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This is to certify that the
dissertation entitled

ATTRACTIONS

presented by

ERIK PHILIP LARSON

has been accepted towards fulfillment
of the requirements for

Ph.D. degree ianlSlLCDMEQSITION

tit/(MW

Major proHsor v

Date F‘Lh Hark via“!

MS U is an Affirmative Action/Equal Opportunity Institution 0-12771

 

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DATE DUE DATE DUE DATE DUE

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

5/08 K:IProj/Acc&Pres/ClRC/DaieDue.indd

ATTRACTIONS

By

Erik Philip Larson

A DISSERTATION

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY

Department of Music

1995

ABSTRACT

ATTRACTIONS

By

Erik Philip Larson

Attractions is a music composition of computer generated sounds. The sounds
were generated with a technique knows as granular synthesis. The musical material and
events, and small- and large-scale relationships, were based on a model drawn from chaos
theory. Specifically, the model employed was the logistic difference equation, an equation
used to model a specific kind of process of change, the most well-known example being
that found in certain aspects of population growth. The technique of sound generation,
granular synthesis, involves the creation of sequences of short tones following each other
in rapid succession resulting in complex densities.

The logistic difference equation is useful in its ability to create events that alternate
between order and disorder. In some events found in Attractions, use of the logistic
difference equation creates a goal-oriented sequence of tones. Depending on the speed of
the sequences of tones, two musical events are produced. If the discrete tones are audible,
a progression of tones is heard that ultimately settles on one tone. If the rate of succession
of these tones is increased, a transient sequence occurs that emulates the fluctuations in
frequency and timbre at the onset of many acoustical events. By altering variables in the
equation, the sequence of tones generated from it will settle on two or more tones and may
progress into a pseudo-random sequence, or chaos.

The sequence of events within sections were created by selecting from the large
number of events generated with various applications of the equation. These events were

then edited and mixed digitally to create the complete work. The result is a musical

composition that deviates from traditional acoustic works in the following ways: the output
of the equation applied to frequencies in Hertz creates no consistent system of tempered
relationships of pitch; the sound synthesis technique of granular synthesis creates timbres
that are difficult to produce on acoustic instruments; the unfolding of events exhibits

proportions that range from simple periodicity to complex periodicity to aperiodicity.

ACKNOWLEDGMENTS

The composer wishes to thank Dr. Mark Sullivan for his inspiration and guidance.
I would also like to thank Pat Callow for listening to my ideas and hearing my works in
progress. Andrea Piotrowski assisted with the layout and copy of this work and deserves
additional recognition. Special thanks to my mother and father for their emotional and

financial support throughout my long academic career.

TABLE OF CONTENTS

LIST OF TABLES ................................................................................... vi
LIST OF FIGURES ................................................................................ vii
DESCRIPTION OF COMPOSITION .............................................................. 1
LISTENING SCORE ................................................................................ 4
APPENDIX ......................................................................................... 37

Table 1.
Table 2.
Table 3.
Table 4.
Table 5.
Table 6.
Table 7.
Table 8.

LIST OF TABLES

Numerical Data of Equation with Rate of Growth = 1.0 ............................ 38
Numerical Data of Equation with Rate of Growth = 1.7 ............................ 39
Numerical Data of Equation with Rate of Growth = 2.2 ............................ 40
Numerical Data of Equation with Rate of Growth = 2.8 ............................ 41
Numerical Data of Equation with Rate of Growth = 3.3 ............................ 42
Numerical Data of Equation with Rate of Growth = 3.5 ............................ 43
Numerical Data of Equation with Rate of Growth = 3.8 ............................ 44
Numerical Data of Equation with Rate of Growth = 4.0 ............................ 45

vi

LIST OF FIGURES

Figure 1. Plot of Data with Rate of Growth = 1.0 ............................................ 38
Figure 2. Plot of Data with Rate of Growth = 1.7 ............................................ 39
Figure 3. Plot of Data with Rate of Growth = 2.2 ............................................ 40
Figure 4. Plot of Data with Rate of Growth = 2.8 ............................................ 41
Figure 5. Plot of Data with Rate of Growth = 3.3 ............................................ 42
Figure 6. Plot of Data with Rate of Growth = 3.5 ............................................ 43
Figure 7. Plot of Data with Rate of Growth = 3.8 ............................................ 44
Figure 8. Plot of Data with Rate of Growth = 4.0 ............................................ 45

vii

DESCRIPTION OF COMPOSITION

Attractions is a music composition of computer generated sounds. The sounds
were generated with a technique knows as granular synthesis. The musical material and
events, and small- and large-scale relationships, were based on a model drawn from chaos
theory. Specifically, the model employed was the logistic difference equation, an equation
used to model a specific kind of process of change, the most well-known example being
that found in certain aspects of population growth. This equation is represented by the

equation x = rx(1 - x), where “x” is the original value and “r” is the rate of growth value.

next
Granular synthesis, in general, involves the creation of a sequence in which
relatively short tones (durations that range between one-tenth of a second to two seconds)
follow each other in relatively rapid succession (time intervals of succession that range
between one-thousandth of a second to one-tenth of a second). Two other variables have a
significant effect on the compositional outcome: the choice of grain waveform (waveforms
that range between a sine wave and waves with rich harmonic content), and the choice of
frequency structure in sequence (frequency content that ranges between random variation
and conjunct or disjunct patterns). Perceptual thresholds determine whether a chosen
sequence of grains will be heard as a single tone with a distinct timbre or as a sequence of
discrete tones (i.e. a sequence of grains, all sine waves with a duration of one second, and
with randomly varying frequencies, that follow one another at the rate of one second, will
be heard as a sequence of discrete tones with randomly varying pitch; another at the rate of
one-thousandth of a second, will be heard as a single pitch with a noise-like tone color).
The compositional process involved the exploration of the consequences of changing these

variables and their relationship to one another within the context of the relevant thresholds

of perception.

2

The logistic difference equation was used in several ways to create events: When
the variable controlling the rate of growth is between one and three, the numerical output
increases rapidly at first, overpasses its eventual value, and finally settles back onto that
value. These values converted into musical frequencies would result in an opening flourish
of frequencies that settles on a single frequency. Depending on the rate of succession of
grains, this progression of frequencies can produce an ascending sequence of discrete tones
settling on a single tone, or a transient sequence of frequencies that characterize timbres in
which there are distinct fluctuations in frequency at the onset of the sound.

If the variable controlling the rate of growth is increased further, the equation settles
on 2 tones, and as increased further, 4, 8, 16 tones, finally resulting in a pseudo-random
sequence, or chaos (see Appendix). The numerical outputs generated by these variables
and subsequently converted to frequencies produce sonic events that, at a slow rate of
succession, create a sequence of tones that form 2—, 4-, 8-, and l6-tone sonorities, and at a
fast rate of succession, create a sequence of sonorities with increasingly dense inharmonic
spectra.

Other implementations of the equation were as follows: four or more outputs of the
equation were layered upon one another to create events having a richer harmonic content
than when used alone; four or more outputs of the equation were staggered in time to create
a type of contrapuntal relationship; the rate of succession of the frequencies was increased
to create short bursts of sounds; each numerical output of the equation was randomized
around itself creating sequences of over 500 tones that all hover around the basic mapping
of the equation.

The small-scale relationships, such as those in the frequency material, follow the
logistic difference equation closely. The large-scale relationships follow the model in a
more general way. Certain sections of the composition can be understood as complex
deviations (that use more complex implementations of the equation) from a single recurring

section, to which they return. The section is one which embodies the simplest

3

implementation of the equation, that is, an implementation in which the output of the
equation begins with a diverse set of values and eventually settles on one value. This
section creates stability in the composition and is represented in the following graph by the
A sections, while the B, C, D, E, and F sections all represent deviations from the stability

of the A sections and use more complex implementations of the equation.

A| B| C I A' I B'|D|EI A" IE—
I | | l l I I I I I I I I I I
O 1 2 3 4 5 6 7 8 9 10 ll 12 13 14

Minutes

The sequence of events within sections were created by selecting from the large
number of events generated with various applications of the equation. These events were
then edited and mixed digitally to create the complete work. The result is a musical
composition that deviates from traditional acoustic works in the following ways: the output
of the equation applied to frequencies in Hertz creates no consistent system of tempered
relationships of pitch; the sound synthesis technique of granular synthesis creates timbres
that are difficult to produce on acoustic instruments; the unfolding of events exhibits

proportions that range from simple periodicity to complex periodicity to aperiodicity.
TECHNICAL NOTE

Attractions was generated using the following method: programs were written using
the C programming language to create CSound score files; these score files were read by
the digital synthesis program CSound which produced CD quality audio files; the sound

files were mixed and processed using SoundEditTM 16.

LISTENING SCORE

The listening score for Attractions was created using the sound editing software
package SoundEditTM 16 for Macintosh. Using the “spectrum” view of the sound file
which plots a Fourier transform of frequency versus time, an image of the screen was
captured and saved to a series of graphics files. These files were then opened in Adobe
IllustratorTM which was used to convert the files into higher quality graphics files and to add
the time scale. The result is a multi-page score that depicts both duration proportions and
harmonic densities of the piece. In the score, the “x” axis displaying time in minutes and

seconds while the y axis shows frequencies in Hertz. In the frequency scale, 5K is

equivalent to 5000 Hertz, 4K is 4000 Hertz, and so on.

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APPENDIX

APPENDIX

The tables and figures on the following pages show typical output of the logistic
difference equation. Using the C programming language, a computer program can easily
compute the output of the equation and print the result each time. Similar to the programs
used in the generation of score files (which are read by the computer to produce sounds),
the C program below computes and prints the equation 100 times, with the output of one

pass becoming the input of the subsequent pass.

#include<stdio.h>
main()

f

1nt X;

float rate;

float start;

fprintf(stderr,"Enter the Starting Value:\n");
scanf("%f",&start);

fprintf(stderr,"Enter the Rate of Growth:\n");
scanf("%f",&rate);

printf("The Beginning Value is: %.2f\n",start);
printf("The Rate of Growth is: %.2f\n",rate);

printf("\n\n");

printf("%.4f\t",start);

for (X = O; X < 99; X++)
{
start = (1.0 — start)*start*rate;
printf("%.4f\t",start);
}
}

The following tables and figures show the output of the equation at various rates of
growth. All start at the same value of 0.2 for consistency and to keep the output in a range
from O to 1. The “x” axis shows the iterations from 1 to 100. Using various multipliers,
these values were used as freqency in Hertz for the generation of the sound files in the

composition.

37

 

0.2000
0.0615
0.0373
0.0269
0.0211
0.0174
0.0148
0.0128
0.01 14
0.0102

38

Table 1. Numerical Data of Equation with Rate of Growth = 1.0

0.1600
0.0577
0.0360
0.0262
0.0207
0.0171
0.0145
0.0127
0.01 12
0.0101

0.1344
0.0544
0.0347
0.0255
0.0202
0.0168
0.0143
0.0125
0.01 l 1
0.0100

0.1163 0.1028 0.0922 0.0837

0.0514
0.0335
0.0249
0.0198
0.0165
0.0141
0.0124
0.0110
0.0099

0.0488
0.0323
0.0243
0.0194
0.0162
0.0139
0.0122
0.0109
0.0098

0.0464
0.0313
0.0237
0.0191
0.0160
0.0137
0.0121
0.0107
0.0097

0.0442
0.0303
0.0231
0.0187
0.0157
0.0135
0.01 19
0.0106
0.0096

0.0767
0.0423
0.0294
0.0226
0.0183
0.0155
0.0134
0.01 18
0.0105
0.0095

Figure 1. Plot of Data with Rate of Growth 2 1.0

0.0708
0.0405
0.0285
0.0221
0.0180
0.0152
0.0132
0.01 16
0.0104
0.0094

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0.0389
0.0277
0.0216
0.0177
0.0150
0.0130
0.01 15
0.0103
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100

0.2000
0.4118
0.4118
0.4118
0.4118
0.4118
0.4118
0.4118
0.4118
0.4118

39

Table 2. Numerical Data of Equation with Rate of Growth = 1.7

0.2720 0.3366

0.4118
0.4118
0.4118
0.4118
0.4118
0.4118
0.4118
0.4118
0.4118

0.4118
0.4118
0.4118
0.4118
0.4118
0.4118
0.4118
0.4118
0.4118

0.3796
0.4118
0.4118
0.4118
0.4118
0.4118
0.4118
0.4118
0.4118
0.4118

0.4004
0.4118
0.4118
0.4118
0.4118
0.4118
0.4118
0.4118
0.4118
0.4118

0.4081
0.4118
0.4118
0.4118
0.4118
0.4118
0.4118
0.4118
0.4118
0.4118

0.4107
0.4118
0.4118
0.4118
0.4118
0.4118
0.4118
0.4118
0.4118
0.4118

0.4114
0.4118
0.4118
0.4118
0.4118
0.4118
0.4118
0.4118
0.4118
0.4118

Figure 2. Plot of Data with Rate of Growth = 1.7

0.4117
0.4118
0.4118
0.4118
0.4118
0.4118
0.4118
0.4118
0.4118
0.4118

0.4117
0.4118
0.4118
0.4118
0.4118
0.4118
0.4118
0.4118
0.4118
0.4118

 

0.9 a

0.8 -

0.7 -

0.6 a

0.5 r

0.4 -

0.3 -

0.2 *9

0.1 -

 

 

 

10

20

30

40

50

60

70

80

90

100

0.2000
0.5455
0.5455
0.5455
0.5455
0.5455
0.5455
0.5455
0.5455
0.5455

40

Table 3. Numerical Data of Equation with Rate of Growth = 2.2

0.3520
0.5455
0.5455
0.5455
0.5455
0.5455
0.5455
0.5455
0.5455
0.5455

0.5018
0.5455
0.5455
0.5455
0.5455
0.5455
0.5455
0.5455
0.5455
0.5455

0.5500
0.5455
0.5455
0.5455
0.5455
0.5455
0.5455
0.5455
0.5455
0.5455

0.5445
0.5455
0.5455
0.5455
0.5455
0.5455
0.5455
0.5455
0.5455
0.5455

0.5456
0.5455
0.5455
0.5455
0.5455
0.5455
0.5455
0.5455
0.5455
0.5455

0.5454
0.5455
0.5455
0.5455
0.5455
0.5455
0.5455
0.5455
0.5455
0.5455

0.5455
0.5455
0.5455
0.5455
0.5455
0.5455
0.5455
0.5455
0.5455
0.5455

Figure 3. Plot of Data with Rate of Growth = 2.2

0.5455
0.5455
0.5455
0.5455
0.5455
0.5455
0.5455
0.5455
0.5455
0.5455

0.5455
0.5455
0.5455
0.5455
0.5455
0.5455
0.5455
0.5455
0.5455
0.5455

 

0.9 4

0.8 ~

0.7 -

0.6 -

0.5- O

0.4 ~

0.3 4

0.2 ‘9

0.1 ‘

 

 

 

10

20

50

60

70

80

100

 

0.2000
0.651 1
0.6438
0.6430
0.6429
0.6429
0.6429
0.6429
0.6429
0.6429

41

Table 4. Numerical Data of Equation with Rate of Growth = 2.8

0.4480
0.6360
0.6421
0.6428
0.6428
0.6429
0.6429
0.6429
0.6429
0.6429

0.6924 0.5963 0.6740 0.6152 0.6628

0.6482
0.6434
0.6429
0.6429
0.6429
0.6429
0.6429
0.6429
0.6429

0.6385
0.6424
0.6428
0.6429
0.6429
0.6429
0.6429
0.6429
0.6429

0.6463
0.6432
0.6429
0.6429
0.6429
0.6429
0.6429
0.6429
0.6429

0.6401
0.6426
0.6428
0.6429
0.6429
0.6429
0.6429
0.6429
0.6429

0.6450
0.6431
0.6429
0.6429
0.6429
0.6429
0.6429
0.6429
0.6429

0.6258
0.641 1
0.6427
0.6428
0.6429
0.6429
0.6429
0.6429
0.6429
0.6429

Figure 4. Plot of Data with Rate of Growth = 2.8

0.6557
0.6443
0.6430
0.6429
0.6429
0.6429
0.6429
0.6429
0.6429
0.6429

0.6321
0.6417
0.6427
0.6428
0.6429
0.6429
0.6429
0.6429
0.6429
0.6429

 

0.9 a

0.8 r

0.7 -

0.6 4 O

0.5 ‘

0.4 a

0.3 ‘

0.2 ‘0

0.1 ‘

 

 

 

20

30

40

50

60

70

80

90

100

0.2000
0.8236
0.8236
0.8236
0.8236
0.8236
0.8236
0.8236
0.8236
0.8236

42

Table 5. Numerical Data of Equation with Rate of Growth = 3.3

0.5280
0.4794
0.4794
0.4794
0.4794
0.4794
0.4794
0.4794
0.4794
0.4794

0.8224
0.8236
0.8236
0.8236
0.8236
0.8236
0.8236
0.8236
0.8236
0.8236

0.4820
0.4794
0.4794
0.4794
0.4794
0.4794
0.4794
0.4794
0.4794
0.4794

0.8239
0.8236
0.8236
0.8236
0.8236
0.8236
0.8236
0.8236
0.8236
0.8236

0.4787
0.4794
0.4794
0.4794
0.4794
0.4794
0.4794
0.4794
0.4794
0.4794

0.8235
0.8236
0.8236
0.8236
0.8236
0.8236
0.8236
0.8236
0.8236
0.8236

0.4796
0.4794
0.4794
0.4794
0.4794
0.4794
0.4794
0.4794
0.4794
0.4794

Figure 5. Plot of Data with Rate of Growth = 3.3

0.8236
0.8236
0.8236
0.8236
0.8236
0.8236
0.8236
0.8236
0.8236
0.8236

0.4794
0.4794
0.4794
0.4794
0.4794
0.4794
0.4794
0.4794
0.4794
0.4794

 

03-
08-
03-
061
O
051
0Aa
03-

02-9

01‘

99909990099000696Q09900900900009999699.6990.06090

990909006060909900000090690000.90.000990006909904

v

 

 

 

m

N

m

40

w

70

80

90

1%

0.2000
0.8683
0.8284
0.8750
0.8269
0.8750
0.8269
0.8750
0.8269
0.8750

43

Table 6. Numerical Data of Equation with Rate of Growth = 3.5

0.5600
0.4002
0.4975
0.3828
0.5009
0.3828
0.5009
0.3828
0.5009
0.3828

0.8624
0.8401
0.8750
0.8269
0.8750
0.8269
0.8750
0.8269
0.8750
0.8269

0.4153
0.4700
0.3829
0.5009
0.3828
0.5009
0.3828
0.5009
0.3828
0.5009

0.8499
0.8719
0.8270
0.8750
0.8269
0.8750
0.8269
0.8750
0.8269
0.8750

0.4465
0.3910
0.5008
0.3828
0.5009
0.3828
0.5009
0.3828
0.5009
0.3828

0.8650
0.8334
0.8750
0.8269
0.8750
0.8269
0.8750
0.8269
0.8750
0.8269

0.4088
0.4859
0.3828
0.5009
0.3828
0.5009
0.3828
0.5009
0.3828
0.5009

Figure 6. Plot of Data with Rate of Growth = 3.5

0.8459
0.8743
0.8269
0.8750
0.8269
0.8750
0.8269
0.8750
0.8269
0.8750

0.4563
0.3846
0.5009
0.3828
0.5009
0.3828
0.5009
0.3828
0.5009
0.3828

 

0.9 -

0.8 -

0.7 -

0.6 '

0.5 a

0.4 -

0.3 ‘

0.2 +9

0.1 1

 

’9

9”.

..oooooooooooooooooooooo
"¢¢oo¢¢o¢¢oooooo¢¢¢ooo

90000909900990.0090.

99
’Oooooooooooooooooooooo

 

 

10

20

30

40

50

60

70

80

90

100

44

Table 7. Numerical Data of Equation with Rate of Growth = 3.8

0.2000 0.6080 0.9057 0.3246 0.8331 0.5283 0.9470 0.1909 0.5869

0.2755 0.7585 0.6960 0.8040 0.5988 0.9129 0.3022 0.8013 0.6050

0.3171 0.8229 0.5539 0.9390 0.2178 0.6474 0.8675 0.4369 0.9349

0.6757 0.8327 0.5295 0.9467 0.1918 0.5890 0.9199 0.2800 0.7660

0.8254 0.5477 0.9414 0.2098 0.6299 0.8859 0.3842 0.8990 0.3450

0.4610 0.9442 0.2001 0.6082 0.9055 0.3252 0.8338 0.5265 0.9473

0.5838 0.9233 0.2691 0.7475 0.7173 0.7706 0.6717 0.8380 0.5160

0.1838 0.5701 0.9313 0.2430 0.6990 0.7995 0.6092 0.9047 0.3278

0.5177 0.9488 0.1846 0.5719 0.9304 0.2462 0.7052 0.7899 0.6306

0.3861 0.9007 0.3398 0.8525 0.4779 0.9481 0.1868 0.5774 0.9273

Figure 7. Plot of Data with Rate of Growth = 3.8

0.9213
0.9081
0.2313
0.6811
0.8587
0.1896
0.9490
0.8373
0.8852
0.2563

 

0.9 - ‘
0.8 - O 6 o .
037+ ..
0.6 4 9 , o 9 ¢ ’ , ' ’
0.5 4
0.4 ‘ 9 O
0.3 ~ 9 .
0.2 -o . . o 9 . . ,

OJ 4

 

 

 

30 40 50 60 70 80 90

100

0.2000
0.1478
0.8349
0.8938
0.3476
0.9993
0.6906
0.4858
0.8396
0.7490

45

Table 8. Numerical Data of Equation with Rate of Growth = 4.0

0.6400 0.9216 0.2890 0.8219 0.5854 0.9708 0.1133 0.4020

0.5039 0.9999 0.0002 0.0010 0.0039 0.0154 0.0606 0.2276

0.5514 0.9894 0.0419 0.1604 0.5387 0.9940 0.0238 0.0929

0.3797 0.9421 0.2181 0.6822 0.8672 0.4606 0.9938 0.0246

0.9070 0.3372 0.8940 0.3789 0.9414 0.2208 0.6883 0.8582

0.0028 0.0113 0.0448 0.1711 0.5673 0.9819 0.0712 0.2645

0.8546 0.4970 1.0000 0.0001 0.0006 0.0023 0.0093 0.0367

0.9992 0.0032 0.0129 0.0508 0.1929 0.6227 0.9398 0.2263

0.5388 0.9940 0.0240 0.0936 0.3392 0.8966 0.3707 0.9332

0.7520 0.7460 0.7579 0.7340 0.7810 0.6842 0.8642 0.4694

Figure 8. Plot of Data with Rate of Growth = 4.0

0.9616
0.7032
0.3371
0.0961
0.4867
0.7781
0.1415
0.7003
0.2495
0.9963

 

0.9 -

0.8 ‘

0.7 ‘

0.6 ~

0.5 .

0.4 a

0.3 ' .

0.2 -O

0.1 -

 

‘1

 

 

 

90

100

llllllllllllllll

[1L 1 4

Illllllllllllllllllllllllll

2
6
0
3
O
3
9
2
1
3

lllllllllillllllll