M500 180 Page 19
Fractal music
Eddie Kent
Everyone knows what a fractal is. My on-board dictionary says it is ‘A
geometric pattern that is repeated at ever smaller scales to produce irreg-
ular shapes and surfaces that cannot be represented by classical geometry’
which sounds more boring than it really is. Briefly, in the dying years
of the nineteenth century an Italian gentleman called Guiseppe Peano be-
gan experimenting with so-called ‘pathological’ curves. He was a man who
delighted in finding counterexamples to other people’s definitions. For in-
stance when some poor fool defined a line as having length but no breadth,
so it is entirely one-dimensional, Peano found a way of bending a line in
such a way that it completely filled a two-dimensional space. This gave him
the idea of a non-integer dimension. He went on to design the ‘snowflake’
curve. Take an equilateral triangle and make the middle third of each line
the base for another equilateral triangle, then do the same again for each
line in the resulting figure, and again, and so on. The curve you end up with
‘at the limit’ has infinite length but encloses a finite area. It has dimension
(log 2)/(log 3) = 0.6309.
Peano had to work theoretically. Any attempt to draw one of his cre-
ations very soon becomes hopelessly entangled. However, a computer is
ideal for this sort of job, and in fact there is a program freely available that
will do it for you. It is called FRACTINT and will work on any PC. No
knowledge of mathematics is required. The program is based on work done
by Benoit Mandelbrot, who noticed, among other things, that in Peano’s
diagrams each iteration is similar to the one before. He experimented with
algebraic rather than geometric objects. He would take a function, like
y = f (z); insert a value for z, find the corresponding value of y and feed it
back into z, and keep on doing this until he could see what was happening to
it. Some initial values would remain stable, some would vanish and others
would go off to infinity (at various ‘speeds’). He then coloured the initial
points (or pixels) accordingly. Surprisingly this produced very strong pat-
terns, the most famous being the well known ‘apple’, from f(z) = z tan z+c.
All of those well-known pretty psychedelic computer art pictures are
generated in this way. Now the BBC has come up with the idea of set-
ting fractals to music. This is not an absolutely new concept: someone
once wrote a set of variations on the New York skyline and Mozart was not
above playing games. But the nature of fractals is that they contain their
own variations. At each magnification the same picture appears, but subtly
altered. To make music a pitch is assigned to each pixel in the same way as
a colour is given in the traditional model. This is called data sonification.
The theme is chosen, then variations are added. Once started the procedure
continues mathematically, but the result is imprecise because of the under-
lying chaotic nature of the material. We are born and live surrounded by
fractals in nature, and thus we inevitably respond at a deep level to fractal
music. To hear some examples, download
http://www.vanderbilt.edu/VUCC/Misc/Art1/Sonify/Mandi.html.
Manymanuals.com
Manymanuals.de
Manymanuals.fr
Manymanuals.it
Manymanuals.pl
Manymanuals.cz
Manymanuals.es
Manymanuals-pt.com
Comments to this Manuals