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Wesleyan Ƈ�8QLYHUVLW\
FROM ADMISSION TO EXITING:
THE GRADUATE STUDENT EXPERIENCE
AT WESLEYAN UNIVERSITY
By
John J. Smith
Faculty Advisor: Dr. Susan Meyer Jones
A Dissertation submitted to the Faculty of Wesleyan University in partial fulfillment of the
requirements for the degree of Master of Arts
Middletown, Connecticut May 2012
THE MUSICAL MAPPINGS OF
L-SYSTEMS
By
Maxwell Tfirn
Faculty Advisor: Dr. Paula Matthusen
A thesis submitted to the Faculty of Wesleyan University in partial fulfillment of the requirements for the degree of Master of Arts
Acknowledgment
! Without the help of my committee members, friends, family and wife
this thesis would have never been able to be completed.
! I would like to thank Dr. Paula Matthusen for two years of help in
research and composition as well as providing a welcoming atmosphere to
work. I would like to express my gratitude towards Ronald Kuivila for
answering all of my programming questions that I was not able to find the
answer to in any text book or help file. I would also like to thank Anthony
Braxton for his critical comments about my compositions and support towards
the creation of new ideas and music. I would also like to thank Rafael Romo
Tavizon and Liz Albee for performing in my composition recital. A special
thanks to Carlos Dominguez for taking the time to compose a piece using an
L-system for me to compare and contrast with on of my pieces.
! Lastly, I would like to thank my wife Seung-Hye for being by my side
during all of the good and bad times and spending the time to learn my music
and perform it.
i
Table of Contents
! ! ! ! ! ! !
Chapter 1 The History of L-systems !
! The Development of String Rewriting in Computer Science ............ 2
! The Creation of L-systems!............................................................... 4
! L-systems in Music Composition ...................................................... 5
! Summary .......................................................................................... 11
! Chapter Preview ............................................................................... 12
Chapter 2 Creating an L-system
! L-system Components ..................................................................... 14
! Context Free L-systems ................................................................... 17
! Context Sensitive L-systems ............................................................ 21
! Stochastic L-systems ....................................................................... 23
! Table-based L-systems .................................................................... 25
! Summary .......................................................................................... 26
Chapter 3 Compositions Using Different Mapping Schemes ..................... 28
! A Meeting of Florets ........................................................................ 28
! 32o(F+F) F[+F][-F]F[-F[-F][+F]F] ..................................................... 38
! Meditation of a Tree ........................................................................ 45
! M101 ............................................................................................... 51
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! Summary ........................................................................................ 58
!
Chapter 4 Process Evaluation ................................................................... 60
! Compositional Emphases ............................................................... 61
! Evolution of Mapping Strategies ..................................................... 63
! Performance Strategies .................................................................. 66
! Environmental Factors .................................................................... 68
! Crossing L-systems ........................................................................ 70
Appendix Compositions
! 32o(F+F) F[+F][-F]F[-F[-F][+F]F] .................................................... 74
! A Meeting of Florets ....................................................................... 77
! M101 .............................................................................................. 125
Reference List .......................................................................................... 135
iii
Chapter 1
The History of L-systems
! The development of rewriting systems in the early 20th century slowly
evolved into linguistic models and later progressed into L-systems. L-
systems, also known as Lindenmayer systems, are a formal grammar that
have been used to model plant growth. In this chapter, the history of the L-
systems and different methods of using them for science, computer
engineering and music will be discussed.
! String rewriting is the most fundamental aspect of L-systems. This
idea predates the formulation of L-systems as they are known today, and was
developed over the course of almost a century by researchers working in a
variety of disciplines, including music, biology, linguistics and computer
science. String rewriting refers to the process of starting with a string or
sequence of characters and substituting some of the characters with others
based on a set of prescribed relationships between them. Figure 1.1 depicts
the initial string of characters “ABBABA”. Underneath the string, rules A = AB
and B=A are the prescribed relationships between the string of characters.
That for every “A” in the string “AB” is substituted and for every “B”, “A” is
substituted. After completing the substitution, “ABBABA” turns into
“ABAAABAAB” and the substitution process starts all over with the new string.
1
! ! ! ! !
This process of rewriting can continue indefinitely and illustrates the
foundation for creating L-systems
! Mathematician Axel Thue created the first approach to string rewriting
in the early 20th century (Manousakis, 2006). Thue sought to create a way to
express mathematical theorems through a formal language that could be
proven automatically (Manousakis, 2006). Decades later in the 1950‘s,
Noam Chomsky applied the ideas of Thue to linguistics as a way to validate
linguistic models. His development was then applied in computer science
because it provided a theory of formal grammars which were later be known
as Chomsky Grammars (Manousakis, 2006).
The Development of String Rewriting in Computer Science
! Shortly after Chomsky’s work in grammars, computer programers
Backus and Naur created a rewriting notation to formally define the computer
language ALGOL. Chomsky Grammars and Backus-Naur’s notation created,
independently from one another, demonstrates the widespread appeal of the
String = ABBABA
Rules:A = ABB = A
New String = ABAAABAAB
Figure 1.1 String and Rules
2
concept of string rewriting across different disciplines. According to
Lindenmayer (1990),
!
Once Backus and Naur created this notation, many researchers studying
string rewriting became interested in applying different syntaxes to grammars
especially in the field computer science. This fascination with string rewriting
blossomed into many different syntaxes, but were all linked through string
rewriting.
! Seymour Papert created a graphic system called Turtle graphics
within his Logo programming environment, that would later be used to graph
and realize L-system images (Fry and Reas, 2007). The concept behind
Turtle graphics is a simple analogy, and, when used to create L-systems,
proves to be very powerful. Imagine a turtle on a piece of paper with a
marker attached to the bottom back of its shell. When the turtle crawls a line
appears. In the case of L-systems the turtle crawls based on a series of
commands. When the turtle receives a command, it can deviate from its
original path, but when done drawing the deviation, it returns to the starting
point of its departure. The resultant image is a tree-like structure. Without
this emphasis on general direction and momentary deviation of these
“The equivalence of the Backus-Naur form (BNF) and the context-free class of Chomsky grammars was soon recognized [52], and a period of fascination with syntax, grammars and their application to computer science began.” (pg. 2)
3
arborescent images would not emerge. These plant-like images are what
researchers today commonly associate with L-systems.
The Creation of L-systems
! L-systems were created in 1968 by Aristid Lindenmayer, a Hungarian
Botanist. Lindenmayer was interested in creating a mathematical
representation of how plants grow that was easily translated on a computer
system (Jurgens, 2004). Lindenmayer’s mathematical implementation of
plant growth led to an increased understanding of plant development at the
cellular level. This representation of growth would later become known as an
L-system or Lindenmayer System. Rewriting defined by Lindenmayer (1990)
is “a technique for defining complex objects by successively replacing parts of
a simple initial object using a set of rewriting rules or productions (pg. 1).”
Lindenmayer defined this system as a parallel rewriting system. As L-
systems continued to be developed, more complex plant structures were able
to be generated.
! The distinction between “parallel” and “sequential” string rewriting
defines L-systems apart from Chomsky grammars. Chomsky grammars are
written sequentially, meaning that the rules are perpetually added to the end
of the previous string. With Lindenmayer’s system, each symbol or character
is replaced in parallel all at once. Lindenmayer created this system with the
intention of being able to accurately show how cells divide simultaneously.
4
This model of cell growth is intuitive to biological behavior since a tree, for
example, does not grow one leaf at a time but many leaves simultaneously
along with other limbs and branches.
L-systems in Music Composition
! Following the development of Lindenmayer’s theories, researchers and
artists continued to expand the initial theory of L-systems. One major
contribution to L-systems was made by Przemyslaw Prusinkiewicz while
working in the field of computer science. Prusinkiewicz is the first scholar to
link music and L-systems in his essay Score Generation with L-systems
(Prusinkiewicz, 1986). Prusinkiewicz first created an L-system and then
visualized it using Turtle graphics. By retracing the graphic, he was able to
map the image directly into musical information. Discussing his proposed
musical system, Prusinkiewicz noted that “the proposed musical interpretation
of L-systems is closely related to their graphic interpretation, which in turn
associates L-systems to fractals” (Manousakis, 2006). In order to make the L-
systems graphic and the music interrelated, Prusinkiewicz used the X-plane
for duration and the Y-plane for pitch. In this particular way of mapping, a
starting note and tonality have to be picked before the musical mapping can
occur. To create rhythm, each time the metaphorical turtle moves in the X-
plane a new note is produced with a duration based on the user’s
specification of the relationship between time and the length travelled.
5
For example, if the turtle moves one space to the right, then the duration
would be “one”. In this case, “one” is an arbitrary number that has to be
determined in terms of durations. Every time the turtle moves in the Y-plane,
the note would go up a note or down a note, based on the tonality picked
(Prusinkiewicz, 1986). Since L-systems create branches, these branches can
be used as polyphony or a hierarchal order that can control multiple voices
with different timbres. Prusinkiewicz also states that other musical functions
can be mapped by interpreting the X- and Y- planes in terms of articulation,
dynamics and tempo (Prusinkiewicz, 1986). Since the produce enormous
amount of information, L-systems can be mapped into numerous sonic events
as well as tonalities and structures.
! Since Prusinkiewicz’s initial developments, numerous other systems
exploring relationships between L-systems and music have evolved. Peter
Worth and Susan Stepney of the University of York Computer Science
Department formulated a creative approach to generating music using L-
systems based on Prusinkiewicz’s work. Worth and Stepney used stochastic
L-systems to create mapping that involved multiple voices parallel to one
another. They used equal probabilities of 1/3 for three different rules for
“F” (Worth and Stepney, 2005). Each rule was mapped to a new voice
resulting in a three voice piece. Similar to Prusinkiewicz, they used a
Cartesian plot and retraced the L-system to generate the notes and rhythms.
However, since they were using a stochastic L-system, the images generated
6
were not as symmetrical as Prusinkiewicz L-system. The asymmetry
emerges due to different rules having probabilities of occurring. Rhythm and
pitch were determined along the X- and Y- planes respectively in a manner
consistent with Prusinkiewicz. However, the growth of multiple voices
simultaneously through their use of stochastic L-systems enabled more
complex rhythms and melodies. These complex rhythms were created in
manor consistent Prusinkiewicz, but with a major difference. Multiple line
segments on the X-axis are interpreted as individual repeated notes in
Prusinkiewicz’s system, whereas in Worth and Stepney’s system all the line
segments are summed together (Prusinkiewicz, 1986).
! Worth and Stepney created not only an elaboration of Prusinkiewicz’s
approach but also a hierarchy based on Heinrich Schenker’s theory on
musical analysis (Worth, 2005). Worth and Stepney only used the data for
the leaves and flowers, ignoring the data for branches and stems that are
created in the L-system. To generate music that follows Schenkerian
analysis, they interpreted the strings differently. The emphasis of structural
hierarchy along the lines of background, middleground and foreground within
Schenkerian analysis resonated with Stepney and Worth’s interpretation of L-
systems. This type of hierarchy can also be seen in the book A Generative
Theory of Tonal Music written by Fred Lerdahl and Ray Jackendorff. As
Lerdahl and Jackendorff and Schenker proposed, the music can be divided
into a hierarchy based on the melodic, harmonic and rhythmic structure. In
7
this hierarchy, however, there cannot be notes that overlap between sections.
In other words, a chord can not be part of section 1 as well as section 2
(Lerdahl and Jackendorff,1996). This idea creates a structure that shows
which parts of the music are the most important for understanding the main
harmonic, melodic or rhythmic structure. Worth and Stepney created a
system, in which they picked apart the structure until only the ending parts of
the tree (leaves and stems) remained. If translated into Schenkerian terms,
the “background” would be seen as the trunk(s) of the tree, the
“middleground” as the limbs and twigs and the “foreground” as the stems and
leaves.
! Worth and Stepney mapped their system differently than Prusinkiewicz
did and created more complex music as a result. The letter “F” was now
interpreted as an increase in the notes duration by a quarter note, “+” or “-” as
the note moved up or down by a scale degree and “[]” to set the duration to 0
and play the note according to the state that it is in (Worth, 2005). The note
duration would restart to its initial value at a change of pitch. This approach
to the generation of music through L-systems creates a musical piece with
more rhythmic complexity and melodies that follow one another.
! In addition to mapping conventional music notation, L-systems can
provide raw data that can be used to generate sound. Stelios Manousakis
developed his own computer environment to use the raw data produced to
created sounds. Manousakis harnessed the power of parallel string rewriting
8
to alter parameters in computer music such as granular synthesis or even
create wave tables. This area of data mapping is highly specialized and
involves not only an understanding of L-systems, but of sound synthesis and
computer programming. Manousakis parses out the data created by an L-
system and forms a hierarchy based on the limbs, branches, stems and
leaves of the L-system. He then interpolates a wavetable using these sets of
data. He devised many different approaches of interpolation that use parts or
all of the hierarchy system (Manousakis, 2006). Developed as part of his
dissertation, Manousakis built a customized Max/MSP library consisting of
many individual modules that can parse, derive, create and interpret the data
from the an L-system (Manousakis, 2006). Manousakis used L-systems to
produce sound and parameters using the generated material.
! Composer Hanspeter Kyburz used L-systems in his work that utilized
pre-generated material derived from an L-system. Kyburz used the data in an
acoustic realm to evolve fully notated music. Kyburz’s work Cells uses a tree-
like structure and the idea of cellular automata to create the form of the piece
(Supper, Computer Music Journal, 2001). Cells is written for a saxophone
ensemble and uses many short motives composed by Kyburz to form the
structure of the piece. Within the piece, the rules that the composer derived
refer to a series of pre-written motives. The rules define the procedure of
substitution thereby creating a piece with self-similar structures. As with
Manousakis, these rules and patterns are not known to the audience and the
9
audience only perceives the music as it sounds. However, unlike
Manousakis’s electronic works, it is possible to perceive the similarity
between musical fragments as well as the overall structure because of the
repeated motives. When Kyburz was designing this piece, he used thirteen
generations of an L-system and marked them in the score as G1, G2, G3....
(Supper, 2001). Martin Supper’s illustration of Kyburz’s work in A Few
Remarks of Algorithmic Music shows how Kyburz went about deriving the
pattern from cellular automata. Each generation is derived from Kyburz’s set
of rules and within each generation a pattern of those rules occur, in which he
then applies the motives (Supper, 2001). Figure 1.2 illustrates the hierarchy
system of Kyburz’s Cells.
! ! !
! ! Figure 1.2 Adapter from Martin Supper Computer Music Journal 2001
In Figure 1.2, one can see how set 1 with four rules splits into another set 1
with four rules and a set 2 with three rules. Then from those sets, set 1 again
10
1(1)1(2)1(3)1(4)
1(1)2(2)3(3)4(4)
2(1)2(2)2(3)
1(1)2(2)3(3)4(4)
2(1)2(2)2(3)
2(1)2(2)2(3)
3(1)3(2)
2(1)2(2)2(3)
3(1)3(2) 2(1)
2(2)2(3)
3(1)3(2)
3(1)3(2)
4
Start
Iteration 1
Iteration 2
Iteration 3
1(1)2(2)3(3)4(4)
2(1)2(2)2(3)
splits into set 1 and 2, but set 2 splits into sets 2 and 3. Set 3 will then split
into set 3 and set 4. Each rule splits into a copy of its self and the next set of
rules. With each new set one rule is taken away, therefore only 4 sets can be
used since there are only 4 rules and with the next set generation 1 rule is
taken away.
Summary
! Through the works of Kyburz, Manousakis, Worth, Stepney, Papert,
Prusinkiewicz and Lindenmayer, L-systems and the idea of string rewriting
have a very rich history that has constantly been evolving through science,
music and computers. As the development of L-systems continues, more
unique uses for them will be found. Manousakis and Kyburz have shown that
L-systems can be utilized in different ways and still have room to expand.
! In this chapter many different themes and techniques have been
covered that encompass what a L-system is and musical ideas developed
from them. As mentioned earlier, L-systems produce long strings of
characters that can be interpreted into Cartesian coordinates to produce
graphics that look like trees or bushes. Worth and Stepney used stochastic
L-systems to create layers of music and created hierarchies to further develop
more complex rhythms. While long strings of characters are used by
Manousakis and Kyburz in their compositions, Manousakis used all of the
data created in the L-system and parsed it out to create different sounds and
11
structures within his pieces, while Kyburz used just the pattern produced by
the L-system and substituted prewritten motives in to create the piece.
Chapter Preview
! This research aims to develop themes and techniques for creating and
interpreting L-systems that can be used in music composition. In this
research, the use of Cartesian mapping, pattern creation and pre-composition
strategies are used as compositional tools by which the composer and
performer can interpret. New ways of mapping the pattern produced and
interpreting the image as musical ideas are themes that will evolve further
throughout this thesis. One example of this can be seen in the composition A
Meeting of Florets, which is pre-composed and uses a series of motives that
are assigned to different symbols of the sequence created. In this
composition, every page reflects a point in the L-system and the performer
makes decisions on what to play based on how the structure of the L-system
looks. For example, if a branch of the tree is created on that page, the
performer can choose to play the branch or continue on its original path. The
motives are then transposed in accordance to how the system grows. “+”, “-”
signs in the system indicate a raising of lowering of the motive. In contrast to
this composition another composition entitled Mediation of a Tree uses only
the sequence of symbols to trigger and manipulate a synthesized sound.
Each sound has the same source but each character in the sequence
12
manipulates the sound differently in order to create points of varying texture.
In chapter 2, various kinds of L-systems will be examined in terms of how
they are created and the components that differentiate them from one
another. This chapter will lay the foundation for creating L-systems which will
then be expanded by looking at compositions written for this research.
Chapter 3 will show the application of L-systems to compositions by analyzing
four different compositions that use different rules, L-systems,
instrumentations and styles. Other compositions which will be explained in
chapter 3 include a piece for computer that uses the image to create “on” and
“off” messages that are interpreted by the performer and a trio for Cello,
Analog Electronics and Computer, that uses different mapping techniques for
each voice and changes mapping techniques when the pattern generated
changes.
13
Chapter 2
Creating an L-system
L-system Components
! In the previous chapter we discussed the theories and developments
of L-systems. This chapter will suggest that one can construct their own L-
systems for musical compositions by understanding how to create L-systems
and their underlying components. This research has focused mainly on three
types of L-systems: Context, Stochastic and Table-based. These three L-
systems are chosen not only because of their ease of use, but because of the
graphics that they produce and data that can be mapped to a variety of
musical parameters and forms. Context Free L-systems are the simplest L-
systems to produce and understand because of their very straight forward
production rules. With the basic understanding of Context Free L-systems,
one can create and expand the development of their own L-systems to fit their
work.
! All L-systems have three common, basic components regardless of the
type of L-system: axiom, rules and angles. The axiom is the starting point,
meaning what rule to start with i.e. “A”. The axiom can also be a series of
rules such as “A+A”. The rules are information that is going to be substituted
into one another to create the final string of characters. For example, if “F =
FG” and the axiom is “F” then iteration one will be “FG” and iteration two will
be “FGG” because “F” gets substituted with “FG” and “G” remains as “G”.
14
When using traditional Turtle graphics, the characters that get interpreted
comprise of “F”, “-”, “+”, “[“, “]”. Along with the development of L-systems,
more characters were added to Turtle graphics such as “f”, “B”, “b”, “@”, “<“
and “>”. Each one of these characters in Turtle graphics directs the turtle to
do a specific task. As seen in the figure 2.1, the instructions for the turtle are
simple, however, can produce natural plant shapes and complex forms.
The combination of these characters form the rules in which L-systems are
created. Other characters may be used to create patterns, however, the
Turtle graphic system will disregard them. The disregarded characters are
still very important because they can still have the Turtle graphic characters in
their rules. These characters can then create very intricate strings that form
F:! Draw a line forward
f: ! Move forward without drawing a line
B: ! Draw line backwards
b:! move backwards without drawing a line
-:! Rotate left
+:! Rotate right
[: ! Start a branch
]: ! End a branch
<:! Move up in the Z dimension
>:! Move down in the Z dimension
@:! Scale the length of the line
Figure 2.1: Turtle Graphic Interpretation of Characters
15
very complex patterns that unfold over the course of hundreds or even
thousands of characters. As more software is developed that utilizes Turtle
graphics, more symbols can be assigned to different functions of the turtle to
fit the composers’ or programmers’ need. By expanding the Turtle graphic
language the composer can denote a “?” to mean change the angle by ten
degrees. By assigning symbols, the composer can create more complicated
systems and mapping schemes.
! The last component of all L-systems is the angle in which the turtle
turns. If the turtle is moving forward and then has to turn right, the angle
assigned to it will tell the turtle, by what angle to move in relation to the
straight line. This factor can drastically change the shape of the L-system
from a flower to a jumble of lines, which is very important when the composer
is trying to create a natural shape to take the data from. Since the angle is
usually a constant this parameter is not evolved through the system and
serves only in creating the graphic. When composing a piece of music that
maps the graphic produced by the L-system, the angle that the graphic is
drawn can change the outcome of the music.
! To create an image or string of characters, one must create rules.
Lindenmayer used the rules A = AB and B = A to model the growth of algae
(Lindenmayer, 1990). If we look at Lindenmayers rules and start with an
axiom of “A” then figure 2.2 shows the first through fourth iterations and how
16
the sequence grows. As one can see using two basic rules, by iteration 4 the
sequence is long and starts to produce a recognizable pattern.
! ! ! ! Figure 2.2 L-system Growth
!
Context Free L-systems
! A Context Free L-system is the most basic type of L-system that can
be formed. From a Context Free L-system, one can see basic plants grow.
Figure 2.3 shows a Context Free L-system interpreted by Turtle Graphics
using the axiom F and a single rule that uses the characters “F”, “-”, “+”, “[“
and “]”.
17
! ! ! ! ! Figure 2.3 Tree
The graphic produced resembles a tree but does not have any leaves or
flowers and is also symmetrical. Context Free L-systems tend to be
symmetrical because the same set of rules are being rewritten over and over
again. If a composer were to map this data into music, it could potentially be
unexciting because of the amount of repetition produced. This could impair
the potential development of the same melody or idea and never really evolve
the way music generally evolves. Context Free L-systems, however, are the
basis for L-systems that uses tables of rules or probabilities to create more
intricate patterns. Context Free L-systems follow a very basic form that can
be explained as predecessor --> successor. An example of this can be seen
in an L-system that starts with the rules “A = AB” and “B = A”. Using these
rules, when an “A” is identified, it is replaced by “AB” and when a “B” is
identified it is replaced by “A”. This means that “B” is the predecessor and “A”
is the successor. “A” then becomes the predecessor and the cycle continues.
18
An example of the replacement and expansion of an L-system can be seen in
Figure 2.4. The initial rule for this system is “F = F-[-F+F]+[+F-F]”. This
means that every “F” will be replaced by the rule “F-[-F+F]+[+F-F]”. The first
three iterations of this system show the rapid growth and complexity
generated by the L-system.
As this L-system shows, iteration 3 is very long and complex. Brackets occur
within large groups of brackets and at certain points there are groupings of
three brackets to close that part of the system. The easiness of Context Free
L-systems was part of the appeal of iterative systems that Lindenmayer
explored. With the aid of a computer, these systems can generate structures
quickly and easily, which can then be interpreted and mapped directly into
assigned musical parameters. These systems are the easiest to map to
music because of their simplicity and pattern. However, there is one major
axiom = F
F=F-[-F+F]+[+F-F]
it1. F-[-F+F]+[+F-F]
it2. F-[-F+F]+[+F-F]-[-F-[-F+F]+[+F-F]+F-[-F+F]+[+F-F]]+[+F-[-F+F]+[+F-F]-F-[-F+F]+[+F-F]]
it3. F-[-F+F]+[+F-F]-[-F-[-F+F]+[+F-F]+F-[-F+F]+[+F-F]]+[+F-[-F+F]+[+F-F]-F-[-F+F]+[+F-F]]-[-F-[-F+F]+[+F-F]-[-F-[-F+F]+[+F-F]+F-[-F+F]+[+F-F]]+[+F-[-F+F]+[+F-F]-F-[-F+F]+[+F-F]]+F-[-F+F]+[+F-F]-[-F-[-F+F]+[+F-F]+F-[-F+F]+[+F-F]]+[+F-[-F+F]+[+F-F]-F-[-F+F]+[+F-F]]]+[+F-[-F+F]+[+F-F]-[-F-[-F+F]+[+F-F]+F-[-F+F]+[+F-F]]+[+F-[-F+F]+[+F-F]-F-[-F+F]+[+F-F]]-F-[-F+F]+[+F-F]-[-F-[-F+F]+[+F-F]+F-[-F+F]+[+F-F]]+[+F-[-F+F]+[+F-F]-F-[-F+F]+[+F-F]]]
Figure 2.4 L-system Growth
19
problem with a Context Free system; that is, the possibility of becoming “stuck
in a loop” or highly repetitive pattern. This can cause a problem when trying
to uses the data produced to control music.
! Conceptually this system creates an internal complexity out of a very
simple procedure. It is this basic procedure however that allows for the
system to create loops. In music, loops that are too long and do not change
may create static music. This looping, in Context Free L-systems will also
always occur after three or four iterations. When a composer uses motives
one or two, looping will not occur because the iterations are still in the early
stages of development. This can also cause the piece to be fairly short
because of the lack of evolution in iterations one and two. To avoid this
problem, one can precompose motives that can be mapped to different rules.
By using precomposed motives and assigning them to rules, characters such
as “+”, “-”, “[“, “]”, “<“, “>” can all be used to transpose or change the tonality of
the pre-composed motives in some prescribed way. Now, a long loop will be
avoided because the internal structure of the system will lend itself to subtle
changes over time. To avoid the problem of having a really short piece of
music, the motives or phrases can be composed to be varying in length and
tempo. The characters may also be used to create a related rhythmic
structure. For example, the character “+” can double the duration of all the
notes while “-” would cut the duration in half. If the motives are brief, then this
technique can create rapidly changing tempo. Both ways of using characters
20
to map tonality and rhythm can be used to create a piece where the notes
and rhythms are constantly changing in relation to one another and the
system still remains internally complex and self similar. A Meeting of Florets
written in conjunction with this thesis uses this type of L-system. Seven
motives were precomposed and assigned to the seven rules of the L-system.
The non-letter characters, then derive how to transpose the motive. For this
piece, “+” was assigned to transpose the motive up one octave, while “-”
transpose the motive down one octave. Also, “<“ and “>” transpose the
system by a whole step up or down. Apart from these rules, all transpositions
are cumulative, so transpositions of initial phrases would remain at that
transposition until it is transposed again. Once transposed beyond the range
of the piano, the motives return to their original state. This piece (discussed
in more detail in chapter 3) illustrates a very simple system that created a
very complex piece of music.
!
Context Sensitive
! Another type of Context L-system is a Context Sensitive L-system
which is the same as a Context Free L-system with one important difference.
In a Context Sensitive L-system, the rules applied to a character are
dependent on the states of their neighbors, thereby emphasizing the “context”
of the character. These Context Sensitive L-systems are called L1 and L2
systems (Lindenmayer, 1990). L1 L-systems have one condition while L2 L-
21
systems have two conditions. An example of an L1 system can be B<A = C.
When interpreted, this illustrates a system that when “A” is next to a “B” in the
string it is replaced by a “C”. Similarly, an L2 system could be B<A>B= C;
when “A” is between “B” and “B”, replace “A” with “C”. With this type of
context, the string can easily generate more variation than a Context Free L-
system due to the increased sensitivity of the rules. The Context Sensitive L-
system produces similar but more complex results than a Context Free L-
system. As seen in figure 2.5, a Context Sensitive L-system can also create a
rule that progresses forward after each iteration (Lindenmayer, 1990).
If one is mapping this information to music the constant shift of F can be used
to create a delayed appearance of a particular sound when consecutive
iterations are used. This gradual displacement is harder to achieve with a
Context Free L-systems because they are not concerned with the order of
characters.
!
! ! ! Axiom = FGGGGGGGG! ! ! Rule! ! ! F < G = F! ! ! F = G
! ! ! it1. FGGGGGGGG! ! ! it2. GFGGGGGGG! ! ! it3. GGFGGGGGG! ! ! it4. GGGFGGGGG
! ! ! Figure 2.5 Progressing Through Iterations
22
Stochastic L-systems
! Stochastic L-systems are created using all of the fundamental
elements of the L-system previously discussed. These systems, however, are
very complex and can potential create images that appear more like nature
because of the broken symmetry that can occur. Stochastic L-systems start
with an axiom and a rule. What differentiates a Stochastic L- system from a
Context L-system is the use of probabilities. The user of the Stochastic L-
system can specify the probability of each rule being used. For example, in
“F1/2 = FF” and “F1/2 = F+F”, each rule can have a fifty percent probability of
occurring. By creating an L-system this way, irregularities can occur, which is
why more accurate models of nature can be created. Stepney and Worth
(2005) illustrated this concept by stating
!
Used in a musical context, these systems can completely remove repetitive
patterns. “Getting stuck in a loop,” as mentioned before, can be a problem
when working with Context Free L-systems. In contrast, any loop occurring in
a stochastic system has the probability of suddenly changing or replacing part
of the loop with something else which in turn, would create something new.
Stochastic L-systems have the potential to bring a change to a piece of music
“ Plants are all different: stochastic L-systems are used to generate plants from the same “family” but with different details” (pg. 3)
23
based on the probabilities of particular rules occurring. As Worth and
Stepney (2005) stated
In their research, Worth and Stepney used three rules for “F” and each rule
had a 33.3% chance of occurring. Also each rule was directly associated with
a different voice in the piece (Worth, 2005).
! Stepney and Worth’s use of multiple voices is very interesting because
it enables the melody and rhythm in each voice to evolve naturally without
deviating from the original system. Because of how Stochastic L-systems are
generated, multiple voice’s rhythms and tonalities will be similar. As the
system grows, each voices‘ notes will raise and lower dependent from one
another, creating a unique contour for each voice.
! Meditation of a Tree, a three movement work for computer is
composed using a stochastic L-system whose sequence is mapped to
different signals and parameters. Besides this L-system used being
stochastic, the mapping of the sequence to different phases, frequencies and
envelopes is also stochastic. Due to the probability of the parameters
changing, the piece naturally evolves through varying textures differently in
every performance. This evolution as, Worth and Stepney stated before will
“A musical rendering should similarly generate a variety of pieces in the same style” (pg. 3)
24
create a different piece, but with the same style every time it is performed
(Worth, 2005). This piece will be analyzed and illustrated in chapter 3.
Table-based L-systems
! Table-based L-systems use a look-up table that specifies different
information to change. An entire set of rules, or completely new vocabulary
can be interjected directly into the L-system and cancel out the rules that
were currently in place. One way to create Tabled L-systems is by starting
with a Context L-system. At a predefined time in the creation of the L-system,
all of the rules will be replaced by a new set of rules. The user of this Tabled
L-system designates at which iteration or time all of the rules are to be
changed. To differentiate between Tabled L-systems and Stochastic L-
systems, Stochastic L-systems have the same rule defined several different
ways with different probabilities of occurring, while Tabled L-systems replace
all of the rules at once. With this Tabled L-system, all initial rules are
substituted at the same time and only reoccur if specified by the user.
! Tabled L-systems can create bizarre or natural shapes that look
mutated or broken which occurs in nature after storms, hurricanes or any
other environmental influences. Since all L-systems are written parallel, the
substitution of new sets of rules in Tabled L-systems will create a very chaotic
system. After one set of rules develops, the interjection of completely new
rules for the next iteration will cause the entire system to change and possibly
25
loose the pattern that was developing. These look-up tables can also be
used in a direct relationship with the look-up tables of sound synthesis. For
example, the sounds being produced from a wave table can be thought of as
“look-up table one”, when the rules are switched, a new sound (look-up table
two) will take the place of “look-up table one”. This is the most basic
example, however an endless amount of possibilities can be used because
the rules and parameters can change in any way. Tabled L-systems also do
not cause the system to get stuck in a loop. Since the rules change all at
once, any loop will also get immediately eliminated. There is no limit to how
many tables can be created. As more tables are used, internal complexities
will arise and influence the overall pattern and shape of the L-system.
!
Summary
! The L-systems previously discussed are easy to produce and also
create complex data. After creating these systems, the composer has to
manipulate the data produced to fit the composition in mind. When using
Context Free L-systems, the mapping of the data has to be more complex in
order to not fall into a loop compared when using a Stochastic L-system. To
completely change a system using different look-up tables is another way of
avoiding a loop and creating more complicated forms and music. L-systems
can be a generative tool not only for individual parameters but also for a
26
musical process. Using an L-system to create music is analogous of rolling
dice or creating a 12-tone music matrix to generate some aspect of the music.
! Using an L-system for music composition is a process that first creates
information and then interprets the information produced. Hurbert Brün’s
piece, Mutatis Mutandis, uses a graphic as a score which the performer
creates as their own musical language to play the piece. The image of L-
systems can be interpreted by the composer to realize as the basis for the
compositions or the can be used as the score for the performer to interpret
just like Mutatis Mutandis. The image created can represent the pitch contour
within specific ranges, or it can be completely abstract and left for the
composer or performers interpretation. The image can be used the same way
that Iannis Xenakis used his UPIC system to create music. Individual lines of
the image can be mapped in order to create a series of events or swooping
sine waves that are read linearly.
! In conclusion, three different types of L-systems were discussed that
all are built in similar fashions. Each L-system has a different character and
outcome that the composer must take into consideration when using it for
music composition. Additionally, each L-system has multiple ways of mapping
the data which can create many pieces of music out of one L-system. The
graphic created can also be used as a source for mapping musical properties.
L-systems are being used as a process to slowly evolve different musical
parameters over a period of time.
27
Chapter 3
Compositions Using Different Mapping Schemes
! This chapter focuses on four pieces composed for this thesis that
examine and explore different strategies for mapping L-systems. These
pieces titled A Meeting of Florets, Meditation of a Tree, 32o(F+F) F[+F][-F]F[-
F[-F][+F]F] and M101 each illustrate different ways of mapping the information
generated by L-systems to different parameters of a composition. These
compositions range in instrumentation, style and type of L-system used to
realize the piece.
! When composing using L-systems, indeterminacies created by human
performance and musical ideas were of the upmost importance and laid the
foundation for the use of the L-system and its algorithmic sequence. In most
cases, the instrumentation was determined prior to the generation of the
musical material. This specification circumscribed the musical parameters to
be affected by the selected L-system. The L-system was carefully sculpted to
create an interesting sequence as well as graphic that could be used as the
foundation of inspiration.
A Meeting of Florets
! A Meeting of Florets is a solo piece for prepared piano that utilizes a
Context Free L-system. This piece consists of a series of pre-composed
28
segments that the performer selects individually from different options that are
given on each page. These motives were through-composed and assigned to
different rules of the L-system. The interaction of the music with the L-system
was designed so the motives were sequenced in the order in which the L-
system evolves. When composing this piece, special care was taken in
designing an L-system that produced a visually pleasing graphic in addition to
a complex sequence of data. Without a complex sequence of data the piece
would not evolve or be interesting enough to fully engage the listener.
! The graphic produced for A Meeting of Florets is a three dimensional
image resembling a cluster of florets. A floret is a closely clustered flower that
makes up the main head of a flower. An example of a plant that is comprised
of florets is a Daisy or Sun Flower. This shape was particularly interesting
because of the interlocking lines within a fairly basic structure. This L-system
resembles a cluster of florets stuck together and intertwined within one
another. Figure 3.1 shows what the graphic of this L-system looks like.
! Figure 3.1 Florets
29
This graphic looked like the same shape repeated and overlaid on top of one
another at different angles and sizes. The composition reflects the idea of
overlaying florets at different angles. Specifically, the composition tries to
reflect how a group of florets look when floating in the wind or sitting on a
plant. Florets, when attached to the plant, look dense and tightly packed
together. Once they are in the air, the wind and environment has a big affect
on them. They float with the wind, but any small breeze from another
direction alters their route, causing them to jump up and down and shift side
to side while continuing on their path. If the floret ever stops and floats to the
ground, there is a chance that a strong gust of wind will come and blow the
floret back in the air and its journey will continue.
! The pieces uses a series of seven rules labeled A - G that, when
substituted in for one another, creates a very long string of characters. These
rules can been seen in figure 3.2.
All the letters that are not “F” or “B” are not interpreted by the Turtle graphic
system and mainly used to create a complex string of characters to interpret
Axiom = AA = A+C[F]B = F+[D]-[G+A]FC = A-[F+C]+BD = F<F[FA]E = F+<A-B[C+D]F = D[A-F]<EG = E<[A]+F[B-D]
Figure 3.2 Rules for A Meeting of Florets
30
into parameters. However, all of the characters are used when writing the
music. The symbols included in this L-system are “-”, “+”, “[“, “]”, “<“. The
symbols are used generating transformations of the motives throughout the
piece. Each symbol describes a transformation that is applied to a motive. A
“-” means to transpose the motive down one octave while a “+” means to
transpose the motive up one octave. Using a minus sign and plus sign lent
itself naturally to those ideas. Unnatural or complicated rules could become
confusing or unclear so a minus sign going down and a plus sign going up
was determined to be the most natural. A “<“ meant to transpose the motive
up one whole step. Transposing by a whole step was to create a sense of a
whole tone scale. As the notes transposed up a whole step, after six
transpositions it would be at its original note but up one octave
enharmonically (B#). By also having whole step transposition, many different
tonalities can occur and interact with one another. These tonalities arise
because of the motives transposing at different rates and originally having
similar notes to all of the other motives. All of the symbols are cumulative and
wrapped to the starting point once the upper or lower limits are reached. For
example, if a motive is transposed beyond the range of the piano, the motive
then returns to its original transposition. This wrapping also applies to the
whole step transpositions. Once the notes transpose back to their starting
notes, up one octave, they are wrapped to the original motive.
31
! The brackets that create the branches in the L-system are used in the
composition as choices for the performer. Each page of the composition has
between two and five staves that are arranged by indentation, to show the
hierarchy branching within the L-system. This indentation tries to keep the
hierarchy of branching intact by having one main line and two indentations.
The third indentation would be the furthest away from the starting “trunk” of
the system. The performer does not know of this hierarchy, but as a result of
these indentations they are able to see groupings of systems and how each
page can have a different flow. It is then up to the performer to choose what
to play. Thus, the piece is different every performance, but still maintains an
overall sonic character. Additionally, the performer must always play the
“main” part, which is not indented and is most prominent within the overall
hierarchy. For example, if there are five systems on the page, systems 1 and
5 have no indentation, 2 and 3 have one indentation and 4 has 2 indentations,
the performer would then have the options to play either staves, “1, 5”, “1, 2,
3, 5”, “1, 4, 5” or “ 1, 2, 3, 4, 5”. In this example systems 1 and 5 would be the
main part and therefore unavoidable to play. If the performer is to select only
the main systems to play then the “G” motive is not played in the piece
because the natural evolution of the system makes “G” either in the first or
second indent. Figure 3.3 is an excerpt of the score that illustrates a page in
which there are two hierarchies. Lines 1, 3, and 4 are in hierarchy two while
line 2 is in hierarchy one.
32
!
The cumulative whole tone transposition and indeterminacy causes certain
motives to transpose slowly over the course of the entire piece while other
motives transpose through a whole tone scale several times. This was not
predesigned, and only arose through the course of working with the L-system.
! The notion of a musical idea unfolding over a long period of time is
something that can be heard in minimalistic works by composers such as
Steve Reich and Terry Riley whose works are designed by the composer to
unfold over a long time. The unfolding of a whole tone scale over a long
period of time was not intended. This is a good example of how an L-system
can influence music in such a way that the composer, until analyzing their
own work can not fully see the outcomes. This effect may not be very
noticeable except to people who are really focused on listening to the pitch
and tonality, however because of this unfolding, one can easily identify that
something is different and changing.
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Figure 3.3 Indentations
33
! This piece is written for prepared piano, such that the preparations
would add timbral variety to the repetitions of the motives as they are
transposed. By preparing the piano, the sense of “looping” is counteracted by
preparations, such as metallic percussive sounds that will be excited when
the motive is in one register, but not when it is transposed to another. The
vibration of the strings on metallic objects also resonates with nature as it
evokes the idea of wind blowing the floret in different directions. Additional
preparations include dampening the strings to produce harmonics and a
combination of dampening the strings and adding metallic objects to create a
metallic harmonic sound. The harmonic created by the dampening is
reminiscent of the weightlessness of the floret as it floats on the wind high up
in the air.
! A Meeting of Florets illustrates restrictions that the composer has to
work around in order to create the imagined piece. In this piece, motive “A” is
repeated frequently and if transposition rules were not employed, the piece
would sound extremely static. Inviting the performer to choose which staves
to play can circumvent motive “A” from being played to many times as well.
Finding the right iteration of the L-system was a difficult choice in this piece as
well. The graphic produced by the third iteration was not very intriguing, and
the image from iteration four looks quite plant-like; however, its sequence is
very long. Figure 3.4 illustrates the difference between iteration three and
four.
34
Looking at these iterations, it is immediately seen that iteration four is more
complex and has more information in it. Composing using iteration four took
a long time, as it was necessary to parse out salient features of the sequence.
A compositional system also had to be created in order to not loose focus in
the large amount of repeated characters. How does one notate seven
motives that are constantly transposing, efficiently and accurately while still
keeping the hierarchy accurate? Breaking the system into little parts and
planning what motives were going to be on each page not only created a
structured form to follow, but also made it easier to see how the lines
transpose. Since notating took more than one day, by separating the systems
into pages, it was a lot easier to come back to the composition and resume
without much confusion. Creating no bar lines, no time signature, written
accelerandos and decelerandos as well as the indentations are specific
formattings that had to be applied to each page and sometimes each system.
! This piece does not follow a particular tonality or rhythmical pattern.
The motives of this piece are independent from one another but have similar
Iteration 3 Iteration 4
Figure 3.4 Iterations
35
ideas within them to allow for a flow between any combination of motives.
The piece is atonal, but favors Gb, Ab, Bb and sometimes Db and Eb;
however, these notes are not always flatted. Certain chords such as quintal
chords can also be seen in various motives of the piece. The use of quintal
chords as well as cluster chords are used to either open up the sound of the
piece (quintal chord) or close the sound and make it sightly more aggressive
(cluster chords). Rhythmically, each motive is quite different; however, long
passages featuring notes of similar duration notes are common. The piece
uses a number of different tuplet groupings in succession to create a shift in
atmosphere that may be heard as slightly chaotic. Groupings of sextuplets
over groupings of quintuplet are used as flourishes while rhythmic groupings
of eleven or nine are used to create a sense of uneasiness. The use of odd
groupings was to musically evoke the lack of balance in a floret as it changes
direction, drops and rises with the wind movement. As with the rest of the
piece, musical decisions are made in part by drawing inspiration from how a
floret behaves in nature.
! Experimental techniques can be seen in two of the seven motives with
one of the motives rarely occuring. Plucking the strings while pedaling is
used to change the sound and pace of the piece. The notation specifying
these actions provides the performer a range of possible notes for each
plucked note. In figure 3.5 many experimental techniques can be seen such
36
as string plucking, indeterminate cluster chords and written accelerandos and
decelerandos.
The performer chooses a note within that particular range to pluck. Vertical
rectangles are used to indicate to the performer to hit all of the natural notes
in the range of the rectangle. These happen between plucked notes to create
a sudden loud dissonant sound that is then resolved quickly by more plucked
notes. Since the plucked sounds are soft in nature, the sudden cluster of
notes that are struck, drown the plucked sound temporarily while the next
passage of plucked notes sneak in from underneath the sound. The plucked
notes, much like the metallic prepared notes, create a gentle counterpoint and
sonically represents the floret as it drifts through the atmosphere. This
composition illustrates how the sequence of the L-system can be used to
create only the form and transpositions of the piece. As one can see, the L-
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37
system created was straight forward but the music became complex. That is
because of the intricate mapping strategies incorporated in the piece to
manipulate the motives and unfold themes over a long period of time.
32o(F+F) F[+F][-F]F[-F[-F][+F]F]
! 32o(F+F) F[+F][-F]F[-F[-F][+F]F] is an interactive computer piece
written using the graphic produced by an L-system as a map for changing
different states of sound production. The graphic produced by this rule looks
like a very elementary, almost cartoon-like tree, that is self similar except for
one part of the tree which forms an anomaly in the self similarity. (The circle
in Figure 3.6 highlights this anomaly) While looking at the graphic for this L-
system, groups of branches can be seen growing at of one point and creates
moments of different densities. This idea sparked the creation of a piece that
layers noises on top of one another. Working with the idea of layering, it was
important to make sure the sounds could interact with one another. As a
byproduct of this layering, the L-system creates different densities simply by
adding or taking away different sustained timbres. This technique created a
perpetually changing atmosphere in which sounds blend and subtle
differences emerge.
! The graphic of this L-system is produced using one rule F = F[+F][-
F]F[-F[-F][+F]F]. The axiom for this L-system is “F+F” and it is drawn at an
angle of 32 degrees. The angle of the graphic is the most important part of
38
this piece, because at any other angle the mapping strategy used becomes
very difficult and overlapping sections become almost impossible to see.
! This piece does not include loops, such as those that can be heard
using Context Free L-systems. Moreover, the mapping strategy employed for
this piece poses few of the challenges encountered in the system used for A
Meeting of Florets. This piece is completely derived from the graphic
produced. The graphic was divided by where a new branch starts to grow.
Each line in the divisions were then given a number between one and twelve.
These divisions were then seen as a group of numbers which would act as
the notation for the performer. For this piece the L-system was separated by
hand, which presented its own challenges. When trying to look and see
which branch comes before another, there can be a very small difference,
almost too small to see with the naked eye.
! Figure 3.6 Bare Tree
39
As one can see in figure 3.7, using an L-system this way eliminates
challenges of sequence mapping or character mapping but creates its own
type of problems that need to be resolved such as how to interpret a very
complex image.
! Each segment that was created was where a line or group of lines
started. After these divisions were made, a number ranging from one to
twelve was assigned to each line within each segment. In total there were
thirty segments that the tree was divided into. Once all of the numbers were
assigned in all of the segments, each segment was written vertically with the
numbers that were in that particular segment. These twelve numbers
corresponded to one of twelve computer sounds that were controllable using
a midi interface. For example, if the first segment had the number "1" in it,
then the performer would trigger "sound 1" to start, or raise the volume to
make the sound audible. If in the next segment, there was another "1," then
the corresponding sound for "number 1" would be turned off. Essentially, the
numbers correspond to "on" or "off" messages for a particular sound.
Figure 3.7 Image Mapping
40
! The aim of using this number system was to create a timbral language
similar to a tonality from which this piece, as well as other pieces, could be
derived. Playfully referring to the convention of dividing the octave into 12
equal parts, this system has twelve timbres. This defines the instrument from
which the laptop performer plays. This system does not only have to be for L-
systems or numbers, but can be used in a serialist fashion. This approach
will obviously produce different outcomes; however, this strategy can easily
be related to earlier compositional techniques. This language can also make
notating a live piece using many timbres easy and intuitive. For this language
to be useful, the sounds being used have to have a similar quality, just like
how all the notes on a violin still sound like a violin, but will sound different
based on how a note is articulated and fingered. When using this language,
having similar but different timbres wil keep a structured and organized
sound throughout the music.
! There is no duration set for each segment or for the entire piece. All
timing is determined by the performer. This is similar to the way a tree grows:
it does not have a time limit, it can grow and then be cut down or it can grow
and last indefinitely depending on its environment. Apart from the timing not
being specified, the sounds produced may be altered in real-time using
parameters specific to each sound, such as equalization. A majority of the
sounds have parameters that can be changed in real-time in order to blend
the sounds better with one another, or highlight a particular sound at
41
moments of high density or sparseness. The reason that each sound can be
changed or improvised, is inspired by plants' abilities to adapt to different
growing conditions. Since environmental factors are so important in the
growth of plants, the performer's personality (environmental factor) can be
reflected in the piece and customized to personal taste. The piece overall,
should be fairly loud and not deviate much in dynamics. The blending of
sounds in the piece should be achieved by changing the timbre, not
dynamics. Changes in dynamics arise naturally in the piece through the
process of layering as well as wave cancellation. Even though it is derived
from an L-system and tree-like graphic, 32o(F+F) F[+F][-F]F[-F[-F][+F]F]
provides a metaphorical contrast by using noisy, aggressive sounds not
commonly thought of as being "natural". The piece reflects a hostile, violent
nature where survival of the fittest is how living things survive or die.
! Each sound used has a variety of different parameters and creates
many different spectrums and timbres. Each sound also posses a slight
challenge to the performer. Some sounds cut through the texture created
while others linger in the background and twinkle. Sometimes certain sounds
will seem to not blend until a parameter is changed. The performer
improvises through these challenges durning the performance. Even though
these challenges might seem overwhelming, it is a byproduct of having as
much control over the sound as possible. The balance of control and
flexibility allows performances to change and gives the piece a mind of its
42
own. It is important in this piece for the performer to know the sounds that
can be produced, in order to effectively adjust them. The sounds in this piece
are also very important because they create their own language for the piece,
just like a key signature. The performer has to learn this piece just as if they
were learning an instrument. By learning this piece, the sounds become
easier to manage, and the performer can play the piece according to his or
her interpretation.
! Each one of the twelve sounds were designed separately with an
underlying concept common amongst them. Creating a complex spectra is of
the upmost importance in this piece. Each sound begins with a pulse or
chaotic sound generators as the initial source material. An FFT analysis of the
sound is taken and split into its phase and amplitude that procure
manipulations in a variety of ways. Convolution is a technique that is used
quite frequently to create the complex spectra in this piece. After convolving
the sound, an impulse wave is used to change the spectrum of the already
existing sound. Buffers are used several times to capture a sound. Once the
sound is captured in a buffer, it can be used to create a periodic signal, which
can then be manipulated into a more complex spectra while retaining some of
the characteristics of both the sources. Video processing is also used to
derive a sound within the piece. Using Jitter, a video matrix can be turned
into an audio signal and vice versa. This is also the only sound that has no
parameters for the user to alter other than playback volume. These processes
43
create complex spectra but the final sound produced by video manipulation is
unpredictable because the direct relationship of video and sound is not given.
The video file is not given and it is up to the performer to create a video file to
use. By combining FFT analysis, convolution, buffer manipulation and
feedback loops, eleven of the twelve sounds are created and altered to create
varying timbres and spectra all derived from one another using the same
techniques.
! Creating many sounds with complex spectra from a few processes is
an important aspect of 32o(F+F) F[+F][-F]F[-F[-F][+F]F]. Since this piece
uses an L-system to derive its form; the process of creating these sounds
were also intended to copy nature. Plants and animals are complex
organisms, that start from simple genetic material. Overtime, the seed grows
into something complex, but not before a variety of events and conditions
occur. The sounds in this piece and the process of creating them represent
the process of growing. A signal is created (the seed) and then analyzed or
manipulated at least once, and sprouts into a new sound. From there the
sound can remain in its present form (bean sprout) or further grow into
something much larger (plant). The new sound (plant) can then supply a
signal that gets manipulated and changed into another signal that has
remnants of the original but is still distinct (evolution into a new species). This
concept cannot be clearly heard in the piece, but this idea of growing a sound
through multiple processes being completed in sequence is something that
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can be directly translated by the L-system theme. Certain sounds have
artifacts that the listener can hear and possibly recognize throughout the
piece, yet the sounds are distinct enough to be recognized individually
(similar plant species but still different plants). The process of growing the
sounds also remains constant with the self-similarity that is found in L-
systems. All of the sounds created in this piece have very similar processes
and spectra. It is the introduction of other minor processes and other sounds
that create a new sound world.
! 32o(F+F) F[+F][-F]F[-F[-F][+F]F] demonstrates a unique way of
mapping an image generated by an L-system. This mapping and language
scheme can be evolved to illustrate musical characteristics that the composer
wants to portray to the performer. This language is the easiest and most
intuitive way of creating a piece the notates “on” and “off” messages without
having to struggle with notational boundaries and discrepancies. This
mapping strategy efficiently specifies what the performer needs to do in order
to make a sound. The mapping and language created for this piece
demonstrates how a composer must manipulate music to fit within the
boundaries of the L-system and musical idea.
Meditation of a Tree
! ! Meditation of a Tree is a fixed media piece that uses a
Stochastic L-system to trigger the playback of different sounds as well as
45
parameters that alter their characteristics. This piece uses a set of three
rules, X = F[+X]F[-X]+X, F = FF, and F = F-F. The two rules for “F” have a fifty
percent chance of being used. By having a fifty percent chance, the piece
has two basic outcomes and sequences that it can follow. One, when F = FF
is used and the other when F = F-F is used. Even though the two rules for “F”
are very similar, the final iteration used is drastically different. Just by adding
the “-”, the beginning sequence of the piece is completely different. When F =
FF is used, the sequence has long sections where “F” is constantly repeated,
which causes the piece to have a long drone in the beginning. When F = F-F
is chosen, because of the ”-” the sequence is broken and the repeated drone
is not heard. The image produced by this L-system, unlike the last pieces, is
fairly asymmetrical and does not look much like a tree or plant. However, the
obvious pattern that is immediately noticed within the L-System reflects
familiar divisions from larger parts into smaller elements. Figure 3.8
illustrates how this Stochastic L-system looks when interpreted by turtle
graphics for both possible outcomes.
!Figure 3.8 X = F[+X]F[-X]+X, F = FF%50, F = F+F
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These images are highly geometric, resembling a blueprint to an office or
cubicles. The idea of a “blueprint” sparked an interest in developing a piece
based on a sound that relies on the acoustical properties of the space it is
being played in to affect the overall sound. The concept of the piece are
centered around the interaction between computational space of the program
and the real, physical space of the listening environment. As the sounds
react within the space, different material will absorb or reflect the sound in a
variety of ways. Sounds will then start to combine or cancel to create new
effects that happen by chance. The final sounds produced rely on the
architecture and acoustics of the space, and therefore, will change from
performance to performance.
! This piece uses a a number of different probabilities to control
parameters such as sequence, frequency, and phase. These probabilities are
representative of weather conditions such as rain or snow. The different
musical parameters that the probabilities control will change the piece and
change the growth of the system The environment in which the piece is
played in changes the sound of the piece. This can be seen in nature, when
a hurricane comes and alters the growth of trees. Both the probabilities to
control parameters and environment that the space is in act like nature,
sculpting and changing the piece.
! The piece is divided into three movements that complement one
another. Each movement uses the same sequence created by the L-system.
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The first movement uses FM synthesis to derive the sustained sounds within
the piece while movement two uses purely sine waves to recast the sounds
produced in the first movement. Specifically, the parameters for frequency in
movement two are derived from a spectral analysis of movement one. Since
the sequence being used is exactly the same, movement two recreates
movement one using sine waves. The use of sine waves exclusively causes
movement two to achieve a completely different character as a result of
differences in phase and amplitude between the two pieces. Since phase
and amplitude are very important in making complex signals, the waves
interact and combine in the space to form intricate sounds. This can be
thought of as a metaphorical acoustic IFFT, in which, the phase and
amplitude components are being combined to form the new spectrum. Since
the space in which the piece is being played will absorb, reflect or damper
certain frequencies, the combination of sine waves will differ between
performances and create new unique timbres within each space. Movement
three features the simultaneous playback of the first two movements. New
timbres emerge as the sounds from FM synthesis meld with the sine waves
from movement two. Because of the accumulation of frequencies, the third
movement is the most intense movement. As a result of these combination of
sounds, high twinkling frequencies seem to pan around the audience, or in
rare cases, a Triangle seems to surface for a moment. The overall structure
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and sound of the three movements are very similar, but seem to grow in
intensity as a result of the combined movements.
! Apart from the stochasticity of the L-system utilized, the sounds also
have probabilities of changing. Every character in the sequence of the
original L-system is assigned to a sound, and each sound has a probability of
one of two sets of parameters being selected. This incorporation of
unpredictability in the system creates variety between successive
performances and movements. Variations that occur between performances
are a function of how these probabilistic rules are applied to frequency,
phases and decay times. The frequencies in movement two though are still
part of the frequency set from movement one. By altering phase and
amplitude, the sounds seem to wobble, as they move in and out of one
another within the room. Since the piece uses very rudimentary sounds that
have long envelopes (even though the decay time changes), sustained
sounds build up and then slow decay, causing shifts in pitch and timbre
throughout the piece. The density of sound also is in constant flux because of
the probability of different release times being chosen. Since there are
probabilities of different parameters occurring, there is also a chance of a
tonality starting to be heard in movement two. Movement two as mentioned
earlier uses sine waves and frequencies taken from a spectral analysis of
movement one. Even though this movement does not have a particular
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tonality that was created, the randomness that is programed into the piece
can cause a pitch center to be heard.
! Meditation of a Tree uses the sequence produced by the L-system to
slowly morph the sounds over a long period of time. Mapping each character
to a sound seemed like the most efficient way to use the characters in their
most basic form but still have the biggest impact on the piece. Since the L-
system as well as the piece is stochastic, having two sounds for each
character using probabilities keeps the stochastic theme. Working with sine
waves and FM Synthesis within SuperCollider is simple to implement. The
challenge in designing the system involved changing the parameters
effectively so the sounds would slowly interact with one another in the
performance space. This involved listening to the piece in a variety of spaces
with different speaker configurations, followed by an analysis of the
frequencies used that were then interpreted. Movement one was listened to
carefully noting how changes in the parameters of the system impacted the
sound in space. Once this part of the system seemed finally tuned, an
analysis of the frequency components were generated as a result of the FM
synthesis. Once the analysis was complete, inputing all of the frequency
information and probabilities of changes in the system was programmed.
! Thinking about what probabilities to use was challenging. Since the
chaos of nature was being evoked, the higher pitches have smaller properties
of occurring, as if they are representing hail, tornado, or a blizzard. Middle
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and lower frequencies have a higher probability of occurring, representing
rain, snow, sun, and fog. The extreme frequencies such as frequencies
below 30Hz that are scarcely audible to most people have tiny probabilities of
occurring, and metaphorically were approached as likely as meteors falling,
volcanoes erupting, and solar winds disrupting satellites. Even though these
probabilities are not exact, while thinking about natural events and how likely
they are to happen over a long period of time, one can think of how common
or uncommon these events are. This piece tries to use natural properties of
the space and acoustics to affect the sound while being a metaphor that tries
to sonify natural events. Meditation of a Tree interacts with the space to
create an audible phenomenon while the process of creating these sounds
are representative of natural events and how likely the average man is to
witnessing different meteorological events.
M101
! M101 is a piece written for cello, computer and custom analog
electronics. This piece explores the properties and sound that a Context Free
L-system can create when using different lookup tables for the notes. M101
is inspired by the shape of a pinwheel galaxy and the evolution of the stars as
they are born, live and die. The specific galaxy is Messier 101 which is a
pinwheel galaxy and resembles the shape of the L-system that was created.
A pinwheel galaxy is a spiral galaxy that can be observed face on as if looking
51
it it from above. It is named as such because it takes on the shape of a
pinwheel. M101 as interpreted by Turtle graphics is shown in Figure 3.9.
! ! !
! ! ! ! Figure 3.9 L-system M101
! M101 references a pinwheel galaxy, from which astronomers recently
observed a supernova exploding. Thinking of a supernova led to the idea of
what it would be like being a planet in that galaxy close to the supernova.
The time leading up to the supernova would be slow and soft just drifting
naturally around a star. All of a sudden, an explosion more powerful than one
hundred atomic bombs goes off and sends what seems to be an endless
stream of radiation in all directions that lasts only seconds. Being close to the
explosion would be the most violent experience for a planet. Remnants of the
supernova however last for millions of years expelling radiation from them.
After the explosion, there is either a neutron star, black hole, or a gap in
space where the supernova was. The idea of this cycle is prevalent in the
piece by the form which begins slowly and quietly, and then suddenly erupts
into a violent fast section. The fast section stops abruptly, and a slow section
slowly fades in. This metaphorically mirrors the life cycle of a star leading up
52
to the point where it goes supernova and then fades over time. This L-system
was chosen because its shape resonated with ideas related to the life and
death of planetary bodies.
! This piece uses a set of six rules to create the L-system, that are
labeled A, B, C, D, F, G (E is excluded). An angle of fifty-one degrees is also
used to create an image that appears to be a star with rays propagating out of
it. The sequence derived is from iteration four. Using the final iteration of this
system, three more L-system sequences were derived. This was done by
assigning a number from one to three for each bracketed system. Then all of
the divisions were written in the order they were produced, alleviating any
hierarchy that the system might have created. Each one of the numbers was
for a voice: sequence one for cello, two for analog electronics and three for
computer. This process facilitated the derivation of notes and patterns since
they were easy to visualize.
! Each letter of the cello parts sequence was mapped to a different table
of notes. These notes, when added together, reflect the harmonic series of a
cello playing a low G. The notes of the spectrum were arranged by writing all
of the notes down and dividing them between multiple tables. The ordering of
the tables were F, G, A, B, C, D. Using this order “F” and “G” get the
fundamental (G2) and the first harmonic (G3). The later rules end up with no
low notes but contrast the low notes with extremely high notes. The “+” and
“-” signs are used to cycle through the notes in the table. “+” would be the
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next higher note or loop to the beginning and “-” would be the next lower note
or loop to the end. The iteration used for the cello part mapped the brackets
to show slurs and ties between groupings of notes. All notes within the
bracket would be slurred together and if there was a repeated note it would
be tied as well as slurred. If there was only one note in the bracket then it
would be played normally. After breaking apart the main iteration into three
separate ones, a pattern emerged between all of the systems. Roughly
around the halfway point, or a little further, the pattern changed and groupings
of notes became longer in the cello part. To interpret this into the piece, the
beginning is very slow with long sustained notes, at the halfway point, their is
a complete change in the music, it is played as fast as possible, the notes do
not have any duration and the dynamics are loud. After the loud section, the
entire piece is played in reverse, creating a palindrome. This palindromic
form resonated with the astronomical ideas that inspired the piece. A star is
born, burns for billions of years, collapses and then transforms into s neutron
star, black hole or completely obliterates itself. This can be seen in the piece
by the use of dynamics and change in pace and style.
! The computer part has six possible sounds that are assigned to the L-
systems rules. All of the sounds that the computer creates are derived from
either the cello, the electronics, or both combined together. The computer
uses an eight channel system to pan the audio in different configurations. All
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of the parameters for the different sounds are controlled, by the “+” and “-”
signs indicate the raising or lowering of a parameter.
! All of the rules are in groups of two and are assigned to a particular
computer process. Figure 3.10 is a reference that shows each rule grouping
and what they are controlling.
! Rules “F” and “G” echo the live signal while “A” and “B” are assigned to
a frequency peak analyzation that outputs only the number of peak
frequencies assigned by the user. Rule “C” and “D” are a combination of
both, that echoes the peak frequency around the eight speakers. At about the
halfway point in the piece, all of the computer process change. “F” and “G”
control the granular synthesis of a buffer recorded from the cello while “A” and
“B” take the buffer, analyze the peak frequencies and then granulate them.
Each grain of the granular synthesis are heard through a different speaker.
Rule “C” and D” do not occur often and because of that the sound is
aggressive and violent. These rules are heard at the exact halfway point of
the sequence which is also when the music becomes fast and loud. For this
reason, rules “C” and “D” are a convolution of the cello and the analog
F, G --> EchoA, B --> Peak FrequenciesC, D --> Echoed Peak Frequencies--Half Way Change--F, G --> Granular Synthesis from BufferA, B --> Peak Frequency Granular SynthesisC, D --> Convolution of Electronics and Cello
Figure 3.10 Computer Mapping
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electronics. In this convolution, the cello and electronics phases and
amplitudes are being multiplied together which creates a very aggressive,
distorted sound. These rules are seen in the sequence at the beginning of
the new section and at the end. Since the piece is a palindrome, the sound is
heard at the beginning of the fast section, middle of the fast section and end
of the fast section leading into the slow section that is from the beginning.
This sound causes the piece to have a distinguishable yet interrelated parts.
! The analog electronics are sectioned into three different modules that
work together to create the sound. The first part of the module is a preamp.
This preamp is not a normal preamp, in that it represents the input signal as
control voltage, that ultimately transforms into a square wave. The signal
used as the initial sound for the electronics comes from a live video of the
cello player. The camera’s AV output is connected via RCA to the preamp.
This initial sound is a noisy sound that changes with the intensity the
performers movement, any changes in light, autofocus or any other change in
the environment that causes the camera to compensate for it. That signal is
then sent to a phase lock loop. A phase lock loop takes the incoming signal
and compares it to an internal voltage. This causes a changing noisy signal
to become slightly more stable and less abrupt, but also can bring out
different parts of the sound and change the timbre. A phase lock loop can be
used for pitch tracking, however, it can also be used to transpose the sound.
The transposition of the sound can also be controlled by manipulating the
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compare voltage by resisting, dividing etc... the signal. The sound from the
phase lock loop is then sent to a binary divider which takes the signal and
divides it by a number which is a power of two i.e. 2, 4, 8, 16, 32 up to 4096.
This signal drastically alters the sound. By dividing the sound too much, the
sound becomes a pulse, or rhythm rather than a steady sound. When the
signal is divided by one of the lower numbers a pitch occurs which is a
harmonic of the sound. When a low and high number both divided the sound
at the same time, the harmonic sound of the low number pulses or changes at
the rate of the division of the high number. The more divisions that take
place, the more distorted and noisy the sound gets, however, the use of timed
switching of pulse can create polyrhythms between pitches.
! Using these electronics, the third part of M101 is a guided
improvisation on the electronics. The performer is to improvise within
parameters given in the score. These parameters are derived from the
sequence created by the L-system. The third part has a repeated pattern that
is changed three times as a natural outcome of the L- system. The performer
improvises within different parameters three times. The first two changes
happen slowly in order to build up into the fast violent section, in which the
performer is allowed to improvise as they choose. Instructions are given to
the performer which states “match the pitch and volume of the piece”. This is
given in order to make the improvisor blend as much as possible with the
computer and Cello.
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! This piece, uses the most deviation from the original system and the
most complex mapping scheme compared to the previous pieces. This piece
demonstrates how one can manipulate an L-systems sequence to create
complex layers and extract parts. This piece also demonstrates another way
of mapping a rule to a sound by using a computer sound instead of a note
and uses “+” and “-” in a literal sense. An L-system that uses tables seemed
more flexible than any of the other L-systems because of the vast amounts of
changing data. Another approach to a tabled L-system could be to not use
notes and sounds as with multiple choices as tables, but use rules to create
the L-system. For example, if “F = F+F” once that rule is used then switch the
rule “F” with “F = F-F”. This would differ from a stochastic approach because
a stochastic L-system does not have a set rule but multiple rules with
probabilities of occurring for each use of the rule. If the composition was
made in this fashion, the pattern that is seen across parts might have
disappeared completely or the piece could be different in terms of pitch and
contour. M101 illustrates how one can use one type of L-system (Context
Free) but use a property of another type of L-system (Tabled) to make a more
complex mapping scheme or piece.
!
Summary
! These pieces aim to show examples of how L-systems can be used to
create detailed music and differentiate from the original note mapping that
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was created by Przemysław Prusinkiewicz. The ways of mapping L-systems
to music were created based on values and metaphorical ideas found in both
the music and L-system. However, anyone using L-systems to map music
can create their own interpretations, guidelines or language that fits their
musical ideas. Mapping L-systems to sound is one of many possible
algorithmic techniques a composer can use as a tool while writing their piece.
These ways of mapping are not intended to write a piece for the composer,
but choose different material or ideas that have already been established to
bring out a pattern in the overall sound of the music.
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Chapter 4
Process Evaluation
! L-systems have an interesting history that starts out in the field of
botany and then slow merges with computer sciences and music. Many of
the early examples of musical applications of L-systems were developed by
computer scientists and then used by musicians. A goal underlying the
development of L-systems was the accurate simulation and representation of
patterns occurring in nature. The integration of probabilities and math
theorems differentiates implementations of L-systems from one another, and
thus the patterns they create. The pieces composed as part of this thesis
explore how three types of L-systems can produce substantial data to
develop a piece. Of particular concern was the examination of how L-systems
can affect the sound and compositional process of the piece. Apart from
exploring the mapping potential of L-systems in each piece, various methods
of performance and notation can be used to compliment the L-system. Each
piece focused on how the L-system is created and tried to evoke the visual
characteristics of the L-system in the composition.
!
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Compositional Emphases
! The mapping strategies employed for the compositions differed based
on the instrumentation and aesthetic goals. As the research for this
composition evolved, subtleties of each system were discovered that altered
the approach and mapping strategies of the compositions. Creative ways of
using L-systems to derive multiple voices developed as a byproduct of
wanting to create more complex music. Many of the pieces are composed for
live performance. These pieces seek to incorporate the performers
interpretation and add elements of chance to the performance. A piece
written for computer focused on the performance space and sought to use the
acoustical properties as part of the composition. By using L-systems to
derive music, numerous possible sonic outcomes were discovered. The
identity of the piece between performances is maintained by allowing the
performer or computer to choose from a delineated field of events, duration,
time and timbre. An example of this can be heard in the performances of
32o(F+F)F[+F][-F]F[-F[-F][+F]F] at the Wesleyan composition seminar concert
and at the Florida International University Electro Bonanza. The duration of
the piece remained approximately consistent between both performances;
however, the parameters of the sounds were different. In the Wesleyan
University performance, the sounds blended with one another creating
multiple different textures and densities throughout the piece. At the Electro
Bonanza, the higher frequencies melded together creating a melody based on
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timbral manipulation. 32o(F+F)F[+F][-F]F[-F[-F][+F]F] at both performances
kept its initial identity but created sonic variances.
! The mapping of L-systems to other musical parameters other than
pitch and duration can be seen in Hanspeter Kyburz’s Cells. In Cells, Kyburz
used pre-composed motives and cellular automata to sequence the material
together. Similarly, A Meeting of Florets used pre-composed motives and
assigned them to different rules in the same fashion as Kyburz’s Cells. The
goal of this mapping was to focus beyond the confines of pitch and rhythm,
and instead concentrate on blocks of sound and how they evolve over the
length of the piece. The technique of mapping specific to A Meeting of Florets
was cumbersome largely do to the amount of time needed to notate all the
musical possibilities. A mistake in a transposition, much like a mutation in a
plant, had ramifications for future processes, such that absolute precision in
notation and mapping was a strong concern. Natural mutations which occur
in nature can not be incorporated into a Context Free L-system, because the
use of probabilities or different rules to substitute into the system suddenly
change a Context Free L-system into a different type of L-system. However,
while notating A Meeting of Florets, the notation of pitches would sometimes
be entered wrong or looked over while editing the score. These cause a
natural mutation of the motives which can be analogues to a natural genetic
mutation in nature.
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Evolution of Mapping Strategies
! Following A Meeting of Florets, finding highly divergent strategies for
mapping became an aesthetic and intellectual pursuit. A visual mapping that
can be applied to any L-system image regardless of the iteration was of great
interest. This mapping would not rely on the amount of data produced by the
L-system, but on the graphic created instead. This visual mapping was used
to create pieces for electronics and laptop ensemble. The mapping of the
image by an L-system involves an additional level of abstraction, when
compared to the mapping of the sequence. The L-system’s image lives in a
Cartesian plane and can be read from any direction. The sequence created
by the L-system is linear and can only be read one way in order to achieve
the natural evolution of the musical parameters. Figure 4.1 shows the image
and division of sets created for 32o(F+F)F[+F][-F]F[-F[-F][+F]F] and the
sequence created for M101. By looking at this image, one can see a pattern
for both pieces, however the amount of data mapped for M101 is much
greater than 32o(F+F)F[+F][-F]F[-F[-F][+F]F].
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Figure 4.1 Comparison of Data 32o(F+F)F[+F][-F]F[-F[-F][+F]F] Left, M101 Right
! In A Meeting of Florets, the iteration of the data itself was important,
which placed the compositional focus on the repetition of motives and
recognizable shifts in transposition. This created a tendency for long
sequences with internal variation. The length of the sequence is irrelevant in
32o(F+F)F[+F][-F]F[-F[-F][+F]F] since the mapping is derived from a visual
image in a Cartesian space. One can create data from the image and/or
divide the image into smaller images which will produce different products.
These images motivates an act of investigation and deciphering of the self-
similar patterns. This way of composing creates a degree of interpretation,
but how does one use the data sequence produced by a large iteration in an
interesting way that is not prohibitively time consuming?
! One way to solve this problem is to use a computer that can sequence
all of the characters together and assign them different gestures, notes and
other musical parameters. This method is convenient because the concerns
+[F][+G-F]+-AA-++C-D[F][+G-F]+[F][+G-F][F][+G-F][F][+G-F]+-AA-++C-D[+G-F][F][+G-F][F][+G-F][F][+G-F][F][+G-F][F][B][F+F+F-G]+[F][+G-F][F][+G-F][F][+G-F]+-AA-++C-D[F][+G-F]-G+G+-[F][+G-F]++B+F-B-FF[F-G][C-D][F-G][C-D][F+F+F-G][F+F+F-G][F+F+F-G][F+F+F-G][F][B][+AA-+C-D]
64
regarding notation or accuracy are mitigated, and one can focus instead on
the manipulation of sounds produced. Since this mapping lives in the
electronic world, a single rule can trigger the sound while all the other
characters and and letters can control different parameters and functions of
the sound. By mapping this way, the evolution of the L-system and a single
sound can be heard and appreciated. In the second movement of Meditation
of a Tree, the sound material is comprised of shifting sine waves. However,
each character of the L-system changes the parameters of these sine waves
in terms of frequency, phase, amplitude and decay time. This, in turn, slowly
evolves the overall sound of the piece.
! Another mapping strategy derives independent voices out of the L-
system or smaller L-systems from the original one. An example of this
involves the creation of a hierarchical system and isolation of data based on
position. By doing this, a new system, similar to the original but with much
less data is created. Also, the composer can use different hierarchical levels
for different movements, compositions or voices within the same composition.
This mapping strategy can be expanded by using the hierarchies to create
different sections of the piece. The evolution of these mapping strategies are
dynamic, and arise from an impetus to control more aspects of the music than
just the motive and transposition of pitch and rhythm. Each piece composed
for this thesis also brings new mapping strategies to light as different L-
systems and instrumentations are being used
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Performance Strategies
! The performative works created with various mapping strategies rely
on the performer to bring the work to fruition. For example, in A Meeting of
Florets, the performer is asked to learn seven, frequently transposed motives.
The transposing patterns require a large amount of concentration and musical
awareness from the performer. In 32o(F+F)F[+F][-F]F[-F[-F][+F]F], the
performer does not have to worry about motives or transpositions, but has to
listen well, blend the timbres together and mold the individual sounds together
into one coherent thought. In M101, the cellist has fewer decisions to make
as the score is highly specific. The cellist simply plays what is written;
however, the use of microtones and harmonics make seemingly easy parts
difficult. Besides the demands on the acoustic instrument, the performers
playing the computer and analog electronics must listen and constantly adjust
to the cello sound in order for all of the voices to be heard. The computer part
is highly interactive, and the performer is asked to quickly change the
parameters of multiple sounds. The computer performer must be comfortable
with computers as well as listening and adjusting to many different sounds
simultaneously. The L-system brings an algorithmic approach to music, that
the performer then creates a musical expression from. If a piece uses solely
an algorithm and computer playback, the tuning and timing will be completely
systematized. When a performer plays a piece, the notes and rhythm will be
66
highly accurate, but have variations. These compositions incorporate human
elements of performance. Inviting these elements of variation often required
unique notational strategies that invoked elements of the L-system while
soliciting input and interpretation from the performer.
! Notation is very important when mapping an L-system to musical
parameters. 32o(F+F)F[+F][-F]F[-F[-F][+F]F] required its own notational
language as it is scored for interactive computer. A series of numbers ranging
from one to twelve were used to indicate to the performer which sound to turn
on and off. This notation is very simple, specifying only what sounds should
be either "on" or "off". The duration of the piece is deliberately unspecified so
as to let the performer alter and manipulate the sounds in real-time. This
method of notation does not feature the L-system's strings of characters
because of the abstract mapping and division of the image. For pieces that
are notated with numbers, the image cannot be seen, but patterns of numbers
reveal that a pattern in the image is present. For pieces with conventional
notation, the L-system aspect is still visible and more tangible than the
number system. The biggest downfall to this is that, only by knowing the rules
of the L-system, one can see the L-system in the music. Creating an
understandable notation with clear performance instructions is the most
important job as a composer. When using L-systems to manipulate or
sequence events, a very clear and easy notation must be created in order to
help the performer read the constantly changing music accurately and
67
effectively. Customized notations to represent sonic characteristics or
computer manipulations are created by the composer, keeping in mind that
that the symbols being used should define gestures. Creating an abstract
and efficient musical language not only makes composing easier, but can be
used in future pieces to represent sonic elements. Using this number system
to represent sonic events will be used in the future for both acoustic and
electronic music.
Environmental Factors
! L-systems are algorithmic systems capable of modeling plant growth
and creating visual representations of patterns found in nature. In nature,
some odd events occur on a regular basis. When composing using an L-
system, natural events most of the time are omitted. These events are not
programed into the L-system and do not affect the system in any way. This
idea is like a tree receiving water from a light rain storm. It does not really
affect the tree in terms of overall structure. If a hurricane comes, the tree's
structure will most likely change from being attacked by strong gusts of wind
and rain. By using indeterminate aspects of the music, natural events can be
slightly alluded to. By preparing a piano or creating an environment in which
the performer can manipulate the sound in real-time, unknown outcomes can
occur and new textures can appear. When adding indeterminacies into a
fixed media piece, the use of probabilities of different sounds occurring or
68
being manipulated in certain ways can drastically alter the piece or bring out
new sounds. The sounds of Meditation of a Tree can cancel, or attenuate
each other depending on the parameters and sounds that are playing.
Composing a piece that reacts differently in each performance space also
adds a layer of unpredictability to the music. The interactions of sound within
a space will always be an experiment and a factor that is hard to predict when
composing music. When a composer takes this uncertainty into account, a
new piece assumes indeterminate elements and, during multiple
performances of the piece, can create completely different perceptions. The
various performance spaces can alter the sounds and create different
densities and textures that were not heard in previous performances. There
is no way to know precisely how a plant is going to be affected by nature. L-
systems create an imaginary world in which weather is not a determining
factor. One has to decide if natural events are going to be represented in the
music and how they will be embodied In these pieces, natural events are not
programed into the L-system, instead natural events manifest in the form of
unpredictable sonic events or changes to the system from an outside source,
such as the performer. The initial inspiration for the piece can develop in
ways that cannot be clearly represented by the L-system. The composer then
has to use sonic metaphors to represent the particular idea that is to be
expressed. These metaphors live in the realm of programatic music. The
audience will not know of these metaphors until told how the music and
69
metaphors sonify an idea. Weather is unpredictable and, when sonified, the
composer can make the metaphor unpredictable. Elements such as
unpredictable sounds can occur in a piece but also have the chance of not
occurring in the piece.
Crossing L-systems
! L-systems are being used as a compositional process to create a
sequence of events or sounds that the composer has precomposed. As more
compositions are created using L-systems, the more complex the mapping
will become because of the composers intention to create or evolve their
music further. Using L-system to create the notes and rhythms as well as the
overall form is an approach that can be used in the future to create more
complex music with intervention from the composer. New mapping schemes
will always be developed to accommodate any instrumentation, inspiration or
medium. These mapping schemes will have a huge range of complexities
that will start at the most simple rule --> motive and continue to the creation of
wavetables and sound using the L-system. Combinations of techniques will
also always be used to try to make the mapping more complex and musical
with emphasis on how the L-system is controlling different musical
parameters. Using L-systems is a very easy process once a computer
generates the image and sequence.
70
! Composing multiple pieces using L-systems can create a variety of
different results. A composer does not have to limit their composition to one
L-system and can use the same L-system to create multiple pieces. This can
be seen in nature when a farmer plants seeds. The plants will grow and
create the same plant but each plant will look differently. By using the same
L-system with different mapping strategies or instrumentation, a new piece of
similar style can emerge. To develop this further, a composer can use a very
traditional mapping scheme such as rule--> motive to write a piece. Then the
composer can take the same mapping scheme and change the motives to a
sound or timbre in an electronic piece. The same L-system can then be used
in terms of mapping the graphic to a number based notational system to
control an interactive computer performance. The same L-system can finally
be used to create a graphic score. Each piece will sound different, though
there is also a chance that certain patterns in common will emerge between
pieces. Even if the pieces all have their own identity, they are still a collection
of pieces derived from one system. The composer created an auditory
evolution of the system and can be thought of as creating new species or
hybrid species of plants.
! In the future, the compositions that were analyzed and discussed will
all have two or three more pieces that use the same L-system but change the
mapping. By using a variety of L-systems, a series of interrelated pieces
emerged that each have a unique identity but are part of a metaphorical
71
family of compositions. This would not serve as a way to compare mapping
and L-systems, but to listen to and identify similarities between related
species of L-systems.
! When composing, many surprises arose due to the pattern and
evolution of the piece that the L-system created. Some of the surprises
encountered were shifts in tonality, the creation of new timbres by the
combinations and cancelations of sounds and overall smoothness that can
occur as the piece is evolving and unfolding. An example of this can be seen
in the composition Meditation of a Tree. As the piece progresses, the
combinations and disturbances of the sine waves form and change timbres
into more complex waveforms. In A Meeting of Florets, motives transpose at
different rates, causing the tonality to shift and the tonal center to diminish or
disappear. Each type of L-system had its weak points that had to be
conquered. None of the pieces were successful or unsuccessful, but showed
different compositional aspects that had to be manipulated by the composer.
Meditation of aTree did not have notes or rhythms that had to be meticulously
transposed or placed in the music, but had to organize many different
frequencies, phases and amplitudes that all had probabilities of occurring.
32o(F+F)F[+F][-F]F[-F[-F][+F]F] did not have instrument limitations or
automated probabilities and sequences, but had twelve sounds that had to be
different from one another and sound good when played in combination in
order. The compositions written for this thesis had indeterminate factors that
72
would cause the pieces to change from one performance to another
performance. Sometimes the performance would be longer, other times
sounds would be created that were not in the previous performances. Each
performance and composition has a similar identity but slightly different
outcomes, as if a rose growing in different colors.
!
73
32o (F
+F)
F[+F
][-F]
F[-F
[-F][+
F]F]
Max
wel
l Tfir
n
Mar
ch 2
011
74
32o (F
+F)
F[+F
][-F]
F[-F
[-F][+
F]F]
32o (F
+F)F
[+F]
[-F]F
[-F[-F
][+F]
F] tr
ies
to c
over
and
sto
p th
e ev
eryd
ay n
oise
s th
at p
eopl
e cr
eate
by
talk
ing
exce
ssiv
ely
loud
, lis
teni
ng to
mus
ic p
laye
rs, t
alki
ng o
n th
e ph
one,
text
mes
sagi
ng, r
evvi
ng th
eir c
ar e
ngin
es, p
layi
ng lo
ud m
usic
in th
eir c
ars,
hum
min
g an
d th
inki
ng o
ut lo
ud.
This
pie
ce h
opes
to e
xagg
erat
e an
d fra
me
a si
lenc
e th
at s
eldo
m o
ccur
s in
a b
ig c
ity b
y ju
xtap
osin
g it
with
ext
rem
e no
ise.
Thi
s pi
ece
is m
eant
to b
e pl
ayed
out
side
for t
he d
urat
ion
of n
ine
to te
n m
inut
es a
nd c
ut o
ff ve
ry a
brup
tly w
ith t
he h
ope
that
a
mom
ent o
f com
plet
e si
lenc
e fo
llow
s be
fore
the
nois
e of
dai
ly li
fe re
sum
es.
This
pie
ce u
ses
12 d
iffer
ent u
niqu
ely
gene
rate
d fo
rms
of c
olor
ed n
oise
that
can
be
man
ipul
ated
by
num
erou
s pa
ram
eter
s sp
ecifi
c to
the
mea
ns b
y w
hich
eac
h no
ise
is g
ener
ated
. Th
e tw
elve
form
s of
noi
ses
use
synt
hesi
s te
chni
ques
rang
ing
from
the
conv
olut
ion
of d
iffer
ent
puls
e ge
nera
tors
to in
terp
retin
g vi
deo
data
as
FFT
fram
es.
The
form
of t
he p
iece
is d
eriv
ed fr
om a
n L-
syst
em w
hich
is d
raw
n at
a 3
20 a
ngle
and
sta
rts w
ith th
e ax
iom
F+F
. U
sing
the
rule
F[+
F][-F
]F[-F
[-F][+
F]F]
, an
imag
e of
a tr
ee is
pro
duce
d w
ith c
ross
ing
limbs
and
num
erou
s sp
littin
g br
anch
es, r
esul
ting
in m
any
com
plex
, int
erw
eavi
ng la
yers
. Th
e in
ters
ectio
ns o
f bra
nche
s an
d lim
bs d
eriv
e th
e fo
rm o
f the
sco
re. W
hen
an in
ters
ectio
n oc
curs
it w
ill ei
ther
act
ivat
e or
mut
e a
parti
cula
r noi
se g
ener
ator
.
Perf
orm
ance
Not
es:
This
pie
ce is
mea
nt to
be
play
ed lo
ud b
ut n
ot lo
se it
s m
usic
al c
hara
cter
istic
. It
is im
porta
nt fo
r the
per
form
er to
list
en c
aref
ully
in o
rder
to m
ake
sure
that
all
of th
e so
unds
are
au
dibl
e. T
his
piec
e re
quire
s a
mix
er, a
t lea
st tw
o sp
eake
rs a
nd a
com
pute
r. I
reco
mm
end
usin
g th
e Ev
olut
ion
UC
33e
Mid
i Con
trol S
urfa
ce b
ecau
se o
f its
man
y sl
ider
s an
d kn
obs
how
ever
oth
er s
imila
r int
erfa
ces
will
wor
k. U
sing
an
inte
rface
mak
es th
e tra
nsiti
ons
betw
een
soun
ds v
ery
fluid
as
wel
l as
man
ipul
atin
g th
e m
any
para
met
ers
mor
e ef
ficie
nt.
The
scor
e is
gra
phic
and
use
s nu
mbe
rs a
nd c
olor
s as
dire
ctio
ns.
The
num
ber i
ndic
ates
whi
ch s
lider
is to
be
mov
ed w
hile
the
colo
r ind
icat
es o
n or
off.
Blu
e m
eans
mak
e th
e so
und
audi
ble
(slid
e sl
ider
up)
whi
le p
ink
mea
ns tu
rn th
e so
und
dow
n/of
f (sl
ide
slid
er d
own)
. Th
e sc
ore
is to
be
read
from
left
to ri
ght a
nd u
p to
dow
n. A
t the
ver
y en
d al
l so
unds
that
are
aud
ible
sho
uld
be im
med
iate
ly tu
rned
dow
n al
l at t
he s
ame
time
to e
nd th
e pi
ece.
Tak
e a
brea
th a
nd li
sten
for a
mom
ent t
o th
e so
unds
aro
und
you.
Ever
yone
of t
he tw
elve
sou
nds
has
a sl
ider
and
a fi
lter.
To
use
the
filte
rs p
ush
the
spac
ebar
to o
pen
up th
e fil
ter s
cree
n. E
ach
filte
r is
labe
led
with
a n
umbe
r fro
m o
ne to
tw
elve
, eac
h nu
mbe
r is
asso
ciat
ed w
ith th
e so
und
assi
gned
to th
at n
umbe
r and
that
num
ber s
lider
.
Exam
ple
of n
otat
ion
4
turn
slid
er 4
up
4
turn
slid
er 4
dow
n
75
32o (F
+F)
F[+F
][-F]
F[-F
[-F][+
F]F]
Max
wel
l Tfir
n
!!
!!
!!
!!
!!
!!
!!
!!
!!
!!
!!
!!
!!
!!
!
!!
!6!
!!
!!
!8!
!!
12!
4!!
!10!
3!!
!!
!3!
!11!
!!
!!
10
!!
!4!
!!
!!
!4!
!!
4!12!
!!
4!1!
7!!
!!
11!
!9!
!!
!!
5
!2!
4!7!
10!
12!
!1!
6!9!
5!5!
1!7!
!8!
11!
4!8!
!1!
7!4!
9!12!
3!3!
!6!
11
1!1!
1!8!
1!!
4!12!
3!10!
6!11!
3!5!
6!6!
12!
5!9!
11!
7!8!
5!8!
1!8!
4!5!
3!8
!
3!5!
5!11!
3!5!
2!7!
5!11!
7!5!
2!!
9!5!
3!2!
12!
2!8!
12!
10!
10!
4!!
!7!
4
!!
!9!
!!
!!
!11!
!!
2!3!
!2!
1!6!
10!
!!
!6!
!2!
!!
!!
9
!!
!!
!!
!!
!!
!!
!!
!!
!!
!!
!!
!!
!!
!!
!
76
A M
eetin
g of
Flo
rets
2010
Max
wel
l Tfir
n
77
A Me
etin
g of
Flo
rets
A Me
etin
g of
Flo
rets
is
a pi
ece
comp
osed
for
pre
pare
d pi
ano.
Th
is p
iece
was
cre
ated
wit
h th
e in
spir
atio
n of
a c
ompu
ter
gene
rate
d gr
aphi
c of
a L
-sys
tem.
Th
e L-
syst
em l
ooke
d li
ke a
sma
ll g
athe
ring
of
flor
ets
blow
ing
in a
gen
tle
bree
ze.
The
L-
syst
em w
as a
lso
used
to
crea
te t
he f
orm
of t
he p
iece
. A
Mee
ting
of
Flor
ets
has
a to
tal
of f
ive
moti
fs t
hat
get
tran
spos
ed i
n di
ffer
ent
ways
bas
ed o
n th
e na
tura
l cr
eati
on o
f th
e L-
syst
em.
Thi
s ca
uses
the
pie
ce t
o un
fold
ver
y sl
owly
and
als
o gi
ves
the
perf
orme
r ma
ny d
iffe
rent
pat
hs i
n wh
ich
he/s
he c
an c
hoos
e to
pla
y or
ski
p.
Each
pag
e is
uni
que
and
each
pag
e ha
s se
vera
l op
tion
s th
at c
an c
ause
the
pie
ce t
o un
wind
in
a va
riet
y of
way
s. Rule
s
-Th
e up
per
left
cor
ner
just
abo
ve t
he f
irst
sys
tem
has
a se
t of
opt
ions
. C
hoos
e on
e fo
r ea
ch p
age.
- Pl
uck
the
stri
ng f
rom
insi
de t
he p
iano
-
Play
not
es s
low,
gra
dual
ly s
peed
ing
up
- Pl
ay n
otes
sta
rtin
g fa
st,
grad
uall
y sl
owin
g do
wn.
Note
s to
Pre
pare
A1,
C2,
A2,
C3,
D3,
F3,
Ab3,
Db4
, Gb
4, B
b4,
C5,
E5,
Eb5,
B5,
D6,
Gb6
, C7
, Db
7, F
7, C
8
To p
repa
re n
otes
put
a l
arge
scr
ew b
etwe
en a
ll o
f th
e pi
ano
stri
ngs
for
that
par
ticu
lar
note
i.e
a p
iano
not
e wi
th 3
str
ings
wo
uld
requ
ire
2 sc
rews
. O
ne f
or t
he l
eft
and
midd
le s
trin
g an
d on
e fo
r th
e ri
ght
and
midd
le s
trin
g.
The
scre
ws s
houl
d no
t we
dge
betw
een
the
stri
ngs,
but
sit
loo
sely
in
orde
r to
let
the
str
ing
vibr
ate
the
scre
w.
For
low
stri
ngs
that
onl
y ha
ve o
ne
stri
ng p
ut a
lar
ge b
olt
or w
ashe
r ar
ound
the
str
ing
if p
ossi
ble.
If
not
pos
sibl
e us
e th
ick
gaug
e me
tal
wire
and
wra
p it
loo
sely
arou
nd t
he s
trin
gs.
It
does
not
mat
ter
wher
e al
ong
the
stri
ng t
he m
ater
ial
is p
lace
s as
lon
g as
it
vibr
ates
suf
fici
entl
y.
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124
M101
2011
Max
wel
l Tfir
n
125
M101
is a
piec
e co
mpo
sed
for C
ello
, Ana
log
Elec
troni
cs, C
ompu
ter a
nd V
ideo
. Th
is pi
ece’
s so
urce
of i
nspi
ratio
n is
the
spira
l gal
axy
nam
ed M
101.
The
stru
ctur
e of
this
piec
e as
wel
l as
the
nam
e an
d m
usica
l qua
lities
are
der
ived
from
an
imag
e th
at lo
oks
like
a sp
iral g
alax
y wi
th a
sta
r in
the
cent
er.
This
piec
e is
split
into
3 s
ectio
ns.
The
first
sec
tion
is ve
ry s
low
and
to b
e pl
ayed
sof
t and
floa
ting
while
th
e se
cond
sec
tion
is to
be
play
ed a
s fa
st a
s po
ssib
le le
adin
g up
to a
sud
den
slow
sect
ion
agai
n. T
he c
ello
give
s al
l of t
he c
ues
for t
he c
ompu
ter p
art t
o fo
llow
while
the
perfo
rmer
pla
ying
the
anal
og
elec
troni
cs im
prov
ises
with
in c
erta
in p
aram
eter
s no
tate
d. T
he v
ideo
is o
ptio
nal a
nd is
don
e in
real
-tim
e us
ing
jitter
.
Inst
ruct
ions
-The
com
pute
r is
to fo
llow
what
the
scor
e sa
ys to
do
and
hold
the
soun
d un
til th
e ne
xt c
omm
and.
The
dot
ted
line
repr
esen
ts th
e ho
ldin
g of
the
soun
d.-T
he C
ello
sho
uld
play
exa
ctly
what
is g
iven.
-I
t is
reco
mm
ende
d th
at th
e ha
rmon
ics b
e pl
ayed
on
the
A st
ring,
how
ever
the
perfo
rmer
can
pla
y th
em o
n wh
ich e
ver s
tring
is m
ost c
omfo
rtabl
e.-G
lissa
ndo
shou
ld b
e ve
ry s
moo
th a
nd s
tead
y.-T
ake
note
of m
icrot
ones
and
cle
f cha
nges
-mm
1 -
26 s
houl
d be
a v
ery
slow
cres
cend
o fro
m p
pp to
fff
-mm
64
- to
the
end
of th
e pi
ece
shou
ld b
e a
slow
decr
esce
ndo
from
fff t
o pp
p-T
he a
nalo
g el
ectro
nics
are
told
to im
prov
ise.
The
impr
ovisa
tions
sho
uld
not s
tand
out
from
wha
t is
goin
g in
the
mus
ic, b
ut s
houl
d re
flect
and
sub
tly a
dd a
noth
er la
yer t
o th
e ov
eral
l sou
nd.
Equi
pmen
t Lis
t
-Max
/MSP
/jitte
r-M
icrop
hone
-At l
east
two
spea
kers
(pie
ce is
des
igne
d fo
r 8)
-Vid
eo c
amer
a wi
th R
CA o
utpu
t-A
udio
Inte
rface
with
at l
east
2 in
puts
-2 M
ixers
. One
with
at l
east
eig
ht in
puts
. O
ctop
honi
c se
tup,
mixe
r nee
ds 8
out
puts
. The
oth
er w
ith a
t lea
st 2
inpu
ts a
nd 2
out
puts
(for a
nalo
g el
ectro
nics
)-I
f usin
g vid
eo, a
pro
ject
or a
nd s
cree
n or
mul
tiple
ana
log
tv’s
scat
tere
d ar
ound
the
stag
e ar
e ac
cept
able
.
126
Cel
lo
Mic
roph
one
Cam
era
Faci
ng C
ello
Anal
og
Elec
troni
csM
ixer
Com
pute
r
Inte
rface
Spea
ker
Spea
ker
Spea
ker
Spea
ker
Spea
ker
Spea
ker
Spea
ker
Spea
ker
Spea
ker c
onfig
urat
ion
can
chan
ge
base
d on
ava
ilabl
e eq
uipm
ent
Mix
er
127
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1q =
75 Áè w8 4w
?w >
wM
101 w
w &è# w#
Max
wel
l Tfir
n w >?&
è wè w
?w >
èá wá &è w
è wèÜ wÜ
w?
ww
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Cho
ose
one
and
play
for 4
cou
nts
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èá Xáèb X
4 4 XXC
hoos
e on
e an
d pl
ay fo
r 4 c
ount
s
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è w
8 4Rai
se E
cho
Time
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wer
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Tim
e(F
-G)
Stop
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nge
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e Ec
ho(F
-G)
(F-G
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wer
Echo
Stop
Cha
nge
Rais
e Ec
ho(F
-G)
Lowe
r Ec
ho(F
-G)
Stop
Cha
nge
4 4(A-B
)on
8 4(A-B
)Ra
ise
Size
, Pe
aks
8 4Mat
ch t
he o
vera
ll s
ound
of
the
musi
c in
reg
arde
s to
pit
ch a
nd v
olum
e
Use
swit
ches
wit
h nu
mmbe
rs 5
12 a
nd l
ower
4 48 4
5 9
128
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2
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13 17 21
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ise
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ise
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wer
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(C-D
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cord
for
(A2
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G2)
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r Ec
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op(F
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onRa
ise
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ze
(A-B
)
Stop
on Rais
e Ec
ho,
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ze
(C-D
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ak,
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ho,
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ze
(C-D
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e Ec
hoEv
eryt
hing
off
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wer
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(F-G
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e Pe
ak,
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Use
swit
ches
fro
m 10
24 a
nd l
ower
129
&8 4 8 4 8 4
? &
3w
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25 27 30
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ak,
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ythi
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ff(F
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on
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on
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more
int
ense
(F2-
G2)
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(C2-
D2)
off
(A2-
B2)
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G2)
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t
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int
ense
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all
swit
ches
sou
nd s
houl
d be
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stan
tly
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lent
Play
as
fast
as
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ible
130
& & & &
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very
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less
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ense
(A2-
B2)
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(F2-
G2)
less
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G2)
more
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131
& & ?
5
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50 54 58
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G2)
more
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(F2-
G2)
less
int
ense
(F2-
G2)
less
int
ense
(F2-
G2)
more
int
ense
(A2-
B2)
onq = 13
5
132
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6
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62 66 70 74
(C-D
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102
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d lo
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133
& & &
7
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ose
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for 4
cou
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ime
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78 82 86
134
Reference List
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Manousakis S. (2006). Musical L-Systems PhD diss., The Royal !
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Prusinkiewicz P. (1986). Score generation with l-systems. University of !
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Reas, C., Fry, B. (2007). Processing: A programing handbook for visual
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135
Whitelaw M. (2004). Metacreation: Art and artificial life. Cambridge, Mass:
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136
