The applications of fractal geometry and self-similarity to art music by

The applications of fractal geometry and self-similarity to art music by
The applications of fractal geometry and self-similarity to art music
by
Ilse Steynberg
A dissertation submitted in fulfilment of the requirements for the degree
Magister Musicae (Musicology)
in the Department of Music at the
UNIVERSITY OF PRETORIA
FACULTY OF HUMANITIES
SUPERVISOR: Dr A.F. Johnson
July 2014
i
Abstract
Title:
The applications of fractal geometry and self-similarity to art music
Student:
Ilse Steynberg
Supervisor:
Dr A.F. Johnson, Department of Music, Faculty of Humanities, University of
Pretoria
Degree:
Magister Musicae (Musicology)
Date:
March 2014
The aim of this research study is to investigate different practical ways in which fractal geometry
and self-similarity can be applied to art music, with reference to music composition and analysis.
This specific topic was chosen because there are many misconceptions in the field of fractal and
self-similar music.
Analyses of previous research as well as the music analysis of several compositions from
different composers in different genres were the main methods for conducting the research.
Although the dissertation restates much of the existing research on the topic, it is (to the
researcher‟s knowledge) one of the first academic works that summarises the many different
facets of fractal geometry and music.
Fractal and self-similar shapes are evident in nature and art dating back to the 16th century,
despite the fact that the mathematics behind fractals was only defined in 1975 by the French
mathematician, Benoit B. Mandelbrot. Mathematics has been a source of inspiration to
composers and musicologists for many centuries and fractal geometry has also infiltrated the
works of composers in the past 30 years. The search for fractal and self-similar structures in
music composed prior to 1975 may lead to a different perspective on the way in which music is
analysed.
Basic concepts and prerequisites of fractals were deliberately simplified in this research in order
to collect useful information that musicians can use in composition and analysis. These include
subjects such as self-similarity, fractal dimensionality and scaling. Fractal shapes with their
defining properties were also illustrated because their structures have been likened to those in
some music compositions.
ii
This research may enable musicians to incorporate mathematical properties of fractal geometry
and self-similarity into original compositions. It may also provide new ways to view the use of
motifs and themes in the structural analysis of music.
iii
Keywords

1/f-noise

Algorithmic composition

Fractal geometry

Fractal music

Fragmentation

Iteration

Mathematics

Repetition

Self-similarity

Symmetry

Transformation
iv
CONTENTS
Title page
Abstract
i
Keywords
iii
Contents
iv
List of figures
ix
CHAPTER 1: INTRODUCTION
1
1.1
Background to the study
1
1.1.1
Music and mathematics through the ages
1
1.1.2
What are fractal geometry and self-similarity?
3
1.1.3
Fractal geometry and self-similarity in music
5
1.2
Problem statement
6
1.3
Motivation for the study
7
1.4
Research question
8
1.5
Objectives of the study
10
1.6
Delimitation of the study
10
1.7
Research design
11
1.8
Research methodology
12
1.9
Summary of chapters
13
CHAPTER 2: THE HISTORY AND MATHEMATICS OF FRACTAL GEOMETRY AND
SELF-SIMILARITY
15
2.1
Introduction
15
2.2
Music and geometry
15
v
2.3
2.4
2.2.1
Transposition and translation
16
2.2.2
Scaling
17
2.2.3
Reflection
19
2.2.4
Rotation and retrograde inversion
21
A short history of fractal geometry
23
2.3.1
Examples of fractals and self-similarity in nature and art
23
2.3.2
The era of “mathematical monsters”
31
2.3.2.1 Cantor set
32
2.3.2.2 Koch snowflake
33
2.3.2.3 Sierpinski triangle and carpet
34
2.3.2.4 Lindenmayer systems
35
2.3.2.5 Turtle interpretation of L-systems
36
Definitions relating to fractals and self-similarity
39
2.4.1
Self-similarity
39
2.4.2
Fractal dimension
41
2.4.3
Fragmentation
43
2.5
Noise
44
2.6
Conclusion
46
CHAPTER 3: LITERATURE REVIEW
48
3.1
Introduction
48
3.2
Mandelbrot and other mathematicians
48
3.3
1/f noise distribution in music
49
3.3.1
Voss and Clarke
49
3.3.1.1 The discovery of 1/f noise distribution in music
50
3.3.1.2 Criticism of Voss and Clarke‟s research
57
vi
3.3.2
3.3.3
3.4
Kenneth and Andrew Hsü
58
3.3.2.1 The fractal geometry of melody
58
3.3.2.2 The fractal geometry of amplitude
62
3.3.2.3 Measurement and reduction of a musical composition
63
3.3.2.4 Criticism of the Hsüs‟ research
68
Spectral density analysis for genre classification
70
3.3.3.1 Ro and Kwon
70
3.3.3.2 Levitin, Chordia and Menon
73
Towards a better understanding of fractal music
75
3.4.1
Harlan J. Brothers
75
3.4.2
Prerequisites for self-similarity and fractality
76
3.4.3
Scaling in music
78
3.4.3.1 Duration scaling
78
3.4.3.2 Pitch scaling
81
3.4.3.3 Structural scaling
84
3.5
Validity of fractal musical analysis
84
3.6
Conclusion
86
CHAPTER 4: FRACTALS AND SELF-SIMILARITY AS A COMPOSITION TOOL
88
4.1
Introduction
88
4.2
Music composition with noise
88
4.2.1
Using spinners to compose music based on noise forms
88
4.2.2
Turning noise voltages into music
92
4.3
Composing with Lindenmayer systems
95
4.3.1
Prusinkiewicz
95
4.3.2
Mason and Saffle
97
vii
4.3.3
Further modifications of L-systems for composition
102
4.4
Self-similar structures in the music of Tom Johnson
104
4.5
Coastlines and mountains
111
4.5.1
Mandelbrot and coastlines
111
4.5.2
Relating music to coastlines and mountains
114
4.5.2.1 Composing with coastlines
115
4.5.2.2 Gary Lee Nelson‟s Fractal Mountains
117
4.6
4.7
Fractal-inspired music of Ligeti
118
4.6.1
Ligeti‟s interest in fractals
118
4.6.2
Fractals in Ligeti‟s compositions
119
4.6.2.1 Désordre
119
4.6.2.2 L’escalier du diable
122
Conclusion
125
CHAPTER 5: FRACTAL GEOMETRY AND SELF-SIMILARITY IN MUSIC ANALYSIS:
MUSIC PRIOR TO 1975
127
5.1
Introduction
127
5.2
Rhythmic self-similarity
127
5.2.1
The mensuration canon
127
5.2.1.1 Ockeghem‟s Missa Prolationum
128
5.2.1.2 Josquin des Prez‟s first Missa l’homme armé
130
Other examples of rhythmic self-similarity
131
5.2.2
5.3
Structural self-similarity
136
5.3.1
Structural scaling in Bach
136
5.3.2
Bach and the Cantor set
138
5.3.3
Beethoven‟s Piano Sonata No. 15 in D major, Op. 28
139
viii
5.3.3.1 Structural analysis of the first movement
139
5.3.3.2 Scaling of the first theme of the first movement
140
5.3.3.3 Interrelationship of material between movements
145
5.3.3.4 Comparison between the third movement and the Sierpinski
triangle
5.4
5.5
Arvo Pärt‟s Fratres für violine und klavier
147
150
5.4.1
The structure
151
5.4.2
Expansion and reduction
156
Conclusion
159
CHAPTER 6: CONCLUSION
160
6.1
Summary of findings
160
6.2
Conclusions
162
6.3
Summary of contributions
163
6.4
Suggestions for further research
164
6.5
Suggestions for further reading
165
REFERENCES
166
ix
List of figures
1.
Kircher‟s two composition “machines”: Arca Musaritmica and Organum
Mathematicum
2
2.
Julia set
4
3.
Outline of the research design
12
4.
Mozart, First movement from Piano Sonata No. 16 in C major, K 545, bars 1-8,
illustrating transposition
16
5.
Geometric translation
17
6.
Geometric scaling
17
7.
Berlioz, Symphonie Fantastique, Fifth movement (Songe d’une Nuit Du
Sabbat), bars 127-162, illustrating diminution of note values
8.
18
Bach, Two-Part Invention No. 14 in B major, BWV 785, right hand, bar 1,
illustrating melodic inversion
19
9.
Reflection around the x-axis creating a horizontal mirror
19
10.
Reflection around the y-axis creating a vertical mirror
20
11.
Haydn, Menuetto al Rovescio from Piano Sonata in A major, Hob. XVI:26,
bars 1-20, illustrating retrograde
20
12.
Hindemith, LudusTonalis, first and last bars, illustrating retrograde inversion
21
13.
Rotation around a fixed point
22
14.
Bach, Contrapunctus VIII from Die Kunst der Fuge, bars 1-4, illustrating
simultaneous use of two transformations, inversion and augmentation
15.
22
Natural fractals: (a) lightning; (b) clouds; (c) fern leaves; and (d) Romanesco
broccoli
24
16.
Da Vinci, The Deluge
25
17.
Hokusaki, The Great Wave
26
18.
Escher, Circle Limit I
27
19.
Bach, Fugue No. 3 in C# major, WTC II, BWV 872, bars 1-2
28
20.
Patterns on the pulpit of the Ravello cathedral resembling the Sierpinski
triangle
29
21.
Dali, Visage of War
30
22.
The first five stages of the Cantor set
32
23.
Steps for creating the Koch snowflake
33
x
24.
Steps in creating the Sierpinski gasket or triangle
34
25.
Steps in creating the Sierpinski carpet
34
26.
Lindenmayer system growth pattern for algae
36
27.
Graphic representation for the first three iterations of the Hilbert curve,
created with turtle graphics
38
28.
First three iterations for the L-system of the Hilbert curve
38
29.
Self-similarity of a fern leaf
39
30.
Self-similarity of the Koch curve
40
31.
Dimension of a cube
42
32.
First iteration of the Koch curve
42
33.
Tangent line drawn on a circle illustrating differentiability
43
34.
White or 1/f0 noise
45
35.
Brown or 1/f2 noise
45
36.
Pink or 1/f noise
46
37.
Spectral density of Bach‟s First Brandenburg Concerto
51
38.
Spectral density of the instantaneous loudness compared to 1/f noise:
(a) Scott Joplin Piano Rags; (b) Classical station; (c) rock station; (d) news
and talk station
39.
52
Power spectrum of pitch fluctuations compared to 1/f noise:
(a) Classical station; (b) jazz and blues station; (c) rock station; (d) news
and talk station
40.
53
2
Audio power fluctuation spectral densities, Sv (f) versus f for
(a) Davidovsky‟s Synchronism I, III and III; (b) Babbit‟s String Quartet, No. 3;
(c) Jolas‟ Quartet, No. 3; (d) Carter‟s Piano Concerto in two movements;
and (e) Stockhausen‟s Momente
41.
54
Pitch fluctuations from different musical cultures:
(a) the Ba-Benzele Pygmies; (b) traditional music of Japan; (c) classical
ragas of India; (d) folk songs of old Russia; (e) American blues
42.
55
Pitch fluctuations in Western music: (a) Medieval music up to 1300;
(b) Beethoven‟s third symphony; (c) Debussy piano works;
(d) Strauss‟ Ein Heldenleben; (e) the Beatles‟ Sgt. Pepper
56
43.
Numerical values assigned to respective note intervals
59
44.
Fractal geometry of sound frequency where
is the percentage
frequency of incidence of note interval : (A) Bach BWV 772; (B) Bach BWV 784;
xi
(C) Bach BWV 910; (D) Mozart KV 533; (E) Six Swiss children‟s songs;
(F) Mozart KV 331; (G) Stockhausen‟s Capricorn.
45.
61
Resemblance between the West Coast of Britain and Bach‟s Invention
No. 1 in C major, BWV 772
64
46.
Digitised score of Bach‟s Invention No. 1 in C major, BWV 772
65
47.
Reductions of Bach‟s Invention No. 1 in C major, BWV 772:
(A) original composition, (B) ⁄ reduction, (C) ⁄ reduction, (D) ⁄ reduction,
(E) ⁄
reduction, (F) ⁄
reduction
66
48.
Bach, Invention No. 1 in C major, BWV 772, first five notes with interval numbers 67
49.
Bach, Invention No. 1 in C major, BWV 772, half-reduction of the first five notes
with interval numbers
67
50.
The degree of 1/f behaviour for seven different genres of music in the region
below 20 Hz
51.
71
Log-log plot of frequency analysis in the Larghetto second movement of
Chopin‟s first Piano Concerto, Op. 11 showing high 1/f correlation in
the low frequency range
52.
Log-log plot of frequency analysis in Song of Simchung displaying very
low 1/f correlation in the low frequency range
53.
71
72
Distribution of spectral exponents for composers ordered from the largest
mean exponent to the smallest
74
54.
Brothers, two melodies with the same distribution of note values
78
55.
Table showing the distribution of note values in Brothers‟ two melodies
79
56.
Log-log plot of the distribution of note values in two melodies by Brothers
79
57.
Brothers, Go for Baroque, illustrating duration scaling
80
58.
Table showing the distribution of note values in Brothers‟ Go for Baroque
80
59.
Brothers, D major melody to examine the possibility of pitch scaling
81
60.
Table showing the distribution of pitches in Brothers‟ D major melody,
arranged from highest to lowest pitch
82
61.
Log-log plot of the distribution of pitches in Brothers‟ D major melody
82
62.
Table of binned data for Brothers‟ D major melody
83
63.
Log-log plot for the binned data of Brothers‟ D major melody
83
64.
Spinner used to generate pitches for white music
89
65.
White music generated with a spinner
89
xii
66.
Spinner used to generate brown music
90
67.
Brown music generated with a spinner
91
68.
1/f music generated with a spinner
92
69.
Music resultant from (a) white, (b) brown and (c) 1/f noise
93
70.
Third iteration of the Hilbert curve
96
71.
Unravelled form of the third iteration of the Hilbert curve
96
72.
Third iteration of the Hilbert curve represented as a melodic line
97
73.
The first iteration of the quadratic Gosper curve read in terms of four
rotations to produce eight different melodic motifs
74.
99
Two transformations of the second iteration of the quadratic Gosper curve
used to compose polyphonic music for flute and piano
100
75.
Self-similarity of higher iterations of the Hilbert curve
101
76.
Warped version of the Hilbert curve used for Nelson‟s Summer Song
102
77.
Scale used in Nelson‟s Summer Song
103
78.
Warped version of the Hilbert curve, unravelled to produce the pitch and time
contour for Nelson‟s Summer Song
103
79.
Three self-similar drawings from Symmetries created with a music typewriter
105
80.
Johnson, excerpt from Rational Melody VIII
106
81.
Johnson, second movement from Counting Keys
107
82.
Johnson, fourth movement from Counting Duets
108
83.
Johnson, 1 2 3, Part I, No. 1
109
84.
Johnson, Tilework for Violin, bars 57-96
110
85.
Richardson‟s data concerning the rate of increase of a coastline‟s length at
decreasing scales
112
86.
Musgrave, a fractal landscape
113
87.
Graphic representation used in Villa-Lobos, New York Skyline
114
88.
Time-plot obtained from a map of Canada for Austin‟s Canadian Coastlines
116
89.
Nelson, Fractal Mountains, different subdivisions used to simulate the outline
of a mountain
117
90.
Ligeti, Étude pour piano No. 1 (Désordre), first page
121
91.
First three steps for creating the devil‟s staircase
122
92.
The complete devil‟s staircase
123
93.
Ligeti, Étude pour piano No. 13 (L’escalier du diable), line 1-4
124
94.
Ockeghem, Kyrie from Missa Prolationum, bars 1-18
128
xiii
95.
Ockeghem, Gloria in Excelsis Deo from Missa Prolationum, bars 1-14
96.
Des Prez, Agnus Dei II from Missa l’homme arme super voces musicales,
129
bars 1-12
130
97.
Subject in Bachs‟ Kunst der Fuge, BWV 1080
131
98.
Bach, Contrapunctus VI from Die Kunst der Fuge, BWV 1080, bars 1-5
132
99.
Bach, Contrapunctus VII a 4 per Augment et Diminu, from Die Kunst der Fuge,
BWV 1080, bars 1-12
100.
133
Bach, Contrapunctus VII a 4 per Augment et Diminu, from Die Kunst der Fuge,
BWV 1080, bars 21-28
134
2
101.
Greene, Allemande, bars 0 -1
135
102.
C.P.E. Bach, Allegro di Molto from Keyboard Sonata No. 6, bars 1-2
135
103.
Bach, Bourrée I from Cello Suite No. 3 in C major, BWV 1009
137
104.
Initiator and first three iterations of the Cantor set created with L-systems
138
105.
Cantor map of the first 16 bars of Bach‟s Bourrée I from the Cello Suite
No. 3 in C major, BWV 1009
106.
138
Analysis of the first movement from Beethoven‟s Piano Sonata No. 15 in
D major, Op. 28
140
107.
Beethoven, Allegro from Piano Sonata No. 15 in D major, Op. 28, bars 1-10
141
108.
Beethoven, Allegro from Piano Sonata No. 15 in D major, Op. 28, comparison
between bars 3-5 and 21-23
142
109.
Beethoven, Allegro from Piano Sonata No. 15 in D major, Op. 28, bars 21-28
143
110.
Beethoven, Allegro from Piano Sonata No. 15 in D major, Op. 28, bars 40-47
144
111.
Beethoven, Allegro from Piano Sonata No. 15 in D major, Op. 28, bars 201-212 144
112.
Beethoven, Andante from Piano Sonata No. 15 in D major, Op. 28, bars 1-2
113.
Beethoven, opening bars of Trio from Scherzo and Trio in Piano Sonata
145
No. 15 in D major, Op. 28
146
114.
Beethoven, Rondo from Piano Sonata No. 15 in D major, Op. 28, bars 1-4
146
115.
Comparison between the melodic contour of (a) Allegro first movement,
(b) Adagio second movement, (c) Trio from the third movement, and
(d) Rondo fourth movement from Beethoven‟s Piano Sonata No. 15 in
D major, Op. 28
147
116.
Beethoven, Scherzo and Trio from Piano Sonata No. 15 in D major, Op. 28
149
117.
Rhythmic structure of Fratres für violine und klavier
151
118.
Pärt, Fratres für violine und klavier, first page
152
xiv
119.
Pärt, Fratres für violine und klavier, reduced copy of bars 1-6
120.
Rotating wheel used for the melodic voices in Pärt‟s Fratres für violine und
klavier
121.
153
154
Rotating wheel used for the tintinnabuli voice in Pärt‟s Fratres für violine und
klavier
154
122.
Pärt, Fratres für violine und klavier, resulting chord sequence in bar 3
155
123.
Pärt, Fratres für violine und klavier, resulting chord sequence in bar 6
155
124.
Pärt, seventh section from Fratres für violine und klavier
157
125.
Pärt, eighth section from Fratres für violine und klavier
158
1
CHAPTER 1: INTRODUCTION
1.1
Background to the study
1.1.1
Music and mathematics through the ages
The integration of music and mathematics is certainly not a new phenomenon – it has existed for
more than two millennia. One of the earliest examples was the discovery that interval ratios are
directly related to the length of an instrument‟s string (frequency). This was determined by the
Greek philosopher and mathematician, Pythagoras (c. 500 BC). Medieval universities taught
what was called the quadrivium, which consisted of four parts: arithmetic, geometry, astronomy
and music (Barbera 2001:642).
One of the first examples of algorithmic composition (i.e. music composed with the use of an
algorithm or formula) dates back to the Medieval Benedictine monk, Guido d‟Arezzo, with his
music treatise entitled Micrologus de disciplina artis musicae (c.1026). His composition method
consisted of assigning a note to each vowel of a text. In addition, it is thanks to D‟Arezzo that we
have music notation on staves as well as solfege (Tufró 2009).
In the middle of the 17th century, the Jesuit polymath, Athanasius Kircher, invented two
“machines” (see Figure 1) that enabled non-musicians to compose four-part hymns. The first,
Arca Musaritmica (music-making ark), was a wooden box containing wooden strips with melodic
and rhythmic sequences on them. Equipped with a set of rules, the “composer” could pull strips
from the box to create polyphonic compositions. A more elaborate version of the machine came
in the form of the Organum Mathematicum, which, in addition to composing music, could also aid
in calculating problems relating to arithmetic, geometry and planetary movement. Bumgardner
(n.d.:2) referred to the Organum Mathematicum as the 17th-century equivalent of today‟s laptop
(Bumgardner n.d.:1-2; Loughridge 2013).
2
Figure 1: Kircher’s two composition “machines”: Arca Musaritmica (left) and Organum
Mathematicum (right)
In 1792, a year after Mozart‟s death, his Musikalisches Würfelspiegel was published. This
“Musical Dice Game” consisted of 272 bars of pre-composed music that could be put together in
different orders to create a minuet and trio. Two six-sided dice are rolled and, adding the
numbers together, the corresponding measure is looked up in a table. This is done for each bar
of the minuet. For the trio, a single dice is rolled and the same procedure followed. By completing
the process, the “composer” will have a minuet and trio of 16 bars each in the style of Mozart.
There are a total of 1116 (or 45,949,729,863,572,161) different possible permutations (Peterson
2001).
Composers began using mathematics as a composition technique or a basis for a certain
composition more frequently during the 20th century. Mathematical properties that were often
used included number sequences, mathematical formulae, geometric figures and statistics.
Twelve-tone music, led by Arnold Schönberg, Alban Berg and Anton Webern, can also be
regarded as mathematical because it relies on the permutations of a series.
The rhythms in Messiaen‟s compositions are often derived from prime numbers or other number
sequences. Xenakis used not only mathematics in his music, but also science and architecture,
3
and many of his compositions are governed by statistics and probability theory, specifically
stochastic theory (Tatlow & Griffiths 2001:231-235).
Twentieth-century musicologists started to do extensive analysis of works by Bach, Mozart and
Bartók, and found significant mathematical structures in some of their compositions. It is said that
Bach‟s music is mathematical, but it is not possible to prove whether or not Bach consciously
used mathematical structures in his compositions. The Fibonacci series and the golden ratio are
two of the most common mathematical properties in music, and feature extensively in the music
of Debussy and Bartók.
Developments in the sciences led to new theories and have since been incorporated into music
composition. Fractal geometry, like algebra, calculus and geometry, is a branch of mathematics
which can be applied to music. This research study explored how the new branch of fractal
geometry and its related self-similarity can be applied in the analysis and composition of music.
1.1.2
What are fractal geometry and self-similarity?
The first question that arises from the topic of the dissertation is: What are fractal geometry and
self-similarity? The term “fractal” is used to describe geometric objects that reveal new detail with
every magnification of that object. If the object is magnified at a specific point, a smaller replica of
its overall shape is seen – a property known as self-similarity (Brothers 2004).
4
As an example to illustrate this, the researcher chose the Julia set (Figure 2). The Julia set is not
merely a beautiful, colourful picture, but also contains many mathematical intricacies. Discovered
by the French mathematician Gaston Julia (1893-1978), this set, when visually represented as
below, contains various levels of self-similarity. Note how the blue spirals as well as the pinkpurple circular shapes occur at different places in the figure on different scales (in other words, in
different sizes). If one zooms into a part of the figure, it will present a copy of the entire object
(Chen 2004).
Figure 2: Julia set (Chen 2004)
5
The term “fractal” was coined in 1975 by the French mathematician Benoit B. Mandelbrot (19242010). In his book, Fractals: form, chance, and dimension, Mandelbrot (1977:4) explained the
etymology of the term as follows:
Fractal comes from the Latin adjective fractus, which has the same root as fraction
and fragment and means “irregular or fragmented;” it is related to frangere which
means “to break”.
The term thus accurately describes the irregular, fragmented yet self-similar nature of fractal
objects.
Towards the end of the 19th century, mathematicians became increasingly aware of nonEuclidean shapes (i.e. shapes that were not perfect circles or squares), but were still not able to
mathematically define all of them. In addition, there was no specific term under which these
shapes could be categorised.
It is necessary to stress that fractals were not identified in 1975, but that a term was finally
conceived to describe specific non-Euclidean shapes with a self-similar structure. Many other
mathematicians had identified irregular shapes prior to Mandelbrot‟s coinage of the term.
Examples are Koch, Cantor, Peano and Sierpinski, to name a few (Mandelbrot 1977:4). The
contributions of these mathematicians as well as the concepts relating to fractal geometry and
self-similarity are discussed in greater detail in Chapter 2. Nevertheless, the researcher used the
year 1975 as a date to separate the conscious use of fractals in art and music from an intuitive
utilisation.
1.1.3
Fractal geometry and self-similarity in music
In the first part of the background earlier in this chapter, it was briefly illustrated how mathematics
has impacted on music composition for hundreds of years. It can thus be accurately assumed
that a new branch of mathematics would result in an altered view of music analysis and new
techniques for music composition.
To analyse a composition by using fractal geometry, one must consider the specific composer‟s
intention with the music. It is highly unlikely that composers such as Bach, Mozart and Beethoven
6
consciously used fractals in their music, since fractals were only defined and understood
centuries later. It is debatable whether or not it is in fact valid to analyse music in this fashion.
Music examples were analysed and discussed in this dissertation to test whether or not fractal
structures exist in some classical compositions. The validity of finding such structures in music
compositions prior to the 20th century was also investigated.
A number of 20th-century composers have intentionally incorporated algorithms or ideas from
fractal geometry in their music compositions. Examples include György Ligeti, Charles Dodge,
Tom Johnson and Gary Lee Nelson. Some of their compositions were built only on the idea of
fractals and chaos theory, while others contain intricate fractal structures. Some works of these
composers and their composition methods are discussed in Chapter 4.
Fractal geometry is still a new concept when compared to other branches of mathematics.
Because mathematical concepts have been used to analyse and compose music for many years,
it is only natural for a new mathematical concept, such as fractal geometry, to be applied to
music in a similar fashion. It was not the researcher‟s intent to define a new method for music
analysis, but rather to investigate a new manner of thinking about music analysis.
1.2
Problem statement
Despite the research that has been conducted on fractal music, the following problems can still
be identified:

Fractal geometry and its different applications to music are not well-known subjects
among musicians or mathematicians. This afforded the researcher the opportunity to
create a better awareness of the topic among individuals in these fields, as well as the
connection between music and fractals.

Preliminary research has shown that there is still some ambiguity to the term “fractal
music”.

Most writings on fractal and self-similar music focus on a single subject, but there are few
sources that give a concise summary of all the developments in fractal music.

There are different types of fractal and self-similar music, but they are not easily
distinguished from one another.
7

Music examples of fractal and self-similar music and their analyses are scarce.

The literature dealing with fractal and self-similar concepts is predominantly explained
from a mathematical or scientific perspective, limiting its understanding for musicians
unfamiliar with mathematical terminology and formulae.

Not all findings from existing research are accurate, which opened up the field for further
research.
It can be concluded that the novelty of fractal geometry inhibits both musicians‟ and
mathematicians‟ understanding of the possible applications of fractal geometry and self-similarity
to music.
1.3
Motivation for the study
The researcher has always been interested in the relationship between music and mathematics,
and how music can be composed or analysed mathematically. Being raised in a family in which
there has always been a fine balance between science and art, the researcher found it easy to
approach an interdisciplinary field such as fractal music. Her father, a medical doctor and
freelance painter, and her mother, a private mathematics tutor and amateur pianist, instilled a
love for both the sciences and the arts from an early age.
The researcher started reading extensively on the applications of various mathematical concepts
to music composition in high school and has been fascinated by it ever since. As an
undergraduate BMus student at the University of Pretoria, she also enrolled for courses in
Algebra and Calculus.
For her BMus essay in 2008, the researcher studied some of the compositions by the Greek
composer, Iannis Xenakis (1922–2001), who incorporated elements of mathematics, science and
architecture in his music. The best example of this is his orchestral work, Metastaseis (1953–
1954), which has the same contour as the Philips Pavilion that Xenakis designed for the Brussels
fair. Making use of graphic notation, Xenakis composed the initial draft of Metastaseis on the
same graph paper used by architects for their designs. The sweeping contour of the music is
similar to that of the pavilion (Matossian 1986:61; Xenakis 1992:10).
8
While researching Xenakis‟s composition methods, the researcher came across information on
fractal music, a topic previously unknown to her. Although Xenakis did not specifically compose
fractal music, many of his composition methods were linked to similar sub-sections in
mathematics and physics such as white noise1 and Brownian motion2. (These noise forms are
discussed in detail in Chapter 2.)
This dissertation is aimed at individuals who, like the researcher, share an interest in both music
and mathematics and would like to learn more about the utilisation of fractal and self-similar
geometry in music. There are numerous compositions that rely on mathematical and scientific
principles and an increasing number of books and journals are being published on this topic. The
researcher‟s preliminary research has shown that there is a general lack of familiarity with fractal
and self-similar music. There are many misconceptions about fractal and self-similar music: How
can fractals be used to compose music? And can fractal or self-similar structures be found in
examples of Western art music through analysis? The researcher‟s aim was to answer such
questions in the dissertation.
1.4
Research question
These questions led directly to the research question of the study, namely How can fractal
geometry and self-similarity be further explained and applied to the composition and analysis of
art music?
The main question led to a number of sub-questions that are discussed in each chapter of the
dissertation:
Chapter 2:
THE HISTORY AND MATHEMATICS OF FRACTAL GEOMETRY AND SELFSIMILARITY
1

How are fractal geometry and self-similarity defined in mathematics?

Do examples of fractals exist in nature and art?
White noise refers to a noise type which is highly uncorrelated and unpredictable, such as the static on a
television.
2
Brownian motion, also called brown noise, is the opposite of white noise as it is correlated and
predictable.
9

Who were the pioneers of the mathematics behind fractal geometry and self-similarity?

What are the different types of fractals that can possibly be used in music composition
and analysis?
Chapter 3:

LITERATURE REVIEW
Who were the predominant researchers in the field of fractal and self-similar music and
how did their approaches to the subject matter differ?

Are these researchers‟ findings still valid today?

What are the different methods used to find fractal or self-similar features in a
composition?

What is the validity of fractals and self-similarity in music?
Chapter 4:

FRACTALS AS A COMPOSITION TOOL
What properties of fractal geometry and self-similarity can be applied to music
composition?

What methods can be used to compose fractal music?

How do different types of fractal music differ?
Chapter 5:
FRACTAL GEOMETRY AND SELF-SIMILARITY IN MUSIC ANALYSIS: MUSIC
PRIOR TO 1975

How can art music be analysed by using fractal geometry?

What is the significance of analysing a composition with fractals?

If fractal structures can be found in music prior to 1975, how can they be explained?
Chapter 6:
CONCLUSION AND SUGGESTIONS FOR FURTHER STUDY
10
1.5
Objectives of the study
From the literature review and the references, it will be clear that much research has already
been conducted on fractal music. Nevertheless, it is hoped that the researcher‟s extended
research will lead to new insights in order to better understand fractal music and create a greater
awareness among musicians and mathematicians of its applications.
The primary objective of the study was to define fractal and self-similar music in order to
compose and analyse such works. Hence the aims of the study are to

establish a better understanding of fractal and self-similar music by summarising the
most important research conducted thus far in an extended literature review

simplify difficult mathematical concepts that are essential in understanding fractals and
self-similarity, but described in a too complex manner in many existing sources

explain the properties of fractal and self-similar music

give examples of how fractal music can be composed

discuss the occurrence of fractal and self-similar properties in selected compositions from
different style periods in Western art music.
1.6
Delimitation of the study
It was assumed that the readers of the study have an understanding of music terminology.
Therefore, the basic musical terms used in the dissertation are not defined in detail unless
absolutely necessary. Similarly, the basic concepts of arithmetic, algebra and geometry used in
mathematics are not defined. However, such concepts are briefly explained in the context of the
dissertation when referred to.
By contrast, the mathematical properties of fractal geometry and self-similarity are defined and
explained in more detail. These definitions and explanations were deliberately simplified from
their complex mathematical definitions and were restricted to the fundamentals – in order to
understand how they can be applied to music.
11
Although there are algebraic applications of fractals, the study only deals with the geometry of
fractals and its subsequent applications to music. Recommendations for further reading on more
scientific applications of fractals are included in the study.
Discussions of fractal compositions from the 20th century were included, but many times without
examples of the scores. This was largely due to the unavailability scores because of copyright. In
other cases, such as the music of Charles Dodge, the music was composed electronically and no
music score could be found.
There are some harmonic and structural analyses of compositions displaying fractal or selfsimilar properties in the dissertation. These discussions focus mainly on fractals in the form
structure and rhythms of the compositions. Owing to the scope of the dissertation, the
compositions that are discussed in the dissertation are limited. Selected works from different time
periods are discussed. Works are divided into two groups: music composed prior to Mandelbrot‟s
coinage of the term “fractal” in 1975 and subsequent works. Music examples before 1975 include
mainly works from the Renaissance, Baroque and Classical eras.
1.7
Research design
Different research designs were used for the study, namely interdisciplinary research, an
extended literature review and analysis of compositions (Mouton 2006:179-180; Hofstee
2006:121-122). These designs were chosen because the researcher deemed them to be the best
way to draw a meaningful conclusion to the research question.
The study of fractal music can already be classified as an interdisciplinary field, because the topic
relies on the application of a specific mathematical concept (in this case, fractal geometry and
self-similarity) to music. This also justifies the use of many non-musical sources for the study.
The purpose of conducting an extended literature review was to summarise as much of the
existing research, in order to establish the current understanding of the topic. In so doing, it was
possible to draw new conclusions on the subject matter. Lastly, small-scale harmonic and
structural analysis of compositions and the discussion thereof show the practical applications of
fractal geometry to music.
12
Figure 3 outlines the research design for the dissertation.
Fractal music
music
Applications of
fractal geometry to
music
The mathematics
of fractal geometry
Extended
literature review
Discussion of
selected music
compositions
Compositions
before 1975
Compositions
since 1975
Figure 3: Outline of the research design
1.8
Research methodology
The data and information necessary for the research were collected from a vast range of books,
journals and internet resources. Similar data and information from different sources were
compared to ensure their validity and reliability. Where necessary, dictionaries were used to find
the apt definitions or explanations of unfamiliar mathematical and scientific concepts.
Since the study is interdisciplinary, scientific books and journals dealing with fractals were
scanned for the word “music”. Likewise, books on music composition and analysis were skimmed
for terms like “fractal”, “self-similarity” and “iteration”3. Internet and library searches were also
conducted with a variety of combinations of these words.
3
Iteration relies on the repetition of a formula or procedure.
13
In this research, a great part of research methodology relied on unifying the findings in existing
literature with their associated mathematical formulae and music examples.
1.9
Summary of chapters
Chapter 2 starts with a discussion of the similarities between music and geometry in general.
Thereafter, the mathematical properties of fractals and self-similarity that support fractal music
are discussed in detail. It also provides a historical overview of the mathematical developments
that led to the classification of fractal geometry as a mathematical discipline. Specific reference is
made to the pioneers in the field of fractal geometry, such as Koch, Cantor, Sierpinski, Peano
and Mandelbrot. This serves as background to understanding how such fractals can be applied
to the composition and analysis of music. Fractal noise forms, such as white, brown and 1/f
noise4, are also be defined because of their close relationship with fractal music.
Chapter 3 provides an extended literature review of the most significant research on fractal and
self-similar music since Mandelbrot‟s coinage of the term in 1975. The first subsection focuses on
the distribution of 1/f noise in music. The researcher also explains how such distributions may be
used to distinguish between different genres of music or even the stylistic differences between
the works of different composers. Furthermore, the prerequisites for fractality are studied, so that
the concept can be applied in subsequent chapters. Finally, the validity for searching for fractal
and self-similar structures in music prior to the 20th century is investigated.
Chapter 4 will describe some of the most common methods for composing fractal music, such as
dice and spinners, 1/f noise and L-systems5. Music examples from research papers as well as a
number of original works by contemporary composers such as Tom Johnson, Gary Lee Nelson
and Charles Dodge are included. It is also explored how fractals were used metaphorically in
some compositions by György Ligeti.
The main focus of Chapter 5 is to see if there are any compositions from previous centuries that
contain elements of fractals or self-similarity. These include compositions from the Renaissance
4
1/f noise (pronounced one-over-f noise) lies between the two extremes of white and brown noise.
L-systems or Lindenmayer systems are methods of rewriting a string of symbols to create increasingly
longer chains.
5
14
to the 21st century. Music examples are included, analysed briefly and discussed with reference
to different fractal elements.
Chapter 6 contains a summary of the entire dissertation. Conclusions and recommendations for
further reading on related topics are also included.
15
CHAPTER 2: THE HISTORY AND MATHEMATICS OF FRACTAL GEOMETRY AND SELFSIMILARITY
2.1
Introduction
Before any of the applications of self-similarity and fractal geometry to music can be investigated,
it is necessary to explain some of the mathematical terms and their properties in greater detail.
This chapter deals with the mathematical concept of fractals and their origins. It is also shown
that fractal shapes occur in many natural phenomena. Concepts such as fractal dimension,
scaling and transformation are defined so that they can be applied to music in subsequent
sections of the dissertation. Some of the most common fractals in mathematics are given as
examples, together with their pioneers.
2.2
Music and geometry
Music and geometry share many of the same terms and concepts which is important throughout
the course of this study. Musical themes or motifs, like geometric figures, can undergo certain
transformations in which the original is discernible but presented differently for variation.
Geometric transformations include scaling, translation, reflection, rotation and shearing. The
geometric and music examples provided below show how each of these geometric
transformations is applicable to music. The image of a treble clef was chosen by the researcher
to illustrate the various transformations a geometric object can undergo.
16
2.2.1
Transposition and translation
One of the simplest tools used by composers to bring variety into music is through transposition.
In the case of large-scale compositions, entire sections are often repeated in a different key.
Alternatively, shorter motifs or themes can be transposed to create sequences. The first 8 bars
from the first movement of Mozart‟s Piano Sonata No. 16 in C major, K 545 (Figure 4) is given as
an example. The C major scale in the right hand in bar 5 was transposed diatonically three times,
each a second lower creating sequences.
Figure 4: Mozart, First movement from Piano Sonata No. 16 in C major, K 545, bars 1-8,
illustrating transposition (Mozart 1938)
17
The geometric equivalent of transposition is known as translation. Translation occurs when all the
coordinates or points of an object are moved by a fixed distance in the same direction, up, down
or sideways. Figure 5 below shows the treble clef being moved to the right.
Figure 5: Geometric translation
2.2.2
Scaling
Scaling (also known as dilation in mathematics) is enlarging or reducing the size of an object
while its dimensions remain the same, as is depicted in Figure 6.
Figure 6: Geometric scaling
18
The musical equivalent of scaling is augmenting or diminishing note values. To illustrate this, an
excerpt from Berlioz‟s Symphonie Fantastique is given below. The fifth movement, entitled Songe
d’une Nuit du Sabbat, changes frequently in tempo, time signature and character. Bar 127 marks
the beginning of the Dies Irae. The bassoons carry the melody in dotted minims. In the second
half of bar 147, the horns take over the melody (in thirds), at double the speed in dotted
crotchets. A further reduction appears from the last beat of bar 156, where the woodwinds bring
out a modified form the melody. Instead of presenting the melody in quavers only (which would
have been four times the speed of the original melody), it is stated in alternating quaver and
crotchet beats for further rhythmic variation (Caltabiano n.d.).
Figure 7: Berlioz, Symphonie Fantastique, Fifth movement (Songe d’une Nuit du Sabbat),
bars 127-162, illustrating the diminution of note values (Adapted from Berlioz 1900)
This shows how the diminution of note values can be used to vary thematic material. The same
can also be done by augmenting the note values.
19
2.2.3
Reflection
Melodic inversion and retrograde are the same as the reflection of an object around the x-axis
and y-axis respectively. This is thus an object‟s mirror image.
Figure 8 shows the right hand from the first bar of Bach‟s Two-Part Invention No. 14 in B major,
BWV 785. This melody consists of a head (marked (a)) and a tail, which is the melodic inversion
of the head.
Figure 8: Bach, Two-Part Invention No. 14 in B major, BWV 785, right hand, bar 1,
illustrating melodic inversion (Adapted from Bach n.d.)
In geometry, inversion can be likened to horizontal mirroring or reflection around the x-axis:
Figure 9: Reflection around the x-axis creating a horizontal mirror
20
A mirror image of an object can also be obtained through reflection around the y-axis or vertical
mirroring:
Figure 10: Reflection around the y-axis creating a vertical mirror
The musical equivalent of vertical mirroring is retrograde, where material is presented
backwards. Figure 11 shows the first 20 bars from Haydn‟s Menuetto al Rovescio from his Piano
Sonata in A major, Hob. XVI:26. Bars 11-20 is an exact retrograde of the first ten bars, thus
creating a perfect vertical mirror and palindrome.
Figure 11: Haydn, Menuetto al Rovescio from Piano Sonata in A major, Hob. XVI:26, bars
1-20, illustrating retrograde (Haydn 1937)
21
2.2.4
Rotation and retrograde inversion
In music, retrograde inversion occurs when a motif or theme is repeated with both a horizontal
and vertical mirror, i.e. retrograde inversion. Hindemith‟s Ludus Tonalis is an example of this. The
Postludium that ends the piece is an exact retrograde of the Praeludium that was heard in the
beginning, except for the added C major chord in the last bar of the composition. Figure 12
shows (a) the opening and (b) closing measures of Ludus Tonalis.
(a.) Opening measures from Hindemith’s Ludus Tonalis
(b.) Last three bars from Hindemith’s Ludus Tonalis
Figure 12: Hindemith, Ludus Tonalis, first and last bars, illustrating retrograde inversion
(Hindemith 1943)
22
Retrograde inversion is similar to spinning an object around a fixed point, known in mathematics
as rotation.
Figure 13: Rotation around a fixed point
If two or more of the aforementioned transformation techniques are combined simultaneously,
this is called shearing in mathematics. Figure 14 shows an excerpt from Bach‟s Contrpunctus VIII
in which the theme is repeated in the top voice in melodic inversion with the note values doubled.
The alto enters in bar 3 also with the theme inverted, but in the same note values as the original
theme. This example illustrates the simultaneous utilisation of melodic inversion and rhythmic
augmentation.
Figure 14: Bach, Contrapunctus VIII from Die Kunst der Fuge, bars 1-4, illustrating
simultaneous use of two transformations, inversion and augmentation (Adapted from
Bach n.d.)
23
From the above transformations it is clear how closely linked geometric construction and music
composition are, and this establishes a connection between the two different fields. Scaling is of
particular significance in this dissertation because it is an important feature of fractals.
2.3
A short history of fractal geometry
Now that a direct connection between music and geometry has been established, the history and
evolution of fractal ideas can be discussed. The easiest way to describe and define fractals is by
looking at examples in nature and art.
2.3.1
Examples of fractals and self-similarity in nature and art
For centuries, Euclidean geometry used to be applied in order to define the physical structures of
objects. It is named after the Greek (Alexandrian) mathematician, Euclid (c.325 BC–c.265 BC),
who wrote The Elements, one of the first known treatises on geometry. He defined shapes such
as circles, triangles, rectangles and squares which formed the foundation for geometry as it is
known today (Hawking 2005:1,7). These theories have served as “a model of what pure
mathematics is about” (Clapham & Nicholson 2005:155). Any geometrical structures that did not
fit the characteristics of Euclidean geometry were simply regarded as “non- Euclidean”.
In the book, Art and physics: parallel visions in space, time and light, Leonard Shlain (1991:3031) pointed out that “Euclid made some … assumptions that he did not state in The Elements.
For example, he organized space as if its points could be connected by an imaginary web of
straight lines that in fact do not exist in nature.” It is exactly such images and objects that cannot
be defined by means of Euclid‟s geometry that form the core of this dissertation.
Nevertheless, examples of fractals and self-similarity have existed in nature around us for
millennia, even though scientists were not able to define them. Figure 15 shows a number of
fractals commonly found in nature: lightning, clouds, fern leaves and Romanesco broccoli.
Mandelbrot is famous for his statement: “Clouds are not spheres, mountains are not cones,
coastlines are not circles, and bark is not smooth, nor does lightning travel in a straight line.”
24
Lightning (Figure 15 (a)) indeed does not move in a straight line, but instead branches out
continuously, with each smaller branch resembling the larger bolt. Similarly, the shape of clouds
(Figure 15(b)) cannot be defined accurately with the use of Euclidean geometry, and selfsimilarity is evident. A fern leaf‟s structure (Figure 15(c)) is also self-similar because it comprises
many smaller leaves with the same shape and structure as itself. Finally, a Romanesco broccoli
(Figure 15(d)) is cone-shaped, but consists of smaller cones with the same shape and level of
irregularity (Frame, Mandelbrot & Neger 2014).
Figure 15: Natural fractals: (a) lightning; (b) clouds; (c) fern leaves; and (d) Romanesco
broccoli (Frame et.al 2014)
25
Intricate self-similar patterns appear to be commonly found in nature. This has inspired many
painters over the centuries. The Italian Renaissance polymath, Leonardo da Vinci, created
paintings with a blotting method that created self-similar cloud-like shapes. The Deluge (Figure
16) is one such example (Frame et.al 2014).
Figure 16: Da Vinci, The Deluge (Kuhn 2009:67)
Da Vinci also wrote about water and sound waves in a way that echoes self-similarity (Shlain
1991:76):
Just as a stone thrown into water becomes the centre and cause of various circles,
sound spreads in circles in the air. Thus everybody placed in the luminous air
spreads out in circles and fills the surrounding space with infinite likeness of itself and
appears all in all in every part.
The phrase in italics can possibly be replaced with “infinite self-similarity on all scales”. This
proves that Da Vinci and probably some of his contemporaries had noticed self-similar and fractal
structures in nature around them.
26
The Japanese artist, Katsushika Hokusai (1760-1849), made woodblock prints inspired by Mount
Fuji during the 1830s and 1840s. His print, In the Hollow of a Wave off the Coast of Kanagawa,
better known as The Great Wave (see Figure 17) presents “fractal aspects of nature with a
sophistication rarely matched, even today” (Frame et.al 2014, Kuhn 2009:67). Many of his prints,
such as this one, are characterised by waves with jagged edges that become increasingly
smaller towards the edges of the waves.
Figure 17: Hokusaki, The Great Wave (Frame et.al 2014)
In the late 19th and early 20th century, there was a greater awareness among artists and
scientists alike of so-called “non-Euclidean shapes and objects” - artists wanted to create them,
while scientists wanted to understand them.
One such example is the Dutch graphic artist, M.C. Escher (1898-1972). While he is best known
for drawings that exploit optical illusions, many of his works are characterised by utmost
precision, symmetry and self-similarity, in particular. His drawings have been analysed
mathematically by mathematicians such as H.S.M. Coxeter (1979). The title of an article by
Coxeter, “The Non-Euclidean Symmetry of Escher‟s Picture „Circle Limit III‟” hints at a
consciousness of the presence of self-similarity in the artist‟s work.
27
In a letter to Coxeter, Escher described his desire to create drawings in which a single image (or
motif) becomes “infinitely smaller”. At first he was not sure how to go about obtaining such
images, but with the help of Coxeter, Escher succeeded in creating a great number of such
drawings and sculptures (Coxeter 1979:19). One such example is Circle Limit I (Figure 18) in
which the same picture or “motif” is presented at different rotations and scales, becoming
increasingly smaller towards the edge of the circle.
Figure 18: Escher, Circle Limit I
28
The symmetry and self-similarity in Escher‟s drawings are fairly obvious. According to Escher, his
greatest inspiration was the music of J.S. Bach, since the composer also used a single theme or
motif and expanded on it by making use of repetition, reflection and other transformation
techniques. In a letter to the art historian E.H. Gombrich, in 1961, Escher compared his drawings
to the works of Bach (Schattschneider 1990:254):
It may be that the canon is near to my (anti)symmetric plane-filling mania. Bach
played with repetition, superposition, inversion, mirroring, acceleration and slowing
down of his themes in a way which is, in many regards, comparable with my
translation and glide-mirroring of my “themes” of recognizable figures. And that‟s
perhaps why I love his music particularly.
The 48 preludes and fugues from Bach‟s Das wohltemperiete Klavier provide many examples of
such transformations in music. Bach also used the same transformations that Escher used in his
art. Take, for instance, the first two bars from the Fugue No. 3 in C# major WTC II, BWV 872:
Figure 19: Bach, Fugue No. 3 in C# major, WTC II, BWV 872, bars 1–2 (Bach 1866)
The fugue starts in the bass with a six-note subject. Before the subject is completed, the soprano
follows with the answer (now in the dominant key), and shortly after, the alto with the subject in
inversion (back in the tonic key). It is thus a single melody that that was transformed by
transposition and inversion to create a variety of different melodic figures.
29
In his studies, Escher also found patterns in architecture that closely resembled some known
fractals. He found that the patterns on the pulpit of the 12th-century Ravello cathedral in Italy
(designed by Nicola di Bartolomeo) resembled the Sierpinski triangle which was only defined
mathematically in the early 1900s. Compare the two images below:
Figure 20: Patterns on the pulpit of the Ravello cathedral (left) resembling the Sierpienski
triangle (right) (Adapted from Peitgen, Jürgens & Saupe 1992:79)
30
Another example of fractals in art is Dali‟s painting, Visage of War (1940), in which each eye
socket and mouth of the faces contains another face with the same construction (Frame et.al
2014).
Figure 21: Dali, Visage of War
All of this is evidence of the existence of self-similar and fractal-like structures as well as
scientists‟ and artists‟ interest in them. Since fractal-like and self-similar shapes were present in
art and architecture before scientists‟ delineation of them, one can anticipate that such structures
can also be found in music. Such music examples are discussed and analysed in subsequent
chapters in this dissertation.
31
2.3.2
The era of “mathematical monsters”
Between 1875 and 1925, mathematicians turned their focus to shapes and objects that were too
irregular to be described by traditional Euclidean geometry. This era was classified as the era of
“mathematical monsters”. Since fractal geometry was only properly defined in 1975 by
Mandelbrot, these early fractals were merely known as self-similar or non-Euclidean shapes at
the time (Mandelbrot & Blumen 1989:4).
Some of the most important fractal structures are explained in the next section. The information
is given in chronological order and includes the dates of the founders, so that the reader can put
each fractal in its specific time frame.
32
2.3.2.1 Cantor set
One of the earliest examples of a fractal is the Cantor set, named after Georg Cantor (18851918). The Cantor set was first published in 1883 to illustrate a perfect, no-where dense subject.
In order to reconstruct the Cantor set, start with a line segment with a length of one unit. Divide
this line into three equal parts and remove the middle third. There are two smaller line segments
each with one-third the length of the original line segment (Figure 22). Continue to divide each of
these lines into thirds and remove their middle thirds. If this process is carried out infinitely, it
results in the Cantor set or Cantor comb, which is essentially only a set of points (Peitgen et al.
1992:67). The Cantor set or comb is regarded as fractal since each subsequent stage is a
smaller replica of its predecessor.
Figure 22: The first five stages of the Cantor set (Adapted from Peigen et al. 1992)
33
2.3.2.2 Koch snowflake
The Koch snowflake was named after the Swedish mathematician, Helge von Koch (1870-1924).
It is an example of a fractal curve that is nowhere smooth and fragmented at all scales.
To construct the Koch snowflake, one starts with an equilateral triangle6. Each of its three sides is
then divided into thirds of equal length. The middle third of each side is replaced with another
equilateral triangle and the base is removed. This process is repeated with each line segment.
The resulting structure is known as the Koch island or snowflake. If only part of the island is
considered, it is known as the Koch curve. The Koch snowflake has a unique property – although
its area can be determined, its length is infinite.
Figure 23: Steps for creating the Koch snowflake (Addison 1997:20)
6
When all three sides of a triangle are of equal length, it is called equilateral.
34
2.3.2.3 Sierpinski triangle and carpet
In the previous section, it was shown how decorations on the pulpit of the 11th-century Ravello
cathedral resemble the Sierpinski triangle. This self-similar shape was first introduced
mathematically in c. 1916 by the Polish mathematician Waclaw Sierpinski (1882-1969), after
whom it was named.
The Sierpinski triangle is constructed as follows: start with a black equilateral triangle. Find the
centre of each of the sides, connect the points to form four smaller triangles and eliminate the
middle triangle. The result is three smaller black triangles, similar to the initial one. When this
process is carried out continuously with each of the black triangles, it results in what is referred to
as the Sierpinski gasket or triangle (Peitgen et al. 1992:78).
Figure 24: Steps in creating the Sierpinski gasket or triangle (Adapted from Peitgen et al.
1992:79)
The Sierpinski carpet is a variation of the triangle, where the same iterative process is carried out
with squares instead of triangles (Peitgen et al. 1992:81).
Figure 25: Steps in creating the Sierpinski carpet (Adapted from Peitgen et al. 1992:81)
35
2.3.2.4 Lindenmayer systems
Lindenmayer systems, also known as L-systems, are a form of string rewriting systems named
after the Hungarian biologist, Aristid Lindenmayer (1925-1989). The method was first introduced
in 1968 to describe “natural growth processes” in plants. It relies on substituting chains of
symbols repeatedly in order to create longer, self-similar chains. (A chain or string in this context
refers to a sequence of symbols.) L-systems had a huge impact in the field known as
morphogenesis, which is concerned with the “development of structural features and patterns in
organisms” (Peitgen et al. 2004:330-331; Prusinkiewicz 1986:445; Jennings 2011).
All L-systems consist of the following three essential elements: an alphabet lists the symbols that
will be used in creating a string; an axiom gives the starting point or “seed” for the algorithm; and
a set of productions provides the rules employed to produce more strings. The set of productions
can be repeatedly applied to each subsequent string in order to create longer strings. According
to Jennings (2011), “the languages of most L-systems […] contain an infinite number or strings.”
An example would be to examine the L-system growth pattern of algae. The alphabet consists of
only two symbols, A and B. The starting point (or axiom) is B and contains two substitution rules:
A → AB and B → A. This means that in each subsequent stage, A is replaced by AB, while B is
replaced by A (Manousakis 2006:22-23; Jennings 2011.)
Figure 26 summarises the different elements necessary for the algae L-system growth pattern
and shows the different stages of reproduction. It is easy to see how the string rapidly increases
in length and that the strings display self-similarity.
36
Alphabet:
A, B
Axiom:
B
Rules:
A → AB
B→A
n=0: B
n=1: A
n=2: AB
n=3: ABA
n=4: ABAAB
n=5: ABAABABA
n=6: ABAABABAABAAB
n=7: ABAABABAABAABABAABABA
Figure 26: Lindenmayer system growth pattern for algae (Adapted from Manousakis
2006:22-23)
The L-system for the growth pattern of algae possesses another interesting quality. If the number
of symbols in each chain is counted, it gives the following number sequence: 1, 1, 2, 3, 5, 8, 13,
21. This series of numbers is known as the Fibonacci sequence, which, like fractals, occurs in
many natural phenomena. Consequently, this particular L-system is also known as a “Fibostring”
(Jennings 2011).
Lindenmayer systems seem abstract when presented as in Figure 26, but can easily be
interpreted graphically through a process known as “turtle graphics”.
2.3.2.5 Turtle interpretation of L-systems
According to Mason and Saffle (1994:32), the term “turtle graphics” comes from Logo ®, a
language developed at Massachusetts Institute of Technology (MIT). The imaginary turtle can be
thought of as a pen that can be turned on and off and can carry out several instructions to create
an image.
37
As an example, the turtle can follow five simple steps in order to create a square (Jennings
2011):
1. Touch the pen to the paper.
2. Move forward 1 step, turn left 90°.
3. Move forward 1 step, turn left 90°.
4. Move forward 1 step, turn left 90°.
5. Move forward 1 step.
This is a simple example to illustrate how the turtle works, but it can perform a vast range of more
complex instructions resulting in equally intricate images. The four basic commands that the
turtle can perform are as follows (Peitgen et al. 2004:351):
F
Move forward by a certain fixed step length and draw a line from the old to the new
position.
f
Move forward by a certain fixed step length, but do not draw a line.
+
Turn left (counter-clockwise) by a fixed angle.
-
Turn right (clockwise) by a fixed angle.
When instructing the Logo® turtle to act on such instructions based on L-systems, there is a vast
array of self-similar images that can result from this. Figures 27 and 28 show the L-system and
turtle interpretation of the Hilbert curve. The axiom and set of rules contain the symbols X and Y
to make substitution possible. For each iteration, a simplified version without these signs is given,
since the turtle only carries out symbols +, -, F and f.
38
Figure 27: Graphic representation for the first three iterations of the Hilbert curve, created
with turtle graphics
Axiom/seed:
-X
Production rules:
X → -YF+XFX+FYY → +XF-YFY-FX+
n=0 :
-X
n=1:
--YF+XFX+FY-
(Simplified:
--F+F+F-)
n=2:
--+XF-YFY-FX+F+-YF+XFX+FY-F-YF+XFX+FY-+F+XF-YFY-FX+-
(Simplified:
--+F-F-F+F+-F+F+F-F-F+F+F-+F+F-F-F+-)
n=3:
--+-YF+XFX+FY-F-+XF-YFY-FX+F+XF-YFY-FX+-F-YF+XFX+FY-+F++XF-YFY-FX+F+-YF+XFX+FY-F-YF+XFX+FY-+F+XF-YFY-FX+-F-+XFYFY-FX+F+-YF+XFX+FY-F-YF+XFX+FY-+F+XF-YFY-FX+-+F+-YFXFX+FY-F--+XF-YFY-FX+YF+XF-YFY-FX+-F-YF+XFX+FY-+-
(Simplified:
--F+F+F-FF-F-F+F+F-F-FF-F+F+FF+F-F-F+FF+F+F-F-F+F+FF+F-F-FFFF-F+FF+F+F-F-F+F+FF+F-F-F+FF-F+F-F-F-F-F+F+F-F-FF-F+F+F-)
Figure 28: First three iterations of the L-system of the Hilbert curve
39
2.4
Definitions relating to fractals and self-similarity
Now that the reader is more familiar with what fractal objects look like, it will be easier to define
many of the terms relating to fractal geometry and self-similarity.
2.4.1
Self-similarity
As mentioned in the introductory chapter, self-similarity is one of the most important properties of
fractal geometry, and is mentioned throughout this dissertation. Self-similarity can be easily
explained with the use of the following illustration of a fern leaf:
Figure 29: Self-similarity of a fern leaf
40
Consider the overall structure of a fern leaf: each of the smaller leaves of the fern is a replica of
the overall shape but on a smaller scale. In theory, this process of magnification can be carried
out infinitely, but with the example of objects in nature, a limit is reached after a certain number of
magnifications. This is logical if one considers that any physical object will decompose into
molecules and atoms after a while, which are not necessarily self-similar or fractal in structure
(Peitgen et al. 2004:138).
Scientists distinguish between two main types of self-similarity, namely statistical and exact selfsimilarity. The iterations of fractal objects in nature will always differ slightly from the large object,
although they are still self-similar (Solomon 2002). This is known as statistical self-similarity.
Etlinger (n.d.:1) also referred to this as a discrete spectrum. The fern leaf in Figure 29 is thus an
example of statistical self-similarity.
Mathematically speaking, it is possible to have an object that is self-similar on an infinite number
of scales without any differences. This is known as exact or strict self-similarity (Solomon 2002).
The best example of exact self-similarity is found in the construction of the Koch curve7. This
shape can be accurately drawn with the use of computers, without any approximations or slight
changes resulting in exact self-similarity.
Figure 30: Self-similarity of the Koch curve
7
Note that the object here is referred to as the Koch curve and not the Koch island or snowflake, as in the
previous section. This is because only a part of the shape is presented and it is not closed.
41
A term closely linked, but often confused with self-similarity, is self-affinity. If an object‟s
reductions are “distorted or skewed” in any way, the object is called self-affine instead of selfsimilar. In self-affine shapes, the “reductions are still linear but the reduction ratios in different
directions are different. For example, a relief is nearly self-affine, in the sense that to go from a
large piece to a small piece one must contract the horizontal and vertical coordinates in different
ratios” (Mandelbrot & Blumen 1989:4, Peitgen et al. 2004:138).
It is also possible for self-similarity to occur only at one specific point (Peitgen et al. 2004:138),
say, in an onion. An onion is not self-similar, but it does consist of concentric, self-similar rings.
Onions are thus self-similar only around their most central part. Another example of this would be
Russian dolls that fit into one another. Brothers (2004) referred to this as limited self-similarity.
Objects or images that display limited self-similarity are not true fractals.
2.4.2
Fractal dimension
From elementary geometry it is general knowledge that a line has a dimension of 1, a square a
dimension of 2 and a cube a dimension of 3. This next section will describe how one can
calculate the dimension of a fractal. This is important because most fractals have dimensions that
are not integers8.
The following is a simple formula to calculate the dimension of any shape:
k =nd ,
where d is the dimension;
is the size of a smaller section of the whole; and k is the number of
copies of n needed to recreate the original shape. Take, for example a cube (Figure 31), and
divide it into 27 smaller cubes, which is rd the size of the original cube. Nine of these smaller
cubes are thus needed to recreate the original cube. Therefore k = 9 and n = 3.
27 = 3d
d=3
The dimension of a cube is therefore 3.
8
Integers, also called whole numbers, refer to the set {...-3,-2,-1,0,1,2,3...}. These numbers do not contain
decimals or fractions.
42
Figure 31: Dimension of a cube (Adapted from Barcellos 1984:99)
The same formula can also be applied to calculate the dimension of a fractal shape, like the Koch
curve. In the first iteration of the Koch curve (Figure 32) there are four smaller line segments,
each rd the length of the initial line segment. Hence k = 4 and n = 3.
Therefore 4 = 3d
d=
= 1.26
The dimension is not an integer and lies between 1 and 2 – one of the characteristics of fractals.
This seems somewhat abstract at first, but a shape‟s fractal dimension is useful to
mathematicians because it shows the degree of fragmentation of the shape. The higher the
fractal dimension, the more fragmented the outline of the fractal will be.
Figure 32: First iteration of the Koch curve
43
2.4.3
Fragmentation
This leads to another important property of fractals, that is, fragmentation on all scales. If one
considers a Euclidean shape like a circle, and continuously enlarges it at a specific point P, the
arc of the circle will tend to look increasingly more like a straight line. That means that it is
possible to draw a tangent at point P on the circle and, in fact, at any point on the circle (Figure
33). In mathematics, it is said that the circle is everywhere differentiable.
Figure 33: Tangent line drawn on a circle illustrating differentiability
The complete opposite is true of fractals. Take, for example, the Koch curve (Figure 30), which
consists entirely of small jagged corners. It is not possible to draw a tangent at a corner, since it
can lie at various angles. If one enlarges the snowflake at a specific point, the same amount of
detail is obtained. One can thus conclude that it is impossible to draw a tangent line and the Koch
curve is therefore nowhere differentiable. This substantiates the contention that fractals have
some type of fragmentation or irregularity, irrespective of the scale (Mandelbrot & Blumen
1989:3-4).
To summarise, fractals adhere to the following three important properties: self-similarity, a fractal
dimension that lies between 1 and 2, and fragmentation on all scales. These properties can be
used and adapted to describe fractal music.
44
2.5
Noise
It is not only shapes or images that can have fractal properties. The next section will focus on
fractal noise forms that had a huge impact on the fractal analysis of compositions.
In the early 1960s, Benoit B. Mandelbrot (1924-2010) was working at the Thomas J. Watson
Research Centre as a research scientist. He was approached by The International Business
Machines Corporation (IBM) to help solve a problem the company had encountered: the flow of
computer data that IBM had tried to transmit through telephone lines was disrupted by a type of
white noise, which caused the signal to break up.
Instead of using traditional analytical techniques, Mandelbrot examined the visual shapes that the
white noise had generated and made a startling discovery: the graph had an internal self-similar
structure on several different time scales. “Regardless of the scale of the graph, whether it
represented data over the course of one day or one hour or one second, the pattern of
disturbance was surprisingly similar” (Anonymous 1994). This marked the discovery of selfsimilar noise, a type of fractal that would be used abundantly in the composition and analysis of
music in the following years.
In the sciences, the term “noise” refers to an unwanted random addition to a signal, which can be
heard as acoustic noise when converted into sound, or can display itself as unwanted random
data in signal processing. There are three main groups in which noise is generally categorised,
namely white, brown and 1/f (pink) noise.
The graph of any type of noise can be converted to a logarithmic graph which presents itself as a
straight line at a specific angle. In so doing, it is much easier to distinguish different noise types
from one another. The graphs in Figures 34-36 display the time-frequency graph of a noise type
on the right and its analogous logarithmic graph on the left (Voss 1988:40).
45
White noise, such as the static on a radio or snow on a television, is completely uncorrelated.
This means that there is no relationship in the fluctuation of frequencies. White noise presents
itself logarithmically as a straight line parallel to the x-axis (Figure 34), since it is a flat type of
noise.
Figure 34: White or 1/f0noise (Voss 1988:40)
The opposite of white noise, namely brown9 or Brownian noise, is highly correlated. Its
logarithmic graph shows a steep negative slope as in Figure 35.
Figure 35: Brown or 1/f2 noise (Voss 1988:40)
9
Brown or Brownian noise was not name after the colour, but the scientist Robert Brown, who discovered
Brownian motion in c. 1827 (Frame et al. 2014)
46
Between these two extremes lies pink or 1/f noise10, which is neither as correlated as brown
noise nor as random as white noise. 1/f noise also displays a negative slope, but not as steep as
that of brown noise (Figure 36). From these graphs it is easy to see that a noise form with high
correlation has a steeper slope than that of a more random noise form (Voss & Clarke 1975:317).
Figure 36: Pink or 1/f noise (Voss 1988:40)
For the purposes of this study, 1/f noise is deemed important because it is a noise form that
displays self-similarity and can be regarded as fractal. Richard Voss and John Clarke as well as
Kenneth and Andrew Hsü conducted some of the most fundamental research on the relationship
between noise and music. Their research and findings are discussed in detail the literature
review.
2.6
Conclusion
The ways in which geometric shapes and musical motifs can be transformed are markedly similar
and suggest that musical motifs and themes may be altered in a way similar to geometric objects
in order to create self-similar or fractal music compositions. These transformations include
transposition (translation), melodic inversion (horizontal mirroring), retrograde (vertical mirroring),
retrograde inversion (rotation) and combinations of two or more of the aforementioned
transformations (shearing).
10
1/f noise is pronounced as “one-over-f noise”.
47
From the historical overview of fractals and self-similarity in nature, art and mathematics, it
becomes clear that we have been surrounded by and fascinated with fractals for centuries.
Thanks to mathematicians like Cantor, Koch, Sierpinski, Lindenmayer and Mandelbrot, we are
also able to better understand such strange shapes and objects. The existence of fractals in art
works created prior to the mathematical definition of fractal geometry, suggests that similar
patterns may be found in music compositions.
In a nutshell, an object must encompass several important features in order to be called fractal.
Firstly, a fractal must adhere to the property of scaling – in other words, it must have the ability to
be enlarged or shrunk. Secondly, the degree of the irregularity and/or fragmentation must be
irregular at all scales and thus also self-similar at all scales. Thirdly, all fractals have a fractal
dimension, which, in most circumstances, is not an integer and lies somewhere between 1 and 2
(Mandelbrot 1982:1).
There are three fractal noise wave forms, namely white, brown and 1/f (pink) noise. However,
only 1/f noise displays self-similarity similar to that of fractals. Its importance and possible
applications to music compositions will be investigated in the chapters to follow.
48
CHAPTER 3: LITERATURE REVIEW
3.1
Introduction
This chapter highlights the most significant research conducted to date on the topic of fractals
and self-similarity and their applications to music. Different scholars‟ published works are
discussed to illustrate how fractal geometry and music are related. The different approaches are
discussed chronologically, starting with a brief overview of sources concerned with fractal
geometry in general. Thereafter, different approaches are analysed to find a connection between
music and 1/f noise. This is followed by a discussion of studies by musicologists at Yale
University who worked closely with Mandelbrot in the past. In conclusion, the validity of
integrating fractal geometry and self-similarity with music is highlighted.
3.2
Fractal geometry in general
In order to fully understand the characteristics of fractal geometry and self-similarity, the
researcher read many books and articles dealing with the mathematics of the subject matter.
Because Mandelbrot is regarded as the so-called “father of fractals”, his works are considered
among of the most important. His book, Fractals: form, chance and dimension, published for the
first time in 1977, summarised all of his early research. In 1982, The fractal geometry of nature
was published, followed seven years later by an article entitled “Fractal geometry: what is it, and
what does it do?” in collaboration with Blumen.
In 1992, Peitgen, Jürgens and Saupe collaborated in a book entitled Chaos and fractals: new
frontiers of science. This is a source of a vast array of subjects connected to fractal geometry and
self-similarity, with explanations of the discoveries of pioneering mathematicians such as Koch,
Sierpinski and Cantor, as discussed in the previous chapter.
In 2012, two years after his death, Mandelbrot‟s memoir The fractalist: memoir of a scientific
maverick was published. The afterword, written by Michael Frame from IBM, contains much
valuable information on Mandelbrot‟s views regarding the use of fractals in music.
49
3.3
1/f noise distribution in music
The first link between music and fractal geometry was made in the late 1970s, when it was found
that the distribution of the some elements in music, such as melody or rhythm, resembled the
structure of 1/f noise. Over the years, many theories have been developed on this subject matter.
This subsection deals with the different approaches by Voss and Clarke in the 1970s, the Hsüs in
the 1990s and, most recently, Ro and Kwon in 2009.
3.3.1
Voss and Clarke
The physicist, Richard F. Voss, was one of the first researchers to study the correlation between
fractals and music and is regarded as “one of the pioneers of the modern study of fractals” (Crilly,
Earnshaw & Jones 1993:1). For his PhD research at the University of California in the 1970s,
Voss experimented specifically with the relationship between music and noise wave forms
(Gardner 1992:3). Together with his supervisor, John Clarke, Voss published their findings in
several journals. Their first article, “„1/f noise‟ in music and speech” was published in the journal,
Nature, in 1975, followed three years later by “„1/f noise‟ in music: music from 1/f noise”. The
main aim of their experiments and research was to detect whether or not music exhibited similar
fractal fluctuations as 1/f noise.
By the end of 1980, fractals had made their way into an article entitled “Science news of the year”
in Science News journal (Anonymous 1980:406). The specific article reviewed some of the
“important news stories” of that year. Among them, two entries relate to fractal geometry and also
music:

“It was found that music can be be analyzed [sic] as a structure with fractal dimensions.”

“Fractals were applied to the geometry of protein structure.”
The first refers to Thomsen‟s article, “Making music – fractally”, published in March 1980 in the
same journal. In the article, Thomsen restated much of the research done by Voss in 1975.
50
3.3.1.1 The discovery of 1/f noise distribution in music
Music can be described as sound that fluctuates in frequency (pitch) and amplitude (loudness or
volume) at different times. Voss and Clarke (1975, 1977) wanted to examine whether the
fluctuation of these elements in different types of music were the same as the fluctuation of 1/f
noise. Since 1/f noise is fractal, a piece of music mimicking 1/f noise could possibly also be
regarded as fractal.
In conducting their research, Voss and Clarke (1977:258) used electronic equipment to measure
the power spectrum or spectral density of different types of music. This showed what the
behaviour of a varying quantity (in this case an audio signal) is over time. Another element that
was considered was the autocorrelation function, which measured how the fluctuating quantities
were related at two different times.
As stated, Voss and Clarke‟s first article dates back to 1975 – the same year in which Mandelbrot
coined the term “fractal”. This article, entitled “„1/f noise‟ in music and speech”, does not contain
any direct reference to fractal geometry, but showed how the distribution of pitches in certain
types of classical music is similar to that of 1/f-noise (Voss & Clarke 1975).
In Voss and Clarke‟s measurements of music and speech, “the fluctuating quantity of interest
was converted to a voltage whose spectral density was measured by an interfaced PDP-11
computer using a Fast-Fourier Transform algorithm that simulates a bank of filters. The most
familiar fluctuating quantity associated with music is the audio signal …” In their first musical
experiment, the audio signal of Bach‟s first Brandenburg Concerto, BWV 1046 was analysed.
They analysed the concerto‟s sound spectrum, averaged over the entire work, but found no
correlations to 1/f noise. The log-log graph in Figure 37(a) shows how the concerto displayed
sharp peaks at high frequencies and was thus not similar to 1/f noise (Voss & Clarke 1975:317).
Despite the initial setback, they decided to test the fluctuations in loudness and pitch separately.
This proved to be a more useful technique, since these quantities fluctuate more slowly
(Thomsen 1980:190). Figure 37(b) shows the spectral density of the “instantaneous loudness” of
the Brandenburg Concerto. This was obtained by measuring the spectral density of the output
voltage after it had been squared, Sv2(f). It is clear that this graph follows the slope of 1/f noise
51
much closer than in the previous example (Voss & Clarke 1975:317). The peaks between 1 and
10 Hz were attributed to the rhythmic structure of the music (Voss & Clarke 1977:260).
Figure 37: Spectral density of Bach’s First Brandenburg Concerto (a) Sv(f) against f (b)
Sv2(f) against f (Voss & Clarke 1975:317)
Voss and Clarke continued with the same method of spectral density analysis to determine
whether the instantaneous loudness in other genres of music would exhibit graphs similar to the
one of the Brandenburg Concerto. The spectral density averaged over an entire recording of
Scott Joplin Piano Rags displayed a downward contour similar to 1/f noise, but still contained
52
many sharp peaks (Figure 38 (a)). They attributed this to the “pronounced rhythm” in the
compositions. In order to test even more types of music, the output from different radio stations
was also analysed and averaged over a period of 12 hours. The radio stations that were used,
included a classical station, a rock station and a news and talk station. The log-log graphs of
each of these can be compared in Figure 38 (Voss and Clarke 1975:318).
Figure 38: Spectral density of the instantaneous loudness compared to 1/f noise: (a) Scott
Joplin Piano Rags; (b) classical station; (c) rock station; (d) news and talk station
53
In order to test the power spectrum of pitch fluctuations, the audio signal of different radio
stations was also measured over a period of 12 hours. The classical station once again showed
the closest resemblance to 1/f noise, while the jazz and blues, rock, and news and talk deviated
from 1/f noise more (Figure 39) (Voss & Clarke 1975:318).
Figure 39: Power spectrum of pitch fluctuations compared to 1/f noise: (a) classical
station; (b) jazz and blues station; (c) rock station; (d) news and talk station
54
From these experiments on the instantaneous loudness of the music as well as the frequency
fluctuations, Voss and Clarke concluded that most melodies fluctuate in a similar manner as 1/f
noise, which in turn shows that they are fractal in nature. There was, however, an exception in
the works of 20th-century composers such as Stockhausen, Jolas and Carter. In their music, the
“melody fluctuations approach white noise at low frequencies”. Figure 40 clearly illustrates that
the spectral densities in the compositions by 20th-century composers deviate greatly from 1/f
noise (Voss 1989:367.)
Figure 40: Audio power fluctuation spectral densities, Sv2(f) versus f for (a) Davidovsky’s
Synchronism I, III and III; (b) Babbit’s String Quartet, No. 3; (c) Jolas’s Quartet, No. 3; (d)
Carter’s Piano Concerto in two movements; and (e) Stockhausen’s Momente (Voss &
Clarke 1975:261)
55
Voss and Clarke‟s research has proven that “almost all musical melodies mimic 1/f noise”. This
was further proven by their analysis of melodies from other cultures. Figure 41 illustrates the
spectral densities found in (a) the Ba-Benzele Pygmies11; (b) traditional music of Japan; (c)
classical ragas of India; (d) folk songs of old Russia; and (e) American blues.
Figure 41: Pitch fluctuations from different musical cultures: (a) the Ba-Benzele Pygmies;
(b) traditional music of Japan; (c) classical ragas of India; (d) folk songs of old Russia; (e)
American blues
11
The Ba-Benzele Pygmies are an indigenous nomadic people of Western Africa. Their music is characterised by
complex polyphony.
56
Figure 42 illustrates the pitch fluctuations in different genres of music over a period of nearly
seven centuries, starting with Medieval music, Beethoven‟s third symphony, Piano compositions
by Debussy, Richard Strauss‟ Ein Heldenleben and finally Sgt. Pepper by the Beatles.
Figure 42: Pitch fluctuations in Western music: (a) Medieval music up to 1300; (b)
Beethoven’s third symphony; (c) Debussy piano works; (d) Strauss’s Ein Heldenleben; (e)
the Beatles’ Sgt. Pepper
57
The downwards slope of the fluctuations seen in the figures above show that many different
genres of music mimic the behaviour of 1/f noise. The scientific community was greatly affected
by Voss and Clarke‟s discovery that music displayed fractal 1/f distributions. Their research made
its way to an entry in Electrical Engineering Times in 1979, entitled “IBM researcher discovers 1/f
distribution in music”. The author of that particular article, whose name was not printed,
summarised the findings as follows:
[Voss] found that the spectral distribution of pitch and volume variations of music
corresponded to the 1/f distribution commonly associated with electrical noise […]
The correlations between notes extended over the whole musical composition; every
note is correlated with every other note. A 1/f composition thus has a unity which,
although subtle, is perceived by listeners […] Voss analysed music from Bach to rock
and roll, and found they all conform closely to 1/f statistics.
3.3.1.2 Criticism of Voss and Clarke’s research
Although Voss and Clarke‟s published works were diligently researched, they failed to include
specific compositions (other than Bach‟s first Brandenburg Concerto), but only mentioned the
composers. In addition, no music examples were ever included to substantiate their research
findings. In the researcher‟s opinion, these should have been included in order to give musicians
a better frame of reference.
In 1992, the Australian musicologist, Nigel Nettheim, published an article in the Journal of New
Music Research as a criticism of Voss and Clarke‟s claim that “much music is well modelled by
1/f noise”. Nettheim (1992) critiqued the aforementioned researchers for not producing “enough
evidence […] for the reliability” of some methods they used in their research. He also believed
that too many assumptions were made in their work (Nettheim 1992:135).
In conclusion, Voss and Clarke‟s extensive research shows that most types of music display
some form of 1/f distribution, which is also a type of fractal. For their experiments, they utilised
music from various genres and time periods, but Classical music (excluding music from the 20th
century) always displayed the closest resemblance to 1/f noise. Numerous other scholars have
used Voss and Clarke‟s findings as the base for their own research and articles.
58
3.3.2
Kenneth and Andrew Hsü
Kenneth J. Hsü and his son, Andrew, have also done research on 1/f noise and music. At the
time of their study, the elder Hsü was an academic geologist at the Zurich Technische
Hochschule and his son a concert pianist at the Zurich Conservatorium (Crilly et al. 1993:2).
Three of their most important articles include “Fractal geometry of music” (1990); “Self-similarity
of the „1/f noise‟ called music” (1990); and “Fractal geometry of music: from bird songs to Bach”
(1993).
From the list of sources in their articles, it is evident that the Hsüs based much of their research
on the works of Voss et al. and Mandelbrot, but their methods and approaches were slightly
modified.
3.3.2.1 The fractal geometry of melody
The Hsüs had a different approach to testing for 1/f fluctuations in music compared to Voss and
Clarke. Where Voss and Clarke tested the pitch fluctuations in compositions, Kenneth and
Andrew Hsü believed that measuring how often a specific interval between successive notes
occurred in a composition would be more accurate (Hsü & Hsü 1990:938). One of the main
reasons for this, was their view that the intervals between successive notes “are the building
blocks of music, not the individual notes” (Hsü 1993:23).
The Hsüs adapted Mandelbrot‟s formula to determine the fractal dimension of an object or set so
that it could be applied to music. Mandelbrot‟s original formula for fractal shapes is as follows:
where
is the frequency;
the fractal dimension.
is the “intensity of events”;
is a constant of proportionality; and
is
59
The adapted formula by the Hsüs is
where
is the incidence frequency of notes and
is the interval between two successive notes.
The variable used in the place of is calculated easily in terms of semitones, where 0 represents
a repeated note, 1 a minor second, and so forth (Figure 43).
Repeated note
0
Minor second
1
Major second
2
Minor third
3
Major third
4
Perfect fourth
5
Tritone
6
Perfect fifth
7
Minor sixth
8
Major sixth
9
Minor seventh
10
Major seventh
11
Octave
12
Figure 43: Numerical values assigned to respective note intervals (Hsü & Hsü 1990:939)
One of the first compositions analysed by the Hsüs using this method was Bach‟s first Invention
in C major, BWV 772. The intervals between successive notes were measured separately for the
right and left hand. After notating their results in a table, they found that “a fractal relation is
established for
”; i.e. there is a fractal relation for the intervals between (and including)
a whole-tone and minor seventh. They also calculated that the fractal dimension is 2.4184, which
results in the following equation:
60
However, the Hsüs did not explain in their article how the constant (2.15) and the fractal
dimension (2.4184) were calculated, or why only a certain range of intervals (major second to
minor seventh) were included (Hsü & Hsü 1990:939). The researcher tried to duplicate the
application of the above formula, but was unable to achieve the same results as in the article.
This may suggest that the article did not supply sufficient information for other researchers to use
or duplicate.
The same method was applied to various other compositions (Hsü & Hsü 1990:940-941) such
as:

Bach: Invention No. 13 in A minor, BWV 784

Bach: Adagio movement from Toccata in F-sharp minor, BWV 910

Mozart: First movement from Sonata in F major, KV 533

Six Swiss children‟s songs

Mozart: First movement from Sonata in A major, KV 331

Stockhausen: Capricorn
The results of their experiments were then plotted on log-log graphs, like Voss and Clarke.
Although their methodologies were different, both groups of researchers came to the same
conclusion: the music of Stockhausen does not imitate 1/f noise in the same fashion as classical
music and folk tunes (Hsü & Hsü 1990:939-940).
Figure 44 shows the log-log graphs for the above-mentioned compositions. From the graphs, it is
clear that the Hsüs‟ research showed a 1/f distribution in all of the compositions analysed, except
for Stockhausen‟s Capricorn. They ultimately reached the conclusion that “the intervals between
successive acoustic frequencies [i.e. pitches] in classical music have a fractal distribution”.
61
Figure 44: Fractal geometry of sound frequency where
is the percentage frequency of
incidence of note interval : (A) Bach BWV 772; (B) Bach BWV 784; (C) Bach BWV 910; (D)
Mozart KV 533; (E) Six Swiss children’s songs; (F) Mozart KV 331; (G) Stockhausen
Capricorn
62
3.3.2.2 The fractal geometry of amplitude
The Hsüs also wanted to determine whether the “loudness of music” had a fractal distribution.
They criticised Voss and Clarke‟s method of using a recording and instead wanted to consider
the intentions of the composer, rather than the interpretation of a performer. Again, they used
note-counting methods, similar to what they had done in the case of melodies. For the amplitude
they measured how many notes were struck simultaneously as this would produce a louder
sound (Hsü & Hsü 1990:941).
Their method, however, is ambiguous. The following is a quotation from their article (Hsü & Hsü
1990:941):
One possibility for evaluating the amplitude is to analyse the number of notes that are
played simultaneously, because more notes sounding together should make the
sound louder. We again chose Bach‟s Toccata, because four melodies are played
simultaneously in that fugue. But we did not find a fractal distribution, which indicates
that this is not an effective way of evaluating loudness. Recognizing, then, that the
intensity of the sound is greatest when a note is first struck on a keyboard, we
analysed the number of notes that are struck simultaneously and found an apparent
fractal distribution of amplitude.
It is clear that the Hsüs hinted towards two different methods of calculating the possible fractal
distribution of loudness, but the researcher could not find any clear distinction between the two
methods in any of their published works. Despite this, the Hsüs concluded that amplitude (or
loudness) in classical music displayed a fractal distribution.
The Hsüs‟ note-counting method was also applied in literature: Ali Eftekhari (2006) from the
Electrochemical Research Centre in Iran, published an article entitled “Fractal geometry of texts:
an initial application to the works of Shakespeare”. He found that “by counting the number of
letters applied in a manuscript, it is possible to study the whole manuscript statistically” (Eftekhari
2006:1). Using a formula similar to the one employed by the Hsüs, Eftekhari was able to show
that the literary works of Shakespeare also have a fractal dimension (Eftekhari 2006:11).
Although some of the methods by the Hsüs appear somewhat strange, it is clear that their
research, in turn, also influenced other scholars.
63
3.3.2.3 Measurement and reduction of a musical composition
In an anecdote about a conversation between Mozart and Emperor Joseph II while discussing
Mozart‟s opera, Die Entführung aus dem Serail, it is said that the Emperor complained that the
music consisted of “too many notes”. Mozart‟s reply was simply that “there are as many notes as
there should be”. While agreeing that all notes in any composition are essential, the Hsüs
wondered if it would be possible to reduce the number of notes in such a way that the remaining
notes would still be reminiscent of the original composition.
In their first article, they had already proven that music exhibits fractal features such as selfsimilarity and scale independency. Taking this into consideration, they argued that it should be
possible to reduce a composition to fewer notes and still recognise the composition or, at least,
the composer (Hsü & Hsü 1991:3507).
To prove this, the Hsüs drew on a particular physics phenomenon known as the coastline
paradox. The British physicist, Lewis F. Richardson, discovered that the length of the common
frontier between Spain and Portugal differed from source to source by as much as 20% (Hsü &
Hsü 1991:3507). Further investigation showed that this was because of the use of different
lengths of “measuring sticks”: a coastline (or any border) will increase in length when smaller
measuring sticks are used. (This is similar to what was explained earlier about the length of the
Koch curve.) Hence a coastline might have an infinite length. This has become known as the
coastline paradox or Richardson effect. Length is thus not an accurate way of measuring a
coastline and the fractal dimension should rather be used (Mandelbrot 1967:636).
64
Since the outline of a coastline looks similar on different scales, owing to its self-similarity, the
Hsüs considered the possibility of applying similar techniques to music. Figure 45 illustrates the
similarity in the fractal distribution between a log-log graph of the West Coast of Britain and
Bach‟s Invention No. 1 in C major, BWV 772. Both graphs have a similar descending slope
characteristic of 1/f noise or fractals (Hsü & Hsü 1991:3507).
Figure 45: Resemblance between (A) the West Coast of Britain and (B) Bach’s Invention
No. 1 in C major, BWV 772 (Hsü & Hsü 1991:3507)
In one of their articles, entitled “Self-similarity of the 1/f noise called music”, they posed some
important questions: “Could we compare the fractal geometry of music to that of a coastline? If a
coastline has no definite length, could we state that Mozart‟s music has no definite number of
notes or note intervals”? (Hsü & Hsü 1991:3507).
65
They tested their research hypothesis by analysing Bach‟s Invention No. 1 in C major, BWV 772.
Firstly, the parts for the right hand and left hand were digitised individually and presented visually
on the same graph (see Figure 46).
Figure 46: Digitised score of Bach’s Invention No. 1 in C major, BWV 772; ○ right hand, ●
left hand (Hsü & Hsü 1991:3508)
Next, the Hsüs eliminated half of the notes in the composition and plotted the results on another
graph. They proceeded to halve the number of notes and plotted the results on different graphs.
The final reduction, ⁄
th of the original notes, resulted in the three key notes on which the
composition is built. As expected, the basic contour of the music remains the same, but it is less
detailed (see Figure 47). They claim that “… to a novice, the half- or quarter-Bach sounds like
Bach…” (Hsü & Hsü 1991:3508).
The Hsüs‟ findings in this experiment are not that surprising, since most compositions are based
on specific harmonies and are further embellished by non-chordal tones. If one were to eliminate
certain notes in a composition, it is highly likely that one would eliminate some of these nonchordal tones and be left with nothing but the basic harmony notes on which the composition is
based. They wrote that “…reduced music gives the skeleton and enhances our understanding of
great music” (Hsü & Hsü 1991:3509).
66
Figure 47: Reductions of Bach’s Invention No. 1 in C major, BWV 772; (A) original
composition, (B) ⁄ reduction, (C) ⁄ reduction, (D) ⁄ reduction, (E) ⁄
⁄
reduction, (F)
reduction (Hsü & Hsü 1991:3508)
Whether or not the different scales of the composition were indeed fractal was another question
that relied on “measuring” the composition. The actual “measurement” of a composition‟s length,
however, leaves much to be desired and contains many ambiguities. The Hsüs stated that the
length of a composition can be expressed as a mathematical formula, just like the length of a
coastline: “… the length of a music score, ∑ , can be represented by the sum of all note intervals
67
in a composition …” (Hsü 1993:29). The numbers used for the calculations should be as in the
table given earlier in Figure 43.
They explained their method with the use of an example: “Take, for example, the first five notes
of the BWV 772 for the right hand; the ∑ value is 1 + 1 + 1 + 2 = 5.” These numbers were,
however, obtained from the digitised version of the score in Figure 46 and not from the original
score. If one tries to apply the same method by using the actual score, and the interval numbers
are as in Figure 43, the answer is different, namely 2 + 2 + 1 + 3 = 8.
Figure 48: Bach, Invention No. 1 in C major, BWV 772, first five notes with interval
numbers
The digitised score used by the Hsüs clearly interpreted the distance between intervals in
diatonic steps, instead of semi-tones as conducted earlier. For a half-reduction, the sum of
intervals was found to be 2 + 1 = 3, but the researcher found that it should be 4 + 2 = 6, as in
Figure 49.
Figure 49: Bach, Invention No. 1 in C major, BWV 772, half-reduction of the first five notes
with interval numbers
This deviation from their original method has a major impact on the Hsüs‟ findings and the
credibility thereof. Hsü calculated the sum of all intervals (without reduction) for the right hand of
the Invention to be 380, but the researcher calculated the total as 585 – a difference of more than
60%.
68
3.3.2.4 Criticism of the Hsüs’ research
In 1993, Henderson-Sellers and Cooper (1993) published an article, “Has classical music a
fractal nature? – A reanalysis”, in which they re-evaluated and criticised Hsüs‟ articles.
Henderson-Sellers and Cooper (1993:277) found that “there is no inherent fractal nature in
classical music; although the converse is not true. In other words, it is feasible to use fractal
ideas to compose musical pieces – an area of much interest in recent years.”
In 1993, Henderson-Sellers and Cooper made a number of negative remarks about the Hsüs‟
interval-counting method to determine the length of a composition. Henderson-Sellers and
Cooper (1993:278) acknowledged the fact that a coastline‟s length is infinite, since its length
depends on the measuring stick used, but they disputed its application to a musical composition:
In contrast, the number of notes in a Mozart sonata is countable (in the mathematical
as well as the practical sense) and therefore does have definite “length,” despite the
Hsüs‟ conjecture that “Mozart‟s music has no definite number of notes.” One major
difference is that notes (or rather their representations) are discrete and therefore
countable whereas a coastline is continuous.
Another error that Henderson-Sellers and Cooper (1993) found in the Hsüs‟ research was
regarding their application of Mandelbrot‟s formula for fractal dimension to a composition. In the
original formula,
the variables on both sides of the equation have the same “underlying basic dimension”, for
example, length. In the Hsüs‟ application of the formula, the left- and right-hand sides of the
formula did not have the same dimension (Henderson-Sellers & Cooper 1993:278).
Henderson-Sellers and Cooper (1993) also stated that although all fractals have a power law
representation, the converse is not necessarily true: not everything with a power law
representation is necessarily fractal, as the Hsüs believed them to be.
In their article, Henderson-Sellers and Cooper (1993:279) asked an important question: “If music
is not fractal in nature, what has the analysis of Hsü and Hsü (1990) taught us?” Their answer
69
was that “an analysis of the use of specific intervals by a composer, and the frequency of use of
specific notes and specific intervals, may be a good guideline to a composer‟s „fingerprint‟.”
In fact, the Hsüs‟ method of counting the distribution of specific intervals is not new: “… the
relative frequency of melodic intervals is often used as a stylistic criterion by ethnomusicologists
…” such as Marcia Herndon and Mervyn McLean. Two of the most common ways in which this
method is applied are to count how often the various intervals appear in the music and to
“compare the distribution of rising and falling intervals” (Cook 1992:191). Note-counting methods
may thus be useful in characterising music from different cultures or genres and comparing them
with one another.
The pianist, Rosalyn Tureck, addressed a letter to the editors of Proceedings of the National
Academy of Sciences of the United States of America, the journal in which the Hsüs published
their first article, “Fractal geometry of music”. In it, Tureck strongly criticised the terminology used
and assumptions made in the Hsüs‟ article. She concluded the letter as follows (Tureck 2009):
The frequency of incidents of note intervals of music is by no means a methodology
for determining the relationships that constitute an artistic composition. If one is
looking for a fractal geometry of music, which I applaud wholeheartedly, then the
steps taken must be based on something immeasurably more solid and accurate than
what appears in this article.
The researcher agrees with Tureck in the sense that the Hsüs made too many assumptions and
that their research methodology was not solid enough. They also failed to use their formulae
uniformly. They also appear to have tried to force some mathematical formulae to fit the music
where they simply did not belong, as is the case with their application of Mandelbrot‟s formula for
determining the fractal dimension.
Similar to Voss and Clarke‟s research, the Hsüs also failed to include any music examples
in their studies. Tureck (2009) wrote as follows:
“… in presenting any kind of claims having to do with musical structure, it is
imperative when dealing with specific motives and compositions to include examples
of those motives and/or compositions … It is as necessary for the reader to have the
musical material for verification as to have the mathematical formulations for
consideration.”
70
Tureck (2009) further criticised the Hsüs‟ article for the incorrect use of some of the music
terminology. For example, they refer to a “small third” or “large third” instead of a minor and
major third. They also referred to “the first movement” of Bach‟s invention, but since
inventions do not consist of different movements, “an informed reader cannot equate their
claims with any part of the Inventions except by guesswork”.
3.3.3
Spectral density analysis for genre classification
Despite their flaws, many of the methods used by Voss, Clarke and the Hsüs are still being used
as the base for the spectral analysis of music. It has already been implied that spectral analysis
may be more useful in the stylistic analysis of compositions. The researcher found an increase in
the amount of research that has been done in classifying genres with the use of spectral analysis
since 2009.
3.3.3.1 Ro and Kwon
In 2009, Wosuk Ro and Younghun Kwon from the Department of Applied Physics of Hanyang
University in South Korea, published an article entitled “1/f noise analysis of songs in various
genre of music”. Ro and Kwon based their research on Voss and Clarke‟s findings that “there is
1/f behaviour in music and speech”. Ro and Kwon‟s research was distinctly different from that of
Voss and Clarke in the sense that they tried to use spectral density analysis to compare different
genres of music. Their hypothesis was that one may be able to distinguish between different
genres of music through the calculation of 1/f behaviour in a composition below 20 Hz. They
analysed 20 “songs” from seven different genres: classical music, hip-hop, newage, jazz, rock,
Korean traditional music (pansori) and Korean modern music (trot) (Ro & Kwon 2009:2305.)
In previous research, 1/f behaviour was often measured across the entire frequency range of a
composition. Instead, Ro and Kwon (2009) were more interested in low frequency range,
specifically frequencies below 20 Hz as this had not been done before. Consequently it was
found that “1/f noise analysis of songs would show the characteristics of each genre of music. So
it implies that we may classify each genre of music by score of 1/f noise analysis in the region
below 20 Hz” (Ro & Kwon 2009:2306).
71
They were highly specific in giving the degree of 1/f behaviour for each genre. This is illustrated
in Figure 50. The data shows that classical music – more than any other genre tested by Ro and
Kwon – mimics 1/f noise closest below 20 Hz.
Rank
1
2
3
4
5
6
7
Genre
Classic
Trot
Jazz
Newage
Rock
Hip-hop
Pansori
Degree of 1/f behaviour in the region below 20 Hz (%)
77.3
65.4
63.9
63.6
36.6
23.5
4.45
Figure 50: The degree of 1/f behaviour for seven different genres of music in the region
below 20 Hz (Adapted from Ro & Kwon 2009:2308)
To illustrate this further, Figure 51 shows the log-log plot for the Larghetto second movement of
Chopin‟s first Piano Concerto, Op. 11. The straight lines indicate a strong 1/f like behaviour below
20 Hz.
Figure 51: Log-log plot of frequency analysis in the Larghetto second movement of
Chopin’s first Piano Concerto, Op. 11 showing high 1/f correlation in the low frequency
range (Ro & Kwon 2009:2306)
72
In stark comparison, Figure 52 shows the log-log plot of frequency analysis for Song of
Simchung, a Pansori song for rice offering. This piece of traditional Korean music, unlike the slow
movement of Chopin‟s first Piano Concerto, shows virtually no 1/f correlation in the low frequency
range.
Figure 52: Log-log plot of frequency analysis in Song of Simchung displaying very low 1/f
correlation in the low frequency range (Ro & Kwon 2009:2307)
Ro and Kwon (2009:2306-2307) also found that “1/f noise analysis of songs in the region below
20 Hz might not show the characteristic of culture but that of each genre of music”. They
concluded that “the score of 1/f noise analysis in that region could classify each genre of music”.
Unlike frequency analysis conducted between the 1970s and 1990s, Ro and Kwon (2009) were
among the first key researchers to use 1/f noise analysis as a tool to identify or classify different
genres of music.
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3.3.3.2 Levitin, Chordia and Menon
The idea of characterising different genres of music was further investigated through the years.
Most recently, in 2012, Daniel J. Levitin, Parag Chordia and Vinod Menon collaborated in the
article, “Musical rhythm spectra from Bach to Joplin obey a 1/f power law”. Since it had already
been proven that musical pitch fluctuations follow a 1/f power law (Voss & Clarke 1975), they
wished to investigate whether the same would be true for the rhythmic fluctuations in music.
Levitin et al. (2012:3716) started their article with the following statement and question:
Musical rhythms, especially those of Western classical music, are considered highly
regular and predictable, and this predictability has been hypothesized to underlie
rhythm‟s contribution to our enjoyment of music. Are musical rhythms indeed entirely
predictable and how do they vary with genre and composer?
While Ro and Kwon (2009) used 1/f distribution in music to distinguish between different genres
of music, Levitin et al. (2012) used it to differentiate specifically between different subgenres in
classical music. They analysed the rhythm spectra of 1 778 movements from 558 compositions
of Western classical music, which covered 16 subgenres, four centuries and a total of 40
composers (Levitin et al. 2012:3716).
They found that musical rhythms also exhibited 1/f spectral structure and that composers “whose
compositions are known to exhibit nearly identical pitch spectra, demonstrated distinctive 1/f
rhythm spectra”. They were of the opinion that these 1/f fluctuations may enhance one‟s
aesthetic appeal to a composition (Levitin et al. 2012:3717).
Levitin et al. (2012) also tried to find a reason behind the fractal structures in the melody and
rhythm of music. They wrote that “… 1/f is not merely an artefact of performance, but exists in the
written scores themselves. Perhaps composers can‟t help but produce 1/f rhythm spectra,
perhaps musical conventions require this. From a psychological standpoint, the finding suggests
that composers have internalized some of the regularities of the physical world as it interacts with
biological systems (including the mind) to recreate self-similarity in works of musical art” (Levitin
et al. 2012:3718).
74
By comparing the rhythm spectra of different composers‟ works, Levitin et al. (2012:3716) found
that “Beethoven‟s rhythms were among the most predictable, and Mozart‟s among the least”.
Figure 53 illustrates the hierarchy of predictability of some of the composers from their study
(Levitin et al. 2012:3719).
Figure 53: Distribution of spectral exponents for composers ordered from the largest
mean exponent to the smallest (Levitin et al. 2012:3719)
Regarding specific classical genres, as opposed to composers, symphonies and quartets were
generally indicated to have the most predictable rhythms. However, mazurkas and ragtime
displayed some of the least predictable rhythms (Levitin et al. 2012:3717-3718).
Levitin et al. (2012:3718) commented that the “1/f structure [of musical rhythms] allows us to
quantify the range of predictability, self-similarity, or fractal-like structure within which listeners
find aesthetic pleasure. Critically, compositions across four centuries and several subgenres of
Western classical music all demonstrated 1/f structure in their rhythm spectra.”
75
According to their research, earlier experiments concerned with the measurement of pitch
fluctuations could not “systematically delineate one composer‟s work from another.” On the
contrary, it was found that “1/f rhythm spectral exponents varied widely and systematically among
composers.” If one re-examines Figure 53, one can see the distinct rhythmic difference between
Haydn, Mozart and Beethoven, despite the fact that these three composers are generally
grouped together in the Classical Era.
3.4
Towards a better understanding of fractal music
3.4.1
Harlan J. Brothers
In the past ten years, some researchers have gradually moved away from the spectral analysis of
music to a more practical approach – one that enables any musician with a rudimentary
understanding of self-similarity and fractals to do a structural analysis of existing compositions or
to compose an original work with the use of these mathematical techniques.
One of the pioneers in making the notion of fractal music more accessible to both musicians and
mathematicians is Harlan J. Brothers. Whereas many of the researchers mentioned earlier were
either scientists or musicians, Brothers is both: he is Director of Technology at The Country
School in Madison, but has also studied composition at the Berklee College of Music in Boston
and is a guitarist. Brothers worked closely with the mathematicians, Michael Frame and Benoit
Mandelbrot, at Yale University where they started Fractal Geometry Workshops in 2004. “During
an informal discussion in 2003 regarding the general lack of understanding associated with the
concept of fractal music, Mandelbrot suggested to Brothers that he undertake a rigorous
mathematical treatment of the subject” (Brothers 2004).
The initial purpose of the workshops at Yale University was “to train educators in the subject of
fractal geometry with the goal of developing fractal-based mathematics curriculum for students in
the middle school through college” (Brothers 2004). Through the years, and with the help of
Mandelbrot and Frame, these workshops have been extended to include fractal music as well.
An entire section of Yale University‟s website (http://.classes.yale.edu/fractals/) is dedicated to
providing information on fractals and self-similarity.
76
3.4.2
Prerequisites for self-similarity and fractality
Brothers‟ courses on fractals and fractal music, which are available on the internet from
www.brotherstechnology.com, include valuable information on what characteristics make
something fractal, as well as some “misconceptions” in this field of study. These are discussed
individually in the next section.
According to Brothers (2004), an object is self-similar when it meets the following three
requirements:
1. It is constructed of a collection of different-size elements whose size distribution
satisfies a power-law relationship spanning at least three scales;
2. It must compromise at least two similar regions in which the arrangement of
elements either mirrors or imitates the structure of the object as a whole; and
3. Its features must possess sufficient detail that the overall structure cannot be
more easily explained in Euclidean terms.
Although not stated directly, it is assumed that a music composition should also adhere to these
three prerequisites in order to be regarded as self-similar. With such a wide array of
prerequisites, the existence of fractal compositions seems almost impossible.
Brothers (2004) also identified some “common misconceptions” regarding self-similar and fractal
music, namely transliteration, all iteration and limited self-similarity. Transliteration in music is the
process in which music is directly derived from another source, for example, the outline of a
graph or geometric object, through a process called mapping. He stressed that the translation of
a self-similar or fractal object into music will not necessarily produce fractal music. He explained
it well with the use of an example in the literature:
This idea [of transliteration] may seem compelling at first and can produce interesting,
if sometimes disconcerting, compositions. However, claiming that these compositions
are fractal is something like claiming that a poem by Robert Frost, if transliterated
using the Cyrillic alphabet, makes sense in Russian.
77
Although transliteration is a valid composition tool, it does not guarantee that the resultant music
will also be fractal. Brothers referred to such examples as “fractal-inspired music” as opposed to
“fractal music”. Examples of fractal-inspired music are given in the subsequent chapter.
The second misconception identified by Brothers (2004), is the idea that a composition created
through iteration constitutes a fractal composition, since fractals rely on iterative processes. It
should be pointed out that although fractal patterns often emerge from some iterative process, an
iterative process will not guarantee a fractal pattern.
There are many shapes that are self-similar around a single point only. Examples include spirals,
onions and Russian dolls whose only point of self-similarity is around the centre of the object.
This is known as limited self-similarity and is not sufficient to classify an object, or a piece of
music, as fractal. In order to be considered fractal, objects must “contain a minimum of two
matching or similar regions in which the arrangement of elements either mirrors or imitates the
structure of the object as a whole” (Brothers 2004).
Brothers (2004) concluded the introductory page of his website with the following words:
Music can exhibit fractality in several ways, from rigid self-similarity of the
composition to a psychological superposition of patterns on different scales. Beware
of books or webpages [sic] about fractal music: many of these are the worst kind of
nonsense.
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3.4.3
Scaling in music
On the basis of all these rules formulated by Brothers, the possibility of fractal music existing is
extremely remote. He went on to describe the different types of scaling that can be found in
music, namely duration, pitch and structural scaling.
3.4.3.1 Duration scaling
One of the clearest examples of scaling in music may be found in the note values or “duration of
notes” in a composition. According to Brothers (2004), if “examination reveals a power law
relationship between a note‟s duration and the total number of such durations, then we can at
least establish that a scaling phenomenon exists”.
To illustrate duration scaling, Brothers provided the following two music examples which contain
the same distribution of note values. That means that both of these examples contain exactly the
same note values in the same distribution throughout each piece:
Figure 54: Brothers, two melodies with the same distribution of note values (Brothers
2004)
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Figure 55 show the distribution of the note values in the two melodies by Brothers (2004).
Note value
Number of occurrences
Quaver (1/8)
16
Crotchet (1/4)
8
Minim (1/2)
4
Semibreve (1)
2
Figure 55: Table showing the distribution of note values in Brothers’ two melodies (2004)
From this data the following log-log plot was generated:
Figure 56: Log-log plot of the distribution of note values in two melodies by Brothers
(2004)
The straight, negative slope of the graph shows that “both melodies obey a power law
distribution”. It does not mean, however, that both melodies are fractal in respect of note
duration. The example in Figure 54(a) “simply states the sorted set of values, starting with the
longest duration, proceeding to the shortest, and then wrapping back around to the longest”. In
addition, Brothers (2004) stressed that the note values are not evenly distributed throughout the
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first melody; instead each bar contains only a single duration value. He contends that the second
melody (Figure 54 (b)) is a more convincing example because each bar (except for bar 4 and bar
8) contains a combination of different note values.
Brothers (2004) used the melody in Figure 54(b) as the top voice in a little counterpoint
composition entitled Go for Baroque. The added voice contains twice as many notes as the top
voice, but with the same distribution of note values (see Figures 57 & 58).
Figure 57: Brothers, Go for Baroque, illustrating duration scaling (2004)
Note value
Number of occurrences
Quaver
32
Crotchet
16
Minim)
8
Semibreve
4
Figure 58: Table showing the distribution of note values in Brothers’ Go for Baroque
(2004)
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3.4.3.2 Pitch scaling
Another element of music to which scaling can be applied is pitch. Brothers (2004) wrote that if
“examination reveals a power law relationship between a note‟s pitch and the total number of
such pitches, then we can in a similar fashion to durations, establish that a scaling phenomenon
exists”.
The following six bar melody in D major (Figure 59) was written by Brothers to examine the
possibility of pitch scaling. (In subsequent text and figures it will be referred to as the D major
melody to avoid confusion with prior music examples.)
Figure 59: Brothers, D major melody to examine the possibility of pitch scaling (2004)
As in the case of testing for duration scaling, a table (Figure 60) was set up to summarise the
relationship between each pitch and the number of occurrences. Pitches were arranged from the
highest to the lowest, with each assigned a numerical value. The researcher also included the
note names for clarification.
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Note
number Number of occurrences
(Pitch)
78 (F#)
2
76 (E)
3
74 (D)
9
73 (C#)
3
71 (B)
2
69 (A)
4
67 (G)
4
66 (F#)
2
62 (D)
3
57 (A)
1
Figure 60: Table showing the distribution of pitches in Brothers’ D major melody, arranged
from highest to lowest pitch (Adapted from Brothers 2004)
In this example, there appears to be no clear relationship in the distribution of pitches. Brothers
(2004) attributed this to “the greater variety of different elements” compared to the examples of
duration scaling. “Here we have 10 different values for only 33 elements; there are relatively few
elements of each size on which to base our analysis” (Brothers 2004). A log-log plot of the data
in the table also did not reveal any power-law relationship:
Figure 61: Log-log plot of the distribution of pitches in Brothers’ D major melody (2004)
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In such cases, Brothers investigated the possibility of binning the data because this might have
revealed patterns that had had not been clear before. Data binning means processing the data
into different intervals, called bins. The data was grouped into four separate bins:
Note number
Bin number
Number of occurrences
62 – 66
1
14
57 – 61
2
9
52 – 56
3
6
45 – 51
4
4
Figure 62: Table of binned data for Brothers’ D major melody (2004)
The binned data obtained in Figure 62 was used to create the following log-log plot:
Figure 63: Log-log plot for the binned data of Brothers’ D major melody (2004)
The table in Figure 62 and the log-log plot of its data in Figure 63 indicate the existence of a
power law relationship in Brothers‟ D major melody.
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3.4.3.3 Structural scaling
The third, and possibly most convincing, place to look for scaling in music is in its structure. Even
if a power law relationship exists in the distribution of the durations or pitches in a composition, it
will not necessarily be audible to the listener; nor will it be of much value to the performer to
utilise in the interpretation of the composition. Structure, however, encompasses an entire piece
of music and it is possible to hear the recurrence of a motif or section; whether presented exactly
the same, slightly altered or transformed. Brothers (2004) found examples of structural scaling in
compositions from the Renaissance and Baroque period, such as the second Agnus Dei in
Josquin Des Prez‟s Missa l’homme armé super voces musicales and the first Bourrée from J.S.
Bach‟s Cello Suite No. 3, BWV 1009 (Brothers 2004, 2007). These are discussed in detail in
Chapter 5.
3.5
Validity of fractal musical analysis
In the article “Numbers and music” in The new Grove dictionary of music and musicians, Tatlow
and Griffiths (2001:231) specifically addressed the credibility of the use of numbers in music
analysis. The following is a quotation from this article:
Musicology is left with a dilemma. Counting notes and pulses frequently reveals a
numerical correlation between the sections of a musical work. This could imply that
the composition was organized numerically at an early stage, and the temptation for
the modern analyst is to assert that the numerical relationships were devised by the
composer. Yet there is slender historical evidence to support this: little is known from
music theory or surviving sketchbooks about the pre-compositional processes of
composers before Beethoven. Without a firm historical basis it is both premature and
irresponsible to draw conclusions about compositional procedure from numbers in the
score. A separation must be maintained between numerical analysis, comment upon
the compositional process and speculative interpretation of the numbers. There is
also a need to consider whether there is any historical justification for the analytical
techniques used to generate the numbers; and if so, whether the numbers in the
score were created consciously by the composer and whether the numbers are
wholly structural or have some further significance.
This slightly places a damper on the mathematical and fractal analysis of compositions, because
there is no sure way to know if a composer thought of fractal or self-similar structures while
composing. There are, however, other theories that can shed light on this use.
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In 1991, Shlain published his book, Art and physics: parallel visions in space, time, and light.
Shlain hypothesised that the arts (particularly the visual arts) anticipated many scientific
discoveries. One example is that the artworks from Picasso‟s blue and rose period anticipated
many of the theories developed by Einstein in his theory of relativity. Shlain (1991:24) stated the
following in this regard:
Art generally anticipates scientific revisions of reality. Even after these revisions have
been expressed in scholarly physics journals, artists continue to create images that
are consonant with these insights. Yet a biographical search of the artists‟ letters,
comments, and conversations reveals that they were almost never aware of how their
works could be interpreted in the light of new scientific insights into the nature of
reality.
Shlain (1991:73) substantiated his hypothesis as follows: “Art reflected the thinking of the times.”
In addition, he believed that “the artist presented society with a new way to see the world before
a scientist discovered a new way to think about the world” (Shlain 1991:73).
Shlain (1991) discussed some of the works by the impressionist painters, Cezanne, Monet and
Manet to illustrate that space could be interpreted in non-Euclidean ways. He wrote:
Their revolutionary assaults upon the conventions of perspective and the integrity of
the straight line forced upon their viewers the idea that the organization of space
along the lines of projective geometry was not the only way it can be envisioned.
Since this created a different way of seeing the world, it enabled them to think about it differently
as well (Shlain 1991:73,118, 135).
Although Shlain‟s (1991) book made little reference to music compositions and fractal geometry,
the researcher posits that a similar hypothesis can be used to substantiate the fractal and selfsimilar structures found in compositions dated prior to Mandelbrot‟s definition of fractals. It was
mentioned that while “…visual art is an exploration of space; music is the art of the permutation
of time. Like his counterpart, the composer has repeatedly expressed forms that anticipated the
paradigms of his age” (Shlain 1991:271).
Shlain (1991) also touched on the philosophies of the ancient Greeks several times. Plato and
Aristotle, for example, postulated that mimesis, the mimicking of nature, “was an innate impulse
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of the human personality” (Shlain 1991:150). He also highlighted the fact that these philosophers
believed that “the essence of beauty was order, proportion and limit” (Shlain 1991:36.)
Many of these ancient philosophies still hold water and are still pursued today. If one considers
the beliefs of Plato and Aristotle, the appearance of fractals and self-similarity in music that dates
back to the 16th century makes sense. Without knowing it, composers, like many visual artists,
mimicked fractal and self-similar structures in their compositions. The aesthetic quality of their
compositions, although subjective, might be because of the level of “order, proportion and limit.”
Plato asked: “Is ugliness anything but lack of measure?” (Shlain 1991:151).
Shlain (1991:270) also quoted Zola: “… art is nature as seen through a temperament; and the
nature of space, time, and light is revealed for those who want to see it through the creations of
the innovative temperaments of the great artists”.
3.6
Conclusion
In this chapter different research methodologies and theories of several researchers over a
period of almost 40 years were examined. All studies relating to the spectral analysis of music
concluded that classical music, more than any other genre of music, resembled fractal 1/f noise
fluctuations. This was true of the melodic and rhythmic elements of various compositions.
Although these findings are interesting, they are of little value to musicians, as was indicated by
Rosalyn Tureck in her criticism of the Hsüs‟ research.
Nicholas Cook (1987), author of A guide to musical analysis, stressed that any type of analysis
should either shed light on how the particular composer went about composing the work, or it
should be an aid to the performer in accurately portraying the music. The researcher is of the
opinion that many of the experiments conducted on the 1/f distribution are of more value to
scientists than to musicians.
More recent studies on the use of 1/f noise distribution to classify genres sound promising.
Scholars like Ro and Kwon (2009) and Levitin et al. (2012) found that 1/f noise distribution of
music can be used to distinguish better between music of different genres and even distinct
composers. This may be helpful, for example, in determining the authenticity of composers‟
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works. To the researcher‟s knowledge, such research has not yet been conducted with the aid of
1/f distribution.
With the assistance of Mandelbrot and other researchers at Yale University, Brothers (2004)
managed to clarify the prerequisites for fractal and self-similar music. These will be useful in the
subsequent chapters where the composition and analysis of music with the help of fractal
geometry and self-similarity will be discussed.
Shlain‟s (1991) hypothesis that the arts anticipated many scientific discoveries, coupled with the
ancient Greeks‟ philosophy that art mimicked nature, justifies the existence of fractal and selfsimilar structures in music that was composed before the 20th century. Examples of such
compositions are given in Chapter 5.
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CHAPTER 4: FRACTALS AND SELF-SIMILARITY AS A COMPOSITION TOOL
4.1
Introduction
The research conducted by other scholars has shown that 1/f noise and fractal distributions can
be found in music. This implies that fractals may be used to compose music. This chapter
investigates exactly how noise forms, self-similarity and fractals can be used as an effective
composition tool.
4.2
Composing music with noise
Voss and Clarke‟s (1975, 1977, 1988) findings that all music has some 1/f distribution has a
bearing on the next section: if 1/f fluctuations could be found in existing music, then a 1/f noise
source could certainly be used to compose an original piece of music. Although stochastic music
has been composed before, it did not show any resemblance to 1/f noise fluctuations. Voss and
Clarke (1988) decided to use sources of white, brown and 1/f noise respectively to see which of
the three noise sources would produce the most interesting and musical compositions. They also
proposed different methods for composing noise-based music, which is discussed in the
subsections below (Voss 1988:42).
4.2.1
Using spinners to compose music based on noise forms
Voss and Clarke‟s (1977, 1988) research on the various methods for composing fractal music
was summarised in the first chapter of Martin Gardner‟s book, Fractal music, hypercards and
more (1992). The chapter entitled “White, brown, and fractal music”, focused specifically on the
implementation of 1/f noise in music composition. Methods included the use of dice and spinners
to determine the different pitches and note values for a composition.
Gardner (1992:4-6) summarised Voss and Clarke‟s manual methods of composing music with
the help of different noise forms. The use of a spinner, similar to one used in a board game, was
one of the manual methods used to compose music based on noise forms.
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In order to compose “white music” the spinner is divided into seven unequal segments; one for
each diatonic note of a scale. The note on which the arrow lands after each spin is recorded on
manuscript paper. A second spinner can be used to determine the note value of each of the
pitches (Gardner 1992:4).
Figure 64: Spinner used to generate pitches for white music
Figure 65: White music generated with a spinner (Gardner 1992:16)
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When using this method, the resultant composition is extremely random and not that musical.
Voss and Clarke‟s research (1977) has shown that music generated by white noise is too
irregular, since there is no correlation between successive notes. This can be seen in the
resultant composition in Figure 65.
In order to compose a piece of music with the use of brown noise, the spinner is once again
divided into seven segments, but this time they are marked with addition and subtraction rules.
+1, +2, and +3 indicate that the next note in the melody should be one, two or three notes higher
than the previous one. The subtraction rules (-1, -2, -3) shows that the next note should be a set
amount of notes lower. The 0 indicates that the subsequent note should be the same as the
previous one. Since each note added to the melody is based on the previous note, the resultant
melody will be highly correlated and uninteresting.
Figure 66: Spinner used to generate brown music
In contrast to music generated from white noise, Brownian motion (brown noise) is too
predictable and will not be able to create interesting music with variation. This can be seen in the
music example in Figure 67.
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Figure 67: Brown music generated with a spinner (Gardner 1992:17)
Between these extremes of white and brown noise lies pink noise, more commonly known as 1/f
noise. It is neither as random as white noise nor as predictable as Brownian motion and is thus
ideal to use for generating music. By using 1/f noise, the music has an element of surprise as
well as recurring ideas to bind a work to a unit (Gardner 1992:6).
Although Gardner (1992) did not explain in detail how the spinner was used to compose 1/f
music, he did include a music example of the resultant music. As in stochastic composition,
transition rules and rejection rules were adopted to make music correlated but interesting
(Gardner 1992:5-6).
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Figure 68: 1/f music generated with a spinner
4.2.2
Turning noise voltages into music
Another method explored by Voss and Clarke (1975) to compose music from noise was to use
the actual noise voltages and then to convert them into music. Samples of three different noise
wave forms were converted into music. So-called “Johnson noise voltage”, across a resistor, was
used to produce the white noise sample. This was later passed through a low-pass filter to obtain
brown noise. The 1/f noise was obtained from fluctuations over a transistor. The different noise
voltages were stored in a computer system and represented as notes in a 12-note chromatic
scale over two octaves. Similarly, the same data was used to decide on the respective note
values to be used for each pitch. The computer that was used could also perform these pieces
and translate them into music notation. Figure 69 shows the music that was created from white,
brown and 1/f noise respectively (Voss & Clarke 1975:262).
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(a)
(b)
(c)
Figure 69: Music resultant from (a) white, (b) brown and (c) 1/f noise (Voss 1988:43)
94
The first melody (Figure 69(a)) was created from white noise and is not that musical in the sense
that the note values cannot be grouped in bars with a single time signature and it contains many
jumps of large intervals and little step-wise movement. The music thus imitates the uncorrelated,
chaotic behaviour of white noise.
Music created from brown noise was notated as in Figure 69(b) and is characterised by many
repeated notes. The melody wavers up and down constantly, but does not have direction. Of the
three examples in Figure 69, the music resulting from brown noise contains the most stepwise
movement and repeated notes, showing a high correlation as with Brownian motion.
As expected, the music resulting from 1/f noise (Figure 69(c)) shows a balance between the two
extremes of the highly correlated music from brown noise and the chaotic melody from white
noise. The 1/f music example contains a mixture of repeated notes, stepwise movement and
some jumps.
It was easy to see that there may be higher or lower correlation between notes in the above
music examples by looking at them, but what do they sound like? Voss (1988:42) remarked that
“although none of the samples […] correspond to a sophisticated composition of a specific type
of music, [Figure 69(c)] generated from 1/f-noise is the closest to real music. Such samples
sound recognizably musical, but from a foreign or unknown culture.” Voss and Clarke (1977:263)
played the resultant music from the above-mentioned experiments to hundreds of people at
several different universities and research laboratories over a period of two years, including
professional musicians and composers as well as listeners with little theoretical knowledge of
music. They came to the following conclusion:
Our 1/f music was judged by most listeners to be far more interesting than either
white music (which was “too random”) or the scalelike 1/f 2 [brown] music (which was
“too correlated”). Indeed the surprising sophistication of the 1/f music (which was
close to being “just right”) suggests that the 1/f noise source is an excellent method
for adding correlations.
After playing the music examples in Figure 69 several times on the piano, the researcher could
not agree fully with Voss and Clarke‟s findings or the opinion of their listeners. Firstly, what
constitutes “real music”? There is also no scale that can measure the level of musicality of a
music example.
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In the conclusion of their article, Voss and Clarke (1977:263) admitted that “there is more to
music than 1/f noise”. Although noise forms can be used to determine the pitch and rhythm in a
music composition, there are other elements of music such as the overall structure, dynamics
and timbre that were not taken into consideration.
Charles Dodge‟s computer-aided composition, Profile (1984), was directly influenced by
Mandelbrot‟s work in fractal geometry. Dodge had been experimenting with ways to compose
computer-generated music that would be aesthetically pleasing, and fractals supplied him with
new ways to compose such music. Profile significantly differs from his earlier works in the sense
that he composed the entire piece from a single computer algorithm. The work‟s success has led
to other compositions written in a similar fashion (Dodge 1988:10). A 1/f-noise algorithm was
used to generate the musical elements such as pitch, duration and volume in Profile. Since
Profile is a computer-aided composition, no music example was available to insert.
4.3
Composing with Lindenmayer systems
In Chapter 2, Lindenmayer systems and their visual representation through the application of
turtle graphics were briefly explained. The next subsection shows how these can also be used to
compose music.
4.3.1
Prusinkiewicz
In 1986, at the International Computer Music Conference, Przemyslaw Prusinkiewicz
demonstrated how the graphic representation of the Hilbert curve can be adapted as a musical
representation. He used the third iteration of the Hilbert curve (as discussed in Section 2.3.2.5) to
illustrate how music can be created from an L-system turtle graphic. When the Hilbert curve is
laid out as in Figure 70 and moves in the direction shown by the arrow, then “the consecutive
horizontal line segments are interpreted as notes” to create a melody. The horizontal line
segments are numbered chronologically from the bottom left corner (Prusinkiewicz 1986:456).
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Figure 70: Third iteration of the Hilbert curve (Prusinkiewicz 1986:456)
Prusinkiewicz (1986) explained that the y-coordinates represent the pitch, while the length of
each horizontal segment is proportional to the note value or duration. Since music occurs in time,
the numbered line segments can be unravelled and presented chronologically.
Figure 71: Unravelled form of the third iteration of the Hilbert curve (Adapted from
Prusinkiewicz 1986:457)
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The scale chosen for the melodic representation was C major with C as starting note (number 1).
For the note values, the shorter lines were represented as quavers and the longer lines (which
are twice as long) as crotchets. The resultant melody is shown in Figure 72:
Figure 72: Third iteration of the Hilbert curve represented as a melodic line (Prusinkiewicz
1986:457)
It is not clear from Prusinkiewicz‟s research why a 4/4 time signature was chosen or why barlines
were positioned as in the example.
All of the curves used by Prusinkiewicz (1986) to generate music were read from left to right, and
the first line of the curve was always horizontal. What would the musical result be if a curve were
to be rotated before it was interpreted as music, or if the curve were to be read from right to left?
Would it be possible to use a curve that begins with a vertical line? These questions were further
investigated by other scholars.
4.3.2
Mason and Saffle
Stephanie Mason and Michael Saffle (1994) built on Prusinkiewicz‟s composition method by
showing how transformations (rotations) of the same curve can be used to create variations of a
melody. Their findings were published in 1994 in an article in the Leonardo Music Journal.
Mason created a program that could automatically draw a curve from any given L-system. The
programme also allowed her to play the resultant music on a synthesiser. Together with Chris
Cianflone, a student at the University of Minnesota at the time of their research, an experimental
music program was developed based on this research (Cipra 1993:37).
Mason and Saffle (1994:32) demonstrated that a total of eight different melodies can be
produced from a single geometric curve. The original curve can be read forwards or backwards
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(giving two “original melodies”). The entire curve can be rotated into three different positions,
each of which can be read forwards and backwards, resulting in six more melodies. To illustrate
this, the first iteration of the quadratic Gosper curve was used. Figure 73 displays each of the
rotations with its corresponding music example. For the purposes of this example, the note
values were interpreted as quavers and semi-quavers, while the C major scale was chosen to
determine the pitches.
Mason and Saffle (1994:33) explained that when a curve starts with a vertical line, “the first note
can be read as the number of forward moves up or down from the first note of a given scale or
mode. Similarly, the number of forward moves used to create the horizontal line determines the
note‟s duration.”
In Figure 73, numbers 1 and 2 show how the original quadratic Gosper curve can be interpreted
both forwards and backwards. The music examples are melodic retrograde from one another
with the second being a third higher than the first.
The curve was then rotated clockwise by 90 degrees to obtain the information for numbers 3 and
4, once again being read forwards and backwards. A 180 degree and 270 degree rotation of the
original are shown in numbers 5-6 and 7-8 respectively. Numbers 5 and 6 are the melodic
inversions of numbers 1 and 2. The same is true when comparing numbers 3 and 4 with 7 and 8.
The given time signatures are unclear from the context of Mason and Saffle‟s (1994) article, but
the resulting melodies are musically sensible. When a higher iteration of the same curve is used,
longer, more interesting melodies can be created. It is also possible to use any scale or mode.
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Figure 73: The first iteration of the quadratic Gosper curve read in terms of four rotations
to produce eight different melodic motifs (Mason & Saffle 1994:32)
100
This method of composition also makes polyphonic composition possible. Two iterations of the
same curve can be played simultaneously. Figure 74 shows two iterations of the quadratic
Gosper curve. The curve on the left was interpreted by Mason as a melody for flute and the curve
on the right as one for piano. Arrows were added to show where each melody starts: the flute
starts in the lower left-hand corner, while the piano starts in the lower right-hand corner.
Underneath, the resulting music is given in music notation (Cipra 1993:37; Mason & Saffle
1994:33).
Figure 74: Two transformations of the second iteration of the quadratic Gosper curve used
to compose polyphonic music for flute and piano (Adapted from Cipra 1993:37; Mason &
Saffle 1994:33)
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Cipra (1993:37) commented as follows on Mason‟s polyphonic composition from the quadratic
Gosper curve:
Composers have long played with the formal structure of music. Bach, for example, is
well known for writing music that could be played backward as well as forward.
Mason had gone a step further, with music that can be played sideways as well, in
what she calls a “right-angle canon”. To do this, she simply takes a curve and rotates
it so that the pitch and duration are interchanged. When both curves are played
together, using separate synthetic “voices” (Mason leans to piano and flute), the
effect is surprisingly musical.
Another aspect of fractals and L-systems that can be brought into context with music
composition are their self-similarity and scaling properties. Figure 75 shows the selfsimilarity of the quadratic Gosper curve. The first iteration of the curve (red) is not fractal.
The second iteration (black) contains copies of the first iteration and is thus self-similar and
fractal. Note how the first iteration of the curve is a larger version of the smaller second
iteration of the curve. This shows that the quadratic Gosper curve in its second and higher
iterations is in fact self-similar and fractal.
Figure 75: Self-similarity of higher iterations of the Hilbert curve
This also indicates the possibility that such scaling can also be applied in resultant music
compositions. The second iteration of the quadratic curve can be interpreted melodically (as in
Figure (75)). The first iteration, which is larger, will produce a melody self-similar to the first, but
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in longer note values. It is thus an example of duration scaling as discussed by Brothers (2004) in
section 3.5.4.
Pursinkiewicz (1986:456) and other scholars such as Mason and Saffle (1994:33) commented
that any FASS curves (space-filling, self-avoiding, simple and self-similar curves), fractal curves
and any other curves containing 90 degree angles can be used to create melodies in a similar
way to the aforementioned method.
4.3.3
Further modifications of L-systems for composition
The use of space filling curves generated from L-systems can lead to interesting music
examples, but for some composers this method was still too dull. One example is Gary Lee
Nelson (born 1940), a pioneer in the field of computer music. He made use of fractals, among
other mathematical ideas, to compose music. His article, “Real time transformation of musical
material with fractal algorithms”, explained how he went about composing such music (Nelson
n.d.).
In Summer Song for solo flute (composed in 1991), Nelson (n.d.) extended some of the ideas
used by Prusinkiewicz (1986) and Mason and Saffle (1994) as discussed above. For this
composition, he used the fourth iteration of the Hilbert curve, but found it to be “too symmetrical”
and that the music it produced contained “rather dull repetitive patterns with little variety in pitch
or rhythm” (Nelson n.d.).
He found a simple solution for this: instead of letting the turtle turn by an angle of 90 degrees, he
widened it to 101 degrees, thus creating a slightly warped version of the original curve (see
Figure 76). He wrote that the result was “much better suited to make the piece [he] envisioned.”
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Figure 76: Warped version of the Hilbert curve used for Nelson’s Summer Song (Nelson
n.d.:6)
Also, instead of using a tonal scale, Nelson used a “projection of the interval series 2 2 3 over
two and a half octaves”. This means that the “scale” consists of two intervals of a major second
followed by a minor third. Figure 77 shows the scale as given by Nelson (n.d.:7) in his article:
Figure 77: Scale used in Nelson’s Summer Song (Nelson n.d.:7)
Once again the vertical position of lines was read as pitch and the length of the lines as duration.
Although Nelson did not include a music example of the music produced, he inserted the
following graph illustrating the contour of the melody:
Figure 78: Warped version of the Hilbert curve, unravelled to produce the pitch and time
contour for Nelson’s Summer Song (Nelson n.d.:7)
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4.4
Self-similar structures in the music of Tom Johnson
The American minimalist composer, Tom Johnson, was interested in creating self-similar
structures in his music. In a lecture presented at the Mathematics, music and other sciences
(MaMuX) seminar in Paris in October 2006, Johnson explained his fascination with self-similarity
and how he went about using such structures in his music. Johnson‟s lecture was entitled “Selfsimilar structures in my music: an inventory”. He defined self-similar music as “music that
somehow contains itself within itself and does so on at least three different levels of time”
(Johnson 2006:2).
Johnson (2006:2) described the three main influences for his self-similar compositions to be
Martin Gardner‟s regular publications in Scientific America in the 1970s; Mandelbrot‟s book on
fractals; and his meeting in 1979 with the mathematician, David Feldman, who taught Johnson
the fundamentals about group theory and self-replicating melodic loops.
Johnson (2006:2-23) stressed that true self-similarity is extremely rare in compositions. Although
most of his music is characterised by minimalism and symmetry, looking back he found only a
few pieces to be self-similar.
One of the earliest examples of self-similarity in Johnson‟s works was Symmetries which consists
of 49 symmetrical “drawings” created with a music typewriter. Figure 79 shows the three selfsimilar drawings in the set. These drawings were not meant solely as visual representations of
symmetry or self-similarity. According to Johnson (2006:5): “At the time I typed these images I
was thinking of a pure visual music, hoping that others would do their own realizations, or would
simply imagine sounds implied by the drawings.”
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Figure 79: Three self-similar drawings from Symmetries created with a music typewriter
(Johnson 2006:4-5)
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Rational Melodies, published in 1983, were “constructed with rational systems, following rigorous
rules” (Johnson 2006:6). Rational Melody VIII (Figure 80) has a clear self-similar structure. The
melody starts with four pitches descending chromatically in minims. In the second line, a
diminution of the melody (in crotchets) is inserted before each minim. The process is repeated
twice in quavers and semi-quavers. Each iteration of the four-note melody thus contains itself in
different note values, finally giving a total of four levels of self-similarity. The process can be
carried out further, but Johnson stated that the semi-quaver iteration was as far as he wanted to
go (Johnson 2006:6-7).
Figure 80: Johnson, excerpt from Rational Melody VIII (Johnson 2006:6)
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The same composition method was applied to the second movement of Counting Keys for piano.
Here the note sequence is C-B-A-F#-D and the notes repeat themselves on a scale of 1:5.
Johnson (2006:9) commented that the score is not necessary and that he performed it for many
years without notating it.
Figure 81: Johnson, second movement from Counting Keys (2006:9)
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Counting Duets, written in 1982, is a piece for speakers12. The first person starts by repeating the
numbers 1-2-3. The second speaker repeats the reverse of the pattern, 3-2-1, but numbers by
the first and second speaker must be in unison –i.e. when they speak together, the same number
must be uttered (see the first two lines in Figure 82). More speakers are added and the 1-2-3 and
3-2-1 patterns alternate with each added speaker. Each added voice is at half the tempo of the
previous – hence the emergence of a different rhythmic pattern. A self-similar structure is also
evident as a result (Johnson 2006:7).
Figure 82: Johnson, Counting Duets, fourth movement (Johnson 2006:7)
In Counting Duets, the slower speakers had to say their number in unison with another speaker,
creating doubling. In other works by Johnson, this simultaneous doubling was deliberately
avoided to create different rhythmic effects and self-similarities. One such example is the nonpitched choral piece 1 2 3, composed in 2002. The composer explained that the “simultaneities
are systematically avoided” in order to create a texture known as “one-dimensional tiling”
(Johnson 2002:13).
12
The word “speaker” here refers to a person speaking.
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In the first part of 1 2 3, it takes three bars (or 24 beats) for each of the eight voices to have its
turn. Johnson explained that the ratio between the voices is 8 : 8 : 4 : 4 : 4 : 4 : 2 : 2 (Figure 83).
Figure 83: Johnson, 1 2 3, Part I, No. 1 (2006:13)
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Tilework for Violin (2003) is the only piece in Johnson‟s Tilework series that has a self-similar
structure. The first note of bars 57-60 gives an ascending melodic pattern moving in whole tones
that continue throughout the rest of the piece: A -B -C-D. The first appearance of the pattern is
marked in red in Figure 84 (bars 57-60). The same pattern is repeated an octave lower, every
three bars (bars 57, 60, 63 and 66) and is marked in blue. An even slower-moving iteration,
another octave lower, is presented every nine bars (bars 59, 68, 77 and 87). It is marked in
yellow in Figure 84. It is evident that the same material is presented in three different octaves and
at three different tempos. There are thus three scale levels present, making the piece self-similar.
Figure 84: Johnson, Tilework for Violin, bars 57-96 (Johnson 2006:14)
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4.5
Coastlines and mountains
The research conducted by the Hsüs (1993) proposed a method for the measurement of a
composition similar to that used to measure irregular coastlines. In the next section, a more
practical application is investigated: how can the irregular contour of coastlines and mountains be
used as a composition tool?
4.5.1
Mandelbrot and coastlines
In 1967, almost a decade prior to the coinage of the word “fractal”, Mandelbrot published an
article entitled “How long is the West Coast of Britain? Statistical self-similarity and fractional
dimension”. This research showed how the structure of coastlines and mountain ranges is selfsimilar and fractal in nature and had a great impact on the view of not only nature, but also art
and music.
In 1961, the British meteorologist, L.F. Richardson, conducted research on the irregularity of the
outline of several coastlines around the world and was further investigated by Mandelbrot in the
1960s. Their research indicated that length was not an accurate way to measure a coastline,
since the length depended on the size of the measuring stick being used as well as the scale of
the map. The more detailed the map is, the greater the length of the coastline is. This has led to
an interesting paradox that the length of a coastline is infinite. This is also known as the
Richardson effect.
Coastlines, like fern leafs, also reveal new detail with each magnification and are self-similar.
This made it possible for scientists to “measure” a coastline – not according to its length, but its
dimension.
A line has a dimension of 1, while a square has a dimension of 2. The dimension of a coastline
lies between 1 and 2, like many other self-similar and fractal objects. Simply by looking at the
west coast of Britain, Mandelbrot could see that it is one of the most irregularly shaped coastlines
in the world and thus chose it for his research in 1967. Its dimension has been calculated to be
1.25. In strong contrast, the dimension of South Africa‟s coastline is 1.02 (close to 1) and is one
of the smoothest in the world (Figure 85.) From this information it is clear that the dimension
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accurately describes the level of irregularity for a coastline or other self-similar shape. The same
was also to be found true for the dimension of mountains. Mandelbrot called this dimension
between 1 and 2 the “fractional dimension” because it is a fraction and not a whole number. This
later became known as the “fractal dimension”.
Figure 85: Richardson’s data concerning the rate of increase of a coastline’s length at
decreasing scales (Kappraff 1986:660)
Mandelbrot discovered that any segment of a coastline is statistically self-similar to the whole.
Objects can be statistically self-similar if they have “the same statistical distribution of their
features, such as ins and outs, under magnification or contraction in their geometric scale …
Statistical self-similarity can never be empirically verified for naturally occurring curves such as
coastlines, since there are an infinite number of features associated with such curves” (Kappraff
1986:657).
Computer graphic artists have used this newly found information useful in reconstructing reallooking mountain ranges on computers. By using a simple, iterative formula it is possible to
“design” a mountain range like in Figure 85, which shows a fractal mountain range that was
created by Ken Musgrave. It is known as a “multifractal forgery” (Frame et al. 2014).
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Figure 86: Musgrave, a fractal landscape (Frame et al. 2014)
This research by Richardson, Mandelbrot and other scientists had a huge impact on the
development of ideas in geo-statistics, physics and even economics (Mandelbrot 1967:356). It is
hard to imagine that it has also impacted on music composition. The remainder of this section
illustrates how this research on the dimension of coastlines and mountains influenced the works
of 20th- and 21st-century composers.
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4.5.2
Relating music to coastlines and mountains
Mountains have been a metaphorical source of inspiration for composers for many years, for
example, Grieg‟s In the Hall of the Mountain King from the Peer Gynt Suite and Mussorgsky‟s
Night on Bald Mountain. From the latter half of the 20th century, composers began using the
structure of mountains more directly in their work.
Composers used what is called “mapping”: a process in which the contour of any shape (in this
case a building, mountain or coastline) is directly translated as the contour for the melodic line(s)
in a composition. One of the best examples in literature of this type of mapping is New York
Skyline by the Brazilian composer, Heitor Villa-Lobos. Although not fractal, this piece illustrates
the basic concept of mapping extremely well.
New York Skyline was composed in 1939 by superimposing the New York skyline on to graph
paper and from there plotted as music notes. It is said that Villa-Lobos used the same method
five years later to compose his sixth symphony. For the symphony, however, he used the contour
of mountains from his homeland, Brazil. Another composition, Melodia da Montanha (Mountain
Melody) was published in 1942 and was based on the horizon of the peaks of Serra da Piedade
in Belo Horizante in the state of Minas Gerais (Melo 2007:3).
Figure 87: Graphic representation used in Villa-Lobos’ New York Skyline (Kater 1984:105)
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If it is possible to map an object as music, the resultant music from a fractal or self-similar object
is expected to produce the same type of music.
4.5.2.1 Composing with coastlines
Music based on coastlines is rare, but the American composer Larry Austin composed an
excellent example of this type of music. Austin‟s composition for instruments and tape, Canadian
Coastlines: Canonic fractals for musicians and computer band (1981), was commissioned by the
Canadian Broadcasting Company and is entirely based on the outline of the coast of Canada.
The composition is for eight instruments and four channels of computer-generated sound (Clark
& Austin 1989:22).
The technique that Austin was using during this time for his compositions relied on the
“processing data by plots over time.” Certain data was extracted from the map and used as a
“seed” for the computer algorithm (Figure 88.) He admitted in an interview with Thomas Clark
that “the fractal result [of the music] could not be understood in terms of the map”. Unlike
composers before him, Austin did not simply plot the contours of the coastline as music notes,
but also made use of a 1/f-noise computer algorithm for the composition (Clark & Austin
1989:22).
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Figure 88: Time-plot obtained from a map of Canada for Austin’s Canadian Coastlines
(Clark & Austin 1989:23)
The subtitle of the composition hints at a type of canon. The eight musicians, guided by click
tracks in their headphones, play at four different tempos. In addition, the four computer tracks
also move at their own tempos. There are eight parts that are “canonic imitations” of one another,
played at different rates. This is thus similar to the prolation or mensuration canons of the
Renaissance period (Clark & Austin 1989:28).
Other compositions by Austin that also incorporate fractals include his Sonata Concertante
(1983-1984) and Beachcombers (1983) in which self-avoiding random walk (similar to Brownian
motion) was used. In the latter work, some actual text from Mandelbrot‟s book was used (Clark &
Austin 1989:23).
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4.5.2.2 Gary Lee Nelson’s Fractal Mountains
Another mountain-inspired composition is Fractal Mountains (1988-1989), composed by Gary
Lee Nelson. Fractal Mountains is an electronic composition in which a fractal algorithm was used
to determine the time, pitch and amplitude of the piece. The piece won first prize in the
international competition for micro tonal music in 1988. This composition marks Nelson‟s first
attempt at composing with fractal algorithms (Nelson n.d.:1-2).
Instead of drawing inspiration from an actual mountain range, the data for Nelson‟s composition
was obtained from the recursive subdivision to simulate a two-dimensional mountain contour.
The points on the simulated “mountain” were plotted as pitch and duration. To construct his
“mountain”, Nelson started by drawing the main peaks. Through an iterative process, similar to
constructing the Koch curve, each line is subdivided to create finer details of the mountain.
Figure 89 shows the subdivision of the mountain‟s contour (Nelson n.d.:3).
Figure 89: Nelson, Fractal Mountains, different subdivisions used to simulate the outline
of a mountain (Nelson n.d.:3)
Each peak of the mountain outline will then represent a different pitch and the distance between
the consecutive will determine the duration of each note. Unfortunately there is no sheet music to
accompany this example because the entire composition was composed digitally with the use of
what is called a MIDI horn. It is performed on a Yamaha TX816 and synthesisers controlled by a
Macintosh computer (Nelson n.d.:1, 3). The “scale” used for this composition is completely
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microtonal: the octave was divided into 96 equally tempered steps, making intervals 12.5 cents
each.
Fractal Mountains, also illustrates another interesting feature of fractals. Nelson (n.d.:3) wrote
that the beginning of each phrase in the music contains most notes in a single channel13. As the
music progresses, “the notes fan out among the eight channels and the richness of the micro
tonal palette is revealed. Toward the end of the each phrase the notes move toward a different
channel and cadence in relative consonance.” It is amazing to think that a computer-generated
piece of music can simulate the same evolution of dissonance to consonance (tension and
release) that is so common in Western music. This is attributed to another characteristic of
fractals, namely strange attractors.
4.6
Fractal-inspired music of Ligeti
According to Beran (2004:92) some contemporary composers employed fractals in their music
“mainly as a conceptual inspiration rather than an exact algorithm”. These composers include, for
example, Harri Vuori and György Ligeti. This section explores how Ligeti incorporated fractals
into his music, although more in a metaphorical sense than his contemporaries.
4.6.1
Ligeti’s interest in fractals
Ligeti started to investigate Mandelbrot‟s work with fractals since the early 1980‟s as well as
Conlon Nancarrow‟s polyrhythmic studies for player piano. Ligeti was also a great admirer of the
optic illusions in Carl Escher‟s art. According to Svard (2000:802) “Ligeti claimed that there are
no direct links to his work from these writers, authors, or musicians, but only speaks about his
interest in them to reveal the intellectual climate in which he composes.”
In 1984, Ligeti‟s friend and former Nobel Prize winner Manfred Eigen showed him some of
Peitgen‟s computer generated pictures of fractals. “Ligeti was one amongst many creative artists
to be inspired by the intricacy of these strange images, and intrigued by the mathematical
13
Eight channels in total were used.
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principles behind them” (Steinitz 2003:273). A year later, Ligeti read Mandelbrot‟s book The
Fractal Geometry of Nature and they finally met in 1986. Later that year, Ligeti also met Peter
Richter and invited him to give a lecture to his composition students at the Musikhochschule.
Thereafter Ligeti met Heinz-Otto Peitgen with whom he had done some collateral work. The
collaborations between Ligeti and these mathematicians continued for a long time. In 1996 two
festivals were held in Lyons and Geneva respectively, where they held conferences on the topics
“Fractals and Music” and “Music and Mathematics” (Steinitz 2003:274).
The 1980s marked an important turning point in Ligeti‟s life and consequently in his
compositions. He came in contact with Simha Aron, who led him to explore polyrhythms of
Central African music; discovered Conlon Nancarrow‟s compositions for player piano; and met
some of the founders of fractal geometry. (Svard 2000:803.)
4.6.2
Fractals in Ligeti’s compositions
4.6.2.1 Désordre
Ligeti‟s first piano etude Désordre is one of two pieces in which he made intentional use of ideas
from self-similarity and fractal geometry, the other being the fourth movement from his Piano
Concerto. In a conversation with Heinz-Otto Peitgen and Richard Steinitz in 1993, Ligeti dubbed
the first etude to be self-similar and consciously based on the structure of the Koch snowflake
(Steinitz 1996:8).
Désordre, meaning “disorder” is based on two prevalent notions from fractal mathematics; firstly
self-similar reductions or contractions characteristic of fractals and, secondly, chaos theory.
Steinitz (2003:280) explained that “Désordre was first conceived as a „pulsation‟ study and an
exercise in segregating the black keys from the white, but as Ligeti worked on it, it also became
an ingenious representation of chaos theory.”
In Désordre the right hand plays only on the white keys (heptatonic) and the left hand on the
black keys (pentatonic). The combination of these two scales is not a chaotic form of polytonality,
but rather a form of “combinatorial tonality” as the two scales together create an “illusion of a third
or resultant tonality” (Steinitz 2003:281-282). That illustrates the principal of chaos theory:
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something that might appear to be chaotic is in fact orderly. Steinitz (1996:8) explained that
“although [the] right and left hands each have independent metrical cycles, as logical processes
they look orderly and deterministic.”
The formal structure for the right hand and left hand are different. For the right hand there are a
total of fourteen iterations of the same melodic statement. Each statement is divided into three
phrases of four, four and six bars each. Each successive statement of the melody is transposed
one step higher on the heptatonic scale. The left hand also has a recurring melodic statement,
but it is longer than the right hand: four phrases of four, four, six and four bars each (Steinitz
1996:8).
Figure 90 shows the first page of Ligeti‟s Désordre. For the first four “bars” the right and left
hands are rhythmically synchronized. At the end of the first line, the right hand starts one quaver
beat earlier with the statement of the melody. Consequently accents between the right and left
hands no longer fall together. Ligeti continued with this process of contracting the statement of
the right hand and starting one quaver beat too early with each statement. This results in an
increased “chaotic” effect. Steinitz (1996:10) was of the opinion that this “resizing” of the phrase
structure in Désordre through iteration is a fractal characteristic and also likened it to the
construction of the Koch snowflake.
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Figure 90: Ligeti, Étude pour piano No. 1 (Désordre), first page (Ligeti 1986)
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The author could not find any evidence that Ligeti based Désordre on a strict mathematical
execution of fractals – no algorithms were used. Therefore, it is suggested that this work rather
be called fractal-inspired music as opposed to fractal music.
4.6.2.2 L’escalier du diable
The thirteenth etude is titled L’escalier du diable (the devil‟s staircase) and serves as another
example of fractal-inspired music. This is the only one of Ligeti‟s etudes that has a direct
reference to chaos theory in the title. The devil‟s staircase is closely related to self-similarity and
the Cantor set. The construction of the devil‟s staircase relies on repeatedly removing the middle
third of a square and pasting that rectangular slither next to the square. With each subsequent
step, the columns get increasingly higher and also increasingly closer to one another (Figure 91).
Figure 91: First three steps for creating the devil’s staircase (Peitgen et.al. 2004:210)
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When this iterative process is carried out an infinite amount of times, it results in the devil‟s
staircase:
Figure 92: The complete devil’s staircase (Peitgen et.al. 2004:210)
Steinitz (1996:18) described the devil‟s staircase as “a secondary phenomenon derived from the
recursive 1/3 to 2/3 proportions within the Cantor set; that is a series of unequal steps produced
by plotting […] the mathematical relationships between the eliminated and surviving segments.”
This is mirrored in this etude in several ways.
Firstly, the notes are grouped together asymmetrically in groups of twos and threes, playing on
the “binary-ternary geometry of the devil‟s staircase” (Figure 93). The music is grouped in “cells”:
2 + 2 + 3 / 2 + 2 + 2 + 3 / 2 + 2 + 2 +2 + 3 / 2 + 2 + 2 + 3 / 2 + 2 +3 / etc. According to Steinitz,
“…the elongated „step‟ of three quavers, emphasized by legato phrasing, makes a small plateau,
whilst the unequal but orderly progression of the subgroups recalls the irregular staircase of the
graphic image” (Steinitz 1996:19).
According to Ligeti, this etude was not directly inspired by fractal mathematics, but instead his
experiences during a bicycle ride in a storm. In 1993, when Ligeti was busy with the composition
of his etudes, he was living in Santa Monica, California. During that time, the coast was struck by
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an El Niño weather system. One day he had to make his way back to his hotel by bicycle. He had
to cycle against the wind and it was during this bicycle ride that the first ideas for L’escalier du
diable came to him. He thought of “an endless climbing, a wild apocalyptic vortex, a staircase it
was almost impossible to ascend.” Despite this, it is almost irrefutable that his interest and
knowledge of fractals influenced this composition as well; even if it was only sub-consciously
(Steinitz 2003:307-308).
Figure 93: Ligeti, Étude pour piano No. 13 (L’escalier du diable), line 1-4 (Ligeti 1998)
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In an interview for the New York Times in 1986, Ligeti admitted that he did not make use of
“direct mathematical translation” in his music. He said that “the influence is more poetic: fractals
are the most complex ornaments ever, in all the arts… They provide exactly what I want to
discover in my own music, a kind of organic development” (Rockwell 1986).
4.7
Conclusion
In this chapter it was explored how music can be composed with the use of fractal and selfsimilar algorithms and ideas. Literal translations of fractal geometry into music can be done with
fractal noise forms, as proposed by Voss and Clarke. Of all noise forms, 1/f noise (or pink noise)
was found to produce the most aesthetically pleasing music. This emphasises one of the keys of
music composition: that there should be a balance between recurring ideas and variation.
Another method that was first presented by Prusinkiewicz (1986) is the musical interpretation of
L-systems and turtle graphics. This was further developed by Mason and Saffle (1994). Tom
Johnson‟s experiments to create self-similar compositions are perhaps more for academic use.
The self-similar nature of his compositions displays characteristics of minimalism, which suggests
that minimalist music might also possess self-similar qualities.
The mapping of mountains and coastlines has also been a source of inspiration to composers
such as Larry Austin and Gary Lee Nelson. György Ligeti made use of fractal mathematics and
chaos theory in his music in a more metaphorical sense and the researcher suggests that it
should be referred to as fractal-inspired music.
Fulkerson (1992:756-757) made some comments regarding fractal composition that is important
to this study:
The model of fractal geometry excites many composers today and it is being
incorporated into their compositional work, but they are still struggling with mapping
problems. We can see the visual applications of this geometry, but mapped onto
music it has not yet shown anything like the same interesting results. One solution is
to transfer the data into timbral generation with greatly reduced pitch or harmonic
movement so that the listener‟s ear has an adequate orientation with respect to pitch
content. When applied to pitches in a chromatic context, fractal geometry has failed to
create the spectacular results that it has generated in the visual domain, because it is
impossible for the listener‟s ear to comprehend the pitch content.
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Most of the compositions discussed in this dissertation cannot be identified as fractal by simply
listening to the music. A diligent analysis of the score is necessary to point out fractal and selfsimilar features in each composition. It can be proposed that one might only be able to
characterise music as fractal if one can hear the fractal structures – just as easily as one can see
the fractal structure of the Julia set (Figure 2). Coupled with Brothers‟ (2004) strict prerequisites,
the existence of true “fractal music” seems even more unlikely.
Nevertheless, the researcher is of the opinion that any composition that was influenced by
fractals should be named appropriately as well. It is again proposed that one should rather refer
to music that was based on fractal ideas as “fractal-based” or “fractal-inspired” music. If an
analysis of a piece of music indeed reveals fractal structures, a term like “quasi-fractal” music
may be used. These terms have been used frequently by several scholars to describe music that
was composed with the use of fractals or that was inspired by fractals.
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CHAPTER 5: FRACTAL GEOMETRY AND SELF-SIMILARITY IN MUSIC ANALYSIS: MUSIC
PRIOR TO 1975
5.1
Introduction
This chapter explores the occurrence of fractal and self-similar structures in art music prior to
1975. Music examples include mensuration canons of the Renaissance, polyphonic compositions
of Bach and, finally, the works of Classical composers like Beethoven. Each composition
discussed will show a unique and different way in which various fractal-like structures are
exhibited.
5.2
Rhythmic self-similarity
5.2.1 The mensuration canon
The mensuration canon is one of the best examples of self-similarity in music. It is named after
the use of mensuration, which refers to the relationship between note values. The system of
mensuration was used in the time period between about 1250 and 1600 (Bent 2001:435).
Mensuration canons (also known as prolation canons) are prevalent in the works of the FrancoFlemish composers of the Renaissance period, such as Johannes Ockeghem and Josquin des
Prez (Hindley 1990:97-98).
In a mensuration canon, the melody appears simultaneously in all voices, but in different
mensurations or note values (Randel 1986:128). This is accomplished by augmentation or
diminution of the note values, which is known as scaling in mathematics (Etlinger n.d.:1).
Because scaling is involved, one can anticipate that some mensuration canons might be selfsimilar in structure.
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5.2.1.1 Ockeghem’s Missa Prolationum
The first example to illustrate the simultaneous use of the same thematic or melodic material in
different voices in different tempos is taken from Johannes Ockeghem‟s Missa Prolationum. As
the title suggests, this mass consists of mensuration canons. Figure 90 shows the Kyrie from the
mass. The two top voices (soprano and contra tenor) share the same melodic material, but the
contra tenor sings the melody in longer note values. A different melodic line is shared between
the tenor and bass, where the bass is in longer note values than the tenor. Note, however, that
the bass returns to the rhythmic values of the tenor from the first time the word eleison is sung.
This gives two distinct melodic lines moving at four different tempos.
Figure 94: Ockeghem, Kyrie from Missa Prolationum, bars 1-18 (Ockeghem 1966)
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The Gloria in excelsis Deo from the same composition has a canonic struture similar to the Kyrie.
Again, the soprano and contra tenor sing the same melody, but the contra tenor‟s part is
augmented in longer note values. However, there is a canon in the same note values between
the tenor and bass, where the bass enters after all of the other voices.
Figure 95: Ockeghem, Gloria in Excelsis Deo from Missa Prolationum, bars 1-14
(Ockeghem 1966)
Further examples of mensuration or prolation can be found in Ockeghem‟s Missa Prolationum.
This composition is a unique example of rhythmic self-similarity obtained through augmenting
and diminishing the note values in the piece.
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5.2.1.2 Josquin Des Prez’s first Missa l’homme armé
A similar example can be found in Josquin des Prez‟s second mass entitled Missa l’homme armé
super voces musicales. The second Agnus Dei consists of a three-part mensuration canon. This
movement will also be used to illustrate rhythmic self-similarity obtained in a mensuration canon.
In the Agnus Dei, the tenor (middle voice) moves in minims and is the slowest of the three
voices. It serves as the cantus firmus or fixed voice of the canon. (It was common in Medieval
and Renaissance music for the tenor to be the slowest-moving voice.) The soprano moves three
times as fast as the tenor, thus giving a ratio of 3:1 between the two voices. The bass moves only
twice as fast as the tenor, giving a ratio of 2:1. Also note that the soprano‟s and bass‟s melodic
line was transposed a fourth higher than the tenor (or fifth lower) starting on a D instead of an A
(Brothers 2004).
Figure 96: Des Prez, Agnus Dei II from Missa l’homme armé super voces musicales, bars
1-12 (Brothers 2004)
Hence, the soprano and bass voices can be seen as smaller copies of the tenor voice, since their
note values will be the same as that of the tenor when augmented. This Agnus Dei can thus be
regarded as a rhythmically self-similar composition. In his research, Brothers (2004) referred to
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this type of self-similarity as rhythmic scaling because the same rhythm is presented in different
levels of augmentation or diminution.
5.2.2
Other examples of rhythmic self-similarity
Rhythmic self-similarity is not found exclusively in mensuration canons, but also appears in the
works of many other composers such as J.S. Bach. The fugues in Die Kunst der Fuge, BWV
1080, contain numerous instances of augmentation and diminution. Although these rhythmic
alterations do not always occur simultaneously, Bach frequently used such rhythmic alterations of
motifs or themes.
Die Kunst der Fuge is a monothematic composition consisting of various fugues in which Bach
exploited various transformations. Figure 97 shows the subject that is used throughout the entire
composition:
Figure 97: Subject in Bach’s Kunst der Fuge, BWV 1080 (Alvira 2013)
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Figure 98 shows the first ten bars of Contrapunctus VI from Die Kunst der Fuge. (Note that Bach
had already altered the subject rhythmically in this fugue by using dotted rhythms.) The fugue
starts in the bass14 (bar 1). The soprano enters in bar 2, but in melodic inversion and diminution.
In the middle of bar 3, the alto enters with the subject also in diminution. This music example
shows the use of the same subject in staggered entries.
Figure 98: Bach, Contrapunctus VI, from Die Kunst der Fuge, BWV 1080, bars 1-5 (Adapted
from Bach n.d.)
14
Bach did not specify instrumentation for the fugues in Die Kunst der Fuge. The researcher therefore
refers to different lines of music as soprano, alto, tenor and bass when discussing fugues from this work.
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In Contrapunctus VII a 4 per Augment et Diminu, the tenor starts the fugue (bar 1), but in
diminution. It is followed by the soprano (bar 2) in the original note values, but melodic inversion.
The alto (bar 3) displays both melodic inversion and diminution. The bass enters in bar 5 with the
note values doubled. In bar 6 and 7 the subject appears in the tenor and alto respectively with
the note values halved.
Figure 99: Bach, Contrapunctus VII a 4 per Augment et Diminu, from Die Kunst der Fuge,
BWV 1080, bars 1-12 (Adapted from Bach n.d.)
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The fourth stretto in Contrapunctus VII is an example of three simultaneous statements of the
fugue. It starts in bar 23 in the alto, where the subject‟s note values are halved. One beat later,
the tenor enters with an augmentation of the subject. Finally, in the middle of bar 24, the soprano
enters with the subject in diminished note values and melodic inversion (Alvira 2013).
Figure 100: Bach, Contrapunctus VII a 4 per Augment et Diminu, from Die Kunst der Fuge,
BWV 1080, bars 21-28 (Adapted from Icking (ed.) n.d.)
The other stretti in Contapunctus VII are only between two voices (Alvira 2013). The excerpt in
Figure 100 is therefore the best example of momentary rhythmic self-similarity that the
researcher could find for this study.
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Shorter examples can be found in countless compositions. In an Allemande by Maurice Greene
(1696-1755), there is a short example of rhythmic augmentation in the anacrusis to the first bar.
The right hand opens with an ascending line from A to D in semi-quavers. The left hand copies
this line starting on a D and ending on an A, but in quavers. The left-hand part is thus a
transposition and augmentation of the right hand for that one moment.
Figure 101: Greene, Allemande, bars 02-1 (Greene 1995)
In the Allegro di molto movement of C.P.E. Bach‟s sixth keyboard sonata there is another
interesting composition technique obtained through the use of augmentation. In the second bar of
the movement, the right hand plays a broken triad on the tonic of f minor. The pattern C-A -C
repeats itself. This jump of a third is inverted in the left hand in the same bar (F-A -F) and the
note values are augmented.
Figure 102: C.P.E. Bach, Allegro di Molto from Keyboard Sonata No. 6, bars 1-2 (Bach
1995)
This play with the same intervals in the right and left hand, but in augmented rhythms,
characterises the entire movement. The same technique can be found in the works of other
composers as well.
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5.3
Structural self-similarity
Some researchers, such as Solomon (2002), Brothers (2004, 2007) and Lee (2004) found the
structures of compositions to be self-similar or fractal-like. Some even went as far as comparing
the structure of such compositions to fractals like the Cantor set and Sierpinski triangle.
5.3.1
Structural scaling in Bach
In the next few music examples, it is shown how a visual representation of a composition can
look similar to the structure of some fractals. In an article in the journal, Fractals, Harlan Brothers
showed how the phrasing of the first Bourrée from Bach‟s Cello Suite No. 3, BWV 1009 displays
structural scaling. That means that the composition‟s structure is fractal on different scales with
regard to the phrasing, and hence its structure (Brothers 2007:91).
The Bourrée is in binary form, with section A from bar 1 to 8 and section B from bar 9 to 28 (see
Figure 106). There are repetition signs at the end of both sections, but according to Brothers
(2007:93), it is customary for cellists to perform only the first repetition, thus giving an AAB
structure to the entire Bourrée. Although the second section is not exactly twice the length of that
of the first, the overall structure is short-short-long.
The first section of the Bourrée, marked A in the score, can be divided into a smaller AAB
structure: bars 1-2 and its sequence in bars 3-4 is followed by longer phrase from bar 5 to 8.
(See the red brackets in Figure 103). Next, bars 1-2 and bars 3-4 can be analysed in a similar
fashion: two similar three note motifs are followed by a longer melodic line, which is yet another
AAB structure. (See the blue blocks in Figure 103.) The smallest scaling unit in the Bourrée is the
three-note rhythmic motif where two quavers are followed by a crotchet (short-short-long), giving
the smallest AAB structure in the movement.
Brothers stated that one can thus see the entire Bourrée as an AAB structure on four different
scales, where A is short and B is twice as long. This is true for the two main sections of the
movement, the phrases and the main rhythmic motifs. Every A section has its own AAB structure,
but on a smaller scale. This is known as structural scaling, as was briefly discussed in section
3.4.4.3, and is also a form of self-similarity.
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Figure 103: Bach, Bourrée I from Cello Suite No. 3 in C major, BWV 1009 (Adapted from
Bach 1911)
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5.3.2
Bach and the Cantor set
In addition to the Bourrée having self-similar structural scaling, it can also be likened to the
structure of a type of fractal, namely the Cantor set. In the second chapter it was explained how
the Cantor set is constructed by repeatedly extracting the middle third of a line segment.
However, the Cantor set can also be created by using L-systems with the following substitution
map: A → ABA, B →BBB. (This means that A is replaced by ABA, and B is replaced by BBB.) In
the case of the Cantor set, A represents a straight line while B represents a gap (see Figure 104)
(Brothers 2007:93).
The structure of the Bourrée can be reproduced by using a similar iteration as the one used for
the Cantor set. The substitution map is changed slightly: A → AAB instead of A → ABA. Now,
compare the Cantor set (Figure 104) to a Cantor map of the first 16 bars (the first eight bars
which are repeated) of the Bourrée (Figure 105).
Figure 104: Initiator and first three iterations of the Cantor set created with L-systems
Figure 105: Cantor map of the first 16 bars of Bach’s Bourrée I from the Cello Suite No. 3
in C major, BWV 1009 (Brothers 2007:93)
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In Figure 105, the red regions each represent four beats (two measures); the blue regions, one
beat; and the yellow regions, quaver beats. The white regions do not represent “nothingness”,
but rather “everything that is not part of the short phrase at each scale measurement”. (See the
score in Figure 103.) From these illustrations it becomes clear how the Cantor set can be used to
give a visual representation of the structure of a piece of music. This visual representation also
accurately displays the recursive nature of this Bourrée (Brothers 2007:89, 93).
5.3.3
Beethoven’s Piano Sonata No. 15 in D major, Op. 28
The next section explores how composers use a single motif or phrase as a building block to
create an entire movement or, in the case of Beethoven‟s Piano Sonata No. 15 in D major, Op.
28, a multi-movement work. The first theme of the first movement, Allegro, contains short melodic
and rhythmic motives which are repeated, expanded and varied throughout all of the sonata‟s
movements.
5.3.3.1 Structural analysis of the first movement
The Allegro first movement is in sonata form. The exposition (bars 1-163) starts with the first
theme in the tonic key, D major. The first theme is ten bars long (bars 1-10) and is characterised
by a pedal point on the note D in the left hand; a step-wise descending melody in the right hand;
and tied notes that create suspensions, anticipations and syncopated rhythms. Bars 11-20 is an
exact repetition of the first theme, transposed an octave higher. Bars 21-28 is a slight variation of
the first theme and is also repeated an octave higher in bars 29-39. The key remains in D major
for the entirety of the first theme.
Bar 40 marks the beginning of an episode which is in A major and ends in the same key in bar
62. This leads to the second theme (bars 63-159) in the dominant key of A major. The first part of
the second theme is characterised by a melody (in octaves) in the outer voices and arpeggiated
harmonies in the inner voices. The melody contains many intervals of a semitone. Bars 160-163
is a single descending line that functions as a link to the development section.
The development section (bars 164-268) starts with the first theme in G major and develops
mainly ideas from the first theme. After the first theme is stated in G major (bars 167-176), it is
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repeated an octave higher, but in G minor (bars 177-186). Thereafter the tail of the first theme is
presented multiple times while the music alternates between G minor and D minor. The
development section ends in the key of B minor. Bars 266-268 is marked Adagio – the only
tempo change in the movement – and serves as a link to the recapitulation.
The recapitulation (bars 269-460) starts with the first theme in the tonic key, D major. Thereafter,
the first episode is again presented in A major. The recapitulation ends with a Coda (bars 438460) that is based on the first theme. The table in Figure 106 summarises the main sections and
keys of the first movement from this sonata.
Exposition
Development
Recapitulation
1-39
First theme
D major
164-268
269-311 First theme
D major
40-62
Episode
A major
Modulations from
312-336 Episode
A major
63-159 Second theme A major
G major through
337-434 Second theme
160-163 Link
G minor, D minor,
435-437 Link
B minor
438-460 Coda
D major
Figure 106: Analysis of the first movement from Beethoven’s Piano Sonata No. 15 in D
major, Op. 28 (Adapted from Harding n.d.:30-31)
5.3.3.2 Scaling of the first theme in the first movement
According to Lee (2004), the first theme of the Allegro is the basic building block for the entire
movement, not only the melodic and rhythmic patterns, but also the underlying harmonies.
Examine the descending melodic line of the first statement of the first theme. Figure 107 shows
the first ten bars of the movement with the harmonic notes circled in red. When only the harmonic
notes are considered and non-harmonic notes are ignored, it results in the following notes: D – A
– G – D – B – A – D. (Note that the first D in the left hand is also regarded as a melodic note,
since subsequent statements of the theme all start on a D.)
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Figure 107: Beethoven, Allegro from Piano Sonata No. 15 in D major, Op. 28, bars 1-10
(Adapted from Beethoven 1975)
Lee (2004) found that the harmonic notes in the first eight bars foreshadow the key changes
throughout the movement. The exposition starts in the tonic key (D major) after which it
modulates to the dominant key (A major) for the episode and the second theme. The
development section starts in G major and then modulates to and fro between D minor and G
minor. In bar 214, the development modulates strongly to B minor, where it stays until the end of
the development. The development closes with a short Adagio section of three bars on an A
major seventh chord (the dominant seventh of D major) before the recapitulation starts in the
tonic key again.
From this short discussion one can see that the main keys through which the first movement
modulates are directly linked to the harmonic notes of the melody in the opening of the
movement.
Lee (2004) showed that there is scaling between the harmonic notes in the first theme of the
movement and the different keys in the entire movement. It was also pointed out that the
rhythmic and melodic motifs of the first theme were altered to create the material for the rest of
the composition. The following examples of the first movement justify this further.
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In bar 21, it appears as if new material is presented, but in fact, old material is presented only in a
different manner. Bars 21-23 (see Figure 108) contains the same material as in bars 3-5. Here it
is in melodic inversion and note augmentation coupled with slightly different harmonies and a
pedal point in the inner voice instead of the bass.
Figure 108: Beethoven, Allegro from Piano Sonata No. 15 in D major, Op. 28, comparison
between bars 3-5 and 21-23 (Adapted from Beethoven 1975)
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From bar 21 onwards Beethoven expands on the idea of tied notes that create suspensions (see
Figure 109). Coupled with the sforzandos, it creates a hemiola-like effect in which the barlines
become vague. Also note the placement of the sforzando markings: on the G, D and A. These
are not only the notes on which the primary chords of D major are built, but also some of the
chordal notes in the first theme discussed above.
Figure 109: Beethoven, Allegro from Piano Sonata No. 15 in D major, Op. 28, bars 21-28
(Adapted from Beethoven 1975)
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In the episode (bars 40-62), Beethoven does away with the pedal point, but still builds on the
idea of a falling second in the melody as well as tied notes (see Figure 110). The opening melody
of the episode (bars 40-43) is presented and then transposed directly to E major (bars 44-47).
Figure 110: Beethoven, Allegro from Piano Sonata No. 15 in D major, Op. 28, bars 40-47
(Adapted from Beethoven 1975)
The tail of the first theme (bars 7-10) also plays a prominent role in the entire sonata. In the
development section, the theme is stated in G major and then modulates to d minor. From bar
187 the tail of the theme is heard three times in full, then four times, but only its end. From bar
207, the last note of the tail becomes the first note of its next appearance, thus allowing the
motifs to overlap. From bar 219, this motif is inverted in the tenor. Figure 111 highlights the tail of
the theme as it occurs in bars 201-212.
Figure 111: Beethoven, Allegro from Piano Sonata No. 15 in D major, Op. 28, bars 201-212
(Adapted from Beethoven 1975)
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From the music examples in Figures 107-111, it is clear that the same thematic and motivic
material was used by Beethoven to create the entire first movement for his Piano Sonata No. 15
in D major, Op. 28.
5.3.3.3 Interrelationship of material between movements
Even more impressive than the similarities among sections in the first movement of Beethoven‟s
Piano Sonata No. 15 in D major, Op. 28, is the use of the same melodic motif for the opening of
the second and fourth movements and the trio of the third movement. Each of these are
discussed in the next section.
The second movement (Andante) starts with the same notes in the melody as the first
movement, only in the tonic minor (D minor). The melodic contour of the first two bars of the
Andante is also the same as the first five bars of the Allegro (Figure 112). In addition, the rhythm
can be seen as an altered diminution of what appeared in the first movement. The D and A are
long notes, the G slightly shorter and the F and E even shorter and dotted.
Figure 112: Beethoven, Andante from Piano Sonata No. 15 in D major, Op. 28, bars 1-2
(Adapted from Beethoven 1975)
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The Trio in the Scherzo and Trio third movement is built on the same descending melodic
contour of the first theme of the Allegro, but in the relative minor, B minor.
Figure 113: Beethoven, Trio from third movement from Piano Sonata No. 15 in D major,
Op. 28, bars 1-2 (Adapted from Beethoven 1975)
The melody in the Rondo (fourth and final movement) again follows the same descending
contour of the theme in the first movement. The countermelody in the alto voice is canonic to this
and presents the same melodic contour as well. There is also a pedal point on the tonic in the left
hand, as in the first movement.
Figure 114: Beethoven, Rondo from Piano Sonata No. 15 in D major, Op. 28, bars 1-4
(Adapted from Schenker (ed.) 1975)
According to Ludwig Misch, “this sonata is probably the only one among Beethoven‟s
instrumental works that carries through the principle of thematic unity with unequivocal clarity”
(Lee 2004).
147
Figure 115 summarises the openings from each movement in Beethoven‟s Piano Sonata No. 15
in D major, Op. 28 to illustrate the similarities between them.
Figure 115: Comparison of the melodic contour of (a) first movement, (b) second
movement, (c) trio from the third movement and (d) fourth movement of Beethoven’s
Piano Sonata No. 15 in D major, Op. 28 (Adapted from Beethoven 1975)
5.3.3.4 Comparison between the third movement and the Sierpinski triangle
Independent of Lee (2004), Larry Solomon‟s analysis of the third movement (Scherzo and Trio)
displays a remarkable similarity to the construction of the Sierpinski triangle. The entire
movement consists of three parts: the Scherzo (A) in the tonic key of D major, the Trio (B) in the
relative key of B minor, followed by a return of the Scherzo (A). According to Solomon (2002),
this can be visually represented by the first step of the Sierpinski triangle, where the three sides
of a solid triangle are drawn.
148
The Scherzo and Trio each have their own inherent structures (see Figure 116). The Scherzo is
clearly divided into two sections, separated by a repeat sign at the beginning of bar 33. Bars 1-32
is A1, while A2 starts in bar 33. There is, however, a return of A1 in bar 49. This makes the
structure of the Scherzo rounded binary:
A1 ||: A2 A1 :|| ||:B1:|| B2 A1 ||: A2 A1 :||
Solomon (2002) argued that since A2 contains A1 within it, the A section can be seen as a
“miniature ABA within the A”. If the Scherzo is then seen as ternary rather than rounded binary in
form structure, it can be represented as yet another step of the Sierpinski triangle.
Solomon (2002) suggested that the structure of the Scherzo and Trio is similar to that of the
Sierpinski triangle when he writes that the movement is “a combination of binary and ternary
schemes similar to the Sierpinski structure”. Although he describes the structural analysis of the
movement in great detail, much of how it relates to the Sierpinski triangle was left to the
discretion and insight of the reader.
149
150
Figure 116: Beethoven, Scherzo and Trio from Sonata No. 15 in D major, Op. 28
(Beethoven 1975)
5.4
Arvo Pärt’s Fratres für violine und klavier
Pärt‟s Fratres für violine und klavier15 exhibits many examples of symmetry and self-similarity.
This is not only evident in the large-scale structure of the composition, but also in its smaller
divisions and subdivisions. The composition is characteristic of Pärt‟s tintinnabuli style that he
started to employ from c.1976 (Hillier 1997:98). Although defined as a style, the tintinnabuli
serves as a formula to create the entire composition.
15
Pärt (1980) arranged many different instrumentations and versions of Fratres. The arrangement for violin
and piano is used as frame of reference for this study, but the structure for all arrangements is the same.
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5.4.1
The structure
This composition has a unique vertical and horizontal structure. Horizontally, the composition is
clearly divided into nine numbered sections of six bars each. Each section is divided by a two-bar
“reprise”. The first section is for the solo violin, whereas the rest of the composition is written for
violin and piano (www.linusakesson.net).
The harmonies in the composition always change according to a specific rhythm throughout the
piece. Figure 117 shows the rhythmic structure of each segment:
Figure 117: Rhythmic structure of Fratres (Adapted from Pärt 1980)
At first glance, it is easy to discern the structure of the piece, but it is more difficult to make sense
of the harmonies or chords being used. Vertically, the piece consists of two melodic voices with a
so-called “tintinnanbuli voice” between them, as well as two drones (on A and E)
(www.linusakesson.net). To better understand the tintinnabulli style used in this composition, the
first section is examined in detail.
152
Figure 118: Pärt, Fratres für violine und klavier, first page (Pärt 1980)
153
The first section stands in contrast to the rest of the work because the violin is not accompanied
by the piano and all chords are arpeggiated. The following is a condensed version of the first
section:
Figure 119: Pärt, Fratres für violine und klavier, reduced copy of bars 1-6 (Adapted from
Pärt 1980)
By looking at this condensed representation, one can see a two characteristics of the piece that
remain constant throughout:
1. The 7/4 bar and 9/4 bar are reductions of the 11/4 bar, in which an eight-chord sequence
appears. This shows some level of fragmentation in the composition.
2. In the first half of each “bar”, the outer voices move downwards, while the second half
moves upwards. In the middle of each bar there is an octave displacement, possibly to
create the illusion of continuous upward or downward movement.
One can also see the parallel movement in tenths between the two outer voices, in this case the
melodic voices (M-voices). Also note that the M-voices move stepwise along the notes of the D
harmonic minor scale (D E F G A B C# D). This is characteristic of Pärt‟s tintinnabuli style.
The middle voice, called the tintinnabuli voice (T-voice) incorporates only notes from the A minor
triad (A-C-E). It is used in such a way that there is never an interval of a unison or octave
between the T-voice and the M-voice(s). Later in the composition, the T-voice is emphasised in
the use of a drone on the notes A and E in the piano part. Although the C natural of the T-voice is
never heard against a C# in the melodic voices, the change between these two notes creates
interesting harmonies.
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Although the key signature of the piece implies D minor, there is no traditional sense of tonality.
Instead, the root of the A minor triad features as the central pitch in the composition. Throughout
the composition there is movement towards and away from the central pitch.
Akesson (2007) showed that the formulation for the entire composition relies on two rotating
wheels, one for the melodic voices and another for the tintinnabuli voice.
C#
D
E
B
A
F
G
Figure 120: Rotating wheel used for the melodic voices in Pärt’s Fratres für violine und
klavier (Akesson 2007)
E
A
E
A
C
C
C
C
C
E
C
E
A
A
Figure 121: Rotating wheel used for the tintinnabuli voice in Pärt’s Fratres für violine und
klavier (Akesson 2007)
155
In the first three bars of the first section, the M-voices start on C# and E, and then move counterclockwise for each consecutive chord. Meanwhile the T-voice starts on A (at the top of the wheel)
and also moves counter-clockwise. This produces the following eight-chord sequence:
Figure 122: Pärt, Fratres für violine und klavier, resulting chord sequence in bar 3
(Adapted from Pärt 1980 and Akesson 2007)
Note how the melodic voices move downwards, except for the octave displacement in the middle
of the chord sequence.
Bars 4-6 of the first section starts on the same chord as the first half, but slightly different
harmonies are produced. This is obtained by letting the M-voices start on C# and E again, but
rotating the wheel clockwise. The T-voice begins on A at the top of the wheel and continues to
move clockwise.
Figure 123: Pärt, Fratres für violine und klavier, resulting chord sequence in bar 6
(Adapted from Pärt 1980 and Akesson 2007)
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5.4.2
Expansion and reduction
A central “theme” that can describe the composition of Fratres is the idea of expansion. Since the
work is a type of variation-form, there is no development. Instead, musical ideas are expanded
on different scales. This gives rise to many self-similar structures in the work. The most obvious
example of this would be the expansion of a 7/4 bar with four chords, to a 9/4 bar with six chords
and finally eight chords in an 11/4 bar. As already mentioned, this is found throughout the
composition.
Another striking example of this is in the seventh section (Figure 124). The violin starts with
single notes in the first bar, then plays double stops in the next bar and finally triple stops in the
third bar. The process is reversed in the second half of the section (Zivanovic 2012:33).
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Figure 124: Pärt, seventh section from Fratres für violin und klavier (Pärt 1980)
158
A similar example is found in the eighth section, but here the expansion is rhythmical and not in
the texture. The rising arpeggiated melody begins in quavers, expands to triplets and further into
semi-quavers in each consecutive bar.
Figure 125: Pärt, eighth section from Fratres für violine und klavier (Pärt 1980)
159
Fratres is an apt example of how a few ideas can be used to create an entire musical work.
Moreover, all of these melodic and harmonic ideas are also fragmented within each section of the
composition. Although no writings on the piece characterised Fratres as fractal or self-similar, the
researcher believes that this composition can be used as another example of symmetry and selfsimilarity in music.
5.5
Conclusion
Polyphonic compositions of the Renaissance and Baroque have supplied some of the best
examples of rhythmic self-similarity, since they enabled the composers to use two or more
transformations of the same motif or theme simultaneously. Just like one is able to see several
transformations of a shape in a fractal object at one glance, one can hear the transformations at
the same moment in time.
Mensuration canons of the Renaissance are an excellent depiction of the same theme or motif
being sung at different speeds, which displays rhythmic scaling. The canons and fugues of J.S.
Bach also supply ample illustrations thereof, although it is only momentarily, mostly in the stretti
of his fugues. Although only a few of Bach‟s compositions were highlighted in this chapter, more
examples can be found in Die Kunst der Fugue and Das Wohltemperierte Klavier.
Structural self-similarity was pointed out in the Bourrée from Bach‟s first Cello Suite. Whether or
not its structure truly represents that of the Cantor set is up for debate, but it is clear that the
structuring of the phrases, sub-phrases and motifs mirror one another.
Beethoven‟s Piano Sonata No. 15 in D major, Op. 28 is a striking example of the way in which a
composer built an entire work from one single theme. Different transformations such as
transposition, retrograde and scaling of note values are also apparent.
The author could not find any evidence that Arvo Pärt was ever directly influenced by fractal
mathematics. Therefore, Fratres was categorised among compositions prior to 1975, despite the
date of his composition (1980); it simply signifies that the work was conceived without a direct
influence of fractal geometry. Fratres displays many examples of expansion, fragmentation and
transformation of the same chord sequences with strong self-similar structures.
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CHAPTER 6: CONCLUSION
6.1
Summary of findings
Throughout this dissertation, several links between music and mathematics and specifically
geometry were highlighted. Furthermore, the researcher illustrated how musical motifs or themes
can be transformed in a similar way to geometric objects, which implies the possibility of applying
fractal geometry to music as well. One transformation that is common to both music and fractals
is that of scaling.
All fractals must adhere to specific prerequisites, namely self-similarity, fragmentation on all
scales and fractal dimensionality. Brothers (2004) translated some of these properties to apply to
music.
Three main noise types have been applied to both music composition and analysis: white noise,
brown noise, and fractal 1/f (pink) noise. Spectral density analysis of compositions has proven
that many classical compositions imitate 1/f noise, making them fractal (Voss & Clarke 1975,
1977; Hsü & Hsü 1990, 1991).
Classical music also showed the greatest resemblance to 1/f noise of a wide spectrum of genres,
including jazz, blues, folk tunes from several cultures, rock and pop. Further developments led
scholars to believe that the level of 1/f noise distribution in music can be used to define a specific
genre in art music and even the stylistic characteristics of different composers (Ro & Kwon 2009;
Levitin et al. 2012).
The application of fractal geometry and self-similarity is not limited to scientific research, but also
has some noteworthy practical implications. The first is the application to music composition.
Many different methods for fractal music composition were investigated in Chapter 4.
The first method is the utilisation of noise forms. Most scholars found that the use of 1/f noise
was the most successful in this regard, since it is midway between chaos and predictability.
Since 1/f noise is fractal, it is reasonable to assume that music created from 1/f noise is also
fractal (Gardner 1992; Voss & Clarke 1977).
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The second fractal that can be used to create music compositions is Lindenmayer systems and
their visual interpretations with the use of turtle graphics. This method was first proposed by
Prusinkiewicz in 1986 and further developed by Mason and Saffle (1994).
The coastline paradox was a source of inspiration, not only for the research of Andrew and
Kenneth Hsü (1991), but also for the music of composers like Larry Austin and Gary Lee Nelson.
Since the scaling of coastlines and mountains is fractal, these composers used these fractal
qualities to map these natural phenomena as music.
Tom Johnson (2006) created many self-similar compositions with the utilisation of iteration and
scaling. His compositions are musical experiments by means of self-similarity and have a
minimalistic feel.
Some of the piano etudes by Ligeti contain metaphoric references to fractal geometry and chaos
theory rather than direct translations of fractals into music; for example Étude pour piano No. 1
(Désordre) and Étude pour piano No. 13 (L’escalier du diable). These types of composition can
be defined as “fractal-inspired”.
All of the compositions discussed in Chapter 5 were referred to as fractal or self-similar by the
composers themselves or musicologists who conducted research on their works. In Chapter 2 it
was shown that Brothers (2004) firmly believed that the translation of a fractal into music does
not necessarily guarantee that a piece of music be fractal. He proposed that such music should
be referred to as “fractal-inspired” music. The researcher also proposed the use of terms such as
“fractal-based” and “quasi-fractal” music.
The appearance of fractal or self-similar structures in music before Mandelbrot defined fractal
mathematics can possibly be attributed to the fact that many fractals appear in nature. Another
possible reason is that the arts, or in this case, music, reflected the scientific thinking of the times
and even anticipated it (Shlain 1991).
While there is no specific method for the analysis of music with regard to fractals and selfsimilarity, such patterns were found in music as early as the Renaissance. A promising type of
self-similar music exists in the form of mensuration canons where the same material is sung
simultaneously, but at different speeds.
162
Solomon (2002) and Brothers (2004) attempted to show that the structure of some musical
compositions can be likened to the structure of well-known fractals like the Sierpinski triangle and
Cantor set. Although this amalgamation may seem somewhat far-fetched by some readers, the
self-similar structures in the works they analysed cannot be refuted.
6.2
Conclusions
From the study it can be concluded that fractal geometry and self-similarity can indeed be applied
in art music in the following ways:

Fractals and self-similarity are a useful composition tools.

Fractals can be used to categorise music in specific genres or to belong to a specific
composer.

Since the development of fractal geometry in the sciences, an increasing number of
musicians have drawn inspiration for their music from fractals and self-similarity.
The different methods in which fractals can be used in music composition include:
 direct translation of noise wave forms, like 1/f noise, into music notation
 direct translation of an L-system or self-similar curve as music
 mapping of the contour of a fractal structure, such as a mountain range or coastline, as
the melodic contour of a composition
 iteration of a musical idea, motif or theme in order to create self-similar musical structures
 metaphoric use of fractals in a composition
Examples of entire works or movements being built on fractals or self-similarity in music prior to
the 20th century are scarce. Nevertheless, “moments” of fractality or self-similarity can be seen in
several compositions, as was shown in Chapter 5. In essence, most classical music can be
defined as fractal in layperson‟s terms: most classical compositions rely on the use of similar
themes and motifs to create uniformity, but also inserting new material to avoid monotony. The
163
research by the Hsüs and Voss and Clarke emphasised that these same qualities can be
obtained when using fractal 1/f noise to compose.
Other than composing or analysing music with fractals and music, the researcher was surprised
by the possibility that fractal distributions in a composition may be used to categorise it into a
specific time period or genre. The prospect that it might be useful to verify the style of a specific
composer mathematically is astounding and could be useful when trying to determine the
authenticity of a specific composer‟s work.
Throughout the 20th century, composers were often influenced by scientific thinking of the time. If
nothing else, the rise of fractal geometry in the sciences served as a new source of inspiration to
composers to create music that had not been composed or heard before. Similarly, fractal
geometry did not bring anything new to the way music is analysed, but it can certainly change
one‟s thinking about music analysis. In the discussions in Chapter 5, certain motifs, themes or
ideas relating to the melody, rhythm or structure of several compositions were highlighted. These
can in turn be used by performers to give different interpretations of the music.
6.3 Summary of contributions
Throughout this dissertation the researcher deliberately simplified the mathematical and scientific
aspects of fractal geometry and self-similarity. This was done to enable musicians who do not
necessarily have a strong mathematical background to understand these concepts in order to
incorporate them into music composition and analysis.
To the researcher‟s knowledge this is one of the first academic works in which several facets of
fractals are discussed together with their musical connotations. Previous scholars addressed only
a single aspect of fractal music, for example, Lindenmayer systems or 1/f noise. In this
dissertation, a number of fractals that can be applied musically were highlighted.
The researcher, like Brothers (2004), posits that one should be careful not to use the term “fractal
music” too loosely. As seen from the works analysed, many compositions can only be
characterised as “fractal-inspired”, “fractal-based” or “quasi-fractal”.
164
Initially, the researcher endeavoured to define the term “fractal music”, but was unsuccessful for
a number of reasons. It is believed that the characteristics of fractal objects should also be valid
for so-called “fractal music”. This implies self-similarity, iteration, scaling and a fractal dimension.
As indicated in the research by the Hsüs, there is still ambiguity surrounding the calculation of the
fractal dimension of a piece of music. Since this calculation largely relies on spectral density
analysis, which could not be duplicated in this study, the fractal dimension of compositions was
not considered.
In addition, spectral density analysis of compositions does not have many practical applications,
except possibly to validate the genre or authenticity of the composer (as pointed out by Ro and
Kwon 2009 and Levitin et al. 2012) Instead, it is an interesting mathematical analysis of music.
The researcher, like Cook (1992), believes that any analysis should be of practical use – in other
words it should be valuable to the performer. When the smaller motifs and themes in a
composition are analysed with regard to the larger structure and similarities between them can
be found, it enables the performer to highlight these in his or her playing. Likewise, such analysis
provides the analyst with a new way of thinking about the music.
The researcher was also successful in pinpointing certain discrepancies in some of the existing
literature in the literature review. Although it was not possible to redo the experiments in order to
correct methods, this should be use to future scholars who wish to investigate the topic further.
6.4
Suggestions for further research
As mentioned earlier in this chapter, the analysis of compositions with 1/f noise distribution may
be helpful in determining the authenticity of the works by specific composers. The earlier
research conducted by Ro and Kwon (2009) and Levitin et al. (2012) indicated that a certain level
of 1/f distribution in a piece of music can be linked to a specific genre or even a specific
composer.
Although the spectral density analyses conducted by scholars like the Hsüs (1990, 1991) and
Voss and Clarke (1975, 1977) do not seem to have many practical applications for music
165
analysts or performers, it is suggested that future musicologists should recreate their
experiments in order to shed some light on aspects that were previously unclear.
6.5
Suggestions for further reading
More can be read about the aesthetic implications of fractal melodies in Michael Beauvois‟
article, “Quantifying aesthetic preference and perceived complexity for fractal melodies” (2007). It
is a further investigation of which type of fractal noise creates the most pleasing melodies
(Beauvois 2007:247).
Different methods for calculating the fractal dimension of a musical composition were discussed
in the article, “Fractal dimension and classification of music” by Bigerelle and Iost16 (2000). The
complexity of the mathematical formulae used in the article prevented its use in this dissertation.
These authors‟ research proved that “different kinds of music could be discriminated by their
fractal dimension” (2007:2191).
Jonathan Foote (n.d.) developed a method for converting the sound waves from musical
compositions into graphs, creating visual representations of the music. If the visualisation is selfsimilar it would mean that the corresponding piece of music is also self-similar.
In the introduction to Applications of fractals and chaos: the shape of things, Crilly et al. (1993:4)
wrote the following on fractal research in the 1990s:
The current wave of interest [in fractals and chaos] may be faddish, but there is
enough solid theoretical material to ensure that the topic‟s importance will extend
beyond any fashionable peak. They continue to have value for both technical and
aesthetic applications. Fractals and Chaos are here to stay.
Today, 34 years later, fractals and chaos are an integral part of scientific research. Possibly, in
another 30 years, one will be able to look back and see that fractals have also played a vital part
in the development of music in the first half of the 21st century. To paraphrase Crilly et al.
(1993:4) “Fractal and self-similar music are here to stay…”
16
The surname Iost is not a spelling error; it starts with a capital i.
166
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