Fractals and Their Use in Music
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 20th 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 20th 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 9th 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.
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:
Mucherino, N. Recursion: A Paradigm for Future Music?
Available online: http://www.ks-rus.uni-
Strohbeen, D. http://www.fractalmusiclab.com
Lanius, C. “Why Study Fractals.” Available Online: