**Fractals and Their Use in
Music **

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** **Two
years ago, at a summer electronic music workshop at Oberlin College in Ohio, I
was presented with some of the most intriguing music I have ever heard. Initially, the music sounded like nothing
more than noise, random short bleeps with no clear purpose or arrangement. I could not help but wonder why anyone would
want to hear this ‘music’ and dismissed it as a failed attempt by someone
wanting to be avant-garde simply for the sake of being avant-garde. I could not have been more wrong. A few days later I had the opportunity to
hear another piece, *The Voyage of the
Golah Iota *explained by its composer, Gary Nelson. While the music did not sound any better to
me aesthetically at the time, I began to understand the reasons behind it and
the concepts it worked under. Since
that time I have come to learn more about this incredible method of writing
music, and in the process have begun to develop an aesthetic taste for the
music

The music I am talking about is fractal
music. Fractal music is based upon the
mathematical concept of the fractal.
Fractals by themselves have no shape, form, or sound, but through
computer generation can be converted to both visual images and audio. Fractal representations are infinitely
detailed and usually, though not necessarily, self-similar in that the larger
image (or main melody) is made up by smaller images of the same form (or the
same melody at a faster speed, etc.). As fractal representations are magnified,
more and more levels of detail appear.
Artists and mathematicians have been using fractals for years to
construct visual images (see Figure 1, note the repetitive nature of the
picture), but have only begin to scratch the surface of their possibilities in
music. (www.fractalmusiclab.com, Fractal Basics).

But why even attempt fractal art and
music? What relevance do fractals have
to everyday life? It is often stated
among those interested in fractals that “fractals are everywhere.”
(www.fractalmusiclab.com, Fractal Basics)
Of course no real object can have the infinite detail of a fractal, but
fractals better represent real objects than do the squares, circles, and other
shapes studied in Euclidean geometry.
Most ‘real’ objects have much more detail than the shapes attributed to
them in classic geometry, and fractals are a way to mathematically represent
this detail. Here is a simple
explanation of a fractal taken from the Cynthia Lanius’ fractal page:

Imagine . . . a picture of the
coastline of Africa. You measure it with mile-long rulers and get a certain
measurement. What if on the next day you measure it with foot-long rulers?
Which measurement would give you a larger measurement? Since the coastline is jagged, you could get
into the nooks and crannies better with the foot-long ruler, so it would yield
a greater measurement. Now what if you measured it with an inch-long ruler? You
could really get into the teeniest and tiniest of crannies there. So the
measurement would be even bigger, that is if the coastline is jagged smaller
than an inch. What if it were jagged at every point on the coastline? You could
measure it with shorter and shorter rulers, and the measurement would get
longer and longer. You could even measure it with infinitesimally short rulers,
and the coastline would be infinitely long. That's fractal.
(http://math.rice.edu/~lanius/fractals/WHY/)

The
world has always refused to act exactly according to the rules of Euclidean
geometry. The discovery of fractal
geometry by scientists allows a new way for scientists to describe the
complexities of the physical universe, such as the irregularities of
coastlines, rocks, leaves, and other aspects of nature. Fractals basically allow a more accurate
mathematical representation of the geometrical objects found everywhere in the
world. They are currently being
researched and developed for use in computer software and simulation.

The application of fractals in music may
seem a bit odd at first, but music is actually highly mathematical. The application of new mathematical concepts
to music has been going on for thousands of years. The mathematical study of music dates back to the Pythagoreans of
ancient Greece. They expressed musical
intervals as numeric proportions by observing the tones produced by plucked
strings of different lengths. They
calculated the intervals for several different scales, including scales as
basic as the 12-tone and as complicated as the enharmonic scale which includes
quartertones.
(www.rose-hulman.edu/~swickape/fmusic.htm) For years music and
mathematics went hand in hand, with new mathematical concepts leading to new
musical ideas and new musical ideas being studied mathematically. For
centuries, music could not stand without numbers. Over time, music theory split from its mathematical background,
explaining itself in terms of ‘inspiration’ and ‘genius’ rather than its
mathematical foundation. Only in the
20^{th} century has there been a serious return to analyzing music in
mathematical terms. Understanding music
mathematically is essential for many of the ways in which we now listen to
music to be possible. For example, compact discs encode and decode music
mathematically, as do multimedia PCs and digital stereo TVs. It was only natural then that the 20^{th}
century concept of fractal be applied to music. (*Recursion, *www-kc.rus.uni-stuttgart.de/people/scgulz/fmusic/recursion.htm)

Fractal theory and music, even classical
and contemporary music, have much in common.
In the 1920s and 1930s, a
physicist names Joseph Schilling developed a new theory of musical rhythm he
called* The Schillinger System of Musical
Composition*. He claimed that
"simple rhythms could be found by superimposing two waves of different
periodicities and forming a new wave that contained the attacks of both
waves." (*Fractals and Chaos in Music*,
http://www.sewanee.edu/Phy_Students/lemarb_0/berenice.html)

He
was able to describe the composition methods of many classical music composers
mathematically and concluded that music is indeed ruled by mathematics.

During the next 40 years sound became
classified in three main categories based on their mathematical elements: white noise, brown noise, and pink noise.
White noise is random noise. The
frequency and amplitude at any given time of white noise has no correlation to
the frequency and amplitude at any other time.
A graph of white noise shows no specific organization and the sound of
white noise is often perceived as irritating and disturbing. Brown noise is very structured and
organized. Every frequency and
amplitude at any point in brown noise is dependent on the frequency and
amplitude of every point before it.
‘Brownian’ music must start with a given note then the following notes
are determined in relation. People
usually hear brown noise as mechanical and rigid. Pink noise is more structured than white noise but not nearly as
rigid as brown music. This noise is
called 1/f, meaning that it is a noise that falls between the two
extremes. Pink noise is usually the
easiest for people to listen to and governs most all human-composed music.

Voss and Clark built upon Schilling’s
research by studying these three types of noise in the 1970s. They analyzed several recordings of both
music and speech and found that human speech and almost all music across genres
displayed 1/f characteristics. Voss and
Clark then reversed the process, composing music through electronic equipment
to be white, brown, and pink. They then
played the compositions for several listeners and asked them to comment on the
pieces. Pink music was hugely
preferred, which Voss and Clark took to be further evidence of the 1/f nature
of music. (*Fractals and Chaos in Music*,
http://www.sewanee.edu/Phy_Students/lemarb_0/berenice.html)

These finding prompted a huge interest
worldwide in algorithmic composition and several different methods were
developed and experimented with. Voss and others used the melodic results of
their algorithms in compositions, most notably a composition by Voss that was
derived from the annual flood records of the Nile River. Voss also experimented with applying 1/f
noise to the pentatonic scale, the result of which sounded very much like
oriental music.

Fractals are quite useful in this
manner of music composition as they are built upon self similarity. Just like classical cannons and fugues,
fractal compositions develop by making new musical material by systematically
transforming previous musical material.
David Clark Little was the first person to use fractals as a
compositional tool. In one of his more
popular pieces, *Fractal Piano 6,* he
used the logistic equation to convert raw data collected from iterating the
function into pitches, durations, and loudnesses. He then edited the result until an aesthetically pleasing piece
of music developed. Little took the
ideas behind fractals one step further using them to develop a live
improvisational technique involving at least three musicians sitting one in
front of the other. The musicians
follow a set of guidelines for what to play based on what the other musicians
are playing. Each musician tries to do
basically the same thing as the players in front are doing and the opposite of
what the players behind them are doing.
Little calls the piece* Brain- Wave
*as it as represents the way that the brain works. The musicians simulate the chaotic inhibitory and excitatory
synapses that increase and decrease the rate at which neurons fire. (*Fractals,
Chaos, and Music, *www.rose-hulman.edu/~swickape/fmusic.htm)

Now that fractals and their
relationship with and use in music have been explained, let us look
specifically at one person’s process involved in writing mathematically based
music, specifically the piece mentioned in the introduction, *The Voyage if the Golah Iota* by Gary
Nelson of Oberlin College. This piece is
particularly interesting as Nelson does not care the least about aesthetic
qualities, focusing rather on building a sonic representation of a specific
concept. Nelson states:

When I work on a piece I often
construct a metaphor that will guide me through the composition process. I use a dictionary, thesaurus, or
encyclopedia to sprout the seed of an initial idea. In this case, I began with
the words ‘grain’ and ‘particle.’ The thesaurus led me to ‘iota’ and the dictionary
told me it was a noun meaning ‘a very small amount, a bit.’ The dictionary also
reminded me that ‘iota’ is the 9^{th} letter of the Greek alphabet and
that it is of Phoenician origin. A
search of the encyclopedia under ‘Phoenicia’ brought me to a picture of a
galley (golah in Phoenician.) Further
reading informed me that there is speculation that the Phoenicians used one of
these ships to cross the Atlantic and visit the mouth of the Amazon many
centuries before the birth of Christ.
If the speculation is correct, the Phoenicians would have predated
Columbus’ voyage to the New World by more than 1000 years. (*Wind, Sand, and Sea Voyages, *www.ks-rus.uni-stuttgart.de/people/schulz/fmusic/gnelson.htm)

Nelson
decided to discover a function that would mathematically mirror a hypothetical
voyage that a Golah might have taken to reach the Amazon. The galley would have to travel from North
Africa across the ocean to what is now Brazil.
The chaotic function X=P*X*(1-X), when visually graphed, roughly maps
this voyage. Nelson was successful in musically representing a theorized voyage
across the ocean that took place thousands of years ago. While this is not the most aesthetically
pleasing piece, it is a brilliant example of how fractals are being used to
create completely new musical ideas.

The use of mathematical ideas in music is
an old practice that is seeing a resurgence thanks to the abilities that new
technology is affording us. Fractals
are particularly interesting in that they produce 1/f noise, the noise most
pleasurable to the human ear, and they work in a similar manner to many
classical styles of composition. The
area of fractal composition is still in its infancy with only a handful of
musicians giving any real time to studying its use, but its potential for
generating new musical ideas is enormous and should not be overlooked.

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Nelson,
G.L. *Wind, Sand, and Sea Voyages: An
Application of *

*Granular
Synthesis and Chaos to Musical Composition*.

Available Online: http://www-ks.rus.uni-
stuttgart.de/people/schulz/fmusic/gnelson.htm

Swickard,
P.E. *Fractals, Chaos, and Music. *Available
online:

http://www/rose-hulman.edu/~swickape/fmusic.html

Mucherino,
N. *Recursion: A Paradigm for Future
Music? *

Available online:
http://www.ks-rus.uni-

stuttgart.de/people/schultz/fmusic/recursion.html

Strohbeen,
D. http://www.fractalmusiclab.com

Lanius,
C. “Why Study Fractals.” Available
Online:

http://math.rice.edu/~lanius/fractals/WHY/