Clouds, Pyramids, and Diamonds- Applying Schrödinger's Equation to Granular Synthesis and Compositional Structure

8/2/2019 Clouds, Pyramids, and Diamonds- Applying Schrdinger's Equation to Granular Synthesis and Compositional Structure

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Clouds, Pyramids, and Diamonds: Applying Schrdinger's Equation to Granular Synthesis andCompositional StructureAuthor(s): Rajmil FischmanReviewed work(s):Source: Computer Music Journal, Vol. 27, No. 2 (Summer, 2003), pp. 47-69Published by: The MIT PressStable URL: http://www.jstor.org/stable/3681609 .Accessed: 23/02/2012 13:50

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Rajmil ischmanMusic DepartmentKeele UniversityKeele, Staffordshire,United Kingdom ST5 [email protected] l o u d s , Pyramids, a n dDiamonds: ApplyingSchrodinger'squationt o Granular Synthesisa n d CompositionalStructure

This article presents the results of research on thegeneration of musical processes resulting from theapplicationof Schrddinger'sEquationfor an atomicpotential with radialsymmetry, a mathematicalmodel of dynamic systems that, like music, maychange and develop in time. (Foran introductorydiscussion of Schr6dinger'sEquation,see Feynman,Leighton,and Sands 1971.) I focus on the creationof musical material and its organization arisingfrom the solutions and implications of the equa-tion. This is appliedto asynchronous granulartechniques (e.g.,Truax 1990b;De Poli, Piccialli,and Roads 1991; Di Scipio 1994), as these are ide-ally suited for stochastic processing of musicalmaterial and allow the manipulation of multi-dimensional timbral characteristics. Work carriedout consisted of three main parts:the developmentof relevant theoretical principles and algorithms forthe generation of music, their implementation ascompositional software, and their realization as acomposition that demonstrates their musical valid-ity and viability.

BackgroundThe relationshipbetween scientific disciplines andmusic is not new in Western culture: suffice it tomention Boethius's concept of "music of thespheres,"the medieval quadrivium, the relation-ship between talea and color in isorhythmic musicof the 14th century, and rules of counterpoint re-garding temporal proportionsbetween voices inFux's Gradus ad Parnassum in the 18th century.During the 20th century, a renewed link between

science and music began to emerge in composition,from the relatively straightforwardprinciples of se-rialism to Schillinger'scompositional theory (Schil-linger 1948), the complex constructs realized byXenakis in works such as AchorripsisandNomosAlpha (Xenakis 1992), the compositional strategiesof Koenig(Koenig1970a and 1970b;Laske 1981)and Hiller (Hiller 1981), and the use of stochasticprocesses by Truax(1982)and others (e.g., Jones1981; Lyon 1995;Ross 1995;Hoffman 2000). Theo-retical and technical advances in information tech-nology propitiatedfurtherexploration in areas suchas generative grammars(Holzman 1981;Beyls1991);self-similar,chaotic, andmore general dy-namic systems (Bolognesi 1983;Dodge 1988;Press-ing 1988;Truax 1990a;Di Scipio 1991;Bidlack1992);and more general algorithmic approaches(Maiaet al. 1999;McAlpine, Miranda,and Gerard1999).The aim of the present work is to investigate andexperiment with the possibilities offeredin thisareaby an equation that contributed to a deepchange in our conception of the world, followingthis path to its ultimate conclusion (i.e., the musi-cal composition). In this sense, this study presentsan additional link between the thought processesthat lead to the mathematical abstraction of reality(i.e., models of the physical world)and those in-volved in the creation of musical works. These pro-cesses mirror each other in the sense that theformerconstruct meaning out of the observationofthe physical world, and the latter transform mean-ing into a physical trace, namely, a piece of music.The currentprojectdevelops previous work onthe derivation of organic structure andgenerationof material offeredby Schr6dinger'sEquation(Bain1990; Fischman 2003), which preservesmusical

ComputerMusic Journal,27:2,pp. 47-69, Summer 2003C 2003 Massachusetts Institute of Technology.

Fischman 47


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logic and coherence. While previous researchbythe authorwas restricted to the derivation of struc-ture in a composition for a single instrument (harp-sichord)and to the generation of pitch material, thegoal of the present study was to derive a strategyfor the creation of structure and musical materialfor a medium which allows greater complexity, in-cludes parametersother than pitch, andprovidesthe basis for the applicationof similar strategies toother media. Thus, the electroacoustic domain,with a specific focus on asynchronous granulartechniques, was chosen, because it allows a signifi-cant degreeof complexity, and it allows thoroughinvestigation of the possibilities inherent in Schr6-dinger'sEquationfor the manipulation of multi-dimensional timbralcharacteristics, with pitchincluded as one of these dimensions (Smalley 1986,1997).In particular,granular echniques areideallysuited for stochastic processingof musical material(Lorrain1980;Xenakis 1992).Furthermore,a workfor tape can providethe immediate auralfeedbacknecessary to evaluate the results.The researchmethodology adopted duringtheproject consisted of three main parts.First,I devel-oped appropriate heoretical principles and algo-rithms for the creation of musical material arisingfrom the solutions and implications of Schra-dinger'sEquation.Next, I developedsoftware thatimplements these algorithms. The software in-cludes a graphicalinterface that enables the manip-ulation of largeamounts of data and musicalstreams, is reasonably self-sufficient for the crea-tion of a complete musical work, and is reasonablyopen-ended,offering possibilities for enhancementand customization via third-partyplug-ins using arecognized standard.The third part of this researchconsisted of testing the validity andviability of thetheoretical principles and the usefulness of thesoftware throughthe creation of a musical work fortape:Erwin'sPlayground.Concerning the last point, it is importanttostress that neither theory nor software are intendedto guarantee satisfactory aesthetic results-or evenmusical coherence. I do believe that these are theresponsibility of the individual who realizes theirpotential through the compositional process, a sys-tem described by Vaggione (2001) as an "action/per-

ception feedback loop." In fact, it is reasonable toassume that most mathematical constructs areneutral to musical logic and that their usefulness isdependenton the actual way they aremappedintomusical processes by the composer. Nevertheless,theoretical principles are vital in providing"formaltools as generative and transformativedevices"(Vaggione2001) within which a specific work maybe realized,projectingthe latter beyond an exclu-sively sensory sphere.Furthermore,decisions re-gardingthe correspondencebetween the model'sparametersand musical attributes,as well as theirrealization by means of the design and functional-ity of the software (which, at first sight, mightseem to be exclusively related to theoretical andtechnical development) already belong to the com-positional process, since they carryspecific aes-thetic presumptionsand aims. (Fora discussion ofthis issue in a more general context, see Vaughan1994.) Forinstance, recent published work presentsdifferentontological approachesto correspondencesbetween quantum mechanics and music (Delatour2000; Sturm2001).After a brief descriptionof Schr6dinger'sEqua-tion, this article presents the frameworkdevelopedfor the correspondencebetween the equation andthe creation of granularclouds and strategies fortheir organization into coherent musical structure.This will be followed by discussion regarding herealization of these principles in Erwin'sPlay-ground, a work for computer-generated ape. Subse-quently, a general descriptionof the software willbe provided.An appendixis offeredto readerswhomay wish to explore the mathematics in more de-tail. (Referenceto equations in the appendixusesthe prefix "A"followed by a number.)

Schr6dinger'squationSchr6dinger'sDifferential Equationforms the basisfor the techniques describedherein, which estab-lish a correspondencebetween the solutions of theequation and musical attributes. We will examinerelevant aspects of these solutions, with a more de-tailed mathematical discussion featured in the ap-pendix (equations A1-A8).

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The particularversion of Schr"dinger'sEquationexamined here results from consideration of theatom as a center that produces a potential. The lat-ter is known as an atomic potential with radialsymmetry, because, if we look at a sphericalsur-face of radiusr, the potential will be symmetricwith respect to the center. Thus, Schr6dinger'sEquationassumes the following form:- V2T + V(r)Y = ih (1)2m at

where V(r) s the atomic potential, m is the mass ofthe electron, r is the distance from the center of thepotential (e.g., the nucleus of an atom) to the elec-tron, and h is the ratio between Plank's constant hand 27r(h = h/27r).The family of functions corresponding o thesolutions of the equation is representedby thesymbol T, also known as the wave function. Exam-ination of the general form of the wave function re-veals that Yconsists of the productof four smallerfunctions (equation A2) that may be treated sepa-rately, because each is dependenton a differentpa-rameter. Three of these were found particularlyuseful: R, which is exclusively dependenton r, thedistance of a point to the center of the potential; 0,which is exclusively dependentof the point's eleva-tion angle 0; and 0, which is exclusively dependenton its horizontal angle 0.Furthermore, he solutions dependon a set ofquantum numbers that restrict the physical attri-butes of the electron to multiples of certain energylevels. The first of these values is n, the principalquantum number (n = 1, 2, 3, ...), which deter-mines the overall energy of the electron (equationA7). The second value is 1, the angularmomentumquantum number (1 = 0, 1, 2, ... , n-l), which gen-erates additionalallowed energies (quantum states)within a single value of n. (The term shell is usedto denote a particularvalue of 1.)So far, four typesof shells have been encounteredin nature and arenamed S, P, D, and F, correspondingto 1 = 0, 1, 2,and 3, respectively. Therefore, a particular energylevel is characterized by a combination of n and 1,which, by convention, is named using the value ofthe principal quantum number and the letter of itsshell (e.g., IS, 2S, and 2P).

The thirdquantum number, m, is the magneticquantum number (-m1 m ? 1).Its values producevariations in the orientation of these shells, knownas orbitals, but m does not affect the overall energyin this model.The physical meaning of an orbital is a regionwithin which there is some probabilityof findingan electron. It is possible to show that this proba-bility is associated with R and O (equationsA9 andA10). As an example of this correlation, Figure 1shows plots of these distributions and the resultingorbital for particularvalues of n, 1,andm, fromwhich we may infer the most probabledistancesand orientations. Forinstance, Figure lb indicatesthat the electron is more likely to be found in thevicinity of 35.2 7 and 144.73'.The quantum numbers were successfully used toexplain the existence of common characteristicsbetween known chemical elements, supportingtheir classification into the Periodic Table of Ele-ments, originallydevised by the chemist DmitriMendeleev in the 19th century but only more re-cently explained by Schr6dinger'sEquation.Roughly speaking, elements that have the samenumber of outermost electrons (electrons withhighest quantum energy)in the same type of shellshare similar physical characteristics, belonging tothe same groupand occupying the same column inthe periodic table. Forinstance, the groupof halo-gens consists of elements that have five electronsin their P shell as follows: fluorine has five elec-trons in shell 2P, chlorine in 3P, bromine in 4P,iodine in 5P, and astatine in 6P.

Generationf Granular loudsMy generation of musical material using Schr6-dinger'sEquationfocuses on granularclouds(sounds composed of a largenumber of short ele-mentary sonic particles, or grains, each lasting afew hundredths of a second) derived from statisticaldistributions obtainedfrom the equation. A parallelis established between these clouds and the chargedensity surrounding the nucleus (electronic clouds).The algorithms affect a variety of time-varying at-tributes of the granular clouds. Special importanceFischman 49


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Figure1. Examples ofSchr6dingerdistributions:r2IRI2or Z = 139, n=4,and I= 1, with radial val-ues normalized to (a) 1.5,

(b) I1(0)12or n = 4, 1= 1,m = 0, and (c) a diagram ofthe orbital resulting fromthese distributions.

(a)

0 0.09 0.31 0.76 1.5(b)

00 35.270 900 144.730 1800

(c) 35.270

. . . . . . . . . . . .

is given to the creation of formal relationships be-tween the inner structure of the source sound-par-ticularly its amplitude envelope-and the structureof the cloud and its constituent grains. Sourcesounds are stored as wave audio files.There are two main types of parameters:thoseaffecting the cloud as a whole and those affectingeach grain individually. These are now each dis-cussed in turn.

Cloud arametersThese parametersrefer to global attributes of thecloud, such as density, overall amplitude, and spa-tial and spectral trajectory.

Cloud DensityTo determine the time-varying cloud density (num-ber of grains per time unit), a correlation betweendensity and the amplitude envelope of the sourcesound is established: the amplitude envelope drivesthe radial and elevation distributions, which, inturn, areproportionalto the density fluctuation. Inpractice, the envelope is extracted from the sourceand stored in a time-amplitude breakpointtable.The cloud density is also stored as pairsof break-points. However, because the cloud duration can bedifferent from that of the envelope, it is necessaryto convert both times and amplitudes from the en-velope table into correspondingvalues for clouddensity (equationAl1).Currently, there are three differentmethods ofestablishing a correspondencebetween source am-plitude envelope and cloud density. The first twocalculate the density by using the envelope to drivethe radial and angulardistributions (equationA12),either by making both r and 0 directly proportionalto the envelope (equationA13) or by making r di-rectly proportionalto the envelope and 0 directlyproportionalto its first derivative (equationA14).The third method establishes a direct proportional-ity between the envelope and the density using astraightforwardproportionrule (equation A15).

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Figure . Valuesofx andy for spatialwidthanddepth.

y>O. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .x=-1 x=O x=l

Listener

x < -1 Beyond he leftspeakerx = -1 Left peaker-1 < x < 1 Between he speakersx = 1 Right peakerx > 1 Beyond he right peakery = 0 On the imaginaryinethat oinsthe speakers.y > 0 Behind he imaginaryinethat oinsthe speakers.When y = 1, the depth is equal to the distancefrom he centreto one of thespeakers.

Cloud Amplitude ScalingThe amplitude of the cloud may be scaled by avalue that fluctuates between a minimum andmaximum gain. The fluctuation follows the enve-lope of the source (equationA16).CloudMotionMotion s carried ut in threedimensions.Two ofthese arespatial widthanddepth),andthe remain-ing one is articulated s a spectraldimension pitchshift).Width and DepthThese are calculated within a stereo image:x indi-cates width, andy indicates depth, as illustrated inFigure2. (See also equation A17.)SpectralMotionThe cloud "moves" in the frequencydomain as aresult of time-varying transpositionwithin limitsspecified by the composer, establishing an analogyto movement across a height axis z. Transpositionis carriedout in the frequencydomain via Short-

Time FourierTransforms,thereby preservingtheduration of the grains.TrajectoryCreationTo implement motion, a three-dimensional trajec-tory is created that is analogous to physical move-ment within the boundariesof a shell (orbital).This consists of the stochastic generation of break-points accordingto Schr6dinger'sradial and angulardistributions.The breakpointsconsist of position-time pairswithin the duration of the cloud, con-verted from r, 0, and 4 into width, depth, andheight (equation A18).The algorithmthat generates the trajectorynor-malizes the values so that the cloud reaches themaximum width, maximum depth, and one of thetranspositionboundaries.(These arenot reachedsimultaneously.) Also, to generate the trajectorybreakpoints,the algorithmrequiresminimum andmaximum values for the time interval providedbythe composer. The intervals between the break-points in the trajectorywill fall randomly betweenthese two values. Forinstance, if the minimum andmaximum intervals are 0.2 and 0.5 sec, respec-tively, and the last breakpointwas generatedattime tc = 1.4 sec, the time tc for the next break-point will fall between 1.6 and 1.9 sec.Finally, trajectoriesmay be repeatedan arbitrarynumber of times so that the sum of the durationsof the repetitions is the overall duration of thecloud. (Ifrequired,this may be equal to the value ofthe orbital quantum number 1plus one.) Repeti-tions may be identical or fluctuate within a per-centage given by the composer.Cloud Volume (Overall Grain Scatter)The volume of the cloud dependson the maximumscatter of the grains in each dimension (width,depth, andpitch shift). As a simple illustration, as-sume that the cloud positioned at the center of thespeakers and does not move. In this case, low scat-ter values will produce a very narrow cloud aroundthe center of the speakers. On the other hand, highscatter values will spread the grains from left toright, producing a wide cloud. This is illustrated inFigure 3.Fischman 51


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Figure3. Examples ofgranularcloud volume:(a) narrowgrain scatterand (b) wide grain scatter.

(a)

(b)

Width and depth scattervalues vary from 0 (noscatter) to any positive numbersx and y, with avalue of 1 indicating a scatter distance equal to thedistance from the center to one of the speakers.Spectralscatter values vary from 0 (no scatter)toany positive number of semitones (orany fre-quency ratio).To implement scatter, a time-varying volumefunction is createdin a similar manner to that ofthe motion trajectorydescribedabove. Breakpointsare also generated stochastically accordingtoSchr6dinger'sradialand angulardistributions(equationA19).

Therefore, a grain created at time t, will movewith the cloud at a fixed "distance"from its center.This distance is generated stochastically withinwidth, depth, andheight boundaryvalues given bythe composer. However, to comply with the repre-sentation of depth (y > 0), the resulting instantane-ous depth of the grain as it moves with the cloud isalways kept positive, which means that the grainmay change its distance with respect to the centerof the cloud.Spectral scatter may be anchored or unanchored.In the formercase, the sign of the currentscatter

value indicates whether the grains will be posi-tioned to the right or left of (ortransposedabove orbelow) the cloud. Forinstance, when the scattervalue is positive, all the grains arepositioned tothe right of the cloud. When the scatter becomesnegative, the grains arepositioned to the left. Un-anchored scatter means that that the grains arepo-sitioned both to the right and left of (andtransposed above andbelow) the cloud.The algorithmthat generates the volume func-tion normalizes the values so that the cloudreaches the maximum width, depth, and transposi-tional scatter. (Again,these arenot reached simul-

taneously.) In a similar fashion to the motiontrajectory,the algorithmalso requiresthe composerto provideminimum and maximum values for thetime interval to generate time-volume breakpoints.The volume function may also be repeatedan arbi-trarynumber of times; this may also be equal tothe value of the orbitalquantumnumber 1plusone. Repetitions may be identical or fluctuatewithin a percentage given by the composer.

GrainParametersThese parameterscontrol the attributes of individ-ual grains,such as their envelope, duration,and in-dividual scatter patterns.GrainAmplitude ScalingThe amplitude of individual grains is scaled by again factor accordingto their onset times relativeto the beginning of the cloud and implemented ac-cordingto various algorithm types. The first optionconsists of a scaling function that follows the enve-lope of the source (equation A20). Other optionsfollow Schr6dinger'sdistribution (r2 in(0)l~'(r,0,I12in equations A21-A23), the complement of Schr6-dinger'sdistribution(equation A24), and Schr&-dinger'scumulative probabilityand itscomplement (equationsA25 and A26). It is worthnoting that if the cumulative probabilityis fol-lowed, amplitude scaling will assume its minimumvalue at the beginning of the cloud and increase to-wards its maximum value at the end of the cloud,producinga nonlinear crescendo. This is becausethe cumulative probabilityalways increases mono-tonically. Similarly, following the complement ofthe cumulative probabilitywill result in a nonlin-ear diminuendo.

Finally, scaling may be generatedfrom the vari-ous distributionsresulting from Schr6dinger'sEquation(equationsA27-A33).Grain DurationThe actual duration of individual grains fluctuatesaccording to their onset times relative to the begin-



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Figure4. Linearenvelopeparameters:(1) start offset,(2) overshoot attack,(3) overshoot decay,(4) steady state, (5) decay,(6) end offset, and(7) overshootpercentage.

o m m r s u t ( 1

2 3 4 5 6ning of the cloud. The function controlling the du-ration fluctuation is calculated in a manner similarto that of the scaling factor, offeringthe same op-tions to the user (equationsA34 and A35). How-ever, in addition to the options available foramplitude scaling, grain duration allows the repeti-tion of the duration function similar to that of thecloud's motion trajectoryand its volume function:repetitions may also be identical or fluctuatewithin a percentage given by the composer. Themain algorithms available consist of the followingschemes: durationproportionalto the source enve-lope, durationproportional to Schradinger'sdistri-bution and its complement, durationproportionalto Schr6dinger'scumulative probabilityand itscomplement, and durationgenerated randomlyfrom various distributions.It is worth noting that because the cumulativeprobability always increases monotonically, grainshave minimum duration at the beginning of thecloud, becoming gradually longer until their dura-tion reaches its maximum value at the end of thecloud. The inverse of the cumulative probabilityproduces the opposite effect, namely, grainsbe-come graduallyshorter until their duration reachesits minimum value at the end of the cloud.

GrainEnvelopeThe envelope applied to each individual grain con-sists of either the envelope extracted from thesource or a linear envelope shapedaccordingto theseven parametersshown in Figure4 and some ofSchr6dinger'sdistributions (equationsA36-A40).

Grain ScatterPathIn addition to the fixed scatter resulting from thevolume of the cloud, each grainis providedwith anindividual trajectory.Grain scatter is given in thesame units discussed above in relation to cloudvolume.The algorithmis implemented by means of atime-varying scatter function created for each di-mension, with normalized duration and maximumvalue of unity (equationsA41-A43). To generatethe trajectorybreakpoints,the algorithm requiresminimum and maximum values for the time inter-val between a pairof breakpoints provided by thecomposer. However, these aregiven as percentagesof the grain duration,as the function is normalized.When a grainis created, the function is scaled to fitits duration and maximum amplitude. It is possibleto change the shape of the function accordingto ajitter value given by the composer, which results ina differentpath for each grain.Finally, Doppler Shift is applied to the overallmotion of the grain, which is the resultant of thefollowing components: individual scatter path, in-dividual scatter position with respect to the cloud,and overall cloud motion (equationA44). The pitchshift relative to the listener is invariant under rota-tion. Therefore, it is possible to simplify the calcu-lation of the DopplerEffect if we rotate thecoordinate system by an angle so that the line join-ing the speakers is kept parallel to the instantane-ous velocity of the grain, as shown in Figure5(equationsA45-A49).

StructureWe will now proceed to examine ways in whichthe generatedmaterial can be structured into a mu-sical work. It is worth noting that these areparticu-lar choices out of a multitude of possibilities andthat these choices, in turn, will have repercussionson the generation of materials (selection of sourcesounds, cloud and graincharacteristics, etc.). Never-theless, it is also important to note that choiceswere made with full consideration and regard ortheir musical viability and formal integrity.Fischman 53


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Figure5. Calculation ofDoppler Shift using a rota-tion of axes.

V9

. ............... _ ' .....................- R/

t??????r??_? .. _L ~ ?r?????1//

".. 1/.,:

...ListeneristenerPyramidalShapeA relatively straightforward trategy for structuringmusical works may be linked to a survey throughthe atomic shells resulting from changes in theoverall energy. Forinstance, the survey may beginin the lowest energy shell and leap to higher shellsas this energy increases. (This case also correspondsto an excursion throughthe periodic table accord-ing to ascending atomic number.) In fact, it isworth adoptingthis particularinstance as a starting

point to develop a basic model that can subse-quently be modified and extended.Following the analogy with the periodic table,majorsectional boundariesmay correspondto thereappearanceof a shell with an increasingvalue ofn (which also correspondsto a change of row in theperiodic table). Because S is the first shell, majorsectional boundaries will start at lS, 2S, 3S,. ...,7S.Subsections within these boundariesmay corre-spond to the shells surveyed in between. Figure6illustrates the resulting structure obtained from theenergy levels in the periodic table; this may becomparedto a pyramid that is surveyed from top tobottom. Examination of this structurebrings us toseveral observations.

First, sections become longer by virtue of thenumber of subsections they include. Thus, section1 contains only one shell (S),sections 2 and 3 con-tain two shells (Sand P), sections 4 and 5 containthree shells (S, P, and D), and sections 6 and 7 con-tain four shells (S, P, D, and F). Second, it is possi-ble to identify four independent threads ofdevelopment correspondingto the reappearanceofeach particularshell in subsequent sections. For in-stance, shell D reappears n sections 4 to 7. Thissuggests the possibility of creatingand additionaldevelopmental dimension that, following the anal-ogy with the pyramid, would consist of four "verti-cal" paths identified by differentgrayscale shadesin Figure6. Forinstance, the reappearanceof ashell could develop the material of previous appear-ances.Two remainingissues must be determined beforethe structure is finalized: section durationand sec-tion differentiation.

Figure6. Pyramidal struc-tureresulting from the en-ergylevels obtained by fol-lowing the periodic tablein ascending order.

SectioniSection2ISection3Section 3DSection 4DD.

Section6 5D'Section 6D. . . . . . , . . . . . .

.,

......?.....:... ...: . ..!...... ... ..........

::.:...:......

... ........

.Ne::S

.MMm,



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Figure7. Schematic viewof pyramidal structurewhen the minimum dura-tion of each subsection isproportionalto the sum ofatomic numbers of its re-lated shell (not to scale).

Section1:Section2:Section3

Section54 5, S

tion 6 5D.:*i;ii~ii~iii!iiii ' 0Sectionon 6 D 4 D

Section DurationWe can follow the path set by the first observationabove (that sections become longer by virtue of thenumber of subsections they include) and establish afurtherprinciple of growth related to the periodictable: the minimum duration of each subsectioncloud is proportionalto the sum of atomic numbersof its related shell by setting a correspondencebe-tween a chosen time unit and atomic numbers asfollows:SectionDuration

= atomic numbers) x time_unit (2)Forexample, if the time unit is 0.05 sec, theminimum duration of the subsection correspondingto shell 2P (atomic numbers 5 to 10)will be(5 + 6 + 7 + 8 + 9 + 10) x 0.05 = 2.25 secApplyingthis principle, we obtain a further re-

finement of the pyramidal shape, whereby a newappearanceof a shell is normally longer than itsprevious appearances.Table 1 shows the minimumduration values for shell D with atomic unit dura-tion of 0.0342 sec, and Figure7 shows a schematicversion of the refined pyramid.To allow some freedom when taking into ac-count musical context, only the minimum dura-tion of each subsection is set. This is notinconsistent with formal assumptions because,accordingto quantum mechanics, an electron re-mains in a given state as long as the energypumpedinto (orextracted from)it is equal to or

Table 1. Minimum duration values for shell D,with atomic unit duration of 0.0342 secShell Duration (sec)3D 8.72104D 14.87705D 25.82106D 36.7650

greaterthan the transition energy requiredto moveto an immediate neighboringstate, as calculated byequation A8.Section DifferentiationIdentification of distinctive threads accordingtoshells requiresthat sections be differentiatedbytheir content. This can be achieved by using differ-ent source sounds in each shell. This will also in-fluence overall cloud andindividual graincharacteristics because, as explained above, manyof these are derived from the attributes-particu-larly the envelope-of the source sound. Anotherway sections may be differentiated is by setting dif-ferent grainattributes (e.g., differentdurations,en-velopes, spatial locations, or by using differentdistributions to generate amplitudes and durations).Extensions of the PyramidalShapeTwo other structural shapes can be derived fromthe pyramid:an inverted pyramid and a diamond.The former is analogous to a survey of the periodictable from the highest to the lowest atomic num-ber. The latter consists of an original pyramidfol-Fischman 55


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Figure . (a)Invertedpyra-midal structure nd(b)di-amondstructure.

(a) (b)?...

lowed by of an inverted one. These are shownschematically in Figure8.Other StructuralPossibilitiesApartfrom the simple trajectoriesrelated to as-cending and descending atomic number order,wecan generate more complex trajectories.For in-stance, shells, or even subdivisions of shells corre-sponding to individual atomic numbers, can beorderedaccordingto stochastic algorithms or gener-ative rules. Alternatively, they may follow attri-butes of a source sound such as the envelope wherethe peak may correspondto the maximum atomicnumber. All these are options beyond the scope ofthe current researchplan;nevertheless, they arewell worth exploring.Another issue that may be examined from a dif-ferent perspective is that of section duration. Forinstance, instead of establishing a correlation be-tween section duration and atomic number, onemay adopt a principle of constant rate of energy ac-cumulation (ordissipation). Accordingto this as-sumption, transition from one section to anothermay occur when the energy reaches a level equal to

the transition energy between the current state andits immediate neighbor.Mathematically, this maybe derived from equation A8 as follows:SectionDurationa-b= abs- - n2 x (3)

where KEis the rate of energy accumulation (ordis-sipation) expressedas a fraction of the ionizationenergy E,.per second. As an illustration, assumingthat KE = 0.01 E,.and that shell 3D progresses toshell 4P, then the duration of the former is3DSectionDuration3D,=bs - =)x

.1E4.86 sec032 42 .01E0.

The problemwith this method is that energytransitions at lower shells are significantly higherthan those at higher shells. Forinstance, the transi-tion from 1S to 2S requiresan energy value overone hundred times higher than a transition from6D to 7P. Consequently, the duration of 1S is inthe same proportionto that of 6D, yielding ex-tremely long durations for 1Sor extremely shortones for 6D. This may be overcome by subdividinglonger sections accordingto proportionssimilar tothose dividing the whole work. Alternatively, therate of energy flow could correspondto a nonlinearfunction, being faster for lower shells and slowerfor higher ones. Another possible strategy may con-sist of generative or stochastic algorithms to regu-late the rate of energy flow. In general, equationA10 may be modified to cater for any energy flowfunction KE(t):{Section-durationa-bE(t)dtabs(1 1 x (4)KEdtt= -bS 2XE (4)

Erwin's laygroundErwin'sPlayground is a work for tape lasting 9 min12 sec. Except for the recordingof the samples andbasic noise and click removal, it was created exclu-sively with AL and ERWIN software (Fischman2003), which was developed to implement the prin-ciples described in this article. As a component ofthe current project,Erwin'sPlayground is intendedto demonstrate the validity and viability of thesetheoretical principles in the generation of a musical



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work as well as the usefulness of the software as aconduit for the realization of these principles.However, its aesthetic value is intended to be inde-pendent of the generating principles, because notheory is capableof guaranteeing satisfactory aes-thetic results or musical coherence. These are ofcourse the responsibility of the composer.The name of this work is an allusion to ErwinSchr6dinger's maginary field of action, namely, theinner shells of the atom. Its structureis modeledon the diamondshape described above and linkedto a survey that begins in the lowest energy state,leaps up until it reaches the highest shell, and thenreturns back to the initial state. This may also becomparedto an excursion throughthe periodic ta-ble that ascends and then descends accordingtoatomic number order.Four differenttypes of sound sources were as-signed to each shell to achieve section differentia-tion. The S shells consisted primarilyof vocalsounds (audiofiles MonoC1, Uba, UlulowL).Pshells were assigned three differenttypes of sound,which, nevertheless, share some common spectralcharacteristics: the "buzz" of a fly, a vocal "buzz,"and the sound of a sparkingwelder (audiofilesDieFly, BZZZ, Welder).D shells were assignedsounds that rangedfrompitched to "squeaky."The pitched sounds were theresult of rubbingthe edge of a glass cup with differ-ent amounts of liquid to obtain differentpitches(audiofiles Crystal E6,Crystal F#,Crystal G#).Squeakswere recorded from the un-oiled wheels ofa trolley (audiofile Squeakl). A sample containinga transition from pitch to squeak was achieved byrubbingthe glass cup with differentdegrees of pres-sure (audiofile Crystal Bb o Squeak). Finally, a re-cordingof the pitched sound of cicadas was alsoincluded toward the middle of the piece (audiofileCicadas).F shells were normally associated with local andglobal climaxes; therefore, they spread through awider bandwidthby means of three spectral types.The first type consists of two noise-based soundsobtained from a busy motorway and sea waves gur-gling through rocks (audio files Motorwayl andWaves 1). The second sound is metallic, consistingof the recording of a frying pan that was hit or

rubbedagainst other objects (audiofile PBglf). Thethird was mainly used to create low-frequencytex-tures and drones from the recordingof a piece ofplywood that was bent at a more or less regularrate (audiofile PBdLow). (Examplesof the sourcefiles, as well as passages characterizingeach shellsand the complete work, are available on the forth-coming ComputerMusic Journalvolume 27 soundexample disc.)Grain durationwas another significant attributeused to characterize the various shells. Finally, theatomic number durationequivalent used in thepiece was 0.0342 sec per atomic number.Table 2 presents a complete list of the structuralhierarchyof Erwin'sPlayground, including starttime, minimum duration, grainduration limits,and the source audio files for each subsection. Asmentioned above, the durationof subsections couldbe extended to allow some freedom when takinginto account musical context. Section extensionsappearbelow the shell name with the letter "X."Also note the bold line denoting the end of the pyr-amid and the beginning of its inversion, which actsas an "axis of reflection." It is worth noting thatthe second half of the piece (sections 8-14, corre-spondingto the inverted pyramid) s more com-pressedthan the first part,as there are fewersubsections that have their minimum durationsextended.

SoftwaremplementationThe software implementation of this procedureconsists of two separate components: AL, a compo-sitional environment for the creation andmanipu-lation of musical events, andERWIN,a plug-inthat implements the applicationof Schr6dinger'smodel described above. These were developedandtested using Visual C ++ 6.0 runningunder Win-dows 2000 and later under Windows XP. Both ap-plications use Microsoft DirectX 8 features andrequirethat DirectX 8 be installed in the host com-puter. For this reason, the AL and ERWIN installa-tion program also installs DirectX 8, if required.Currently, AL and ERWIN are distributed as freesoftware under the GNU public license and areFischman 57


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Table 2. Structuralhierarchylist of Erwin's Playground ("X"denotes section extensions)DurationSection Shell Start Time (sec) Grain Duration (msec) Source Audio Files

1 1S 0:00.0000 0.1026 5-25 MonoC1X 0:00.1026 3.72582 2S 0:03.7384 0.2394 5-50 MonoC12P 0:03.9778 1.5390 5-50 DieFlyX 0:05.5168 2.15723 3S 0:07.6740 0.7866 5-503P 0:08.4606 3.1806 7.5-75, 7.5-20 DieFly4 4S 0:11.6412 1.3338 5-100 MonoC13D 0:12.9750 8.7210 10-60, 30-60 CrystalEb4P 0:21.6960 6.8742 5-60 DieFlyX 0:28.5702 3.1777

5 5S 0:31.7479 2.5650 5-25 MonoC110-160 Uba4D 0:34.3129 14.8770 10-150, 100-150 CrystalEbSP 0:49.1899 10.5678 10-60, 30-60 DieFly30-60 WelderX 0:59.7577 0.5818

6 6S 1:00.3395 3.7962 5-25 MonoC110-160 Uba100-500 UlulowLX 1:04.1357 2.95954F 1:07.0952 30.4038 100-5000 Waves11000-4000, 2000-6000 PBdLow100-300, 4000-8000 PBglfX 1:37.4990 2.6599SD 1:40.1589 25.8210 1000-4500 CrystalEb10-50, 10-4000, 20-150 CrystalF#10-50 CrystalG#30-50 SqueakIX 2:05.9799 0.84946P 2:06.8293 17.1342 30-60 DieFly20-80, 20-150 Welder5-1000 BZZZX 2:23.9635 5.0250

7 7S 2:28.9885 5.9850 5-25 MonoC110-200 Uba5-25, 200-700 UlulowX 2:34.9735 4.34425F 2:39.3177 45.7254 200-8000 Waves11000-4000, 2000-5000, 2000-6000 PBdLow100-300, 4000-8000 PBglf5-25, 40-150 Motorwayl

X 2:25.0431 10.51106D 3:35.5541 36.7650 200-350 CrystalEb10-50, 10-4000, 20-150 CrystalF#10-50 CrystalG#50-150, 50-250 SqueaklX 4:12.3191 3.04657P 4:15.3656 23.7006 7.5-20, 7.5-75, 10-60, 30-60 DieFly30-60 WelderX 4:39.0662 1.20908 7P 4:40.2752 23.7006 7.5-20, 20-40 DieFly6D 5:03.9758 36.7650 10-50, 200-350 CrystalEb10-50 CrystalF#50-250 Squeakl5-5000 Cicadas

(continued)58 Computer Music Journal


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Table 2. (Continued)DurationSection Shell Start Time (sec) Grain Duration (msec) SourceAudio Files

5F 5:40.7408 45.7254 100-5000 Waves12000-8000 PBdLow100-300, 4000-8000 PBglf5-25, 40-150, 150-200 Motorwayl7S 6:26.4662 5.9850 5-25 MonoC110-200 Uba5-25, 90-700, 200-700 UlulowL

9 6P 6:32.4512 17.1342 30-60 DieFly20-80, 100-250 Welder5-1000 BZZZ5D 6:49.5854 25.8210 200-350, 1000-4500 CrystalEb20-150 CrystalF#10-50 CrystalG#50-250 Squeakl4F 7:15.4064 30.4038 100-5000 Waves11000-4000 PBdLow100-300, 4000-8000 PBglf100-10000, 150-200 MotorwaylX 7:45.8102 1.00006S 7:46.8102 3.7962 5-25 MonoC110-160, 10-200 Uba100-500 UlulowL

10 5P 7:50.6064 10.5678 10-60, 30-60 DieFly30-60 Welder5-1000, 10-60 BZZZX 8:01.1742 3.15524D 8:04.3294 14.8770 200-350, 100-150 CrystalE?20-60, 50-250 Squeakl50-150 CrystalB6 o SqueakSS 8:19.2064 2.5650 5-25 MonoC110-160 UbaX 8:21.7714 8.0546

11 4P 8:29.8260 6.8742 5-60, 10-30 DieFly5-60 BZZZ3D 8:36.7002 8.7210 80-450, 200-350 CrystalEb40-400, 50-250, 100-350 SqueaklX 8:45.4212 0.02684S 8:45.4480 1.3338 5-25, 5-100 MonoC1X 8:46.7818 3.669612 3P 8:50.4514 3.1806 7.5-20, 7.5-75 DieFly3S 8:53.6320 0.7866 5-25, 5-50 MonoC1X 8:54.4186 4.604613 2P 8:59.0232 1.5390 5-50 DieFly25 9:00.5622 0.2394 15-50 MonoC1X 9:00.8016 4.597214 1S 9:05.3988 0.1026 5-25 MonoC1

10-200, 70-200 UbaX 9:05.5014 6.1339END 9:11.6353

Fischman 59


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Figure9. The AL user in-terface, including themenubar, ool bar,and aview of a document.

i "w

s e c f i o n 4 - 8 G F ? ? ( O s e c "

. .. . . . . ...... . ... ... . . . . . . .... .

-....................i@ @:....... 00:JJlilii~ i~ Ii~i!!;N,; j!j![!N..

M ' 0 0 : 3 1 A 1 9 A section4-30G (000:00:42.E

.:iilJ.':..:"..:.i 4'i:i.i.u.:....". ..................

..... ............... . . . . . . . . . . . . .

. ........ ........ . . . . . . s e c f l o t ; ? 4 -5GFG:::s ( 0 0 IO~...... ........ ..... s e t i o n..:.....:..:......_..:...... :: _:.~i:i..~~~~.~I::. ~~ ~ I( ii~:i~i~~-;~l~:i:

*t e c t io n 4 - 3 G F S W G F

available online at www.keele.ac.uk/depts/mu/staff/A1/Al_software.htm. The programsare alsodistributed with the Composer's Desktop Project.ALAL is a multiple document interface (MDI)applica-tion, with frame windows that present views of dif-ferent documents and/orseveral views of the samedocument. It is designedfor the creation and ma-nipulation of sonic events and their organizationinto a structuredmusical work.To aid user familiaritywith the software and pro-mote a reasonably short learningcurve, care wastaken to develop an interface within an accepted

general standard or Windows applications, includ-ing a menu bar,a tool bar,common window com-ponents, and typical mouse "drag-and-drop"operations.It also contains detailed online helppages, including a tutorial. Figure9 shows an in-stance of the user interface.Another important considerationin the design ofAL was the desire to providea reasonably open sys-tem with the option for development of third-partyplug-ins, thus avoiding the constraints imposed bya single type of musical process. Futuredevelop-ments by the author and others will ideally imple-ment a variety of compositional approaches naddition to the application of Schrodinger'sEqua-tion in the existing plug-in (see the description ofERWINbelow).



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Documents and ViewsAL documents store the information necessary todefine and characterize the composed musicalwork. They consist of a set of orderedevents, eachwith its own data and attributes,as well as generaldata such as creation time, session number, overallduration,sampling rate, etc. When a document isviewed, events are representedby rectangles on thescreen, as shown in Figure9. These rectangles maybe draggedand stretched with the mouse.

Events may be empty or they may representanactual audio file. The audio file can be a pre-existing file or the result of a plug-in operation,inwhich case the event stores the parametersex-ported by the plug-in. Events associated with anaudio file adopt its name and aregiven a three-dimensional edge in the graphicaluser interface.Events may be locked so that they do not changetheir onset times when dragged,allowing only ver-tical movement. It is also possible to mute eventsso that they are not performedduringplayback andnot included in any mix.Main FeaturesIn addition to standardWindows features (such assaving and loading events and their generating pa-rameters, zooming, scrolling, undo, redo, cutting,copying, andpasting),a number of operations canbe performedon an event by clicking on its rectan-gle with the right mouse button.It is possible to classify the items in this menuaccording to four categories: playback;modificationof event status, including propertiessuch as onsettime, color, locking, and muting; simple audio fileoperations, including association with an existingaudio file, gain, fade, and channel swappingofstereo events; and finally plug-ins consisting of ex-ternal processes, which create an audio file and as-sociate it with the event. Users can also auditionan audio mix of events or bounce the latter to anaudio file.ERWINPlug-inThis applicationis providedas a separate dynamic-link library(DLL)conforming to the Microsoft

Component ObjectModel (COM)standard.It maybe called by AL through the right-buttoneventmenu to create a granularcloud accordingto thealgorithms described above. The source sound forgranulationis read from a wave file, and the cloudis stored in an output wave file that AL associateswith the calling event.The user can supply the necessary parametersviapropertypages, which fall into four categories. Thefirst category consists of input/output andgeneralsettings, including the name of the input file, thename of the output file, and parametersfor enve-lope extraction, grainextraction, cloud density cal-culation, Doppler Shift, and FFToperations. Thesecond categoryis section attributes, consisting ofthe values of quantum numbers,atomic number,and the minimum and maximum radii for a partic-ular atomic shell. The third category includescloud attributes,that is, parametersfor duration,grain density, amplitude scaling, motion, andvol-ume. Finally, grainattributesconsists of parame-ters for amplitude fluctuation, durationfluctuation,envelope, and scatter.Except for the input and output names, the prop-erty pages present a set of defaults.The default forcloud duration is taken from that of the callingevent.ERWIN'sgraphicalinterfacepresents the resultsof the process as shown in Figure 10, including thefollowing data:plots of the distributions and theirmaxima and minima; a plot of the extractedinputenvelope, the location and value of its maximum,and the analysis window duration;plots of cloudattributes including density, motion, and scatter;and plots of the grain attributes, including the en-velope andmotion functions ("archetypes").

Future oftwareDevelopmentAs with any piece of software, AL could be devel-oped almost ad infinitum, and a variety of plug-inscould be created to enhance its capabilities. Wewill therefore focus on some of the developmentsthat are relevant to the aesthetic path chosen here.

An immediate improvement might consist of al-Fischman 61


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Figure10. The ERWINuser interface.

.. .. .. ......Z.Mn:ii;iiiiiis~~li::l~j ii~~E~~~~lu'?~~I~i~'L'"'""'s~'~ i~'~P~p?~EP~~~~~nm~~i~p~'iX, lii~ifiiiirsi~iriiicldi%i::l: :i?;:'.'i~d~l*:8:BF':"iY Qi;EjPr-oessigchror-kingeqiahoR a i a l S y n - m e t y . , _ ` ? A t i d o F l e s ? f l o d l i a v

X aL~R~ 1Q~ dur - -- ------ ----------:4: . 4z -Ni:,. WT.W... x.:?-M::. ...... .LV.S., Heg?NN?'Mp . . 1-n: :M." 4 wulE'aNs N Ml'"""""'""ii~iii~?a~::. . .P s AM-- -- --. ................... W-UMiffiv.5::4 trl, . ..... xvis!,"i:,-??N.0-44,'N 0--...N--.. ,.-, .-Nl"..........-N.. RMili:ii:'l:::I..........N N'illN. N.~lll~i~ii'~:4,HIIIB E:,:::::::.LOM..:::,:.::::...:iflii~r~iira3wi,M,,,,,,,, i~ii*~iii~~i:?iidii: 5!Q n.. -122~E~i~~i~~~il~~~i~i~i,~~B~%;~~lsa~~6~iss _N.......M~~ji~~!.: _~1 NM:I::..X

"'NI-Im.is:aul.4: ......-M ,- h8 ; 11. . -UNow :U UNii;:? - %?,-, R ?I~li. 2-x Emmiil iZMiii-zu FSrilrgi85iisiN-mim'i L T-Ml8

gorithmic automation of structure. Forinstance,AL could include a menu option for plug-ins thatcreate structural hierarchies that could then be dis-played in the document view. Specific colors couldbe assigned to events in differentsections, andviews could display dividing lines and sectionnames. Furthermore, hese algorithms could imple-ment structuralhierarchiesin addition to those de-scribedabove, where it has alreadybeen suggestedthat it may be possible to generate more complextrajectorieswith random or generative algorithmsor by following the attributes of a source sound.The generation of events could also be auto-mated using, for instance, randomalgorithms basedon Schrodinger'sdistribution or other distributionsfor the determination of the number of events,their onsets, and their durations.Finally, it mightbe possible to develop software that applies thesame principles to visual grains, integratingsoundswith corresponding"animated" clouds seen on ascreen or in a virtual reality environment.

ConclusionA work of art may be articulated at various levels.At one level, articulation takes place througha me-dium that carries with it a set of principles result-ing from its physical properties.Forinstance, theuse of stone in sculpture leads to an organization ofspace ruledby visual proportion,texture, and thephysical constraints resulting from gravitationalforces (such as those that determine whether astructuremay collapse or not).A second level from which propertiesand func-tionality are derived is more specific to particularcultures, periods, or genres. In the case of most mu-sic, it dependson the particular aspects of sound apiece articulates. Forinstance, Western music inthe 18th and 19th centuries is structuredaccordingto harmonic relationships between keys (e.g., fugueand sonata forms).One may find a third level relevant to music thatconsists of an extra-musical dimension and corre-



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spondingto the concepts of intrinsic and extrinsicreferraldiscussed by Nattiez (1990).In this case, anadditional set of principles operates by establishinga relationshipbetween musical articulationandnon-musical concepts, or, in other words, by map-ping extra-musical models to the generation of mu-sical discourse and structure.This is typical, butnot exclusive, of programmaticmusic. As an exam-ple of work that is articulated at an extra-musicallevel without becoming programmatic,we may citeXenakis's use of parabolictrajectoriesfor the de-sign of string glissandi in Metastasis. The princi-ples describedin this article, and their realizationin Erwin's Playground, fall within this category.It is hoped that the realization of the extra-musical dimension proposedhere is reasonablyfaithful to the essence of a quantum mechanicalmodel and the non-deterministic view of the uni-verse emanating from the model, while allowingthe creation of musically coherent works. Never-theless, it is vital to point out that mappingtheequations to musical parameterspresentedhere isneither unique nor exhaustive. It is only one of amultitude of alternatives, which, to the author,makes musical sense. Furthermore, he model de-veloped here can still be enhanced and explored, forexample with improvements to the construction ofstructuralhierarchiesother than those presentedabove and the algorithmicgeneration of largerstructuralunits, such as the number of events in asection and their individual attributes (e.g., onsets,durations,and granularparameters).Also, the specific direction assumed duringthisresearchmay lead to other paths. One of the mostobvious consists of the application of the theoreti-cal model to other media, including animation andvirtual reality. The possibility of applyingothermathematical models (e.g., those describingrelativ-istic phenomena, heat convection, behavior offluids, etc.) or alternative interpretationsof Schr6-dinger's Equationmay be worth investigating.Finally, I hope that Erwin'sPlayground is able tocontribute in some constructive way to the alreadyconsiderablebody of compositions derivingfrommathematical logic and that the principles devel-oped here-and their software implementations--may be of use to other composers, resulting inother musical works.

AcknowledgmentsI am grateful to the Arts and Humanities ResearchBoard(AHRB),UK, for its supportin the form of aResearch LeaveAward,which made possible therealization of this project. Special thanks to Dalit,my partner n life, for her patience and moral sup-port. Many thanks to my colleagues in the KeeleUniversity Music Department for their effectivecoverage of my duties duringthe periodof researchleave. Also, thanks to Richard Dobson and ArcherEndrichof the Composers' Desktop Projectfortheir technical advice.

ReferencesBain,R. 1990."Algorithmic omposition:QuantumMe-chanics & the Musical Domain." Proceedingsof the1990 International ComputerMusic Conference.SanFrancisco:InternationalComputerMusic Association,pp. 276-279.Beyls, P. E. 1991. "Self-OrganizingControl StructuresUsing Multiple Cellular Automata."Proceedingsofthe 1991 International ComputerMusic Conference.San Francisco:InternationalComputerMusic Associa-tion, pp. 254-257.Bidlack,R. 1992. "Chaotic Systems As Simple (ButCom-plex) Compositional Algorithms." ComputerMusicJournal 16(3):33-47.Bolognesi,T. 1983. "Automatic Composition:Experi-ments with Self-SimilarMusic." ComputerMusicJournal 7(1):35-36.De Poli, G., A. Piccialli, and C. Roads. 1991. Representa-tion of Musical Signals. Cambridge,Massachusetts:

MIT Press,pp. 137-186.Delatour, T. 2000. "Molecular Music: The Acoustic Con-version of MolecularVibrationalSpectra."ComputerMusic Journal24(3):48-68.Di Scipio,A. 1991. "Composition by Explorationof Non-LinearDynamic Systems." Proceedings of the 1991International ComputerMusic Conference.SanFrancisco:InternationalComputerMusic Association,pp. 324-327.Di Scipio, A. 1994. "Micro-TimeSonic Design and Tim-bre Formation."ContemporaryMusic Review 10(2):135-148.Dodge, C. 1988. "Profile:A Musical Fractal."Computer

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FeynmanR. P., R. B. Leighton,and M. Sands. 1971. TheFeynman Lectures on Physics, VolumeIII. Reading,Massachusetts:Addison-Wesley,pp. 19.1-19.20.Fischman, R. 2003. "Derivation of OrganicMusicalStructureand Materials from the Solution of Differen-tial Equations."Intercultural Music 5. Point Rich-mond, CA: MRIPress,pp. 105-130.Hiller, L. 1981. "Composingwith Computers:A ProgressReport." ComputerMusic Journal5(4):7-21.Hoffman, P. 2000. "The New GENDYN Program."Com-puter Music Journal24(2):31-38.Holzman, S. R. 1981. "Using Generative Grammars orMusic Composition." ComputerMusic fournal5(1):51-64.Jones,K. 1981. "CompositionalApplications of Stochas-tic Processes."ComputerMusic Journal5(2):45-61.Koenig,G. M. 1970a. "Project1."Electronic Music Re-ports 2. Utrecht: Institute of Sonology. (Reprinted1977, Amsterdam: Swets and Zeitlinger.)Koenig,G. M. 1970b. "Project2: A Program or MusicalComposition." Electronic Music Reports3. Utrecht:Institute of Sonology. (Reprinted1977, Amsterdam:Swets and Zeitlinger.)Laske,0. 1981. "Composition Theory in Koenig'sProjectOne and ProjectTwo." ComputerMusic Journal5(4):54-65.Lorrain,D. 1980. "A Panoplyof Stochastic Canons."ComputerMusic Journal4(1 :53-81.Lyon,D. 1995. "Using Stochastic Petri Nets for Real-time Nth-Order Stochastic Composition." ComputerMusic Journal 19(4):13-22.Maia, A., P. Do Valle, J. Manzolli, and L. N. S. Pereira.1999. "A ComputerEnvironment for PolymodalMu-sic." OrganisedSound 5(2):111-114.McAlpine, K., E. R. Miranda,and H. Gerard.1999. "Mak-ing Music with Algorithms:A Case-StudySystem."ComputerMusic Journal23(2):19-30.Nattiez, J. J. 1990. Music and Discourse: Towards a Se-miology of Music, trans. C. Abbate.Princeton,NewJersey:PrincetonUniversity Press,pp. 102-129.Pressing,J. 1988. "Nonlinear Maps as Generatorof Musi-cal Design." ComputerMusic Journal 12(2):35-46.Ross, B. J. 1995. "A ProcessAlgebrafor Stochastic MusicComposition." Proceedings of the1995 InternationalComputerMusic Conference.San Francisco: nterna-tional ComputerMusic Association, pp. 448-451.Schillinger,J. 1948. TheMathematical Basis of the Arts.New York:The Philosophical Library. Reprinted1976.Cambridge,Massachussetts: Da Capo Press.)Smalley D. 1986. "Spectro-Morphology nd Structuring

Processes."In S. Emmerson,ed. TheLanguage of Elec-

troacoustic Music. Basingstoke,UK:Macmillan,pp. 61-93.Smalley D. 1997. "Spectromorphology:ExplainingSound-Shapes."OrganisedSound 2(2):107-120.Sturm,B. L. 2001. "Composingfor an Ensemble of At-oms: The Metamorphosisof Scientific ExperimentIntoMusic." OrganisedSound 6(2):131-145.Truax,B. 1982. "TimbralConstruction in Arras as a Sto-chastic Process." ComputerMusic Journal6(3):72-77.Truax,B. 1990a. "Chaotic or Non-LinearSystems andDigital Synthesis:An ExploratoryStudy."Proceedingsof the 1990 International ComputerMusic Confer-ence. San Francisco:InternationalComputerMusicAssociation, pp. 100-103.Truax,B. 1990b."Composingwith Real-TimeGranularSound."Perspectives of New Music 28(2):120-134.Vaggione,H. 2001. "Some Ontological RemarksaboutMusic Composition Processes."ComputerMusic Jour-nal 25(1):54-61.Vaughan,M. 1994. "The Human-Machine Interface nElectroacoustic Music Composition." ContemporaryMusic Review 10:111-128.Xenakis I., 1992. FormalizedMusic, revised edition.Stuyvesant, New York:PendragonPress.

AppendixSchroedinger'sEquationfor an Atomic Potentialwith Radial Symmetry

h V2fT + V(r)T = ih (Al)2m atUsing spherical coordinates,

'(r,O,4,t) = R(r) - 0(0). D(4) - T(t) (A2)wherer is the distance of a point to the center of thepotential,0 is the elevation angle,k is the horizontal angle,and t is the time. R, 0, (), and T are given below:

R(r) = (Zr)1 ezr/nr?[ bk2-)Z(A3)L k=O nro

0(0) = P"(coso) (A4)64 Computer Music Journal


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In equation A12, r and 0 may be determined ac-cordingto equations A13 or A14 below at the com-poser's discretion. If r and 0 are both directlyproportionalto the envelope, then we haveamp(te)r(tc) = rmin + (rmax - rmin) X ampmax (A13)

(tOmax-- min) amp(te)0(tc) = x2 amPmaxIf r is directly proportionalto the envelope and 0 isdirectly proportionalto its slope, we have

amp(te)r(tc) = rmin + (rmax - rmin)amp(tamPmax (A14)

0(tc) = arctan(amp'(te))wherermin, rmax arethe minimum and maximum radialvalues given by the composer,Omin, Omax re the minimum and maximum eleva-tion angle values (set to 0 andnt),amp(te) is the correspondingenvelope amplitude

value at time te,ampmax s the maximum amplitude extractedfrom the envelope, andamp'(te) is the derivative of the envelope attime te.Alternatively, we can requiredirectproportional-ity between the source envelope and the density:

density(tc) = densitymin+ (densitymax - densitymin) xamp (A15)ampmax

Cloud Amplitude ScalingWe can scale the cloud amplitudes accordingto thefollowing equation:gain(tc) = gainmin + (gainmax- gainmin)

(amp(t) - ampm1,)(ampma,,- ampmn) (A6)where gainin, and gainmx are the values of the min-imum and maximum gain (provided by the com-poser), and ampmin and ampmax are the minimum

and maximum amplitudes extractedfrom the enve-lope.

CloudMotionWidth and Depth

2 (1 - X)AL - 2(1

y)4 + X2

(A17)?_2AL A=R 0 x< -1V(l + y)4+X4V2AL = 0; AR= (1+ y)4 + x4 >l

wherex is the width,y is the depth (see Figure 2), andALand AR areamplitude factors that multiplythe left and right channels, respectively.

TrajectoryCreationx(tc) = mWidthmax x r sin0 cosormax -min

y(tc)= mDepthmax - sine abs(sino)rmax rminz(tc) = mSemitonesmin + (mSemitonesmax - mSemitonesmin)

rX cos 0rmax-rmin (A18)

where r, 0, and0 aregeneratedfrom distributionsR(r),0(0), and0(40).The other parametersaregivenby the composer: rin, rmax are the minimum andmaximum radial values, m Widthmax and mDepthm~xare the maximum absolute value of the width andthe maximum depth, and mSemitonesmi, and



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mSemitones max re the low and high transpositionboundaries.

Cloud Volume (Overall Grain Scatter)

xScatter(tc) = sWidthmax X abs(sin 0 cos 4)max rminr

yScatter(tc) = sDepthmax abs(sin 0 sin 4)rmax rminzScatter(tj) = sSemitonesmax -r cos 0rmax rmin (A19)where r, 0, and 4 aregeneratedrandomly from dis-tributions R(r), 0(0), and0(4P).The other parame-ters aregiven by the composer:rmin,max are theminimum and maximum radialvalues, andsWidthmax, sDepthmax,and sSemitonesmax are thewidth, depth, and height (in semitones) of thescatter.

GrainAmplitude ScalingScaling proportional to Source EnvelopegGain(tc) = gGainmin amp(te)+ (gGainmax - gGainmin) x (A20)ampmaxwhere

gGainmin ndgGain,,x are the minimum andmaximum gain (given by the composer),amp(te) is the correspondingenvelope amplitudevalue at time te, andampmax is the maximum amplitude extractedfrom the envelope.

Scaling Proportional to Schrodinger's Distribution

gGain(tc) = gGainmi+ (gGainmax- gGainmin) x F(tj) (A21)where F(te) is given by

F(t) = ',, / sin(0(tc))lIRr(t))I121((t j))2 (A22)Lrmax

andr(t,) = rmin + (rmax r- min) X

0(tc) = Omin ? (Omax min) Xt

cloudDurationHere, rminand rmax are the minimum and maximumradial values (given by the composer), Omin and Omaxare the minimum and maximum elevation angles.

Scaling Proportionalto the Complement ofSchr6dinger'sDistributionReplace the following definition of F(tj) n equationA22:

F(tc)= 1 - r(tj)12sin(0(tj)R(r(tj))l2(Otj))12 (A24)Lrmax]

Scaling Proportional to Schr6dinger'sCumulativeProbabilityReplace the following definition of F(tc)in equationA22:

4oftc) O(tc) r(tc)F(t,)r [(t)2 (A25)frmax-Pmin 0min rminsin(O(tc))lR(r(tj))121(O())l2 drd0dp

Scaling Proportionalto the Complement ofSchr6dinger'sCumulative ProbabilityReplace the following definition of F(tj) n equationA22:

0(tc)O(tc)

r(tc)

F(tc) = 1- f [r(tc) 2 (A26)'min 9min rmin

sin(0(tc))iR(r(tc))l2mn(0(t))12drd7Fischman 67


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Scaling generated randomly from variousdistributionsReplaceF(tc)in equation A22 with values generatedby one of the following distributions:

r2lR(r)12 (A27)sin(O)lI0)12 (A28)

kI(4)l12 (A29)r2 sin(O)IR(r)1210(O)12 (A30)r2 sin(O)IR(r)12k1(4p)12 (A31)

sin(0)IO(0)21(4 )12 (A32)r2 sin(O)IR(r)121?(O)12)(P)12 (A33)

GrainDuration

Duration Proportional to SourceEnvelopegDuration(tj) = gDurationm,,+ (gDurationmax - gDurationmin) (A34)amp(te)X ampmax

where gDurationmin andgDurationmaxare the mini-mum and maximum durations (given by the com-poser).Duration Proportional to Schr6dinger'sDistributionsgDuration(tj) = gDurationmin+ (gDurationmax - gDurationmin) x F(tj) (A35)The various functions used are analogous to thoseused for grain amplitude, with F(tj)given in equa-tions A22, A24, A25, and A26.Duration Generated from Various DistributionsThe distributions used are the same as those inequations A27-A33.

GrainEnvelopeDistributions Used as Envelopes

r2R(r)12 (A36)Reverse of r2IR(r)12 (A37)

sin()1(OI0)12 (A38)rz sin(O)lYP(r,, t)l12 (A39)

Reverse of r2 sin(O)tP(r,O,t)l2 (A40)GrainScatterPathgxScatter= gsWidthmin + (gsWidth - gsWidthmin)gyScatter= gsDepthm,, + (gsDepth - gsDepthmin)gzScatter= gsSemitonesmax

+ (gsSemitones - gsSemitones,,,min)(A41)where

gsWidthmi,,,gsWidthm,_ re the minimum andmaximum width scatter,gsDepth,,, gsDepthm,.are the minimum andmaximum depth scatter, andgsSemitonesm?,, sSemitonesmaxre the minimumand maximum semitone scatter.All of these values are given by the composer.The values gsWidth, gsDepth, andgsSemitones arederived from the mean and standarddeviation ofthe distributionfunctions:

gxScatter(t,)= rScatter x sin(OScatter)cos(4Scatter)gyScatter(tj)= rScatter x sin(OScatter)sin(4Scatter)gzScatter(tc) rScatter x cos(OScatter) (A42)Here,

rScatter = r - rmeanrstdDeviation

OScatter = mean (A43)OstdDeviation4 - kmean4Scatter =

PstdDeviationwhere



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r, 0, and 4 aregeneratedrandomly from the dis-tributions R(r),0(4P),and 0(4),rmean and rstdDeviation arethe mean and standarddeviation of R(r),Omean and stdDeviation are the mean and standarddeviation of 0(0), and

mean and 4stdDeviation are the mean andstandarddeviation of 0(o).

Grain Overall Motionx,(tc) = gxScatter(tc) + xScatter + x(tJ) (A44)y,(tc) = gyScatter(tj) + yScatter + y(tC)

The values gxScatter(tj)andgyScatter(tj)are de-fined in equation A42, xScatter and yScatter are de-fined in equation A19, andx(tJ)andy(tJ)are definedin equation A18.GrainDoppler ShiftThe pitch shift relative to the listener is invariantunder rotation. Therefore, it is possible to simplifythe calculation of the DopplerEffect if we rotatethe coordinate system by an angle a(t,), so that theline joining the loudspeakersis kept parallel to theinstantaneous velocity of the grain, as shown inFigure5. The angle a(tJ) s then

a(tc) = arctg(dydx)(A45)X-gI

Also, to maintain consistency, a(t,) must be posi-tive if it produces a counterclockwise rotation.The DopplerShift may be expressed as a time-varying frequencyratio calculated accordingto thefollowing equation:frequencyRatio(tc) = (A46)1 V,(t,)cosO

whereV,(tc) is the instantaneous velocity of the grain,V, is the velocity of sound in air, and0 is the angle of the grain'svelocity relative tothe listener (see Figure 5).In the rotatedsystem, V, and a(tj)can be calcu-lated from the rotated coordinates xa and y,,which can be obtained from the standardequationsfor a rotation:

x,(tc) = xg(tc)cosa + (y,(tc) + 1)sinayg(tc) = - x,(tc)sina + (yg(tc)+ 1)cosa - 1

(A47)Thus we haveV, dxag (A48)dt

andcos = - Xg (A49)(j1 + yag)2 + X g