Fantastic images of the Mandelbrot set and the mathematics it represents recently revolutionised the scientific establishment. Now fractal maths is finding its way into music.

IN THE 1970's, scientists in various disciplines and in various parts of the world began to discover new ways of understanding the apparent disorder of the natural universe. It was the beginning of a movement against classical science. Now this movement has emerged as a new science in its own right, and has named itself. It is the science of chaos.

Chaos has found wide application, influencing the study of everything from snowflake formation to weather forecasting, from the flow patterns around Jupiter's Great Red Spot to the epidemiology of chicken pox. If you're wondering what any of this has to do with music technology, read on.

The science of chaos is not so much about chaos as it is about seeing order and patterns within what is apparently chaos. To take an example from physics: when a liquid is heated from below, convection causes it to move in a particular way. The liquid rises up the container at its hottest parts, and descends where it is coolest, causing a steady circular motion. But at certain temperatures, the motion suddenly ceases to be predictable, and turbulence takes over. The liquid begins to behave in a chaotic way despite the fact that it is governed only by ordinary physical laws. This is an example of what the science of chaos calls a "dynamical system", and it illustrates an important principle of chaos - that a simple set of rules can create incredibly complex results.

There probably isn't a more wondrous example of how simplicity can create complexity, than in the case of fractals. Fractals can be understood as computer-generated images of chaos. They consist of intriguing and often very beautiful patterns created entirely from simple mathematical formulae. Fractals also have an extraordinary property - the more closely they are examined, the more detail emerges, so that every level of magnification can reveal an entirely new pattern, each as detailed as the last, and so on ad infinitum. With fractal-generating software now freely available for home computers, exploring fractals has already become, for many, something of an obsession.

So much so that chaos now seems to be acquiring the status of a bona fide subculture. The lure of chaos has spilled out of the universities and onto the streets. Chaos fanzines now abound in the trendier bookshops, and a place called Strange Attractions has opened in West London which is - yes, a chaos shop, selling chaos literature, fractal images and posters, and - here it comes at last - fractal music.

For fractal music, read fractal-generated music. But why would anyone want to use fractals to generate music? The perhaps rather unlikely-sounding answer is that for some, fractals symbolise an important connection between mathematics and aesthetics. Fractal patterns seem to imitate the forms and dynamical systems found in nature, in that they appear to combine structure with unpredictability.

German physicist Gert Eilenberger, quoted in James Gleick's 1988 book Chaos - Making a New Science, puts it this way: "Why is it that the silhouette of a storm-bent leafless tree against an evening sky in winter is perceived as beautiful, but the corresponding silhouette of any multi-purpose university building is not, in spite of all the efforts of the architect? The answer seems to me, even if somewhat speculative, to follow from the new insights into dynamical systems. Our feeling for beauty is inspired by the harmonious arrangement of order and disorder as it occurs in natural objects - in clouds, trees, mountain ranges, or snow crystals. The shapes of all of these are dynamical processes jelled into physical forms, and particular combinations of order and disorder are typical for them."

Analysis of almost any music reveals another parallel. Music too consists essentially of patterns - rhythmic and melodic - which are not entirely predictable. If repetition was incessant and absolutely exact, music would not appeal. Neither would music do anything for (most of) us if it were entirely disordered. This, then, is the connection: In nature, music and fractals, there is a happy blend of form and irregularity, structure and surprise - or, if you like, theme and variation.

TO REALLY UNDERSTAND fractal music, you need to understand fractals, and for this, a little maths is required. But don't worry, the mathematical equations used to generate fractals are surprisingly simple, despite the fact that they sometimes use what are known as complex numbers. Complex numbers consist of two parts, one "real" and one "imaginary". The real part is an ordinary number, whereas the imaginary part uses the symbol "i" to stand for the square root of minus one. (There is no real square root of minus one, because any number multiplied by itself yields a positive result). By convention, 2i stands for the square root of minus two, 3i the square root of minus 3, and so on. For example, the complex number 3+7i means 3 plus the square root of minus 7. Imaginary numbers are useful to mathematicians, so they can be forgiven for not really existing.

The most famous fractal of all is the Mandelbrot set, discovered in 1979 by mathematician Benoit Mandelbrot, and described by enthusiasts as the most complex object in mathematics. It uses the equation z=z²+ c, where z and c are complex numbers. The key to any fractal equation is the concept of "iteration". Iteration is a kind of mathematical feedback. In this case, take a complex number, square it, add a constant and then do the same with the result, and then do the same to that result, as many times as you like. Each time around counts as one iteration.

Complex numbers don't just get larger and larger when multiplied by themselves over and over again, because one term of the result collapses into another, as the following example shows:

In the Mandelbrot equation, although not all values get larger and larger, some do indeed speed off into infinity. The ones that do are deemed not to belong to the Mandelbrot-set. A Mandelbrot set fractal image is simply a graph of the complex numbers that belong to the set. The real part is along one axis, the imaginary part along the other. A monochrome fractal image is created as follows: If the number is in the set its co-ordinates are coloured black, otherwise they are coloured white. More impressive pictures are obtained by colour-coding the number of iterations before the running total clearly starts to head off to infinity.

"If repetition was incessant and absolutely exact, music would not appeal. Neither would music do anything for (most of) us if it were entirely disordered."

That, then, is how to calculate and plot a fractal. Making fractals into music is another matter, and there is no real consensus as to the best way to do it.

SOFTWARE PACKAGES WHICH can generate music have existed for some years. Most of them, such as Intelligent Music's M and Jam Factory, use random numbers to generate variety in the music, leaving the musician to apply order to the randomness by choosing pitch and duration limits, time signature, tempo, and so on. These programs also allow the musician to vary the probabilities of the occurrence of particular pitches, timings and durations of the notes. This kind of interactive computer-aided music making, where the computer actually contributes to the music, is sometimes known as algorithmic composition.

By taking the principles of existing algorithmic composition software, but replacing the random element with a fractal element, the importance of pattern in music can be re-asserted, whilst still retaining interest and surprise. This is one of the ideas that led musician and programmer Hugh McDowell to write fractal music MIDI software for the Atari ST, which he will be making commercially available very soon.

"By replacing the random element within algorithmic composition software with a fractal element, the importance of pattern can be re-asserted, whilst still retaining interest and surprise."

It all began when, after reading an article in Scientific American, he acquired a computer and began to experiment, plotting diagrams on the screen, and then using the plotted points to decide pitches and loudnesses. Encouraged by pleasing musical results, he eventually developed two user-friendly fractal music programs.

The first is based on the Mandelbrot set. The program displays the set, (or any magnified part of it) on the screen, and allows the musician to select small rectangular areas of the fractal by clicking and moving the mouse. The values held in the chosen boxes will then be used by the program to generate the pitches and durations of notes for up to four instruments. All the generated music conforms to preliminary choices made by the composer - the range of acceptable pitches and durations for each instrument, the tone maps to be used at various points in the music (major, minor, chromatic etc), time signature, key changes and tempo.

The second program works on the principle that the Mandelbrot set can be seen as a catalogue of Julia sets (another type of fractal), so that each point on the Mandelbrot image can generate a corresponding Julia fractal. Here, four Julia set patterns corresponding to the four instruments are simultaneously drawn, dot by dot, as the music plays.

With both programs, the fractal image itself operates as the key interface between the musician and the music, with other important musical options also available. The musician can decide how much control to assume. In the Mandelbrot program, the musician is able to "read" the fractal, to find out in advance the exact pitch and duration values that each area of the fractal would generate. This is useful if the more stable areas of the fractal are to be used to generate specific notes. Alternatively, by using the fractal's "chaotic" parts, a more experimental approach is implied. In practice, though, good musical results are often achieved when one or two instruments are set to behave in a fairly strictly-controlled way, whilst others are left to behave unpredictably.

Hugh McDowell isn't the only musician to take the chaos road. London composer Chris Sansom has also written a fractal music MIDI program for the ST which, according to the manual, "began as a tool for use in his own compositions... and grew". Instead of complex numbers, it uses a formula involving trigonometrical functions to generate three-dimensional fractals corresponding to the three fundamental elements of melody: pitch, loudness and time. Although this program doesn't actually produce fractal images on the screen, it is more advanced in other ways. Basic operation involves first choosing some starting numbers, and then pressing a "tractate'' button. Music for up to 16 instruments is immediately generated, and the program then allows further mathematical processes to be performed on the music. Called simply Fractal Music, it is now on sale for £65.

In Oxford, two musicians, Gavin Davis and Tim Fenn, have already recorded and released an LP and a cassette of fractal music. The LP consists of a live performance of a computer-generated fractal music score, whilst the cassette was produced using a program which generated the music in real-time. The software was written for the Apple Macintosh by Tim Fenn and Adam McLean, and works in a similar way to Hugh McDowell's. It consists of two parts - the Fractal Pattern Generator, which creates saveable images based on various different fractal sets, and the Fractal Music Generator, which creates music through MIDI. The Music Generator offers a choice of algorithms for translating screen data into MIDI data on up to four channels. The package is on sale now for £35. For those who are unwilling or unable to splash out on what may seem like an abstract concept, a working demonstration version is also available. The demo disk has no Fractal Pattern Generator but does include one sample pattern to get you started.

Fierce Recordings, a Swansea-based independent record company, are planning to release a CD of fractal music later this year.

WHAT DOES FRACTAL music sound like? Well, clearly it depends on a number of factors - the software, the choices made within the software, and the choice of instrumentation. Fractal music does seem to have its own general characteristics, though. From my own experiments, I've found that it seems to be in the nature of fractals to produce music that rarely has abrupt changes, but which may undergo gradual shifts of mood or emphasis. This accords with a basic property of fractals - the property of "self-similarity", or the tendency for shapes to repeat themselves, with variations, at larger and smaller scales of magnification.

Both Hugh McDowell's and Chris Sansom's programs support Standard MIDI File, so the software can be employed simply to generate and store musical ideas to be worked on later with the aid of a sequencer, rather than being used to create finished pieces of fractal music. The door is thus open to anyone who uses MIDI, regardless of musical style.

It remains to be seen, then, whether fractal music will succeed as a new music in its own right, or simply as a useful composer's tool. Of course, it may all just be a passing fad. On the other hand, maybe fractal music is not only a soundtrack for a strong and growing "chaos culture", but perhaps it really does herald a new and radically important way of understanding musical form. Either way, it's nice to think that when our children go playing on their virtual reality three-dimensional fractal climbing frames, they'll have something to listen to at the same time.

Further information:

Datamusic Ltd: (Fractal Music for the Atari ST by Chris Sansom - demo disk £5), (Contact Details).

Hugh McDowell (Two fractal music programs shortly to be released for the Atari ST), c/o CFM, (Contact Details).

Fractal Records (LP, Cassette and Macintosh software by Tim Fenn, Gavin Davis and Adam McLean), (Contact Details).

Fierce Recordings (Planned Fractal Music CD), (Contact Details).

Strange Attractions (Chaos shop), (Contact Details).