Chaos Report - Heather Elko

Heather Elko

Chaos and the Logistic Map

February 2001

Chaotic Music

Fractals are applications of chaos theory. They are derived by entering an equation into a computer. The picture that results, while seemingly chaotic, actually repeats over and over within itself. Thus, chaos theory emerges, which shows that seemingly chaotic systems actually do have order.

These fractals are important because they are found over and over throughout nature. Coastlines have been described as chaotic, as they can be analyzed on many different levels producing different results. A coastline can be analyzed on the astronomic, topographic, or even microscopic level, each producing very different results. Fractals can also be seen in plants such as the fern, which is found to repeat itself over and over. Michaech is found to repeat itself over and over. Michael Barnsley was able to create a fractal that looked extremely similar to a fern.

But one of the most interesting uses of fractals that I have discovered is that they are being used to create music. Music from chaos. Music from an algorithm. It's extremely compelling.

There are many different types of fractals, and thus many different ways to transfer them to music. One example is the Julia set, below.

When this picture was generated, each pixel on the computer screen was assigned a color. Each color is then assigned a note on the musical scale. A computer program then moves through the fractal, playing the notes for each color as it passes. The result is a very intriguing techno-type musical sound.

One interesting thing that I have discovered in listening to this fractal music, is that while it seems to continue to repeat itself over and over, at the same time, it keeps changing. So while sounding the same, it is actually different. This seems to represent very well, order and chaos at the same time.

Another type of fractal is the Gosper Curve seen below.

This fractal is quite different from the Julia set. It does not have the colors that would allow fractal music to be created. Its intriguing quality is its single line that repeats, yet at closer inspection also does not completely repeat. Musiclso does not completely repeat. Music is created from this curve by following the line. A melody line follows the direction of the curve. In this case you follow the line starting from the lower left corner, the horizontal lines representing notes. The higher the line, the higher the pitch; the longer the line, the longer the note.

Each fractal functions differently and correspondingly has a different style of creating music. And each can produce a much different sound. While the harmony is very dictated in the Julia set, as all tones that are played are represented by the fractal image, the Gosper Curve produces only a melody. The accompanying chords and harmony are at the whim of the composer.

In the 1970s, a mathematical study of music was performed at the University of California by Richard Voss and John Clarke. They researched the actual audio physical sound of music as it was played. There are three types of noise: white noise, 1/f noise, and Brownian motion noise (1/f2). By taking a musical score and connecting the notes dot to dot you get a graph very similar to 1/f noise. Consequently, 1/f noise is most pleasing to human ears. White noise is too random and Brownian noise is to correlated. And interestingly, when most fractals are converted to music, they are found to create 1/f noise.

There are currently some ongoing studies that are studying all music, and the claim is that all music is fractylic. Iat all music is fractylic. It is a very interesting concept and hopefully more information will be available on such studies soon. But for now, it is just fun to find that mathematical algorithms can function more than just numbers and graphs and can actually produce aesthetic music where one might not even think to look.

Works Cited