However, this is just the shiny facade of a revolution that has left the international community of scientists of most diciplines at first totally dumbfounded by seeing central parts of their theoretical basis crumble, then completely frantic to investigate to what extent these scientific Columbi eggs can be applicable on their particular field. Fluid mechanics, geology, medicine, meteorology, in most fields many unsolvable problems have been solved by applying some part of this manyfaceted and bewildering bag of theories. Visual artists has also been attracted by the organical shapes that appear on the computer screens.

So how about Music? Will fractal mathematics come to rescue in a time of apparent stagnation, this time in the guise of simple mathematical algorithms that yield delicious patterns that can be further processed, either as succession of musical events or as complex frequency spectra? The attempts to find an answer to this has barely begun. In the field of music the answers are not as selfevident as for the visual field. But this can actually turn out to be an advantage, if not technically, then at least aesthetically.

**" - But Is It Art?"**

The totally predominating figure in the above-mentioned articles and books is the so-called Mandelbrot Set, discovered by Benoit Mandelbrot, and flamboyantly and exquisitely coloured by H.-O. Peitgen and P. H. Richter. This totally amazing figure, derived from the simple equation y = x2 + c solved in the complex number plane, has a greater resemblance with a phantasy ocean than with a mathematical graph. You can dive into it, enlarge it by up to a couple of millions, and always discover fascinating forms that in one way or another are interrelated, but still never completely identical.

The aesthetical problem, however, is that these forms are so alarmingly like the 'psychedelic art' of the 60's that one quite easily can imagine the Yellow Submarine puffing around among the swirls of the 'Sea-horse Valley', the illustrating nickname for one of the regions in the Mandelbrot set. One is seriously tempted to believe that God, after all, is a hippie.

In the cases that the fractal colourprints bear marks of manipulations of the forms themselves, not only in the selection of colours, one is increasingly doubtful. The aesthetics is often taken from the visual world of either tourist brochures (fake snowy mountains and lakes) or Science Fiction, which in its best case is entertaining: a mountain plateau shaped like a Mandelbrot set majestetically rising over the barren lowland of a planet, while a galaxy shaped like a Julia set hovers in the sky, and a Julia set lightning pierces the gloomy night. A ring of moons with mysterious-looking fractal patterns parades along the equator.

Other pictures show more mature choises and suggest fascinating possibilities for visual artists. But the fact that fascinating forms emanate so easily from relatively simple formulas (easily for those who have the necessary mathematical know-how and access to large numbercrunching computers, that is) has attracted computer freaks with a taste for nice colours and 'weird' shapes in a "So Simple that Even the Computer Science Professor Can Make Art" fever.

Luckily, the work so far with musical exploitation of chaotic systems does not indicate a similar sticky embrace. Furthermore, the patterns that are formed are promising, but will hardly be of any interest for composers of Science Fiction film scores.

György Ligeti has in interviews stated his strong interest in chaos theory, and his friendship with some of the scientists that work with it. In Musiktexte 28/29 (March 1989), however, after a lengthy description of the scientific background of the Mandelbrot set, he cuts the whole thing off by declaring his distance to the 'scientific' ("Naturwissenschaftlige") music of Xenakis, and that he rather utilizes the underlying idea of chaos theory as an inspiration for his compositional techniques. He points out that the idea of small, simple 'cells' that form complex and interesting patterns when iterated many times, reoccurs in many kinds of music. The idea of a transition from order to chaos and back again is also very relevant to Ligeti's own music, at least from the 60's and 70's.

These observations are of course correct. But when one investigates Ligeti's supposedly 'fractal' music, for instance his piano etudes from 1985, one finds that this music could very well be achieved without having heard of the existence of chaos theory or having seen the Peitgen-Richter pictures. The techniques he uses are derived exclusively from traditional compositional practise, and the music is neither more nor less fractal than African drum rhythms or a Bartók string quartet. After all, the concept of complexity from the interaction of simple particles and of the parts resembling the whole can be traced back to the early history of mankind, because it is so obvious to us that the world is constructed that way. Chaos theory therefore brings nothing strictly new to our perception of the world, it only confirms it scientifically in a shockingly simple way, and it also - for the first time - gives us a possibility to simulate some of Nature's own structures by mathematical means.

This is of course no "artistic proof" against Ligeti's music, as the aesthetical value lies in the music itself, not in its origin. But there is something defeatistic in denying the possibilities of actually using fractals in music while admiring the Peitgen-Richter pictures for their artistic value, well knowing that they are made in a much more 'scientific' way than Xenakis makes his music.

So the question some composers ask themselves these days is: Can my music benefit from chaos theory in a truly innovative way? If, after serious examining, this question must be answered negatively, there is no point in using 'fractal' as a trendy tag on one's music.

I will try to give a short description of the investigations the programmer Øyvind Hammer and I have done on the subject, and of how I have tried to use these experiences in my own music.

**Chaos and music**

One of the first serious turn-downs that appear in an attempt to find viable ways of using Chaos in music is of course the fact that music is serial information, flowing into our ears in one dimension, degree of air pressure, in relation to another dimension, time. The Mandelbrot pictures also have two dimentions, horizontal and vertical, but these two dimensions are much more closely combined and caught by the eye in one glimpse. One could represent the vertical dimension in the Mandelbrot set with frequency and the horizontal with time, but the aural information would be almost as meaningless as to scan Mona Lisa and put the information into a synthesizer: visual and aural information are seldom directly compatible.

But there are also fractal formulas that can be represented as an evolution in time, especially the socalled dynamical systems. One of these classical formulas was originally used for simulating the growth of population (xn+1), a value that is dependent on the size preceding "generation" (xn) and some "environmental conditions" (r):

xn+1 = r * xn * (1 - xn) ( 1 )

where '*' stands for multiplication, and x varies between 0 and 1. A printout of the result of letting r grow from 1 to 4 is shown in Fig. 1, which will be easily recognized by anyone who has opened one of the abovementioned books.