READING COMPLEXITY IN CHUA’S OSCILLATOR THROUGH...

Tutorials and Reviews

International Journal of Bifurcation and Chaos, Vol. 15, No. 2 (2005) 253–382c© World Scientific Publishing Company

READING COMPLEXITY IN CHUA’S OSCILLATORTHROUGH MUSIC. PART I: A NEW WAY OF

UNDERSTANDING CHAOS

ELEONORA BILOTTADepartment of Linguistics, University of Calabria, Italy

STEFANIA GERVASI and PIETRO PANTANODepartment of Mathematics, University of Calabria, Italy

Received June 7, 2004; Revised August 1, 2004

Modern Science is finding new methods of looking at biological, physical or social phenomena.Traditional methods of quantification are no longer sufficient and new approaches are emerging.These approaches make it apparent that the phenomena the observer is looking at are not clas-sifiable by conventional methods. These phenomena are complex. A complex system, as Chua’soscillator, is a nonlinear configuration whose dynamical behavior is chaotic. Chua’s oscillatorequations allow to define the basic behavior of a dynamical system and to detect the changesin the qualitative behavior of a system when bifurcation occurs, as parameters are varied. Thetypical set of behavior of a dynamical system can be detailed as equilibrium points, limit cycles,strange attractors. The concepts, methods and paradigms of Dynamical Systems Theory canbe applied to understand human behavior. Human behavior is emergent and behavior patternsemerge thanks to the way the parts or the processes are coordinated among themselves. Infact, the listening process in humans is complex and it develops over time as well. Sound andmusic can be both inside and outside humans. This tutorial concerns the translation of Chua’soscillators into music, in order to find a new way of understanding complexity by using music.By building up many computational models which allow the translation of some quantitativefeatures of Chua’s oscillator into sound and music, we have created many acoustical and musicalcompositions, which in turn present the characteristics of dynamical systems from a perceptualpoint of view. We have found interesting relationships between dynamical systems behavior andtheir musical translation since, in the process of listening, human subjects perceive many of thestructures as possible to perceive in the behavior of Chua’s oscillator. In other words, humancognitive abilities can analyze the large and complicated patterns produced by Chua’s systemstranslated into music, achieving the cognitive economy and the coordination and synthesis ofcountless data at our disposal that occur in the perception of dynamic events in the real world.Music can be considered the semantics of dynamical systems, which gives us a powerful methodfor interpreting complexity.

Keywords : Chaos; Chua’s oscillator; sounds and music.

Contents

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2542. Basic Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256

2.1. Historical introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2562.2. Physical and psychological aspects of sounds and music . . . . . . . . . . . . . . 260

253

254 E. Bilotta et al.

2.2.1. Physics of music . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2602.2.2. Music psychology basics . . . . . . . . . . . . . . . . . . . . . . . . . . 263

2.3. Acoustical and musical spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 2642.4. Current trends in Computer Music . . . . . . . . . . . . . . . . . . . . . . . . 2662.5. Languages and codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267

3. Chua’s Circuit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2713.1. Complexity and strange attractors in Chua’s circuit . . . . . . . . . . . . . . . . 2713.2. Experimental observations on chaos . . . . . . . . . . . . . . . . . . . . . . . . 2723.3. A garden of attractors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274

4. Codes and Software for Translating Chua’s Attractors into Sound and Music . . . . . . 3004.1. Sound and musical codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 300

4.1.1. Sonification codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3004.1.2. Musical codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3014.1.3. Systems based on the attractor’s shape . . . . . . . . . . . . . . . . . . . 3024.1.4. Color code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3024.1.5. Multiresolution analysis code . . . . . . . . . . . . . . . . . . . . . . . . 3024.1.6. WFSound . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3064.1.7. Chaos generator: Generating music from Chua’s oscillator . . . . . . . . . . 3094.1.8. Chua’s harmonies: Generation of harmonies from Chua’s oscillator . . . . . . 3094.1.9. On-line music: applet and web site for the exploration of music from

Chua’s oscillator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3125. Exploring Chua’s Attractors through Music . . . . . . . . . . . . . . . . . . . . . . . 332

5.1. Musical examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3325.2. Musical complexity and emergence of rhythm in Chua’s attractor . . . . . . . . . 3375.3. In the garden of chaos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 354

6. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 367

1. Introduction

Music is a complex, biological and cultural phe-nomenon, whose development, in the humansubjects, can be compared to the acquisition ofa natural language. The essential connection thatlinks up language and music has been demonstratedby some researchers [Parsons, 2001; Parsons &Thaut, 2001], who, using brain imaging instru-ments, have analyzed the influence of music on thebrain and have found that cerebral structures ofmusical cognition are similar to the cerebral struc-tures humans use for linguistic functions.

These results underline that there is a similar-ity between music and language [Swain, 1997] andthe evolution of musical cognition is connected tothe development of language [Wallin et al., 2000].Considering that there are millions of spoken lan-guages, humans have developed different musicalsystems in different cultures. Human subjects love

music, they look for it actively, they have typi-cal listening behaviors, they can listen to it like apleasant background also when they are engaged inother activities. The acquisition of musical languagedevelops along with time. Such process implies, likein the case of natural languages, a competence inthe identification and production of a richly struc-tured sequence of distinctions, such as happens forthe acquisition of musical language. The subjectslearn simply by listening and reproducing themesand this implies that the human brain has a special-ization, genetically determined, which is devoted tomusical language.

Musical abilities (as other cognitive competen-cies) are a complex combination of nature andculture, since they are acquired by the dynamicinteraction between the subject and the environ-ment [Brooks, 1991; Varela et al., 1991]. The int-eraction with the environment is complementary to

Reading Complexity in Chua’s Oscillator through Music 255

the genetic equipment. A child acquires the abil-ity to manage a complex phenomenon like languageand to assimilate it in a relatively short time.

Besides, Chomsky [1965] has formulated aUniversal Grammar for the acquisition of language,codified in the genes, Lerdahl and Jackendoff [1983]have conceptualized that for music the same processcan happen: in spite of the differences of the cul-turally determined musical languages, the presenceof universal structures in music allows to “tune”musical pieces which are distant from the culturaltraditions to which a subject belongs.

Unfortunately, there is no witness of the phylo-genetic evolution of language and the musical evo-lution, except for the fossil relics of our ancestors’ancient instruments [Kunej & Turk, 2000], while thefirst written documents manifest at a first glancean expressive variety and grammatical complexitywhich are peculiar to the modern languages.

Therefore, the way through which languageevolved is still today an enigma, even if currentlynumerous studies are trying to simulate the evo-lution, both through dynamical systems [Elman,1995; Nowak & Komarova, 2001], and throughneural networks inserted into agents interactingwith the environment which, in their turn, trans-mit to their descendents the meaning of suchinteractions, encapsulated in linguistic structures[Cangelosi & Parisi, 2002].

Is it possible to recreate by means of a com-puter the productivity and the grammar structurestypical of musical language? Is it possible to simu-late a generator of music pieces that are in someway structured or structurable according to par-ticular rules and that are acceptable for a listenerfrom the harmonic and melodic point of view? Isit possible to evolve musical languages, startingfrom the sounds and arriving at the notes and themelodies?

Adopting as a metaphor a musification triangle(Fig. 1) that joins, at one vertex, the mathematicalequations of Chua’s oscillators, used as contents tobe translated into music, on the other vertex variouscoding systems, we have created many musical arte-facts, creating a musical language (third vertex).

Such a language has the following charac-teristics:

• production rules, that can generate an infiniteand diversified amount of musical phrases;

• abstract and general character of the produc-tions;

• arbitrariness of coding and representations;• representations that have a semantic.

The main aim of this tutorial is to translate Chua’soscillator model into music, in order to discover andinterpret complexity through unexpected media:sound and music.

Technology, music and scientific models areclosely interrelated and currently a wide produc-tion of musical instruments and music does exist,which is based on such models, using various typesof algorithmic codings [Schwanauer & Levitt, 1993;Roads, 1996].

Computers have carried out a prominent rolein the production of digital music and the com-puter programs that have been implemented in themusical field are based on algorithms or procedureswhich simulate some modalities of musical produc-tion, but also on artificial biological models [Leach& Fitch, 1995], and neural networks [Todd, 2000].

Various theories of science can be representedin the domain of the computer-generated music.In particular, new approaches based on the use ofiterations [Gogins, 1991], on the theory of chaos[Bidlack, 1992; Bilotta et al., 2003; Pressing, 1988;Rodet & Vergez, 1999; Witten, 1996], on fractals[Bolognesi, 1983; Dodge, 1988; Leach & Fitch, 1995;Manaris et al., 2003; Voss & Clarke, 1975; Voss &Clarke, 1978], on cellular automata [Bilotta et al.,2000; Bilotta & Pantano 2002a; Miranda, 2001],on artificial life [Bentley and Corne, 2002; Bilottaet al., 2002a] allow composers to increase their cre-ative possibilities in the musical production, con-senting some useful and profitable connections tothe fields of electronic art [Barry, 2001], computergraphics [Spector & Klein, 2002] or auditory inter-faces [Kramer, 1994].

As Wolfram [1984a] says, computer has becomea virtual environment where the experimentationand the computation allow to recreate natural phe-nomena. Computers are the place where Artifi-cial Life is “growing”, or it is the environment inwhich it is possible to construct a synthetic biology[Langton, 1995] or to simulate natural and/or bio-logical phenomena to be studied with times andmodalities that the real world does not allowus. Therefore, we have used mathematical modelsbecause of their ability to simulate salient featuresof biological processes, suitable to be translated intomusical structures, adopting the idea of Pythagorasof a close correlation between music and mathemat-ics. The Greek philosopher discovered that some


Fig. 1. The musification triangle we have adopted in order to translate Chua’s oscillators into music.

particular relationships lead the notes to fuse, inorder to merge into melodies pleasant to the ear[Frova, 1999]. According to Pythagoras, the wholeuniverse was endowed with these numerical proper-ties, and man had to do nothing but to read them.

In the researches carried out in these years[Bilotta & Pantano, 2000; Bilotta et al., 2000;Bilotta & Pantano, 2001a, 2001b; Bilotta et al.,2002; Bilotta & Pantano, 2002a, 2002b; Bilottaet al., 2003; Bilotta & Pantano, 2004; Bilotta et al.,2004], the authors have tried to develop some theo-retical hypotheses about the construction of musicallanguages, the sound and musical quality of pro-duced musical pieces, in terms of acceptability byhuman subjects, musicians and not, and the con-struction of software instruments that implementthe proposed theoretical assumptions. In this tuto-rial we present the realization of some musical codesin order to translate into music the mathematicalconfigurations produced by Chua’s oscillator, withsome remarks on the composition modalities thatwe have used in order to produce music and to readthe complexity at various levels of organization.

The work is organized in four main parts. Thefirst one deals with the principal concepts of musictheory, the physics of sounds and music, and a his-torical perspective on the evolution of the westernmusical system, from the Pythagorean scholars tothe well-tempered scale realized by Bach. Further-more, we report about current trends in ComputerMusic and the key-concepts we have used in order

to translate dynamical systems as Chua’s attractorsinto music. In the second part, we analyze the mainfeatures of Chua’s circuit and the possible sounds itcan produce, giving an idea on how chaotic systemscan be explored using the physical system. In thethird part, we give an account of the codes and thecomputer programs we have constructed in order toachieve our goals.

In the fourth part, musical examples obtainedby the musification of the typical aspects of thedynamical systems are described, together with theexamples that refer to the musical exploration ofchaos in its numerous aspects: sensitivity to ini-tial conditions and bifurcations, analyzing the influ-ences that the behavior of these systems produceon musical realizations. The examples are also ana-lyzed from a perceptual point of view.

2. Basic Concepts

2.1. Historical introduction

The first attempt to comprehend nature, interpret-ing it in terms of numbers and measures, was madeby Pythagoras in the VI century B.C. But why isPythagoras’ philosophy so important? And, aboveall, what has it to do with music?

Pythagoras was interested in the properties ofinteger numbers, and discovered several theoremsthat are important for arithmetics and the ModernTheory of Numbers. Following Pythagoras, we can


argue that the number (and its qualities) is inher-ent in all the physical objects and is the principleof everything. The universe is composed of mate-rial substance, and whatsoever substance, it alwayshas the form of a number, in a certain geometri-cal or arithmetic combination suitable for a numer-ical or mathematical formulation. Each substancethat can be perceived by our senses appears in acertain numerical order, and each physical entity,that could be a substance or the state of a sys-tem or of its mutations, can always be expressedin numerical terms or through a mathematicalformula of the universe. Another key concept inthe Pythagorean philosophy is that of harmony.Harmony is the fundamental quality of numbers,by which they unify the opposites, substance andform. The Greek philosopher Philolaos (fragmentD44B6) affirms that “the substance of which thecosmos is composed is made of different things, andthat in order to understand them we should havedivine rather than human capacities. For this reasonthese elements are unified by an ordering processrepresented by harmony. If the elements had beensimilar, there would be no need for harmony: onlyharmony can unify the cosmos”.

“Nothing would be comprehensible, neitherthings in themselves nor their relations, if therewould not be the number and its substances.But this, harmonizing everything in the soulthrough perception, makes them and their rela-tions intelligible” (Philolaos, fragment D44B4)[Geymonat, 1971]. In this fragment Philolaos pointsout the relationship between the number and thesensorial world, which is found in the Pythagoreanworld and that could constitute the origin of themathematical explanation of the physical world andof the sensorial and perceptual phenomena, as cur-rently intended by the modern neurophysiologicaland neuropsychological theories.

In addition, Pythagoras noticed that harmonyof numbers was surprisingly manifested in themathematical relations between the various num-bers. These relations assumed the form both ofproportions and geometric relationships (these rela-tionships have been rediscovered by Hofstatder[1979]). The laws of musical harmony were deducedby Pythagoras’ disciples by a series of experiments,particularly noticeable if compared to the lack ofsystematic experimentation in the Greek world.

The concept of “Harmony of the spheres”deeply influenced the way Greeks conceived the

cosmos, with a profound faith in the union of geom-etry, music and astronomy that makes man a partof the universe. From this assumption arose the ideaof “Music of the spheres”, continued by Kepler thatintuitively realized the harmony of the spheres inhis conjectures on the relationship existing betweenthe planetary orbits and the Platonic solids (and bythe modern Physics).

In addition, the Pythagoreans believed that aprofound link existed between the laws of musi-cal harmony, developed on simple arithmetic orgeometrical proportions, and a fundamental phys-iological and psychological event as the sense ofeuphoria, that leads Leibniz to affirm: “Music is thearithmetic of the soul, that uses numbers withoutrealizing that”.

Some of the Pythagoreans narrate that theirmaster and his disciples performed some experi-ments with strings of different length and width,varying their tension rotating the screws thatfastened them. Other experiments were carriedout using wind instruments of different lengthsand containers of different forms, filled with dif-ferent amounts of water, thus producing air col-umn vibrations of different lengths. While some ofthese experiments were simply qualitative, thoseconcerning wind instruments and different lengthstrings were authentic quantitative measurements.The turning point was the monochord experience.The monochord is a single string stretched over asound box, with two fixed end points, as shown inFig. 2.

It uses a movable bridge to change the degreeof tension and the length of the string, placing itin each intermediate point. The obtained result, ata fixed tension of the chord, is that the pitch ofa sound, according to the experimenter, is inverselyproportional to the length of the string that is oscil-lating. The monochord consented to make the fol-lowing observations:

• As long as the ratio between the lengths of theoscillating strings remains constant, the distancebetween two pitches, nowadays called interval, isconstant too.

• The interval between the sound emitted by thestring without bridge and the one emitted withthe bridge placed in the central point of thestring, that corresponds to a 2 : 1 ratio betweenthe oscillating parts of the string, identifiableas the maximum consonance, is said an octaveinterval.


Fig. 2. In this image, a 3D reconstruction of the ancient monochord is displayed.

• Other remarkable intervals apart from the octaveare: perfect fifth, that corresponds to a ratio of3 : 2 between the oscillating string lengths (C-G);perfect fourth (C-F), a ratio of 4 : 3. It is interest-ing to observe that the notes that seem to formconsonant chords correspond to integer-fractionfrequency ratios. The results are more agreeablethe smaller the numbers.

• It was possible to notice that all the intervals ofinterest in the tonal scale correspond to noteswhose frequency is a multiple of a fundamentalf1. Namely, they are said to be in harmonic ratio.

Pythagoras proposed a diatonic musical scale(i.e. composed of seven notes, the white keys ofa piano) based on the concepts emerged from theexperience in harmony made with the monochord.However we have to clarify that at that time musicwas strictly monophonic, so the consonance of thenotes did not have great relevance. This makesthe result obtained by Pythagoras more consider-able, because it was achieved following, more thana practical need, the aspiration to a mathematicaland conceptual formal elegance. However his studieshave been of fundamental importance for the com-plex polyphonic music of the following centuries,that could not subsist without the harmonic struc-ture. Afterwards Ptolemy, established a new con-sonance ratio that did not appear in Pythagoreanscale, but nowadays is of fundamental importancein the natural scale: the major third chord C-E.

Among the different new scales that in the-ory could be built with Ptolemy’s interval, the oneobtained choosing as starting note C was preferred,and from it other two scales, respectively E and G,with a major third and a perfect fifth interval (fre-quency ratios with C = 5 : 4 and 3 : 2), were gener-ated. This obtained scale is the so-called “Naturalor Just Intonation scale” and it is also called scaleof the simple ratios or Zarlinian scale, for the reasonthat, around the half of the XVI century, GioseffoZarlino’s writings promoted its use. So the Greekculture gave birth to something profoundly insight-ful: the evidence of the interconnection betweenmusic and science, the perfect permeation amongharmony, physics and mathematics.

During the Middle Ages, the Liberal Arts weredivided into the three-fold Trivium and the four-fold Quadrivium of Arithmetic, Geometry, Musicand Astronomy. Music, thus, is once again consid-ered as a “scientific art”. So it is evident how art canadvance if supported by the physical-mathematicalprinciples that sustain it.

The musical scales proposed by Pythagoras andPtolemy present however some weak points that jus-tify the successive introduction of scales of com-promise: the so-called tempered scales, conceivedto overcome the problems of the historical scalesand to satisfy important practical and instrumentalrequirements.

Pythagoras’ approach suggests to generate theaccidental notes of the chromatic scale (sharp and


flat) putting together the circle of the fifths. How-ever some problems may arise, so that the semitoneinterval F-F# (chromatic semitone) does not cor-respond with the semitone interval B-C (diatonicsemitone). Thus the Pythagorean scale, opposing tothe natural one, has the benefit of having only onevalue for the whole tone interval (9 : 8), but has thedisadvantage of requiring two values for the semi-tone. The Just intonation scale, based on the C-E-G chord, made of a major third (interval ratio of5 : 4), followed by a minor third (interval ratio of6 : 5), presents some problems, in that the fourthsand the fifths are not all perfect (D-A, of interval40 : 27 = 1.481, is smaller than 3 : 2 = 1.5). Anotherdifficulty resides in the fact that there are two wholetones (one larger, C-D, equivalent to 9 : 8 and onesmaller, D-E, equal to 10 : 9). This characteristicinhibits the execution of a piece in different tonal-ities using the same instrument without a previoustuning of it. To surmount this difficulty, the SpanishFrancisco Salinas, during the XVI century, built akeyboard instrument containing twenty-four notesper octave, that gave the possibility to execute acertain number of tonalities, but not all. However,he realized that the solution of a huge keyboardwas not resolutive, so he proposed to adopt a tem-pered scale, in which the notes where “adjusted” tovalues that did not correspond to the requirementsof the maximum consonance, but put an end to theproblem of changes in tonality. Among all the formsof temperament of the chromatic scale, the equaltemperament, since the Baroque age, has becomeof universal use.

This temperament creates equal-spaced musi-cal intervals for each chosen tonic. In the chromaticscale, every semitone interval is fixed to 100 cent(the cent, introduced by the English Alexander Ellisin 1885, has been defined starting from the loga-rithm of the frequencies), thus the five whole toneintervals of the diatonic scale are equal to 200 cents:doing so, the effortless passage from one tonalityto the other becomes infinite. The procedure to bepursued to build a well-tempered scale is the follow-ing: the octave C-C, equal to 1200 cent, is dividedinto 12 intervals, obtaining the principal seven notesand the five accidental notes, that can be conse-quently represented in a cyclic path. In the equallytempered scale all semitone intervals are exactlythe same ratio, irrespective of frequency location.This implies that all whole-tone intervals are like-wise equal. In order to accomplish this the octaveis divided into twelve equal semitone intervals.

At that time several musicians refused thetempered scales, reproving them for the excessivemechanistic features and affirming that each formof temperament did not respect harmony. JohannSebastian Bach, instead, was of the opposite advice:he believed that the possibility to pass from atonality to another one was an essential feature ofthe vitality of music and that symmetry and reg-ularity in the notes sequence were not deplorable,but a warranty of order and transparency. He favor-ably accepted the tempered scales, that allowed touse all the tonalities. To confirm that, he composedthe first and the second books of Well TemperedClavier, a well-known collection of 48 preludes andfugues, in which each piece is executed in all thetonalities of the tempered scale, both in major andminor modes.

Bach was actually very sensible to the mathe-matical structure of music, and some of his com-positions, such as Goldberg variations, MusicalOffering and Art of Fugue, make use in a system-atic way of geometric transformations that invert,reverse and expand musical themes. Western musi-cians in the last few hundred years made manyadjustments to these scales to satisfy three mainrequirements:

1. True intonation — that is exact thirds, fourths,fifths, etc.

2. Complete freedom of modulation — namely fromany one key to any other.

3. Convenience in the practical use of keyboardinstruments — that is, to make it possible forinstruments which have been tuned differentlyto play together without retuning.

The same transformations, fundamental for poly-phonic compositions, have been explicitly formu-lated at the beginning of the XX century, as rules fordodecaphonic music. The presence of symmetriesinto the artistic production generally contributes tothe aesthetic value of the composition or, rather, itis an essential element of it. Complex musical sym-metries are present in the classical compositions,especially during the Renaissance and the Baroque.

The studies of Galileo Galilei on the pendu-lum’s oscillations were of fundamental importancefor the development of modern musical science, aswell as the consequent observations on the relation-ship between the number of vibrations, the musicalintervals and the importance of the parameters(length, diameter, density and tension) of an


oscillating string. Other well known scientists of theXVII century gave their contribution to the scien-tific foundations of musical research.

Marin Mersenne, a French mathematician andthe closest friend of Descartes, considered theinitiator of modern Acoustics, proved that thefrequency of a stiff oscillating string is inverselyproportional to its length and to the square rootof its linear density; and it is directly propor-tional to the square root of its tension. His mostfamous book on music and musical instruments wasHarmonie Universelle. Afterwards Descartes inCompendium Musicae dealt with consonance,whose scientific explanation is arranged throughArithmetic, the same as Zarlino, even if Descartesdivides the numerical quantities of Zarlino intosegments. Starting from a fixed length segment,that from a musical point of view represents thegravest sound, and subdividing it into equal parts(as in Pythagoras’ monochord), all the highest con-sonances of the original sound are obtained. In addi-tion, Isaac Newton in his Philosophiae naturalisPrincipia Mathematica, besides expressing the prin-ciples of Rational Mechanics and the Universal Lawof Gravitation, reformed the procedures in musicalacoustics; affirming that all the surrounding worldis held by laws that can be subject to mathemati-cal calculations and empirically known, he declared:“God created everything by number, weight andmeasure”.

Furthermore we have to mention the fundamen-tal contributions by D’Alembert and Fourier. Thefirst one discovered the general solution of the waveequation, while the second one developed the theoryof harmonic analysis. Fourier’s series, in particular,were of fundamental importance in the FrequencyModulation (FM) synthesis, through the so-calledBessel functions.

Fourier invented a type of mathematical analy-sis by which it can be proved that any periodic wavecan be represented as a sum of sine waves havingappropriate amplitude, frequency and phase. Fur-thermore, for harmonic spectra, the frequencies ofthe component waves are integer multiples of a sin-gle frequency, f0, 2f0, 3f0, and so on.

A square wave requires the sum of an infinitenumber of sine components whose frequencies areodd integer multiples of the fundamental fre-quency (f0, 3f0, 5f0, 7f0, . . . ) and whose ampli-tudes decrease in proportion to the inverse ofthe harmonic number (1, 1 : 3, 1 : 5, 1 : 7, . . . )and the proper phases. A Fourier representation

of a complex wave of finite duration (as musicalsounds are) requires an infinite number of differentharmonics.

Many “Noisy” sounds such as “sh” or “s”sounds of speech can be represented as the sumof sine and cosine waves (a Fourier integral) thathave slightly different frequencies. When the soundis repeated the waveforms will not be exactly thesame. Nevertheless the two different “sh” soundswill sound the same; we will hear them as beingidentical.

At the end of the sixties of the XX century,Chowning and Bristow [1986] proposed just theFrequency Modulation (FM) synthesis as an alter-native to be preferred to the additive synthesis. So,we arrive at the last century, remembering Xenakis[1971] as one of the avant-garde interpreters, amusician whose art is bound to scientific obser-vation of the world and to its relationship withmathematics. Xenakis worked to the binomialmusic-mathematics experimenting the new perspec-tives offered by Applied Mathematics. He wasamong the first ones to use computer and computerscience for musical composition.

A great contribution to the modern acousticsarea was by Pierce [1983], Mathews and Pierce[1989] and Risset [1969], scientists and musicians,pioneers in the electronic sound processing field.

2.2. Physical and psychologicalaspects of sounds and music

2.2.1. Physics of music

In this section we introduce some basic conceptsof the physical aspect of sound. Our life is fullof sound and noise. The ear has the function tocollect this sound and a very considerable num-ber of neurons are devoted to the processing ofacoustic stimuli. It would seem that the proper-ties of the ear determined all the peculiarities ofhearing, or that every such peculiarity had a phys-ical basis in the structure and functioning of theear. In fact, the ear is very important, since itis the intermediate between acoustic disturbancesand the brain, but the ear only provides data, themind interprets. Now, what is sound and how is itproduced? Generally, a vibrating object produces adisturbance in the air, related to the difference ofpressure. This disturbance arrives at the ear andis experienced as sound. So the aural sensation isrealized by means of wave motion arriving from


distant points, giving us information of things notimmediately in contact with us. The waves thatbring this disturbance are called acoustic waves.The physical stimulus of a sinusoidal acoustic wave,or simple tone, can be specified by two parameters,the frequency f in Hz and the overpressure ampli-tude p in dyne/cm2 or µbar. This stimulus playsan important role in hearing. Ohm’s Law, statedby Ohm in 1843, says that the audible sensation ofa simple tone cannot be analyzed further, and theear analyzes any aural sensation into simple tones.Hermann von Helmholtz proved this phenomenonvery well. For a simple tone of any frequency f ,sensation begins when the pressure reaches alevel called the threshold of audibility. As pressureincreases, the sensation becomes louder and louderuntil it becomes unpleasant at a level called thethreshold of feeling.

It is difficult to determine the curves close to thefrequency limits of audibility of any human subject,even though the curve that represents the lower andthe upper threshold shows a general tendency. Somescholars say there is aural sensation from as low as16 Hz to as high as 28 000 Hz, but the limits are verydifficult to determine since a great variability inhuman subjects does exist. The lower limit, in par-ticular, is subject to effects resulting from the largeamplitude of vibration and possible nonlinearities.The threshold of hearing rises at higher frequen-cies for older people, significantly above 8000 Hz.However a much smaller bandwidth of 100 Hz to5000 Hz includes most speech and music and fewears would notice restriction to this bandwidth. Oursenses handle such enormous dynamic ranges bylogarithmic response. When we speak about theheight of a sound generally we refer to its frequency.Instead, the pressure of the sound is related tothe amount of wave energy propagating in the unittime. There are three parameters which character-ize the basic features of any periodic motion:

1. The period T , is the time required for onecomplete cycle.

2. The frequency f , is the number of cyclesoccurring in a given time period.

3. The Amplitude A.

Most sound generators or vibrating bodies producerecurrent waves which are generally similar to eachother. There is a difference between frequency andpitch. Frequency is a measure of a certain propertyof an acoustic disturbance, while pitch is relatedto what the brain perceives of this phenomenon.

The definition of frequency, the concept of timehas a fundamental importance. The wavelength ofa sound wave is the distance that the sound coversthrough to complete one cycle. Noise is the resultof irregular vibration patterns. Tones are the resultof regular vibrations.

Pitch refers to how high or low a note sounds.High pitches are on the right side of the pianokeyboard and low pitches are on the left side ofthe piano keyboard. Also the human singing voicehas pitch ranges, with different qualities for maleand female subject. For example, Soprano is a highwoman’s voice. Alto is a low woman’s voice. Tenoris a high man’s voice. Bass is a low man’s voice. Thenote A above middle C has a frequency of 440 Hzand it is considered the pitch standard. The musi-cal alphabet consists of the letters A, B, C, D, E, F,and G. These letters represent musical pitches andcorrespond to the white keys of the piano. As onemoves forward through the musical alphabet thepitch of each note gets higher. The musical alpha-bet is repeated cyclically on the 88 keys of the pianokeyboard.

Music involves such highly complex sounds thatit is very difficult to analyze what kind of phe-nomenon occurs from the point of view of psychol-ogy of perception.

Why do some tonal combinations sound goodand some do not? If we combine a tone of 500 Hertzat a comfortable listening level, with a second tone,an identical 500 Hertz of the same amplitude, wenotice that there is an increase in loudness. Thishappens because the two sine waves have a timerelationship described as being “in phase”. Sincethe two tones are in phase, their resulting combina-tion has twice the amplitude of either single tone,when heard alone. The same happens adding twowaves of the same frequency, amplitude and phasewhich will produce an increase in the signal level.What happens if the same two 500 Hertz sine wavesare combined “out of phase” or in “phase opposi-tion”, that is, when one waveform goes positive, theother goes negative and vice versa? When the twosignals are in “phase opposition”, one cancels theother and the resulting output is zero.

Instead, when two tones of different frequen-cies are combined, some interesting facts happenthat have a direct influence on our sense of musicalsounds. The result may be considered as eitherpleasant or unpleasant. If we keep two tones, onlydiffering for 1Hz, they alternately combine in phaseand phase opposition to produce a 1 Hz beat. By


holding the first tone constant and changing thefrequency of the second tone, the beat frequencycan be varied in a regular way. The frequency ofthe beats is determined by the difference betweenthe frequencies of the two tones which are heardtogether. Thus, if a tone of either 490 Hertz or510 Hertz is combined with the 500 Hertztone, a beat of 10 Hertz is produced. As thedifference between the two tones is increased toabout 20 Hertz, the ear becomes unable to dis-cern the individual beats. Beyond 20 Hertz, a roughsound is heard. This phenomenon is the basis ofwhat we consider to be unpleasant musical effects.

Another way of describing these sensationsis by using the terms consonant or dissonant. Ina psycho-acoustic context, the term “consonance”means tonal, or sensory consonance, referring tothe human perception. This is distinguished fromthe way musicians use the word, a meaning whichis ultimately defined by culture and is dependenton Helmholtz’s frequency ratios and Music Theory.Helmholtz, in his fundamental 19th century workon tone perception, recognized that the true causeof consonance and dissonance was conflict betweenovertones, not fundamentals; tones with fundamen-tal frequencies in a simple harmonic ratio sharemany overtones. Helmholtz viewed the sharing ofovertones as the origin of the pleasing sound ofconsonant intervals. The work of Terhardt [1974]illustrated the important distinction between musi-cal consonance and psychoacoustic consonance.

Summarizing, if we combine two tones we canobtain the following results:

• If their frequencies are separated by a criticalbandwidth, or more, the effect is consonant.

• If their frequencies are separated by less than thecritical bandwidth, varying degrees of dissonanceare heard.

The most dissonant/least consonant spacing of twotones is about 1/4 of a critical bandwidth.

When two areas on the basilar membrane whichare close to each other are stimulated simultane-ously, an interference or roughness is heard, and wecall it tonal roughness, or sensory dissonance.

As we have said, while the term tonal con-sonance is used in reference to the sense of“smoothness” or “pleasantness” that results whentwo sounds with certain properties are playedtogether, the term tonal fusion refers to the senseof two sounds “merging” into a single sound in a

musical context. This concept is very important inthe theory of orchestration in music. The Germanword “Verschmelzung” (which literally translatedmeans “melting”) denotes the phenomenon of inter-mingled character of fused sounds.

One of the primary ways in which bothmusicians and nonmusicians understand music isthrough the perception of musical structure. Thisterm refers to the understanding received by a lis-tener that a piece of music is not static, but evolvesand changes over time. Perception of musical struc-ture is deeply interwoven with memory for musicand music understanding at the highest levels, yet itis not clear what features are used to convey struc-ture in the acoustic signal or what representationsare used to maintain it mentally.

Our ears are only partially sensitive to the over-all amount of acoustic energy which reaches them.They are however sensitive to the rate at which theenergy arrives. This rate is what determines loud-ness. The intensity of a sound wave is the powertransmitted through an area of 1m2 oriented per-pendicular to the propagation direction of the wave.If we move away from a constant sound source,we perceive a decrease in loudness. Thus, the sen-sation of loudness is determined by the intensity.The greater the intensity, the greater is the per-ceived loudness [Cook, 2001; Cope & Hofstadter,2001].

The rate at which an instrument radiatesacoustic energy is the acoustic power output of theinstrument. Our ears are very sensitive to the per-centage at which energy arrives at the eardrum (avery important part of our internal ear). Thus earsare sensitive to acoustic power. The sensation ofloudness is realized by the amplitude of the eardrumoscillations and this amplitude is directly relatedto the average pressure variation of the incomingsound wave.

Timbre is a musical word which refers to thespectral characteristics of sounds. Usually describedas color, brightness, rugosity, etc., the spectral char-acteristics of a heard sound can give us impor-tant clues of how far away a sound source is, thetype of environment in which the sound sourceis produced and through which the sound reachesus. A wave can assume different forms and thecharacters of the sound that are related to thesewaveforms are called timbres. Sounds with thesame frequencies and the same intensity or pres-sure can be perceived as different entities if theyhave different waveforms, as their timbres change.


A pure tone is a sound produced by a sinusoidalwaveform. Given a basic frequency, called funda-mental harmonic, other sounds with a predefinedheight can be produced, which are called harmon-ics. The harmonics are the following:

• II harmonic, settled one octave over the funda-mental;

• III harmonic, settled on the fifth note over theII harmonic;

• IV harmonic, settled on the fourth note over theIII harmonic;

• V harmonic, settled on the third note over theIV harmonic;

and so on. Each musical instrument has a particu-lar waveform, which results from the combinationof the fundamental harmonic and the higher har-monics. The ultimate waveform, which comes outof this process, determines the timbre of the instru-ment [Rissett & Wessel, 1982]. By means of Fourieranalysis it is possible to extract all the harmonicsof a waveform. Vice versa by means of the synthesisit is possible to create predefined waveforms, start-ing from some particular frequencies. The SignalTheory deals with these problems [Roads, 1996;Luise & Vitetta, 1999].

Summarizing, it is possible to say that the phys-ical features of sound are:

• Loudness, related to the energy of the acousticwave which propagates in the air and is sensedby the ear as volume;

• Pitch, which is the frequency of the acoustic wave;• Timbre, which is the feature that is related to the

waveform. Each musical instrument has a timbre;• Length, that is the time interval in which the

sound is emitted.

2.2.2. Music psychology basics

There is a vast literature on sound and music psy-chology, reporting many investigations that havebeen carried out. Briefly, we want to give some basicinformation on the major topics of these relatedfields.

Bregman [1990] has developed a theory accord-ing to which Gestalt laws is heuristics that weemploy in making sense of our auditory envi-ronment. Starting from the work realized byMarr [1982] on the Visual Scene Analysis (VSA),Bregman refers to the processes whereby we make

sense of the world of sound as Auditory Scene Anal-ysis (ASA). ASA processes operate on sound sig-nals, employing principles that enable the makingof valid inferences about the existence and the char-acter of the sources of sounds in the real world.The goal of this field is to understand the way theauditory system and brain process work in com-plex sound environments, where multiple sourceswhich change independently over time are present.Two subsectors are dominant in the ASA approach:auditory grouping theory, which attempts to explainhow multiple simultaneous sounds are partitionedto form multiple “auditory images”; and auditorystreaming theory, which tries to explain how mul-tiple sequential sounds are associated over timeinto individual coherent entities. In the generally-accepted Bregman model, the sound organizationsystem groups primitive components of sound intosources, and sources into streams. These groupingprocesses utilize rules such as “good continuation”,“common fate”, and “old plus new” to decide whatcomponents belong together in time and frequency.In many ways, Bregman’s articulation of perceptualgrouping cues can be seen as a formalization andprincipled evaluation of the scientific ideas proposedby the school of Gestalt Psychology in the earlypart of the last century [Kohler, 1925]. The work ofBregman and colleagues influenced the ComputerScience community, especially for the applicationof this theory in the field of Linguistics analysisand segmentation, and there is a vast amount ofpapers on this topic. Brown and Cooke [1994], start-ing from the ASA theory, called the discipline ofconstructing computer models to perform auditorysource segregation, computational auditory sceneanalysis (CASA). McAdams [1984, 1989] showedthrough psychoacoustic experiments that frequencymodulation applied to one source from a mixtureof synthetic vowels makes it “pop out” perceptu-ally. Also, complex tones in which all partials aremodulated coherently are perceived as more “fused”than tones in which partials are modulated inco-herently. McAdams subsumes these results into ageneral interpretation and model of the formationof auditory images. Recent work in psychologicalauditory scene analysis has led to a reexaminationof the “classical” view of perceptual properties ofsounds, such as loudness, pitch and timbre, in anattempt to understand how such sound qualitiesare influenced by the perceptual grouping context.Misdariis et al. [1998] found evidence that loudnessis not a property of an overall sound, but rather


of each sound object in a perceived scene, and fur-ther, that fundamental auditory organization (thedivision of the scene into streams or objects) mustprecede the determination of loudness.

One of the central topics of research into thePsychology of Music has regarded the use of pitch inmusic. This includes the way multiple notes grouphorizontally into melodies, vertically into chords,and in both directions into larger-scale structuressuch as “harmonies” and “keys”. These latter con-cepts are crucial in the theory of Western music; thepreponderance of formal Music Theory deals withthe formation of notes into melodies and harmonicstructures, and harmonic structures into areas of“key” or “tonality”. A great deal of works have beendone in this area from the early Shepard’s helicalmodel of pitch [1964] to the study on pitch rela-tionships [Krumhansl, 1979]. It has been demon-strated that the tonal hierarchy characterizing therelatedness of pitch and harmonic context turnsout to be a very stable percept in human subjects.Under a variety of context stimuli, including chords,scales, triads, harmonic sequences, and even indi-vidual notes, very similar response functions for thetonal hierarchy can be measured. A very extensivebook by Krumhansl [1990] summarized her work onthis topic.

More recent studies in these directions explorethe relevance of the rhythmic relations among notesin the formation of a sense of key [Schmuckler &Boltz, 1994; Bigand et al., 1996], the relationshipsbetween melody, harmony, and key [Povel & vanEgmond, 1993; Thompson, 1993; Holleran et al.,1995] and the development of these processes ininfants. Narmour [1990], used a different approachto melody perception developing what he calls an“implication-realization” model of melodic under-standing. Narmour’s model is based on Gestalt the-ories of psychology. He proposed several rules thatdescribe what listeners prefer to hear in melodies,based on the principles of good continuation,closure, and return-to-origin. He claimed that theserules represent universal guidelines for note-to-note transitions in melodic motion and that assuch, they apply to atonal and non-Western musicas well as to Western tonal music. From therules, he developed an extensive symbolic-analysismodel of melody and proposed several experimentsto analyze its predictions. All these works haveattracted computer scientists who realized ArtificialIntelligence (AI) models of music perception.Longuet-Higgins [1994] realized a number of models

of musical melody and rhythm around phrase-structure grammars. Steedman [1994] worked onsimilar ideas and on the use of various distance met-rics in “tonality space” to describe the relationshipbetween pitch, key, chord and tonality.

2.3. Acoustical and musical spaces

Let us introduce the concept of a three-dimensionalacoustical space, whose axes are Intensity, Durationand Frequency. An acoustical sequence is a set ofpoints in this space (Fig. 3).

In this representation timbre is not present,as timbre is related to the presence of higherharmonics according to the fundamental. If weconsidered the timbre, the acoustical space shouldcontain infinite dimensions. Generally, the way toovercome this difficulty is to consider a set ofpredefined timbres, labeled by a number, addingonly a further dimension to the acoustical space.This dimension assumes usually discrete values. Letus introduce the musical space, starting from theacoustical space. Musical notes are obtained choos-ing sounds of predefined frequencies. They can begenerated starting from the fundamental A, labeledby A4, which is used as standard international note:its frequency is today fixed at 440 Hz. The othervalues of A, belonging to the other octaves, canbe obtained by dividing or multiplying the frequen-cies (Fig. 4). The octave intervals (distance betweena note and another note) grow exponentially. So,using the logarithmic scale, it is possible to real-ize neighborhoods and equal distances among theintervals, going from one octave to the immediatlysubsequent one.

As we have said in the previous paragraphs,in the past, several attempts have been made inorder to divide the interval of the octave in moreparts, obtaining therefore other notes and creatingmusical scales; besides the classic contributions ofPythagoras and Euclid, a fundamental contribu-tion was supplied by Bach, who created the tem-pered musical scale by dividing the octave intervalinto 12 equal parts in the logarithmic scale, andby approximating the notes to the nearest ones.Therefore, making reference to the sound spacepreviously introduced, the musical space presentsa discretization on the axis of the frequencies and,operating with a logarithmic scale, the step of thediscretization is constant. In the musical notes, afurther discretization is carried out on the durationof the notes, that will now assume the values 4/4,


Fig. 3. Representation of an acoustical sequence in an ideal acoustical space. Here timbre and other acoustical features arenot considered.

Fig. 4. In this image, the frequencies of the note A, considering different octaves, are represented.

2/4, 1/4, 1/8, 1/16, 1/32, 1/64. Maintaining theintensity constant, the musical space can be thenrepresented as a two-dimensional discrete space,where we have inserted on one axis the durationof the note and on the other axis the value of thesame note. As it happens for example in the MIDIcode [Roads, 1996], we can associate to each musi-cal instrument a number between 1 and 128 andtherefore, our musical space is transformed into adiscrete three-dimensional space. Then, a note willcorrespond to a point of such a space, from nowon denoted by M , and it will be characterized bya tuple where the first number represents the pitchof the note, the second number its length and thethird one the instrument with which the note mustbe executed:

note = (pitch, length, instrument);

a melody then becomes:

f : N −→ N3

where f is an application that associates to asequence of natural numbers a tuple of naturalnumbers, which characterize, in an appropriate

coding, the musical note. Therefore, a melody Mecan also be characterized by succession of M ele-ments:

Me = m1,m2, . . . ,mi, . . . ,mn/mi ∈ MSummarizing, we can assert that:

• Musical spaces can be considered as three-dimensional spaces, analogous to the soundspaces, but in this case the values of all the vari-ables will be discrete.

• These spaces are also bounded.• The MIDI code codifies exactly, in binary format,

the values (pitches) of notes, the length and thetimbre.

Once we have rigorously characterized the soundor musical space and the corresponding sound ormelodic sequences, we can operate some transfor-mations on them. For instance, if we consider amelody Me, we can construct a new melody oper-ating a translation on the pitch of the note, forexample. This will be equivalent to play the melodyon a different melodic scale. The construction ofcanons is based on this process. The most frequent


transformations that can be made on the musicalsequences are the following [Scimemi, 2001]:

(a) translations (movements of the melodies inpitch or in time);

(b) reflections (both in pitch and in time);(c) half turns (punctual symmetries).

To this point we can already establish some cor-respondences between mathematical structures andmusical structures in a quite natural way. For exam-ple we can take into account some curves in theplane or the space characterized by a parame-ter λ: considering the values λ at discrete inter-vals, a sonification process can make the pointsP1, P2, . . . , Pn of the curve characterized by thesubsequent λ values to correspond to a successionof points in the sound space, so creating a corre-sponding sound sequence.

In the subsequent sections we will use an anal-ogous process for the sonification of trajectoriesin the phase spaces of Chua’s attractors. Musicalsequences can be obtained in the same way.

2.4. Current trends in ComputerMusic

The main idea that lies beyond the algorithmic com-position is the application of a rigid, well-definedalgorithm to construct the process of composingmusic. This method has been criticized, as musicproduced by algorithmic composition is consideredof lower quality, because it belongs to the designerof the algorithm and not to the user of the sys-tem in which the algorithm runs. Composing musicin a traditional way is of course a different pro-cess from algorithmic composition. Since humancomposers have some capacities such as creativityand emotion, they can break the rules while algo-rithms are fixed procedures. In order to break therules it is necessary to device creative solutions thatmight not otherwise have been chosen. The maindifference is that composers can exhibit much moreadaptability and creativity. However there are somereasons why well-defined structures are used to cre-ate music, art, sculpture, dance, poetry. Notwith-standing computers can help musicians in findingnew creative solutions to realize new approaches inthe musical composition domain. Currently, com-puters are programmed to perform a variety of tasksthat were unimaginable only few years ago, andthey are used in ever more differing domains. Theuse of computers allows us to perform tasks that

have never been considered before; as defining andprogramming new function, according to the user’srequirements as well as creating for example vir-tual worlds and music, that lead to new fields asComputer Music. The explorations into ComputerMusic can be divided into two broad categories:what information goes into a computer and how it ismanaged (input), what information comes out of acomputer and how it is generated (output). Thesetwo categories are closely correlated and interde-pendent, and this corresponds to musical cognitionand musical composition. The attempts to modelmusical knowledge through Artificial Intelligenceare usual approaches to the increasing of our knowl-edge of human Psychology and mental processes.At the moment implemented cognition at computerlevel (computer cognition of music) faces four seri-ous problems. First, how music has to be measuredin order to provide information to the computerizedsystem. Second, how the information has to be pre-sented to the computer. Third, how it has to be rep-resented in the computerized program, so that theprogram can understand its meaning. Fourth, whatthe computer has to do with such knowledge. Thepractical problem of measuring music is not easy.It involves making basic decisions from the begin-ning on what is important in the musical domain.What are we trying to measure? Our society hasculturally established notions about what is impor-tant in musical cognition, without knowing why wehold such ideas, or why different cultures have com-pletely different ideas on the subject. For example,it can be said, that western notation of music andthe theory of music itself, tell us what is impor-tant in music is that we have to understand it asa set of simultaneous parametric dimensions (manyof which are measured in fixed and discrete uni-ties): height, duration, intensity, timbre, etc. Notonly does the method of measuring the parame-ters depend on culture, but also the idea of phys-ical separation into parameters is peculiar in thewestern tradition. On the other hand theorists ofGestalt Psychology maintain that at a perceptuallevel we perceive a melody, which is not the sumof individual notes, but a more complex structure,a model. When this model is realized in ComputerMusic, it necessarily is a parametric model of musi-cal perception. The choice of these parameters isalways based on culture, on musical style and evenon the programmer’s preferences. Once it has beendecided what to measure, it is necessary to dealwith the problem of how to measure it. In general,


it would be convenient if the measurement of theinput and its parametric description had the maxi-mum detail allowed by the system of representation,on the assumption that the program will deducethe necessary information algorithmically. Histori-cally there have been three approaches to algorith-mic composition with computers:

• algorithms for the synthesis of sound;• algorithms for the construction of structures for

musical composition;• algorithms that correlate the synthesis of sound

with the realization of musical structures.

The algorithms that generate sound synthesis havebeen used in different ways by musicians, fromthe calculation of waveforms to the evolution ofthe timbre over time. Several algorithmic programsfor composition specify only pieces of informa-tion on the beats of the musical composition, suchas height, duration and other dynamic qualities.In other words, in some contexts the composerworks with computerized musical programs to gen-erate raw musical pieces. He could also be inter-ested in building acoustic instruments for the soundsynthesis. As we have already said, different pro-cesses of algorithmic musical production have beendeveloped over time:

1. stochastic [Xenakis, 1971];2. chaotic and fractal [Pressing, 1988; Bidlack,

1992; Rodet & Vergez, 1999; Voss & Clarke,1978];

3. rule based [Schottstaedt, 1989];4. based on grammars [Roads, 1996];5. based on methods deduced from Artificial

Intelligence [Todd & Werner, 1999; Bentley &Corne, 2002; Bilotta et al., 2002];

6. based on dynamical systems [Gogins, 1991;Witten, 1996];

7. based on cellular automata [Bilotta et al.,2000; Bilotta & Pantano, 2001a; Miranda, 2001;Bilotta & Pantano, 2002a; Bilotta & Pantano,2002b; Bilotta et al., 2004].

When we use the sound synthesis to explore char-acteristics of dynamical systems, a correspondencebetween the variables of the dynamical systemand physical characteristics of the sound (fre-quency, amplitude, timbre, etc. [Frova, 1999]) hasto be established. Using the musification trianglemetaphor [Bilotta & Pantano 2001a, 2002a] (Fig. 1),we have developed music from dynamical systems.

At one vertex of this triangle might be the resultsof the numerical integration, which would be con-nected to musification processes, which, in turn,would be the origin of the real musical pieces. Fol-lowing this method, we produced a series of codes.We analyzed the musical pieces produced from aperceptual point of view; for example, in the musi-fication of Chua’s attractors, the passage from highto intermediate to low octaves can be heard, thenagain from low to intermediate to high octaves andso on. The perceptual effect is dynamic and variableand it is possible to clearly distinguish the courseof the curves.

Results obtained with human subjects demon-strates the possibility of recognizing the typicaltendency of the curves that dynamical systemsproduce in the space of phases. The auditory per-ception process let the subject create a “men-tal/sonorous image” of the behavior of the system.Indeed the basic elements of dynamical systembehavior (focuses, sources, etc.), translated intomusic, are always recognized by the subjects, even ifthe data are disguised as a modified auditory scene.Therefore, besides generating a specific melodicline, music produced by dynamical systems canalso be identified as a sonorous object within morecomplex musical contexts. Furthermore we noticedthat the musical notation allows us to analyze thedynamics that occur over time, particularly forchaotic systems. Musifying different trajectories, itcan be noted how they follow different tendencies,producing organized musical movements and createdifferent melodies. Such melodic lines, consonant ornonconsonant, provide a complex musical scene, inwhich it is possible, from time to time, to focuson one line, leaving the others in the background[Bregman, 1990]. In such compositions, other typesof auditory effects are traceable, linked to the lawsof gestaltic perception, such as similarity, differ-ences, common destiny, etc. [Wertheimer, 1958].This allows us to assume that organizational laws,similar to those of the physical stimuli that we per-ceive in the world around us, also exist in dynamicalsystems.

2.5. Languages and codes

The idea to produce music from dynamical systems,in fact, is not new. Witten [1996] translated intomusic a two-dimensional dynamic process creatingcorrespondences between the points of the dynamicsystem in a certain instant and some musical


attributes (frequency, amplitude, timbre, envelope).Experimentations have been realized also withother systems. For example, the circuit of Chuaoscillates with audible frequencies, rather near tothose of typical musical notes, while for deter-mined values of the control parameters it has somesonorities which are typical of the chaotic attrac-tors [Rodet & Vergez, 1999]. The above cited cor-respondence between the variables of the dynamicsystem and some physical characteristics, are notjust those related to intensity and frequency, butalso related to the production of higher harmonics,with the possibility of modifying the timbre, the fre-quency modulation, etc. This correspondence allowsus to transform the dynamic variables into sounds,which, in turn, can be analyzed in order to inquirethe behavior of the considered dynamical system.

This procedure is applicable to whicheverdynamical system, by creating a correspondencebetween musical notes and the intervals of thevalues obtained by the variables of the dynamic sys-tem. In this way the sound flow has been trans-formed into a structured sonorous flow (music).This process allows us of:

1. producing artificial sounds of chosen frequency;2. overlapping sounds;3. modifying sounds;4. producing continuous sound flows, supplying a

basic sound and a generative algorithm.

But this does not mean realizing acceptable con-structions from the musical point of view. Gener-ally we refer to these compositions as coarse musicalbases, which are normally modified by the artists,in order to obtain refined musical pieces. A way toface this problem is to consider the musical prod-uct like a language and to apply to it the approachindicated by Morris [1971] for the examination oflanguages. According to this scholar, a system ofsigns is a language if it respects the rules of thesyntax, of the semantics and of the pragmatics.

The musical syntax deals with rules allowingto arrange the signs into a composition. Every cul-ture, every age has had, and has its own musicalsyntax. The rise and fall of musical styles are dueto their development during the time. As variousauthors assumed, the musical experience is a partic-ular organization of the musical time: that means areorganization of the sound in time. Imberty [1987]has shown, with examples from Brahms, Debussyand Berio, that music is a symbolic representation

of the human experience of time. It seems that thefocus on temporal organization in the musical syn-tax is a preferential way to operate, and that itexpresses itself in terms of speed and succession(rhythm), and in terms of tonal length (interval).

With the term interval we indicate the soundcontinuum into which it is allowed to choose theelements of the musical composition, while thesemantics is a sector of the Semiotics which stud-ies the relationships the signs have with the thingsthey refer to. To become a language it is necessarythat the signs that compose a system have a definitereference to objects or events of which they becomethe substitutes. Musical semantics is represented bythe emotions it refers to [Imberty, 2000].

The Pragmatics studies the relationshipsbetween the signs and the people using them; thisimplies that their interpretation is the knowledgeof their context. The musical behavior, supportedand programmed by a human subject, finds in therules of the syntax, semantics and pragmatics theelements which allow its measurability. The latterconcerns the communicative features of language.The understanding of a musical message relies onthe understanding of the interplay among the abovementioned components, but it is subordinated tothe condition that the exchange takes place betweenpeople accepting and using the same rules andparticipating to the same pragmatics. It followsthat the communicative feature of this language isrelated in the series of rules and to the conditionsthat are part of the musical system interpreted asa code.

According to Eco [1975], a code expresses fourdifferent phenomena:

(a) a series of signals connected by inner combina-torial laws;

(b) a series of states of the system;(c) a series of possible behavioral answers from a

receiver;(d) a rule that associates some elements of the

system (a) to elements of the system (b) or ofthe system (c).

Eco calls (a)–(c) the S-Code name (System-CODE)and he considers CODE the rule that associates theelements of S-Code when an association rule exists,as in the case (d). In our case, the codes we havedeveloped for translating into music Chua’s attrac-tors belong to the (d) category. In fact we usedthe configuration of Chua’s attractors which can be


interpreted by another system (Fig. 5). We define as“musification” the semiotic process of translating adynamical system like Chua’s oscillator into music,using a specific code. We define as sonification“a mapping of numerically represented relations inthe same domain under study to relations in anacoustic domain for the purposes of interpreting,understanding, or communicating relations in thesame domain under study” [Scaletti, 1994].

Using this semiotic approach for Chua’s attrac-tors, the process follows these steps:

1. The space configurations created by Chua’sattractors and the solutions drawn by a dynamicsystem in the plane of the phases representa continuum of physical-mathematical possibili-ties, used as discrete and pertinent elements, andas possible occurrences that can be translatedinto music.

2. From such a continuum, concrete occurrences ofsuch dynamic systems can be selected, as theunits to be used in the translation.

3. The units are represented by mathematicalstructures (points 2 and 3 represent the plane ofthe expression which articulates into units andsystems).

4. In the plane of the content three musical mean-ings are associated to the units, through differentcodes. Such a process will realize concrete units,which will be transformed into a physical contin-uum (Fig. 6).

In order to musify the trajectories of Chua’sattractors in the phases space, we have to associateto the continuous values of the several quantities ofthe dynamical system the discrete values related tothe musical parameters. The first step in this direc-tion consists in the determination of the interval ofvalues included between the minimum and the max-imum of the quantities. The second step foresees theassociation of a musical note when the value of aquantity is included in a certain interval.

In the realization of musical compositions gen-erated by Chua’s attractors we set the followinggoals:

1. recognition of the points of equilibrium of thesystem;

2. detection of the behavior of stable attractorsgives more solutions, starting from differentinitial conditions;

3. investigation of the nature of chaos, by observingthe critical behaviors of the system at differentparameters values;

4. investigation of how changing the parameters ofthe system changes musical compositions;

5. perceptive analysis of the compositions byhuman subjects.

A complete description of such problems for discreteand continuous dynamical systems can be foundin [Bilotta & Pantano, 2004] and [Bilotta et al.,2004].

Fig. 5. The elements of the coding system we have created in translating into music Chua’s attractor behavior.

Fig. 6. In this figure, a semiotic methodology for building up a code is reported. The semiotic matters we have considered are Chua’s attractors, which have concreteoccurrences when they are visualized as trajectories, and expressed by marked functions (the system’s parameters). These tokens are then detected in order to translatethem into musical marked occurrences, which in turn are transformed again in semiotic matter (musical meanings).

270


In the course of our research, we have translatedinto music some typical behaviors of a dynamic sys-tem. At the web site http://galileo.cincom.unical.itsection “Art and galleries”, several sound and musi-cal compositions, generated from dynamic systemstrajectories, are available. A musification modal-ity considers many trajectories of the system atthe same time. In this case, in addition to makingthe notes vary in time, to each variable a differ-ent instrument is assigned, so that it can be distin-guished from others, also from the point of view oftimbre. With this type of musification we were ableto point out the strong sensitivity to variations ofthe starting conditions, which is typical of chaoticsystems.

3. Chua’s Circuit

3.1. Complexity and strangeattractors in Chua’s circuit

Chua’s oscillator is a nonlinear circuit whosedynamical behavior is chaotic and which can giverise to a great family of strange attractors. Thissystem has been widely studied [Chua, 1993; Chuaet al., 1993a; Chua et al., 1993b], for its charac-teristics both of simplicity in the realization andease of use, since it is possible to visualize manytypical chaotic behaviors. There is an extensiveamount of literature devoted to Chua’s circuit andits applications [Madan, 1993]. As Chua reports inone of his papers [Chua, 1993], the first evidenceof the chaotic nature of this system was observedby Matsumoto [1984], using a computer simulation.Then other studies confirmed the presence of chaosin Chua’s circuit [Zhong & Ayrom, 1985a, 1985b],obtained experimental results [Matsumoto et al.,1985], visualized the global bifurcation landscape[Komuro et al., 1991], realized rigorous proofs ofthe presence of chaos [Chua et al., 1986]. Otherresearchers dealt with the complexity of Chua’scircuit and found that there are many qualitativecircuits equivalent to it [Chua & Lin, 1990]. Thename Chua’s oscillator was given by Madan [1992,1993], with the intention of distinguishing the globalunfolded circuit [Chua et al., 1993b] from the orig-inal Chua’s circuit. Many other papers aimed atconfirming Chua’s circuit as a universal paradigmfor studying chaos [Shil’nikov, 1993; Madan, 1992,1993], as this is “the first real physical system wherechaos is observed in laboratory, confirmed by com-puter simulation, proven mathematically by two

independent methods” [Matsumoto et al., 1988].Chua’s oscillator equations allows to define thebasic behavior of a dynamical system and to detectthe changes in the qualitative behavior of a systemwhen bifurcations occurs, as parameters are varied.

Let us briefly analyze the main componentsof Chua’s oscillator and its dynamical behavior.Chua’s oscillator is based on Chua’s circuit to whicha resistor has been added in series. This resistoris composed of five linear elements (two capacitorsC1 and C2, one inductor L and two resistors Rand R0) and a nonlinear resistor NR, called Chua’sdiode, which are organized in the pattern displayedin Fig. 7.

The resistor NR provides the nonlinearityand acts as energy source to drive the system.The unfolded Chua’s circuit state equations thatregulate the dynamical behavior of the system arethe following:

dv1

dt=

1C1

[G(v2 − v1) − f(v1)]

dv2

dt=

1C2

[G(v1 − v2) + i3]

di3dt

= − 1L

(v2 + R0i3)

(1)

where v1, v2, and i3 denote the voltage across C1,the voltage across C2 and the current through L,respectively;

G =1R

and

f(v1) = Gbv1 +12(Ga − Gb)|v1 + E| − |v1 − E|,

(2)

is the v − i characteristic of the nonlinear resistorNR with a slope equal to Ga in the inner region andGb in the outer region (Fig. 8).

As reported in many papers (see [Kennedy,1993a, 1993b] for an exhaustive tutorial on thistopic), for a fixed set of parameters, Chua’s oscil-lator equations identify a dynamical system, witha typical set of behaviors, which we can detail asequilibrium points, limit cycles or strange attrac-tors. In the following section, we demonstrateexperimentally many of the key features of Chua’soscillator.


Fig. 7. In this image, a representation of Chua’s oscillator is reported.

Fig. 8. Typical v − i characteristic of Chua’s diode.

3.2. Experimental observationson chaos

Chua’s oscillator is easy to implement and to usein many different ways. We have used the schemeintroduced by Arena et al. [1995], based on Cellu-lar Neural Networks (CNNs). Chua and Yang intro-duced the CNN paradigm in [1988]. The CNN is ananalog dynamic processor array with the propertyof processing elements interacting directly within

a finite local neighborhood [Chua & Roska, 1993;Chua, 1998], which can be obtained as VSLI chipand can operate at a very high speed and levelof complexity. Arena et al. [1995] demonstratedthat Chua’s oscillator can be obtained as a CNN,achieved by the proper connection of three general-ized CNN cells, as each single CNN cell “representsthe primitive for realizing high complex dynamics”.A PSpice representation of this model, with therelated component values, is displayed in Fig. 9.

Fig. 9. The PSpice scheme of Chua’s oscillator implemented by Arena et al. [1995], by means of Cellular Neural Networks (CNNs).

273


Using the potential differences v1 and v2, it ispossible to project on an oscilloscope their valuesover tie (Fig. 10). By varying the resistance value, itis possible to visualize in the plane v1v2 the variousbehaviors of the system. The observed behaviors arevisualized in Fig. 11.

In this way, modifying the resistance values, itis possible to investigate how the system goes froman ordered behavior to a chaotic one, passing from

a fixed point to a limit cycle [Fig. 11(a)], tomany stable limit cycles [Figs. 11(b) and 11(c)],to Chua’s spiral [Fig. 11(d)], to the strange attrac-tors called double scrolls [Figs. 11(e) and 11(f)]. Fora more detailed version of the routes to chaos see[Chua et al., 1993a]. These behaviors can be ana-lyzed also by numerical integration of the system(Fig. 12).

The behavior represented in Fig. 12 has beenobtained by the following parameters:

C1 = 5.75 nF C2 = 21.32 nF R0 = 30.86Ω E = 1V

L = 12mH Ga = −0.879mS Gb = −0.4124mS

Varying R: it is possible to pass again from afixed point (R = 1.9) [Fig. 12(a)] to a limit cycle(R = 1.59) [Fig. 12(b)], to many stable limit cycles(R = 1.575 and R = 1.57) [Figs. 12(c) and 12(d)],to the Chua’s spiral (R = 1.565) [Fig. 12(e)], to thestrange attractors called double scrolls (R = 1.44and R = 1.4399) [Figs. 12(f) and 12(g)].

We tried to send the signal directly to thespeakers. In this way it is possible to hear thesounds that present many interesting structures(Fig. 13). In the ordered regions (limit cycle behav-ior), sound presents very interesting structures andthe presence of a basic frequency, with many higher

harmonics, allows for the emergence of timbreswhich can be used for creating virtual instruments[Rodet, 1993; Rodet & Vergez, 1999]. As soon as thesystem goes through chaotic behavior, sound losesthese structures and it seems noise. In this experi-mental context, sonorous analysis allows to realizeobservations that can support visual observation ofthe behavior of chaotic systems.

3.3. A garden of attractors

If we introduce the following dimensionless vari-ables:

x v1

E, y v2

E, z i3

(R

E

),

α C2

C1, β R2C2

L, γ RR0C2

L,

a RGa, b RGb, τ t

|RC2| ,

k = 1, if RC2 > 0,

k = −1, if RC2 < 0,

(3)

Eq. (1) becomes:

dx

dτ= kα(y − x − f(x))

dy

dτ= k(x − y + z)

dz

dτ= k(−βy − γz)

(4)

where

f(x) = bx +12(a − b)|x + 1| − |x − 1|.

The configuration identified by Eq. (4) is adynamical system with three grades of freedom andsix control parameters: α, β, γ, a, b, k. Slight mod-ifications of these parameters identify many attrac-tors. A collection of these attractors is displayed inFig. 14.

Details of each attractor parameters will begiven in Table 1, while each attractor configura-tion will be presented in the following gallery (fromFigs. 15–33). These attractors have been producedwith 3D STUDIOMAX.

Fig. 10. In this image, Chua’s oscillator implemented by a CNN and projected on an oscilloscope.

275

Fig. 11. In this image, we collected by a camera the behaviors of Chua’s oscillator, projected on an oscilloscope.

276

Fig. 12. Simulation of the typical behaviors of Chua’s oscillator, integrated numerically. In the images, the parameter R has been varied as follows: in Fig. 12(a)R = 1.9; in Fig. 12(b) R = 1.59; in Fig. 12(c) R = 1.575; in Fig. 12(d) R = 1.57; in Fig. 12(e) R = 1.565; in Fig. 12(f) R = 1.4400258052624301990673093072878;in Fig. 12(g) R = 1.4399470099500336936134164233122.

277

Fig. 13. Spectrograms of the typical behaviors of Chua’s circuit.

278

Fig. 14. In this image, we collected all the 19 Chua’s attractors we have considered for this tutorial, with their related number, as the starting point of the gallery weare going to present in the following pages. In this way, the reader can exploit many configurations of the same system and can understand the complexity of Chua’sattractors.

279

Table 1. Parameter values of 19 attractors in Chua’s oscillator.

Attractor α β γ a b k

1 9.3515908493 14.7903198054 0.0160739649 −1.1384111956 −0.7224511209 1

2 −1.5590535687 0.0156453845 0.1574556102 −0.2438532907 −0.0425189943 −1

3 −4.898979 −3.624135 −0.0011808881 −2.501256 −0.9297201 1

4 −1.458906 −0.09307192 −0.3214346 1.218416 −0.5128436 −1

5 −6.69191 −1.52061 0 −1.142857 −0.7142857 16 −1.3184010525 0.0125741900 0.1328593070 −0.2241328978 −0.028110195 −1

7 −1.301814 −0.0136073 −0.02969968 0.1690817 −0.4767822 1

8 −1.3635256878 −0.0874054928 −0.3114345114 1.292150 −0.49717 −1

9 −1.2331692348 0.0072338195 0.0857850567 −0.1767031151 −0.0162669575 −1

10 8.4562218418 12.0732335925 0.0051631393 −0.7056296732 −1.1467573476 1

11 6.5792294673 10.8976626192 −0.0447440294 −1.1819730746 −0.6523354182 1

12 3.7091002664 24.0799705758 −0.8592556780 −2.7647222013 0.1805569489 1

13 −4.08685 −2 0 −1.142857 −0.7142857 1

14 15.6 28.58 0 −1.142857 −0.7142857 1

15 −75 31.25 −3.125 −2.4 −0.98 1

16 −75 31.25 −3.125 −0.98 −2.4 1

17 −1.7327033212 0.0421159445 0.2973436607 −0.0974632164 −0.2623276484 −1

18 −2.0073661199 0.0013265482 0.0164931244 −0.5112930674 0.0012702165 −1

19 −1.0837792952 0.0000969088 0.0073276247 −0.0941189549 0.0001899298 −1

280


Fig. 15. In this image, three views of attractor 1 are displayed. The parameters are the following: α = 9.3515908493,β = 14.7903198054, γ = 0.0160739649, a = −1.1384111956, b = −0.7224511209, k = 1.


Fig. 16. In this image, three views of attractor 2 are displayed. The parameters are the following: α = −1.5590535687,β = 0.0156453845, γ = 0.1574556102, a = −0.2438532907, b = −0.0425189943, k = −1.


Fig. 17. In this image, three views of attractor 3 are displayed. The parameters are the following: α = −4.898979,β = −3.624135, γ = −0.0011808881, a = −2.501256, b = −0.9297201, k = 1.


Fig. 18. In this image, three views of attractor 4 are displayed. The parameters are the following: α = −1.458906,β = −0.09307192, γ = −0.3214346, a = 1.218416, b = −0.5128436, k = −1.


Fig. 19. In this image, three views of attractor 5 are displayed. The parameters are the following: α = −6.69191, β = −1.52061,γ = 0, a = −1.142857, b = −0.7142857, k = 1.


Fig. 21. In this image, three views of attractor 7 are displayed. The parameters are the following: α = −1.301814,β = −0.0136073, γ = −0.02969968, a = 0.1690817, b = −0.4767822, k = 1.


Fig. 22. In this image, three views of attractor 8 are displayed. The parameters are the following: α = −1.3635256878,β = −0.0874054928, γ = −0.3114345114, a = 1.292150, b = −0.49717, k = −1.


Fig. 24. In this image, three views of attractor 10 are displayed. The parameters are the following: α = 8.4562218418,β = 12.0732335925, γ = 0.0051631393, a = −0.7056296732, b = −1.1467573476, k = 1.


Fig. 25. In this image, three views of attractor 11 are displayed. The parameters are the following: α = 6.5792294673,β = 10.8976626192, γ = −0.0447440294, a = −1.1819730746, b = −0.6523354182, k = 1.


Fig. 26. In this image, three views of attractor 12 are displayed. The parameters are the following: α = 3.7091002664,β = 24.0799705758, γ = −0.8592556780, a = −2.7647222013, b = 0.1805569489, k = 1.


Fig. 27. In this image, three views of attractor 13 are displayed. The parameters are the following: α = −4.08685, β = −2,γ = 0, a = −1.142857, b = −0.7142857, k = 1.


Fig. 28. In this image, three views of attractor 14 are displayed. The parameters are the following: α = 15.6, β = 28.58,γ = 0, a = −1.142857, b = −0.7142857, k = 1.


Fig. 29. In this image, three views of attractor 15 are displayed. The parameters are the following: α = −75, β = 31.25,γ = −3.125, a = −2.4, b = −0.98, k = 1.


Fig. 30. In this image, three views of attractor 16 are displayed. The parameters are the following: α = −75, β = 31.25,γ = −3.125, a = −0.98, b = −2.4, k = 1.


Fig. 32. In this image, three views of attractor 18 are displayed. The parameters are the following: α = −2.0073661199,β = 0.0013265482, γ = 0.0164931244, a = −0.5112930674, b = 0.0012702165, k = −1.


Fig. 33. In this image, three views of attractor 19 are displayed. The parameters are the following: α = −1.0837792952,β = 0.0000969088, γ = 0.0073276247, a = −0.0941189549, b = 0.0001899298, k = −1.


4. Codes and Software forTranslating Chua’s Attractorsinto Sound and Music

4.1. Sound and musical codes

The process of communication between humansubjects, animals or artificial machines is basedon conveying information that should be specificand comprehensible for each system. According toJakobson [1963], the main elements of a specificcommunication system are generally the sender, thereceiver, the communication channel, the context inwhich communication happens, the message and thecode in which each message is encoded. In order tounderstand the meaning of the message, the receivershares the same language and the same behavioralrules. The code is the language or the conventionalsigns system by which a sender expresses a mes-sage to be conveyed to the receiver. According toLewin [1936], the code is not only a mechanism thatallows communication but a process that allowsfor meaningful transformations between two sys-tems. This is the distinctive meaning of the codeswe have developed to produce music from Chua’soscillators. Generally the code is implemented in acomputational system or algorithm. What gener-ative algorithms are most appropriate in generat-ing music and how should they be used? In realityany repetitive rule or dynamic rule can be used inthis sense. Through initial parameters, these algo-rithms are capable of generating musical sequenceswhich then can be associated with musical physicalparameters. Consequently, with these same rules,varying the initial data, it is possible to generatenumerous differing yet related numerical sequences.The problem in this sector is tied to the abundancerather than to the lack of generative rules. Gen-erative music is based on two separate processes:an algorithm generating numerical sequences anda process of translating numerical sequences intomusic (which we have called musification or cod-ing process). Many different algorithms can be usedto generate numerical sequences; algorithms derivedfrom recursive functions give many examples. Thesealgorithms, starting from one or two initial val-ues produce a numerical sequence. There is a greatnumber of algorithms that can be used to generatesound or musical sequences. The process of musifi-cation is also crucial and significantly influences themusic produced. As mentioned in the above section,in Musical Acoustics some physical parameters existwhich regulate the characteristics of the sounds and

their composition. The process of musification cor-relates one or more physical parameters to variousmathematical functions. The process of musifica-tion or musical rendering will associate the values ofthe generated numeric sequence to the values of oneor more physical parameters, leading to an audibleevent. For example, we can generate an acousticalevent using the values of the Fibonacci sequence aspure sound frequencies. We will then find a seriesof sounds of growing pitch. The result of this pro-cess cannot be considered music in so much as thepitches corresponding to the sequence are not thoseof the notes of the scale and, besides, the sequencecan be objectively boring starting from very lowpitches and generating sounds of growing pitch. Itis obvious that the process of musical rendering iscompletely arbitrary and established in a conven-tional way, and the results depend substantially onthe choices made by musicians. In any case, trans-lating the physical parameters of a mathematicalsystem does not mean generating music. This is amore complex human expression, which has got agrammar and an aesthetic: melody, harmony, con-sonance, canons and fugues and rhythm, transform-ing a sequence of sounds into music. Unlike physicalfacts whose laws are unchangeable, musical expres-sion depends on rules that evolve during time andare strongly bound to their historical period; thismakes the generation of music a difficult and verycomplex process. The use of algorithmic and ofmusification processes allows us to create sequencesof sounds and notes, but only the use of a pro-cess similar to that of the evolution of the musicalsystem during time, through natural selection, willpermit us to select productive algorithms and musi-fication processes useful to produce music. Thecodes we have developed for translating Chua’sattractors into sound and music belong to two maincategories:

1. sonification codes;2. musical codes.

Let us analyze in detail these codes.

4.1.1. Sonification codes

The sounds are generated starting from the solu-tions x(t), y(t), z(t) of the system. The generatedsound is given from the sum of the values of thesolutions at step t. It is even possible to combinetwo solutions or to play only one of them.


The md(t) data that will be directly sent tothe sound card are expressed by the followingexpression:

md(t) = a1x(t) + a2y(t) + a3z(t) (5)

in which a1, a2, a3 are activation parameters thatassume value 1 or 0 if the respective component isor is not considered in the sonification process.

In this sonification process, the solutions ofChua’s oscillator are used to modulate a basic fre-quency. Indicating with SR the number of sam-ples per second that will be sent to the soundcard (SR = 11050 in the softwares described lateron), we can define a temporal parameter τ , asfollows:

τ =SR

16.

To each value x(t), y(t), z(t) of the path of a solu-tion of Chua’s oscillator we will associate a soundof duration τ . Indicating by ν the basic frequencythat will be modulated by the x(t), y(t), z(t) values,at each time t, the samples, whose total durationis τ , that will be sent to the sound card, are thefollowing:

md(t) = a1

τ∑i=1

Sin(πi/τ) Sin(2πνix(t)/SR)

+ a2

τ∑i=1

Sin(πi/τ) Sin(2πνiy(t)/SR)

+ a3

τ∑i=1

Sin(πi/τ) Sin(2πνiz(t)/SR)

in which a1, a2, a3 are, as in the previous exam-ple, the activation parameters. The first sinusoidalterm of the sum represents an envelope whose formis the positive part of a sinusoid that is used toseparate the different sounds of duration τ in thesonorous flow.

4.1.2. Musical codes

The musical codes associated to the solution x(t),y(t), z(t) of the system realize a predefined suc-cession of musical notes. Indicating with Sk a setcomposed by musical notes defined by the user:

Sk = note1,note2, . . . ,notekthe process of association occurs as follows:

1. for the given solution x(t), y(t), z(t), we find themaximum and minimum values, M and m;

2. we decompose the volume that contains the pathin h3 cubes of side l = (M − m)/h;

3. in this way three integer numbers (nx(t), ny(t),nz(t)) will correspond to each point of the path;

4. we will associate them to three sequences ofmusical notes:

seq(1) = notenx(t), t = 1, 2, . . . , r

seq(2) = noteny(t), t = 1, 2, . . . , r

seq(3) = notenz(t), t = 1, 2, . . . , r

in which r is the numerical integration time.

Figure 34 shows a schematic representation of thiscode to define the association between numericalintervals and musical notes in the case in which the

Fig. 34. Schematic representation we used for the associations between numeric intervals and musical notes. The intervalsare organized on five octaves of the piano keyboard.


set of notes Sk is composed of the notes of threeoctaves of the piano.

This musification code known as basic musi-fication code presents however some problems inits practical application. In particular the musicalsequences are very long, and, even if the duration ofthe notes is short, due to the complex form of theattractor, it is very hard to understand them dur-ing the listening process. This curve in fact needsseveral numerical integration steps. In addition,the resulting musical sequences are presented asscales. A way of overcoming this problem is select-ing only some points of the path to generate thesound sequence. This choice could be done in twoways:

• automatically selecting these points, for example:after a defined temporal interval;

• selecting the local maximum and minimum pointsof the curve;

• choosing the values the curve reaches in deter-mined points;

• selecting random points, according to particularvisual structures;

• using a Poincare map.

These approaches, less formal than the precedingones, result in better melodies.

4.1.3. Systems based on the attractor’sshape

As seen in the above sections, Chua’s oscillator cre-ates a great number of attractors that build up dif-ferent curves in three-dimensional space. From thesecurves we can obtain different curves on a plane.The musification codes we have elaborated are oftwo types: the first one is based on a color code,while the second one is based on a process of mul-tiresolution analysis, through a system of waveletsfunctions.

4.1.4. Color code

Given an attractor, we can obtain different curveson a plane. For example, apart from the projectionson the planes xy, xz, yz, we can obtain a curve ina determined projective plane or a Poincare map(Fig. 35). The musification process occurs along thefollowing steps.

1. The square containing the plane curve is dis-cretized by means of a grid of variable dimension(Fig. 36).

2. The points contained in each square are cal-culated. The maximum value Nmax (the min-imum value is 0), is determined, obtaining mintervals

Ir =[Nmax

mr,

Nmax

m(r + 1)

], r = 0, . . . ,m− 1.

We can associate to each interval an appropri-ate color, thus, each square will assume a deter-mined color, depending on the points it contains(Fig. 37).

3. After that, through an appropriate association,we can obtain a single note for each interval. Weshould point out that, since the space discretiza-tion is arbitrary, this process lets us understandthe attractor’s fractal nature, thus generatingfractal music.

4. The musification process (namely the passagefrom a matrix to a melody) is not unique. In thiscase we can refer to the local and global musifica-tion codes that have been developed for CellularAutomata [Bilotta & Pantano, 2001a].

4.1.5. Multiresolution analysis code

Two musification processes to generate music fromChua’s circuit have been created, using a waveletdecomposition algorithm. Wavelets are useful toolsin approximation, analysis, denoising both signaland image. The program, developed in Matlab,considers the two-dimensional images obtained byChua’s systems.

One of these images is saved in an appropri-ate format (.raw 128 × 128 pixel for Chuawave2.mand 256 × 256 pixel for Chuawave.m) and under-goes a process of multiresolution analysis, througha system of wavelets (Fig. 38). Using this tool, theimage, considered as a function f(x, y) ∈ L2(R2),can be processed using its projections in the appro-priate Hilbert’s subspaces. Once these subspaceshave been determined, it is possible to decomposethe initial image in a set of values that:

• determines an approximation of the sameimage;

• detects details of horizontal orientation;• detects details of vertical orientation;• detects details of oblique orientation.

Fig. 35. Plane curves obtained by Chua’s attractor number 1. In particular, in (a) the projection xy, (b) the projection xz, (c) the projection yz, (d) a 3D visualization,(e) a Poincare map are displayed.

303


Fig. 36. In this image, a grid which contains the visualization of the plane curve of Chua’s attractor is shown.

Fig. 37. In this image, Chua’s attractor shape, discretized by means of a color code is visualized.


Fig. 38. In this figure, an image of Chua’s attractor processed by the multiresolution analysis is displayed.

These numerical values are inserted into a matrix as shown in the following table:

Approximation coefficients Vertical details coefficient

Horizontal details coefficient Oblique details coefficient

Starting from this matrix it is possible to gener-ate music creating an appropriate correspondencebetween numerical codes and musical notes. Twomusification processes have been realized using twoMatlab codes: chuawave.m e chuawave2.m and acode that uses the extension .sta and is then trans-formed into .mid files. The program ChuaWave.mcreates four text files with .sta extension related to:

1. the vector obtained summing up all the ele-ments in the columns of the submatrix contain-ing the approximation coefficients of the image(main.met);

2. the vector obtained summing up all theelements in the columns of the submatrixcontaining the vertical details of the image(accordo1.met);

3. the vector obtained summing up all the elementsin the columns of the submatrix containing thehorizontal details of the image;

4. the vector obtained summing up all the elementsin the columns of the submatrix containing theoblique details of the image.

Each of these values was appropriately matched tothe 35 notes of the diatonic scale. The resultingsequence of notes has been inserted, with the correct


syntax, into the text file. All the notes have thesame duration (1/32). The program ChuaWave2.mcreates only three text files, having .met extension,with the following contents:

1. a first file related to the submatrix of theapproximation coefficients (following the proce-dure above described) (voice1.met);

2. a second file containing the oblique details(voice2.met);

3. a third file containing the vertical and horizontaldetails (accordo.met); the notes in this file havehalf duration of the notes generated in the othertwo files.

Thus, following this procedure two voices areobtained, accompanied by a third melody thathas notes of different duration; in particular, eachnote related to the submatrix of the approximationcoefficients corresponds to a note of the subma-trix of the oblique details and to two notes relatedrespectively to the submatrix of vertical and hori-zontal details.

4.1.6. WFSound

WFSound is a system that allows the exploration ofsounds and noises of the complexity of Chua’s oscil-lator. It has been realized as a Visual ProgrammingLanguage (VPL), which uses icons that hide therelated programming structure which the icon refersto, in order to allow an easy-to-use tool for userswho are not programmers. These icons are manipu-lated by the user. They appear in the workspace andcan be combined in a flowchart, linked using arrowsin the direction of the signal flow, from the source tothe output. To insert the objects into the workspace,the user operates a double-click on the object or aclick on the object and then on the workspace. TheInput category comprises three types of icons:

1. pitched: creates a sound with a defined pitch;from the objects menu it is possible to set thewaveform type (sinusoidal, square, triangular) orimport .raw waveforms, and set the frequency,gap and duration;

2. buzzer, a pitched oscillator that produces asound with a dense spectrum, i.e. containingharmonics; the user can select the number of har-

monics, the odd/even ones, the fundamental fre-quency, the gap and duration;

3. noiser, an unpitched oscillator that generates asound containing all the audible frequencies; theuser can set the gap and the duration.

The Scripted Input icon allows to import inputscreated by the user. These scripts are based onWindows Visual Basic Script language and can bewritten using Notepad, saving the file with .wbsextension. In Fig. 39, the WFSound environmentis presented.

We have realized a plug-in for WFSound inorder to analyze the pure sounds emitted by theoscillator of Chua and in order to use the vari-ous states to carry out frequency modulations. Thefigure shows the use of Chua’s oscillator in orderto generate a frequency modulation, starting froma configuration that generates Chua’s attractornumber 1. The three icons are useful to generatean audible sound, to visualize the sonogram andthe corresponding spectrogram. In Fig. 40 it is pos-sible to see, on the left side of the interface, thespectrogram realized by Chua’s attractor, and onthe bottom the correspondent waveform, while onthe right side of the window the process to real-ize the sound in the WFSound environment by link-ing the input, the operator and the output icons areshown.

In the waveform it is possible to notice the char-acteristic envelope generated by using the sinusoidalshape in a half-period. We have realized the puresounds and the frequency modulations generatedin correspondence to several Chua’s strange attrac-tors, which are available at the following web site:

http://galileo.cincom.unical.it/ESG/playchaos/index.htm

The plug-in allows to set the various parameters inan appropriate way. In particular, the six controlparameters α, β, γ, a, b, k, to which is added E,in this context can also be modified together withk, the starting data (x0, y0, z0), the integration stepand the Runge–Kutta iterations number. The sys-tem allows:

• to generate both the pure sound and thefrequency modulation;

• to load the data of the various attractors bymeans of an import menu, and to load or savethe setup parameters.

Fig. 39. In this image, an example of how it is possible to produce visual example of chords in the WFSound environment.

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Fig. 40. In this image, it is possible to see how to produce a visual scripting for creating sound from Chua’s attractor in the WFSound environment.

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4.1.7. Chaos generator: Generating musicfrom Chua’s oscillator

Chaos Generator is a software package created inMatlab. It allows to generate images and musicfrom Chua’s oscillator, starting both from a seriesof attractors whose data are contained in the pro-gram itself, and from a set of set-up data, analo-gous to the software previously described. Also forthis software, in fact, it is possible to set the Chua’sattractors parameters, together with the integrationstep and the Runge–Kutta iterations number. Thesystem allows:

• to generate both the pure sound and the fre-quency modulation;

• to load the data of the various attractors bymeans of an import menu, and to load or savethe setup parameters;

• to visualize the result of the numerical integrationin a 3D graphical window, which in turn builds upthe visualization of the image from several pointsof view;

• to generate a sound flow in a .wav format.

In Fig. 41, the interface of the system is dis-played.

The user can choose the type of the envelopeto employ according to the ADSR model (Attack,Delay, Sustain, Release), by choosing the value ofthe points and the correspondent intensity in theenvelope curve. Finally the user can select betweenan amplitude modulation and a frequency modula-tion and choose a particular chromatic scale. Thesoftware can moreover generate a .sta file that canbe, through an appropriate program, transformedinto a .mid file.

4.1.8. Chua’s harmonies: Generation ofharmonies from Chua’s oscillator

This software allows to generate harmonies fromChua’s oscillator, by using several codes. We willshortly describe this software and the producedharmonies. The system has a first dialog box bywhich it is possible to set the parameters and theinitial data of the system, to define the step ofintegration and the Runge–Kutta iterations num-ber, to recall or save sets of data. Moreover, itis possible to define the equations of a plane inwhich the Poincare map will be displayed. Oncethe system has been set, it is possible to show a 3D

image, which is the result of the integration process(Fig. 42).

This software, based on OPENGL libraries,allows different visualizations of the objects, byvarying the background, the diffusion and the posi-tion of the light, as well as the reflection of theobject properties, thus obtaining stunning images,as shown in Figs. 43–47.

After the user has set up the system’s param-eters and the results of the numerical integrationhas been obtained, it is possible to open the win-dow that allows to start the automatic processof translating Chua’s oscillator curves into music(Fig. 48).

The system allows to musify in MIDI formatthe three variables of the system in the time. Theallowed choices refer to:

1. setting the starting and the ending time of thecomposition;

2. setting to play notes only for each T temporalstep;

3. defining the musical parameters of the com-position;

4. defining the starting note, according to thecorrespondent MIDI code;

5. setting the number of the considered octaves(this obviously is needed to define the dis-cretization step in the space of musicalfrequencies);

6. setting the instruments by which the com-position will be played.

Once these choices have been operated, the usercan choose to play a single curve or many curvesof Chua’s oscillator at the same time. Furthermore,for each curve he/she can define both the chan-nel and the instrument. The produced music canbe directly listened to or recorded in MIDI format,in order to be listened to subsequently. The sys-tem supports the realization of the Poincare map(Fig. 49), from which it is possible to obtain thecorresponding music. The result is a two voices com-position, in which the user can define the type ofinstrument used for each voice.

The system also supports the above describedcolor code that operates on two-dimensional imagesin the following way (Fig. 50):

1. starting from an attractor’s image, the space isdivided in a chosen number of discrete cells;

2. by counting the number of points in each cell, acorresponding color can be defined;

Fig. 41. The interface of the Chaos Generator, a software programme developed using MATLAB libraries.

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Fig. 42. Software’s main windows: the workspace, in the background, visualizes a modified attractor; on the left, the attractor’s parameters window, in which thevalues can be loaded and edited; on the right, the visualization window, by which the user operates the choices for the graphical visualization of the generated pattern.

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Fig. 43. In this image, visualizations of the attractors 1, 2, 3 and 4 are reported. The corresponding values of the sixparameters can be found in Table 1.

3. the color code associates a musical note to eachcolor;

4. the computational algorithm scans the imagefrom left to right and from top to bottom thusrealizing a melody;

5. the image resolution can be increased ordecreased according to the user’s requirements.

This approach allows to transform into musicthe structure of Chua’s attractors. The systemallows a creative composition of melodies as well,exploiting the spatial organization of the curvesin the plane, starting from a visualization of timeaxis, appropriately magnified (Fig. 51). The bluelines represent the pentagram lines. With drag-ging and dropping actions, some notes selected

by the user can be hung up on the pentagramto compose a melody (Fig. 52). This methodallows to detect structures and symmetries inthe attractor’s curves, obtaining pleasant musicalcompositions.

4.1.9. On-line music: applet and web sitefor the exploration of music fromChua’s oscillator

Another software we have realized is a JAVAapplet that was purposely planned in order toallow students and researchers of the field to makedirect online hands-on experiments on Chua’s oscil-lators. This applet is available at the following


Fig. 44. In this image, visualizations of the attractors 5, 6, 7 and 8 are reported. The corresponding values of the sixparameters can be found in Table 1.

web site:

http://galileo.cincom.unical.it/esg/playchaos/index.htm

When the user accesses the applet, the inter-face appears. It is divided into four sub-windows(Fig. 53). In the lower part, on the right, there is aseries of instruments that a user can choose in orderto obtain music from the trajectories; in the upperpart, on the right, by clicking on one of the but-tons, the user can select one of Chua’s attractors inthe library of the system; on the left top, the tra-jectory of Chua’s system in a 3D space is shown,

emerging from the numerical integration of the sys-tem, based on the choice of the set-up parametersfrom the user. A series of icons placed on the top ofthis sub-window allows the exploration of the tra-jectory through rotation, zoom, etc. One of theseicons allows also the visualization of the data fromthe numerical integration.

On the lower part of the interface windowthere is a piano keyboard, and when the simu-lation of the chosen Chua’s attractor is runningon the choice operated by the user, the notes aresimultaneously played. The system’s interface also


Fig. 45. In this image, visualizations of the attractors 9, 10, 11 and 12 are reported. The corresponding values of thesix parameters can be found in Table 1.

allows the setting of the musical parameters andoutputs MIDI files. The Java applet consents amore spectacular 3D exploration of Chua’s attrac-tors, using Java 3D as graphical engine (Fig. 54).The user can explore the stunning images producedby the attractors, moving, rotating and navigat-ing into them, through simple visual user-friendlyinterfaces. The points of the trajectories are visu-alized through spheres, and when they are very

close they intermingle, forming interesting emer-gent patterns. In Figs. 55–59 some 3D imagesof Chua’s attractors created using the applet areshown. It is possible to notice that these 3D imagespresent some particulars and various emergent pat-terns are formed when the spheres are very closetogether.

On the same repository, devoted to Chua’sattractors, it is possible to find many other contents


Fig. 46. In this image, visualizations of the attractors 13, 14, 15 and 16 are reported. The corresponding values of thesix parameters can be found in Table 1.


Fig. 47. In this image, visualizations of the attractors 17, 18, and 19 are reported. The corresponding values of the sixparameters can be found in Table 1.

(Fig. 60). In fact there are a lot of visual and audi-tory resources, generated from the dimensionlessChua’s systems. In this repository it is also possibleto download the software described above.

The site is organized into various sections.The first consents an hands-on investigation of theattractors, the other collects a series of images,sounds and music realized using Chua’s attrac-tors. The Images section includes two categories(Fig. 61):

• Images and forms;• Escher-like illusions.

Figures 62 and 63 show some image collec-tions related respectively to the first and secondcategories. The characteristics of the images andthe related music will be described in a tutorial tofollow.

As previously observed, the site consents theaccess to a series of sound and musical resources.In particular, for each attractor there is a web pageshowing a 3D visualization of each attractor andits projections on the planes xt, yt, zt e xy, xz e yz(Fig. 64). From these pages it is possible to listen tothe different sounds produced using the previouslydescribed Chua’s harmonies software.

Fig. 48. In this window of the Chua’s Harmonies program it is possible to visualize the curves x, y and z of the system, to choose the musical parameters, to listento the music and to save the related compositions.

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Fig. 49. In this image a 3D view of the attractor number 1 in 49(a) together with a projection of Chua’s attractor in the plane xy 49(b), a projection in the plane xz49(c), and finally a visualization of a Poincare’s map 49(d).

Fig. 50. In this image, the process the algorithm realizes is represented by the following steps: visualization of the attractor 1 (a); division of the space by a grid of30 × 30 cells (b); application of the color code through the numeration of the points contained in each cell (c); the color code is then applied into a 50 × 50 grid (d).

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Fig. 51. In this image a representation of a state function in the time of Chua’s attractor is displayed. On it a musical pentagram has been projected, represented bythe blue lines.

Fig. 52. In this image, a process of creating melodies by using the point of intersection of the curve with the virtual pentagram is displayed.

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Fig. 53. The interface of the PlayChaos applet. The system is available and real time operating in Internet, at the url given in the text.

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Fig. 54. From the PlayChaos3D page it is possible to choose one of the attractors, and visualize it in a JAVA 3D applet, with different settings.

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Fig. 55. In this image, on the top a visualization of the pattern generated by Chua’s attractor number 7. On the bottom twoparticulars of the same system. These images have been produced by using the PlayChaos applet in JAVA 3D environment.

Fig. 60. Home page of the web site devoted to Chua’s attractors. From this page it is possible to access many hands-on activities, to see beautiful images and tolisten to sounds and music produced starting from Chua’s systems.

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Fig. 61. Images section access web page, including the two categories: Images and Forms and Escher-like illusions.

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Fig. 62. Images and forms: a catalog of art-works based on Chua’s oscillator. Starting from Chua’s attractors, some images were created adding colors and effects tothe basic forms. Surprisingly appeared are images of flowers, stars and other forms.

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Fig. 63. Escher-like illusions. Starting from Chua’s attractors, some images were created varying some visualization parameters. The resulting images show someinteresting effects that are similar to Escher’s illusions.

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Fig. 64. Access web page to the sound and musical resources of the attractors. On the left, a list of the attractors; in the middle, a 3D visualization of the attractor;on the right, the attractor’s projections on the planes xt, yt, zt and xy, xz and yz. On the bottom, the links to the sound and musical resources.

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5. Exploring Chua’s Attractorsthrough Music

5.1. Musical examples

The need of finding new methods of looking at com-plex phenomena is strictly connected to a new visionof Science and Mathematics. Traditional meth-ods are no longer sufficient and new approachesare emerging. These approaches, which are moreobservational-qualitative than quantitative, provethat complex phenomena are not classifiable bytraditional mathematical and geometric methods.The science of complexity aims at linking tradi-tional sciences, across different domains. Complexphenomena, which emerge from the interaction ofa multitude of simple elements, occur in manyfields and at many levels of description. A com-plex system is a set of functions that consists ofindependent and variable parts, in which, contraryto conventional systems, the parts have neitherfixed relationship, nor a fixed behavior or quantity.Their individual functions cannot be defined follow-ing traditional scientific methods [Wolfram, 1984b;Kauffman, 1989]. In spite of the vagueness of theseconcepts, these models are found in the biological,social and economic world. The principal objectivesof these approaches are:

1. explanation of the emerging structure (self-organization);

2. measure of the relative complexity;3. supply of control methods for complex systems;4. generation of functional models;5. solution of the most important issues;6. demonstration of possible new applications;7. quantification of the laws of order and organiza-

tion present in these systems.

In this part of the tutorial, we would like tostudy the complexity of Chua’s attractors by music.Since “the double scroll circuit is the first realphysical system where chaos is observed in lab-oratory, confirmed by computer simulation andproven mathematically by two independent meth-ods” [Matsumoto et al., 1988], we try to demon-strate how it is possible to find out the routes tochaos [Kennedy, 1993a] using the musical approach.Possibly, we would explain the emerging struc-tures in Chua’s attractor translating them intomusic, thus demonstrating new ways of analyzingand entertaining, giving students new approachesin the study of this important topic of mod-ern Science. Furthermore, many modifications inthe musical compositions realized by means ofChua’s attractors show different auditory waysof interpreting and understanding complexity andemergence in dynamical systems. In the musicalexamples we are going to present, we analyze theattractor shown in Fig. 65, whose parameters arethe following:

Runge–Kutta Step ofiterations: 10.000 integration: 0.04 α: 9.3515908493

β: 14.7903198054 γ: 0.016739649 a: −1.1384111956

b: −0.7224511209 k: 1 e: 1

x0: 0.2 y0: 0 z0: 0

The system presents three different waveforms(Fig. 66).

The simplest code we have developed inorder to translate the complex behavior of Chua’sattractor into music allows the discretization of eachwaveform. Following this procedure, each curve iscomposed by points, which are projected on a carte-sian plane, and to each point of the axis y cor-responds a note of the piano octaves, while thecurve evolves in time (axis x), as shown in Fig. 67.C4 is the central octave of the piano arrangement.In this way, the frequencies of the notes of each

curve will depend on their settling up or down thiscentral octave, which in the mathematical languagecorrespond respectively to positive (above the cen-tral octave) or negative values (under the centraloctave). The number of points of the curve whichwill be translated into music can be set up accordingto the choice to read exactly the waveform or to readonly the more interesting parts of the curve. Thecode allows the user to determine the rate of pointshe/she wants to pick up and to translate them intomusic. The default rate is 1.

Fig. 65. Double scroll computer simulation of Chua’s attractor number 1. Runge–Kutta integration routine was iterated 10 000 times.

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Fig. 66. Simulated waveforms from Chua’s attractor. On the horizontal axis t (expressed by the time steps of the computer simulation), in the vertical axis x, y andz are represented.

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Fig. 67. Representation of what we have called the basic musification code. The curve of Chua’s attractor has been discretizedand to each point of the y axis corresponds a note, having the frequency of one of the eight octaves of the piano arrangement.

Adopting this code, we obtained musical com-positions consistent with the physical character-istics of the waveforms of Chua’s attractor. Thenumber of octaves ofeach composition, the instru-ments, the number of notes the code will pick upfrom the curve (the sample rate), the duration ofthe notes, the sonata tempo and some other auto-matic options can be chosen by the user, in the soft-ware we have developed. The curve x describes anoscillatory tendency, going rhythmically from neg-ative to positive values, with very well organizedtraits, which seem repetitive in time, interruptedby nonorganized traits that allow for the reitera-tion of the first organization. The music follows therhythmic movements of this curve, with organizedstructures, recurring on different octaves, generallypassing from low to high ones. This organization,that goes on for many time-steps, is interrupted bya speedy transient from low to high notes. From theacoustical point of view, the passages from low to

high octaves is very distinctive of this composition.A musical page of this composition is visualized inFig. 68. To realize this composition, we have usedan instrument available in the MIDI language (theinstrument number 10, Music box), a note length of4/4. The first note the code has picked up from thecurve is the number 36, in order to exclude from themusification process the transient part of the curve,which depends on the initial starting conditions ofthe attractor, before it develops its typical behav-ior. Four are the octaves which have been chosen. Atfirst glance, the curve y seems not so organized. Butif we magnify it, both horizontally and vertically, itis possible to note that some specific arrangementsare present. Translating this curve into music, itseems to oscillate between low and high notes, butthe differences are not so typical as in the curve x.The resulting melodic organization still maintainsthe same octave, with some slight modifications. Torealize this composition, we have used the MIDI

CompositionX. mid

Fig. 68. In this image two pages of the musical composition produced by the curve x of Chua’s attractor are displayed.

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instrument number 44, Tremolo Strings and a notelength of 4/4. Two pages of this composition areshown in Fig. 69.

The curve z appears to be the inverse of thecurve x, with different scale and patterns, but coor-dinated and probably synchronized in time. Themusic pursues the rate of the curve, going fromhigh to low notes, but the intervals between highand low octaves are longer, with abrupt changesgoing down and immediately growing in high tones.The MIDI instrument which has been chosen is thenumber 46, Orchestral Harp, the length of the noteis 4/4. A sample of this composition is visualized inFig. 70.

In order to investigate whether there are anyrelations among the curves, which can give us inter-esting musical compositions and consequently tounderstand the nature of the complexity of thisattractor, we tried to put together in a single com-position, made up of three instruments or voices,the music we have obtained for each curve. Theacoustical effect is surprising as the music respectsthe structural organization of each curve, but theharmonization is unexpected: the different scaleof the behavior of each curve becomes coupled,sometimes amplifying sometimes dissipating thedifferences. Choosing the default rate of the sys-tem, in other words, picking up all the notes ofthe three curves, it is possible to listen to threemelodic rhythms, coordinated together. The firstvoice (curve x) is synchronized on the oppositeoctave with the third one (curve z), while the sec-ond voice (curve y) is still on the same octave.The result is similar to a contrapunctual composi-tion, where there are many organized voices playingtogether. Here the word emergence refers to under-standing how collective properties arise from theproperties of parts. More generally, it refers to howbehavior at a larger scale of the system arises fromthe detailed structure, behavior and relationshipson a finer scale; it is about how macroscopic behav-ior arises from microscopic behavior. Each voice ofthe composition has proper vigorous musical move-ments, but the acoustical effect is an emergent com-pound dynamics.

5.2. Musical complexity and emergenceof rhythm in Chua’s attractor

In the compositions it is possible to detect musicalstructures, which are repeated over time. Perception

of music essentially depends on a structuringprocess operated by the listener, since dealing withmusic is an active and constructive process, asGestalt’s psychologists pointed out [Wertheimer,1938, 1958]. The basic assumption of this approachis that perception is an emergent process, whichis realized by understanding global qualities ofthe objects of knowledge. The holistic vision andthe biological approach of the world state thatglobal structures resist destruction of some of theirparts. Individual elements are better perceivedwithin a context than in isolation. In the com-positions we have realized, the musical structures,perceived as global organizations, have a strongstability. In this context a structure is defined asa geometrical figure (a set of points, correspondingto a group of notes). If we alter the sample rate ofthe code, what happens to the musical compositionand to musical recognition? Since we have definedthe sample rate of the code, in other words, thenumber of notes the system will pick up from eachcurve of the attractor, we noted that the melodicdevelopment of the first composition changes instrict relationship with the alteration of this rate.We made many investigations using the composi-tion with the default rate as prototype, and thenrealizing compositions at rates 5, 10, 15, 20, and 25until 55, for all the curves of attractor 1. In Figs. 71and 72 there are two samples of the musical compo-sitions created. The first one corresponds to a rate5 on the curve x of this system, the second using 55as sample rate, again on the same curve. The orig-inal prototype of the basic composition seems tobreak down. Anyway, it can be said that progres-sively the musical structure decays and the recogni-tion seems to be correlated with the increase of thetransformations that the rate algorithm produces inthe musical pieces, which is obtained by varying theset of selected points: operating a relevant changein many points of a configuration, the whole dis-tribution is affected and, in a listener, the relatedprocess of pattern recognition is altered. Interestingexperiments could be created with human subjects,testing the rate at which the perception of the musi-cal structure of each curve decays.

A larger set of variations can be realized alter-ing the parameters of the attractor. If we modifythe parameter of the attractor, what happens in therelated musical compositions? We found that slightvariations in the structure of the curves, due to themodification of its parameters, do not alter sub-stantially the motion of the musical composition.

CompositionY. mid

Fig. 69. A sample of the musical composition obtained by the curve y of Chua’s attractor is visualized.

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CompositionZ. mid

Fig. 70. Representation of the musical composition obtained by the curve z of the considered Chua’s attractor.

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CompositionA. rate5. mid

Fig. 71. An example of the musical pages created with 5 as sample rate of the basic musification code. The completecomposition is 18 pages long, since from each interval of five the system picks one note from the attractor’s curve.

CompositionA, rate55. mid

Fig. 72. Musical pages created with 55 as sample rate of the basic musification code. Since from each interval of fifty-five the system picks one note from the attractor’scurve, this composition is only two pages.

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For example, if we use the following data:

Runge–Kutta Step ofiterations 10.000 integration: 0.04 α: 9.3515908493

β: 14.3903198054instead of 14.7903198054 γ: 0.016739649 a: −1.1384111956

b: −0.7224511209 k: 1 e: 1

x0: 0.2 y0: 0 z0: 0

we get the curves represented in Fig. 73.The compositions we obtain by translating these modifications into music are quite similar to the

previous one. But if we greatly alter one of the attractors’ parameters, or many of them, using for examplethe following changes:

Runge–Kutta Step of α: 8.3515908493iterations 10.000 integration: 0.04 instead of 9.3515908493

β: 13.9903198054 γ: 0.2256739649 a: −12.3584111956instead of 14.7903198054 instead of 0.016739649 instead of −1.1384111956

b: −0.6924511209instead of −0.7224511209 k: 1 e: 1

x0: 0.2 y0: 0 z0: 0

then the attractor configuration is extremely mod-ified, the same happens to the curves x, y and z(Fig. 74).

Consequently, also the musical compositionresults are profoundly transformed: the melodicorganization is varied, the rhythm is different,while the repetitive structures are not present,or existing with different sequences and/or peri-ods over time. In this way, we found that bychanging the parameters of a single attractor itis possible to obtain many alternatives to the pro-totype composition, which might present eitherslight modifications or structural diversity inthe musical motion. In between there are wide-ranging collections of assorted musical pieces.In order to highlight the complexity of Chua’sattractor and the possibility to take advantageof that, by creating infinite musical compositions,we have used other kinds of codes to translateChua’s attractors into music. We employed a geo-metrical code, which exploits the geometries ofthe curves x, y and z and their Pythagoreanrelationships. The curves of this attractorrealize different patterns. Pattern formation

(understanding how patterns are formed) and pat-tern recognition (how the human brain recognizespatterns) are two important areas of study inComplex Systems. In this context, a patternmight be:

1. a set of relationships that are detected by obser-vations of the behavior of this attractor, or acollection of attractors which present the samepattern related to their common ancestor;

2. a simple kind of emergent property of the attrac-tor, where a pattern is a property of the systemas a whole but is not a property of small partsof the system;

3. a property of the attractor that allows itsdescription to be shortened if compared to thelist of the descriptions of its parts.

Generally, a pattern can accomplish repetitionsboth in space and time [Stewart & Golubitsky,1992]. Given these definitions, we can say that thecurves x, y and z realize patterns in the three-dimensional space in which they are visualized andin time, as magnifications of the periodic patternsas shown in Fig. 75. In a possible description of this

Fig. 73. In this image, from left to right, the curves x, y and z, resulting after changing the parameter β, are displayed.

Fig. 74. In this image, from left to right, the curves x, y and z, resulting after changing the parameters α, β, γ, a and b, are displayed. On the right side of the image,a visualization of the new form generated by these modifications is reported.

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Fig. 75. This image displays the patterns the curves x, y and z produce in time. The curves have been magnified in order to make evident the structural organizationof each one.

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attractor, repetition corresponds to redundancy:saying the same thing again and again. When thereis redundancy we can shorten the description bysaying the repeating part only once and saying howmany times it should be repeated.

It is also possible to think of patterns as proto-types or exemplars. This is the meaning we used todescribe a pattern in order to realize slightly differ-ent attractors, from the prototypical parameters ofthe attractor 1. In this situation, the pattern is notabout the relationships within the other attractors,but about the possibility of repeating the processmany times. The relationship between the patternas repetition and the pattern as prototype is justlike the relationship between two types of emergentproperties: emergent properties that arise from rela-tionships between parts of a system, and emergentproperties that arise from relationships between asystem and its environment (the larger system ofwhich it is a part). So the behavior of this attractoris emergent, since the structure arises from relation-ships between the curves x, y and z. In the case ofthe attractor number 1, the macroscopic behaviorof this system arises from the synchronization of thecurves x, y and z, as displayed in Fig. 76.

It is possible to take advantage of these relation-ships by using a superimposed pentagram, in orderto see whether, in turn, the curves form patterns onthe musical compositions (Fig. 77).

We observed that, according to the way theuser puts the notes on the pentagram, it is pos-sible to realize different, but related musical com-positions. Let us illustrate this code by a graphicalexample, referring to attractor 1. In Fig. 78, dif-ferent ways of exploiting the patterns of the curvex are shown. In the first part of the image, thenotes are placed on the pentagram according to thehigh and low peaks, while in the second part of thisimage, the notes are located in line with the pointsof intersection of the curve with the pentagram. Aslightly different process has been repeated in thecase of the curves y and z, illustrated in Figs. 79and 80.

In Figs. 81–83 the musical pages for thesethree compositions are visualized. If we listen tothe music, it is possible to note that, in contrastwith the basic musification code, these composi-tions present an intense melodic organization whichis repeated over time. It is as if the picking up ofthe salient points of the curves builds up an orga-nization, which is reflected in the musical melodies.While in the experiment referred to above, the use

of different paces varies profoundly the structureof the organization of the curves, detecting thepatterns by hand (for example, choosing only thehigh and low peaks, or finding out simply the pointsof intersection of the curves with the pentagram)does not produce a rough musical piece. On the con-trary, these compositions are organized and melodicand they seem to realize a basic, easy to the earmodel, which is repeated over time with some slightvariations. Detecting the geometrical organizationof the curves means that we can impose to the roughmusical material a defined organization, in the sameway as musicians do, when from the musical sub-stance they cut off the melody, according to theirparticular compositional needs and their emotions.Harmony is an ordering process that organizesmusical thoughts. With an analogous method, thenotes the user puts on the curves make evident theirstructures, which in turn are transformed into har-monies. The harmonies of the curves x, y and z arerelated.

We obtained the following notes for each curveas it is possible to see, in Table 2, that orders stepby step the notes we have chosen to settle on eachcurve, there are horizontal and vertical relation-ships. The horizontal ordering refers to the presenceof loops that are repeated, after a defined number oftemporal steps, for each curve. For example, aftera transient that lasts about 1500 temporal steps,the loop of the curve x is realized by: F5; A5; B5;F4; B5; E4; C5; B5; and is repeated every eighttemporal steps. The curve y follows another pro-cess, in which a note, B5 is reiterated at differentintervals, one, two, three, four, seven times and thenagain, without a rule, but changing each time. Aftera transient that lasts about 1500 temporal steps, thecurve z realizes a loop in eight temporal steps, madeof the following notes: F4; C5; E5; B5; E5; A5; F4;A5. The vertical correspondences could be relatedboth to the Pythagorean consonance/dissonancethat each note from each curve has in relationshipto the other notes and to the repetition of the sametuple in different time steps. If we modify the lengthon the note, it is possible to obtain a different kindof music. We know that the sonata tempo is deter-mined by the length of the note (whose usual valuesare 4/4, 2/4, 1/4, 1/8, 1/16, 1/32, 1/64), which aregrouped together to form the scores of the musi-cal composition. In this artificial context, the pro-cess of shortening this variable means that it ispossible to assign real values to each note of thecurve, as the user likes, thus realizing compositions

Fig. 76. The behavior of this attractor arises from the coordination among the curves x, y and z, which in this image are represented together.

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Fig. 77. On the curves x, y and z a pentagram has been projected. In the software Chua’s Harmonies, the user can drag and drop the notes on the curves manually,setting previously the note duration in the options menu.

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Fig. 78. In this image, the curve x filled with 4/4 duration notes is shown. In Chua’s Harmonies program this action can be also produced automatically.

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Fig. 79. In this image, the curve y magnified filled with notes, manually selecting the high and the low peaks, is shown.

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Fig. 80. In this image, the magnified curve z filled with notes, manually selecting high and low peaks, is shown.

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Fig. 81. The musical composition, produced filling the curve x with notes, is reported in this image. From the musical pointof view, the melody is very different from the composition obtained automatically.


Fig. 82. The musical composition, produced filling the curve y with notes, is displayed in this image. From the musical pointof view, the melody is very different from the composition obtained automatically.


Fig. 83. By filling the curve z with notes is possible to obtain the musical piece, presented in this figure. From the musicalpoint of view, the melody is very different from the composition obtained automatically.


Table 2. Musical structures generated by the curves x, y, and z of Chua’s attractor number 1.

Step 1 Step 2 Step 3 Step 4 Step 5 Step 6 Step 7 Step 8 Step 9 Step 10 Step 11 Step 12

Curve X F5 A5 B5 F4 B5 E4 C5 B5 F5 B5 G5 A5

Curve Y C5 B5 C5 C5 C5 A5 B5 B5 C5 A5 C5 B5

Curve Z F4 C5 E5 B5 E5 A5 F4 A5 F4 C5 E5 C5

Step 13 Step 14 Step 15 Step 16 Step 17 Step 18 Step 19 Step 20 Step 21 Step 22 Step 23

Curve X B5 F4 E4 C5 B5 F5 B5 G5 A5 B5 F4

Curve Y B5 B5 C5 A5 B5 B5 B5 B5 C5 B5 C5

Curve Z E5 A5 E4 A5 F4 C5 E5 C5 E5 A5 E4

with noncanonical tempos. We realized that if thenote length is very short (for example: 1/12, 1/121,4/271), the curve is translated into music realizingvery strange compositions. From the perceptualpoint of view, it seems an auditory stream settlesin between music and the acoustical dimension ofsound. The perceptual knowledge of Chua’s attrac-tor is on the whole and not dissolved in a multitudeof notes, which we normally reorganize and per-ceive as melodic organization of the musical com-position. The curves of the attractor evolve in timewhile this music develops with an increase in thevolume intensity, with different degrees of volume,according to each tendency of the curve. It seemsthat music increases in speed and volume when eachcurve intercepts high octaves and fades over manymeasures, maintaining speed when the curve is inthe region of the central and the low octaves.A sample of this musical variation is shown inFig. 84. Note that the traditional software pro-grams do not have this possibility of shorteningthe note’s length. So the musical page we pro-pose will have a default tempo of 4/4 and also themusical notation (which is conform to the canon-ical music) should be changed, or created since itdoes not exist. The MIDI interface realizes this phe-nomenon by using the value 1/64. Changing thelength of the note, consequently the sonata tempoalters the rhythm of the composition as well, espe-cially when we use fractions that are near to 1 (forexample, 11/21). These compositions are similar tojazz improvisation pieces. A sample is reported inFig. 84.

The translation of Chua’s attractors into musicyields the emergence of rhythms and melodies whichcan be realized by using different codes and produc-ing a vast amount of musical compositions.

5.3. In the garden of chaos

Our aim is to traverse in the complexity of the bifur-cation map realized by Chua’s attractor using musicas a new possible tool of comprehension of com-plex phenomena. We will try to discover the keyelements of complexity, according to the literatureon this topic. In other words we will try to discoverhow it is possible to find out the routes to chaos[Kennedy, 1993a, 1993b] using the musical approachand the global dynamics concepts, which allow usto determine the basic behavior of a dynamical sys-tem. We constructed a bifurcation map (Fig. 85),starting from the value 6.4 of the parameter α anddeveloping it for 600 total steps, with an integra-tion step of 0.01, and magnification scale 4. In thisway, we realized the space of all possible behav-iors of Chua’s attractor in the discretized space ofparameter α. The bifurcation map is highly com-plex. It is possible to detect the frontier betweenordered and chaotic behavior, the “edge of chaos”[Langton, 1986, 1990], characterized by period dou-bling, followed by quadrupling, etc., even thoughother routes to chaos are also possible [Abarbanelet al., 1993; Hilborn, 1994; Strogatz, 1994].

Let us analyze the main behavior in this spaceby music. At the value 6.55 of this parameter, wefind a fixed point, where the system presents astate of equilibrium. At this value of the parame-ter, Chua’s attractor is in one of these states, and itwill remain there indefinitely. The curves x, y andz of Chua’s attractor and their visualizations in therespective planes are displayed in Fig. 86.

The musical compositions we have realizedpresent interesting tendencies in the octaves theyselect and in the collective behavior of the threecurves. The curve x begins from high tones and


Fig. 84. Musical example that is similar to a jazz improvisation piece.

Fig. 85. Bifurcation map of the attractor number 1, starting from the value 6.4 of the parameter α and developing it for 600 total steps, with an integration stepof 0.01, 4 as magnification scale. From this map, it is possible to detect all the changes in the qualitative behavior of the system, as parameters are varied.

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Fig. 86. In this table, the curves x, y and z and their visualizations in the respective planes for the parameter α equal to 6.55 are displayed. Runge–Kutta integrationroutine was iterated 3000 times.

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rapidly slows down to low ones, maintaining anoscillatory tendency, before collapsing into the fixedpoint. The extension of the tones the curve y inter-cepts is wider, but the behavior is the same, whilethe curve z is permanently on high tones, aftera start from the low ones. The composition thatcollects the curves x, y and z presents attractivemovements in the musical dynamics, since it is pos-sible to listen to their collective behavior whichseems coordinated, before reaching together thestable behavior, which presents an iterated repe-tition of the same notes, at different octaves butsynchronized in time (Fig. 87). From the acousti-cal point of view, or the auditory scene analysisapproach [Bregman, 1990], the music realized bythe curve y seems to be in the foreground, whilethe pieces of music produced by the curves x and zstay in the background, producing complex move-ments, which interlace with the primary musicalsource.

Neighborhood parameter values in the bifur-cation map give similar musical compositions. Wehave found that going from 6.55 to 6.98 the curvesx, y and z amplify their internal oscillatory period,but the musical movements are quite the same,although the pattern begins to change (Fig. 88).This type of behavior is due to the presence of thelimit cycle.

The translation of curve x into music presentsa descending motion. The curve z on the con-trary is produced by an ascending one, while thecurve y presents both descending and ascendingmotion rhythmically. The composition created join-ing together the music obtained by the three curves,alternates in a synchronized way these kind of musi-cal movements. The only difference we can roughlyunderline is that the music obtained by the threecurves does not converge on the same group of notesat the end, but it seems that the motive is repeatedmany times in a circular way. This last phenomenonis amplified in the music developed by the curvesfor the values 7.11 of the parameter α of Chua’sattractor (Fig. 89). The musical motions are thesame of those described above and the enlargingof the oscillating period of each curve produces alarger difference in the group of the generated notes.A repetition of the same melodic pattern character-izes the music composed by the curve y, at differentoctaves, as it is possible to note in the first pageof the composition represented in Fig. 90. This rep-etition seems to create an echoing effect from theacoustical point of view, which expands the force of

the musical dynamics and develops greater rhyth-mic and melodic complexity.

The value 7.11 is on the first division of thebifurcation map presented in Fig. 85. In fact, sincethis region is a sensitive section of transformation,from the value 6.99 to 7.11 it is possible to noteinteresting cycles of changes in the visual config-uration of the attractor’s patterns. In Fig. 91, weshow a magnification of this region of the bifur-cation map. The range of values goes from 7.012to 7.507. The number of points for each pixel ofthe figure is very high, so that it is possible toget a very good definition of the changes in theattractor’s patterns. The configuration gets thin-ner (parameter’s value 7.08) while the curves x, yand z undergo relevant enlarging modifications intheir oscillations; then the attractor begins to grow(parameter’s value 7.11), becoming larger at the val-ues 7.15 (Fig. 92). The music produced by usingthe curve x uses the central octaves of the pianokeyboard, without any significant movements in thechanges of tones. The curve y uses central octaveswith a descending tendency, which produces verylow tones. The tendency that the musified curve zmaintains is constant and it stays constantly at thesame two octaves. The curve selects high notes ofthe octaves, but lower in comparison to the preced-ing values of the parameter α of this system. Thecomposition created mixing the music accomplishedby these curves presents some interesting phenom-ena. The first one is the general lowering of the highpeaks and an arrangement on central octaves of thepiano keyboard. It seems that the system tends toconverge towards an overlapping of the curves. Thislast effect is clearly audible by taking into consider-ation the curves x and y, which share many sets ofnotes for central octaves, and then considering thecurves y and z, which share sets of notes on highoctaves. Going until the values 7.46 of the bifurca-tion map, we find the configurations and the curvesreported in Fig. 93.

The first outstanding feature is the change ofthe configurations of the curves and the almost totaloverlapping of the curve x on the curve y, whilethe visual patterns are changed as well. The musi-cal composition obtained by the curve x selects thecentral octaves of the piano keyboard, with somejumps on the higher octaves, and its tendency isconstant in time. The same happens for the curvey and partially for the curve z, with some jumpson the low octaves (Fig. 94). Merging these com-positions produces a synchronization or a fusion

Fig. 87. The first and last pages of the composition realized by choosing the value 6.55 of the parameter α of the attractor. After a transient, the music ends in aniterated repetition of the same notes, at different octaves.

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Fig. 88. In this table, the curves x, y and z and their visualizations in the respective planes for the parameter α equal to 6.98 are displayed. Runge–Kutta integrationroutine was iterated 3000 times.

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Fig. 89. The curves x, y and z and their visualizations in the respective planes for the parameter α equal to 7.11 are displayed. As it is possible to note, the oscillationvalues of the curves is elevated in relation to the changes in the attractor’s configuration. Runge–Kutta integration routine was iterated 3000 times.

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Fig. 90. In this image, a page of the musical composition, produced by translating into music the curve y and using thevalues 7.11 of the parameter α of Chua’s attractor, is presented.

Fig. 91. In this figure, a particular of the bifurcation map. The values of the parameter α range from 7.012 to 7.507 in order to detect, at a finer scale, the changesin the qualitative behavior of the system.

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Fig. 92. The visual configurations and the curves x, y and z of Chua’s attractor for the value 7.15 of the parameter α are displayed in this image. Runge–Kuttaintegration routine was iterated 3000 times.

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7,46 xyz

Fig. 94. In this image, the first two pages of the composition realized by joining together the pieces realized by the curves x, y and z are shown.

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between the pieces produced by the curves y and z,while the melody of the curves x creates a contra-punctual voice.

From the region of the bifurcation map of theparameter α that goes from 7.46 to 8.50, interestingvariations take place both in the patterns of thecurves x, y and z and in the attractors’ configura-tions. At the value 7.72 the curves x and y overlapmore or less completely. The length of their oscilla-tions is very high. If we join together the pieces ofthe three curves, the curves x and y seem to per-form as one, while the curve z works as a contra-punctual voice. The melody of the pieces is a longcycle with high and low notes that repeat forever.The visual configurations enlarge as they approx-imate the pattern of the first scroll. At the value7.90 the curves x and y overlap for the major partand the length of the oscillations of the curve zincreases.

From the value 8.10 to the value 8.50, also thecurve z overlaps almost 3/4 of the other two curves.The cycles last as the simulation time. We are inthe region of the “period-double route to chaos”[Kennedy, 1993a, 1993b] in the bifurcation map ofChua’s attractor. The patterns and the curves forthe value 8.51 of the parameter α are presented inFig. 95.

The curves x and y have slightly different setsof notes in the cycle they produce and in the octavesof the piano keyboard they intercept, but from theacoustical point of view they seem to work as agroup. The curve z is slightly different as it inter-cepts different octaves of the piano organization.The three melodies together create a constantly par-allel motion which may also include contrary andoblique motion. The region of the bifurcation mapthat range from the value 8.51 to the value 8.87 isdisplayed in Fig. 96. As it is possible to note, also inthis region there are important modifications bothin the pattern of the attractor and in the curves x,y and z.

Many variations take place in this region at dif-ferent levels of details. The finer the scale, the morethe cycles of changes in the attractor pattern canbe displayed. These metrics can define the fractalcharacteristics of Chua’s attractors. In Fig. 97, asequence of images of the region that goes from 8.52to 8.81 is shown.

At the value 8.81 of the parameter α, Chua’sattractor realizes different kinds of curves and pat-terns, as can be noted in Fig. 98. Speaking aboutthe previous example, we have said that if two parts

of a musical piece always move in similar motion,from the psycho-acoustical point of view, the earregards them as one part enriched by doubling,but if their melodic direction diverges, they takeon separate identities. This happens for the musicalcomposition created by combining the pieces for thecurves x, y and z for the value 8.81 of the param-eter α in this system (Fig. 99). From the musicalpoint of view, at the beginning the curve x pro-duces ascending–descending movements, with longintervals of notes in the intermediate octaves. Thecurve y generates a descending motion, using thelower octaves of the piano keyboard. The curve zcreates very slow ascending–descending movements.The texture of the harmonic progression is disso-nant as the parts are in contrary motion. The samekind of musical texture is present in the melodiesrealized by the patterns and curves presented inFig. 100 and in the related musical composition,of which Fig. 101 presents two pages. The threevoices build up different motions while the intervalsresulting from their relation to each other seem tobe dissonant. In this composition, harmonies seemto be created for their sonorous and expressivequalities and not to underline the melody, whilethe rhythm becomes quite irregular. The symme-tries of the preceding configuration are broken andthis composition is permeated by the principle ofcontinual variation, which replaces the sequentialrepetition of the musical pieces obtained before.The pattern grows according to unknown modelsof morphogenesis. The system duplicates the firstconfiguration with a symmetrical and inversemodality, but this process does not resemble Tur-ing’s model of action-reaction [1952] nor D’ArcyThompson’s mathematical growth model [D’ArcyThompson, 1917]. In fact at the value 9.16 wefind the configuration which presents the clas-sical double scroll (Fig. 102). Music offers thepossibility to discover fundamental proprieties ofnatural and artificial objects, and to discover inthem unexpected organizations. We think that arti-ficial and natural systems share the same lawsof organization. The music of Chua’s attractorsshows the inner measure of time, the beauty oflife.

6. Conclusions

This tutorial is about how to translate into soundand music the behavior of Chua’s attractors by

Fig. 95. In this image, the patterns and the curves for the value 8.51 of the parameter α are presented. Runge–Kutta integration routine was iterated 3000 times.

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Fig. 96. In this image, a magnification of a region of the bifurcation map of Chua’s attractor for the parameter α from the value 8.51 to the value 8.87 is displayed.The scale of magnification is 0.001.

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Fig. 97. A sequence of images is visualized in order to detail the cycles of changes in the attractor pattern, picking up key values in critical regions, of the bifurca-tion map.

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8,81 xyz

Fig. 99. The first two pages of the composition, produced by joining together the pieces created by using the curves x, y and z, are shown.

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Fig. 100. The visual configurations and the curves x, y and z of Chua’s attractor for the value 8.91 of the α parameter are displayed in this image. Runge–Kuttaintegration routine was iterated 3000 times. The pattern which is visualized by using these values of the bifurcation map is called Chua’s single scroll.

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Fig. 101. The first two pages of the composition, generated by joining together the compositions obtained from the curves x, y and z, are shown.

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Fig. 102. Double visualization of the attractor at α = 9.16. The visual configuration and the curves x, y and z of Chua’s attractor for the value 9.16 of the parameterα is displayed in this image. The resulting pattern is called Chua’s double scroll.

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using music. The ideas we intended to investigatewere the following:

1. Is it possible to translate the behavior of Chua’sattractors into music?

2. What kind of sound and music does Chua’sattractor produce?

3. Can we use the music which interprets the behav-ior of these dynamical systems as a tool forunderstanding chaos?

4. What kind of relationship exists between dynam-ical systems, their musical translation and theirperception in the process of listening by humansubjects?

5. Is it possible to detect emergent patterningphenomena in music in strict correlation with thepatterns present in Chua’s oscillators?

The exploration of the complexity, in continuousor discrete dynamic systems, through music, hasallowed us to generate many types of musical andsound pieces, supplying new points of view for theanalysis of such systems and the possibility, almostwithout limits, to generate many rough composi-tions, starting from Chua’s attractors. The goalwas that of supplying Chua’s oscillators with asemantics (sound and music), through which onecan understand, from the acoustical point of view,the behavior of such complex systems. This typeof experimentation, that favors the auditory senseand therefore the sonification rather than the sci-entific visualization, joins the search for significantelements inside the complex systems or more struc-tured categories (their global behavior) to semiot-ical languages that, by means of a computationalprocess, transform eminently numerical meaningsinto sonorous and musical ones. Such semioticalprocess is not always faithful and the effort tobe made consists in creating a good computa-tional model (code) that, interpreting the inputdata (some variables of Chua’s oscillators) pro-duces a significant translation process, creating anoutput which maintains in some way the semioti-cal characteristics of the dynamical systems them-selves. From data reported by human listeners, thisprocess seems to have been obtained partially forthe continuous dynamical systems, paradoxicallythrough their discretization (transformation intonotes), that find their continuity only in the tempo-ral execution of the musical piece. In short, we haveused a piece of simulated reality or one of its rep-resentations, constituted by Chua’s oscillators, we

have fragmented it in many small parts, in order toreassemble it in continuous sound and/or musicalflow. The perceptual data reported by human lis-teners with various codes differ. They do not likeor they like little the musical pieces produced bysome codes, while the musical pieces produced byother codes are considered sometimes with favourand sometimes in a negative way. The semioticprocess has been carried out to make a piece ofreality discrete (or its representation through a sim-ulated model) when it appears as being continu-ous. Therefore, from the perceptual point of viewwe can say that the human brain considers pleas-ant the sequences where the continuity prevails incomparison with those in which the discretizationprevails, that means a lack of organization, withthe absence of sound structures and the correlatedlack of sound dynamic behaviors in the listenedpieces.

We have summarized in Table 3 the phenom-ena we have investigated. Our hypothesis is thatlistening is a dynamic interaction process withthe sound environment, through which the subjectsynchronizes with other dynamical systems, as ithappens with clocks hung on the same wall whichafter a period of time mark all the same hour. Inparticular, the cerebral dynamics of the subjectorganizes with the dynamics produced by the soundor musical flow, creating pleasant feelings, if suchsynchronization concretely happens, and unpleas-antness in the opposite case. Table 3 emphasizesthat the subjects perceive the presence of sonorousor musical Gestalten in the musical pieces generatedusing Chua’s oscillators which create some emerg-ing modalities of the dynamic systems. The basicidea is that Chua’s attractors are models of realityand we can understand complexity by means of aprocess of comprehension, only because complexityuses the same organizational laws we normally usein the perceptual interpretation of reality. In theprocess of codifying and classifying Chua’s attrac-tors, we can use the same laws to detect the maincharacteristics of complexity. We can say that thebifurcation maps produced by the parameters of aChua’s attractor are the ideal spaces of all possi-ble musical compositions we can obtain. Then bychoosing some of the parameters of these maps, wecan generate real occurrences of musical pieces thatwe have analyzed above, according to their group-ing and metrical structures which give to the lis-teners the perception that in the musical piecesthere are dynamics and the possibility to detect

Table 3. Summary table of the perceptual responses of human subjects to the produced compositions. This table underlines that the subjectsperceive the presence of sonorous and musical Gestalten from the compositions produced using Chua’s attractors according to the specific dynamicsystem we have translating into music and to the codes we have used.

PleasantnessPerception = YES Presence of Identification of

Dynamic Sound Flow Unpleasantness Sound and Dynamic SoundSystems Used Code Perception Perception = NO Music Gelstalten and Music Behaviour

Continuous Discrete Continuous YES Organization YES

Discrete Discrete (local) Continuous NO Disorganization NO

Discrete Discrete (global) Continuous YES and NO Organization NO

Discrete/Continuous Discrete/Continuous Discrete/Continuous YES Disorganization YES

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these dynamics. The grouping structure reflectsthe listener’s recursive segmentation of the musicalcontinuum into progressively larger units. At largerlevels, grouping boundaries are articulated by struc-tural accents, and it is these structural accents thatare heard as rhythmically important, related tothe changes in the structures of Chua’s attractors,produced by different values of the bifurcationmaps. The analysis of the musical pieces producedby means of Chua’s attractor allows us to detectthree main regions of grouping structures in musi-cal compositions. The first is an ordered region. Inthis portion of the space of musical composition,the behavior of Chua’s oscillator goes toward fixedpoints. The structural organization of the music wehave obtained is ordered, according to ascendingand descending movements and with well organizedrelationship among the melodies, which reflect thethree curves of the system. We can roughly iden-tify this region as going from the values 6.55 to7.46. In the musical compositions obtained by thesevalues it seems that there exists a “musical gram-mar” [Lerdahl & Jackendoff, 1983] which consists ofboth well-formed rules, which describe the minimalconditions for an intuitively understandable struc-ture, and preference rules, which correspond to theintuitions that allow a listener to choose the favoriteinterpretation of the structure from all of the possi-ble ones that conform to the well-formed rules. Thesecond region is complex, from about 7.46 to 8.80.The musical compositions obtained by these vari-ous values still have well organized structures, evenif with composite melodic movements and they tendto vary faster as the parameters change. Well struc-tured musical patterns persist, even if it seems thatthere is no unifying grammar. The third region ischaotic and goes roughly from about 8.90 to 9.16.In this region, the ordered organization which waspeculiar of the preceding compositions seems lostand lacking coherence. At the same time a musi-cal grammar does not exist, the minimal condi-tions for an intuitively understandable structure.The rhythmical organization seems to vary and topass from an accented and regular structure to theabrupt change of this structure. Furthermore, thestructures present in these compositions are veryshort, so it is impossible to perceive them at thesurface of the musical organization while the rela-tionships among the three curves are usually notconsonant, producing in the listeners the idea ofunpleasant music. The fourth region ranging from9.20 to 11.00 values is complex.

In the fifth region, from 11.01 to 11.08, thesystem tends to disappear and it is necessary tochange other parameters of the system (jumpinginto another dimension of the parameters space?) inorder to allow the system to reappear. These con-clusions are only at a preliminary stage, as we didnot have the opportunity to explore all the otherparameters of the systems, translating them intomusic, as well as analyzing them from the per-ceptual point of view. Further developments willinclude the translation into music of Chua’s n-scroll systems, hyperchaotic systems, coupled sys-tems together with the production of artworks andartificial worlds. These topics will be extensivelydealt with in the second part of this tutorial.

Acknowledgments

The authors are grateful to Prof. Leon O. Chua, forkindly inviting them to write this contribution andfor the enriching intellectual opportunities beingcontinuously provided to their group. The authorswould also like to express their gratitude to the fol-lowing people, for their helpful support:

• Anna Rosa Gabriele and Fausto Stranges,Psychology of programming and Artificial Intelli-gence PhD students, and Stefano Vena, engineer-ing student, for their assistance with the graphics,the software development and interfaces;

• Professor Marcello Anile and Dr. Giuseppe Alıwho read the draft and made many constructivesuggestions;

• Rossano Campolo, engineering student andMarcella G. Lorenzi, Psychology of programmingand Artificial Intelligence PhD student, for thedesign and implementation of the website devotedto Chua’s attractors;

• Adriano Talarico, Arts and Humanities stu-dent, for the cover illustration and the graphicalorganization of this tutorial;

• Professor Bertacchini, under whose auspices theInterdepartmental Center of Communication,environment of innovation and growth of theirgroup, was created, for pushing them beyondtheir intellectual potentialities.

References

Abarbanel, H. D. I., Brown, R., Sidorowich, J. J. &Tsimring, L. Sh. [1993] “The analysis of observedchaotic data in physical systems,” Rev. Mod. Phys.65, 1331–1392.


Arena, P., Baglio, S., Fortuna, L. & Manganaro, G.[1995] “Chua’s circuit can be generated by CNN cell,”IEEE Trans. Circuits Syst. I: Fund. Th. Appl. 42,123–125.

Barry, R. [2001] “Unifinished symphonies — songs of3 1

2 worlds,” in Proc. ECAL 2001 Workshop ArtificialLife Models for Musical Applications, eds. Bilotta, E.,Miranda, E. R. & Todd, P. M. (Springer-Verlag,Heidelberg), pp. 51–64.

Bentley, P. J. & Corne, D. W. [2002] Creative Evo-lutionary Systems (Morgan Kaufmann Publishers,San Francisco).

Bidlack, R. [1992] “Chaotic systems as simple (butcomplex) compositional algorithms,” Comput. MusicJ. 16, 33–47.

Bigand, E., Parncutt, R. & Lerdahl, F. [1996] “Percep-tion of musical tension in short chord sequences: Theinfluence of harmonic function, sensory dissonance,horizonal motion, and musical training,” Percep.Psychophys. 58, 125–141.

Bilotta, E. & Pantano, P. [2000] “In search for musi-cal fitness on consonance,” Electron. Musicol. Rev.,Special Issue 5(3).

Bilotta, E., Pantano, P. & Talarico, V. [2000] “Syn-thetic harmonies: An approach to musical semiosis bymeans of cellular automata,” in Artificial Life VII,eds. Bedau, M. A., McCaskill, J. S., Packard, N. H. &Rasmussen, S. (MIT Press, Cambridge), pp. 537–546.

Bilotta, E. & Pantano, P. [2001a] “Artificial life musictells complexity,” in Proc. ECAL 2001 Workshop onArtificial Life Models for Musical Applications, eds.Bilotta, E., Miranda, E. R. & Todd, P. M. (Springer-Verlag, Heidelberg), pp. 17–29.

Bilotta, E. & Pantano, P. [2001b] “Observations oncomplex multi-state CAs,” in Advances in Artifi-cial Life: 6th European Conf., ECAL 2001, eds.Kelemen, J. & Sosik, P., Lecture Notes inComputer Science (Springer-Verlag, Heidelberg),pp. 226–235.

Bilotta, E. & Pantano, P. [2002a] “Synthetic harmonies:Recent results,” Leonardo 35, 35–42.

Bilotta, E. & Pantano, P. [2002b] “Self-reproducersuse contrapuntal means,” in Proc. ALife VIIIWorkshops, eds. Bilotta, E., Groß, D., Smith, T.,Leanerts, T., Bullock, S., Lund, H. H., Bird, J.,Watson, R., Pantano, P., Pagliarini, L., Abbass, H.,Standish, R. & Bedau, M. (University of New SouthWales, Sydney), pp. 3–8.

Bilotta, E., Miranda, E. R., Pantano, P. & Todd, P. M.[2002] “Artificial life models for musical applications,”Artif. Life 8, 83–86.

Bilotta, E., Di Bianco, E., Pantano, P. & Vena, S. [2003]“A visual programming language for sound synthesisand analysis,” Cogn. Process. 4, in press.

Bilotta, E. & Pantano, P. [2004] “Matematica, musicae tecnologie: Un trinomio possibile,” in Atti del

Convegno Nazionale Matematica Senza Frontiere,Dipartimento di Matematica, Universita degli Studidi Lecce, Quaderno No. 2, pp. 309–324.

Bilotta, E., Di Bianco, E., Gervasi, S. & Pantano,P. [2004] “Discovering complexity and emergence inCellular Automata using music,” submitted.

Bolognesi, T. [1983] “Automatic composition: Experi-ments with self-similar music,” Comput. Music J. 7,25–36.

Bregman, A. S. [1990] Auditory Scene Analysis (MITPress, Cambridge, MA).

Brooks, R. A. [1991] “Intelligence without representa-tion,” Artif. Intell. 47, 139–159.

Brown, G. J. & Cooke, M. P. [1994] “Computationalauditory scene analysis,” Comput. Speech Lang. 8,297–336.

Cangelosi, A. & Parisi, D. [2002] “Computer simulation:A new scientific approach to the study of languageevolution,” in Simulating the Evolution of Language,eds. Cangelosi, A. & Parisi, D. (Springer-Verlag,London), Chap. 1, pp. 3–28.

Chomsky, N. [1965] Aspects of the Theory of Syntax(MIT Press, Cambridge MA).

Chowning, J. & Bristow, D. [1986] FM Theory andApplications (Yamaha Music Foundation, Tokyo).

Chua, L. O., Komuro, M. & Matsumoto, T. [1986] “Thedouble scroll family,” IEEE Trans. Circuits Syst. 33,1072–1118.

Chua, L. O. & Yang L. [1988] “Cellular neural network:Theory and applications,” IEEE Trans. Circuits Syst.35, 1257–1290.

Chua, L. O. & Lin, G. N. [1990] “Canonical realizationof Chua’s circuit family,” IEEE Trans. Circuits Syst.37, 885–902.

Chua, L. O. [1993] “Global unfolding of Chua circuits,”IEICE Trans. Fundam. Electron. Commun. Comput.Sci. E76-A, 704–734.

Chua, L. O. & Roska, T. [1993] “The CNN paradigm,”IEEE Trans. Circuits Syst.-I: Fund. Th. Appl. 40,147–156.

Chua, L. O., Wu, C. W., Huang, A. & Zhong, G.[1993a] “A universal circuit for studying and generat-ing chaos. I. Routes to chaos,” IEEE Trans. CircuitsSyst.-I: Fundam. Th. Appl. 40, 732–744.

Chua, L. O., Wu, C. W., Huang, A. & Zhong,G. Q. [1993b] “A universal circuit for studyingand generating chaos. II. Strange attractors,” IEEETrans. Circuits Syst.-I: Fundam. Th. Appl. 40,745–761.

Chua, L. O. [1998] CNN: A Paradigm for Complexity(World Scientific, Singapore).

Cook, P. R. [2001] Music, Cognition, and ComputerizedSound (MIT Press, Cambridge, MA).

Cope, D. & Hofstadter, D. R. [2001] Virtual Music:Computer Synthesis of Musical Style (MIT Press,Cambridge, MA).


D’Arcy Thompson, W. [1917] On Growth and Form(Cambridge University Press, Cambridge, MA).

Dodge, C. [1988] “Profile: A musical fractal,” Comput.Music J. 12, 10–14.

Eco, U. [1975] Trattato di Semiotica Generale(Bompiani, Milano).

Elman, J. L. [1995] “Language as a dynamical system,”in Mind as Motion: Explorations in the Dynamics ofCognition, eds. Port, R. F. & van Gelder, T. (MITPress, Cambridge, MA), pp. 195–223.

Frova, A. [1999] Fisica nella Musica (Zanichelli,Bologna).

Geymonat, L. [1971] Storia del Pensiero Filosofico eScientifico (Garzanti, Milano).

Gogins, M. [1991] “Iterated function systems music,”Comput. Music J. 15, 40–48.

Hilborn, R. [1994] Chaos and Nonlinear Dynamics: AnIntroduction for Scientists and Engineers (OxfordUnivesity Press, Oxford).

Hofstadter, D. R. [1979] Godel, Escher, Bach: AnEternal Golden Braid (Basic Books, NY).

Holleran, S., Jones, M. R. & Butler, D. [1995] “Perceiv-ing musical harmony: The influence of melodic andharmonic context,” J. Experim. Psychol.: Learning,Mem. Cogn. 21, 737–753.

Imberty, M. [1987] “La psicologia della musica: Problemigenerali, percezione ed educazione,”beQuadro 27/28,5–9.

Imberty, M. [2000] “The question of innate compe-tencies in musical communication,” in The Ori-gins of Music, eds. Wallin, N. L., Merker, B.& Brown, S. (MIT Press, Cambridge, MA),pp. 449–462.

Jakobson, R. [1963] Essais de Linguistique Generale(Les editions de minuit, Paris).

Kauffman, S. A. [1989] Origins of Order : Self-Organization and Selection in Evolution (OxfordUniversity Press, Oxford).

Kennedy, M. P. [1993a] “Three steps to chaos. I.Evolution,” IEEE Trans. Circuits Syst.-I: Fundam.Th. Appl. 40, 640–656.

Kennedy, M. P. [1993b] “Three steps to chaos. II. AChua’s circuit primer,” IEEE Trans. Circuits Syst.-I:Fundam. Th. Appl. 40, 657–674.

Kohler, W. [1925] The Mentality of Apes (HarcourtBrace, NY).

Komuro, M., Tokunaga, R., Matsumoto, T., Chua, L. O.& Hotta, A. [1991] “Global bifurcation analysis of thedouble scroll circuit,” Int. J. Bifurcation and Chaos1, 139–182.

Kramer, G. [1994] Auditory Display: Sonification, Aud-ification and Auditory Interfaces (Addison-Wesley,NY).

Krumhansl, C. L. [1979] “The psychological represen-tation of musical pitch in a tonal context,” Cogn.Psychol. 11, 346–374.

Krumhansl, C. L. [1990] Cognitive Foundationsof Musical Pitch (Cambridge University Press,Cambridge).

Kunej, D. & Turk, I. [2000] “New perspectives onthe beginnings of music: Archaeological and musi-cological analysis of a Middle Paleolithic bone‘flute’,” in The Origins of Music, eds. Wallin, N. L.,Merker, B. & Brown, S. (MIT Press, Cambridge,MA), pp. 234–268.

Langton, C. G. [1986] “Studying artificial life withcellular automata,” Physica D22, 120–149.

Langton, C. G. [1990] “Computation at the edge ofchaos: Phase transition and emergent computation,”Physica D42, 12–37.

Langton, C. G. [1995] Artificial Life: An Introduction(MIT Press, Cambridge, MA).

Leach, J. & Fitch, J. [1995] “Nature, music, and algo-rithmic composition,” Comput. Music J. 19, 23–33.

Lerdahl, F. & Jackendoff, R. [1983] A Generative Theoryof Tonal Music (MIT Press, Cambridge, MA).

Lewin, K. [1936] Principles of Topological Psychology(McGraw-Hill, NY).

Longuet-Higgins, H. C. [1994] “Artificial intelligence andmusical cognition,” Philos. Trans. Roy. Soc. LondonA349, 103–113.

Luise, M. & Vitetta, G. M. [1999] Teoria dei Segnali(McGraw-Hill, Milano).

Madan, R. N. [1992] “Observing and learning chaoticphenomena from Chua’s circuit,” in Proc. 35th Mid-west Symp. Circuits and Syst. (Washington, D.C.),pp. 736–745.

Madan, R. N. [1993] Chua’s Circuit: A Paradigm forChaos (World Scientific, Singapore).

Manaris, B., Vaughan, D., Wagner, C., Romero, J.& Davis, R. [2003] “Evolutionary music and theZipf–Mandelbrot law: Developing fitness functionsfor pleasant music,” in Applications of Evolution-ary Computing, Lecture Notes in Computer Science(Springer-Verlag, Heidelberg), pp. 522–534.

Marr, D. [1982] Vision (Freeman and Company, NY).Mathews, M. V. & Pierce, J. R. [1989] Current Directions

in Computer Music Research (MIT Press, Cambridge,MA).

Matsumoto, T. [1984] “A chaotic attractor from Chua’scircuit,” IEEE Trans. Circuits Syst. 31, 1055–1058.

Matsumoto, T., Chua, L. O. & Komuro, M. [1985]“The double scroll,” IEEE Trans. Circuits Syst. 32,797–818.

Matsumoto, T., Chua, L. O. & Ayaki, K. [1988] “Realityof chaos in the double scroll circuit: A computer ass-isted proof,” IEEE Trans. Circuits Syst. 35, 909–925.

McAdams, S. [1984] “The auditory image: A metaphorfor musical and psychological research on auditoryorganisation,” in Cognitive Processes in the Percep-tion of Art, eds. Crozier, W. R. & Chapman, A. J.(North-Holland, Amsterdam), pp. 289–323.


McAdams, S. [1989] “Segregation of concurrent sounds.I: Effects of frequency modulation coherence,” J.Acoust. Soc. Amer. 86, 2148–2159.

Miranda, E. R. [2001] “Evolving cellular automatamusic: From sound synthesis to composition,” in Proc.ECAL 2001 Workshop on Artificial Life Models forMusical Applications, eds. Bilotta, E., Miranda, E. R.& Todd, P. M. (Springer-Verlag, Heidelberg),pp. 87–99.

Misdariis, N., Smith, B., Pressnitzer, D., Susini, P.,& McAdams, S. [1998] “Validation and multidimen-sional distance model for perceptual dissimilaritiesamong musical timbres,” in Proc. 16th Int. Con-gress on Acoustics, eds. Kuhl, P. K. & Crum, L.(Acoustical Society of America, Seattle), pp. 2213–2214.

Morris, C. [1971] Writings on the General Theory ofSigns (The Hague, Mouton).

Narmour, E. [1990] The Analysis and Cognition of BasicMelodic Structures (University of Chicago Press,Chicago).

Nowak, M. A. & Komarova, N. L. [2001] “Towards anevolutionary theory of language,” Trends in Cogn.Sci. 5, 288–295.

Parsons, L. M. [2001] “Exploring the functionalneuroanatomy of music performance, perception,and comprehension,” Ann. NY Acad. Sci. 930,211–231.

Parsons, L. M. & Thaut, M. H. [2001] “Functionalneuroanatomy of the perception of musical rhythmin musicians and non-musicians,” Neuroimage 13,p. 925.

Pierce, J. R. [1983] The Science of Musical Sound(Freeman and Company, NY).

Povel, D. J. & van Egmond, R. [1993] “The functionof accompanying chords in the recognition of melodicfragments,” Music Percep. 11, 101–115.

Pressing, J. [1988] “Nonlinear maps as generators ofmusical design,” Comput. Music J. 12, 35–45.

Risset, J. C. [1969] Introductory Catalogue of Com-puter Synthesized Sounds (Bell Telephone Laborato-ries, Murray Hill, NJ).

Risset, J. C. & Wessel, D. [1982] “Exploration of timbreby analysis and synthesis,” in Psychology of Music, ed.Deutsch, D. (Academic Press, Orlando), pp. 26–58.

Roads, C. [1996] The Computer Music Tutorial (MITPress, Cambridge, MA).

Rodet, X. [1993] “Sound and music from Chua’s cir-cuit,” in Chua’s Circuit: A Paradigm for Chaos, ed.Madan, R. (World Scientific, Singapore), pp. 434–446.

Rodet, X. & Vergez, C. [1999] “Nonlinear dynamics inphysical models: Simple feedback-loop systems andproperties,” Comput. Music J. 23, 18–34.

Scaletti, C. [1994] “Sound synthesis algorithms forauditory data rapresentations,” in Auditory Display:Sonification, Audification and Auditory Interfaces,ed. Kramer, G. (Addison-Wesley, NY), pp. 223–251.

Schmuckler, M. A. & Boltz, M. G. [1994] “Harmonicand rhythmic influences on musical expectancy,”Percep. Psychophys. 56, 313–325.

Schottstaedt, W. [1989] “Automatic counterpoint,” inCurrent Directions in Computer Music Research,System Development Foundation Benchmark, eds.Matthews, V. & Pierce, J. R. (MIT Press, Cambridge,MA), pp. 199–214.

Schwanauer, S. M. & Levitt, D. A. [1993] MachineModels of Music (MIT Press, Cambridge, MA).

Scimemi, B. [2001] “Contrappunto musicale,” in Mate-matica e Cultura 2001, ed. Emmer, M. (Springer-Verlag, Milano), pp. 119–134.

Shepard, R. N. [1964] “Circularity in judgmentsof relative pitch,” J. Acoust. Soc. Am. 36, 2346–2353.

Shil’nikov, L. P. [1993] “Chua’s circuit: Rigorous resultsand future problems,” IEEE Trans. Circuits Syst. I:Fundam. Th. Appl. 40, 784–786.

Spector, L. & Klein, J. [2002] “Complex adaptive musicsystems in the BREVE simulation environment,”in Proc. ALife VIII Workshops, eds. Bilotta, E.,Groß, D., Smith, T., Leanerts, T., Bullock, S.,Lund, H. H., Bird, J., Watson, R., Pantano, P.,Pagliarini, L., Abbass, H., Standish, R. & Bedau, M.(University of New South Wales, Sydney), pp. 17–24.

Steedman, M. [1994] “The well-tempered computer,”Philos. Trans. Roy. Soc. London A349, 115–131.

Stewart, I. & Golubitsky, M. [1992] Fearful Symmetry:Is God a Geometer? (Blackwell Publishers, Oxford).

Strogatz, S. [1994] Nonlinear Dynamics and Chaos(Addison-Wesley, MA).

Swain, J. P. [1997] Musical Languages (NortonCompany, NY).

Terhardt, E. [1974] “Pitch, consonance and harmony,”J. Acoust. Soc. Amer. 55, 1061–1069.

Thompson, W. F. [1993] “Modeling perceived relation-ships between melody, harmony, and key,” Percep.Psychophy. 53, 13–24.

Todd, P. M. & Werner, G. M. [1999] “Frankensteinianmethods for evolutionary music composition,” inMusical Networks: Parallel Distributed Perceptionand Performance, eds. Griffith, N. & Todd, P. M.(MIT Press, Cambridge, MA), pp. 313–339.

Todd, P. M. [2000] “The ecological rationality of mecha-nisms evolved to make up minds,” Amer. Behav. Sci.43, 940–956.

Turing, A. M. [1952] “The chemical basis of morphogen-esis,” Philos. Trans. Roy. Soc. London B237, 37–72.

Varela, F. J., Thompson, E. & Rosch, E. [1991]The Embodied Mind: Cognitive Science and HumanExperience (MIT Press, Cambridge, MA).

Voss, R. F. & Clarke, J. [1975] “1/F Noise in music andspeech,” Nature 258, 317–318.

Voss, R. F. & Clarke, J. [1978] “Music from 1/F noise,”J. Acoust. Soc. Amer. 63, 258–263.

Wallin, N. L., Merker, B. & Brown, S. [2000] The Originsof Music (MIT Press, Cambridge, MA).


Wertheimer, M. [1938] “Laws of organization in percep-tual forms,” in A Source Book of Gestalt Psychology,ed. Ellis, W. D. (Routledge & Kegan Paul, London),pp. 71–88.

Wertheimer, M. [1958] “Principles of perceptualorganization,” in Readings in Perception, eds.Beardslee, D. C. & Wertheimer, M. (van Nostrand,NY), pp. 115–135.

Witten, M. [1996] “The sounds of science: Listening todynamical systems towards a musical exploration ofcomplexity,” Comput. Math. Appl. 32, 145–173.

Wolfram, S. [1984a] “Computer software in science andmathematics,” Sci. Amer. 251, 188–203.

Wolfram, S. [1984b] “Universality and complexity incellular automata,” Physica D10, 1–35.

Xenakis, I. [1971] Formalized Music: Thought and Math-ematics in Composition (Indiana University Press,Bloomington/London).

Zhong, G. Q. & Ayrom, F. [1985a] “Periodicity and chaosin Chua’s circuits,” IEEE Trans. Circuits Syst. 32,501–503.

Zhong, G. Q. & Ayrom, F. [1985b] “Experimental confir-mation of chaos from Chua’s circuit,” Int. J. CircuitTh. Appl. 13, 93–98.

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