGA Poetry Generation Algorithm
RIA Random Iteration Algorithm
rms root mean square
SDS Shift Dynamical System
SNR signal to noise ratio
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Chapter 1
Introduction
This thesis is about applying science and technology to the arts. In particular, the
science is that of chaos theory, which includes fractal geometry, the technology is the
computer, and the medium of interest, sound. Fractals and chaos are recent
developments which are revolutionising our understanding of the complex and
irregular nature of the world. Chaos theory is concerned specifically with the
behaviour of nonlinear dynamical systems. It is about the realisation that simple,
deterministic systems can exhibit complex, unpredictable behaviour. Fractal geometry
deals with a class of forms that are not accounted for by conventional, Euclidean
geometry. The two overlap with the concept of a strange attractor which both
embodies the nature of chaotic systems and is itself a fractal object. The relevance and
use of chaos and fractals is currently spreading through a diverse range of subjects. A
number of developing areas of interest are characterised by the overlap of both
scientific and artistic concerns. In particular, two subjects have emerged that have
considerable popularity: visual art and music. Both combine fractal and chaotic
models with computer technology to provide powerful tools for artistic
experimentation. The aim of this work is to seek a parallel to this, but involving
sound.
Consider the images shown in Figure Error! Bookmark not defined..1. These
are examples of the power of fractals and chaos. Using only very simple models it is
possible to create images that can be either complex abstract forms or realistic replicas
of natural objects. The question is, can the same be found in the acoustic domain? For
example, could a complex, naturally occurring sound be represented with a simple
model? Does there exist an aural equivalent of the Julia set?
Figure 1.1 A synthetic cloud, fern and a Julia set [frac90].
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Interest in fractal music has concentrated on the arrangement of sequences of notes
with reference to fractal or chaotic models. Although the end product is audio, the
actual sounds used are conventional natural or synthetic ones (for example see
[pres88, gogi91 and jone90] ). The time scale on which fractals and chaos are being
used for music, then, is different to that of the sounds themselves. Musical
fluctuations range from thousandths of Hertz up to several Hertz. Audio fluctuations,
however, range from hundreds of Hertz to tens of thousands. An important discovery
that supports the use of fractals and chaos for music composition is that, when
analysed, music from a wide range of cultures and historical periods is found to have
fractal properties [voss78, hsu90 and hsu91]. It has been suggested, however, by
Benoit Mandelbrot, the inventor of the term fractal, that such properties should not
extend beyond the musical structure to the sounds themselves as these are governed
by different mechanisms [mand83].
But why should this necessarily be the case? What about the complex and
irregular side of musical sound, for example the hiss of a breathy saxophone, or the
crash of a cymbal? Also, what about non-musical sound? All around us there are
complex and irregular sounds generated by our environments: a burbling brook,
splashing water, the roaring of the wind, the rumble of thunder and the variety of
screeching, scraping, buzzing and humming noises made by machinery. Is it, perhaps,
that these sounds represent an aural equivalent to the shapes found in nature that have
been neglected by Euclidean geometry and then rediscovered as fractals? Criticising
the conventional Fourier approach to modelling musical sound, the contemporary
composer Iannis Xenakis has said:
"It is as though we wanted to express a sinuous mountain silhouette by portions of
circles." [xena71]
Compare this to what Mandelbrot says in the introduction to his 'The Fractal
Geometry of Nature':
"Clouds are not spheres, mountains are not cones, coastlines are not circles, andbark is not smooth, nor does lightning travel in straight lines." [mand83]
This thesis, then, presents an exploratory study into the idea of using chaos theory
and fractal geometry to model sound. Apart from the interest in this as a research
topic, the work is practically motivated with the aim of developing computerised tools
that would allow control over complex and irregular sounds for creative uses. The
potential applications for such tools include computer music composition and the
generation of sound effects for film and television.
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The overall design of this thesis is as follows: Chapters 2, 3 and 4 present the
background to this thesis and develop specific problems on which to work. Then
Chapters 5, 6, 7 and 8 present original contributions towards the solution of these
problems. Each of these chapters contains its own conclusions and a discussion of
further work where relevant. Chapter 9 contains a summary of the thesis and some
general conclusions. An appendix is included which contains copies of previously
published papers on this work and the thesis ends with a full list of references.
Throughout the thesis, references are made to sound examples which are presented on
an accompanying cassette tape. The sound examples are listed, along with all figures
and tables, after the contents pages. Also included is a summary of acronyms for
reference. The content of each chapter is previewed below.
Chapter 2 defines what is meant by a sound model. It considers what sound is, and
the general concept of its representation via the procedures of analysis and synthesis.
Some specific applications are described, including 'the roomtone problem', which
allows a functional description of a model to be developed. Brief reviews of some
well known models fitting this description are given including some of their
advantages and limitations.
Chapter 3 presents a review of chaos theory and fractal geometry. This includes an
outline of some main features and their significance. The emphasis is on
understanding how complex behaviour arises from simple systems, the importance of
strange attractors, and the introduction of Iterated Function Systems (IFS), which
provide a useful practical framework for manipulating strange attractors.
In Chapter 4 the issue of applying the ideas of chaos theory and fractal geometry
to the problem of modelling sound is considered. It is argued that both appear to have
potential use, but that two main questions are raised. Firstly, on a diagnostic level: are
sounds chaotic or fractal? Positive evidence is collected both from the literature and
from original work. The second question is then a practical one: in what way can
sound be represented with chaos or fractals? The conclusion is to concentrate on using
strange attractors in two different ways with an emphasis on involving IFS.
Chapter 5 is concerned with using IFS strange attractors to produce synthetic
sound by generating waveforms with Fractal Interpolation Functions (FIF), a class of
IFS. A basic technique is designed that is then advanced in several ways. The most
important result is the discovery of a new class of sounds that are simultaneously
rhythms and timbres. With these techniques complex sounds may be generated with
small amounts of data and are demonstrated to have potential for musical applications.
Chapter 6 keeps the theme of FIF, but considers the analysis and synthesis of agiven sound. An algorithm is taken from the literature which appears suitable for this
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task. It is shown, however, to be inadequate, a reason found, and the algorithm
improved. Results indicate that some degree of data compression may be obtained for
certain sounds.
Chapter 7 is concerned with the problem of modelling the dynamics of a sound viaa strange attractor. The assumption is made that a chaotic system is responsible for a
digital audio time series. The system may then be reconstructed from the time series
with a technique known as embedding. Because of the properties preserved by
embedding, the construction of another chaotic system that approximates the
embedded one should produce a time series that is statistically similar to the original.
An approach to this problem is considered which combines techniques taken from
work on the nonlinear prediction of time series with an original method inspired by
the Shift Dynamical System (SDS) version of an IFS. An analysis/synthesis algorithm
is developed and a number of experiments performed. The algorithm is shown to be
capable of modelling known chaotic systems from their time series. Also, despite
some difficulties, the algorithm is capable of successfully reproducing some natural
sound so that it is perceptually similar to the original.
Chapter 8 is also concerned with the problem of modelling the dynamics of a
sound in an embedded state space setting. The model considered, however, is the
Random Iteration Algorithm (RIA) version of an IFS where a Markov chain is used to
model the embedded invariant measure. In the course of this investigation, an
algorithm is developed which solves the roomtone problem for certain ambient
sounds.
Chapter 9 presents a summary of the thesis and some general conclusions on the
subjects of inverse problems, algorithmic complexity and developments of the work.
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Chapter 2
Modelling Sound
This chapter develops a working definition of a sound model. It will consider what
sound is and its representation within an analysis/synthesis framework. Some possible
applications of such a model will be discussed including a specific one concerning
film sound-track editing, known as 'the roomtone problem'. This leads to a set of
useful functions that define the model. Also, a brief review of established modelling
techniques, their advantages and limitations is included.
2.1. Sound and its Representation
What is sound? It can be defined as either an auditory sensation perceived by the
mind, or as the physical disturbance that gives rise to such a sensation [ross82]. A
practical model for sound has, in some way, to represent it in an appropriate form.
Starting from this definition of sound there are a number of levels on which this
representation could take place. Consider these as ordered from the outside in: on the
outside level, a model could be made of the complete physical system that isresponsible for the sound. This might include the source of the disturbance and its
reverberant environment. A list of possible disturbances is shown in Table 2.1.
Secondly, this model may be simplified to include only that which is relevant to
describing the pressure fluctuations in the air at a single point; for example at the ear
or a microphone. Next, a model could be made for the time waveform created by
recording those pressure fluctuations at a single point without any, or little,
consideration of the physical system that created it. The waveform is then an abstract
pattern which is to be modelled. Finally, the model may account for just the
perception of the sound, so that an accurate representation of the time waveform is not
necessary, but a representation is needed that just contains the relevant information to
capture the essential characteristics of the sound.
At whatever level the representation is made, a useful framework within which to
test its validity is provided by the analysis-synthesis scheme shown in Figure 2.1
[riss82]. The important feature is that a listener judges how good the representation is
at capturing the characteristics of the sound. In order to refine this modelling
framework, it will be useful to consider some of the applications where sound models
are, or might be used.
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synthesisanalysis
representation sousound
listener
Figure 2.1 The analysis-synthesis scheme.
Physical Disturbance Example
vibrating solid bodies metal bar, speaker cone, violin body
vibrating air column pipe organ, woodwind instrument
flow noise in fluids due to
turbulence
jet engines, air leaking under
pressure, wind noise
interaction of
moving solid with fluid
or
moving fluid with solid
rotating propeller or fan blade
air flow in duct or through grill,
water in pipe, waves breaking on sea
shore
rapid changes in temperature or
pressure
thunder and other sounds caused by
electrical discharge, chemical explosion
shock waves caused by motion or
flow at supersonic speed
supersonic boom caused by jet
aircraft
Table 2.1 A summary of possible sound types. After [ross82].
2.2. Music composition.
An important aspect of music composition is, obviously, the control over the type
and quality of sound used. This century has seen the use of electronic and, more
recently, computer based techniques grow from the experimental to the mainstream.
Typically, such techniques involve obtaining musical sound and processing it to
modify it, or generating it entirely synthetically. Of importance are the degrees of
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musical usefulness and flexibility that are offered by a technique coupled with the
ease and efficiency with which it can be executed.
Imagine the example of a drum synthesiser. What might be its attractive features
for a composer? It might be able to take the recording of an original drum sound andreproduce it so as to retain its relevant characteristics, discarding any perceptually
unimportant information in the process. It might then allow the sound to be modified
in a way related to its physical attributes, for example, to be able to change the sound
as if it came from a larger version of the same drum, or one that had a tighter skin and
has been struck with a different beater. Furthermore, the synthesiser might allow drum
sounds to be generated that it would not be possible to create with real instruments.
A more detailed discussion of sound modelling techniques used for music
composition is given in the forthcoming sections 2.5 - 2.8.
2.3. The Roomtone Problem
Another area of creative sound use is film sound-track editing. This, as with music
composition, generally involves manipulating sound in a number of ways except that
often the sound is non-musical. A good example of this is the use of sound effects.
Here, the desire is to add certain sounds to a film to enhance or complement what is
taking place visually. Traditionally, this is done by simulating the appropriate soundswith a variety of acoustic devices or making use of large reference libraries of
recordings. It is, however, often problematic and time consuming to get exactly the
desired sound. A specific example of this is the roomtone problem which was posed
by the company that sponsored this research.
The roomtone problem arises during post-production editing of a film sound-track.
Often, due to problems that have occurred with the location filming, it is necessary to
replace sections of the original sound-track at a later date. For example, this can
involve having them dubbed by the original actors in an acoustically dry sound studio.The problem occurs when the new pieces of sound track are inserted into the original
as there is often a noticeable lack of background sound. As these background sounds
tend to be characteristics of internal locations, they are known as roomtones. One
traditional solution to this problem involves referring to libraries of roomtone
recordings to find a matching sound. It is often difficult, however, to find exactly the
right sound and the process can also be time consuming. Another solution is to make
use of small snippets of the roomtone found in places on the original recording, for
example between lines of dialogue. These may be spliced together, or looped to form
as long a piece as is necessary. As with the other solution, this can be an intricate and
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time consuming process, the results often not good enough because the splices and
loops are audible.
An ideal solution to this problem, then, would be some form of sound model that
is able to capture certain essential characteristics of the roomtone from a smalloriginal sample and then produce greater quantities of a synthetic version.
Both the examples of the drum synthesiser and the roomtone problem illustrate a
certain type of creative application for sound models. Generally, the need is for the
model to capture essential characteristics of the sound; for it to allow useful
manipulation of the sound; and/or for it to generate synthetic sound. An important
aspect of such models is that the representation involves a set of parameters. These are
the variables of the model that, with the particular representation, form all the
information extracted by the analysis, and/or used by the synthesis. So for the drum
model, the parameters might include the physical attributes of the drum, or for the
roomtone model, the extract of original sound.
2.4. Digital Audio
Being more specific about the sound model, it is assumed that it will operate
within a computer and therefore rely on digital audio as an intermediate
representation. This brings the enormous advantage that the modelling process may be
implemented as a computer program, which makes it highly flexible, and convenient
to develop [math82]. Digital audio satisfies the definition of a representation for
sound that has been given already. It is a discrete time, discrete amplitude model for
the time waveform generated from recording sound at a single point in space. It
preserves perceived information in the form of all frequencies contained within the
sound up to one half of the sampling frequency. This is guaranteed by Nyquist's
sampling theorem [nyqu28]. It is, however, unwieldy, in that a large amount of data isrequired for good quality representation. For example, the industry standard of a
48kHz sampling rate and 16 bits per sample [aes85] means that approximately one
million bytes of data are required to represent ten seconds of sound; this data not
being in a form that is obviously related to the perceived characteristics of the sound.
This is therefore another reason for further representation of the sound waveform: so
as to reduce the amount of parameter data. Assuming the use of digital audio and
therefore computers also means that the model has to perform its desired functions
within the constraints imposed by the processing ability of the computing devices
used.
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2.5. The Modelling Framework
Following the discussion developed within this chapter, then, a working functional
description of a sound model is summarised as follows. A sound model is of use if:
1) it can represent the essential perceived characteristics of the sound;
2) there is less parameter data than there is original sound data;
3) the parameter data is of a form such that its manipulation has a useful or
interesting effect on the sound;
4) it can generate new sounds, or replicas of naturally occurring ones, from a little
data and/or a simple model.
Although much is known for particular situations, it is very difficult to say, in
general, what physical attributes of the sound it is sufficient to preserve in the
representation so as to satisfy 1). This is still an open question in psychoacoustics [see
deut82]. Point 2) on its own may also be described as data compression. Although this
tends to be an attractive feature of a model in terms of reducing the amount of storage
required, it is considered here also in combination with 3) in the sense that the
parameters are more manageable if there are less of them. The synthesis capability of
the model, 4), may be derived from the analysis model and used by supplying it
modified, or artificial parameters, or it may exist on its own as a synthesis-only
technique.
It has also been assumed that the model will operate on a digital audio
representation so that it can operate within a computer. A more detailed diagram of
the sound modelling framework, then, is shown in Figure 2.2.
representation
parameters
modify etc.
operator
microphone loudsp
analysis synthesis
reconsandamplif
13741587
1745
1956
....
....
....
....
digital audio
time waveform
13741587
1745
1956
....
....
....
....
digital audi
sample and quantise
sound
Figure 2.2 The sound modelling framework.
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Now that a general modelling framework has been defined, the next section gives
some brief reviews of particular, well known representations that fit this description.
These serve to illustrate the points made so far, and act as a reference when the issue
of modelling sound using chaos theory is discussed in Chapter 4.
2.6. Conventional Models
2.6.1. Physical Modelling
Physical modelling is a synthesis-only technique that is used to generate musical
sound from a computer representation of the physical system responsible for that
sound. The system can include the action of the musician on the instrument, and the
instrument itself. The system is usually partitioned according to physical, functional orcomputational criteria which in fact often coincide. So for example, a violin may be
divided into the bow, strings, bridge and soundboard as separate coupled physical
systems; or into an excitation part (bow on string) that feeds a resonator (string,
bridge, sound board); or into a nonlinear oscillator (exciter) that is input to a linear
filter (resonator).
The appeal of physical modelling is that sounds may be created from a purely
theoretical basis and that the models and parameters are in a form that can be
intuitively understood by the user. The main disadvantage is that despite much basictheory being known about the physics of musical sound generation, often the models
resulting from a direct implementation of the equations produce sounds that are flat
and lifeless [riss82]. This suggests that there are therefore many subtle aspects of
sound production that are important to the highly sensitive perceptual mechanisms of
the ear and brain that are not included in the basic theory. This is an area of current
research [cmj92].
2.6.2. Additive and Subtractive Synthesis
Additive and subtractive synthesis are terms used to cover a range of analysis-
synthesis techniques used for modelling musical instrument and voice sounds and
which rely on spectral representations of the time waveform. As mentioned above, a
number of such sounds can be presumed to be the product of some form of excitation
feeding a resonator. A time-varying spectral analysis of the sound can reveal these
components in a form that then suggests suitable further representations. For example,
such an analysis shows a bowed violin sound to consist of an approximately periodic
excitation, revealed as a set of harmonically related spectral lines, or partials, within
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an overall spectral envelope, which is attributed to the resonances of the violin body.
A similar result can be found for voiced speech sounds, where the resonances, also
called formants, vary in time. Unvoiced speech sounds, however, show a broad-band
spectrum modulated by the formant envelope.
Additive synthesis seeks to regenerate the sound by adding together a set of
sinewaves whose frequency and amplitude 'trajectories' vary in time [serr90, riss82].
A diagram of this is shown in Figure 2.3. The trajectories are extracted from the
spectral analysis using a variety of methods. In this form, however, a large amount of
parameter data can be generated. It has been shown, however, that it is the overall
trend of the trajectories that is of greatest perceptual importance and their
approximation with simple piece-wise linear functions allows a considerable degree of
data reduction while maintaining the quality of the reproduced sound [grey75].
Modification of these functions then also allows musically interesting transformations
of the sound.
output
.
.
.
.
.
.
.
.
amp 1
+
freq 1
amp 2
freq 2
amp 3
freq 3
amp 4
freq 4
amp 5
freq 5
amp 6
freq 6
control
sinewave
generators
trajectories
Figure 2.3 A schematic diagram for additive synthesis.
Additive synthesis works well at representing certain sounds to a high degree of
perceptual accuracy. These are ones with a well defined partial structure arising from
periodic excitation and/or systems with simple vibrational modes. It is, however,
limited in its capability to represent complex or noisy sounds, i.e. ones with broad-
band spectral structures.
Subtractive synthesis also seeks to regenerate the sound using the spectral
information. It does this in the opposite sense to additive synthesis by starting with a
spectrally rich input that is then refined with a time varying filter. The excitation may
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be periodic or noise-like, to give harmonic or wide-band spectral structure
respectively. The filter then shapes this to provide the formant envelope.
A powerful method for estimating suitable filters is linear prediction [makh75,
moor90]. This encompasses a number of techniques that allow the estimate ofparameters for a digital, recursive linear filter from the original time series. These
filters are of the form,
y x b yn n i n ii
M
1
where x is the excitation input, y the output, b the filter coefficients, and Mthe filter
order which corresponds to one half the number of formant peaks.
This technique is used widely for speech modelling where between 3-7 formants
are required to adequately represent the sound, and so provides a considerable degreeof data reduction. Attempts at modelling drum sounds suggest that approximately 100
are necessary [sand89]. This technique offers the potential for modification of the
individual resonances or the excitation so as to transform the sound in an intuitive
way. There are difficulties, however, associated with the numerical manipulation and
implementation of the high order filters required [sand92].
A much simplified synthesis-only derivative of the recursive filter model, known
as the Karplus-Strong algorithm, has been found to generate certain sounds very
effectively. These include plucked string, drum and electric guitar timbres [karp83,jaff83, sull90]. The simplification is in having high order filter models, but with all
the coefficients set to zero except the higher index ones. Variants include the insertion
of other elements, for example randomly controlled switches and nonlinearities, in the
feedback path. It is therefore equivalently described as a delay-line with feedback via
some kind of modifier. Both these views are shown in Figure 2.4. Typically, the sound
is generated by inputting a burst of noise, or a simple periodic waveform to the delay
line.
modifier
output
z-1 z-1 z-1 z-1 z-1 z-1......
+
delay of samplesDoutput
delay of samplesD
coefficients
input
input
Figure 2.4 Karplus-Strong algorithm. Top, simplified recursive linear filter and
bottom, general delay-line view.
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Finally, a technique for combining both additive and subtractive synthesis has also
been proposed [serr90].
2.6.3. Frequency Modulation and Waveshaping
Frequency modulation (FM) and waveshaping are related synthesis-only
techniques that allow the generation of sounds with complex line spectra using simple
models [chow73 and lebr79]. A basic unit of each technique is shown in Figure 2.5.
The units are then combined by either adding several outputs together, or nested so
that the output of one forms the input to another. The parameters inputted to the
model are accessed directly by the user, and/or controlled by simple functions to
generate time-varying sounds.
To their advantage, the sounds produced by these models are often approximate
replicas of musical ones. Both harmonic and inharmonic sounds may be simulated that
are like those generated from string or wind, and percussive instruments, respectively.
It is also possible to generate a wide range of abstract sounds. The relatively small
number of parameters involved allows for easy experimentation by the user and the
simplicity of the models enables them to be easily implemented.
+
amp
freq
amp
freq
out
output
x(t) f [x(t)]f
nonlinearfunction
input
carrier frequency
modulationfrequencyandintensity
outputamplitude
Figure 2.5 The basic units used within the FM (top) and waveshaping (bottom)synthesis techniques.
The disadvantages of these models are that no analysis methods exist that can
produce a set of parameters from a given sound and that, as with physical modelling,
the sounds can lack certain 'natural' qualities [moor90].
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2.7. Summary
This chapter has developed the concept of a model for sound with which to work.
The principal idea is that of representation. There are many levels on which a
representation for sound can take place, from the physical to the perceptual. Also,several representations may be used together. An example is the chain of
representations that exists within the additive synthesis model: physical system;
pressure fluctuations at microphone; time waveform; digital audio time series; time-
varying spectrum; set of variable amplitude and frequency sinewaves; set of piece-
wise linear functions.
From a consideration of the types of creative applications where such a model
might be used, a functional description has been advanced. Central to this description
is the idea of a parameterised representation, where the parameters consist of less datathan the modelled sound and are of a form that facilitates manipulation of the sound in
useful ways.
Finally, several well known models fitting this description have been reviewed. These
models are primarily for music and speech sounds and, consequently, focus on
representing those elements that characterise such sounds, both physically and
perceptually, for example spectral lines and formant envelopes. The models, therefore,
concentrate mainly on the top two categories of Table 2.1. No models fitting the
description given in this chapter have been found in the literature which have beenfound for sounds that are outside these categories.
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Chapter 3
Chaos Theory and Fractal Geometry
3.1. Introduction
This chapter presents an overview of chaos theory and fractal geometry. The
intention is to present a theoretical basis for the forthcoming chapters. Theory relevant
to each experimental chapter is then presented in that chapter. The emphasis is
therefore on the following subjects: the significance of chaos and fractals; strange
attractors; Iterated Function Systems; and several other relevant ideas and tools. The
chapter may be read in its entirety as a concise introduction to chaos and fractals, or
referred to as and when needed during later chapters. Sources for the general theory of
chaos and fractals include [stew92, farm90, glei87, laut88, deva89, schr91, peit88,
moon87, hao84, mand83, barn88].
Chaos theory is about a new understanding of dynamics, the way in which systems
behave through time. It concerns the realisation that deterministic systems which obey
fixed laws, can exhibit unpredictable behaviour. This runs contrary to the established
viewpoint, dating back to Newton, that the behaviour of deterministic systems can be
predicted for all future time. Also, chaotic behaviour, characterised by being irregularand complex, may be found in very simple systems. This, again, apparently
contradicts the traditional scientific expectation that complex behaviour arises only in
complex systems.
The theory of fractals, however, provides a new understanding of geometry. It is
based on a realisation that there exists a large class of geometric objects not
encompassed by the traditional Euclidean geometry of points, lines and circles, or the
forms of differential calculus, for example smooth curves. Fractal objects have
properties unlike those of their traditional counterparts because of the way they fill
space. For example, they typically have dimensions which are not integers and curves
with infinite length can be contained within a finite volume. Many fractals have the
same form when viewed on different scales, a property known as self-similarity. Like
chaos, it is also possible to construct complex fractal forms using only simple rules.
Of greatest importance, perhaps, is that both chaos and fractals can accurately
represent naturally occurring phenomena. Advances in abstract theory have been
paralleled with discoveries of real-world phenomena which confirm the relevance and
usefulness of chaos and fractals. A selection of the subjects in which this has taken
place are: architecture, art, astrophysics, biology, chemistry, communications,
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computing, data compression, economics, electronics, fluid dynamics, geology,
geophysics, linguistics, meteorology, music, physics, signal processing. See [glei87,
pick90, schr91, peit88, cril91, stew90 and moon87] and references therein.
3.2. The Significance of Chaos
Chaos theory concerns the dynamic behaviour of simple nonlinear systems.
Traditionally, the problem of dynamics has been approached in two different ways -
deterministic dynamics and stochastic processes. The deterministic approach assumes
that fixed laws govern the behaviour of a system. These laws may be written down
with linear differential equations, a solution found, and so the behaviour of the system
is known for all time. Such an approach applies to systems with a few degrees of
freedom and where linear relationships, or approximations, exist between thecomponent parts. The advantage to this approach is that the resulting solution gives
complete, predictive knowledge about the behaviour of the system. The main
disadvantage, however, lies also with the solution - it is not always possible to find
one. Analytic techniques do not provide a universal means of solution to systems of
differential equations, especially if they contain nonlinearities.
The alternative, stochastic, approach makes the assumption that the system under
investigation is too complex to be able to describe explicitly with fixed laws. This is
either because there are too many degrees of freedom, or it is not possible to measure
all the relevant aspects of the system. In this case, a partial description of the system
may be given using probability. That is, the degree of uncertainty about a system's
present state, or future behaviour may be quantified. Instead of describing the dynamic
behaviour of every degree of freedom with an explicit solution, only the likelihoods of
expected behaviour are known. These correspond to the average or typical behaviour
found by empirically accumulating information about the system. This is also a
powerful approach as, for example in thermodynamics, the average properties of
particles in a gas provides a useful desc