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Holo-Fractal Music

An organic connection between Music and Physics

Eufrasio Prates

There is always the possibility of proving any definite theory wrong;
but notice that we can never prove it right.
Richard Feynman (1992, 157)


This essay intends to show some possibilities for using the Fractal Geometry and Holonomic Physics fundamental concepts to create a new musical aesthetic language. After identifying those main concepts, the composer has many ways to put them into his composing practice. Some of them are born with an extremely technical and mathematical approach, what results in a rigid, mechanical and cold music. We try to show with this analysis a path to a more organic perspective, through the use of Semiotics for connecting the "Music as culture" and "Physics as nature" knowledge fields, emphasizing the transversality of communicational dimension.

One of the main differences between pre-socratic and post-modernist era is maybe the velocity of social changes. Despite the enormous distance between the ancient greek way of thinking and our own, the major difficulty defying contemporary science is to keep our production moving fast enough. That is one of the reasons why the knowledge increases geometrically, asking from us a constant up-to-dating. Therefore we're actually moving from the habitual changes in paradigm to the paradigm of changes.

Assuming this constant mutation attitude, although uncomfortable, would present many advantages for making productive researches in experimental fields. That is the reason why Semiotics, as a new tool for comprehending and interconnecting the languages world, becomes stronger each day. The annoyance of the dispute among Peirce and Greimas Semiotics tends to be outdone by the growing demand for solving cultural signification questions, without losing the links to reality (whatever each one means with this term).

Thus, the aim of this essay is to present, in a short and therefore superficial analysis, a perspective of producing music based upon new paradigmatic ideas - as holonomy and fractals - keeping the organic-intuitive cultural and communicative approach.

HoloFractal paradigm: Model under construction

Ph?sis is the Greek word for nature, rooting the term Physics. This century witnessed the emergence of a new vision of the nature, by means of the Quantum Physics. Thus, the new Physics reached and connected two invisible, and supposedly distant, universes: the subatomic particles and the sidereal space. It also yielded surprisingly discoveries, as the paradoxality and unpredictability of nature behavior, showing the impossibility of making any definite statement about it.

The most important theories produced upon this quantical view share the concept of h?los, that is totality, integrality, wholeness. Since the sixties, some physicists began to talk about holographic or holonomic paradigm, emphasizing the significance of the inter-relation between the part and the whole (Prates, 1997, 17-19). All this structural changes brought us new fields of research, specially one that seems to be extremely fruitful: the Chaos Science.

The definition of the word "chaos", considered by common sense as the absolute opposite of the concept of "order", put us in a difficult position: as we saw in the work presented by Floyd Merrel in the 3rd International Latin American Congress of Semiotics (III CILAS), there is some kind of order inside the chaos itself that, paradoxically, shows the falsity of the "absolute chaos". Actually, the supposed disorder of chaos is nothing more than our incapability to understand the non-periodical systems, typical of nature and society, as for instance the turbulence of fluids, variations in economy, meteorological transformations, population growth and a large number of phenomena that escapes from the famous Temperature and Pressure Normal Conditions - inexistent, as far as we know, in the universe. In its ancient Greek roots, the term ch?os means "origin" or "originary opening from where everything comes", meaning therefore "indeterminate". It was only with the Roman sense that the word became known as disorder, as opposite of k?smos or order, which nowadays became anachronic.

Let us take a simple example: the Second Law of Thermodynamics, that says the universe walks with no return for ever more chaotical states, is usually explained to the lay through the water and ink mixture experiment. The entropy of that system is intended to be proved by the fact that the component substances will never come back, naturally, to their original states. However, this very same experiment can be used to prove and explain a new organic, homogeneous and, therefore, entalpic organization. This is one of the first problems some scientists began to think about in the sixties. We will find, in the origins of non-periodical systems studies, names as Benoit Mandelbrot and Christopher Scholz, the creators of a new mathematics and geometry much closer to nature and society. Those systems are also known as non-linear, because of their unpredictable development and their initial conditions sensitive dependence. The famous "butterfly effect" exemplifies them: it says the air moved by a butterfly wings can be responsible for a tornado in a distant part of the planet. That led us for a technical definition of chaos, as a phenomenon dependent of its initial conditions (Lorenz, 1996:39).

As the world always refused to behave according to euclidian rules, those scientists began to build a new way, the Fractal Geometry, to describe the complexity of our coastlines and hillsides, the micro-irregularities in metal surfaces, the porosity of petroleum ladden rocks etc. Mandelbrot, joining a relativistic approach to the profound study of the dimensions, concluded that in order to be better understood, nature demands the traffic by the fractions between the point and the line, the line and the plan, the plan and the volume. How should we calculate the quantity of petroleum in a porous rock? Finding a number between the second and the third dimension that comes close to that reality. Something like, for instance, 2.4 dimensions. Based upon this new logic, it was possible to calculate the degree of irregularity or fragmentation of a phenomenon. From this fractionation, this fracture, Mandelbrot invented the word fractal in 1975.

A more recent approach of Fractal Science is sustained by the ideas of Edward Lorenz, Mitchell Feigenbaum, Stephen Smale and David Ruelle. These ideas were developed upon natural phenomena and designed a much more complex conception of nature. Collecting data from the turbulence of viscous fluids or from the changing of the weather, and finding equations for representing their hidden complex patterns, those scientists reached the conclusions that led to the construction of concepts like strange atractor, non-linearity, randomicity and self-similarity. This last one shows that nature presents some kind of strong inter-relation between the big and the small. Hans Lauwerier asserts "a fractal is a geometrical figure in which an identical motif repeats itself on an ever diminishing scale" (1991:XI), despite his mention that original Mandelbrot s definition refers to a more restrict and complex concept of fractal dimension. Good examples of the singularity and importance of this organic geometry are the Eiffel Tower - built upon a thin and resistant auto-reproductive fractal scale structure - or the human lungs - that fractally compacts a surface that would occupy an area bigger than a tennis court.

Summarizing, what we call holofractal paradigm is a model based upon concepts like wholeness, relativity, paradoxality, unpredictability and similarity between part and whole. As Feynman's citation asserts, this theoretical model cannot be proved right. Nevertheless, it brings us a view much closer to nature than the previous ones.

Models, of course, are useful for many things. Some of them are potentially dangerous, because of the misuse of its reductionism over reality. But one of them is to apply its concepts and internal relationship to other knowledge fields. That is what we have been doing: building paradigmatic connections between music and physics.

Bridging our subject to the Aesthetics field, we will find many composers applying those fractal equations to a new music composing method. In some way, they believe we can make a more natural music translating the equation generated numbers to musical parameters. We are exactly interested in which way we could reach this goal without immoderately mechanizing and automating that process.

From nature to culture: a semiotic path

... the aesthetic experience (...) suggests the codes from which it starts could be submitted to successive segmentation.
Umberto Eco (1976:227)

It is largely known that Johann Sebastian Bach's music is highly and mathematically organized. This quality had never generated negative critics upon his music because it is clear that aesthetical characteristics are privileged. Differently, in some dodecaphonic and serial contemporary works, we find the opposite approach. Serial method tried to overrule tradition by mechanizing composition process and, despite its technical advances, it soon proved to be excessively rigid and resulted in a cold and inhuman music. Nowadays we must search for the middle way, fitting the demand for a creative path that, simultaneously, shows the main ideas of our times without losing the human aesthetics touch. A rich way to respond to that, is the use of the sign as a bridge between nature and culture. In other words, Semiotics seems to be the tool to fill up the vacuum left by the abandon of traditional composing parameters.

Beyond early definitions, we see Semiotics as an intuition path through the languages with the aim of understanding the complex multiple layers that compounds a meaning phenomenon. If it seems to be an unusual way of looking to this science, we must remember that most creative explorations of new fields find their path under the peircian concept of abduction. With this term, Peirce designed the instinctive and suggestive "reasoning", typical of insights eruption (Sebeok, 1991:9; Peirce, 1995:221). In fact, the abductive method is described by words as perception, trial and error, experimentation and evolution by chance,.

That vision, on the other hand, may risk losing its richness, if the communicational dimension is put aside. We must never forget that music is made for others to listen. Obviously, no contemporary composer is deluded about his work's comprehension, but that does not means the absence of concern with the codes used. If Mukarovsky is not wrong about the autonomy of aesthetic sign, specially about contemporary art, that does not liberate the poetical creation from its communicational role.

This alert is directed to some experimental fractal music trials, born under an excessively mathematical approach. Of course it is necessary to make a lot of calculation for working with fractal parameters to produce art. Computers are indispensable tools for managing the large amount of calculations of fractal equations. Nevertheless, it is dangerous to believe they can solve aesthetical problems. This automation process must handle exclusively the repertoire development and the material manipulation of sound. All those tasks must be done under a major organizational plan, sometimes forgotten by fractal-fascinated composers. Everyone can easily feed some random numbers in a fractal music software and it will generate a highly redundant or a somewhat strange sequence of sounds. The distance between this rigid result and what we call music - even chaotic and fractal music - must be fulfilled by the expertise of a musician, who owns the necessary knowledge for transforming sounds into a meaningful complex.

The game between form and element, whole and part, is just the first process to join new music and new paradigms. Actually, the holofractal music composer has to join all those cited concepts with the finest techniques in sampling, calculating, editing and sequencing sounds. But this is not everything: without a concern for language - bridging nature and culture - that process can become schizophrenic, divided into theory and practice, aesthetic and technique. Just a semiotic perspective, explicit or not, can avoid this serious problem. That is why a critical positioning is required for materializing new paradigmatic ideas into communicative practices.

In short, the best possibility of testing the limits and exploring new fields of musical language, with an aesthetic concern, rests upon a semiotic approach. That is the direction we took for designing a musical experiment called coVeranS, analyzed as follows for elucidating a possibility of holofractal creation.

coVeranS: a holofractal music case study

This 3 20" experimental music was concluded in November/96, after four months of work. Its realization can be divided into three parts, for didactical aims: the project designing, the sounds fractal treatment and the samples edition. coVeranS was conceived with the background idea of linking the "synthetic" to the "organic", interconnecting culture and nature through a semiotic bridge.

For obvious reasons, it was avoided a discursive causal form, because semiosis is a complex, non-linear and unpredictable phenomenon. We used a new form of organization, developed by Koellreutter, called sineretic form. He describes sineresis as "result of a transrational perception process that produces the sense of unity, integrating the elements in a whole" (1990:120). For organizing music sinereticaly, it is necessary at first to dispose the selected musical signs in the drawing board to create a sketch. After the wholeness becomes perceptible, it is time for designing multiple interconnections between the elements, followed by the choose of a temporal version that meets the schemed aesthetic purposes. We used for this work the sequencing audiofiles software DECK II:

As for the musical signs, we decided to choose two basic elements: a percussive synthesized sound, similar to the snare, and a human voice, speaking the words "fractal geometry". This choice was made to build up a repertoire with high contrast, where variation intends to fill up the spectrum between the synthetic snare and the organic voice sounds. Both units were digitalized and wrote down on the computer hard-disk with the names "snare" and "fracvoc". They were the only source for generating all the other mutant sounds of this music, at first called SnareVoc and, finally, reversed to coVeranS.

That is the point where fractal parameters enters the scene. To create new sounds from those sources we used a computer program that, differently from traditional fractal music softwares, penetrates the entrails of the sound. Researching on the Internet, we discovered that all the references found are based upon a generation of numbers that will play a MIDI synthesizer. This approach, although interesting in many ways, avoids internal dimensions of sound, managing only basic and macro parameters as duration, frequency and, in a few cases, intensity. Beyond that, its music is often highly redundant, rigid and mechanical, because of its characteristic difficulty of managing global structure and form. Actually, this music is chained to the pre-manufactured synthesizers sounds.

That is the reason why we preferred to work out the microdimensions of music. SoundHack, a software developed by Tom Erbe in 1995, brings us digital sound processing that includes spectral mutation, spectral dynamics and soundfile convolution, between many other sound processing tools. Beyond that, it opens a door for entering fractal generated parameters that will be submitted to mutational internal functions.

To create coVeranS elementary sounds it was necessary, after choosing a source and a target sound, to input parameters with the aim of defining how they would interact to generate a mutated descendant. Like the natural pregnancy, we spent the development time thinking how the newborn would sound like. Despite the use of a PowerPC Macintosh, with a fast RISC processor, sometimes a 38 seconds sound processing spent more than one hundred hours. And it was not rare to have surprisingly strange or fantastic results. Other times, we have got a big silence soundfile. It happened this way because there is a strong relationship between this fractal processing and the "butterfly effect": initial conditions sensitive dependence.

It is noticeable how the mutated sounds appear to be, paradoxically, of high contrast and self-similar. That strange quality reflects the natural and organic inter-relation between part and whole, as all the little sounds are generated from the main samples or from inter-mutation between the derived mutated sounds. Under the fractal logic, it was done one derivation over another, translating the inter-relation part-whole.

This process, richer than the ordinary ones, allows the manipulation of the sound itself, emphasizing the semiotic human role in the composing of these little sound elements in a whole that makes an aesthetical sense. With the mentioned trial-and-error process, typical of abductive thinking, a couple of months was spent generating the variations and mutations that were, simultaneously, worked out as parts of the main structural work. The guidelines of coVeranS have been weaved as web-like forms. Like the spider, looking for the balance between technique and intuition.

Each new sound position in the piece was submitted to a non-linear ordering, reinforcing the unpredictability of the composition. As one ascendant sound may appear after its derivatives, it was thus avoided the cause-effect relation. Also the complexity of those sounds may hamper the understanding of this music, especially for the rationalist ear. However, it cannot be forgotten that we are dealing with new transrationalist concepts. Intuition is the key for transcending from understanding to comprehension.

As the triadic peircian conception, our secondness is found on the polishing of the rude samples, born under the mutation processing. This struggle was undertaken through the use of another program called SoundEdit, that provides an enormous variety of finishing tools. This phase was extremely delicate, because applying effects and filters on a sound can easily destroy its personality and, worst than that, can disconnect it from its fractal ancestors. The most used editing tools were the equalizer, normalizer, pitch shifter, reverser, enveloper and bender.

This software was very useful for reversing some sounds, as for cutting and pasting parts of sounds to create cubist-like assemblage effects, breaking the temporal linearity of the music even on its basic elements. It mediated the come-and-go process between the firstness of sound elementary qualities and the thirdness of symbolic relations structure. Therefore, there was an inter-reflecting non-linear process of creation, from sound creation to structure and vice versa.

To finish this short exposition of a holofractal essay music, it is desirable to mention that especial attitude, necessary for a fruitful hearing experience. The major part of the difficulties in the listening of new music does not proceed from the low exposition of the listeners to it or from their ignorance of new technical framing but, much more than that, from the inherited deep-rooted preconception, reinforced by the massive capitalist media strategies. Instead of refusing the new for its strangeness, it is desirable to assume an intuitive position, being ready for an aesthetic experience.

After that, remembering that music is a strongly semiotic phenomenon, it will be easier to occupy the place of the creative interpreting mind, the one who is capable of fulfilling, as the cosmogonical ancient greek man used to do, the path between nature and culture.


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Eco, Umberto e Sebeok, Thomas. O signo de tr?s. S?o Paulo: Perspectiva, 1991.

Feynman, Richard P. The character of physical law. London: Penguin Books, 1992.

Koellreutter, H. J. Terminologia de uma nova est?tica da m?sica. Porto Alegre: Novas Metas, 1990.

Lauwerier, Hans. Fractals: Images of chaos. London: Penguin Books, 1991.

Lorenz, Edward N. A ess?ncia do caos. Bras?lia: EdunB, 1996.

Peirce, Charles S. Semi?tica. S?o Paulo: Perspectiva, 1995.

Prates, Eufrasio. M?sica qu?ntica: de um novo paradigma est?tico-f?sico-musical. Bras?lia: Universidade de Bras?lia, Master degree thesis, n/e, 1997.

Ruelle, David. Chance and chaos. London: Penguin Books, 1993.

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