Strange Attractors

I went down to the local bookshop, and I picked up a book on chaos,
called (aptly) "INTRODUCEING CHAOS" by Ziauddin Sardar and Iwona Abrams,
and I was looking through the diagrams.

Three in particulat caught my attneton, these were;

The Lorenz attractor (the one that looks like a butterfly),
Lyupanov graphs,
The Mandelbrot set (naturally),
Bifurcations,

and what really cuaght my attention was for the possibilitys of making
Just Intonation lattices out of these.

Any comments?

I have done a little experimentation with Verhulst dynamics, which is
related to the Mandelbrot set. I represented the attractor X values as time
and the Y values as pitch scaled to an octave. Time has been normalized to
10 measures of 4/4 between bifurcations and continues until the period
exceeds 16. Each value in a period is assigned to a MIDI track, and then to
a channel with note values and pitch bends assigned to produce the
appropriate pitch. The result is not a tuning, per se, but a series of
tunings gradually changing from 1 to 16 tones. This is microtonal but not
just intonation. The result is available on my website at
http://www.geocities.com/bmames.geo/

The function used is: y -> x(1-y)y

Verhulst dynamics are fairly simple to use for the purpose of creating
musical interpretations of fractals. Since Sarkovskii's theorem assures us
that periods of all whole numbers exist in this function, one only need to
find the appropriate value of X to produce the number of tones desired, then
provide an interpretation of the Y values.

Of the items you mentioned, studies of the Lorenz attractor, Mandelbrot set,
and bifurcations should be productive, but graphs of the Lyapunov exponent
would not be since it is a measure of the stability of an attractor, not an
attractor itself. Other possibilities include Newton's method for real
equations and Volterra-Lotka systems.

References:

The Beauty of Fractals, H.-O. Peitgen and P.H. Richter, Springer-Verlag, 1986
Chaotic Dynamic Systems, Robert L. Devaney, Addison-Wesley, 1989

> Date: Tue, 30 May 2000 21:24:06 +1200
> From: Sarn Richard Ursell <thcdelta@ihug.co.nz>
>Subject: Strange Attractors
>
>I went down to the local bookshop, and I picked up a book on chaos,
>called (aptly) "INTRODUCEING CHAOS" by Ziauddin Sardar and Iwona Abrams,
>and I was looking through the diagrams.
>
>Three in particulat caught my attneton, these were;
>
>The Lorenz attractor (the one that looks like a butterfly),
>Lyupanov graphs,
>The Mandelbrot set (naturally),
>Bifurcations,
>
>and what really cuaght my attention was for the possibilitys of making
>Just Intonation lattices out of these.
>
>Any comments?
Brian M. Ames
Visit The Ames Hymn Collection
http://junior.apk.net/~bmames
The Ames Classical Collection
http://www.geocities.com/Vienna/Strasse/7398
The St Patrick Palestrina Collection
http://www.geocities.com/Athens/Troy/6080