What is Fractal Music?
Self similarity is the essence of
fractal geometry and human DNA . Fractal music is one slice of the study of
fractals in general. ‘Fractal’ is the name given to images, landscapes, sounds,
and any other pattern that is selfsimilar in nature, that is, if you look at
one small part, no matter how small, you get a sense of the whole picture .
Twigs look like branches, look like trees. Same principle has to be used to
create a music masterpiece with self similar, recursive themes repeated on top
of each other over and over again with minor, random variations in the rhythm which
create Fractal Music as per the Mandlebrot equation . The effect of this is
divine since it fills one with life energy by triggering the self healing loops
of higher consciousness . It is something beyond the Fibonocci Series that was
used by musicians like Mozart .
The possibility of using fractal
methods to create music was first mentioned upon the discovery that all music,
regardless of culture, followed the patterns of fractal motion or ‘pink noise.’
Pink noise is somewhere in between white noise (complete chaos, far too
disorderly to be considered musical) and brown noise (very orderly, and too
dull to sound like music). Thus, the question arose: if fractal methods produce
pink noise, and all music resembles pink noise, is it possible to produce music
using fractal methods? Some of the methods that we used successfully are
discussed beneath .
How is Fractal Music Made?
We used two basic methods to create
our fractal music.
The first method is called LSystems.
To briefly describe LSystems, they create the selfsimilarity of fractals,
starting with a short string of symbols and replacing the symbols with
corresponding rules (which are their own strings of symbols that can be
replaced). The symbols are then interpreted as notes, chords, and several other
things.
The second method involves Fractal
Motion. Fractal motion (pink noise) is generated with various random number
methods, starting with a straight line and repeatedly altering portions of the
line.
What does Fractal Music Sound Like?
The fractal music methods we have
used produce a wide variety of songs. Many of our pieces follow the conventions
of Western music, but that is only due to constraints we placed on them to add
an air of familiarity to our tunes. In several of our pieces, the keen ear can
detect some repeated themes and selfsimilarity inherent in the music.
We did experiment as to what methods
and what parameters sounded the best, especially with the fractal motion
method. Our findings were that music with a fractal dimension near 1.4 sounded
the best (the fractal dimension is a measure of how close to white or brown
noise the fractal is). The authors' personal favorite tunes are those generated
by the single note LSystems and the chord progression LSystems.
An LSystem is a recursive method of
generating long strings of symbols from a short initial string (or axiom) and a
set of production rulesone for each symbol considered in the system. From
here, we generate longer strings by replacing each symbol with its respective
rule, and repeat this process until we have a string of a desired length.
Here is a simple example:
Axiom: A B
Production Rules:
 A > A B C
 B > C A D
 C > D C
 D > B D B
Start out with the axiom, "A
B". After one iteration, the string would become "A B C C A D".
Now, we would use the same production rules with this new string. It becomes
"A B C C A D D C D C A B C B D B". The results of LSystems are
characterized by repeated themes and selfsimilarity. LSystems tend to grow
fast, so a small number of iterations is usually sufficient.
After we have generated some data,
the next step is to interpret that data as music, which we did in the following
three ways:
Strings of Notes
The most straightforward
interpretation is to assign each symbol directly to its corresponding note on
the musical staff  "A" to A, "B" to B, etc., as well as
throwing some extra symbols in such as "R" for a rest. Our above
string of letters would become this:
Variations on this method include
assigning each of the letters to a percussion instrument. To pick our axioms
and production rules, we resorted to spelling out words and pulling short
excerpts from humancomposed pieces.
Another interpretation of the
LSystem strings is as a chord progression... consider this following system.
Axiom: I
Production Rules:
 I > I IV V I
 ii > ii V I IV
 IV > IV V I ii
 V > V I ii V
After two iterations, this produces
a chord progression:
I IV V I IV V I ii V I ii V I IV V I
This method is not very effective
when used alone, but is excellent when used in conjunction with other methods,
either as background chords or to further constrain melodies to give them more
of a traditional Western music feel.
Turtle Graphics
In the Turtle Graphics interpretation, there are four basic
symbols, "F", "f", "+", and "".
Instead of being assigned directly to notes or chords, they are interpreted as
instructions to "draw" a melody. In the traditional visual
interpretation, these control a virtual "turtle" holding a pencil.
The commands mean "Move forward one unit, and draw a line",
"Move forward one unit, but don"t draw a line", "turn d
degrees to your left", and "turn d degrees to your right",
respectively. The result is a picture like the one at right.
In our musical interpretation,
horizontal motion was seen as note length, while vertical motion was seen as
change in pitch. The above drawing would be interpreted like this:
This method also had its variations,
such as a "trill next note" command as well as a system of marking
pitches and returning to that pitch at a later time.
White Noise
White noise consists of random
values the entire distance across. It is of f 0.
Pink Noise
Pink noise is between brown noise
and white noise, and is closest to the differences in music patterns, and can
be generated by various fractal methods. It is of f 1.
Brown Noise
Brown noise is a "random
walk." Each point is displaced a Gaussian (distributed) random amount from
the previous point. It is of f 2.
Midpoint Displacement
One method of Brownian motion
generation is called random midpoint displacement. This is done by taking the
middle of a line segment and randomly displacing it up or down, then each of
the line segments made by the displacement are taken in half and displaced by a
random amount. This is repeated as far as possible to create Brownian motion.
Random Cuts
This method of Brownian motion
generation is called the random cuts method. A random point is picked in the
line, and half of the line segment from that point is displaced by a random
amount. The more cuts that are made, the better the approximation to Brownian
motion.
MIDI stands for musical instrument
digital interface. It was designed to produce a synthesizer on personal
computers, and has made much progress in quality since its beginning. MIDI is
essentially a format of sheet music, which a PC sound card interprets.
The Midi data is all stored in one
file, chronologically.
Here is an example of how the data
is put in a midi file:
Time to Wait

Command

Parameters

0 ms

Turn on note

Play key of middle C fairly loud

400 ms

Turn note off

Turn off key middle C

These two commands together will
play a note of C for 400 ms. When translated to data for the computer in base
16, these 2 lines look like this:
0 90 63 70
400 90 63 0