Sp&Hrg 803
Complex Systems in Neurobiology, Language, and Speech

Day 6
12 March 1998
Self-Organized Criticality

Summary of this Day's Class

We discussed power laws and self-organized criticality.
For an excellent bibliography on power laws consult:
A Bibliography on one-over-f noise by Wentian Li.

There are also JAVA applets for chaos and fractals from the Dynamical Systems and Technology Project at Boston University, directed by Robert Devaney
Also look at software created by Robert Devaney.

Also I found a site where one can download software for making fractals and playing with chaos:
Sprott's Chaos and Fractal Software by J.C. Sprott

I made an attempt to make music from white noise, 1/f noise, and 1/f^2 noise.
For details please refer to:
Gardner, Martin. 1978. Mathematical Games: White and brown music, fractal curves and one-over-f fluctuations, Scientific American, April 1978, 238:16-32.

The paper by Richard Voss and John Clarke is an excellent summary of their first work on 1/f noise in music.
Voss, R.F. and Clarke, J. 1975. '1/f noise' in music and speech, Nature, 258:317-318.

Manfred Schroeder's book also has quite a bit on power laws and 1/f noise:
Schroeder, M. 1991. Fractals, Chaos, Power Laws: Minutes from an Infinite Paradise, W.H. Freeman and Co. pp: 1-25.


Now for my attempt at making music from various types of noise.
First I generated a list of 200 numbers with values ranging from 500 to 5000. These numbers will represent the frequencies of the 200 notes in the music sample.
A plot of these numbers looks like this:

The horizontal axis represents the ith note in the sequence and the vertical axis denotes its frequency in Hz. This isn't very fancy. I am not worrying about particular notes on the scale.
From this list of numbers I generated a sound file using tone pips according to the frequencies in the graph above. Again this is not fancy. Tone pips are not very pleasant to listen to and I didn't vary their duration at all.
Click on the sound icon next to the graph to hear the corresponding sound.

The next step was to use these notes to create sounds with a 1/f spectrum. One could try what Richard Voss suggests. I guess I took a more difficult approach. I took the Fourier Transform of the list of random numbers. This allows me to see the frequencies present in the list. Now remember these frequencies are not the frequencies of the notes that I will later map these numbers to, but are instead are frequencies representing the rates at which these notes are varying. I then applied a 1/f filter so that the high frequencies (rapid variations in the note transitions) are attenuated or suppressed. I then transformed back to obtain a new list of numbers with a 1/f spectrum (note that I had to rescale them to fit from 50 to 5000 Hz).

You can see that the spikiness of the list of numbers is gone (rapid variations are suppressed). If you compare them closely you can see that the 1/f series has the same slowly varying pattern as the original list of random numbers. Also the waveform looks self-similar in the sense that if you were to take a small piece and blow it up, it would look similar to the original.
When you listen to the corresponding tones it is a bit more pleasant since there are now correlations, but the correlations are not over-powering.

Finally I made 1/f^2 (one over f squared) music. I played the same game as before. I took the Fourier Transform of the original list of numbers and I applied a 1/f^2 filter. This filter will wipe out rapid variations much more strongly than before. Another way to do this would be to take the 1/f list of numbers and apply the 1/f filter again! Transforming back and rescaleing the amplitudes to ft in from 50-5000, I have a new list of numbers with the desired variations. Note this time that the slow variations dominates the scene. Everything is highly correlated - too highly correlated to be pleasant.

Homework Assignment 6 - Due 19 March 98
Calculate the Hausdorff (fractal) Dimensions of the following self-similar figures.
Notice that the figures with similar fractal dimension have a similar density of points.
Also notice that the fractal dimension can be the same for very different self-similar shapes.

I regret that I do not have online versions of these fractals.
They can be obtained from Robert Devaney's book, pages 197-198:
Devaney, R.L. 1992. A First Course in Chaotic Dynamical Systems: Theory and Experiment, Addison-Wesley Pub. Co. Inc., Reading, MA., pp. 197-198.