Fractal Music – Red Herring or Promised Land?

…or “Just Another of those Boring Papers on Chaos”

This article is based on a lecture given at the Nordic Symposium for Computer Assisted Composition Stockholm 1989.

If the Vietnam war was the first TV war, the so-called Chaos theory must be the first commercially available scientific revolution, brought to a large public step by step. Chaos, fractals, Mandelbrot Sets, strange attractors, Cantor dust – these and some dozens other magical words have totally invaded popular science magazines during the 80’s, accompanied by glossy colourprints of strange, disturbingly organical computergenerated shapes. 

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.

Figure 1

This figure is not much to look at, apart from that it, just like the Mandelbrot set, shows the same astonishing selfsimilarity when smaller regions are investigated. By letting the x values control the pitches of a MIDI synthesizer, the chaotic regions appeared much more ordered than they look: Three melodic “motives” seem to dominate:1) A generally ascending zig-zag pattern, the pitches of the motif slightly varying, but leaving the profile recognizable, the length of the motif varying unpredictably from 3 to 7 notes.2) A “spiral”, meandering in increasing loops around a middle point until motif 1 takes over. 3) Small and large ‘windows of order’ where 3 to 6 notes repeat a number of times before motif 2 brings them out of balance again.(Note that these figures have up to 30 000 points in order to let the patterns be visually recognizable. The same evolution can be represented by significantly fewer musical events – 50-500 – and still show clear patterns.) 

This was quite entertaining for a while, until we understood that the selfsimilarity in this system is so selfsimilar that it soon gets boring – the same three motives bite oneanother’s tail ad infinitum. No new motives appear when “diving” into the graph by enlarging smaller regions. 

Then Jan Frøyland from Institute of Physics at the University of Oslo made us aware of so-called “symmetrically coupled nonlinear systems”, which have been quite closely examined at the institute, and that he thought might be of interest for creating musical patterns. Instead of having one “population value” as in the formula above, these can have from 2 up to a large number of “dimensions” (x, y, z etc.) that mutually feed back into eachother, their evolution governed by two “parameters” (c, d). A “3 dimension version” might look like this: 

xn+1 = 2 * d * (yn + zn) + 2 * xn * (c + xn)yn+1 = 2 * d * (xn + zn) + 2 * yn * (c + yn)zn+1 = 2 * d * (xn + yn) + 2 * zn * (c + zn) 

This formula, though only slightly more complicated than the first one, reveals a dramatically wider gamut of behaviour, from totally periodic through constantly changing patterns to totally chaotic. Instead of a one-dimensional parameter axis (the horizontal axis in Fig. 1), one can simultaneously change the values of both c (always negative) and d (positive), resulting in a two-dimentional parameter plane, where each point gives certain conditions for the evolution of the x, y, and z values.

Figure 2

Fig. 2 shows a printout of the x value (vertical axis) using the formula above, letting c move (horisontal axis) from -0.57 to -0.8 and d is constant at 0.2. The plot shows that the points jump alternately below high and low values, where the low values create a ‘diminished mirror image’ of the high values. The evolution resembles that of Fig. 1, with windows of order between patterned regions, and the overall tendency to move from perfect order to chaos. But here the patterns are longer and therefore more clearly distinguishable. 

Please note that the printout here, and my further work with the formula, uses a slightly different, and assymetrical formula, due to a (blush, blush!) programming error. Oddly enough, this one works also fine, and in my opinion, the error makes an “improvement” for my musical use:xn+1 = 2 * d * (yn) + 2 * xn * (c + xn)yn+1 = 2 * d * (xn + zn) + 2 * yn * (c + yn)zn+1 = 2 * d * (xn + yn) + 2 * zn * (c + zn) 

So what happened when we let the three dimensions (x, y, z) control respectively pitch, duration (exponentially weighted), and dynamics (MIDI attack velocity) on a MIDI synthesizer? The first thing we did was to wrap the values around x = 0.05 (x = abs(x – 0.05)), to avoid the ping-pong movement between high and low pitches, durations and attack velocities. And out came an astonishingly vivid lattice of melodical patterns, rhythmically punctuated and with idiosyncratic accents, that slowly evolved or suddenly changed behaviour as the c and d values very slowly changed. It was like listening to a skilled improvisator, at the same time extremely consistent and shamelessly surprising, so “human-like” was the behaviour of this simple formula. And even better: by moving in the c/d plane, or by changing the initial values of x, y and z, the musical patterns changed to something very different, not like the simpler formula 1, that stayed within basically three modes of behaviour. 

Then what?

Then a new question occurs: This is all very fine, but how can I use this material in a composition? The patterns that come out are of course nothing more than raw material, that must be processed in some way or other. And the simple projection of the three values on pitch, duration and dynamics was primarily done to give a simple and clear picture of the dynamic behaviour applied on music, as an experimental tool. 

After having investigated these two and other formulas, I decided that the material and its inherent ‘logic’ was so interesting and aesthetically appealing that I wanted to use it in a piece I am currently writing for the Oslo Sinfonietta. I also found out that both formulas would be needed in a hierarchic form structure, where Formula 1 controls the overall form, and Formula 2 creates the melodical, rhythmical and dynamical microstructure. Until now I have composed the single musical events ‘by hand’ according to specifications from a stochastic form program, because I couldn’t find any algorithm that generated the microstructure better than I did myself. This has usually worked out fine, but it is slightly frustrating to know there is a discrepancy of origin between the different layers of a composition. 

The hierarchical structure in the piece works like this: Using Formula 1, letting r decrease from 3.855 to 3.55, the different ‘fields’ of the piece (the term ‘field’ is used for a collective of notes that are percepted as one musical entity) are defined with these criteria: 
1) Distance to the start of next ‘field’ in seconds.2) Duration of the field in seconds.3) Central pitch.4) Pitch range, symmetrically around central pitch.5) General timbre.6) Initial average speed of the field.7) Final average speed of the field.8) Initial rhythmic influence.9) Final rhythmic influence.10) Initial average dynamics.11) Final average dynamics.12) Value of c and d (determining degree of order) in Formula 2, which controls melodic microstructure. 

Each of the criteria has its own initial x value, and this gives each of them a unique path in the chaotic regions, resembling the others, but never quite identical, so that they go in and out of phase with oneanother. This produces a rich lattice of fields with attributes that change sometimes collectively, sometimes individually. Furthermore, the progression of each criterion is weighted in time, either by limiting the possible values with a border (usually a very slow sinusoidal), or by a logarithmic ‘distortion’ formula that weighs the values to a higher or lower value. The first technique is shown in Fig. 3 and Fig. 4.

Figure 3

Figure 3Fig. 3 shows the evolution of parameter 1 (distance to the start of next ‘field’ in seconds). The sinusoidal shape reduces the maximum length of the ‘fields’ in the middle of the piece, and gives a more restless feeling than in the start and end. (Note again that the graph contains many more points than there are fields in the piece, this is for making the illustration clearer.)

Figure 4

Fig. 4 shows the evolution of parameter 10 and 11, (initial and final average dynamics). This gives the piece a relatively soft start and ending, with the possibility of strong dynamical levels near the ‘Golden Section’ of the piece. Because the ‘Initial average dynamics’ and the ‘Final average dynamics’ parameter have two different initial x values, the different fields will make a pattern of crescendi and diminuendi from/to different dynamic levels. 

Points 8 and 9, initial and final rhythmical influence, means that the values assigned to rhythm in the microstructure can have a certain influence on the durations, from none (resulting in completely even notes with a value determined by point 6 and 7, speed) to maximum (resulting in a jerky rhythm with big difference between the longest and shortest durations, with an average duration determined by point 6 and 7). 

The parameters that are represented by an initial and final value, go through a transition (linear or logarithmic) in the timespan of the field (point 2), this resulting in small or large accelerandi/ritardandi, crescendi/diminuendi, and transitions between smooth and jagged rhythms. Point 5, general timbre, is very openly stated, and creates the main opportunity for me to colour the resulting form with instrumentation, besides the open parameters of phrasing, glissandi, tremoli etc., and sustaining notes to accumulate chords. 

This slightly dry run-through of my first attempt to assimilate fractal mathematics into my musical language is done less to focus on the piece itself than to suggest that this is only one (and quite private) of countless ways of applying fractals on music. I have not even mentioned the work that goes on concerning fractal frequency spectras, a field that is equally exiting. 

One of my main reasons for using a computer in my compositional work is that I want to be surprised. Like C.P.E. Bach suggested for musicians that to move their audience they must be moved themselves, I have found out that to surprise the listeners, I have to be surprised myself. The computer has had that function, which means that I give it a task where the direction is clearly defined, but where the details are unpredictable, and therefore generating a balance between global order and local disorder that has been very fruitful for me and has kicked me out of some tracks that I was stuck in. What intrigues me about the dynamic systems is that the order and the disorder have the very same origin, not like in classical stochastical computer music where Euclidian geometry has been ‘ragged’ by a random generator, a prosess that can be likened to mixing oil and water. With chaos theory, the gap between total order and entropy is about to be filled in, and exactly this can turn out to be interesting for musicians. After all, this gap has been our playground for some thousands of years…. 

James Gleick: Chaos – Making a New Science (Viking Penguin) 
H.-O. Peitgen and P. H. Richter: The Beauty of Fractals (Springer Verlag) 
Jan Frøyland: Symmetrically coupled nonlinear oscillators as a function of their numbers. 
Øyvind Hammer: Fraktalmusikk – noen anvendelser for fraktaler og ulineære dynamiske systemer i algoritmisk komposisjon.