47Fischman

Rajmil FischmanMusic DepartmentKeele UniversityKeele Staffordshire United Kingdom ST5 5BGraFischmanmuskeeleacuk

Clouds Pyramids andDiamonds ApplyingSchrodingerrsquos Equationto Granular Synthesisand CompositionalStructure

Computer Music Journal 272 pp 47ndash69 Summer 2003q 2003 Massachusetts Institute of Technology

This article presents the results of research on thegeneration of musical processes resulting from theapplication of Schrodingerrsquos Equation for an atomicpotential with radial symmetry a mathematicalmodel of dynamic systems that like music maychange and develop in time (For an introductorydiscussion of Schrodingerrsquos Equation see FeynmanLeighton and Sands 1971) I focus on the creationof musical material and its organization arisingfrom the solutions and implications of the equa-tion This is applied to asynchronous granulartechniques (eg Truax 1990b De Poli Piccialliand 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

Background

The relationship between scienti c disciplines andmusic is not new in Western culture suf ce it tomention Boethiusrsquos concept of lsquolsquomusic of thespheresrsquorsquo the medieval quadrivium the relation-ship between talea and color in isorhythmic musicof the 14th century and rules of counterpoint re-garding temporal proportions between voices inFuxrsquos Gradus ad Parnassum in the 18th centuryDuring the 20th century a renewed link between

science and music began to emerge in compositionfrom the relatively straightforward principles of se-rialism to Schillingerrsquos compositional theory (Schil-linger 1948) the complex constructs realized byXenakis in works such as Achorripsis and NomosAlpha (Xenakis 1992) the compositional strategiesof Koenig (Koenig 1970a and 1970b Laske 1981)and Hiller (Hiller 1981) and the use of stochasticprocesses by Truax (1982) and others (eg Jones1981 Lyon 1995 Ross 1995 Hoffman 2000) Theo-retical and technical advances in information tech-nology propitiated further exploration in areas suchas generative grammars (Holzman 1981 Beyls1991) self-similar chaotic and more general dy-namic systems (Bolognesi 1983 Dodge 1988 Press-ing 1988 Truax 1990a Di Scipio 1991 Bidlack1992) and more general algorithmic approaches(Maia et al 1999 McAlpine Miranda and Gerard1999)

The aim of the present work is to investigate andexperiment with the possibilities offered in thisarea by an equation that contributed to a deepchange in our conception of the world followingthis path to its ultimate conclusion (ie the musi-cal composition) In this sense this study presentsan additional link between the thought processesthat lead to the mathematical abstraction of reality(ie 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 theformer construct meaning out of the observation ofthe physical world and the latter transform mean-ing into a physical trace namely a piece of music

The current project develops previous work onthe derivation of organic structure and generationof material offered by Schrodingerrsquos Equation (Bain1990 Fischman 2003) which preserves musical

48 Computer Music Journal

logic and coherence While previous research bythe author was 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 parameters other than pitch and providesthe basis for the application of similar strategies toother media Thus the electroacoustic domainwith a speci c focus on asynchronous granulartechniques was chosen because it allows a signi -cant degree of complexity and it allows thoroughinvestigation of the possibilities inherent in Schro-dingerrsquos Equation for the manipulation of multi-dimensional timbral characteristics with pitchincluded as one of these dimensions (Smalley 19861997) In particular granular techniques are ideallysuited for stochastic processing of musical material(Lorrain 1980 Xenakis 1992) Furthermore a workfor tape can provide the immediate aural feedbacknecessary to evaluate the results

The research methodology adopted during theproject consisted of three main parts First I devel-oped appropriate theoretical principles and algo-rithms for the creation of musical material arisingfrom the solutions and implications of Schro-dingerrsquos Equation Next I developed software thatimplements these algorithms The software in-cludes a graphical interface that enables the manip-ulation of large amounts of data and musicalstreams is reasonably self-suf cient for the crea-tion of a complete musical work and is reasonablyopen-ended offering possibilities for enhancementand customization via third-party plug-ins using arecognized standard The third part of this researchconsisted of testing the validity and viability of thetheoretical principles and the usefulness of thesoftware through the creation of a musical work fortape Erwinrsquos Playground

Concerning the last point it is important tostress that neither theory nor software are intendedto guarantee satisfactory aesthetic resultsmdashor 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 lsquolsquoactionper-

ception feedback looprsquorsquo In fact it is reasonable toassume that most mathematical constructs areneutral to musical logic and that their usefulness isdependent on the actual way they are mapped intomusical processes by the composer Neverthelesstheoretical principles are vital in providing lsquolsquoformaltools as generative and transformative devicesrsquorsquo(Vaggione 2001) within which a speci c work maybe realized projecting the latter beyond an exclu-sively sensory sphere Furthermore decisions re-garding the correspondence between the modelrsquosparameters and musical attributes as well as theirrealization by means of the design and functional-ity of the software (which at rst sight mightseem to be exclusively related to theoretical andtechnical development) already belong to the com-positional process since they carry speci c aes-thetic presumptions and aims (For a discussion ofthis issue in a more general context see Vaughan1994) For instance recent published work presentsdifferent ontological approaches to correspondencesbetween quantum mechanics and music (Delatour2000 Sturm 2001)

After a brief description of Schrodingerrsquos Equa-tion this article presents the framework developedfor the correspondence between the equation andthe creation of granular clouds and strategies fortheir organization into coherent musical structureThis will be followed by discussion regarding therealization of these principles in Erwinrsquos Play-ground a work for computer-generated tape Subse-quently a general description of the software willbe provided An appendix is offered to readers whomay wish to explore the mathematics in more de-tail (Reference to equations in the appendix usesthe pre x lsquolsquoArsquorsquo followed by a number)

Schrodingerrsquos Equation

Schrodingerrsquos Differential Equation forms the basisfor the techniques described herein which estab-lish a correspondence between 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 A1ndashA8)

49Fischman

The particular version of Schrodingerrsquos Equationexamined 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 spherical sur-face of radius r the potential will be symmetricwith respect to the center Thus SchrodingerrsquosEquation assumes the following form

]Y21 sup1 Y ` V(r)Y 4 i (1)2m ]t

where V(r) is the atomic potential m is the mass ofthe electron r is the distance from the center of thepotential (eg the nucleus of an atom) to the elec-tron and is the ratio between Plankrsquos constant hand 2p ( 4 h2p)

The family of functions corresponding to thesolutions of the equation is represented by thesymbol W also known as the wave function Exam-ination of the general form of the wave function re-veals that W consists of the product of four smallerfunctions (equation A2) that may be treated sepa-rately because each is dependent on a different pa-rameter Three of these were found particularlyuseful R which is exclusively dependent on r thedistance of a point to the center of the potential Hwhich is exclusively dependent of the pointrsquos eleva-tion angle h and U which is exclusively dependenton its horizontal angle f

Furthermore the solutions depend on a set ofquantum numbers that restrict the physical attri-butes of the electron to multiples of certain energylevels The rst of these values is n the principalquantum number (n 4 1 2 3 ) which deter-mines the overall energy of the electron (equationA7) The second value is l the angular momentumquantum number (l 4 0 1 2 nndash1) which gen-erates additional allowed energies (quantum states)within a single value of n (The term shell is usedto denote a particular value of l) So far four typesof shells have been encountered in nature and arenamed S P D and F corresponding to l 4 0 1 2and 3 respectively Therefore a particular energylevel is characterized by a combination of n and lwhich by convention is named using the value ofthe principal quantum number and the letter of itsshell (eg 1S 2S and 2P)

The third quantum number m is the magneticquantum number (ndashl m l) 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 probability of ndingan electron It is possible to show that this proba-bility is associated with R and H (equations A9 andA10) As an example of this correlation Figure 1shows plots of these distributions and the resultingorbital for particular values of n l and m fromwhich we may infer the most probable distancesand orientations For instance Figure 1b indicatesthat the electron is more likely to be found in thevicinity of 35278 and 144738

The quantum numbers were successfully used toexplain the existence of common characteristicsbetween known chemical elements supportingtheir classi cation into the Periodic Table of Ele-ments originally devised by the chemist DmitriMendeleev in the 19th century but only more re-cently explained by Schrodingerrsquos EquationRoughly 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 group and occupying the same column inthe periodic table For instance the group of halo-gens consists of elements that have ve electronsin their P shell as follows uorine has ve elec-trons in shell 2P chlorine in 3P bromine in 4Piodine in 5P and astatine in 6P

Generation of Granular Clouds

My generation of musical material using Schro-dingerrsquos Equation focuses on granular clouds(sounds composed of a large number of short ele-mentary sonic particles or grains each lasting afew hundredths of a second) derived from statisticaldistributions obtained from 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 importance

50 Computer Music Journal

009 031 076 150

(a)

3527o 90o 14473o0o 180o

(b)

3527o(c)

is given to the creation of formal relationships be-tween the inner structure of the source soundmdashpar-ticularly its amplitude envelopemdashand the structureof the cloud and its constituent grains Sourcesounds are stored as wave audio les

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 Parameters

These parameters refer to global attributes of thecloud such as density overall amplitude and spa-tial and spectral trajectory

Cloud Density

To 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 are proportional to the density uctuation Inpractice the envelope is extracted from the sourceand stored in a timendashamplitude breakpoint tableThe cloud density is also stored as pairs of 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 corresponding values for clouddensity (equation A11)

Currently there are three different methods ofestablishing a correspondence between source am-plitude envelope and cloud density The rst twocalculate the density by using the envelope to drivethe radial and angular distributions (equation A12)either by making both r and h directly proportionalto the envelope (equation A13) or by making r di-rectly proportional to the envelope and h directlyproportional to its rst derivative (equation A14)The third method establishes a direct proportional-ity between the envelope and the density using astraightforward proportion rule (equation A15)

Figure 1 Examples ofSchrodinger distributionsr2|R|2 for Z 4 139 n4 4and l4 1 with radial val-ues normalized to (a) 15

(b) |H(h)|2 for n 4 4 l 4 1m 4 0 and (c) a diagram ofthe orbital resulting fromthese distributions

51Fischman

x=-1 x=1

RL y=0

ygt0

xgt1xlt-1

Listener

x lt -1 Beyond the left speakerx = -1 Left speaker-1 lt x lt 1 Between the speakersx = 1 Right speakerx gt 1 Beyond the right speaker

y = 0 On the imaginary line that joins the speakersy gt 0 Behind the imaginary line that joins the speakers

When y = 1 the depth is equal to the distancefrom the centre to one of the speakers

x=0

Cloud Amplitude Scaling

The amplitude of the cloud may be scaled by avalue that uctuates between a minimum andmaximum gain The uctuation follows the enve-lope of the source (equation A16)

Cloud Motion

Motion is carried out in three dimensions Two ofthese are spatial (width and depth) and the remain-ing one is articulated as a spectral dimension (pitchshift)

Width and Depth

These are calculated within a stereo image x indi-cates width and y indicates depth as illustrated inFigure 2 (See also equation A17)

Spectral Motion

The cloud lsquolsquomovesrsquorsquo in the frequency domain as aresult of time-varying transposition within limitsspeci ed by the composer establishing an analogyto movement across a height axis z Transpositionis carried out in the frequency domain via Short-

Time Fourier Transforms thereby preserving theduration of the grains

Trajectory Creation

To implement motion a three-dimensional trajec-tory is created that is analogous to physical move-ment within the boundaries of a shell (orbital)This consists of the stochastic generation of break-points according to Schrodingerrsquos radial and angulardistributions The breakpoints consist of position-time pairs within the duration of the cloud con-verted from r h and f into width depth andheight (equation A18)

The algorithm that generates the trajectory nor-malizes the values so that the cloud reaches themaximum width maximum depth and one of thetransposition boundaries (These are not reachedsimultaneously) Also to generate the trajectorybreakpoints the algorithm requires minimum andmaximum values for the time interval provided bythe composer The intervals between the break-points in the trajectory will fall randomly betweenthese two values For instance if the minimum andmaximum intervals are 02 and 05 sec respec-tively and the last breakpoint was generated attime tc 4 14 sec the time tc for the next break-point will fall between 16 and 19 sec

Finally trajectories may be repeated an arbitrarynumber of times so that the sum of the durationsof the repetitions is the overall duration of thecloud (If required this may be equal to the value ofthe orbital quantum number l plus one) Repeti-tions may be identical or uctuate within a per-centage given by the composer

Cloud Volume (Overall Grain Scatter)

The volume of the cloud depends on the maximumscatter of the grains in each dimension (widthdepth and pitch 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

Figure 2 Values of x andy for spatial width anddepth

52 Computer Music Journal

RL(a)

RL(b)

Width and depth scatter values vary from 0 (noscatter) to any positive numbers x and y with avalue of 1 indicating a scatter distance equal to thedistance from the center to one of the speakersSpectral scatter values vary from 0 (no scatter) toany positive number of semitones (or any fre-quency ratio)

To implement scatter a time-varying volumefunction is created in a similar manner to that ofthe motion trajectory described above Breakpointsare also generated stochastically according toSchrodingerrsquos radial and angular distributions(equation A19)

Therefore a grain created at time tc will movewith the cloud at a xed lsquolsquodistancersquorsquo from its centerThis distance is generated stochastically withinwidth depth and height boundary values 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 unanchoredIn the former case the sign of the current scattervalue indicates whether the grains will be posi-tioned to the right or left of (or transposed above orbelow) the cloud For instance when the scattervalue is positive all the grains are positioned tothe right of the cloud When the scatter becomesnegative the grains are positioned to the left Un-anchored scatter means that that the grains are po-sitioned both to the right and left of (andtransposed above and below) the cloud

The algorithm that generates the volume func-tion normalizes the values so that the cloudreaches the maximum width depth and transposi-tional scatter (Again these are not reached simul-

taneously) In a similar fashion to the motiontrajectory the algorithm also requires the composerto provide minimum and maximum values for thetime interval to generate time-volume breakpointsThe volume function may also be repeated an arbi-trary number of times this may also be equal tothe value of the orbital quantum number l plusone Repetitions may be identical or uctuatewithin a percentage given by the composer

Grain Parameters

These parameters control the attributes of individ-ual grains such as their envelope duration and in-dividual scatter patterns

Grain Amplitude Scaling

The amplitude of individual grains is scaled by again factor according to their onset times relativeto the beginning of the cloud and implemented ac-cording to various algorithm types The rst optionconsists of a scaling function that follows the enve-lope of the source (equation A20) Other optionsfollow Schrodingerrsquos distribution (r2 sin(h)|Y(rhf|2

in equations A21ndashA23) the complement of Schro-dingerrsquos distribution (equation A24) and Schro-dingerrsquos cumulative probability and itscomplement (equations A25 and A26) It is worthnoting that if the cumulative probability is 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 cloudproducing a nonlinear crescendo This is becausethe cumulative probability always increases mono-tonically Similarly following the complement ofthe cumulative probability will result in a nonlin-ear diminuendo

Finally scaling may be generated from the vari-ous distributions resulting from SchrodingerrsquosEquation (equations A27ndashA33)

Grain Duration

The actual duration of individual grains uctuatesaccording to their onset times relative to the begin-

Figure 3 Examples ofgranular cloud volume(a) narrow grain scatterand (b) wide grain scatter

53Fischman

ning of the cloud The function controlling the du-ration uctuation is calculated in a manner similarto that of the scaling factor offering the same op-tions to the user (equations A34 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 thecloudrsquos motion trajectory and its volume functionrepetitions may also be identical or uctuatewithin a percentage given by the composer Themain algorithms available consist of the followingschemes duration proportional to the source enve-lope duration proportional to Schrodingerrsquos distri-bution and its complement duration proportionalto Schrodingerrsquos cumulative probability and itscomplement and duration generated 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 grains be-come gradually shorter until their duration reachesits minimum value at the end of the cloud

Grain Envelope

The envelope applied to each individual grain con-sists of either the envelope extracted from thesource or a linear envelope shaped according to theseven parameters shown in Figure 4 and some ofSchrodingerrsquos distributions (equations A36ndashA40)

Grain Scatter Path

In addition to the xed scatter resulting from thevolume of the cloud each grain is provided with anindividual trajectory Grain scatter is given in thesame units discussed above in relation to cloudvolume

The algorithm is implemented by means of atime-varying scatter function created for each di-mension with normalized duration and maximumvalue of unity (equations A41ndashA43) To generatethe trajectory breakpoints the algorithm requiresminimum and maximum values for the time inter-val between a pair of breakpoints provided by thecomposer However these are given as percentagesof the grain duration as the function is normalizedWhen a grain is created the function is scaled to tits duration and maximum amplitude It is possibleto change the shape of the function according to ajitter value given by the composer which results ina different path 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 cloudand overall cloud motion (equation A44) The pitchshift relative to the listener is invariant under rota-tion Therefore it is possible to simplify the calcu-lation of the Doppler Effect 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 Figure 5(equations A45ndashA49)

Structure

We will now proceed to examine ways in whichthe generated material can be structured into a mu-sical work It is worth noting that these are particu-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 grain characteristics etc) Never-theless it is also important to note that choiceswere made with full consideration and regard fortheir musical viability and formal integrity

Figure 4 Linear envelopeparameters (1) start offset(2) overshoot attack(3) overshoot decay(4) steady state (5) decay(6) end offset and(7) overshoot percentage

54 Computer Music Journal

x

q

1

RL

Listener

y

a

R

L

Vg

Pyramidal Shape

A relatively straightforward strategy for structuringmusical works may be linked to a survey throughthe atomic shells resulting from changes in theoverall energy For instance the survey may beginin the lowest energy shell and leap to higher shellsas this energy increases (This case also correspondsto an excursion through the periodic table accord-ing to ascending atomic number) In fact it isworth adopting this particular instance as a starting

point to develop a basic model that can subse-quently be modi ed and extended

Following the analogy with the periodic tablemajor sectional boundaries may correspond to thereappearance of a shell with an increasing value ofn (which also corresponds to a change of row in theperiodic table) Because S is the rst shell majorsectional boundaries will start at 1S 2S 3S 7SSubsections within these boundaries may corre-spond to the shells surveyed in between Figure 6illustrates the resulting structure obtained from theenergy levels in the periodic table this may becompared to a pyramid that is surveyed from top tobottom Examination of this structure brings 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 (S and 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 corresponding to the reappearance ofeach particular shell in subsequent sections For in-stance shell D reappears in sections 4 to 7 Thissuggests the possibility of creating and additionaldevelopmental dimension that following the anal-ogy with the pyramid would consist of four lsquolsquoverti-calrsquorsquo paths identi ed by different grayscale shadesin Figure 6 For instance the reappearance of ashell could develop the material of previous appear-ances

Two remaining issues must be determined beforethe structure is nalized section duration and sec-tion differentiation

Figure 5 Calculation ofDoppler Shift using a rota-tion of axes

Figure 6 Pyramidal struc-ture resulting from the en-ergy levels obtained by fol-lowing the periodic tablein ascending order

55Fischman

7S 5F 6D 7P

6S 4F 5D 6P

5S 4D 5P

4S 4P

3S

2S 2P

1SSection 2

Section 1

Section 3

Section 4

Section 5

Section 6

Section 7

Section Duration

We can follow the path set by the rst observationabove (that sections become longer by virtue of thenumber of subsections they include) and establish afurther principle of growth related to the periodictable the minimum duration of each subsectioncloud is proportional to the sum of atomic numbersof its related shell by setting a correspondence be-tween a chosen time unit and atomic numbers asfollows

SectionDuration

4 atomic_numbers 2 time_unit (2)o1 2For example if the time unit is 005 sec the

minimum duration of the subsection correspondingto shell 2P (atomic numbers 5 to 10) will be

(5 ` 6 ` 7 ` 8 ` 9 ` 10) 2 005 4 225 sec

Applying this principle we obtain a further re- nement of the pyramidal shape whereby a newappearance of a shell is normally longer than itsprevious appearances Table 1 shows the minimumduration values for shell D with atomic unit dura-tion of 00342 sec and Figure 7 shows a schematicversion of the re ned 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 becauseaccording to quantum mechanics an electron re-mains in a given state as long as the energypumped into (or extracted from) it is equal to or

greater than the transition energy required to moveto an immediate neighboring state as calculated byequation A8

Section Differentiation

Identi cation of distinctive threads according toshells requires that sections be differentiated bytheir content This can be achieved by using differ-ent source sounds in each shell This will also in- uence overall cloud and individual graincharacteristics because as explained above manyof these are derived from the attributesmdashparticu-larly the envelopemdashof the source sound Anotherway sections may be differentiated is by setting dif-ferent grain attributes (eg different durations en-velopes spatial locations or by using differentdistributions to generate amplitudes and durations)

Extensions of the Pyramidal Shape

Two other structural shapes can be derived fromthe pyramid an inverted pyramid and a diamondThe former is analogous to a survey of the periodictable from the highest to the lowest atomic num-ber The latter consists of an original pyramid fol-

Figure 7 Schematic viewof pyramidal structurewhen the minimum dura-tion of each subsection isproportional to the sum ofatomic numbers of its re-lated shell (not to scale)

Table 1 Minimum duration values for shell Dwith atomic unit duration of 00342 sec

Shell Duration (sec)

3D 872104D 1487705D 2582106D 367650

56 Computer Music Journal

(a) (b)

lowed by of an inverted one These are shownschematically in Figure 8

Other Structural Possibilities

Apart from the simple trajectories related 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 beordered according to stochastic algorithms or gener-ative rules Alternatively they may follow attri-butes of a source sound such as the envelope wherethe peak may correspond to the maximum atomicnumber All these are options beyond the scope ofthe current research plan 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 (or dissipation) According to 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

1 1 E`SectionDuration 4 abs 1 2 (3)a U b 1 22 2n n Ka b E

where KE is the rate of energy accumulation (or dis-sipation) expressed as a fraction of the ionizationenergy E` per second As an illustration assumingthat KE 4 001 E` and that shell 3D progresses toshell 4P then the duration of the former is

3DSectionDuration3D U 4P

1 1 E`4 abs 1 2 4 486 sec1 22 23 4 001E`

The problem with this method is that energytransitions at lower shells are signi cantly higherthan those at higher shells For instance the transi-tion from 1S to 2S requires an energy value overone hundred times higher than a transition from6D to 7P Consequently the duration of 1S is inthe same proportion to that of 6D yielding ex-tremely long durations for 1S or extremely shortones for 6D This may be overcome by subdividinglonger sections according to proportions similar tothose dividing the whole work Alternatively therate of energy ow could correspond to 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 ow In general equationA10 may be modi ed to cater for any energy owfunction KE(t)

Section_durationa U b 1 1K (t)dt 4 abs 1 2 E (4)E ` 1 22 20 n na b

Erwinrsquos Playground

Erwinrsquos Playground is a work for tape lasting 9 min12 sec Except for the recording of 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 Erwinrsquos Playground is intendedto demonstrate the validity and viability of thesetheoretical principles in the generation of a musical

Figure 8 (a) Inverted pyra-midal structure and (b) di-amond structure

57Fischman

work as well as the usefulness of the software as aconduit for the realization of these principlesHowever its aesthetic value is intended to be inde-pendent of the generating principles because notheory is capable of 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 ErwinSchrodingerrsquos imaginary eld of action namely theinner shells of the atom Its structure is modeledon the diamond shape described above and linkedto a survey that begins in the lowest energy stateleaps up until it reaches the highest shell and thenreturns back to the initial state This may also becompared to an excursion through the periodic ta-ble that ascends and then descends according toatomic number order

Four different types of sound sources were as-signed to each shell to achieve section differentia-tion The S shells consisted primarily of vocalsounds (audio les MonoC1 Uba UlulowL) Pshells were assigned three different types of soundwhich nevertheless share some common spectralcharacteristics the lsquolsquobuzzrsquorsquo of a y a vocal lsquolsquobuzzrsquorsquoand the sound of a sparking welder (audio lesDieFly BZZZ Welder)

D shells were assigned sounds that ranged frompitched to lsquolsquosqueakyrsquorsquo The pitched sounds were theresult of rubbing the edge of a glass cup with differ-ent amounts of liquid to obtain different pitches(audio les Crystal E Crystal F Crystal G )Squeaks were recorded from the un-oiled wheels ofa trolley (audio le Squeak1) A sample containinga transition from pitch to squeak was achieved byrubbing the glass cup with different degrees of pres-sure (audio le Crystal B to Squeak) Finally a re-cording of the pitched sound of cicadas was alsoincluded toward the middle of the piece (audio leCicadas)

F shells were normally associated with local andglobal climaxes therefore they spread through awider bandwidth by means of three spectral typesThe rst type consists of two noise-based soundsobtained from a busy motorway and sea waves gur-gling through rocks (audio les Motorway1 andWaves1) The second sound is metallic consistingof the recording of a frying pan that was hit or

rubbed against other objects (audio le PBg1f) Thethird was mainly used to create low-frequency tex-tures and drones from the recording of a piece ofplywood that was bent at a more or less regularrate (audio le PBdLow) (Examples of the source les as well as passages characterizing each shellsand the complete work are available on the forth-coming Computer Music Journal volume 27 soundexample disc)

Grain duration was another signi cant attributeused to characterize the various shells Finally theatomic number duration equivalent used in thepiece was 00342 sec per atomic number

Table 2 presents a complete list of the structuralhierarchy of Erwinrsquos Playground including starttime minimum duration grain duration limitsand the source audio les for each subsection Asmentioned above the duration of subsections couldbe extended to allow some freedom when takinginto account musical context Section extensionsappear below the shell name with the letter lsquolsquoXrsquorsquoAlso note the bold line denoting the end of the pyr-amid and the beginning of its inversion which actsas an lsquolsquoaxis of re ectionrsquorsquo It is worth noting thatthe second half of the piece (sections 8ndash14 corre-sponding to the inverted pyramid) is more com-pressed than the rst part as there are fewersubsections that have their minimum durationsextended

Software Implementation

The software implementation of this procedureconsists of two separate components AL a compo-sitional environment for the creation and manipu-lation of musical events and ERWIN a plug-inthat implements the application of Schrodingerrsquosmodel described above These were developed andtested using Visual C ` ` 60 running under Win-dows 2000 and later under Windows XP Both ap-plications use Microsoft DirectX 8 features andrequire that DirectX 8 be installed in the host com-puter For this reason the AL and ERWIN installa-tion program also installs DirectX 8 if requiredCurrently AL and ERWIN are distributed as freesoftware under the GNU public license and are

58 Computer Music Journal

Table 2 Structural hierarchy list of Erwinrsquos Playground (lsquolsquoXrsquorsquo denotes section extensions)

Section Shell Start TimeDuration

(sec) Grain Duration (msec) Source Audio Files

1 1S 0000000 01026 5ndash25 MonoC1X 0001026 37258

2 2S 0037384 02394 5ndash50 MonoC12P 0039778 15390 5ndash50 DieFlyX 0055168 21572

3 3S 0076740 07866 5ndash503P 0084606 31806 75ndash75 75ndash20 DieFly

4 4S 0116412 13338 5ndash100 MonoC13D 0129750 87210 10ndash60 30ndash60 Crystal E4P 0216960 68742 5ndash60 DieFlyX 0285702 31777

5 5S 0317479 25650 5ndash25 MonoC110ndash160 Uba

4D 0343129 148770 10ndash150 100ndash150 Crystal E5P 0491899 105678 10ndash60 30ndash60 DieFly

30ndash60 WelderX 0597577 05818

6 6S 1003395 37962 5ndash25 MonoC110ndash160 Uba100ndash500 UlulowL

X 1041357 295954F 1070952 304038 100ndash5000 Waves1

1000ndash4000 2000ndash6000 PBdLow100ndash300 4000ndash8000 PBg1f

X 1374990 265995D 1401589 258210 1000ndash4500 Crystal E

10ndash50 10ndash4000 20ndash150 Crystal F10ndash50 Crystal G30ndash50 Squeak1

X 2059799 084946P 2068293 171342 30ndash60 DieFly

20ndash80 20ndash150 Welder5ndash1000 BZZZ

X 2239635 50250

7 7S 2289885 59850 5ndash25 MonoC110ndash200 Uba5ndash25 200ndash700 Ululow

X 2349735 434425F 2393177 457254 200ndash8000 Waves1

1000ndash4000 2000ndash5000 2000ndash6000 PBdLow100ndash300 4000ndash8000 PBg1f5ndash25 40ndash150 Motorway1

X 2250431 1051106D 3355541 367650 200ndash350 Crystal E

10ndash50 10ndash4000 20ndash150 Crystal F10ndash50 Crystal G50ndash150 50ndash250 Squeak1

X 4123191 304657P 4153656 237006 75ndash20 75ndash75 10ndash60 30ndash60 DieFly

30ndash60 WelderX 4390662 12090

8 7P 4402752 237006 75ndash20 20ndash40 DieFly6D 5039758 367650 10ndash50 200ndash350 Crystal E

10ndash50 Crystal F50ndash250 Squeak15ndash5000 Cicadas

(continued )

59Fischman

Table 2 (Continued)

Section Shell Start TimeDuration

(sec) Grain Duration (msec) Source Audio Files

5F 5407408 457254 100ndash5000 Waves12000ndash8000 PBdLow100ndash300 4000ndash8000 PBg1f5ndash25 40ndash150 150ndash200 Motorway1

7S 6264662 59850 5ndash25 MonoC110ndash200 Uba5ndash25 90ndash700 200ndash700 UlulowL

9 6P 6324512 171342 30ndash60 DieFly20ndash80 100ndash250 Welder5ndash1000 BZZZ

5D 6495854 258210 200ndash350 1000ndash4500 Crystal E20ndash150 Crystal F10ndash50 Crystal G50ndash250 Squeak1

4F 7154064 304038 100ndash5000 Waves11000ndash4000 PBdLow100ndash300 4000ndash8000 PBg1f100ndash10000 150ndash200 Motorway1

X 7458102 100006S 7468102 37962 5ndash25 MonoC1

10ndash160 10ndash200 Uba100ndash500 UlulowL

10 5P 7506064 105678 10ndash60 30ndash60 DieFly30ndash60 Welder5ndash1000 10ndash60 BZZZ

X 8011742 315524D 8043294 148770 200ndash350 100ndash150 Crystal E

20ndash60 50ndash250 Squeak150ndash150 Crystal B to Squeak

5S 8192064 25650 5ndash25 MonoC110ndash160 Uba

X 8217714 80546

11 4P 8298260 68742 5ndash60 10ndash30 DieFly5ndash60 BZZZ

3D 8367002 87210 80ndash450 200ndash350 Crystal E40ndash400 50ndash250 100ndash350 Squeak1

X 8454212 002684S 8454480 13338 5ndash25 5ndash100 MonoC1X 8467818 36696

12 3P 8504514 31806 75ndash20 75ndash75 DieFly3S 8536320 07866 5ndash25 5ndash50 MonoC1X 8544186 46046

13 2P 8590232 15390 5ndash50 DieFly2S 9005622 02394 15ndash50 MonoC1X 9008016 45972

14 1S 9053988 01026 5ndash25 MonoC110ndash200 70ndash200 Uba

X 9055014 61339

END 9116353

60 Computer Music Journal

available online at wwwkeeleacukdeptsmustaffAlAl_softwarehtm The programs are alsodistributed with the Composerrsquos Desktop Project

AL

AL is a multiple document interface (MDI) applica-tion with frame windows that present views of dif-ferent documents andor several views of the samedocument It is designed for the creation and ma-nipulation of sonic events and their organizationinto a structured musical work

To aid user familiarity with the software and pro-mote a reasonably short learning curve care wastaken to develop an interface within an accepted

general standard for Windows applications includ-ing a menu bar a tool bar common window com-ponents and typical mouse lsquolsquodrag-and-droprsquorsquooperations It also contains detailed online helppages including a tutorial Figure 9 shows an in-stance of the user interface

Another important consideration in the design ofAL was the desire to provide a reasonably open sys-tem with the option for development of third-partyplug-ins thus avoiding the constraints imposed bya single type of musical process Future develop-ments by the author and others will ideally imple-ment a variety of compositional approaches inaddition to the application of Schrodingerrsquos Equa-tion in the existing plug-in (see the description ofERWIN below)

Figure 9 The AL user in-terface including themenu bar tool bar and aview of a document

48 Computer Music Journal

logic and coherence While previous research bythe author was 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 parameters other than pitch and providesthe basis for the application of similar strategies toother media Thus the electroacoustic domainwith a speci c focus on asynchronous granulartechniques was chosen because it allows a signi -cant degree of complexity and it allows thoroughinvestigation of the possibilities inherent in Schro-dingerrsquos Equation for the manipulation of multi-dimensional timbral characteristics with pitchincluded as one of these dimensions (Smalley 19861997) In particular granular techniques are ideallysuited for stochastic processing of musical material(Lorrain 1980 Xenakis 1992) Furthermore a workfor tape can provide the immediate aural feedbacknecessary to evaluate the results

The research methodology adopted during theproject consisted of three main parts First I devel-oped appropriate theoretical principles and algo-rithms for the creation of musical material arisingfrom the solutions and implications of Schro-dingerrsquos Equation Next I developed software thatimplements these algorithms The software in-cludes a graphical interface that enables the manip-ulation of large amounts of data and musicalstreams is reasonably self-suf cient for the crea-tion of a complete musical work and is reasonablyopen-ended offering possibilities for enhancementand customization via third-party plug-ins using arecognized standard The third part of this researchconsisted of testing the validity and viability of thetheoretical principles and the usefulness of thesoftware through the creation of a musical work fortape Erwinrsquos Playground

Concerning the last point it is important tostress that neither theory nor software are intendedto guarantee satisfactory aesthetic resultsmdashor 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 lsquolsquoactionper-

ception feedback looprsquorsquo In fact it is reasonable toassume that most mathematical constructs areneutral to musical logic and that their usefulness isdependent on the actual way they are mapped intomusical processes by the composer Neverthelesstheoretical principles are vital in providing lsquolsquoformaltools as generative and transformative devicesrsquorsquo(Vaggione 2001) within which a speci c work maybe realized projecting the latter beyond an exclu-sively sensory sphere Furthermore decisions re-garding the correspondence between the modelrsquosparameters and musical attributes as well as theirrealization by means of the design and functional-ity of the software (which at rst sight mightseem to be exclusively related to theoretical andtechnical development) already belong to the com-positional process since they carry speci c aes-thetic presumptions and aims (For a discussion ofthis issue in a more general context see Vaughan1994) For instance recent published work presentsdifferent ontological approaches to correspondencesbetween quantum mechanics and music (Delatour2000 Sturm 2001)

After a brief description of Schrodingerrsquos Equa-tion this article presents the framework developedfor the correspondence between the equation andthe creation of granular clouds and strategies fortheir organization into coherent musical structureThis will be followed by discussion regarding therealization of these principles in Erwinrsquos Play-ground a work for computer-generated tape Subse-quently a general description of the software willbe provided An appendix is offered to readers whomay wish to explore the mathematics in more de-tail (Reference to equations in the appendix usesthe pre x lsquolsquoArsquorsquo followed by a number)

Schrodingerrsquos Equation

Schrodingerrsquos Differential Equation forms the basisfor the techniques described herein which estab-lish a correspondence between 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 A1ndashA8)

49Fischman

The particular version of Schrodingerrsquos Equationexamined 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 spherical sur-face of radius r the potential will be symmetricwith respect to the center Thus SchrodingerrsquosEquation assumes the following form

]Y21 sup1 Y ` V(r)Y 4 i (1)2m ]t

where V(r) is the atomic potential m is the mass ofthe electron r is the distance from the center of thepotential (eg the nucleus of an atom) to the elec-tron and is the ratio between Plankrsquos constant hand 2p ( 4 h2p)

The family of functions corresponding to thesolutions of the equation is represented by thesymbol W also known as the wave function Exam-ination of the general form of the wave function re-veals that W consists of the product of four smallerfunctions (equation A2) that may be treated sepa-rately because each is dependent on a different pa-rameter Three of these were found particularlyuseful R which is exclusively dependent on r thedistance of a point to the center of the potential Hwhich is exclusively dependent of the pointrsquos eleva-tion angle h and U which is exclusively dependenton its horizontal angle f

Furthermore the solutions depend on a set ofquantum numbers that restrict the physical attri-butes of the electron to multiples of certain energylevels The rst of these values is n the principalquantum number (n 4 1 2 3 ) which deter-mines the overall energy of the electron (equationA7) The second value is l the angular momentumquantum number (l 4 0 1 2 nndash1) which gen-erates additional allowed energies (quantum states)within a single value of n (The term shell is usedto denote a particular value of l) So far four typesof shells have been encountered in nature and arenamed S P D and F corresponding to l 4 0 1 2and 3 respectively Therefore a particular energylevel is characterized by a combination of n and lwhich by convention is named using the value ofthe principal quantum number and the letter of itsshell (eg 1S 2S and 2P)

The third quantum number m is the magneticquantum number (ndashl m l) 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 probability of ndingan electron It is possible to show that this proba-bility is associated with R and H (equations A9 andA10) As an example of this correlation Figure 1shows plots of these distributions and the resultingorbital for particular values of n l and m fromwhich we may infer the most probable distancesand orientations For instance Figure 1b indicatesthat the electron is more likely to be found in thevicinity of 35278 and 144738

The quantum numbers were successfully used toexplain the existence of common characteristicsbetween known chemical elements supportingtheir classi cation into the Periodic Table of Ele-ments originally devised by the chemist DmitriMendeleev in the 19th century but only more re-cently explained by Schrodingerrsquos EquationRoughly 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 group and occupying the same column inthe periodic table For instance the group of halo-gens consists of elements that have ve electronsin their P shell as follows uorine has ve elec-trons in shell 2P chlorine in 3P bromine in 4Piodine in 5P and astatine in 6P

Generation of Granular Clouds

My generation of musical material using Schro-dingerrsquos Equation focuses on granular clouds(sounds composed of a large number of short ele-mentary sonic particles or grains each lasting afew hundredths of a second) derived from statisticaldistributions obtained from 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 importance

50 Computer Music Journal

009 031 076 150

(a)

3527o 90o 14473o0o 180o

(b)

3527o(c)

is given to the creation of formal relationships be-tween the inner structure of the source soundmdashpar-ticularly its amplitude envelopemdashand the structureof the cloud and its constituent grains Sourcesounds are stored as wave audio les

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 Parameters

These parameters refer to global attributes of thecloud such as density overall amplitude and spa-tial and spectral trajectory

Cloud Density

To 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 are proportional to the density uctuation Inpractice the envelope is extracted from the sourceand stored in a timendashamplitude breakpoint tableThe cloud density is also stored as pairs of 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 corresponding values for clouddensity (equation A11)

Currently there are three different methods ofestablishing a correspondence between source am-plitude envelope and cloud density The rst twocalculate the density by using the envelope to drivethe radial and angular distributions (equation A12)either by making both r and h directly proportionalto the envelope (equation A13) or by making r di-rectly proportional to the envelope and h directlyproportional to its rst derivative (equation A14)The third method establishes a direct proportional-ity between the envelope and the density using astraightforward proportion rule (equation A15)

Figure 1 Examples ofSchrodinger distributionsr2|R|2 for Z 4 139 n4 4and l4 1 with radial val-ues normalized to (a) 15

(b) |H(h)|2 for n 4 4 l 4 1m 4 0 and (c) a diagram ofthe orbital resulting fromthese distributions

51Fischman

x=-1 x=1

RL y=0

ygt0

xgt1xlt-1

Listener

x lt -1 Beyond the left speakerx = -1 Left speaker-1 lt x lt 1 Between the speakersx = 1 Right speakerx gt 1 Beyond the right speaker

y = 0 On the imaginary line that joins the speakersy gt 0 Behind the imaginary line that joins the speakers

When y = 1 the depth is equal to the distancefrom the centre to one of the speakers

x=0

Cloud Amplitude Scaling

The amplitude of the cloud may be scaled by avalue that uctuates between a minimum andmaximum gain The uctuation follows the enve-lope of the source (equation A16)

Cloud Motion

Motion is carried out in three dimensions Two ofthese are spatial (width and depth) and the remain-ing one is articulated as a spectral dimension (pitchshift)

Width and Depth

These are calculated within a stereo image x indi-cates width and y indicates depth as illustrated inFigure 2 (See also equation A17)

Spectral Motion

The cloud lsquolsquomovesrsquorsquo in the frequency domain as aresult of time-varying transposition within limitsspeci ed by the composer establishing an analogyto movement across a height axis z Transpositionis carried out in the frequency domain via Short-

Time Fourier Transforms thereby preserving theduration of the grains

Trajectory Creation

To implement motion a three-dimensional trajec-tory is created that is analogous to physical move-ment within the boundaries of a shell (orbital)This consists of the stochastic generation of break-points according to Schrodingerrsquos radial and angulardistributions The breakpoints consist of position-time pairs within the duration of the cloud con-verted from r h and f into width depth andheight (equation A18)

The algorithm that generates the trajectory nor-malizes the values so that the cloud reaches themaximum width maximum depth and one of thetransposition boundaries (These are not reachedsimultaneously) Also to generate the trajectorybreakpoints the algorithm requires minimum andmaximum values for the time interval provided bythe composer The intervals between the break-points in the trajectory will fall randomly betweenthese two values For instance if the minimum andmaximum intervals are 02 and 05 sec respec-tively and the last breakpoint was generated attime tc 4 14 sec the time tc for the next break-point will fall between 16 and 19 sec

Finally trajectories may be repeated an arbitrarynumber of times so that the sum of the durationsof the repetitions is the overall duration of thecloud (If required this may be equal to the value ofthe orbital quantum number l plus one) Repeti-tions may be identical or uctuate within a per-centage given by the composer

Cloud Volume (Overall Grain Scatter)

The volume of the cloud depends on the maximumscatter of the grains in each dimension (widthdepth and pitch 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

Figure 2 Values of x andy for spatial width anddepth

52 Computer Music Journal

RL(a)

RL(b)

Width and depth scatter values vary from 0 (noscatter) to any positive numbers x and y with avalue of 1 indicating a scatter distance equal to thedistance from the center to one of the speakersSpectral scatter values vary from 0 (no scatter) toany positive number of semitones (or any fre-quency ratio)

To implement scatter a time-varying volumefunction is created in a similar manner to that ofthe motion trajectory described above Breakpointsare also generated stochastically according toSchrodingerrsquos radial and angular distributions(equation A19)

Therefore a grain created at time tc will movewith the cloud at a xed lsquolsquodistancersquorsquo from its centerThis distance is generated stochastically withinwidth depth and height boundary values 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 unanchoredIn the former case the sign of the current scattervalue indicates whether the grains will be posi-tioned to the right or left of (or transposed above orbelow) the cloud For instance when the scattervalue is positive all the grains are positioned tothe right of the cloud When the scatter becomesnegative the grains are positioned to the left Un-anchored scatter means that that the grains are po-sitioned both to the right and left of (andtransposed above and below) the cloud

The algorithm that generates the volume func-tion normalizes the values so that the cloudreaches the maximum width depth and transposi-tional scatter (Again these are not reached simul-

taneously) In a similar fashion to the motiontrajectory the algorithm also requires the composerto provide minimum and maximum values for thetime interval to generate time-volume breakpointsThe volume function may also be repeated an arbi-trary number of times this may also be equal tothe value of the orbital quantum number l plusone Repetitions may be identical or uctuatewithin a percentage given by the composer

Grain Parameters

These parameters control the attributes of individ-ual grains such as their envelope duration and in-dividual scatter patterns

Grain Amplitude Scaling

The amplitude of individual grains is scaled by again factor according to their onset times relativeto the beginning of the cloud and implemented ac-cording to various algorithm types The rst optionconsists of a scaling function that follows the enve-lope of the source (equation A20) Other optionsfollow Schrodingerrsquos distribution (r2 sin(h)|Y(rhf|2

in equations A21ndashA23) the complement of Schro-dingerrsquos distribution (equation A24) and Schro-dingerrsquos cumulative probability and itscomplement (equations A25 and A26) It is worthnoting that if the cumulative probability is 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 cloudproducing a nonlinear crescendo This is becausethe cumulative probability always increases mono-tonically Similarly following the complement ofthe cumulative probability will result in a nonlin-ear diminuendo

Finally scaling may be generated from the vari-ous distributions resulting from SchrodingerrsquosEquation (equations A27ndashA33)

Grain Duration

The actual duration of individual grains uctuatesaccording to their onset times relative to the begin-

Figure 3 Examples ofgranular cloud volume(a) narrow grain scatterand (b) wide grain scatter

53Fischman

ning of the cloud The function controlling the du-ration uctuation is calculated in a manner similarto that of the scaling factor offering the same op-tions to the user (equations A34 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 thecloudrsquos motion trajectory and its volume functionrepetitions may also be identical or uctuatewithin a percentage given by the composer Themain algorithms available consist of the followingschemes duration proportional to the source enve-lope duration proportional to Schrodingerrsquos distri-bution and its complement duration proportionalto Schrodingerrsquos cumulative probability and itscomplement and duration generated 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 grains be-come gradually shorter until their duration reachesits minimum value at the end of the cloud

Grain Envelope

The envelope applied to each individual grain con-sists of either the envelope extracted from thesource or a linear envelope shaped according to theseven parameters shown in Figure 4 and some ofSchrodingerrsquos distributions (equations A36ndashA40)

Grain Scatter Path

In addition to the xed scatter resulting from thevolume of the cloud each grain is provided with anindividual trajectory Grain scatter is given in thesame units discussed above in relation to cloudvolume

The algorithm is implemented by means of atime-varying scatter function created for each di-mension with normalized duration and maximumvalue of unity (equations A41ndashA43) To generatethe trajectory breakpoints the algorithm requiresminimum and maximum values for the time inter-val between a pair of breakpoints provided by thecomposer However these are given as percentagesof the grain duration as the function is normalizedWhen a grain is created the function is scaled to tits duration and maximum amplitude It is possibleto change the shape of the function according to ajitter value given by the composer which results ina different path 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 cloudand overall cloud motion (equation A44) The pitchshift relative to the listener is invariant under rota-tion Therefore it is possible to simplify the calcu-lation of the Doppler Effect 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 Figure 5(equations A45ndashA49)

Structure

We will now proceed to examine ways in whichthe generated material can be structured into a mu-sical work It is worth noting that these are particu-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 grain characteristics etc) Never-theless it is also important to note that choiceswere made with full consideration and regard fortheir musical viability and formal integrity

Figure 4 Linear envelopeparameters (1) start offset(2) overshoot attack(3) overshoot decay(4) steady state (5) decay(6) end offset and(7) overshoot percentage

54 Computer Music Journal

x

q

1

RL

Listener

y

a

R

L

Vg

Pyramidal Shape

A relatively straightforward strategy for structuringmusical works may be linked to a survey throughthe atomic shells resulting from changes in theoverall energy For instance the survey may beginin the lowest energy shell and leap to higher shellsas this energy increases (This case also correspondsto an excursion through the periodic table accord-ing to ascending atomic number) In fact it isworth adopting this particular instance as a starting

point to develop a basic model that can subse-quently be modi ed and extended

Following the analogy with the periodic tablemajor sectional boundaries may correspond to thereappearance of a shell with an increasing value ofn (which also corresponds to a change of row in theperiodic table) Because S is the rst shell majorsectional boundaries will start at 1S 2S 3S 7SSubsections within these boundaries may corre-spond to the shells surveyed in between Figure 6illustrates the resulting structure obtained from theenergy levels in the periodic table this may becompared to a pyramid that is surveyed from top tobottom Examination of this structure brings 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 (S and 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 corresponding to the reappearance ofeach particular shell in subsequent sections For in-stance shell D reappears in sections 4 to 7 Thissuggests the possibility of creating and additionaldevelopmental dimension that following the anal-ogy with the pyramid would consist of four lsquolsquoverti-calrsquorsquo paths identi ed by different grayscale shadesin Figure 6 For instance the reappearance of ashell could develop the material of previous appear-ances

Two remaining issues must be determined beforethe structure is nalized section duration and sec-tion differentiation

Figure 5 Calculation ofDoppler Shift using a rota-tion of axes

Figure 6 Pyramidal struc-ture resulting from the en-ergy levels obtained by fol-lowing the periodic tablein ascending order

55Fischman

7S 5F 6D 7P

6S 4F 5D 6P

5S 4D 5P

4S 4P

3S

2S 2P

1SSection 2

Section 1

Section 3

Section 4

Section 5

Section 6

Section 7

Section Duration

We can follow the path set by the rst observationabove (that sections become longer by virtue of thenumber of subsections they include) and establish afurther principle of growth related to the periodictable the minimum duration of each subsectioncloud is proportional to the sum of atomic numbersof its related shell by setting a correspondence be-tween a chosen time unit and atomic numbers asfollows

SectionDuration

4 atomic_numbers 2 time_unit (2)o1 2For example if the time unit is 005 sec the

minimum duration of the subsection correspondingto shell 2P (atomic numbers 5 to 10) will be

(5 ` 6 ` 7 ` 8 ` 9 ` 10) 2 005 4 225 sec

Applying this principle we obtain a further re- nement of the pyramidal shape whereby a newappearance of a shell is normally longer than itsprevious appearances Table 1 shows the minimumduration values for shell D with atomic unit dura-tion of 00342 sec and Figure 7 shows a schematicversion of the re ned 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 becauseaccording to quantum mechanics an electron re-mains in a given state as long as the energypumped into (or extracted from) it is equal to or

greater than the transition energy required to moveto an immediate neighboring state as calculated byequation A8

Section Differentiation

Identi cation of distinctive threads according toshells requires that sections be differentiated bytheir content This can be achieved by using differ-ent source sounds in each shell This will also in- uence overall cloud and individual graincharacteristics because as explained above manyof these are derived from the attributesmdashparticu-larly the envelopemdashof the source sound Anotherway sections may be differentiated is by setting dif-ferent grain attributes (eg different durations en-velopes spatial locations or by using differentdistributions to generate amplitudes and durations)

Extensions of the Pyramidal Shape

Two other structural shapes can be derived fromthe pyramid an inverted pyramid and a diamondThe former is analogous to a survey of the periodictable from the highest to the lowest atomic num-ber The latter consists of an original pyramid fol-

Figure 7 Schematic viewof pyramidal structurewhen the minimum dura-tion of each subsection isproportional to the sum ofatomic numbers of its re-lated shell (not to scale)

Table 1 Minimum duration values for shell Dwith atomic unit duration of 00342 sec

Shell Duration (sec)

3D 872104D 1487705D 2582106D 367650

56 Computer Music Journal

(a) (b)

lowed by of an inverted one These are shownschematically in Figure 8

Other Structural Possibilities

Apart from the simple trajectories related 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 beordered according to stochastic algorithms or gener-ative rules Alternatively they may follow attri-butes of a source sound such as the envelope wherethe peak may correspond to the maximum atomicnumber All these are options beyond the scope ofthe current research plan 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 (or dissipation) According to 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

1 1 E`SectionDuration 4 abs 1 2 (3)a U b 1 22 2n n Ka b E

where KE is the rate of energy accumulation (or dis-sipation) expressed as a fraction of the ionizationenergy E` per second As an illustration assumingthat KE 4 001 E` and that shell 3D progresses toshell 4P then the duration of the former is

3DSectionDuration3D U 4P

1 1 E`4 abs 1 2 4 486 sec1 22 23 4 001E`

The problem with this method is that energytransitions at lower shells are signi cantly higherthan those at higher shells For instance the transi-tion from 1S to 2S requires an energy value overone hundred times higher than a transition from6D to 7P Consequently the duration of 1S is inthe same proportion to that of 6D yielding ex-tremely long durations for 1S or extremely shortones for 6D This may be overcome by subdividinglonger sections according to proportions similar tothose dividing the whole work Alternatively therate of energy ow could correspond to 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 ow In general equationA10 may be modi ed to cater for any energy owfunction KE(t)

Section_durationa U b 1 1K (t)dt 4 abs 1 2 E (4)E ` 1 22 20 n na b

Erwinrsquos Playground

Erwinrsquos Playground is a work for tape lasting 9 min12 sec Except for the recording of 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 Erwinrsquos Playground is intendedto demonstrate the validity and viability of thesetheoretical principles in the generation of a musical

Figure 8 (a) Inverted pyra-midal structure and (b) di-amond structure

57Fischman

work as well as the usefulness of the software as aconduit for the realization of these principlesHowever its aesthetic value is intended to be inde-pendent of the generating principles because notheory is capable of 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 ErwinSchrodingerrsquos imaginary eld of action namely theinner shells of the atom Its structure is modeledon the diamond shape described above and linkedto a survey that begins in the lowest energy stateleaps up until it reaches the highest shell and thenreturns back to the initial state This may also becompared to an excursion through the periodic ta-ble that ascends and then descends according toatomic number order

Four different types of sound sources were as-signed to each shell to achieve section differentia-tion The S shells consisted primarily of vocalsounds (audio les MonoC1 Uba UlulowL) Pshells were assigned three different types of soundwhich nevertheless share some common spectralcharacteristics the lsquolsquobuzzrsquorsquo of a y a vocal lsquolsquobuzzrsquorsquoand the sound of a sparking welder (audio lesDieFly BZZZ Welder)

D shells were assigned sounds that ranged frompitched to lsquolsquosqueakyrsquorsquo The pitched sounds were theresult of rubbing the edge of a glass cup with differ-ent amounts of liquid to obtain different pitches(audio les Crystal E Crystal F Crystal G )Squeaks were recorded from the un-oiled wheels ofa trolley (audio le Squeak1) A sample containinga transition from pitch to squeak was achieved byrubbing the glass cup with different degrees of pres-sure (audio le Crystal B to Squeak) Finally a re-cording of the pitched sound of cicadas was alsoincluded toward the middle of the piece (audio leCicadas)

F shells were normally associated with local andglobal climaxes therefore they spread through awider bandwidth by means of three spectral typesThe rst type consists of two noise-based soundsobtained from a busy motorway and sea waves gur-gling through rocks (audio les Motorway1 andWaves1) The second sound is metallic consistingof the recording of a frying pan that was hit or

rubbed against other objects (audio le PBg1f) Thethird was mainly used to create low-frequency tex-tures and drones from the recording of a piece ofplywood that was bent at a more or less regularrate (audio le PBdLow) (Examples of the source les as well as passages characterizing each shellsand the complete work are available on the forth-coming Computer Music Journal volume 27 soundexample disc)

Grain duration was another signi cant attributeused to characterize the various shells Finally theatomic number duration equivalent used in thepiece was 00342 sec per atomic number

Table 2 presents a complete list of the structuralhierarchy of Erwinrsquos Playground including starttime minimum duration grain duration limitsand the source audio les for each subsection Asmentioned above the duration of subsections couldbe extended to allow some freedom when takinginto account musical context Section extensionsappear below the shell name with the letter lsquolsquoXrsquorsquoAlso note the bold line denoting the end of the pyr-amid and the beginning of its inversion which actsas an lsquolsquoaxis of re ectionrsquorsquo It is worth noting thatthe second half of the piece (sections 8ndash14 corre-sponding to the inverted pyramid) is more com-pressed than the rst part as there are fewersubsections that have their minimum durationsextended

Software Implementation

The software implementation of this procedureconsists of two separate components AL a compo-sitional environment for the creation and manipu-lation of musical events and ERWIN a plug-inthat implements the application of Schrodingerrsquosmodel described above These were developed andtested using Visual C ` ` 60 running under Win-dows 2000 and later under Windows XP Both ap-plications use Microsoft DirectX 8 features andrequire that DirectX 8 be installed in the host com-puter For this reason the AL and ERWIN installa-tion program also installs DirectX 8 if requiredCurrently AL and ERWIN are distributed as freesoftware under the GNU public license and are

58 Computer Music Journal

Table 2 Structural hierarchy list of Erwinrsquos Playground (lsquolsquoXrsquorsquo denotes section extensions)

Section Shell Start TimeDuration

(sec) Grain Duration (msec) Source Audio Files

1 1S 0000000 01026 5ndash25 MonoC1X 0001026 37258

2 2S 0037384 02394 5ndash50 MonoC12P 0039778 15390 5ndash50 DieFlyX 0055168 21572

3 3S 0076740 07866 5ndash503P 0084606 31806 75ndash75 75ndash20 DieFly

4 4S 0116412 13338 5ndash100 MonoC13D 0129750 87210 10ndash60 30ndash60 Crystal E4P 0216960 68742 5ndash60 DieFlyX 0285702 31777

5 5S 0317479 25650 5ndash25 MonoC110ndash160 Uba

4D 0343129 148770 10ndash150 100ndash150 Crystal E5P 0491899 105678 10ndash60 30ndash60 DieFly

30ndash60 WelderX 0597577 05818

6 6S 1003395 37962 5ndash25 MonoC110ndash160 Uba100ndash500 UlulowL

X 1041357 295954F 1070952 304038 100ndash5000 Waves1

1000ndash4000 2000ndash6000 PBdLow100ndash300 4000ndash8000 PBg1f

X 1374990 265995D 1401589 258210 1000ndash4500 Crystal E

10ndash50 10ndash4000 20ndash150 Crystal F10ndash50 Crystal G30ndash50 Squeak1

X 2059799 084946P 2068293 171342 30ndash60 DieFly

20ndash80 20ndash150 Welder5ndash1000 BZZZ

X 2239635 50250

7 7S 2289885 59850 5ndash25 MonoC110ndash200 Uba5ndash25 200ndash700 Ululow

X 2349735 434425F 2393177 457254 200ndash8000 Waves1

1000ndash4000 2000ndash5000 2000ndash6000 PBdLow100ndash300 4000ndash8000 PBg1f5ndash25 40ndash150 Motorway1

X 2250431 1051106D 3355541 367650 200ndash350 Crystal E

10ndash50 10ndash4000 20ndash150 Crystal F10ndash50 Crystal G50ndash150 50ndash250 Squeak1

X 4123191 304657P 4153656 237006 75ndash20 75ndash75 10ndash60 30ndash60 DieFly

30ndash60 WelderX 4390662 12090

8 7P 4402752 237006 75ndash20 20ndash40 DieFly6D 5039758 367650 10ndash50 200ndash350 Crystal E

10ndash50 Crystal F50ndash250 Squeak15ndash5000 Cicadas

(continued )

59Fischman

Table 2 (Continued)

Section Shell Start TimeDuration

(sec) Grain Duration (msec) Source Audio Files

5F 5407408 457254 100ndash5000 Waves12000ndash8000 PBdLow100ndash300 4000ndash8000 PBg1f5ndash25 40ndash150 150ndash200 Motorway1

7S 6264662 59850 5ndash25 MonoC110ndash200 Uba5ndash25 90ndash700 200ndash700 UlulowL

9 6P 6324512 171342 30ndash60 DieFly20ndash80 100ndash250 Welder5ndash1000 BZZZ

5D 6495854 258210 200ndash350 1000ndash4500 Crystal E20ndash150 Crystal F10ndash50 Crystal G50ndash250 Squeak1

4F 7154064 304038 100ndash5000 Waves11000ndash4000 PBdLow100ndash300 4000ndash8000 PBg1f100ndash10000 150ndash200 Motorway1

X 7458102 100006S 7468102 37962 5ndash25 MonoC1

10ndash160 10ndash200 Uba100ndash500 UlulowL

10 5P 7506064 105678 10ndash60 30ndash60 DieFly30ndash60 Welder5ndash1000 10ndash60 BZZZ

X 8011742 315524D 8043294 148770 200ndash350 100ndash150 Crystal E

20ndash60 50ndash250 Squeak150ndash150 Crystal B to Squeak

5S 8192064 25650 5ndash25 MonoC110ndash160 Uba

X 8217714 80546

11 4P 8298260 68742 5ndash60 10ndash30 DieFly5ndash60 BZZZ

3D 8367002 87210 80ndash450 200ndash350 Crystal E40ndash400 50ndash250 100ndash350 Squeak1

X 8454212 002684S 8454480 13338 5ndash25 5ndash100 MonoC1X 8467818 36696

12 3P 8504514 31806 75ndash20 75ndash75 DieFly3S 8536320 07866 5ndash25 5ndash50 MonoC1X 8544186 46046

13 2P 8590232 15390 5ndash50 DieFly2S 9005622 02394 15ndash50 MonoC1X 9008016 45972

14 1S 9053988 01026 5ndash25 MonoC110ndash200 70ndash200 Uba

X 9055014 61339

END 9116353

60 Computer Music Journal

available online at wwwkeeleacukdeptsmustaffAlAl_softwarehtm The programs are alsodistributed with the Composerrsquos Desktop Project

AL

AL is a multiple document interface (MDI) applica-tion with frame windows that present views of dif-ferent documents andor several views of the samedocument It is designed for the creation and ma-nipulation of sonic events and their organizationinto a structured musical work

To aid user familiarity with the software and pro-mote a reasonably short learning curve care wastaken to develop an interface within an accepted

general standard for Windows applications includ-ing a menu bar a tool bar common window com-ponents and typical mouse lsquolsquodrag-and-droprsquorsquooperations It also contains detailed online helppages including a tutorial Figure 9 shows an in-stance of the user interface

Another important consideration in the design ofAL was the desire to provide a reasonably open sys-tem with the option for development of third-partyplug-ins thus avoiding the constraints imposed bya single type of musical process Future develop-ments by the author and others will ideally imple-ment a variety of compositional approaches inaddition to the application of Schrodingerrsquos Equa-tion in the existing plug-in (see the description ofERWIN below)

Figure 9 The AL user in-terface including themenu bar tool bar and aview of a document

49Fischman

The particular version of Schrodingerrsquos Equationexamined 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 spherical sur-face of radius r the potential will be symmetricwith respect to the center Thus SchrodingerrsquosEquation assumes the following form

]Y21 sup1 Y ` V(r)Y 4 i (1)2m ]t

where V(r) is the atomic potential m is the mass ofthe electron r is the distance from the center of thepotential (eg the nucleus of an atom) to the elec-tron and is the ratio between Plankrsquos constant hand 2p ( 4 h2p)

The family of functions corresponding to thesolutions of the equation is represented by thesymbol W also known as the wave function Exam-ination of the general form of the wave function re-veals that W consists of the product of four smallerfunctions (equation A2) that may be treated sepa-rately because each is dependent on a different pa-rameter Three of these were found particularlyuseful R which is exclusively dependent on r thedistance of a point to the center of the potential Hwhich is exclusively dependent of the pointrsquos eleva-tion angle h and U which is exclusively dependenton its horizontal angle f

Furthermore the solutions depend on a set ofquantum numbers that restrict the physical attri-butes of the electron to multiples of certain energylevels The rst of these values is n the principalquantum number (n 4 1 2 3 ) which deter-mines the overall energy of the electron (equationA7) The second value is l the angular momentumquantum number (l 4 0 1 2 nndash1) which gen-erates additional allowed energies (quantum states)within a single value of n (The term shell is usedto denote a particular value of l) So far four typesof shells have been encountered in nature and arenamed S P D and F corresponding to l 4 0 1 2and 3 respectively Therefore a particular energylevel is characterized by a combination of n and lwhich by convention is named using the value ofthe principal quantum number and the letter of itsshell (eg 1S 2S and 2P)

The third quantum number m is the magneticquantum number (ndashl m l) 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 probability of ndingan electron It is possible to show that this proba-bility is associated with R and H (equations A9 andA10) As an example of this correlation Figure 1shows plots of these distributions and the resultingorbital for particular values of n l and m fromwhich we may infer the most probable distancesand orientations For instance Figure 1b indicatesthat the electron is more likely to be found in thevicinity of 35278 and 144738

The quantum numbers were successfully used toexplain the existence of common characteristicsbetween known chemical elements supportingtheir classi cation into the Periodic Table of Ele-ments originally devised by the chemist DmitriMendeleev in the 19th century but only more re-cently explained by Schrodingerrsquos EquationRoughly 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 group and occupying the same column inthe periodic table For instance the group of halo-gens consists of elements that have ve electronsin their P shell as follows uorine has ve elec-trons in shell 2P chlorine in 3P bromine in 4Piodine in 5P and astatine in 6P

Generation of Granular Clouds

My generation of musical material using Schro-dingerrsquos Equation focuses on granular clouds(sounds composed of a large number of short ele-mentary sonic particles or grains each lasting afew hundredths of a second) derived from statisticaldistributions obtained from 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 importance

50 Computer Music Journal

009 031 076 150

(a)

3527o 90o 14473o0o 180o

(b)

3527o(c)

is given to the creation of formal relationships be-tween the inner structure of the source soundmdashpar-ticularly its amplitude envelopemdashand the structureof the cloud and its constituent grains Sourcesounds are stored as wave audio les

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 Parameters

These parameters refer to global attributes of thecloud such as density overall amplitude and spa-tial and spectral trajectory

Cloud Density

To 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 are proportional to the density uctuation Inpractice the envelope is extracted from the sourceand stored in a timendashamplitude breakpoint tableThe cloud density is also stored as pairs of 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 corresponding values for clouddensity (equation A11)

Currently there are three different methods ofestablishing a correspondence between source am-plitude envelope and cloud density The rst twocalculate the density by using the envelope to drivethe radial and angular distributions (equation A12)either by making both r and h directly proportionalto the envelope (equation A13) or by making r di-rectly proportional to the envelope and h directlyproportional to its rst derivative (equation A14)The third method establishes a direct proportional-ity between the envelope and the density using astraightforward proportion rule (equation A15)

Figure 1 Examples ofSchrodinger distributionsr2|R|2 for Z 4 139 n4 4and l4 1 with radial val-ues normalized to (a) 15

(b) |H(h)|2 for n 4 4 l 4 1m 4 0 and (c) a diagram ofthe orbital resulting fromthese distributions

51Fischman

x=-1 x=1

RL y=0

ygt0

xgt1xlt-1

Listener

x lt -1 Beyond the left speakerx = -1 Left speaker-1 lt x lt 1 Between the speakersx = 1 Right speakerx gt 1 Beyond the right speaker

y = 0 On the imaginary line that joins the speakersy gt 0 Behind the imaginary line that joins the speakers

When y = 1 the depth is equal to the distancefrom the centre to one of the speakers

x=0

Cloud Amplitude Scaling

The amplitude of the cloud may be scaled by avalue that uctuates between a minimum andmaximum gain The uctuation follows the enve-lope of the source (equation A16)

Cloud Motion

Motion is carried out in three dimensions Two ofthese are spatial (width and depth) and the remain-ing one is articulated as a spectral dimension (pitchshift)

Width and Depth

These are calculated within a stereo image x indi-cates width and y indicates depth as illustrated inFigure 2 (See also equation A17)

Spectral Motion

The cloud lsquolsquomovesrsquorsquo in the frequency domain as aresult of time-varying transposition within limitsspeci ed by the composer establishing an analogyto movement across a height axis z Transpositionis carried out in the frequency domain via Short-

Time Fourier Transforms thereby preserving theduration of the grains

Trajectory Creation

To implement motion a three-dimensional trajec-tory is created that is analogous to physical move-ment within the boundaries of a shell (orbital)This consists of the stochastic generation of break-points according to Schrodingerrsquos radial and angulardistributions The breakpoints consist of position-time pairs within the duration of the cloud con-verted from r h and f into width depth andheight (equation A18)

The algorithm that generates the trajectory nor-malizes the values so that the cloud reaches themaximum width maximum depth and one of thetransposition boundaries (These are not reachedsimultaneously) Also to generate the trajectorybreakpoints the algorithm requires minimum andmaximum values for the time interval provided bythe composer The intervals between the break-points in the trajectory will fall randomly betweenthese two values For instance if the minimum andmaximum intervals are 02 and 05 sec respec-tively and the last breakpoint was generated attime tc 4 14 sec the time tc for the next break-point will fall between 16 and 19 sec

Finally trajectories may be repeated an arbitrarynumber of times so that the sum of the durationsof the repetitions is the overall duration of thecloud (If required this may be equal to the value ofthe orbital quantum number l plus one) Repeti-tions may be identical or uctuate within a per-centage given by the composer

Cloud Volume (Overall Grain Scatter)

The volume of the cloud depends on the maximumscatter of the grains in each dimension (widthdepth and pitch 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

Figure 2 Values of x andy for spatial width anddepth

52 Computer Music Journal

RL(a)

RL(b)

Width and depth scatter values vary from 0 (noscatter) to any positive numbers x and y with avalue of 1 indicating a scatter distance equal to thedistance from the center to one of the speakersSpectral scatter values vary from 0 (no scatter) toany positive number of semitones (or any fre-quency ratio)

To implement scatter a time-varying volumefunction is created in a similar manner to that ofthe motion trajectory described above Breakpointsare also generated stochastically according toSchrodingerrsquos radial and angular distributions(equation A19)

Therefore a grain created at time tc will movewith the cloud at a xed lsquolsquodistancersquorsquo from its centerThis distance is generated stochastically withinwidth depth and height boundary values 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 unanchoredIn the former case the sign of the current scattervalue indicates whether the grains will be posi-tioned to the right or left of (or transposed above orbelow) the cloud For instance when the scattervalue is positive all the grains are positioned tothe right of the cloud When the scatter becomesnegative the grains are positioned to the left Un-anchored scatter means that that the grains are po-sitioned both to the right and left of (andtransposed above and below) the cloud

The algorithm that generates the volume func-tion normalizes the values so that the cloudreaches the maximum width depth and transposi-tional scatter (Again these are not reached simul-

taneously) In a similar fashion to the motiontrajectory the algorithm also requires the composerto provide minimum and maximum values for thetime interval to generate time-volume breakpointsThe volume function may also be repeated an arbi-trary number of times this may also be equal tothe value of the orbital quantum number l plusone Repetitions may be identical or uctuatewithin a percentage given by the composer

Grain Parameters

These parameters control the attributes of individ-ual grains such as their envelope duration and in-dividual scatter patterns

Grain Amplitude Scaling

The amplitude of individual grains is scaled by again factor according to their onset times relativeto the beginning of the cloud and implemented ac-cording to various algorithm types The rst optionconsists of a scaling function that follows the enve-lope of the source (equation A20) Other optionsfollow Schrodingerrsquos distribution (r2 sin(h)|Y(rhf|2

in equations A21ndashA23) the complement of Schro-dingerrsquos distribution (equation A24) and Schro-dingerrsquos cumulative probability and itscomplement (equations A25 and A26) It is worthnoting that if the cumulative probability is 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 cloudproducing a nonlinear crescendo This is becausethe cumulative probability always increases mono-tonically Similarly following the complement ofthe cumulative probability will result in a nonlin-ear diminuendo

Finally scaling may be generated from the vari-ous distributions resulting from SchrodingerrsquosEquation (equations A27ndashA33)

Grain Duration

The actual duration of individual grains uctuatesaccording to their onset times relative to the begin-

Figure 3 Examples ofgranular cloud volume(a) narrow grain scatterand (b) wide grain scatter

53Fischman

ning of the cloud The function controlling the du-ration uctuation is calculated in a manner similarto that of the scaling factor offering the same op-tions to the user (equations A34 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 thecloudrsquos motion trajectory and its volume functionrepetitions may also be identical or uctuatewithin a percentage given by the composer Themain algorithms available consist of the followingschemes duration proportional to the source enve-lope duration proportional to Schrodingerrsquos distri-bution and its complement duration proportionalto Schrodingerrsquos cumulative probability and itscomplement and duration generated 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 grains be-come gradually shorter until their duration reachesits minimum value at the end of the cloud

Grain Envelope

The envelope applied to each individual grain con-sists of either the envelope extracted from thesource or a linear envelope shaped according to theseven parameters shown in Figure 4 and some ofSchrodingerrsquos distributions (equations A36ndashA40)

Grain Scatter Path

In addition to the xed scatter resulting from thevolume of the cloud each grain is provided with anindividual trajectory Grain scatter is given in thesame units discussed above in relation to cloudvolume

The algorithm is implemented by means of atime-varying scatter function created for each di-mension with normalized duration and maximumvalue of unity (equations A41ndashA43) To generatethe trajectory breakpoints the algorithm requiresminimum and maximum values for the time inter-val between a pair of breakpoints provided by thecomposer However these are given as percentagesof the grain duration as the function is normalizedWhen a grain is created the function is scaled to tits duration and maximum amplitude It is possibleto change the shape of the function according to ajitter value given by the composer which results ina different path 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 cloudand overall cloud motion (equation A44) The pitchshift relative to the listener is invariant under rota-tion Therefore it is possible to simplify the calcu-lation of the Doppler Effect 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 Figure 5(equations A45ndashA49)

Structure

We will now proceed to examine ways in whichthe generated material can be structured into a mu-sical work It is worth noting that these are particu-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 grain characteristics etc) Never-theless it is also important to note that choiceswere made with full consideration and regard fortheir musical viability and formal integrity

Figure 4 Linear envelopeparameters (1) start offset(2) overshoot attack(3) overshoot decay(4) steady state (5) decay(6) end offset and(7) overshoot percentage

54 Computer Music Journal

x

q

1

RL

Listener

y

a

R

L

Vg

Pyramidal Shape

A relatively straightforward strategy for structuringmusical works may be linked to a survey throughthe atomic shells resulting from changes in theoverall energy For instance the survey may beginin the lowest energy shell and leap to higher shellsas this energy increases (This case also correspondsto an excursion through the periodic table accord-ing to ascending atomic number) In fact it isworth adopting this particular instance as a starting

point to develop a basic model that can subse-quently be modi ed and extended

Following the analogy with the periodic tablemajor sectional boundaries may correspond to thereappearance of a shell with an increasing value ofn (which also corresponds to a change of row in theperiodic table) Because S is the rst shell majorsectional boundaries will start at 1S 2S 3S 7SSubsections within these boundaries may corre-spond to the shells surveyed in between Figure 6illustrates the resulting structure obtained from theenergy levels in the periodic table this may becompared to a pyramid that is surveyed from top tobottom Examination of this structure brings 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 (S and 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 corresponding to the reappearance ofeach particular shell in subsequent sections For in-stance shell D reappears in sections 4 to 7 Thissuggests the possibility of creating and additionaldevelopmental dimension that following the anal-ogy with the pyramid would consist of four lsquolsquoverti-calrsquorsquo paths identi ed by different grayscale shadesin Figure 6 For instance the reappearance of ashell could develop the material of previous appear-ances

Two remaining issues must be determined beforethe structure is nalized section duration and sec-tion differentiation

Figure 5 Calculation ofDoppler Shift using a rota-tion of axes

Figure 6 Pyramidal struc-ture resulting from the en-ergy levels obtained by fol-lowing the periodic tablein ascending order

55Fischman

7S 5F 6D 7P

6S 4F 5D 6P

5S 4D 5P

4S 4P

3S

2S 2P

1SSection 2

Section 1

Section 3

Section 4

Section 5

Section 6

Section 7

Section Duration

We can follow the path set by the rst observationabove (that sections become longer by virtue of thenumber of subsections they include) and establish afurther principle of growth related to the periodictable the minimum duration of each subsectioncloud is proportional to the sum of atomic numbersof its related shell by setting a correspondence be-tween a chosen time unit and atomic numbers asfollows

SectionDuration

4 atomic_numbers 2 time_unit (2)o1 2For example if the time unit is 005 sec the

minimum duration of the subsection correspondingto shell 2P (atomic numbers 5 to 10) will be

(5 ` 6 ` 7 ` 8 ` 9 ` 10) 2 005 4 225 sec

Applying this principle we obtain a further re- nement of the pyramidal shape whereby a newappearance of a shell is normally longer than itsprevious appearances Table 1 shows the minimumduration values for shell D with atomic unit dura-tion of 00342 sec and Figure 7 shows a schematicversion of the re ned 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 becauseaccording to quantum mechanics an electron re-mains in a given state as long as the energypumped into (or extracted from) it is equal to or

greater than the transition energy required to moveto an immediate neighboring state as calculated byequation A8

Section Differentiation

Identi cation of distinctive threads according toshells requires that sections be differentiated bytheir content This can be achieved by using differ-ent source sounds in each shell This will also in- uence overall cloud and individual graincharacteristics because as explained above manyof these are derived from the attributesmdashparticu-larly the envelopemdashof the source sound Anotherway sections may be differentiated is by setting dif-ferent grain attributes (eg different durations en-velopes spatial locations or by using differentdistributions to generate amplitudes and durations)

Extensions of the Pyramidal Shape

Two other structural shapes can be derived fromthe pyramid an inverted pyramid and a diamondThe former is analogous to a survey of the periodictable from the highest to the lowest atomic num-ber The latter consists of an original pyramid fol-

Figure 7 Schematic viewof pyramidal structurewhen the minimum dura-tion of each subsection isproportional to the sum ofatomic numbers of its re-lated shell (not to scale)

Table 1 Minimum duration values for shell Dwith atomic unit duration of 00342 sec

Shell Duration (sec)

3D 872104D 1487705D 2582106D 367650

56 Computer Music Journal

(a) (b)

lowed by of an inverted one These are shownschematically in Figure 8

Other Structural Possibilities

Apart from the simple trajectories related 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 beordered according to stochastic algorithms or gener-ative rules Alternatively they may follow attri-butes of a source sound such as the envelope wherethe peak may correspond to the maximum atomicnumber All these are options beyond the scope ofthe current research plan 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 (or dissipation) According to 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

1 1 E`SectionDuration 4 abs 1 2 (3)a U b 1 22 2n n Ka b E

where KE is the rate of energy accumulation (or dis-sipation) expressed as a fraction of the ionizationenergy E` per second As an illustration assumingthat KE 4 001 E` and that shell 3D progresses toshell 4P then the duration of the former is

3DSectionDuration3D U 4P

1 1 E`4 abs 1 2 4 486 sec1 22 23 4 001E`

The problem with this method is that energytransitions at lower shells are signi cantly higherthan those at higher shells For instance the transi-tion from 1S to 2S requires an energy value overone hundred times higher than a transition from6D to 7P Consequently the duration of 1S is inthe same proportion to that of 6D yielding ex-tremely long durations for 1S or extremely shortones for 6D This may be overcome by subdividinglonger sections according to proportions similar tothose dividing the whole work Alternatively therate of energy ow could correspond to 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 ow In general equationA10 may be modi ed to cater for any energy owfunction KE(t)

Section_durationa U b 1 1K (t)dt 4 abs 1 2 E (4)E ` 1 22 20 n na b

Erwinrsquos Playground

Erwinrsquos Playground is a work for tape lasting 9 min12 sec Except for the recording of 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 Erwinrsquos Playground is intendedto demonstrate the validity and viability of thesetheoretical principles in the generation of a musical

Figure 8 (a) Inverted pyra-midal structure and (b) di-amond structure

57Fischman

work as well as the usefulness of the software as aconduit for the realization of these principlesHowever its aesthetic value is intended to be inde-pendent of the generating principles because notheory is capable of 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 ErwinSchrodingerrsquos imaginary eld of action namely theinner shells of the atom Its structure is modeledon the diamond shape described above and linkedto a survey that begins in the lowest energy stateleaps up until it reaches the highest shell and thenreturns back to the initial state This may also becompared to an excursion through the periodic ta-ble that ascends and then descends according toatomic number order

Four different types of sound sources were as-signed to each shell to achieve section differentia-tion The S shells consisted primarily of vocalsounds (audio les MonoC1 Uba UlulowL) Pshells were assigned three different types of soundwhich nevertheless share some common spectralcharacteristics the lsquolsquobuzzrsquorsquo of a y a vocal lsquolsquobuzzrsquorsquoand the sound of a sparking welder (audio lesDieFly BZZZ Welder)

D shells were assigned sounds that ranged frompitched to lsquolsquosqueakyrsquorsquo The pitched sounds were theresult of rubbing the edge of a glass cup with differ-ent amounts of liquid to obtain different pitches(audio les Crystal E Crystal F Crystal G )Squeaks were recorded from the un-oiled wheels ofa trolley (audio le Squeak1) A sample containinga transition from pitch to squeak was achieved byrubbing the glass cup with different degrees of pres-sure (audio le Crystal B to Squeak) Finally a re-cording of the pitched sound of cicadas was alsoincluded toward the middle of the piece (audio leCicadas)

F shells were normally associated with local andglobal climaxes therefore they spread through awider bandwidth by means of three spectral typesThe rst type consists of two noise-based soundsobtained from a busy motorway and sea waves gur-gling through rocks (audio les Motorway1 andWaves1) The second sound is metallic consistingof the recording of a frying pan that was hit or

rubbed against other objects (audio le PBg1f) Thethird was mainly used to create low-frequency tex-tures and drones from the recording of a piece ofplywood that was bent at a more or less regularrate (audio le PBdLow) (Examples of the source les as well as passages characterizing each shellsand the complete work are available on the forth-coming Computer Music Journal volume 27 soundexample disc)

Grain duration was another signi cant attributeused to characterize the various shells Finally theatomic number duration equivalent used in thepiece was 00342 sec per atomic number

Table 2 presents a complete list of the structuralhierarchy of Erwinrsquos Playground including starttime minimum duration grain duration limitsand the source audio les for each subsection Asmentioned above the duration of subsections couldbe extended to allow some freedom when takinginto account musical context Section extensionsappear below the shell name with the letter lsquolsquoXrsquorsquoAlso note the bold line denoting the end of the pyr-amid and the beginning of its inversion which actsas an lsquolsquoaxis of re ectionrsquorsquo It is worth noting thatthe second half of the piece (sections 8ndash14 corre-sponding to the inverted pyramid) is more com-pressed than the rst part as there are fewersubsections that have their minimum durationsextended

Software Implementation

The software implementation of this procedureconsists of two separate components AL a compo-sitional environment for the creation and manipu-lation of musical events and ERWIN a plug-inthat implements the application of Schrodingerrsquosmodel described above These were developed andtested using Visual C ` ` 60 running under Win-dows 2000 and later under Windows XP Both ap-plications use Microsoft DirectX 8 features andrequire that DirectX 8 be installed in the host com-puter For this reason the AL and ERWIN installa-tion program also installs DirectX 8 if requiredCurrently AL and ERWIN are distributed as freesoftware under the GNU public license and are

58 Computer Music Journal

Table 2 Structural hierarchy list of Erwinrsquos Playground (lsquolsquoXrsquorsquo denotes section extensions)

Section Shell Start TimeDuration

(sec) Grain Duration (msec) Source Audio Files

1 1S 0000000 01026 5ndash25 MonoC1X 0001026 37258

2 2S 0037384 02394 5ndash50 MonoC12P 0039778 15390 5ndash50 DieFlyX 0055168 21572

3 3S 0076740 07866 5ndash503P 0084606 31806 75ndash75 75ndash20 DieFly

4 4S 0116412 13338 5ndash100 MonoC13D 0129750 87210 10ndash60 30ndash60 Crystal E4P 0216960 68742 5ndash60 DieFlyX 0285702 31777

5 5S 0317479 25650 5ndash25 MonoC110ndash160 Uba

4D 0343129 148770 10ndash150 100ndash150 Crystal E5P 0491899 105678 10ndash60 30ndash60 DieFly

30ndash60 WelderX 0597577 05818

6 6S 1003395 37962 5ndash25 MonoC110ndash160 Uba100ndash500 UlulowL

X 1041357 295954F 1070952 304038 100ndash5000 Waves1

1000ndash4000 2000ndash6000 PBdLow100ndash300 4000ndash8000 PBg1f

X 1374990 265995D 1401589 258210 1000ndash4500 Crystal E

10ndash50 10ndash4000 20ndash150 Crystal F10ndash50 Crystal G30ndash50 Squeak1

X 2059799 084946P 2068293 171342 30ndash60 DieFly

20ndash80 20ndash150 Welder5ndash1000 BZZZ

X 2239635 50250

7 7S 2289885 59850 5ndash25 MonoC110ndash200 Uba5ndash25 200ndash700 Ululow

X 2349735 434425F 2393177 457254 200ndash8000 Waves1

1000ndash4000 2000ndash5000 2000ndash6000 PBdLow100ndash300 4000ndash8000 PBg1f5ndash25 40ndash150 Motorway1

X 2250431 1051106D 3355541 367650 200ndash350 Crystal E

10ndash50 10ndash4000 20ndash150 Crystal F10ndash50 Crystal G50ndash150 50ndash250 Squeak1

X 4123191 304657P 4153656 237006 75ndash20 75ndash75 10ndash60 30ndash60 DieFly

30ndash60 WelderX 4390662 12090

8 7P 4402752 237006 75ndash20 20ndash40 DieFly6D 5039758 367650 10ndash50 200ndash350 Crystal E

10ndash50 Crystal F50ndash250 Squeak15ndash5000 Cicadas

(continued )

59Fischman

Table 2 (Continued)

Section Shell Start TimeDuration

(sec) Grain Duration (msec) Source Audio Files

5F 5407408 457254 100ndash5000 Waves12000ndash8000 PBdLow100ndash300 4000ndash8000 PBg1f5ndash25 40ndash150 150ndash200 Motorway1

7S 6264662 59850 5ndash25 MonoC110ndash200 Uba5ndash25 90ndash700 200ndash700 UlulowL

9 6P 6324512 171342 30ndash60 DieFly20ndash80 100ndash250 Welder5ndash1000 BZZZ

5D 6495854 258210 200ndash350 1000ndash4500 Crystal E20ndash150 Crystal F10ndash50 Crystal G50ndash250 Squeak1

4F 7154064 304038 100ndash5000 Waves11000ndash4000 PBdLow100ndash300 4000ndash8000 PBg1f100ndash10000 150ndash200 Motorway1

X 7458102 100006S 7468102 37962 5ndash25 MonoC1

10ndash160 10ndash200 Uba100ndash500 UlulowL

10 5P 7506064 105678 10ndash60 30ndash60 DieFly30ndash60 Welder5ndash1000 10ndash60 BZZZ

X 8011742 315524D 8043294 148770 200ndash350 100ndash150 Crystal E

20ndash60 50ndash250 Squeak150ndash150 Crystal B to Squeak

5S 8192064 25650 5ndash25 MonoC110ndash160 Uba

X 8217714 80546

11 4P 8298260 68742 5ndash60 10ndash30 DieFly5ndash60 BZZZ

3D 8367002 87210 80ndash450 200ndash350 Crystal E40ndash400 50ndash250 100ndash350 Squeak1

X 8454212 002684S 8454480 13338 5ndash25 5ndash100 MonoC1X 8467818 36696

12 3P 8504514 31806 75ndash20 75ndash75 DieFly3S 8536320 07866 5ndash25 5ndash50 MonoC1X 8544186 46046

13 2P 8590232 15390 5ndash50 DieFly2S 9005622 02394 15ndash50 MonoC1X 9008016 45972

14 1S 9053988 01026 5ndash25 MonoC110ndash200 70ndash200 Uba

X 9055014 61339

END 9116353

60 Computer Music Journal

available online at wwwkeeleacukdeptsmustaffAlAl_softwarehtm The programs are alsodistributed with the Composerrsquos Desktop Project

AL

AL is a multiple document interface (MDI) applica-tion with frame windows that present views of dif-ferent documents andor several views of the samedocument It is designed for the creation and ma-nipulation of sonic events and their organizationinto a structured musical work

To aid user familiarity with the software and pro-mote a reasonably short learning curve care wastaken to develop an interface within an accepted

general standard for Windows applications includ-ing a menu bar a tool bar common window com-ponents and typical mouse lsquolsquodrag-and-droprsquorsquooperations It also contains detailed online helppages including a tutorial Figure 9 shows an in-stance of the user interface

Another important consideration in the design ofAL was the desire to provide a reasonably open sys-tem with the option for development of third-partyplug-ins thus avoiding the constraints imposed bya single type of musical process Future develop-ments by the author and others will ideally imple-ment a variety of compositional approaches inaddition to the application of Schrodingerrsquos Equa-tion in the existing plug-in (see the description ofERWIN below)

Figure 9 The AL user in-terface including themenu bar tool bar and aview of a document

50 Computer Music Journal

009 031 076 150

(a)

3527o 90o 14473o0o 180o

(b)

3527o(c)

is given to the creation of formal relationships be-tween the inner structure of the source soundmdashpar-ticularly its amplitude envelopemdashand the structureof the cloud and its constituent grains Sourcesounds are stored as wave audio les

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 Parameters

These parameters refer to global attributes of thecloud such as density overall amplitude and spa-tial and spectral trajectory

Cloud Density

To 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 are proportional to the density uctuation Inpractice the envelope is extracted from the sourceand stored in a timendashamplitude breakpoint tableThe cloud density is also stored as pairs of 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 corresponding values for clouddensity (equation A11)

Currently there are three different methods ofestablishing a correspondence between source am-plitude envelope and cloud density The rst twocalculate the density by using the envelope to drivethe radial and angular distributions (equation A12)either by making both r and h directly proportionalto the envelope (equation A13) or by making r di-rectly proportional to the envelope and h directlyproportional to its rst derivative (equation A14)The third method establishes a direct proportional-ity between the envelope and the density using astraightforward proportion rule (equation A15)

Figure 1 Examples ofSchrodinger distributionsr2|R|2 for Z 4 139 n4 4and l4 1 with radial val-ues normalized to (a) 15

(b) |H(h)|2 for n 4 4 l 4 1m 4 0 and (c) a diagram ofthe orbital resulting fromthese distributions

51Fischman

x=-1 x=1

RL y=0

ygt0

xgt1xlt-1

Listener

x lt -1 Beyond the left speakerx = -1 Left speaker-1 lt x lt 1 Between the speakersx = 1 Right speakerx gt 1 Beyond the right speaker

y = 0 On the imaginary line that joins the speakersy gt 0 Behind the imaginary line that joins the speakers

When y = 1 the depth is equal to the distancefrom the centre to one of the speakers

x=0

Cloud Amplitude Scaling

The amplitude of the cloud may be scaled by avalue that uctuates between a minimum andmaximum gain The uctuation follows the enve-lope of the source (equation A16)

Cloud Motion

Motion is carried out in three dimensions Two ofthese are spatial (width and depth) and the remain-ing one is articulated as a spectral dimension (pitchshift)

Width and Depth

These are calculated within a stereo image x indi-cates width and y indicates depth as illustrated inFigure 2 (See also equation A17)

Spectral Motion

The cloud lsquolsquomovesrsquorsquo in the frequency domain as aresult of time-varying transposition within limitsspeci ed by the composer establishing an analogyto movement across a height axis z Transpositionis carried out in the frequency domain via Short-

Time Fourier Transforms thereby preserving theduration of the grains

Trajectory Creation

To implement motion a three-dimensional trajec-tory is created that is analogous to physical move-ment within the boundaries of a shell (orbital)This consists of the stochastic generation of break-points according to Schrodingerrsquos radial and angulardistributions The breakpoints consist of position-time pairs within the duration of the cloud con-verted from r h and f into width depth andheight (equation A18)

The algorithm that generates the trajectory nor-malizes the values so that the cloud reaches themaximum width maximum depth and one of thetransposition boundaries (These are not reachedsimultaneously) Also to generate the trajectorybreakpoints the algorithm requires minimum andmaximum values for the time interval provided bythe composer The intervals between the break-points in the trajectory will fall randomly betweenthese two values For instance if the minimum andmaximum intervals are 02 and 05 sec respec-tively and the last breakpoint was generated attime tc 4 14 sec the time tc for the next break-point will fall between 16 and 19 sec

Finally trajectories may be repeated an arbitrarynumber of times so that the sum of the durationsof the repetitions is the overall duration of thecloud (If required this may be equal to the value ofthe orbital quantum number l plus one) Repeti-tions may be identical or uctuate within a per-centage given by the composer

Cloud Volume (Overall Grain Scatter)

The volume of the cloud depends on the maximumscatter of the grains in each dimension (widthdepth and pitch 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

Figure 2 Values of x andy for spatial width anddepth

52 Computer Music Journal

RL(a)

RL(b)

Width and depth scatter values vary from 0 (noscatter) to any positive numbers x and y with avalue of 1 indicating a scatter distance equal to thedistance from the center to one of the speakersSpectral scatter values vary from 0 (no scatter) toany positive number of semitones (or any fre-quency ratio)

To implement scatter a time-varying volumefunction is created in a similar manner to that ofthe motion trajectory described above Breakpointsare also generated stochastically according toSchrodingerrsquos radial and angular distributions(equation A19)

Therefore a grain created at time tc will movewith the cloud at a xed lsquolsquodistancersquorsquo from its centerThis distance is generated stochastically withinwidth depth and height boundary values 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 unanchoredIn the former case the sign of the current scattervalue indicates whether the grains will be posi-tioned to the right or left of (or transposed above orbelow) the cloud For instance when the scattervalue is positive all the grains are positioned tothe right of the cloud When the scatter becomesnegative the grains are positioned to the left Un-anchored scatter means that that the grains are po-sitioned both to the right and left of (andtransposed above and below) the cloud

The algorithm that generates the volume func-tion normalizes the values so that the cloudreaches the maximum width depth and transposi-tional scatter (Again these are not reached simul-

taneously) In a similar fashion to the motiontrajectory the algorithm also requires the composerto provide minimum and maximum values for thetime interval to generate time-volume breakpointsThe volume function may also be repeated an arbi-trary number of times this may also be equal tothe value of the orbital quantum number l plusone Repetitions may be identical or uctuatewithin a percentage given by the composer

Grain Parameters

These parameters control the attributes of individ-ual grains such as their envelope duration and in-dividual scatter patterns

Grain Amplitude Scaling

The amplitude of individual grains is scaled by again factor according to their onset times relativeto the beginning of the cloud and implemented ac-cording to various algorithm types The rst optionconsists of a scaling function that follows the enve-lope of the source (equation A20) Other optionsfollow Schrodingerrsquos distribution (r2 sin(h)|Y(rhf|2

in equations A21ndashA23) the complement of Schro-dingerrsquos distribution (equation A24) and Schro-dingerrsquos cumulative probability and itscomplement (equations A25 and A26) It is worthnoting that if the cumulative probability is 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 cloudproducing a nonlinear crescendo This is becausethe cumulative probability always increases mono-tonically Similarly following the complement ofthe cumulative probability will result in a nonlin-ear diminuendo

Finally scaling may be generated from the vari-ous distributions resulting from SchrodingerrsquosEquation (equations A27ndashA33)

Grain Duration

The actual duration of individual grains uctuatesaccording to their onset times relative to the begin-

Figure 3 Examples ofgranular cloud volume(a) narrow grain scatterand (b) wide grain scatter

53Fischman

ning of the cloud The function controlling the du-ration uctuation is calculated in a manner similarto that of the scaling factor offering the same op-tions to the user (equations A34 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 thecloudrsquos motion trajectory and its volume functionrepetitions may also be identical or uctuatewithin a percentage given by the composer Themain algorithms available consist of the followingschemes duration proportional to the source enve-lope duration proportional to Schrodingerrsquos distri-bution and its complement duration proportionalto Schrodingerrsquos cumulative probability and itscomplement and duration generated 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 grains be-come gradually shorter until their duration reachesits minimum value at the end of the cloud

Grain Envelope

The envelope applied to each individual grain con-sists of either the envelope extracted from thesource or a linear envelope shaped according to theseven parameters shown in Figure 4 and some ofSchrodingerrsquos distributions (equations A36ndashA40)

Grain Scatter Path

In addition to the xed scatter resulting from thevolume of the cloud each grain is provided with anindividual trajectory Grain scatter is given in thesame units discussed above in relation to cloudvolume

The algorithm is implemented by means of atime-varying scatter function created for each di-mension with normalized duration and maximumvalue of unity (equations A41ndashA43) To generatethe trajectory breakpoints the algorithm requiresminimum and maximum values for the time inter-val between a pair of breakpoints provided by thecomposer However these are given as percentagesof the grain duration as the function is normalizedWhen a grain is created the function is scaled to tits duration and maximum amplitude It is possibleto change the shape of the function according to ajitter value given by the composer which results ina different path 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 cloudand overall cloud motion (equation A44) The pitchshift relative to the listener is invariant under rota-tion Therefore it is possible to simplify the calcu-lation of the Doppler Effect 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 Figure 5(equations A45ndashA49)

Structure

We will now proceed to examine ways in whichthe generated material can be structured into a mu-sical work It is worth noting that these are particu-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 grain characteristics etc) Never-theless it is also important to note that choiceswere made with full consideration and regard fortheir musical viability and formal integrity

Figure 4 Linear envelopeparameters (1) start offset(2) overshoot attack(3) overshoot decay(4) steady state (5) decay(6) end offset and(7) overshoot percentage

54 Computer Music Journal

x

q

1

RL

Listener

y

a

R

L

Vg

Pyramidal Shape

A relatively straightforward strategy for structuringmusical works may be linked to a survey throughthe atomic shells resulting from changes in theoverall energy For instance the survey may beginin the lowest energy shell and leap to higher shellsas this energy increases (This case also correspondsto an excursion through the periodic table accord-ing to ascending atomic number) In fact it isworth adopting this particular instance as a starting

point to develop a basic model that can subse-quently be modi ed and extended

Following the analogy with the periodic tablemajor sectional boundaries may correspond to thereappearance of a shell with an increasing value ofn (which also corresponds to a change of row in theperiodic table) Because S is the rst shell majorsectional boundaries will start at 1S 2S 3S 7SSubsections within these boundaries may corre-spond to the shells surveyed in between Figure 6illustrates the resulting structure obtained from theenergy levels in the periodic table this may becompared to a pyramid that is surveyed from top tobottom Examination of this structure brings 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 (S and 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 corresponding to the reappearance ofeach particular shell in subsequent sections For in-stance shell D reappears in sections 4 to 7 Thissuggests the possibility of creating and additionaldevelopmental dimension that following the anal-ogy with the pyramid would consist of four lsquolsquoverti-calrsquorsquo paths identi ed by different grayscale shadesin Figure 6 For instance the reappearance of ashell could develop the material of previous appear-ances

Two remaining issues must be determined beforethe structure is nalized section duration and sec-tion differentiation

Figure 5 Calculation ofDoppler Shift using a rota-tion of axes

Figure 6 Pyramidal struc-ture resulting from the en-ergy levels obtained by fol-lowing the periodic tablein ascending order

55Fischman

7S 5F 6D 7P

6S 4F 5D 6P

5S 4D 5P

4S 4P

3S

2S 2P

1SSection 2

Section 1

Section 3

Section 4

Section 5

Section 6

Section 7

Section Duration

We can follow the path set by the rst observationabove (that sections become longer by virtue of thenumber of subsections they include) and establish afurther principle of growth related to the periodictable the minimum duration of each subsectioncloud is proportional to the sum of atomic numbersof its related shell by setting a correspondence be-tween a chosen time unit and atomic numbers asfollows

SectionDuration

4 atomic_numbers 2 time_unit (2)o1 2For example if the time unit is 005 sec the

minimum duration of the subsection correspondingto shell 2P (atomic numbers 5 to 10) will be

(5 ` 6 ` 7 ` 8 ` 9 ` 10) 2 005 4 225 sec

Applying this principle we obtain a further re- nement of the pyramidal shape whereby a newappearance of a shell is normally longer than itsprevious appearances Table 1 shows the minimumduration values for shell D with atomic unit dura-tion of 00342 sec and Figure 7 shows a schematicversion of the re ned 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 becauseaccording to quantum mechanics an electron re-mains in a given state as long as the energypumped into (or extracted from) it is equal to or

greater than the transition energy required to moveto an immediate neighboring state as calculated byequation A8

Section Differentiation

Identi cation of distinctive threads according toshells requires that sections be differentiated bytheir content This can be achieved by using differ-ent source sounds in each shell This will also in- uence overall cloud and individual graincharacteristics because as explained above manyof these are derived from the attributesmdashparticu-larly the envelopemdashof the source sound Anotherway sections may be differentiated is by setting dif-ferent grain attributes (eg different durations en-velopes spatial locations or by using differentdistributions to generate amplitudes and durations)

Extensions of the Pyramidal Shape

Two other structural shapes can be derived fromthe pyramid an inverted pyramid and a diamondThe former is analogous to a survey of the periodictable from the highest to the lowest atomic num-ber The latter consists of an original pyramid fol-

Figure 7 Schematic viewof pyramidal structurewhen the minimum dura-tion of each subsection isproportional to the sum ofatomic numbers of its re-lated shell (not to scale)

Table 1 Minimum duration values for shell Dwith atomic unit duration of 00342 sec

Shell Duration (sec)

3D 872104D 1487705D 2582106D 367650

56 Computer Music Journal

(a) (b)

lowed by of an inverted one These are shownschematically in Figure 8

Other Structural Possibilities

Apart from the simple trajectories related 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 beordered according to stochastic algorithms or gener-ative rules Alternatively they may follow attri-butes of a source sound such as the envelope wherethe peak may correspond to the maximum atomicnumber All these are options beyond the scope ofthe current research plan 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 (or dissipation) According to 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

1 1 E`SectionDuration 4 abs 1 2 (3)a U b 1 22 2n n Ka b E

where KE is the rate of energy accumulation (or dis-sipation) expressed as a fraction of the ionizationenergy E` per second As an illustration assumingthat KE 4 001 E` and that shell 3D progresses toshell 4P then the duration of the former is

3DSectionDuration3D U 4P

1 1 E`4 abs 1 2 4 486 sec1 22 23 4 001E`

The problem with this method is that energytransitions at lower shells are signi cantly higherthan those at higher shells For instance the transi-tion from 1S to 2S requires an energy value overone hundred times higher than a transition from6D to 7P Consequently the duration of 1S is inthe same proportion to that of 6D yielding ex-tremely long durations for 1S or extremely shortones for 6D This may be overcome by subdividinglonger sections according to proportions similar tothose dividing the whole work Alternatively therate of energy ow could correspond to 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 ow In general equationA10 may be modi ed to cater for any energy owfunction KE(t)

Section_durationa U b 1 1K (t)dt 4 abs 1 2 E (4)E ` 1 22 20 n na b

Erwinrsquos Playground

Erwinrsquos Playground is a work for tape lasting 9 min12 sec Except for the recording of 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 Erwinrsquos Playground is intendedto demonstrate the validity and viability of thesetheoretical principles in the generation of a musical

Figure 8 (a) Inverted pyra-midal structure and (b) di-amond structure

57Fischman

work as well as the usefulness of the software as aconduit for the realization of these principlesHowever its aesthetic value is intended to be inde-pendent of the generating principles because notheory is capable of 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 ErwinSchrodingerrsquos imaginary eld of action namely theinner shells of the atom Its structure is modeledon the diamond shape described above and linkedto a survey that begins in the lowest energy stateleaps up until it reaches the highest shell and thenreturns back to the initial state This may also becompared to an excursion through the periodic ta-ble that ascends and then descends according toatomic number order

Four different types of sound sources were as-signed to each shell to achieve section differentia-tion The S shells consisted primarily of vocalsounds (audio les MonoC1 Uba UlulowL) Pshells were assigned three different types of soundwhich nevertheless share some common spectralcharacteristics the lsquolsquobuzzrsquorsquo of a y a vocal lsquolsquobuzzrsquorsquoand the sound of a sparking welder (audio lesDieFly BZZZ Welder)

D shells were assigned sounds that ranged frompitched to lsquolsquosqueakyrsquorsquo The pitched sounds were theresult of rubbing the edge of a glass cup with differ-ent amounts of liquid to obtain different pitches(audio les Crystal E Crystal F Crystal G )Squeaks were recorded from the un-oiled wheels ofa trolley (audio le Squeak1) A sample containinga transition from pitch to squeak was achieved byrubbing the glass cup with different degrees of pres-sure (audio le Crystal B to Squeak) Finally a re-cording of the pitched sound of cicadas was alsoincluded toward the middle of the piece (audio leCicadas)

F shells were normally associated with local andglobal climaxes therefore they spread through awider bandwidth by means of three spectral typesThe rst type consists of two noise-based soundsobtained from a busy motorway and sea waves gur-gling through rocks (audio les Motorway1 andWaves1) The second sound is metallic consistingof the recording of a frying pan that was hit or

rubbed against other objects (audio le PBg1f) Thethird was mainly used to create low-frequency tex-tures and drones from the recording of a piece ofplywood that was bent at a more or less regularrate (audio le PBdLow) (Examples of the source les as well as passages characterizing each shellsand the complete work are available on the forth-coming Computer Music Journal volume 27 soundexample disc)

Grain duration was another signi cant attributeused to characterize the various shells Finally theatomic number duration equivalent used in thepiece was 00342 sec per atomic number

Table 2 presents a complete list of the structuralhierarchy of Erwinrsquos Playground including starttime minimum duration grain duration limitsand the source audio les for each subsection Asmentioned above the duration of subsections couldbe extended to allow some freedom when takinginto account musical context Section extensionsappear below the shell name with the letter lsquolsquoXrsquorsquoAlso note the bold line denoting the end of the pyr-amid and the beginning of its inversion which actsas an lsquolsquoaxis of re ectionrsquorsquo It is worth noting thatthe second half of the piece (sections 8ndash14 corre-sponding to the inverted pyramid) is more com-pressed than the rst part as there are fewersubsections that have their minimum durationsextended

Software Implementation

The software implementation of this procedureconsists of two separate components AL a compo-sitional environment for the creation and manipu-lation of musical events and ERWIN a plug-inthat implements the application of Schrodingerrsquosmodel described above These were developed andtested using Visual C ` ` 60 running under Win-dows 2000 and later under Windows XP Both ap-plications use Microsoft DirectX 8 features andrequire that DirectX 8 be installed in the host com-puter For this reason the AL and ERWIN installa-tion program also installs DirectX 8 if requiredCurrently AL and ERWIN are distributed as freesoftware under the GNU public license and are

58 Computer Music Journal

Table 2 Structural hierarchy list of Erwinrsquos Playground (lsquolsquoXrsquorsquo denotes section extensions)

Section Shell Start TimeDuration

(sec) Grain Duration (msec) Source Audio Files

1 1S 0000000 01026 5ndash25 MonoC1X 0001026 37258

2 2S 0037384 02394 5ndash50 MonoC12P 0039778 15390 5ndash50 DieFlyX 0055168 21572

3 3S 0076740 07866 5ndash503P 0084606 31806 75ndash75 75ndash20 DieFly

4 4S 0116412 13338 5ndash100 MonoC13D 0129750 87210 10ndash60 30ndash60 Crystal E4P 0216960 68742 5ndash60 DieFlyX 0285702 31777

5 5S 0317479 25650 5ndash25 MonoC110ndash160 Uba

4D 0343129 148770 10ndash150 100ndash150 Crystal E5P 0491899 105678 10ndash60 30ndash60 DieFly

30ndash60 WelderX 0597577 05818

6 6S 1003395 37962 5ndash25 MonoC110ndash160 Uba100ndash500 UlulowL

X 1041357 295954F 1070952 304038 100ndash5000 Waves1

1000ndash4000 2000ndash6000 PBdLow100ndash300 4000ndash8000 PBg1f

X 1374990 265995D 1401589 258210 1000ndash4500 Crystal E

10ndash50 10ndash4000 20ndash150 Crystal F10ndash50 Crystal G30ndash50 Squeak1

X 2059799 084946P 2068293 171342 30ndash60 DieFly

20ndash80 20ndash150 Welder5ndash1000 BZZZ

X 2239635 50250

7 7S 2289885 59850 5ndash25 MonoC110ndash200 Uba5ndash25 200ndash700 Ululow

X 2349735 434425F 2393177 457254 200ndash8000 Waves1

1000ndash4000 2000ndash5000 2000ndash6000 PBdLow100ndash300 4000ndash8000 PBg1f5ndash25 40ndash150 Motorway1

X 2250431 1051106D 3355541 367650 200ndash350 Crystal E

10ndash50 10ndash4000 20ndash150 Crystal F10ndash50 Crystal G50ndash150 50ndash250 Squeak1

X 4123191 304657P 4153656 237006 75ndash20 75ndash75 10ndash60 30ndash60 DieFly

30ndash60 WelderX 4390662 12090

8 7P 4402752 237006 75ndash20 20ndash40 DieFly6D 5039758 367650 10ndash50 200ndash350 Crystal E

10ndash50 Crystal F50ndash250 Squeak15ndash5000 Cicadas

(continued )

59Fischman

Table 2 (Continued)

Section Shell Start TimeDuration

(sec) Grain Duration (msec) Source Audio Files

5F 5407408 457254 100ndash5000 Waves12000ndash8000 PBdLow100ndash300 4000ndash8000 PBg1f5ndash25 40ndash150 150ndash200 Motorway1

7S 6264662 59850 5ndash25 MonoC110ndash200 Uba5ndash25 90ndash700 200ndash700 UlulowL

9 6P 6324512 171342 30ndash60 DieFly20ndash80 100ndash250 Welder5ndash1000 BZZZ

5D 6495854 258210 200ndash350 1000ndash4500 Crystal E20ndash150 Crystal F10ndash50 Crystal G50ndash250 Squeak1

4F 7154064 304038 100ndash5000 Waves11000ndash4000 PBdLow100ndash300 4000ndash8000 PBg1f100ndash10000 150ndash200 Motorway1

X 7458102 100006S 7468102 37962 5ndash25 MonoC1

10ndash160 10ndash200 Uba100ndash500 UlulowL

10 5P 7506064 105678 10ndash60 30ndash60 DieFly30ndash60 Welder5ndash1000 10ndash60 BZZZ

X 8011742 315524D 8043294 148770 200ndash350 100ndash150 Crystal E

20ndash60 50ndash250 Squeak150ndash150 Crystal B to Squeak

5S 8192064 25650 5ndash25 MonoC110ndash160 Uba

X 8217714 80546

11 4P 8298260 68742 5ndash60 10ndash30 DieFly5ndash60 BZZZ

3D 8367002 87210 80ndash450 200ndash350 Crystal E40ndash400 50ndash250 100ndash350 Squeak1

X 8454212 002684S 8454480 13338 5ndash25 5ndash100 MonoC1X 8467818 36696

12 3P 8504514 31806 75ndash20 75ndash75 DieFly3S 8536320 07866 5ndash25 5ndash50 MonoC1X 8544186 46046

13 2P 8590232 15390 5ndash50 DieFly2S 9005622 02394 15ndash50 MonoC1X 9008016 45972

14 1S 9053988 01026 5ndash25 MonoC110ndash200 70ndash200 Uba

X 9055014 61339

END 9116353

60 Computer Music Journal

available online at wwwkeeleacukdeptsmustaffAlAl_softwarehtm The programs are alsodistributed with the Composerrsquos Desktop Project

AL

AL is a multiple document interface (MDI) applica-tion with frame windows that present views of dif-ferent documents andor several views of the samedocument It is designed for the creation and ma-nipulation of sonic events and their organizationinto a structured musical work

To aid user familiarity with the software and pro-mote a reasonably short learning curve care wastaken to develop an interface within an accepted

general standard for Windows applications includ-ing a menu bar a tool bar common window com-ponents and typical mouse lsquolsquodrag-and-droprsquorsquooperations It also contains detailed online helppages including a tutorial Figure 9 shows an in-stance of the user interface

Another important consideration in the design ofAL was the desire to provide a reasonably open sys-tem with the option for development of third-partyplug-ins thus avoiding the constraints imposed bya single type of musical process Future develop-ments by the author and others will ideally imple-ment a variety of compositional approaches inaddition to the application of Schrodingerrsquos Equa-tion in the existing plug-in (see the description ofERWIN below)

Figure 9 The AL user in-terface including themenu bar tool bar and aview of a document

51Fischman

x=-1 x=1

RL y=0

ygt0

xgt1xlt-1

Listener

x lt -1 Beyond the left speakerx = -1 Left speaker-1 lt x lt 1 Between the speakersx = 1 Right speakerx gt 1 Beyond the right speaker

y = 0 On the imaginary line that joins the speakersy gt 0 Behind the imaginary line that joins the speakers

When y = 1 the depth is equal to the distancefrom the centre to one of the speakers

x=0

Cloud Amplitude Scaling

The amplitude of the cloud may be scaled by avalue that uctuates between a minimum andmaximum gain The uctuation follows the enve-lope of the source (equation A16)

Cloud Motion

Motion is carried out in three dimensions Two ofthese are spatial (width and depth) and the remain-ing one is articulated as a spectral dimension (pitchshift)

Width and Depth

These are calculated within a stereo image x indi-cates width and y indicates depth as illustrated inFigure 2 (See also equation A17)

Spectral Motion

The cloud lsquolsquomovesrsquorsquo in the frequency domain as aresult of time-varying transposition within limitsspeci ed by the composer establishing an analogyto movement across a height axis z Transpositionis carried out in the frequency domain via Short-

Time Fourier Transforms thereby preserving theduration of the grains

Trajectory Creation

To implement motion a three-dimensional trajec-tory is created that is analogous to physical move-ment within the boundaries of a shell (orbital)This consists of the stochastic generation of break-points according to Schrodingerrsquos radial and angulardistributions The breakpoints consist of position-time pairs within the duration of the cloud con-verted from r h and f into width depth andheight (equation A18)

The algorithm that generates the trajectory nor-malizes the values so that the cloud reaches themaximum width maximum depth and one of thetransposition boundaries (These are not reachedsimultaneously) Also to generate the trajectorybreakpoints the algorithm requires minimum andmaximum values for the time interval provided bythe composer The intervals between the break-points in the trajectory will fall randomly betweenthese two values For instance if the minimum andmaximum intervals are 02 and 05 sec respec-tively and the last breakpoint was generated attime tc 4 14 sec the time tc for the next break-point will fall between 16 and 19 sec

Finally trajectories may be repeated an arbitrarynumber of times so that the sum of the durationsof the repetitions is the overall duration of thecloud (If required this may be equal to the value ofthe orbital quantum number l plus one) Repeti-tions may be identical or uctuate within a per-centage given by the composer

Cloud Volume (Overall Grain Scatter)

The volume of the cloud depends on the maximumscatter of the grains in each dimension (widthdepth and pitch 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

Figure 2 Values of x andy for spatial width anddepth

52 Computer Music Journal

RL(a)

RL(b)

Width and depth scatter values vary from 0 (noscatter) to any positive numbers x and y with avalue of 1 indicating a scatter distance equal to thedistance from the center to one of the speakersSpectral scatter values vary from 0 (no scatter) toany positive number of semitones (or any fre-quency ratio)

To implement scatter a time-varying volumefunction is created in a similar manner to that ofthe motion trajectory described above Breakpointsare also generated stochastically according toSchrodingerrsquos radial and angular distributions(equation A19)

Therefore a grain created at time tc will movewith the cloud at a xed lsquolsquodistancersquorsquo from its centerThis distance is generated stochastically withinwidth depth and height boundary values 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 unanchoredIn the former case the sign of the current scattervalue indicates whether the grains will be posi-tioned to the right or left of (or transposed above orbelow) the cloud For instance when the scattervalue is positive all the grains are positioned tothe right of the cloud When the scatter becomesnegative the grains are positioned to the left Un-anchored scatter means that that the grains are po-sitioned both to the right and left of (andtransposed above and below) the cloud

The algorithm that generates the volume func-tion normalizes the values so that the cloudreaches the maximum width depth and transposi-tional scatter (Again these are not reached simul-

taneously) In a similar fashion to the motiontrajectory the algorithm also requires the composerto provide minimum and maximum values for thetime interval to generate time-volume breakpointsThe volume function may also be repeated an arbi-trary number of times this may also be equal tothe value of the orbital quantum number l plusone Repetitions may be identical or uctuatewithin a percentage given by the composer

Grain Parameters

These parameters control the attributes of individ-ual grains such as their envelope duration and in-dividual scatter patterns

Grain Amplitude Scaling

The amplitude of individual grains is scaled by again factor according to their onset times relativeto the beginning of the cloud and implemented ac-cording to various algorithm types The rst optionconsists of a scaling function that follows the enve-lope of the source (equation A20) Other optionsfollow Schrodingerrsquos distribution (r2 sin(h)|Y(rhf|2

in equations A21ndashA23) the complement of Schro-dingerrsquos distribution (equation A24) and Schro-dingerrsquos cumulative probability and itscomplement (equations A25 and A26) It is worthnoting that if the cumulative probability is 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 cloudproducing a nonlinear crescendo This is becausethe cumulative probability always increases mono-tonically Similarly following the complement ofthe cumulative probability will result in a nonlin-ear diminuendo

Finally scaling may be generated from the vari-ous distributions resulting from SchrodingerrsquosEquation (equations A27ndashA33)

Grain Duration

The actual duration of individual grains uctuatesaccording to their onset times relative to the begin-

Figure 3 Examples ofgranular cloud volume(a) narrow grain scatterand (b) wide grain scatter

53Fischman

ning of the cloud The function controlling the du-ration uctuation is calculated in a manner similarto that of the scaling factor offering the same op-tions to the user (equations A34 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 thecloudrsquos motion trajectory and its volume functionrepetitions may also be identical or uctuatewithin a percentage given by the composer Themain algorithms available consist of the followingschemes duration proportional to the source enve-lope duration proportional to Schrodingerrsquos distri-bution and its complement duration proportionalto Schrodingerrsquos cumulative probability and itscomplement and duration generated 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 grains be-come gradually shorter until their duration reachesits minimum value at the end of the cloud

Grain Envelope

The envelope applied to each individual grain con-sists of either the envelope extracted from thesource or a linear envelope shaped according to theseven parameters shown in Figure 4 and some ofSchrodingerrsquos distributions (equations A36ndashA40)

Grain Scatter Path

In addition to the xed scatter resulting from thevolume of the cloud each grain is provided with anindividual trajectory Grain scatter is given in thesame units discussed above in relation to cloudvolume

The algorithm is implemented by means of atime-varying scatter function created for each di-mension with normalized duration and maximumvalue of unity (equations A41ndashA43) To generatethe trajectory breakpoints the algorithm requiresminimum and maximum values for the time inter-val between a pair of breakpoints provided by thecomposer However these are given as percentagesof the grain duration as the function is normalizedWhen a grain is created the function is scaled to tits duration and maximum amplitude It is possibleto change the shape of the function according to ajitter value given by the composer which results ina different path 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 cloudand overall cloud motion (equation A44) The pitchshift relative to the listener is invariant under rota-tion Therefore it is possible to simplify the calcu-lation of the Doppler Effect 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 Figure 5(equations A45ndashA49)

Structure

We will now proceed to examine ways in whichthe generated material can be structured into a mu-sical work It is worth noting that these are particu-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 grain characteristics etc) Never-theless it is also important to note that choiceswere made with full consideration and regard fortheir musical viability and formal integrity

Figure 4 Linear envelopeparameters (1) start offset(2) overshoot attack(3) overshoot decay(4) steady state (5) decay(6) end offset and(7) overshoot percentage

54 Computer Music Journal

x

q

1

RL

Listener

y

a

R

L

Vg

Pyramidal Shape

A relatively straightforward strategy for structuringmusical works may be linked to a survey throughthe atomic shells resulting from changes in theoverall energy For instance the survey may beginin the lowest energy shell and leap to higher shellsas this energy increases (This case also correspondsto an excursion through the periodic table accord-ing to ascending atomic number) In fact it isworth adopting this particular instance as a starting

point to develop a basic model that can subse-quently be modi ed and extended

Following the analogy with the periodic tablemajor sectional boundaries may correspond to thereappearance of a shell with an increasing value ofn (which also corresponds to a change of row in theperiodic table) Because S is the rst shell majorsectional boundaries will start at 1S 2S 3S 7SSubsections within these boundaries may corre-spond to the shells surveyed in between Figure 6illustrates the resulting structure obtained from theenergy levels in the periodic table this may becompared to a pyramid that is surveyed from top tobottom Examination of this structure brings 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 (S and 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 corresponding to the reappearance ofeach particular shell in subsequent sections For in-stance shell D reappears in sections 4 to 7 Thissuggests the possibility of creating and additionaldevelopmental dimension that following the anal-ogy with the pyramid would consist of four lsquolsquoverti-calrsquorsquo paths identi ed by different grayscale shadesin Figure 6 For instance the reappearance of ashell could develop the material of previous appear-ances

Two remaining issues must be determined beforethe structure is nalized section duration and sec-tion differentiation

Figure 5 Calculation ofDoppler Shift using a rota-tion of axes

Figure 6 Pyramidal struc-ture resulting from the en-ergy levels obtained by fol-lowing the periodic tablein ascending order

55Fischman

7S 5F 6D 7P

6S 4F 5D 6P

5S 4D 5P

4S 4P

3S

2S 2P

1SSection 2

Section 1

Section 3

Section 4

Section 5

Section 6

Section 7

Section Duration

We can follow the path set by the rst observationabove (that sections become longer by virtue of thenumber of subsections they include) and establish afurther principle of growth related to the periodictable the minimum duration of each subsectioncloud is proportional to the sum of atomic numbersof its related shell by setting a correspondence be-tween a chosen time unit and atomic numbers asfollows

SectionDuration

4 atomic_numbers 2 time_unit (2)o1 2For example if the time unit is 005 sec the

minimum duration of the subsection correspondingto shell 2P (atomic numbers 5 to 10) will be

(5 ` 6 ` 7 ` 8 ` 9 ` 10) 2 005 4 225 sec

Applying this principle we obtain a further re- nement of the pyramidal shape whereby a newappearance of a shell is normally longer than itsprevious appearances Table 1 shows the minimumduration values for shell D with atomic unit dura-tion of 00342 sec and Figure 7 shows a schematicversion of the re ned 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 becauseaccording to quantum mechanics an electron re-mains in a given state as long as the energypumped into (or extracted from) it is equal to or

greater than the transition energy required to moveto an immediate neighboring state as calculated byequation A8

Section Differentiation

Identi cation of distinctive threads according toshells requires that sections be differentiated bytheir content This can be achieved by using differ-ent source sounds in each shell This will also in- uence overall cloud and individual graincharacteristics because as explained above manyof these are derived from the attributesmdashparticu-larly the envelopemdashof the source sound Anotherway sections may be differentiated is by setting dif-ferent grain attributes (eg different durations en-velopes spatial locations or by using differentdistributions to generate amplitudes and durations)

Extensions of the Pyramidal Shape

Two other structural shapes can be derived fromthe pyramid an inverted pyramid and a diamondThe former is analogous to a survey of the periodictable from the highest to the lowest atomic num-ber The latter consists of an original pyramid fol-

Figure 7 Schematic viewof pyramidal structurewhen the minimum dura-tion of each subsection isproportional to the sum ofatomic numbers of its re-lated shell (not to scale)

Table 1 Minimum duration values for shell Dwith atomic unit duration of 00342 sec

Shell Duration (sec)

3D 872104D 1487705D 2582106D 367650

56 Computer Music Journal

(a) (b)

lowed by of an inverted one These are shownschematically in Figure 8

Other Structural Possibilities

Apart from the simple trajectories related 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 beordered according to stochastic algorithms or gener-ative rules Alternatively they may follow attri-butes of a source sound such as the envelope wherethe peak may correspond to the maximum atomicnumber All these are options beyond the scope ofthe current research plan 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 (or dissipation) According to 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

1 1 E`SectionDuration 4 abs 1 2 (3)a U b 1 22 2n n Ka b E

where KE is the rate of energy accumulation (or dis-sipation) expressed as a fraction of the ionizationenergy E` per second As an illustration assumingthat KE 4 001 E` and that shell 3D progresses toshell 4P then the duration of the former is

3DSectionDuration3D U 4P

1 1 E`4 abs 1 2 4 486 sec1 22 23 4 001E`

The problem with this method is that energytransitions at lower shells are signi cantly higherthan those at higher shells For instance the transi-tion from 1S to 2S requires an energy value overone hundred times higher than a transition from6D to 7P Consequently the duration of 1S is inthe same proportion to that of 6D yielding ex-tremely long durations for 1S or extremely shortones for 6D This may be overcome by subdividinglonger sections according to proportions similar tothose dividing the whole work Alternatively therate of energy ow could correspond to 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 ow In general equationA10 may be modi ed to cater for any energy owfunction KE(t)

Section_durationa U b 1 1K (t)dt 4 abs 1 2 E (4)E ` 1 22 20 n na b

Erwinrsquos Playground

Erwinrsquos Playground is a work for tape lasting 9 min12 sec Except for the recording of 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 Erwinrsquos Playground is intendedto demonstrate the validity and viability of thesetheoretical principles in the generation of a musical

Figure 8 (a) Inverted pyra-midal structure and (b) di-amond structure

57Fischman

work as well as the usefulness of the software as aconduit for the realization of these principlesHowever its aesthetic value is intended to be inde-pendent of the generating principles because notheory is capable of 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 ErwinSchrodingerrsquos imaginary eld of action namely theinner shells of the atom Its structure is modeledon the diamond shape described above and linkedto a survey that begins in the lowest energy stateleaps up until it reaches the highest shell and thenreturns back to the initial state This may also becompared to an excursion through the periodic ta-ble that ascends and then descends according toatomic number order

Four different types of sound sources were as-signed to each shell to achieve section differentia-tion The S shells consisted primarily of vocalsounds (audio les MonoC1 Uba UlulowL) Pshells were assigned three different types of soundwhich nevertheless share some common spectralcharacteristics the lsquolsquobuzzrsquorsquo of a y a vocal lsquolsquobuzzrsquorsquoand the sound of a sparking welder (audio lesDieFly BZZZ Welder)

D shells were assigned sounds that ranged frompitched to lsquolsquosqueakyrsquorsquo The pitched sounds were theresult of rubbing the edge of a glass cup with differ-ent amounts of liquid to obtain different pitches(audio les Crystal E Crystal F Crystal G )Squeaks were recorded from the un-oiled wheels ofa trolley (audio le Squeak1) A sample containinga transition from pitch to squeak was achieved byrubbing the glass cup with different degrees of pres-sure (audio le Crystal B to Squeak) Finally a re-cording of the pitched sound of cicadas was alsoincluded toward the middle of the piece (audio leCicadas)

F shells were normally associated with local andglobal climaxes therefore they spread through awider bandwidth by means of three spectral typesThe rst type consists of two noise-based soundsobtained from a busy motorway and sea waves gur-gling through rocks (audio les Motorway1 andWaves1) The second sound is metallic consistingof the recording of a frying pan that was hit or

rubbed against other objects (audio le PBg1f) Thethird was mainly used to create low-frequency tex-tures and drones from the recording of a piece ofplywood that was bent at a more or less regularrate (audio le PBdLow) (Examples of the source les as well as passages characterizing each shellsand the complete work are available on the forth-coming Computer Music Journal volume 27 soundexample disc)

Grain duration was another signi cant attributeused to characterize the various shells Finally theatomic number duration equivalent used in thepiece was 00342 sec per atomic number

Table 2 presents a complete list of the structuralhierarchy of Erwinrsquos Playground including starttime minimum duration grain duration limitsand the source audio les for each subsection Asmentioned above the duration of subsections couldbe extended to allow some freedom when takinginto account musical context Section extensionsappear below the shell name with the letter lsquolsquoXrsquorsquoAlso note the bold line denoting the end of the pyr-amid and the beginning of its inversion which actsas an lsquolsquoaxis of re ectionrsquorsquo It is worth noting thatthe second half of the piece (sections 8ndash14 corre-sponding to the inverted pyramid) is more com-pressed than the rst part as there are fewersubsections that have their minimum durationsextended

Software Implementation

The software implementation of this procedureconsists of two separate components AL a compo-sitional environment for the creation and manipu-lation of musical events and ERWIN a plug-inthat implements the application of Schrodingerrsquosmodel described above These were developed andtested using Visual C ` ` 60 running under Win-dows 2000 and later under Windows XP Both ap-plications use Microsoft DirectX 8 features andrequire that DirectX 8 be installed in the host com-puter For this reason the AL and ERWIN installa-tion program also installs DirectX 8 if requiredCurrently AL and ERWIN are distributed as freesoftware under the GNU public license and are

58 Computer Music Journal

Table 2 Structural hierarchy list of Erwinrsquos Playground (lsquolsquoXrsquorsquo denotes section extensions)

Section Shell Start TimeDuration

(sec) Grain Duration (msec) Source Audio Files

1 1S 0000000 01026 5ndash25 MonoC1X 0001026 37258

2 2S 0037384 02394 5ndash50 MonoC12P 0039778 15390 5ndash50 DieFlyX 0055168 21572

3 3S 0076740 07866 5ndash503P 0084606 31806 75ndash75 75ndash20 DieFly

4 4S 0116412 13338 5ndash100 MonoC13D 0129750 87210 10ndash60 30ndash60 Crystal E4P 0216960 68742 5ndash60 DieFlyX 0285702 31777

5 5S 0317479 25650 5ndash25 MonoC110ndash160 Uba

4D 0343129 148770 10ndash150 100ndash150 Crystal E5P 0491899 105678 10ndash60 30ndash60 DieFly

30ndash60 WelderX 0597577 05818

6 6S 1003395 37962 5ndash25 MonoC110ndash160 Uba100ndash500 UlulowL

X 1041357 295954F 1070952 304038 100ndash5000 Waves1

1000ndash4000 2000ndash6000 PBdLow100ndash300 4000ndash8000 PBg1f

X 1374990 265995D 1401589 258210 1000ndash4500 Crystal E

10ndash50 10ndash4000 20ndash150 Crystal F10ndash50 Crystal G30ndash50 Squeak1

X 2059799 084946P 2068293 171342 30ndash60 DieFly

20ndash80 20ndash150 Welder5ndash1000 BZZZ

X 2239635 50250

7 7S 2289885 59850 5ndash25 MonoC110ndash200 Uba5ndash25 200ndash700 Ululow

X 2349735 434425F 2393177 457254 200ndash8000 Waves1

1000ndash4000 2000ndash5000 2000ndash6000 PBdLow100ndash300 4000ndash8000 PBg1f5ndash25 40ndash150 Motorway1

X 2250431 1051106D 3355541 367650 200ndash350 Crystal E

10ndash50 10ndash4000 20ndash150 Crystal F10ndash50 Crystal G50ndash150 50ndash250 Squeak1

X 4123191 304657P 4153656 237006 75ndash20 75ndash75 10ndash60 30ndash60 DieFly

30ndash60 WelderX 4390662 12090

8 7P 4402752 237006 75ndash20 20ndash40 DieFly6D 5039758 367650 10ndash50 200ndash350 Crystal E

10ndash50 Crystal F50ndash250 Squeak15ndash5000 Cicadas

(continued )

59Fischman

Table 2 (Continued)

Section Shell Start TimeDuration

(sec) Grain Duration (msec) Source Audio Files

5F 5407408 457254 100ndash5000 Waves12000ndash8000 PBdLow100ndash300 4000ndash8000 PBg1f5ndash25 40ndash150 150ndash200 Motorway1

7S 6264662 59850 5ndash25 MonoC110ndash200 Uba5ndash25 90ndash700 200ndash700 UlulowL

9 6P 6324512 171342 30ndash60 DieFly20ndash80 100ndash250 Welder5ndash1000 BZZZ

5D 6495854 258210 200ndash350 1000ndash4500 Crystal E20ndash150 Crystal F10ndash50 Crystal G50ndash250 Squeak1

4F 7154064 304038 100ndash5000 Waves11000ndash4000 PBdLow100ndash300 4000ndash8000 PBg1f100ndash10000 150ndash200 Motorway1

X 7458102 100006S 7468102 37962 5ndash25 MonoC1

10ndash160 10ndash200 Uba100ndash500 UlulowL

10 5P 7506064 105678 10ndash60 30ndash60 DieFly30ndash60 Welder5ndash1000 10ndash60 BZZZ

X 8011742 315524D 8043294 148770 200ndash350 100ndash150 Crystal E

20ndash60 50ndash250 Squeak150ndash150 Crystal B to Squeak

5S 8192064 25650 5ndash25 MonoC110ndash160 Uba

X 8217714 80546

11 4P 8298260 68742 5ndash60 10ndash30 DieFly5ndash60 BZZZ

3D 8367002 87210 80ndash450 200ndash350 Crystal E40ndash400 50ndash250 100ndash350 Squeak1

X 8454212 002684S 8454480 13338 5ndash25 5ndash100 MonoC1X 8467818 36696

12 3P 8504514 31806 75ndash20 75ndash75 DieFly3S 8536320 07866 5ndash25 5ndash50 MonoC1X 8544186 46046

13 2P 8590232 15390 5ndash50 DieFly2S 9005622 02394 15ndash50 MonoC1X 9008016 45972

14 1S 9053988 01026 5ndash25 MonoC110ndash200 70ndash200 Uba

X 9055014 61339

END 9116353

60 Computer Music Journal

available online at wwwkeeleacukdeptsmustaffAlAl_softwarehtm The programs are alsodistributed with the Composerrsquos Desktop Project

AL

AL is a multiple document interface (MDI) applica-tion with frame windows that present views of dif-ferent documents andor several views of the samedocument It is designed for the creation and ma-nipulation of sonic events and their organizationinto a structured musical work

To aid user familiarity with the software and pro-mote a reasonably short learning curve care wastaken to develop an interface within an accepted

general standard for Windows applications includ-ing a menu bar a tool bar common window com-ponents and typical mouse lsquolsquodrag-and-droprsquorsquooperations It also contains detailed online helppages including a tutorial Figure 9 shows an in-stance of the user interface

Another important consideration in the design ofAL was the desire to provide a reasonably open sys-tem with the option for development of third-partyplug-ins thus avoiding the constraints imposed bya single type of musical process Future develop-ments by the author and others will ideally imple-ment a variety of compositional approaches inaddition to the application of Schrodingerrsquos Equa-tion in the existing plug-in (see the description ofERWIN below)

Figure 9 The AL user in-terface including themenu bar tool bar and aview of a document

52 Computer Music Journal

RL(a)

RL(b)

Width and depth scatter values vary from 0 (noscatter) to any positive numbers x and y with avalue of 1 indicating a scatter distance equal to thedistance from the center to one of the speakersSpectral scatter values vary from 0 (no scatter) toany positive number of semitones (or any fre-quency ratio)

To implement scatter a time-varying volumefunction is created in a similar manner to that ofthe motion trajectory described above Breakpointsare also generated stochastically according toSchrodingerrsquos radial and angular distributions(equation A19)

Therefore a grain created at time tc will movewith the cloud at a xed lsquolsquodistancersquorsquo from its centerThis distance is generated stochastically withinwidth depth and height boundary values 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 unanchoredIn the former case the sign of the current scattervalue indicates whether the grains will be posi-tioned to the right or left of (or transposed above orbelow) the cloud For instance when the scattervalue is positive all the grains are positioned tothe right of the cloud When the scatter becomesnegative the grains are positioned to the left Un-anchored scatter means that that the grains are po-sitioned both to the right and left of (andtransposed above and below) the cloud

The algorithm that generates the volume func-tion normalizes the values so that the cloudreaches the maximum width depth and transposi-tional scatter (Again these are not reached simul-

taneously) In a similar fashion to the motiontrajectory the algorithm also requires the composerto provide minimum and maximum values for thetime interval to generate time-volume breakpointsThe volume function may also be repeated an arbi-trary number of times this may also be equal tothe value of the orbital quantum number l plusone Repetitions may be identical or uctuatewithin a percentage given by the composer

Grain Parameters

These parameters control the attributes of individ-ual grains such as their envelope duration and in-dividual scatter patterns

Grain Amplitude Scaling

The amplitude of individual grains is scaled by again factor according to their onset times relativeto the beginning of the cloud and implemented ac-cording to various algorithm types The rst optionconsists of a scaling function that follows the enve-lope of the source (equation A20) Other optionsfollow Schrodingerrsquos distribution (r2 sin(h)|Y(rhf|2

in equations A21ndashA23) the complement of Schro-dingerrsquos distribution (equation A24) and Schro-dingerrsquos cumulative probability and itscomplement (equations A25 and A26) It is worthnoting that if the cumulative probability is 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 cloudproducing a nonlinear crescendo This is becausethe cumulative probability always increases mono-tonically Similarly following the complement ofthe cumulative probability will result in a nonlin-ear diminuendo

Finally scaling may be generated from the vari-ous distributions resulting from SchrodingerrsquosEquation (equations A27ndashA33)

Grain Duration

The actual duration of individual grains uctuatesaccording to their onset times relative to the begin-

Figure 3 Examples ofgranular cloud volume(a) narrow grain scatterand (b) wide grain scatter

53Fischman

ning of the cloud The function controlling the du-ration uctuation is calculated in a manner similarto that of the scaling factor offering the same op-tions to the user (equations A34 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 thecloudrsquos motion trajectory and its volume functionrepetitions may also be identical or uctuatewithin a percentage given by the composer Themain algorithms available consist of the followingschemes duration proportional to the source enve-lope duration proportional to Schrodingerrsquos distri-bution and its complement duration proportionalto Schrodingerrsquos cumulative probability and itscomplement and duration generated 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 grains be-come gradually shorter until their duration reachesits minimum value at the end of the cloud

Grain Envelope

The envelope applied to each individual grain con-sists of either the envelope extracted from thesource or a linear envelope shaped according to theseven parameters shown in Figure 4 and some ofSchrodingerrsquos distributions (equations A36ndashA40)

Grain Scatter Path

In addition to the xed scatter resulting from thevolume of the cloud each grain is provided with anindividual trajectory Grain scatter is given in thesame units discussed above in relation to cloudvolume

The algorithm is implemented by means of atime-varying scatter function created for each di-mension with normalized duration and maximumvalue of unity (equations A41ndashA43) To generatethe trajectory breakpoints the algorithm requiresminimum and maximum values for the time inter-val between a pair of breakpoints provided by thecomposer However these are given as percentagesof the grain duration as the function is normalizedWhen a grain is created the function is scaled to tits duration and maximum amplitude It is possibleto change the shape of the function according to ajitter value given by the composer which results ina different path 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 cloudand overall cloud motion (equation A44) The pitchshift relative to the listener is invariant under rota-tion Therefore it is possible to simplify the calcu-lation of the Doppler Effect 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 Figure 5(equations A45ndashA49)

Structure

We will now proceed to examine ways in whichthe generated material can be structured into a mu-sical work It is worth noting that these are particu-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 grain characteristics etc) Never-theless it is also important to note that choiceswere made with full consideration and regard fortheir musical viability and formal integrity

Figure 4 Linear envelopeparameters (1) start offset(2) overshoot attack(3) overshoot decay(4) steady state (5) decay(6) end offset and(7) overshoot percentage

54 Computer Music Journal

x

q

1

RL

Listener

y

a

R

L

Vg

Pyramidal Shape

A relatively straightforward strategy for structuringmusical works may be linked to a survey throughthe atomic shells resulting from changes in theoverall energy For instance the survey may beginin the lowest energy shell and leap to higher shellsas this energy increases (This case also correspondsto an excursion through the periodic table accord-ing to ascending atomic number) In fact it isworth adopting this particular instance as a starting

point to develop a basic model that can subse-quently be modi ed and extended

Following the analogy with the periodic tablemajor sectional boundaries may correspond to thereappearance of a shell with an increasing value ofn (which also corresponds to a change of row in theperiodic table) Because S is the rst shell majorsectional boundaries will start at 1S 2S 3S 7SSubsections within these boundaries may corre-spond to the shells surveyed in between Figure 6illustrates the resulting structure obtained from theenergy levels in the periodic table this may becompared to a pyramid that is surveyed from top tobottom Examination of this structure brings 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 (S and 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 corresponding to the reappearance ofeach particular shell in subsequent sections For in-stance shell D reappears in sections 4 to 7 Thissuggests the possibility of creating and additionaldevelopmental dimension that following the anal-ogy with the pyramid would consist of four lsquolsquoverti-calrsquorsquo paths identi ed by different grayscale shadesin Figure 6 For instance the reappearance of ashell could develop the material of previous appear-ances

Two remaining issues must be determined beforethe structure is nalized section duration and sec-tion differentiation

Figure 5 Calculation ofDoppler Shift using a rota-tion of axes

Figure 6 Pyramidal struc-ture resulting from the en-ergy levels obtained by fol-lowing the periodic tablein ascending order

55Fischman

7S 5F 6D 7P

6S 4F 5D 6P

5S 4D 5P

4S 4P

3S

2S 2P

1SSection 2

Section 1

Section 3

Section 4

Section 5

Section 6

Section 7

Section Duration

We can follow the path set by the rst observationabove (that sections become longer by virtue of thenumber of subsections they include) and establish afurther principle of growth related to the periodictable the minimum duration of each subsectioncloud is proportional to the sum of atomic numbersof its related shell by setting a correspondence be-tween a chosen time unit and atomic numbers asfollows

SectionDuration

4 atomic_numbers 2 time_unit (2)o1 2For example if the time unit is 005 sec the

minimum duration of the subsection correspondingto shell 2P (atomic numbers 5 to 10) will be

(5 ` 6 ` 7 ` 8 ` 9 ` 10) 2 005 4 225 sec

Applying this principle we obtain a further re- nement of the pyramidal shape whereby a newappearance of a shell is normally longer than itsprevious appearances Table 1 shows the minimumduration values for shell D with atomic unit dura-tion of 00342 sec and Figure 7 shows a schematicversion of the re ned 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 becauseaccording to quantum mechanics an electron re-mains in a given state as long as the energypumped into (or extracted from) it is equal to or

greater than the transition energy required to moveto an immediate neighboring state as calculated byequation A8

Section Differentiation

Identi cation of distinctive threads according toshells requires that sections be differentiated bytheir content This can be achieved by using differ-ent source sounds in each shell This will also in- uence overall cloud and individual graincharacteristics because as explained above manyof these are derived from the attributesmdashparticu-larly the envelopemdashof the source sound Anotherway sections may be differentiated is by setting dif-ferent grain attributes (eg different durations en-velopes spatial locations or by using differentdistributions to generate amplitudes and durations)

Extensions of the Pyramidal Shape

Two other structural shapes can be derived fromthe pyramid an inverted pyramid and a diamondThe former is analogous to a survey of the periodictable from the highest to the lowest atomic num-ber The latter consists of an original pyramid fol-

Figure 7 Schematic viewof pyramidal structurewhen the minimum dura-tion of each subsection isproportional to the sum ofatomic numbers of its re-lated shell (not to scale)

Table 1 Minimum duration values for shell Dwith atomic unit duration of 00342 sec

Shell Duration (sec)

3D 872104D 1487705D 2582106D 367650

56 Computer Music Journal

(a) (b)

lowed by of an inverted one These are shownschematically in Figure 8

Other Structural Possibilities

Apart from the simple trajectories related 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 beordered according to stochastic algorithms or gener-ative rules Alternatively they may follow attri-butes of a source sound such as the envelope wherethe peak may correspond to the maximum atomicnumber All these are options beyond the scope ofthe current research plan 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 (or dissipation) According to 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

1 1 E`SectionDuration 4 abs 1 2 (3)a U b 1 22 2n n Ka b E

where KE is the rate of energy accumulation (or dis-sipation) expressed as a fraction of the ionizationenergy E` per second As an illustration assumingthat KE 4 001 E` and that shell 3D progresses toshell 4P then the duration of the former is

3DSectionDuration3D U 4P

1 1 E`4 abs 1 2 4 486 sec1 22 23 4 001E`

The problem with this method is that energytransitions at lower shells are signi cantly higherthan those at higher shells For instance the transi-tion from 1S to 2S requires an energy value overone hundred times higher than a transition from6D to 7P Consequently the duration of 1S is inthe same proportion to that of 6D yielding ex-tremely long durations for 1S or extremely shortones for 6D This may be overcome by subdividinglonger sections according to proportions similar tothose dividing the whole work Alternatively therate of energy ow could correspond to 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 ow In general equationA10 may be modi ed to cater for any energy owfunction KE(t)

Section_durationa U b 1 1K (t)dt 4 abs 1 2 E (4)E ` 1 22 20 n na b

Erwinrsquos Playground

Erwinrsquos Playground is a work for tape lasting 9 min12 sec Except for the recording of 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 Erwinrsquos Playground is intendedto demonstrate the validity and viability of thesetheoretical principles in the generation of a musical

Figure 8 (a) Inverted pyra-midal structure and (b) di-amond structure

57Fischman

work as well as the usefulness of the software as aconduit for the realization of these principlesHowever its aesthetic value is intended to be inde-pendent of the generating principles because notheory is capable of 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 ErwinSchrodingerrsquos imaginary eld of action namely theinner shells of the atom Its structure is modeledon the diamond shape described above and linkedto a survey that begins in the lowest energy stateleaps up until it reaches the highest shell and thenreturns back to the initial state This may also becompared to an excursion through the periodic ta-ble that ascends and then descends according toatomic number order

Four different types of sound sources were as-signed to each shell to achieve section differentia-tion The S shells consisted primarily of vocalsounds (audio les MonoC1 Uba UlulowL) Pshells were assigned three different types of soundwhich nevertheless share some common spectralcharacteristics the lsquolsquobuzzrsquorsquo of a y a vocal lsquolsquobuzzrsquorsquoand the sound of a sparking welder (audio lesDieFly BZZZ Welder)

D shells were assigned sounds that ranged frompitched to lsquolsquosqueakyrsquorsquo The pitched sounds were theresult of rubbing the edge of a glass cup with differ-ent amounts of liquid to obtain different pitches(audio les Crystal E Crystal F Crystal G )Squeaks were recorded from the un-oiled wheels ofa trolley (audio le Squeak1) A sample containinga transition from pitch to squeak was achieved byrubbing the glass cup with different degrees of pres-sure (audio le Crystal B to Squeak) Finally a re-cording of the pitched sound of cicadas was alsoincluded toward the middle of the piece (audio leCicadas)

F shells were normally associated with local andglobal climaxes therefore they spread through awider bandwidth by means of three spectral typesThe rst type consists of two noise-based soundsobtained from a busy motorway and sea waves gur-gling through rocks (audio les Motorway1 andWaves1) The second sound is metallic consistingof the recording of a frying pan that was hit or

rubbed against other objects (audio le PBg1f) Thethird was mainly used to create low-frequency tex-tures and drones from the recording of a piece ofplywood that was bent at a more or less regularrate (audio le PBdLow) (Examples of the source les as well as passages characterizing each shellsand the complete work are available on the forth-coming Computer Music Journal volume 27 soundexample disc)

Grain duration was another signi cant attributeused to characterize the various shells Finally theatomic number duration equivalent used in thepiece was 00342 sec per atomic number

Table 2 presents a complete list of the structuralhierarchy of Erwinrsquos Playground including starttime minimum duration grain duration limitsand the source audio les for each subsection Asmentioned above the duration of subsections couldbe extended to allow some freedom when takinginto account musical context Section extensionsappear below the shell name with the letter lsquolsquoXrsquorsquoAlso note the bold line denoting the end of the pyr-amid and the beginning of its inversion which actsas an lsquolsquoaxis of re ectionrsquorsquo It is worth noting thatthe second half of the piece (sections 8ndash14 corre-sponding to the inverted pyramid) is more com-pressed than the rst part as there are fewersubsections that have their minimum durationsextended

Software Implementation

The software implementation of this procedureconsists of two separate components AL a compo-sitional environment for the creation and manipu-lation of musical events and ERWIN a plug-inthat implements the application of Schrodingerrsquosmodel described above These were developed andtested using Visual C ` ` 60 running under Win-dows 2000 and later under Windows XP Both ap-plications use Microsoft DirectX 8 features andrequire that DirectX 8 be installed in the host com-puter For this reason the AL and ERWIN installa-tion program also installs DirectX 8 if requiredCurrently AL and ERWIN are distributed as freesoftware under the GNU public license and are

58 Computer Music Journal

Table 2 Structural hierarchy list of Erwinrsquos Playground (lsquolsquoXrsquorsquo denotes section extensions)

Section Shell Start TimeDuration

(sec) Grain Duration (msec) Source Audio Files

1 1S 0000000 01026 5ndash25 MonoC1X 0001026 37258

2 2S 0037384 02394 5ndash50 MonoC12P 0039778 15390 5ndash50 DieFlyX 0055168 21572

3 3S 0076740 07866 5ndash503P 0084606 31806 75ndash75 75ndash20 DieFly

4 4S 0116412 13338 5ndash100 MonoC13D 0129750 87210 10ndash60 30ndash60 Crystal E4P 0216960 68742 5ndash60 DieFlyX 0285702 31777

5 5S 0317479 25650 5ndash25 MonoC110ndash160 Uba

4D 0343129 148770 10ndash150 100ndash150 Crystal E5P 0491899 105678 10ndash60 30ndash60 DieFly

30ndash60 WelderX 0597577 05818

6 6S 1003395 37962 5ndash25 MonoC110ndash160 Uba100ndash500 UlulowL

X 1041357 295954F 1070952 304038 100ndash5000 Waves1

1000ndash4000 2000ndash6000 PBdLow100ndash300 4000ndash8000 PBg1f

X 1374990 265995D 1401589 258210 1000ndash4500 Crystal E

10ndash50 10ndash4000 20ndash150 Crystal F10ndash50 Crystal G30ndash50 Squeak1

X 2059799 084946P 2068293 171342 30ndash60 DieFly

20ndash80 20ndash150 Welder5ndash1000 BZZZ

X 2239635 50250

7 7S 2289885 59850 5ndash25 MonoC110ndash200 Uba5ndash25 200ndash700 Ululow

X 2349735 434425F 2393177 457254 200ndash8000 Waves1

1000ndash4000 2000ndash5000 2000ndash6000 PBdLow100ndash300 4000ndash8000 PBg1f5ndash25 40ndash150 Motorway1

X 2250431 1051106D 3355541 367650 200ndash350 Crystal E

10ndash50 10ndash4000 20ndash150 Crystal F10ndash50 Crystal G50ndash150 50ndash250 Squeak1

X 4123191 304657P 4153656 237006 75ndash20 75ndash75 10ndash60 30ndash60 DieFly

30ndash60 WelderX 4390662 12090

8 7P 4402752 237006 75ndash20 20ndash40 DieFly6D 5039758 367650 10ndash50 200ndash350 Crystal E

10ndash50 Crystal F50ndash250 Squeak15ndash5000 Cicadas

(continued )

59Fischman

Table 2 (Continued)

Section Shell Start TimeDuration

(sec) Grain Duration (msec) Source Audio Files

5F 5407408 457254 100ndash5000 Waves12000ndash8000 PBdLow100ndash300 4000ndash8000 PBg1f5ndash25 40ndash150 150ndash200 Motorway1

7S 6264662 59850 5ndash25 MonoC110ndash200 Uba5ndash25 90ndash700 200ndash700 UlulowL

9 6P 6324512 171342 30ndash60 DieFly20ndash80 100ndash250 Welder5ndash1000 BZZZ

5D 6495854 258210 200ndash350 1000ndash4500 Crystal E20ndash150 Crystal F10ndash50 Crystal G50ndash250 Squeak1

4F 7154064 304038 100ndash5000 Waves11000ndash4000 PBdLow100ndash300 4000ndash8000 PBg1f100ndash10000 150ndash200 Motorway1

X 7458102 100006S 7468102 37962 5ndash25 MonoC1

10ndash160 10ndash200 Uba100ndash500 UlulowL

10 5P 7506064 105678 10ndash60 30ndash60 DieFly30ndash60 Welder5ndash1000 10ndash60 BZZZ

X 8011742 315524D 8043294 148770 200ndash350 100ndash150 Crystal E

20ndash60 50ndash250 Squeak150ndash150 Crystal B to Squeak

5S 8192064 25650 5ndash25 MonoC110ndash160 Uba

X 8217714 80546

11 4P 8298260 68742 5ndash60 10ndash30 DieFly5ndash60 BZZZ

3D 8367002 87210 80ndash450 200ndash350 Crystal E40ndash400 50ndash250 100ndash350 Squeak1

X 8454212 002684S 8454480 13338 5ndash25 5ndash100 MonoC1X 8467818 36696

12 3P 8504514 31806 75ndash20 75ndash75 DieFly3S 8536320 07866 5ndash25 5ndash50 MonoC1X 8544186 46046

13 2P 8590232 15390 5ndash50 DieFly2S 9005622 02394 15ndash50 MonoC1X 9008016 45972

14 1S 9053988 01026 5ndash25 MonoC110ndash200 70ndash200 Uba

X 9055014 61339

END 9116353

60 Computer Music Journal

available online at wwwkeeleacukdeptsmustaffAlAl_softwarehtm The programs are alsodistributed with the Composerrsquos Desktop Project

AL

AL is a multiple document interface (MDI) applica-tion with frame windows that present views of dif-ferent documents andor several views of the samedocument It is designed for the creation and ma-nipulation of sonic events and their organizationinto a structured musical work

To aid user familiarity with the software and pro-mote a reasonably short learning curve care wastaken to develop an interface within an accepted

general standard for Windows applications includ-ing a menu bar a tool bar common window com-ponents and typical mouse lsquolsquodrag-and-droprsquorsquooperations It also contains detailed online helppages including a tutorial Figure 9 shows an in-stance of the user interface

Another important consideration in the design ofAL was the desire to provide a reasonably open sys-tem with the option for development of third-partyplug-ins thus avoiding the constraints imposed bya single type of musical process Future develop-ments by the author and others will ideally imple-ment a variety of compositional approaches inaddition to the application of Schrodingerrsquos Equa-tion in the existing plug-in (see the description ofERWIN below)

Figure 9 The AL user in-terface including themenu bar tool bar and aview of a document

53Fischman

ning of the cloud The function controlling the du-ration uctuation is calculated in a manner similarto that of the scaling factor offering the same op-tions to the user (equations A34 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 thecloudrsquos motion trajectory and its volume functionrepetitions may also be identical or uctuatewithin a percentage given by the composer Themain algorithms available consist of the followingschemes duration proportional to the source enve-lope duration proportional to Schrodingerrsquos distri-bution and its complement duration proportionalto Schrodingerrsquos cumulative probability and itscomplement and duration generated 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 grains be-come gradually shorter until their duration reachesits minimum value at the end of the cloud

Grain Envelope

The envelope applied to each individual grain con-sists of either the envelope extracted from thesource or a linear envelope shaped according to theseven parameters shown in Figure 4 and some ofSchrodingerrsquos distributions (equations A36ndashA40)

Grain Scatter Path

In addition to the xed scatter resulting from thevolume of the cloud each grain is provided with anindividual trajectory Grain scatter is given in thesame units discussed above in relation to cloudvolume

The algorithm is implemented by means of atime-varying scatter function created for each di-mension with normalized duration and maximumvalue of unity (equations A41ndashA43) To generatethe trajectory breakpoints the algorithm requiresminimum and maximum values for the time inter-val between a pair of breakpoints provided by thecomposer However these are given as percentagesof the grain duration as the function is normalizedWhen a grain is created the function is scaled to tits duration and maximum amplitude It is possibleto change the shape of the function according to ajitter value given by the composer which results ina different path 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 cloudand overall cloud motion (equation A44) The pitchshift relative to the listener is invariant under rota-tion Therefore it is possible to simplify the calcu-lation of the Doppler Effect 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 Figure 5(equations A45ndashA49)

Structure

We will now proceed to examine ways in whichthe generated material can be structured into a mu-sical work It is worth noting that these are particu-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 grain characteristics etc) Never-theless it is also important to note that choiceswere made with full consideration and regard fortheir musical viability and formal integrity

Figure 4 Linear envelopeparameters (1) start offset(2) overshoot attack(3) overshoot decay(4) steady state (5) decay(6) end offset and(7) overshoot percentage

54 Computer Music Journal

x

q

1

RL

Listener

y

a

R

L

Vg

Pyramidal Shape

A relatively straightforward strategy for structuringmusical works may be linked to a survey throughthe atomic shells resulting from changes in theoverall energy For instance the survey may beginin the lowest energy shell and leap to higher shellsas this energy increases (This case also correspondsto an excursion through the periodic table accord-ing to ascending atomic number) In fact it isworth adopting this particular instance as a starting

point to develop a basic model that can subse-quently be modi ed and extended

Following the analogy with the periodic tablemajor sectional boundaries may correspond to thereappearance of a shell with an increasing value ofn (which also corresponds to a change of row in theperiodic table) Because S is the rst shell majorsectional boundaries will start at 1S 2S 3S 7SSubsections within these boundaries may corre-spond to the shells surveyed in between Figure 6illustrates the resulting structure obtained from theenergy levels in the periodic table this may becompared to a pyramid that is surveyed from top tobottom Examination of this structure brings 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 (S and 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 corresponding to the reappearance ofeach particular shell in subsequent sections For in-stance shell D reappears in sections 4 to 7 Thissuggests the possibility of creating and additionaldevelopmental dimension that following the anal-ogy with the pyramid would consist of four lsquolsquoverti-calrsquorsquo paths identi ed by different grayscale shadesin Figure 6 For instance the reappearance of ashell could develop the material of previous appear-ances

Two remaining issues must be determined beforethe structure is nalized section duration and sec-tion differentiation

Figure 5 Calculation ofDoppler Shift using a rota-tion of axes

Figure 6 Pyramidal struc-ture resulting from the en-ergy levels obtained by fol-lowing the periodic tablein ascending order

55Fischman

7S 5F 6D 7P

6S 4F 5D 6P

5S 4D 5P

4S 4P

3S

2S 2P

1SSection 2

Section 1

Section 3

Section 4

Section 5

Section 6

Section 7

Section Duration

We can follow the path set by the rst observationabove (that sections become longer by virtue of thenumber of subsections they include) and establish afurther principle of growth related to the periodictable the minimum duration of each subsectioncloud is proportional to the sum of atomic numbersof its related shell by setting a correspondence be-tween a chosen time unit and atomic numbers asfollows

SectionDuration

4 atomic_numbers 2 time_unit (2)o1 2For example if the time unit is 005 sec the

minimum duration of the subsection correspondingto shell 2P (atomic numbers 5 to 10) will be

(5 ` 6 ` 7 ` 8 ` 9 ` 10) 2 005 4 225 sec

Applying this principle we obtain a further re- nement of the pyramidal shape whereby a newappearance of a shell is normally longer than itsprevious appearances Table 1 shows the minimumduration values for shell D with atomic unit dura-tion of 00342 sec and Figure 7 shows a schematicversion of the re ned 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 becauseaccording to quantum mechanics an electron re-mains in a given state as long as the energypumped into (or extracted from) it is equal to or

greater than the transition energy required to moveto an immediate neighboring state as calculated byequation A8

Section Differentiation

Identi cation of distinctive threads according toshells requires that sections be differentiated bytheir content This can be achieved by using differ-ent source sounds in each shell This will also in- uence overall cloud and individual graincharacteristics because as explained above manyof these are derived from the attributesmdashparticu-larly the envelopemdashof the source sound Anotherway sections may be differentiated is by setting dif-ferent grain attributes (eg different durations en-velopes spatial locations or by using differentdistributions to generate amplitudes and durations)

Extensions of the Pyramidal Shape

Two other structural shapes can be derived fromthe pyramid an inverted pyramid and a diamondThe former is analogous to a survey of the periodictable from the highest to the lowest atomic num-ber The latter consists of an original pyramid fol-

Figure 7 Schematic viewof pyramidal structurewhen the minimum dura-tion of each subsection isproportional to the sum ofatomic numbers of its re-lated shell (not to scale)

Table 1 Minimum duration values for shell Dwith atomic unit duration of 00342 sec

Shell Duration (sec)

3D 872104D 1487705D 2582106D 367650

56 Computer Music Journal

(a) (b)

lowed by of an inverted one These are shownschematically in Figure 8

Other Structural Possibilities

Apart from the simple trajectories related 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 beordered according to stochastic algorithms or gener-ative rules Alternatively they may follow attri-butes of a source sound such as the envelope wherethe peak may correspond to the maximum atomicnumber All these are options beyond the scope ofthe current research plan 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 (or dissipation) According to 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

1 1 E`SectionDuration 4 abs 1 2 (3)a U b 1 22 2n n Ka b E

where KE is the rate of energy accumulation (or dis-sipation) expressed as a fraction of the ionizationenergy E` per second As an illustration assumingthat KE 4 001 E` and that shell 3D progresses toshell 4P then the duration of the former is

3DSectionDuration3D U 4P

1 1 E`4 abs 1 2 4 486 sec1 22 23 4 001E`

The problem with this method is that energytransitions at lower shells are signi cantly higherthan those at higher shells For instance the transi-tion from 1S to 2S requires an energy value overone hundred times higher than a transition from6D to 7P Consequently the duration of 1S is inthe same proportion to that of 6D yielding ex-tremely long durations for 1S or extremely shortones for 6D This may be overcome by subdividinglonger sections according to proportions similar tothose dividing the whole work Alternatively therate of energy ow could correspond to 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 ow In general equationA10 may be modi ed to cater for any energy owfunction KE(t)

Section_durationa U b 1 1K (t)dt 4 abs 1 2 E (4)E ` 1 22 20 n na b

Erwinrsquos Playground

Erwinrsquos Playground is a work for tape lasting 9 min12 sec Except for the recording of 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 Erwinrsquos Playground is intendedto demonstrate the validity and viability of thesetheoretical principles in the generation of a musical

Figure 8 (a) Inverted pyra-midal structure and (b) di-amond structure

57Fischman

work as well as the usefulness of the software as aconduit for the realization of these principlesHowever its aesthetic value is intended to be inde-pendent of the generating principles because notheory is capable of 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 ErwinSchrodingerrsquos imaginary eld of action namely theinner shells of the atom Its structure is modeledon the diamond shape described above and linkedto a survey that begins in the lowest energy stateleaps up until it reaches the highest shell and thenreturns back to the initial state This may also becompared to an excursion through the periodic ta-ble that ascends and then descends according toatomic number order

Four different types of sound sources were as-signed to each shell to achieve section differentia-tion The S shells consisted primarily of vocalsounds (audio les MonoC1 Uba UlulowL) Pshells were assigned three different types of soundwhich nevertheless share some common spectralcharacteristics the lsquolsquobuzzrsquorsquo of a y a vocal lsquolsquobuzzrsquorsquoand the sound of a sparking welder (audio lesDieFly BZZZ Welder)

D shells were assigned sounds that ranged frompitched to lsquolsquosqueakyrsquorsquo The pitched sounds were theresult of rubbing the edge of a glass cup with differ-ent amounts of liquid to obtain different pitches(audio les Crystal E Crystal F Crystal G )Squeaks were recorded from the un-oiled wheels ofa trolley (audio le Squeak1) A sample containinga transition from pitch to squeak was achieved byrubbing the glass cup with different degrees of pres-sure (audio le Crystal B to Squeak) Finally a re-cording of the pitched sound of cicadas was alsoincluded toward the middle of the piece (audio leCicadas)

F shells were normally associated with local andglobal climaxes therefore they spread through awider bandwidth by means of three spectral typesThe rst type consists of two noise-based soundsobtained from a busy motorway and sea waves gur-gling through rocks (audio les Motorway1 andWaves1) The second sound is metallic consistingof the recording of a frying pan that was hit or

rubbed against other objects (audio le PBg1f) Thethird was mainly used to create low-frequency tex-tures and drones from the recording of a piece ofplywood that was bent at a more or less regularrate (audio le PBdLow) (Examples of the source les as well as passages characterizing each shellsand the complete work are available on the forth-coming Computer Music Journal volume 27 soundexample disc)

Grain duration was another signi cant attributeused to characterize the various shells Finally theatomic number duration equivalent used in thepiece was 00342 sec per atomic number

Table 2 presents a complete list of the structuralhierarchy of Erwinrsquos Playground including starttime minimum duration grain duration limitsand the source audio les for each subsection Asmentioned above the duration of subsections couldbe extended to allow some freedom when takinginto account musical context Section extensionsappear below the shell name with the letter lsquolsquoXrsquorsquoAlso note the bold line denoting the end of the pyr-amid and the beginning of its inversion which actsas an lsquolsquoaxis of re ectionrsquorsquo It is worth noting thatthe second half of the piece (sections 8ndash14 corre-sponding to the inverted pyramid) is more com-pressed than the rst part as there are fewersubsections that have their minimum durationsextended

Software Implementation

The software implementation of this procedureconsists of two separate components AL a compo-sitional environment for the creation and manipu-lation of musical events and ERWIN a plug-inthat implements the application of Schrodingerrsquosmodel described above These were developed andtested using Visual C ` ` 60 running under Win-dows 2000 and later under Windows XP Both ap-plications use Microsoft DirectX 8 features andrequire that DirectX 8 be installed in the host com-puter For this reason the AL and ERWIN installa-tion program also installs DirectX 8 if requiredCurrently AL and ERWIN are distributed as freesoftware under the GNU public license and are

58 Computer Music Journal

Table 2 Structural hierarchy list of Erwinrsquos Playground (lsquolsquoXrsquorsquo denotes section extensions)

Section Shell Start TimeDuration

(sec) Grain Duration (msec) Source Audio Files

1 1S 0000000 01026 5ndash25 MonoC1X 0001026 37258

2 2S 0037384 02394 5ndash50 MonoC12P 0039778 15390 5ndash50 DieFlyX 0055168 21572

3 3S 0076740 07866 5ndash503P 0084606 31806 75ndash75 75ndash20 DieFly

4 4S 0116412 13338 5ndash100 MonoC13D 0129750 87210 10ndash60 30ndash60 Crystal E4P 0216960 68742 5ndash60 DieFlyX 0285702 31777

5 5S 0317479 25650 5ndash25 MonoC110ndash160 Uba

4D 0343129 148770 10ndash150 100ndash150 Crystal E5P 0491899 105678 10ndash60 30ndash60 DieFly

30ndash60 WelderX 0597577 05818

6 6S 1003395 37962 5ndash25 MonoC110ndash160 Uba100ndash500 UlulowL

X 1041357 295954F 1070952 304038 100ndash5000 Waves1

1000ndash4000 2000ndash6000 PBdLow100ndash300 4000ndash8000 PBg1f

X 1374990 265995D 1401589 258210 1000ndash4500 Crystal E

10ndash50 10ndash4000 20ndash150 Crystal F10ndash50 Crystal G30ndash50 Squeak1

X 2059799 084946P 2068293 171342 30ndash60 DieFly

20ndash80 20ndash150 Welder5ndash1000 BZZZ

X 2239635 50250

7 7S 2289885 59850 5ndash25 MonoC110ndash200 Uba5ndash25 200ndash700 Ululow

X 2349735 434425F 2393177 457254 200ndash8000 Waves1

1000ndash4000 2000ndash5000 2000ndash6000 PBdLow100ndash300 4000ndash8000 PBg1f5ndash25 40ndash150 Motorway1

X 2250431 1051106D 3355541 367650 200ndash350 Crystal E

10ndash50 10ndash4000 20ndash150 Crystal F10ndash50 Crystal G50ndash150 50ndash250 Squeak1

X 4123191 304657P 4153656 237006 75ndash20 75ndash75 10ndash60 30ndash60 DieFly

30ndash60 WelderX 4390662 12090

8 7P 4402752 237006 75ndash20 20ndash40 DieFly6D 5039758 367650 10ndash50 200ndash350 Crystal E

10ndash50 Crystal F50ndash250 Squeak15ndash5000 Cicadas

(continued )

59Fischman

Table 2 (Continued)

Section Shell Start TimeDuration

(sec) Grain Duration (msec) Source Audio Files

5F 5407408 457254 100ndash5000 Waves12000ndash8000 PBdLow100ndash300 4000ndash8000 PBg1f5ndash25 40ndash150 150ndash200 Motorway1

7S 6264662 59850 5ndash25 MonoC110ndash200 Uba5ndash25 90ndash700 200ndash700 UlulowL

9 6P 6324512 171342 30ndash60 DieFly20ndash80 100ndash250 Welder5ndash1000 BZZZ

5D 6495854 258210 200ndash350 1000ndash4500 Crystal E20ndash150 Crystal F10ndash50 Crystal G50ndash250 Squeak1

4F 7154064 304038 100ndash5000 Waves11000ndash4000 PBdLow100ndash300 4000ndash8000 PBg1f100ndash10000 150ndash200 Motorway1

X 7458102 100006S 7468102 37962 5ndash25 MonoC1

10ndash160 10ndash200 Uba100ndash500 UlulowL

10 5P 7506064 105678 10ndash60 30ndash60 DieFly30ndash60 Welder5ndash1000 10ndash60 BZZZ

X 8011742 315524D 8043294 148770 200ndash350 100ndash150 Crystal E

20ndash60 50ndash250 Squeak150ndash150 Crystal B to Squeak

5S 8192064 25650 5ndash25 MonoC110ndash160 Uba

X 8217714 80546

11 4P 8298260 68742 5ndash60 10ndash30 DieFly5ndash60 BZZZ

3D 8367002 87210 80ndash450 200ndash350 Crystal E40ndash400 50ndash250 100ndash350 Squeak1

X 8454212 002684S 8454480 13338 5ndash25 5ndash100 MonoC1X 8467818 36696

12 3P 8504514 31806 75ndash20 75ndash75 DieFly3S 8536320 07866 5ndash25 5ndash50 MonoC1X 8544186 46046

13 2P 8590232 15390 5ndash50 DieFly2S 9005622 02394 15ndash50 MonoC1X 9008016 45972

14 1S 9053988 01026 5ndash25 MonoC110ndash200 70ndash200 Uba

X 9055014 61339

END 9116353

60 Computer Music Journal

available online at wwwkeeleacukdeptsmustaffAlAl_softwarehtm The programs are alsodistributed with the Composerrsquos Desktop Project

AL

AL is a multiple document interface (MDI) applica-tion with frame windows that present views of dif-ferent documents andor several views of the samedocument It is designed for the creation and ma-nipulation of sonic events and their organizationinto a structured musical work

To aid user familiarity with the software and pro-mote a reasonably short learning curve care wastaken to develop an interface within an accepted

general standard for Windows applications includ-ing a menu bar a tool bar common window com-ponents and typical mouse lsquolsquodrag-and-droprsquorsquooperations It also contains detailed online helppages including a tutorial Figure 9 shows an in-stance of the user interface

Another important consideration in the design ofAL was the desire to provide a reasonably open sys-tem with the option for development of third-partyplug-ins thus avoiding the constraints imposed bya single type of musical process Future develop-ments by the author and others will ideally imple-ment a variety of compositional approaches inaddition to the application of Schrodingerrsquos Equa-tion in the existing plug-in (see the description ofERWIN below)

Figure 9 The AL user in-terface including themenu bar tool bar and aview of a document

54 Computer Music Journal

x

q

1

RL

Listener

y

a

R

L

Vg

Pyramidal Shape

A relatively straightforward strategy for structuringmusical works may be linked to a survey throughthe atomic shells resulting from changes in theoverall energy For instance the survey may beginin the lowest energy shell and leap to higher shellsas this energy increases (This case also correspondsto an excursion through the periodic table accord-ing to ascending atomic number) In fact it isworth adopting this particular instance as a starting

point to develop a basic model that can subse-quently be modi ed and extended

Following the analogy with the periodic tablemajor sectional boundaries may correspond to thereappearance of a shell with an increasing value ofn (which also corresponds to a change of row in theperiodic table) Because S is the rst shell majorsectional boundaries will start at 1S 2S 3S 7SSubsections within these boundaries may corre-spond to the shells surveyed in between Figure 6illustrates the resulting structure obtained from theenergy levels in the periodic table this may becompared to a pyramid that is surveyed from top tobottom Examination of this structure brings 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 (S and 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 corresponding to the reappearance ofeach particular shell in subsequent sections For in-stance shell D reappears in sections 4 to 7 Thissuggests the possibility of creating and additionaldevelopmental dimension that following the anal-ogy with the pyramid would consist of four lsquolsquoverti-calrsquorsquo paths identi ed by different grayscale shadesin Figure 6 For instance the reappearance of ashell could develop the material of previous appear-ances

Two remaining issues must be determined beforethe structure is nalized section duration and sec-tion differentiation

Figure 5 Calculation ofDoppler Shift using a rota-tion of axes

Figure 6 Pyramidal struc-ture resulting from the en-ergy levels obtained by fol-lowing the periodic tablein ascending order

55Fischman

7S 5F 6D 7P

6S 4F 5D 6P

5S 4D 5P

4S 4P

3S

2S 2P

1SSection 2

Section 1

Section 3

Section 4

Section 5

Section 6

Section 7

Section Duration

We can follow the path set by the rst observationabove (that sections become longer by virtue of thenumber of subsections they include) and establish afurther principle of growth related to the periodictable the minimum duration of each subsectioncloud is proportional to the sum of atomic numbersof its related shell by setting a correspondence be-tween a chosen time unit and atomic numbers asfollows

SectionDuration

4 atomic_numbers 2 time_unit (2)o1 2For example if the time unit is 005 sec the

minimum duration of the subsection correspondingto shell 2P (atomic numbers 5 to 10) will be

(5 ` 6 ` 7 ` 8 ` 9 ` 10) 2 005 4 225 sec

Applying this principle we obtain a further re- nement of the pyramidal shape whereby a newappearance of a shell is normally longer than itsprevious appearances Table 1 shows the minimumduration values for shell D with atomic unit dura-tion of 00342 sec and Figure 7 shows a schematicversion of the re ned 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 becauseaccording to quantum mechanics an electron re-mains in a given state as long as the energypumped into (or extracted from) it is equal to or

greater than the transition energy required to moveto an immediate neighboring state as calculated byequation A8

Section Differentiation

Identi cation of distinctive threads according toshells requires that sections be differentiated bytheir content This can be achieved by using differ-ent source sounds in each shell This will also in- uence overall cloud and individual graincharacteristics because as explained above manyof these are derived from the attributesmdashparticu-larly the envelopemdashof the source sound Anotherway sections may be differentiated is by setting dif-ferent grain attributes (eg different durations en-velopes spatial locations or by using differentdistributions to generate amplitudes and durations)

Extensions of the Pyramidal Shape

Two other structural shapes can be derived fromthe pyramid an inverted pyramid and a diamondThe former is analogous to a survey of the periodictable from the highest to the lowest atomic num-ber The latter consists of an original pyramid fol-

Figure 7 Schematic viewof pyramidal structurewhen the minimum dura-tion of each subsection isproportional to the sum ofatomic numbers of its re-lated shell (not to scale)

Table 1 Minimum duration values for shell Dwith atomic unit duration of 00342 sec

Shell Duration (sec)

3D 872104D 1487705D 2582106D 367650

56 Computer Music Journal

(a) (b)

lowed by of an inverted one These are shownschematically in Figure 8

Other Structural Possibilities

Apart from the simple trajectories related 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 beordered according to stochastic algorithms or gener-ative rules Alternatively they may follow attri-butes of a source sound such as the envelope wherethe peak may correspond to the maximum atomicnumber All these are options beyond the scope ofthe current research plan 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 (or dissipation) According to 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

1 1 E`SectionDuration 4 abs 1 2 (3)a U b 1 22 2n n Ka b E

where KE is the rate of energy accumulation (or dis-sipation) expressed as a fraction of the ionizationenergy E` per second As an illustration assumingthat KE 4 001 E` and that shell 3D progresses toshell 4P then the duration of the former is

3DSectionDuration3D U 4P

1 1 E`4 abs 1 2 4 486 sec1 22 23 4 001E`

The problem with this method is that energytransitions at lower shells are signi cantly higherthan those at higher shells For instance the transi-tion from 1S to 2S requires an energy value overone hundred times higher than a transition from6D to 7P Consequently the duration of 1S is inthe same proportion to that of 6D yielding ex-tremely long durations for 1S or extremely shortones for 6D This may be overcome by subdividinglonger sections according to proportions similar tothose dividing the whole work Alternatively therate of energy ow could correspond to 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 ow In general equationA10 may be modi ed to cater for any energy owfunction KE(t)

Section_durationa U b 1 1K (t)dt 4 abs 1 2 E (4)E ` 1 22 20 n na b

Erwinrsquos Playground

Erwinrsquos Playground is a work for tape lasting 9 min12 sec Except for the recording of 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 Erwinrsquos Playground is intendedto demonstrate the validity and viability of thesetheoretical principles in the generation of a musical

Figure 8 (a) Inverted pyra-midal structure and (b) di-amond structure

57Fischman

work as well as the usefulness of the software as aconduit for the realization of these principlesHowever its aesthetic value is intended to be inde-pendent of the generating principles because notheory is capable of 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 ErwinSchrodingerrsquos imaginary eld of action namely theinner shells of the atom Its structure is modeledon the diamond shape described above and linkedto a survey that begins in the lowest energy stateleaps up until it reaches the highest shell and thenreturns back to the initial state This may also becompared to an excursion through the periodic ta-ble that ascends and then descends according toatomic number order

Four different types of sound sources were as-signed to each shell to achieve section differentia-tion The S shells consisted primarily of vocalsounds (audio les MonoC1 Uba UlulowL) Pshells were assigned three different types of soundwhich nevertheless share some common spectralcharacteristics the lsquolsquobuzzrsquorsquo of a y a vocal lsquolsquobuzzrsquorsquoand the sound of a sparking welder (audio lesDieFly BZZZ Welder)

D shells were assigned sounds that ranged frompitched to lsquolsquosqueakyrsquorsquo The pitched sounds were theresult of rubbing the edge of a glass cup with differ-ent amounts of liquid to obtain different pitches(audio les Crystal E Crystal F Crystal G )Squeaks were recorded from the un-oiled wheels ofa trolley (audio le Squeak1) A sample containinga transition from pitch to squeak was achieved byrubbing the glass cup with different degrees of pres-sure (audio le Crystal B to Squeak) Finally a re-cording of the pitched sound of cicadas was alsoincluded toward the middle of the piece (audio leCicadas)

F shells were normally associated with local andglobal climaxes therefore they spread through awider bandwidth by means of three spectral typesThe rst type consists of two noise-based soundsobtained from a busy motorway and sea waves gur-gling through rocks (audio les Motorway1 andWaves1) The second sound is metallic consistingof the recording of a frying pan that was hit or

rubbed against other objects (audio le PBg1f) Thethird was mainly used to create low-frequency tex-tures and drones from the recording of a piece ofplywood that was bent at a more or less regularrate (audio le PBdLow) (Examples of the source les as well as passages characterizing each shellsand the complete work are available on the forth-coming Computer Music Journal volume 27 soundexample disc)

Grain duration was another signi cant attributeused to characterize the various shells Finally theatomic number duration equivalent used in thepiece was 00342 sec per atomic number

Table 2 presents a complete list of the structuralhierarchy of Erwinrsquos Playground including starttime minimum duration grain duration limitsand the source audio les for each subsection Asmentioned above the duration of subsections couldbe extended to allow some freedom when takinginto account musical context Section extensionsappear below the shell name with the letter lsquolsquoXrsquorsquoAlso note the bold line denoting the end of the pyr-amid and the beginning of its inversion which actsas an lsquolsquoaxis of re ectionrsquorsquo It is worth noting thatthe second half of the piece (sections 8ndash14 corre-sponding to the inverted pyramid) is more com-pressed than the rst part as there are fewersubsections that have their minimum durationsextended

Software Implementation

The software implementation of this procedureconsists of two separate components AL a compo-sitional environment for the creation and manipu-lation of musical events and ERWIN a plug-inthat implements the application of Schrodingerrsquosmodel described above These were developed andtested using Visual C ` ` 60 running under Win-dows 2000 and later under Windows XP Both ap-plications use Microsoft DirectX 8 features andrequire that DirectX 8 be installed in the host com-puter For this reason the AL and ERWIN installa-tion program also installs DirectX 8 if requiredCurrently AL and ERWIN are distributed as freesoftware under the GNU public license and are

58 Computer Music Journal

Table 2 Structural hierarchy list of Erwinrsquos Playground (lsquolsquoXrsquorsquo denotes section extensions)

Section Shell Start TimeDuration

(sec) Grain Duration (msec) Source Audio Files

1 1S 0000000 01026 5ndash25 MonoC1X 0001026 37258

2 2S 0037384 02394 5ndash50 MonoC12P 0039778 15390 5ndash50 DieFlyX 0055168 21572

3 3S 0076740 07866 5ndash503P 0084606 31806 75ndash75 75ndash20 DieFly

4 4S 0116412 13338 5ndash100 MonoC13D 0129750 87210 10ndash60 30ndash60 Crystal E4P 0216960 68742 5ndash60 DieFlyX 0285702 31777

5 5S 0317479 25650 5ndash25 MonoC110ndash160 Uba

4D 0343129 148770 10ndash150 100ndash150 Crystal E5P 0491899 105678 10ndash60 30ndash60 DieFly

30ndash60 WelderX 0597577 05818

6 6S 1003395 37962 5ndash25 MonoC110ndash160 Uba100ndash500 UlulowL

X 1041357 295954F 1070952 304038 100ndash5000 Waves1

1000ndash4000 2000ndash6000 PBdLow100ndash300 4000ndash8000 PBg1f

X 1374990 265995D 1401589 258210 1000ndash4500 Crystal E

10ndash50 10ndash4000 20ndash150 Crystal F10ndash50 Crystal G30ndash50 Squeak1

X 2059799 084946P 2068293 171342 30ndash60 DieFly

20ndash80 20ndash150 Welder5ndash1000 BZZZ

X 2239635 50250

7 7S 2289885 59850 5ndash25 MonoC110ndash200 Uba5ndash25 200ndash700 Ululow

X 2349735 434425F 2393177 457254 200ndash8000 Waves1

1000ndash4000 2000ndash5000 2000ndash6000 PBdLow100ndash300 4000ndash8000 PBg1f5ndash25 40ndash150 Motorway1

X 2250431 1051106D 3355541 367650 200ndash350 Crystal E

10ndash50 10ndash4000 20ndash150 Crystal F10ndash50 Crystal G50ndash150 50ndash250 Squeak1

X 4123191 304657P 4153656 237006 75ndash20 75ndash75 10ndash60 30ndash60 DieFly

30ndash60 WelderX 4390662 12090

8 7P 4402752 237006 75ndash20 20ndash40 DieFly6D 5039758 367650 10ndash50 200ndash350 Crystal E

10ndash50 Crystal F50ndash250 Squeak15ndash5000 Cicadas

(continued )

59Fischman

Table 2 (Continued)

Section Shell Start TimeDuration

(sec) Grain Duration (msec) Source Audio Files

5F 5407408 457254 100ndash5000 Waves12000ndash8000 PBdLow100ndash300 4000ndash8000 PBg1f5ndash25 40ndash150 150ndash200 Motorway1

7S 6264662 59850 5ndash25 MonoC110ndash200 Uba5ndash25 90ndash700 200ndash700 UlulowL

9 6P 6324512 171342 30ndash60 DieFly20ndash80 100ndash250 Welder5ndash1000 BZZZ

5D 6495854 258210 200ndash350 1000ndash4500 Crystal E20ndash150 Crystal F10ndash50 Crystal G50ndash250 Squeak1

4F 7154064 304038 100ndash5000 Waves11000ndash4000 PBdLow100ndash300 4000ndash8000 PBg1f100ndash10000 150ndash200 Motorway1

X 7458102 100006S 7468102 37962 5ndash25 MonoC1

10ndash160 10ndash200 Uba100ndash500 UlulowL

10 5P 7506064 105678 10ndash60 30ndash60 DieFly30ndash60 Welder5ndash1000 10ndash60 BZZZ

X 8011742 315524D 8043294 148770 200ndash350 100ndash150 Crystal E

20ndash60 50ndash250 Squeak150ndash150 Crystal B to Squeak

5S 8192064 25650 5ndash25 MonoC110ndash160 Uba

X 8217714 80546

11 4P 8298260 68742 5ndash60 10ndash30 DieFly5ndash60 BZZZ

3D 8367002 87210 80ndash450 200ndash350 Crystal E40ndash400 50ndash250 100ndash350 Squeak1

X 8454212 002684S 8454480 13338 5ndash25 5ndash100 MonoC1X 8467818 36696

12 3P 8504514 31806 75ndash20 75ndash75 DieFly3S 8536320 07866 5ndash25 5ndash50 MonoC1X 8544186 46046

13 2P 8590232 15390 5ndash50 DieFly2S 9005622 02394 15ndash50 MonoC1X 9008016 45972

14 1S 9053988 01026 5ndash25 MonoC110ndash200 70ndash200 Uba

X 9055014 61339

END 9116353

60 Computer Music Journal

available online at wwwkeeleacukdeptsmustaffAlAl_softwarehtm The programs are alsodistributed with the Composerrsquos Desktop Project

AL

AL is a multiple document interface (MDI) applica-tion with frame windows that present views of dif-ferent documents andor several views of the samedocument It is designed for the creation and ma-nipulation of sonic events and their organizationinto a structured musical work

To aid user familiarity with the software and pro-mote a reasonably short learning curve care wastaken to develop an interface within an accepted

general standard for Windows applications includ-ing a menu bar a tool bar common window com-ponents and typical mouse lsquolsquodrag-and-droprsquorsquooperations It also contains detailed online helppages including a tutorial Figure 9 shows an in-stance of the user interface

Another important consideration in the design ofAL was the desire to provide a reasonably open sys-tem with the option for development of third-partyplug-ins thus avoiding the constraints imposed bya single type of musical process Future develop-ments by the author and others will ideally imple-ment a variety of compositional approaches inaddition to the application of Schrodingerrsquos Equa-tion in the existing plug-in (see the description ofERWIN below)

Figure 9 The AL user in-terface including themenu bar tool bar and aview of a document

55Fischman

7S 5F 6D 7P

6S 4F 5D 6P

5S 4D 5P

4S 4P

3S

2S 2P

1SSection 2

Section 1

Section 3

Section 4

Section 5

Section 6

Section 7

Section Duration

We can follow the path set by the rst observationabove (that sections become longer by virtue of thenumber of subsections they include) and establish afurther principle of growth related to the periodictable the minimum duration of each subsectioncloud is proportional to the sum of atomic numbersof its related shell by setting a correspondence be-tween a chosen time unit and atomic numbers asfollows

SectionDuration

4 atomic_numbers 2 time_unit (2)o1 2For example if the time unit is 005 sec the

minimum duration of the subsection correspondingto shell 2P (atomic numbers 5 to 10) will be

(5 ` 6 ` 7 ` 8 ` 9 ` 10) 2 005 4 225 sec

Applying this principle we obtain a further re- nement of the pyramidal shape whereby a newappearance of a shell is normally longer than itsprevious appearances Table 1 shows the minimumduration values for shell D with atomic unit dura-tion of 00342 sec and Figure 7 shows a schematicversion of the re ned 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 becauseaccording to quantum mechanics an electron re-mains in a given state as long as the energypumped into (or extracted from) it is equal to or

greater than the transition energy required to moveto an immediate neighboring state as calculated byequation A8

Section Differentiation

Identi cation of distinctive threads according toshells requires that sections be differentiated bytheir content This can be achieved by using differ-ent source sounds in each shell This will also in- uence overall cloud and individual graincharacteristics because as explained above manyof these are derived from the attributesmdashparticu-larly the envelopemdashof the source sound Anotherway sections may be differentiated is by setting dif-ferent grain attributes (eg different durations en-velopes spatial locations or by using differentdistributions to generate amplitudes and durations)

Extensions of the Pyramidal Shape

Two other structural shapes can be derived fromthe pyramid an inverted pyramid and a diamondThe former is analogous to a survey of the periodictable from the highest to the lowest atomic num-ber The latter consists of an original pyramid fol-

Figure 7 Schematic viewof pyramidal structurewhen the minimum dura-tion of each subsection isproportional to the sum ofatomic numbers of its re-lated shell (not to scale)

Table 1 Minimum duration values for shell Dwith atomic unit duration of 00342 sec

Shell Duration (sec)

3D 872104D 1487705D 2582106D 367650

56 Computer Music Journal

(a) (b)

lowed by of an inverted one These are shownschematically in Figure 8

Other Structural Possibilities

Apart from the simple trajectories related 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 beordered according to stochastic algorithms or gener-ative rules Alternatively they may follow attri-butes of a source sound such as the envelope wherethe peak may correspond to the maximum atomicnumber All these are options beyond the scope ofthe current research plan 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 (or dissipation) According to 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

1 1 E`SectionDuration 4 abs 1 2 (3)a U b 1 22 2n n Ka b E

where KE is the rate of energy accumulation (or dis-sipation) expressed as a fraction of the ionizationenergy E` per second As an illustration assumingthat KE 4 001 E` and that shell 3D progresses toshell 4P then the duration of the former is

3DSectionDuration3D U 4P

1 1 E`4 abs 1 2 4 486 sec1 22 23 4 001E`

The problem with this method is that energytransitions at lower shells are signi cantly higherthan those at higher shells For instance the transi-tion from 1S to 2S requires an energy value overone hundred times higher than a transition from6D to 7P Consequently the duration of 1S is inthe same proportion to that of 6D yielding ex-tremely long durations for 1S or extremely shortones for 6D This may be overcome by subdividinglonger sections according to proportions similar tothose dividing the whole work Alternatively therate of energy ow could correspond to 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 ow In general equationA10 may be modi ed to cater for any energy owfunction KE(t)

Section_durationa U b 1 1K (t)dt 4 abs 1 2 E (4)E ` 1 22 20 n na b

Erwinrsquos Playground

Erwinrsquos Playground is a work for tape lasting 9 min12 sec Except for the recording of 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 Erwinrsquos Playground is intendedto demonstrate the validity and viability of thesetheoretical principles in the generation of a musical

Figure 8 (a) Inverted pyra-midal structure and (b) di-amond structure

57Fischman

work as well as the usefulness of the software as aconduit for the realization of these principlesHowever its aesthetic value is intended to be inde-pendent of the generating principles because notheory is capable of 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 ErwinSchrodingerrsquos imaginary eld of action namely theinner shells of the atom Its structure is modeledon the diamond shape described above and linkedto a survey that begins in the lowest energy stateleaps up until it reaches the highest shell and thenreturns back to the initial state This may also becompared to an excursion through the periodic ta-ble that ascends and then descends according toatomic number order

Four different types of sound sources were as-signed to each shell to achieve section differentia-tion The S shells consisted primarily of vocalsounds (audio les MonoC1 Uba UlulowL) Pshells were assigned three different types of soundwhich nevertheless share some common spectralcharacteristics the lsquolsquobuzzrsquorsquo of a y a vocal lsquolsquobuzzrsquorsquoand the sound of a sparking welder (audio lesDieFly BZZZ Welder)

D shells were assigned sounds that ranged frompitched to lsquolsquosqueakyrsquorsquo The pitched sounds were theresult of rubbing the edge of a glass cup with differ-ent amounts of liquid to obtain different pitches(audio les Crystal E Crystal F Crystal G )Squeaks were recorded from the un-oiled wheels ofa trolley (audio le Squeak1) A sample containinga transition from pitch to squeak was achieved byrubbing the glass cup with different degrees of pres-sure (audio le Crystal B to Squeak) Finally a re-cording of the pitched sound of cicadas was alsoincluded toward the middle of the piece (audio leCicadas)

F shells were normally associated with local andglobal climaxes therefore they spread through awider bandwidth by means of three spectral typesThe rst type consists of two noise-based soundsobtained from a busy motorway and sea waves gur-gling through rocks (audio les Motorway1 andWaves1) The second sound is metallic consistingof the recording of a frying pan that was hit or

rubbed against other objects (audio le PBg1f) Thethird was mainly used to create low-frequency tex-tures and drones from the recording of a piece ofplywood that was bent at a more or less regularrate (audio le PBdLow) (Examples of the source les as well as passages characterizing each shellsand the complete work are available on the forth-coming Computer Music Journal volume 27 soundexample disc)

Grain duration was another signi cant attributeused to characterize the various shells Finally theatomic number duration equivalent used in thepiece was 00342 sec per atomic number

Table 2 presents a complete list of the structuralhierarchy of Erwinrsquos Playground including starttime minimum duration grain duration limitsand the source audio les for each subsection Asmentioned above the duration of subsections couldbe extended to allow some freedom when takinginto account musical context Section extensionsappear below the shell name with the letter lsquolsquoXrsquorsquoAlso note the bold line denoting the end of the pyr-amid and the beginning of its inversion which actsas an lsquolsquoaxis of re ectionrsquorsquo It is worth noting thatthe second half of the piece (sections 8ndash14 corre-sponding to the inverted pyramid) is more com-pressed than the rst part as there are fewersubsections that have their minimum durationsextended

Software Implementation

The software implementation of this procedureconsists of two separate components AL a compo-sitional environment for the creation and manipu-lation of musical events and ERWIN a plug-inthat implements the application of Schrodingerrsquosmodel described above These were developed andtested using Visual C ` ` 60 running under Win-dows 2000 and later under Windows XP Both ap-plications use Microsoft DirectX 8 features andrequire that DirectX 8 be installed in the host com-puter For this reason the AL and ERWIN installa-tion program also installs DirectX 8 if requiredCurrently AL and ERWIN are distributed as freesoftware under the GNU public license and are

58 Computer Music Journal

Table 2 Structural hierarchy list of Erwinrsquos Playground (lsquolsquoXrsquorsquo denotes section extensions)

Section Shell Start TimeDuration

(sec) Grain Duration (msec) Source Audio Files

1 1S 0000000 01026 5ndash25 MonoC1X 0001026 37258

2 2S 0037384 02394 5ndash50 MonoC12P 0039778 15390 5ndash50 DieFlyX 0055168 21572

3 3S 0076740 07866 5ndash503P 0084606 31806 75ndash75 75ndash20 DieFly

4 4S 0116412 13338 5ndash100 MonoC13D 0129750 87210 10ndash60 30ndash60 Crystal E4P 0216960 68742 5ndash60 DieFlyX 0285702 31777

5 5S 0317479 25650 5ndash25 MonoC110ndash160 Uba

4D 0343129 148770 10ndash150 100ndash150 Crystal E5P 0491899 105678 10ndash60 30ndash60 DieFly

30ndash60 WelderX 0597577 05818

6 6S 1003395 37962 5ndash25 MonoC110ndash160 Uba100ndash500 UlulowL

X 1041357 295954F 1070952 304038 100ndash5000 Waves1

1000ndash4000 2000ndash6000 PBdLow100ndash300 4000ndash8000 PBg1f

X 1374990 265995D 1401589 258210 1000ndash4500 Crystal E

10ndash50 10ndash4000 20ndash150 Crystal F10ndash50 Crystal G30ndash50 Squeak1

X 2059799 084946P 2068293 171342 30ndash60 DieFly

20ndash80 20ndash150 Welder5ndash1000 BZZZ

X 2239635 50250

7 7S 2289885 59850 5ndash25 MonoC110ndash200 Uba5ndash25 200ndash700 Ululow

X 2349735 434425F 2393177 457254 200ndash8000 Waves1

1000ndash4000 2000ndash5000 2000ndash6000 PBdLow100ndash300 4000ndash8000 PBg1f5ndash25 40ndash150 Motorway1

X 2250431 1051106D 3355541 367650 200ndash350 Crystal E

10ndash50 10ndash4000 20ndash150 Crystal F10ndash50 Crystal G50ndash150 50ndash250 Squeak1

X 4123191 304657P 4153656 237006 75ndash20 75ndash75 10ndash60 30ndash60 DieFly

30ndash60 WelderX 4390662 12090

8 7P 4402752 237006 75ndash20 20ndash40 DieFly6D 5039758 367650 10ndash50 200ndash350 Crystal E

10ndash50 Crystal F50ndash250 Squeak15ndash5000 Cicadas

(continued )

59Fischman

Table 2 (Continued)

Section Shell Start TimeDuration

(sec) Grain Duration (msec) Source Audio Files

5F 5407408 457254 100ndash5000 Waves12000ndash8000 PBdLow100ndash300 4000ndash8000 PBg1f5ndash25 40ndash150 150ndash200 Motorway1

7S 6264662 59850 5ndash25 MonoC110ndash200 Uba5ndash25 90ndash700 200ndash700 UlulowL

9 6P 6324512 171342 30ndash60 DieFly20ndash80 100ndash250 Welder5ndash1000 BZZZ

5D 6495854 258210 200ndash350 1000ndash4500 Crystal E20ndash150 Crystal F10ndash50 Crystal G50ndash250 Squeak1

4F 7154064 304038 100ndash5000 Waves11000ndash4000 PBdLow100ndash300 4000ndash8000 PBg1f100ndash10000 150ndash200 Motorway1

X 7458102 100006S 7468102 37962 5ndash25 MonoC1

10ndash160 10ndash200 Uba100ndash500 UlulowL

10 5P 7506064 105678 10ndash60 30ndash60 DieFly30ndash60 Welder5ndash1000 10ndash60 BZZZ

X 8011742 315524D 8043294 148770 200ndash350 100ndash150 Crystal E

20ndash60 50ndash250 Squeak150ndash150 Crystal B to Squeak

5S 8192064 25650 5ndash25 MonoC110ndash160 Uba

X 8217714 80546

11 4P 8298260 68742 5ndash60 10ndash30 DieFly5ndash60 BZZZ

3D 8367002 87210 80ndash450 200ndash350 Crystal E40ndash400 50ndash250 100ndash350 Squeak1

X 8454212 002684S 8454480 13338 5ndash25 5ndash100 MonoC1X 8467818 36696

12 3P 8504514 31806 75ndash20 75ndash75 DieFly3S 8536320 07866 5ndash25 5ndash50 MonoC1X 8544186 46046

13 2P 8590232 15390 5ndash50 DieFly2S 9005622 02394 15ndash50 MonoC1X 9008016 45972

14 1S 9053988 01026 5ndash25 MonoC110ndash200 70ndash200 Uba

X 9055014 61339

END 9116353

60 Computer Music Journal

available online at wwwkeeleacukdeptsmustaffAlAl_softwarehtm The programs are alsodistributed with the Composerrsquos Desktop Project

AL

AL is a multiple document interface (MDI) applica-tion with frame windows that present views of dif-ferent documents andor several views of the samedocument It is designed for the creation and ma-nipulation of sonic events and their organizationinto a structured musical work

To aid user familiarity with the software and pro-mote a reasonably short learning curve care wastaken to develop an interface within an accepted

general standard for Windows applications includ-ing a menu bar a tool bar common window com-ponents and typical mouse lsquolsquodrag-and-droprsquorsquooperations It also contains detailed online helppages including a tutorial Figure 9 shows an in-stance of the user interface

Another important consideration in the design ofAL was the desire to provide a reasonably open sys-tem with the option for development of third-partyplug-ins thus avoiding the constraints imposed bya single type of musical process Future develop-ments by the author and others will ideally imple-ment a variety of compositional approaches inaddition to the application of Schrodingerrsquos Equa-tion in the existing plug-in (see the description ofERWIN below)

Figure 9 The AL user in-terface including themenu bar tool bar and aview of a document

56 Computer Music Journal

(a) (b)

lowed by of an inverted one These are shownschematically in Figure 8

Other Structural Possibilities

Apart from the simple trajectories related 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 beordered according to stochastic algorithms or gener-ative rules Alternatively they may follow attri-butes of a source sound such as the envelope wherethe peak may correspond to the maximum atomicnumber All these are options beyond the scope ofthe current research plan 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 (or dissipation) According to 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

1 1 E`SectionDuration 4 abs 1 2 (3)a U b 1 22 2n n Ka b E

where KE is the rate of energy accumulation (or dis-sipation) expressed as a fraction of the ionizationenergy E` per second As an illustration assumingthat KE 4 001 E` and that shell 3D progresses toshell 4P then the duration of the former is

3DSectionDuration3D U 4P

1 1 E`4 abs 1 2 4 486 sec1 22 23 4 001E`

The problem with this method is that energytransitions at lower shells are signi cantly higherthan those at higher shells For instance the transi-tion from 1S to 2S requires an energy value overone hundred times higher than a transition from6D to 7P Consequently the duration of 1S is inthe same proportion to that of 6D yielding ex-tremely long durations for 1S or extremely shortones for 6D This may be overcome by subdividinglonger sections according to proportions similar tothose dividing the whole work Alternatively therate of energy ow could correspond to 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 ow In general equationA10 may be modi ed to cater for any energy owfunction KE(t)

Section_durationa U b 1 1K (t)dt 4 abs 1 2 E (4)E ` 1 22 20 n na b

Erwinrsquos Playground

Erwinrsquos Playground is a work for tape lasting 9 min12 sec Except for the recording of 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 Erwinrsquos Playground is intendedto demonstrate the validity and viability of thesetheoretical principles in the generation of a musical

Figure 8 (a) Inverted pyra-midal structure and (b) di-amond structure

57Fischman

work as well as the usefulness of the software as aconduit for the realization of these principlesHowever its aesthetic value is intended to be inde-pendent of the generating principles because notheory is capable of 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 ErwinSchrodingerrsquos imaginary eld of action namely theinner shells of the atom Its structure is modeledon the diamond shape described above and linkedto a survey that begins in the lowest energy stateleaps up until it reaches the highest shell and thenreturns back to the initial state This may also becompared to an excursion through the periodic ta-ble that ascends and then descends according toatomic number order

Four different types of sound sources were as-signed to each shell to achieve section differentia-tion The S shells consisted primarily of vocalsounds (audio les MonoC1 Uba UlulowL) Pshells were assigned three different types of soundwhich nevertheless share some common spectralcharacteristics the lsquolsquobuzzrsquorsquo of a y a vocal lsquolsquobuzzrsquorsquoand the sound of a sparking welder (audio lesDieFly BZZZ Welder)

D shells were assigned sounds that ranged frompitched to lsquolsquosqueakyrsquorsquo The pitched sounds were theresult of rubbing the edge of a glass cup with differ-ent amounts of liquid to obtain different pitches(audio les Crystal E Crystal F Crystal G )Squeaks were recorded from the un-oiled wheels ofa trolley (audio le Squeak1) A sample containinga transition from pitch to squeak was achieved byrubbing the glass cup with different degrees of pres-sure (audio le Crystal B to Squeak) Finally a re-cording of the pitched sound of cicadas was alsoincluded toward the middle of the piece (audio leCicadas)

F shells were normally associated with local andglobal climaxes therefore they spread through awider bandwidth by means of three spectral typesThe rst type consists of two noise-based soundsobtained from a busy motorway and sea waves gur-gling through rocks (audio les Motorway1 andWaves1) The second sound is metallic consistingof the recording of a frying pan that was hit or

rubbed against other objects (audio le PBg1f) Thethird was mainly used to create low-frequency tex-tures and drones from the recording of a piece ofplywood that was bent at a more or less regularrate (audio le PBdLow) (Examples of the source les as well as passages characterizing each shellsand the complete work are available on the forth-coming Computer Music Journal volume 27 soundexample disc)

Grain duration was another signi cant attributeused to characterize the various shells Finally theatomic number duration equivalent used in thepiece was 00342 sec per atomic number

Table 2 presents a complete list of the structuralhierarchy of Erwinrsquos Playground including starttime minimum duration grain duration limitsand the source audio les for each subsection Asmentioned above the duration of subsections couldbe extended to allow some freedom when takinginto account musical context Section extensionsappear below the shell name with the letter lsquolsquoXrsquorsquoAlso note the bold line denoting the end of the pyr-amid and the beginning of its inversion which actsas an lsquolsquoaxis of re ectionrsquorsquo It is worth noting thatthe second half of the piece (sections 8ndash14 corre-sponding to the inverted pyramid) is more com-pressed than the rst part as there are fewersubsections that have their minimum durationsextended

Software Implementation

The software implementation of this procedureconsists of two separate components AL a compo-sitional environment for the creation and manipu-lation of musical events and ERWIN a plug-inthat implements the application of Schrodingerrsquosmodel described above These were developed andtested using Visual C ` ` 60 running under Win-dows 2000 and later under Windows XP Both ap-plications use Microsoft DirectX 8 features andrequire that DirectX 8 be installed in the host com-puter For this reason the AL and ERWIN installa-tion program also installs DirectX 8 if requiredCurrently AL and ERWIN are distributed as freesoftware under the GNU public license and are

58 Computer Music Journal

Table 2 Structural hierarchy list of Erwinrsquos Playground (lsquolsquoXrsquorsquo denotes section extensions)

Section Shell Start TimeDuration

(sec) Grain Duration (msec) Source Audio Files

1 1S 0000000 01026 5ndash25 MonoC1X 0001026 37258

2 2S 0037384 02394 5ndash50 MonoC12P 0039778 15390 5ndash50 DieFlyX 0055168 21572

3 3S 0076740 07866 5ndash503P 0084606 31806 75ndash75 75ndash20 DieFly

4 4S 0116412 13338 5ndash100 MonoC13D 0129750 87210 10ndash60 30ndash60 Crystal E4P 0216960 68742 5ndash60 DieFlyX 0285702 31777

5 5S 0317479 25650 5ndash25 MonoC110ndash160 Uba

4D 0343129 148770 10ndash150 100ndash150 Crystal E5P 0491899 105678 10ndash60 30ndash60 DieFly

30ndash60 WelderX 0597577 05818

6 6S 1003395 37962 5ndash25 MonoC110ndash160 Uba100ndash500 UlulowL

X 1041357 295954F 1070952 304038 100ndash5000 Waves1

1000ndash4000 2000ndash6000 PBdLow100ndash300 4000ndash8000 PBg1f

X 1374990 265995D 1401589 258210 1000ndash4500 Crystal E

10ndash50 10ndash4000 20ndash150 Crystal F10ndash50 Crystal G30ndash50 Squeak1

X 2059799 084946P 2068293 171342 30ndash60 DieFly

20ndash80 20ndash150 Welder5ndash1000 BZZZ

X 2239635 50250

7 7S 2289885 59850 5ndash25 MonoC110ndash200 Uba5ndash25 200ndash700 Ululow

X 2349735 434425F 2393177 457254 200ndash8000 Waves1

1000ndash4000 2000ndash5000 2000ndash6000 PBdLow100ndash300 4000ndash8000 PBg1f5ndash25 40ndash150 Motorway1

X 2250431 1051106D 3355541 367650 200ndash350 Crystal E

10ndash50 10ndash4000 20ndash150 Crystal F10ndash50 Crystal G50ndash150 50ndash250 Squeak1

X 4123191 304657P 4153656 237006 75ndash20 75ndash75 10ndash60 30ndash60 DieFly

30ndash60 WelderX 4390662 12090

8 7P 4402752 237006 75ndash20 20ndash40 DieFly6D 5039758 367650 10ndash50 200ndash350 Crystal E

10ndash50 Crystal F50ndash250 Squeak15ndash5000 Cicadas

(continued )

59Fischman

Table 2 (Continued)

Section Shell Start TimeDuration

(sec) Grain Duration (msec) Source Audio Files

5F 5407408 457254 100ndash5000 Waves12000ndash8000 PBdLow100ndash300 4000ndash8000 PBg1f5ndash25 40ndash150 150ndash200 Motorway1

7S 6264662 59850 5ndash25 MonoC110ndash200 Uba5ndash25 90ndash700 200ndash700 UlulowL

9 6P 6324512 171342 30ndash60 DieFly20ndash80 100ndash250 Welder5ndash1000 BZZZ

5D 6495854 258210 200ndash350 1000ndash4500 Crystal E20ndash150 Crystal F10ndash50 Crystal G50ndash250 Squeak1

4F 7154064 304038 100ndash5000 Waves11000ndash4000 PBdLow100ndash300 4000ndash8000 PBg1f100ndash10000 150ndash200 Motorway1

X 7458102 100006S 7468102 37962 5ndash25 MonoC1

10ndash160 10ndash200 Uba100ndash500 UlulowL

10 5P 7506064 105678 10ndash60 30ndash60 DieFly30ndash60 Welder5ndash1000 10ndash60 BZZZ

X 8011742 315524D 8043294 148770 200ndash350 100ndash150 Crystal E

20ndash60 50ndash250 Squeak150ndash150 Crystal B to Squeak

5S 8192064 25650 5ndash25 MonoC110ndash160 Uba

X 8217714 80546

11 4P 8298260 68742 5ndash60 10ndash30 DieFly5ndash60 BZZZ

3D 8367002 87210 80ndash450 200ndash350 Crystal E40ndash400 50ndash250 100ndash350 Squeak1

X 8454212 002684S 8454480 13338 5ndash25 5ndash100 MonoC1X 8467818 36696

12 3P 8504514 31806 75ndash20 75ndash75 DieFly3S 8536320 07866 5ndash25 5ndash50 MonoC1X 8544186 46046

13 2P 8590232 15390 5ndash50 DieFly2S 9005622 02394 15ndash50 MonoC1X 9008016 45972

14 1S 9053988 01026 5ndash25 MonoC110ndash200 70ndash200 Uba

X 9055014 61339

END 9116353

60 Computer Music Journal

available online at wwwkeeleacukdeptsmustaffAlAl_softwarehtm The programs are alsodistributed with the Composerrsquos Desktop Project

AL

AL is a multiple document interface (MDI) applica-tion with frame windows that present views of dif-ferent documents andor several views of the samedocument It is designed for the creation and ma-nipulation of sonic events and their organizationinto a structured musical work

To aid user familiarity with the software and pro-mote a reasonably short learning curve care wastaken to develop an interface within an accepted

general standard for Windows applications includ-ing a menu bar a tool bar common window com-ponents and typical mouse lsquolsquodrag-and-droprsquorsquooperations It also contains detailed online helppages including a tutorial Figure 9 shows an in-stance of the user interface

Another important consideration in the design ofAL was the desire to provide a reasonably open sys-tem with the option for development of third-partyplug-ins thus avoiding the constraints imposed bya single type of musical process Future develop-ments by the author and others will ideally imple-ment a variety of compositional approaches inaddition to the application of Schrodingerrsquos Equa-tion in the existing plug-in (see the description ofERWIN below)

Figure 9 The AL user in-terface including themenu bar tool bar and aview of a document

57Fischman

work as well as the usefulness of the software as aconduit for the realization of these principlesHowever its aesthetic value is intended to be inde-pendent of the generating principles because notheory is capable of 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 ErwinSchrodingerrsquos imaginary eld of action namely theinner shells of the atom Its structure is modeledon the diamond shape described above and linkedto a survey that begins in the lowest energy stateleaps up until it reaches the highest shell and thenreturns back to the initial state This may also becompared to an excursion through the periodic ta-ble that ascends and then descends according toatomic number order

Four different types of sound sources were as-signed to each shell to achieve section differentia-tion The S shells consisted primarily of vocalsounds (audio les MonoC1 Uba UlulowL) Pshells were assigned three different types of soundwhich nevertheless share some common spectralcharacteristics the lsquolsquobuzzrsquorsquo of a y a vocal lsquolsquobuzzrsquorsquoand the sound of a sparking welder (audio lesDieFly BZZZ Welder)

D shells were assigned sounds that ranged frompitched to lsquolsquosqueakyrsquorsquo The pitched sounds were theresult of rubbing the edge of a glass cup with differ-ent amounts of liquid to obtain different pitches(audio les Crystal E Crystal F Crystal G )Squeaks were recorded from the un-oiled wheels ofa trolley (audio le Squeak1) A sample containinga transition from pitch to squeak was achieved byrubbing the glass cup with different degrees of pres-sure (audio le Crystal B to Squeak) Finally a re-cording of the pitched sound of cicadas was alsoincluded toward the middle of the piece (audio leCicadas)

F shells were normally associated with local andglobal climaxes therefore they spread through awider bandwidth by means of three spectral typesThe rst type consists of two noise-based soundsobtained from a busy motorway and sea waves gur-gling through rocks (audio les Motorway1 andWaves1) The second sound is metallic consistingof the recording of a frying pan that was hit or

rubbed against other objects (audio le PBg1f) Thethird was mainly used to create low-frequency tex-tures and drones from the recording of a piece ofplywood that was bent at a more or less regularrate (audio le PBdLow) (Examples of the source les as well as passages characterizing each shellsand the complete work are available on the forth-coming Computer Music Journal volume 27 soundexample disc)

Grain duration was another signi cant attributeused to characterize the various shells Finally theatomic number duration equivalent used in thepiece was 00342 sec per atomic number

Table 2 presents a complete list of the structuralhierarchy of Erwinrsquos Playground including starttime minimum duration grain duration limitsand the source audio les for each subsection Asmentioned above the duration of subsections couldbe extended to allow some freedom when takinginto account musical context Section extensionsappear below the shell name with the letter lsquolsquoXrsquorsquoAlso note the bold line denoting the end of the pyr-amid and the beginning of its inversion which actsas an lsquolsquoaxis of re ectionrsquorsquo It is worth noting thatthe second half of the piece (sections 8ndash14 corre-sponding to the inverted pyramid) is more com-pressed than the rst part as there are fewersubsections that have their minimum durationsextended

Software Implementation

The software implementation of this procedureconsists of two separate components AL a compo-sitional environment for the creation and manipu-lation of musical events and ERWIN a plug-inthat implements the application of Schrodingerrsquosmodel described above These were developed andtested using Visual C ` ` 60 running under Win-dows 2000 and later under Windows XP Both ap-plications use Microsoft DirectX 8 features andrequire that DirectX 8 be installed in the host com-puter For this reason the AL and ERWIN installa-tion program also installs DirectX 8 if requiredCurrently AL and ERWIN are distributed as freesoftware under the GNU public license and are

58 Computer Music Journal

Table 2 Structural hierarchy list of Erwinrsquos Playground (lsquolsquoXrsquorsquo denotes section extensions)

Section Shell Start TimeDuration

(sec) Grain Duration (msec) Source Audio Files

1 1S 0000000 01026 5ndash25 MonoC1X 0001026 37258

2 2S 0037384 02394 5ndash50 MonoC12P 0039778 15390 5ndash50 DieFlyX 0055168 21572

3 3S 0076740 07866 5ndash503P 0084606 31806 75ndash75 75ndash20 DieFly

4 4S 0116412 13338 5ndash100 MonoC13D 0129750 87210 10ndash60 30ndash60 Crystal E4P 0216960 68742 5ndash60 DieFlyX 0285702 31777

5 5S 0317479 25650 5ndash25 MonoC110ndash160 Uba

4D 0343129 148770 10ndash150 100ndash150 Crystal E5P 0491899 105678 10ndash60 30ndash60 DieFly

30ndash60 WelderX 0597577 05818

6 6S 1003395 37962 5ndash25 MonoC110ndash160 Uba100ndash500 UlulowL

X 1041357 295954F 1070952 304038 100ndash5000 Waves1

1000ndash4000 2000ndash6000 PBdLow100ndash300 4000ndash8000 PBg1f

X 1374990 265995D 1401589 258210 1000ndash4500 Crystal E

10ndash50 10ndash4000 20ndash150 Crystal F10ndash50 Crystal G30ndash50 Squeak1

X 2059799 084946P 2068293 171342 30ndash60 DieFly

20ndash80 20ndash150 Welder5ndash1000 BZZZ

X 2239635 50250

7 7S 2289885 59850 5ndash25 MonoC110ndash200 Uba5ndash25 200ndash700 Ululow

X 2349735 434425F 2393177 457254 200ndash8000 Waves1

1000ndash4000 2000ndash5000 2000ndash6000 PBdLow100ndash300 4000ndash8000 PBg1f5ndash25 40ndash150 Motorway1

X 2250431 1051106D 3355541 367650 200ndash350 Crystal E

10ndash50 10ndash4000 20ndash150 Crystal F10ndash50 Crystal G50ndash150 50ndash250 Squeak1

X 4123191 304657P 4153656 237006 75ndash20 75ndash75 10ndash60 30ndash60 DieFly

30ndash60 WelderX 4390662 12090

8 7P 4402752 237006 75ndash20 20ndash40 DieFly6D 5039758 367650 10ndash50 200ndash350 Crystal E

10ndash50 Crystal F50ndash250 Squeak15ndash5000 Cicadas

(continued )

59Fischman

Table 2 (Continued)

Section Shell Start TimeDuration

(sec) Grain Duration (msec) Source Audio Files

5F 5407408 457254 100ndash5000 Waves12000ndash8000 PBdLow100ndash300 4000ndash8000 PBg1f5ndash25 40ndash150 150ndash200 Motorway1

7S 6264662 59850 5ndash25 MonoC110ndash200 Uba5ndash25 90ndash700 200ndash700 UlulowL

9 6P 6324512 171342 30ndash60 DieFly20ndash80 100ndash250 Welder5ndash1000 BZZZ

5D 6495854 258210 200ndash350 1000ndash4500 Crystal E20ndash150 Crystal F10ndash50 Crystal G50ndash250 Squeak1

4F 7154064 304038 100ndash5000 Waves11000ndash4000 PBdLow100ndash300 4000ndash8000 PBg1f100ndash10000 150ndash200 Motorway1

X 7458102 100006S 7468102 37962 5ndash25 MonoC1

10ndash160 10ndash200 Uba100ndash500 UlulowL

10 5P 7506064 105678 10ndash60 30ndash60 DieFly30ndash60 Welder5ndash1000 10ndash60 BZZZ

X 8011742 315524D 8043294 148770 200ndash350 100ndash150 Crystal E

20ndash60 50ndash250 Squeak150ndash150 Crystal B to Squeak

5S 8192064 25650 5ndash25 MonoC110ndash160 Uba

X 8217714 80546

11 4P 8298260 68742 5ndash60 10ndash30 DieFly5ndash60 BZZZ

3D 8367002 87210 80ndash450 200ndash350 Crystal E40ndash400 50ndash250 100ndash350 Squeak1

X 8454212 002684S 8454480 13338 5ndash25 5ndash100 MonoC1X 8467818 36696

12 3P 8504514 31806 75ndash20 75ndash75 DieFly3S 8536320 07866 5ndash25 5ndash50 MonoC1X 8544186 46046

13 2P 8590232 15390 5ndash50 DieFly2S 9005622 02394 15ndash50 MonoC1X 9008016 45972

14 1S 9053988 01026 5ndash25 MonoC110ndash200 70ndash200 Uba

X 9055014 61339

END 9116353

60 Computer Music Journal

available online at wwwkeeleacukdeptsmustaffAlAl_softwarehtm The programs are alsodistributed with the Composerrsquos Desktop Project

AL

AL is a multiple document interface (MDI) applica-tion with frame windows that present views of dif-ferent documents andor several views of the samedocument It is designed for the creation and ma-nipulation of sonic events and their organizationinto a structured musical work

To aid user familiarity with the software and pro-mote a reasonably short learning curve care wastaken to develop an interface within an accepted

general standard for Windows applications includ-ing a menu bar a tool bar common window com-ponents and typical mouse lsquolsquodrag-and-droprsquorsquooperations It also contains detailed online helppages including a tutorial Figure 9 shows an in-stance of the user interface

Another important consideration in the design ofAL was the desire to provide a reasonably open sys-tem with the option for development of third-partyplug-ins thus avoiding the constraints imposed bya single type of musical process Future develop-ments by the author and others will ideally imple-ment a variety of compositional approaches inaddition to the application of Schrodingerrsquos Equa-tion in the existing plug-in (see the description ofERWIN below)

Figure 9 The AL user in-terface including themenu bar tool bar and aview of a document

58 Computer Music Journal

Table 2 Structural hierarchy list of Erwinrsquos Playground (lsquolsquoXrsquorsquo denotes section extensions)

Section Shell Start TimeDuration

(sec) Grain Duration (msec) Source Audio Files

1 1S 0000000 01026 5ndash25 MonoC1X 0001026 37258

2 2S 0037384 02394 5ndash50 MonoC12P 0039778 15390 5ndash50 DieFlyX 0055168 21572

3 3S 0076740 07866 5ndash503P 0084606 31806 75ndash75 75ndash20 DieFly

4 4S 0116412 13338 5ndash100 MonoC13D 0129750 87210 10ndash60 30ndash60 Crystal E4P 0216960 68742 5ndash60 DieFlyX 0285702 31777

5 5S 0317479 25650 5ndash25 MonoC110ndash160 Uba

4D 0343129 148770 10ndash150 100ndash150 Crystal E5P 0491899 105678 10ndash60 30ndash60 DieFly

30ndash60 WelderX 0597577 05818

6 6S 1003395 37962 5ndash25 MonoC110ndash160 Uba100ndash500 UlulowL

X 1041357 295954F 1070952 304038 100ndash5000 Waves1

1000ndash4000 2000ndash6000 PBdLow100ndash300 4000ndash8000 PBg1f

X 1374990 265995D 1401589 258210 1000ndash4500 Crystal E

10ndash50 10ndash4000 20ndash150 Crystal F10ndash50 Crystal G30ndash50 Squeak1

X 2059799 084946P 2068293 171342 30ndash60 DieFly

20ndash80 20ndash150 Welder5ndash1000 BZZZ

X 2239635 50250

7 7S 2289885 59850 5ndash25 MonoC110ndash200 Uba5ndash25 200ndash700 Ululow

X 2349735 434425F 2393177 457254 200ndash8000 Waves1

1000ndash4000 2000ndash5000 2000ndash6000 PBdLow100ndash300 4000ndash8000 PBg1f5ndash25 40ndash150 Motorway1

X 2250431 1051106D 3355541 367650 200ndash350 Crystal E

10ndash50 10ndash4000 20ndash150 Crystal F10ndash50 Crystal G50ndash150 50ndash250 Squeak1

X 4123191 304657P 4153656 237006 75ndash20 75ndash75 10ndash60 30ndash60 DieFly

30ndash60 WelderX 4390662 12090

8 7P 4402752 237006 75ndash20 20ndash40 DieFly6D 5039758 367650 10ndash50 200ndash350 Crystal E

10ndash50 Crystal F50ndash250 Squeak15ndash5000 Cicadas

(continued )

59Fischman

Table 2 (Continued)

Section Shell Start TimeDuration

(sec) Grain Duration (msec) Source Audio Files

5F 5407408 457254 100ndash5000 Waves12000ndash8000 PBdLow100ndash300 4000ndash8000 PBg1f5ndash25 40ndash150 150ndash200 Motorway1

7S 6264662 59850 5ndash25 MonoC110ndash200 Uba5ndash25 90ndash700 200ndash700 UlulowL

9 6P 6324512 171342 30ndash60 DieFly20ndash80 100ndash250 Welder5ndash1000 BZZZ

5D 6495854 258210 200ndash350 1000ndash4500 Crystal E20ndash150 Crystal F10ndash50 Crystal G50ndash250 Squeak1

4F 7154064 304038 100ndash5000 Waves11000ndash4000 PBdLow100ndash300 4000ndash8000 PBg1f100ndash10000 150ndash200 Motorway1

X 7458102 100006S 7468102 37962 5ndash25 MonoC1

10ndash160 10ndash200 Uba100ndash500 UlulowL

10 5P 7506064 105678 10ndash60 30ndash60 DieFly30ndash60 Welder5ndash1000 10ndash60 BZZZ

X 8011742 315524D 8043294 148770 200ndash350 100ndash150 Crystal E

20ndash60 50ndash250 Squeak150ndash150 Crystal B to Squeak

5S 8192064 25650 5ndash25 MonoC110ndash160 Uba

X 8217714 80546

11 4P 8298260 68742 5ndash60 10ndash30 DieFly5ndash60 BZZZ

3D 8367002 87210 80ndash450 200ndash350 Crystal E40ndash400 50ndash250 100ndash350 Squeak1

X 8454212 002684S 8454480 13338 5ndash25 5ndash100 MonoC1X 8467818 36696

12 3P 8504514 31806 75ndash20 75ndash75 DieFly3S 8536320 07866 5ndash25 5ndash50 MonoC1X 8544186 46046

13 2P 8590232 15390 5ndash50 DieFly2S 9005622 02394 15ndash50 MonoC1X 9008016 45972

14 1S 9053988 01026 5ndash25 MonoC110ndash200 70ndash200 Uba

X 9055014 61339

END 9116353

60 Computer Music Journal

available online at wwwkeeleacukdeptsmustaffAlAl_softwarehtm The programs are alsodistributed with the Composerrsquos Desktop Project

AL

AL is a multiple document interface (MDI) applica-tion with frame windows that present views of dif-ferent documents andor several views of the samedocument It is designed for the creation and ma-nipulation of sonic events and their organizationinto a structured musical work

To aid user familiarity with the software and pro-mote a reasonably short learning curve care wastaken to develop an interface within an accepted

general standard for Windows applications includ-ing a menu bar a tool bar common window com-ponents and typical mouse lsquolsquodrag-and-droprsquorsquooperations It also contains detailed online helppages including a tutorial Figure 9 shows an in-stance of the user interface

Another important consideration in the design ofAL was the desire to provide a reasonably open sys-tem with the option for development of third-partyplug-ins thus avoiding the constraints imposed bya single type of musical process Future develop-ments by the author and others will ideally imple-ment a variety of compositional approaches inaddition to the application of Schrodingerrsquos Equa-tion in the existing plug-in (see the description ofERWIN below)

Figure 9 The AL user in-terface including themenu bar tool bar and aview of a document

59Fischman

Table 2 (Continued)

Section Shell Start TimeDuration

(sec) Grain Duration (msec) Source Audio Files

5F 5407408 457254 100ndash5000 Waves12000ndash8000 PBdLow100ndash300 4000ndash8000 PBg1f5ndash25 40ndash150 150ndash200 Motorway1

7S 6264662 59850 5ndash25 MonoC110ndash200 Uba5ndash25 90ndash700 200ndash700 UlulowL

9 6P 6324512 171342 30ndash60 DieFly20ndash80 100ndash250 Welder5ndash1000 BZZZ

5D 6495854 258210 200ndash350 1000ndash4500 Crystal E20ndash150 Crystal F10ndash50 Crystal G50ndash250 Squeak1

4F 7154064 304038 100ndash5000 Waves11000ndash4000 PBdLow100ndash300 4000ndash8000 PBg1f100ndash10000 150ndash200 Motorway1

X 7458102 100006S 7468102 37962 5ndash25 MonoC1

10ndash160 10ndash200 Uba100ndash500 UlulowL

10 5P 7506064 105678 10ndash60 30ndash60 DieFly30ndash60 Welder5ndash1000 10ndash60 BZZZ

X 8011742 315524D 8043294 148770 200ndash350 100ndash150 Crystal E

20ndash60 50ndash250 Squeak150ndash150 Crystal B to Squeak

5S 8192064 25650 5ndash25 MonoC110ndash160 Uba

X 8217714 80546

11 4P 8298260 68742 5ndash60 10ndash30 DieFly5ndash60 BZZZ

3D 8367002 87210 80ndash450 200ndash350 Crystal E40ndash400 50ndash250 100ndash350 Squeak1

X 8454212 002684S 8454480 13338 5ndash25 5ndash100 MonoC1X 8467818 36696

12 3P 8504514 31806 75ndash20 75ndash75 DieFly3S 8536320 07866 5ndash25 5ndash50 MonoC1X 8544186 46046

13 2P 8590232 15390 5ndash50 DieFly2S 9005622 02394 15ndash50 MonoC1X 9008016 45972

14 1S 9053988 01026 5ndash25 MonoC110ndash200 70ndash200 Uba

X 9055014 61339

END 9116353

60 Computer Music Journal

available online at wwwkeeleacukdeptsmustaffAlAl_softwarehtm The programs are alsodistributed with the Composerrsquos Desktop Project

AL

AL is a multiple document interface (MDI) applica-tion with frame windows that present views of dif-ferent documents andor several views of the samedocument It is designed for the creation and ma-nipulation of sonic events and their organizationinto a structured musical work

To aid user familiarity with the software and pro-mote a reasonably short learning curve care wastaken to develop an interface within an accepted

general standard for Windows applications includ-ing a menu bar a tool bar common window com-ponents and typical mouse lsquolsquodrag-and-droprsquorsquooperations It also contains detailed online helppages including a tutorial Figure 9 shows an in-stance of the user interface

Another important consideration in the design ofAL was the desire to provide a reasonably open sys-tem with the option for development of third-partyplug-ins thus avoiding the constraints imposed bya single type of musical process Future develop-ments by the author and others will ideally imple-ment a variety of compositional approaches inaddition to the application of Schrodingerrsquos Equa-tion in the existing plug-in (see the description ofERWIN below)

Figure 9 The AL user in-terface including themenu bar tool bar and aview of a document

60 Computer Music Journal

available online at wwwkeeleacukdeptsmustaffAlAl_softwarehtm The programs are alsodistributed with the Composerrsquos Desktop Project

AL

AL is a multiple document interface (MDI) applica-tion with frame windows that present views of dif-ferent documents andor several views of the samedocument It is designed for the creation and ma-nipulation of sonic events and their organizationinto a structured musical work

To aid user familiarity with the software and pro-mote a reasonably short learning curve care wastaken to develop an interface within an accepted

general standard for Windows applications includ-ing a menu bar a tool bar common window com-ponents and typical mouse lsquolsquodrag-and-droprsquorsquooperations It also contains detailed online helppages including a tutorial Figure 9 shows an in-stance of the user interface

Another important consideration in the design ofAL was the desire to provide a reasonably open sys-tem with the option for development of third-partyplug-ins thus avoiding the constraints imposed bya single type of musical process Future develop-ments by the author and others will ideally imple-ment a variety of compositional approaches inaddition to the application of Schrodingerrsquos Equa-tion in the existing plug-in (see the description ofERWIN below)

Figure 9 The AL user in-terface including themenu bar tool bar and aview of a document

61Fischman

Documents and Views

AL documents store the information necessary tode ne and characterize the composed musicalwork They consist of a set of ordered events 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 represented by rectangles on thescreen as shown in Figure 9 These rectangles maybe dragged and stretched with the mouse

Events may be empty or they may represent anactual audio le The audio le can be a pre-existing le or the result of a plug-in operation inwhich case the event stores the parameters ex-ported by the plug-in Events associated with anaudio le adopt its name and are given a three-dimensional edge in the graphical user interfaceEvents 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 performed during playback andnot included in any mix

Main Features

In addition to standard Windows features (such assaving and loading events and their generating pa-rameters zooming scrolling undo redo cuttingcopying and pasting) a number of operations canbe performed on 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 modi cationof event status including properties such as onsettime color locking and muting simple audio leoperations including association with an existingaudio le gain fade and channel swapping ofstereo events and nally plug-ins consisting of ex-ternal processes which create an audio le and as-sociate it with the event Users can also auditionan audio mix of events or bounce the latter to anaudio le

ERWIN Plug-in

This application is provided as a separate dynamic-link library (DLL) conforming to the Microsoft

Component Object Model (COM) standard It maybe called by AL through the right-button eventmenu to create a granular cloud according to thealgorithms described above The source sound forgranulation is read from a wave le and the cloudis stored in an output wave le that AL associateswith the calling event

The user can supply the necessary parameters viaproperty pages which fall into four categories The rst category consists of inputoutput and generalsettings including the name of the input le thename of the output le and parameters for enve-lope extraction grain extraction cloud density cal-culation Doppler Shift and FFT operations Thesecond category is section attributes consisting ofthe values of quantum numbers atomic numberand the minimum and maximum radii for a partic-ular atomic shell The third category includescloud attributes that is parameters for durationgrain density amplitude scaling motion and vol-ume Finally grain attributes consists of parame-ters for amplitude uctuation duration uctuationenvelope 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

ERWINrsquos graphical interface presents the resultsof the process as shown in Figure 10 including thefollowing data plots of the distributions and theirmaxima and minima a plot of the extracted inputenvelope the location and value of its maximumand the analysis window duration plots of cloudattributes including density motion and scatterand plots of the grain attributes including the en-velope and motion functions (lsquolsquoarchetypesrsquorsquo)

Future Software Development

As with any piece of software AL could be devel-oped almost ad in nitum 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-

62 Computer Music Journal

gorithmic automation of structure For instanceAL could include a menu option for plug-ins thatcreate structural hierarchies that could then be dis-played in the document view Speci c colors couldbe assigned to events in different sections andviews could display dividing lines and sectionnames Furthermore these algorithms could imple-ment structural hierarchies in addition to those de-scribed above where it has already been suggestedthat it may be possible to generate more complextrajectories with random or generative algorithmsor by following the attributes of a source sound

The generation of events could also be auto-mated using for instance random algorithms basedon Schrodingerrsquos distribution or other distributionsfor the determination of the number of eventstheir onsets and their durations Finally it mightbe possible to develop software that applies thesame principles to visual grains integrating soundswith corresponding lsquolsquoanimatedrsquorsquo clouds seen on ascreen or in a virtual reality environment

Conclusion

A work of art may be articulated at various levelsAt one level articulation takes place through a me-dium that carries with it a set of principles result-ing from its physical properties For instance theuse of stone in sculpture leads to an organization ofspace ruled by visual proportion texture and thephysical constraints resulting from gravitationalforces (such as those that determine whether astructure may collapse or not)

A second level from which properties and func-tionality are derived is more speci c to particularcultures periods or genres In the case of most mu-sic it depends on the particular aspects of sound apiece articulates For instance Western music inthe 18th and 19th centuries is structured accordingto harmonic relationships between keys (eg fugueand sonata forms)

One may nd a third level relevant to music thatconsists of an extra-musical dimension and corre-

Figure 10 The ERWINuser interface

63Fischman

sponding to the concepts of intrinsic and extrinsicreferral discussed by Nattiez (1990) In this case anadditional set of principles operates by establishinga relationship between musical articulation andnon-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 programmatic music As an exam-ple of work that is articulated at an extra-musicallevel without becoming programmatic we may citeXenakisrsquos use of parabolic trajectories for the de-sign of string glissandi in Metastasis The princi-ples described in this article and their realizationin Erwinrsquos Playground fall within this category

It is hoped that the realization of the extra-musical dimension proposed here 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 mapping theequations to musical parameters presented here isneither unique nor exhaustive It is only one of amultitude of alternatives which to the authormakes musical sense Furthermore the model de-veloped here can still be enhanced and explored forexample with improvements to the construction ofstructural hierarchies other than those presentedabove and the algorithmic generation of largerstructural units such as the number of events in asection and their individual attributes (eg onsetsdurations and granular parameters)

Also the speci c direction assumed during thisresearch may 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 applying othermathematical models (eg those describing relativ-istic phenomena heat convection behavior of uids etc) or alternative interpretations of Schro-dingerrsquos Equation may be worth investigatingFinally I hope that Erwinrsquos Playground is able tocontribute in some constructive way to the alreadyconsiderable body of compositions deriving frommathematical logic and that the principles devel-oped heremdashand their software implementationsmdashmay be of use to other composers resulting inother musical works

Acknowledgments

I am grateful to the Arts and Humanities ResearchBoard (AHRB) UK for its support in the form of aResearch Leave Award which made possible therealization of this project Special thanks to Dalitmy partner in life for her patience and moral sup-port Many thanks to my colleagues in the KeeleUniversity Music Department for their effectivecoverage of my duties during the period of researchleave Also thanks to Richard Dobson and ArcherEndrich of the Composersrsquo Desktop Project fortheir technical advice

References

Bain R 1990 lsquolsquoAlgorithmic Composition Quantum Me-chanics amp the Musical Domainrsquorsquo Proceedings of the1990 International Computer Music Conference SanFrancisco International Computer Music Associationpp 276ndash279

Beyls P E 1991 lsquolsquoSelf-Organizing Control StructuresUsing Multiple Cellular Automatarsquorsquo Proceedings ofthe 1991 International Computer Music ConferenceSan Francisco International Computer Music Associa-tion pp 254ndash257

Bidlack R 1992 lsquolsquoChaotic Systems As Simple (But Com-plex) Compositional Algorithmsrsquorsquo Computer MusicJournal 16(3)33ndash47

Bolognesi T 1983 lsquolsquoAutomatic Composition Experi-ments with Self-Similar Musicrsquorsquo Computer MusicJournal 7(1)35ndash36

De Poli G A Piccialli and C Roads 1991 Representa-tion of Musical Signals Cambridge MassachusettsMIT Press pp 137ndash186

Delatour T 2000 lsquolsquoMolecular Music The Acoustic Con-version of Molecular Vibrational Spectrarsquorsquo ComputerMusic Journal 24(3)48ndash68

Di Scipio A 1991 lsquolsquoComposition by Exploration of Non-Linear Dynamic Systemsrsquorsquo Proceedings of the 1991International Computer Music Conference SanFrancisco International Computer Music Associationpp 324ndash327

Di Scipio A 1994 lsquolsquoMicro-Time Sonic Design and Tim-bre Formationrsquorsquo Contemporary Music Review 10(2)135ndash148

Dodge C 1988 lsquolsquoPro le A Musical Fractalrsquorsquo ComputerMusic Journal 12(2)35ndash46

62 Computer Music Journal

gorithmic automation of structure For instanceAL could include a menu option for plug-ins thatcreate structural hierarchies that could then be dis-played in the document view Speci c colors couldbe assigned to events in different sections andviews could display dividing lines and sectionnames Furthermore these algorithms could imple-ment structural hierarchies in addition to those de-scribed above where it has already been suggestedthat it may be possible to generate more complextrajectories with random or generative algorithmsor by following the attributes of a source sound

The generation of events could also be auto-mated using for instance random algorithms basedon Schrodingerrsquos distribution or other distributionsfor the determination of the number of eventstheir onsets and their durations Finally it mightbe possible to develop software that applies thesame principles to visual grains integrating soundswith corresponding lsquolsquoanimatedrsquorsquo clouds seen on ascreen or in a virtual reality environment

Conclusion

A work of art may be articulated at various levelsAt one level articulation takes place through a me-dium that carries with it a set of principles result-ing from its physical properties For instance theuse of stone in sculpture leads to an organization ofspace ruled by visual proportion texture and thephysical constraints resulting from gravitationalforces (such as those that determine whether astructure may collapse or not)

A second level from which properties and func-tionality are derived is more speci c to particularcultures periods or genres In the case of most mu-sic it depends on the particular aspects of sound apiece articulates For instance Western music inthe 18th and 19th centuries is structured accordingto harmonic relationships between keys (eg fugueand sonata forms)

One may nd a third level relevant to music thatconsists of an extra-musical dimension and corre-

Figure 10 The ERWINuser interface

63Fischman

sponding to the concepts of intrinsic and extrinsicreferral discussed by Nattiez (1990) In this case anadditional set of principles operates by establishinga relationship between musical articulation andnon-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 programmatic music As an exam-ple of work that is articulated at an extra-musicallevel without becoming programmatic we may citeXenakisrsquos use of parabolic trajectories for the de-sign of string glissandi in Metastasis The princi-ples described in this article and their realizationin Erwinrsquos Playground fall within this category

It is hoped that the realization of the extra-musical dimension proposed here 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 mapping theequations to musical parameters presented here isneither unique nor exhaustive It is only one of amultitude of alternatives which to the authormakes musical sense Furthermore the model de-veloped here can still be enhanced and explored forexample with improvements to the construction ofstructural hierarchies other than those presentedabove and the algorithmic generation of largerstructural units such as the number of events in asection and their individual attributes (eg onsetsdurations and granular parameters)

Also the speci c direction assumed during thisresearch may 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 applying othermathematical models (eg those describing relativ-istic phenomena heat convection behavior of uids etc) or alternative interpretations of Schro-dingerrsquos Equation may be worth investigatingFinally I hope that Erwinrsquos Playground is able tocontribute in some constructive way to the alreadyconsiderable body of compositions deriving frommathematical logic and that the principles devel-oped heremdashand their software implementationsmdashmay be of use to other composers resulting inother musical works

Acknowledgments

I am grateful to the Arts and Humanities ResearchBoard (AHRB) UK for its support in the form of aResearch Leave Award which made possible therealization of this project Special thanks to Dalitmy partner in life for her patience and moral sup-port Many thanks to my colleagues in the KeeleUniversity Music Department for their effectivecoverage of my duties during the period of researchleave Also thanks to Richard Dobson and ArcherEndrich of the Composersrsquo Desktop Project fortheir technical advice

References

Bain R 1990 lsquolsquoAlgorithmic Composition Quantum Me-chanics amp the Musical Domainrsquorsquo Proceedings of the1990 International Computer Music Conference SanFrancisco International Computer Music Associationpp 276ndash279

Beyls P E 1991 lsquolsquoSelf-Organizing Control StructuresUsing Multiple Cellular Automatarsquorsquo Proceedings ofthe 1991 International Computer Music ConferenceSan Francisco International Computer Music Associa-tion pp 254ndash257

Bidlack R 1992 lsquolsquoChaotic Systems As Simple (But Com-plex) Compositional Algorithmsrsquorsquo Computer MusicJournal 16(3)33ndash47

Bolognesi T 1983 lsquolsquoAutomatic Composition Experi-ments with Self-Similar Musicrsquorsquo Computer MusicJournal 7(1)35ndash36

De Poli G A Piccialli and C Roads 1991 Representa-tion of Musical Signals Cambridge MassachusettsMIT Press pp 137ndash186

Delatour T 2000 lsquolsquoMolecular Music The Acoustic Con-version of Molecular Vibrational Spectrarsquorsquo ComputerMusic Journal 24(3)48ndash68

Di Scipio A 1991 lsquolsquoComposition by Exploration of Non-Linear Dynamic Systemsrsquorsquo Proceedings of the 1991International Computer Music Conference SanFrancisco International Computer Music Associationpp 324ndash327

Di Scipio A 1994 lsquolsquoMicro-Time Sonic Design and Tim-bre Formationrsquorsquo Contemporary Music Review 10(2)135ndash148

Dodge C 1988 lsquolsquoPro le A Musical Fractalrsquorsquo ComputerMusic Journal 12(2)35ndash46

63Fischman

sponding to the concepts of intrinsic and extrinsicreferral discussed by Nattiez (1990) In this case anadditional set of principles operates by establishinga relationship between musical articulation andnon-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 programmatic music As an exam-ple of work that is articulated at an extra-musicallevel without becoming programmatic we may citeXenakisrsquos use of parabolic trajectories for the de-sign of string glissandi in Metastasis The princi-ples described in this article and their realizationin Erwinrsquos Playground fall within this category

It is hoped that the realization of the extra-musical dimension proposed here 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 mapping theequations to musical parameters presented here isneither unique nor exhaustive It is only one of amultitude of alternatives which to the authormakes musical sense Furthermore the model de-veloped here can still be enhanced and explored forexample with improvements to the construction ofstructural hierarchies other than those presentedabove and the algorithmic generation of largerstructural units such as the number of events in asection and their individual attributes (eg onsetsdurations and granular parameters)

Also the speci c direction assumed during thisresearch may 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 applying othermathematical models (eg those describing relativ-istic phenomena heat convection behavior of uids etc) or alternative interpretations of Schro-dingerrsquos Equation may be worth investigatingFinally I hope that Erwinrsquos Playground is able tocontribute in some constructive way to the alreadyconsiderable body of compositions deriving frommathematical logic and that the principles devel-oped heremdashand their software implementationsmdashmay be of use to other composers resulting inother musical works

Acknowledgments

I am grateful to the Arts and Humanities ResearchBoard (AHRB) UK for its support in the form of aResearch Leave Award which made possible therealization of this project Special thanks to Dalitmy partner in life for her patience and moral sup-port Many thanks to my colleagues in the KeeleUniversity Music Department for their effectivecoverage of my duties during the period of researchleave Also thanks to Richard Dobson and ArcherEndrich of the Composersrsquo Desktop Project fortheir technical advice

References

Bain R 1990 lsquolsquoAlgorithmic Composition Quantum Me-chanics amp the Musical Domainrsquorsquo Proceedings of the1990 International Computer Music Conference SanFrancisco International Computer Music Associationpp 276ndash279

Beyls P E 1991 lsquolsquoSelf-Organizing Control StructuresUsing Multiple Cellular Automatarsquorsquo Proceedings ofthe 1991 International Computer Music ConferenceSan Francisco International Computer Music Associa-tion pp 254ndash257

Bidlack R 1992 lsquolsquoChaotic Systems As Simple (But Com-plex) Compositional Algorithmsrsquorsquo Computer MusicJournal 16(3)33ndash47

Bolognesi T 1983 lsquolsquoAutomatic Composition Experi-ments with Self-Similar Musicrsquorsquo Computer MusicJournal 7(1)35ndash36

De Poli G A Piccialli and C Roads 1991 Representa-tion of Musical Signals Cambridge MassachusettsMIT Press pp 137ndash186

Delatour T 2000 lsquolsquoMolecular Music The Acoustic Con-version of Molecular Vibrational Spectrarsquorsquo ComputerMusic Journal 24(3)48ndash68

Di Scipio A 1991 lsquolsquoComposition by Exploration of Non-Linear Dynamic Systemsrsquorsquo Proceedings of the 1991International Computer Music Conference SanFrancisco International Computer Music Associationpp 324ndash327

Di Scipio A 1994 lsquolsquoMicro-Time Sonic Design and Tim-bre Formationrsquorsquo Contemporary Music Review 10(2)135ndash148

Dodge C 1988 lsquolsquoPro le A Musical Fractalrsquorsquo ComputerMusic Journal 12(2)35ndash46

64 Computer Music Journal

Feynman R P R B Leighton and M Sands 1971 TheFeynman Lectures on Physics Volume III ReadingMassachusetts Addison-Wesley pp 191ndash1920

Fischman R 2003 lsquolsquoDerivation of Organic MusicalStructure and Materials from the Solution of Differen-tial Equationsrsquorsquo Intercultural Music 5 Point Rich-mond CA MRI Press pp 105ndash130

Hiller L 1981 lsquolsquoComposing with Computers A ProgressReportrsquorsquo Computer Music Journal 5(4)7ndash21

Hoffman P 2000 lsquolsquoThe New GENDYN Programrsquorsquo Com-puter Music Journal 24(2)31ndash38

Holzman S R 1981 lsquolsquoUsing Generative Grammars forMusic Compositionrsquorsquo Computer Music Journal5(1)51ndash64

Jones K 1981 lsquolsquoCompositional Applications of Stochas-tic Processesrsquorsquo Computer Music Journal 5(2)45ndash61

Koenig G M 1970a lsquolsquoProject 1rsquorsquo Electronic Music Re-ports 2 Utrecht Institute of Sonology (Reprinted1977 Amsterdam Swets and Zeitlinger)

Koenig G M 1970b lsquolsquoProject 2 A Program for MusicalCompositionrsquorsquo Electronic Music Reports 3 UtrechtInstitute of Sonology (Reprinted 1977 AmsterdamSwets and Zeitlinger)

Laske O 1981 lsquolsquoComposition Theory in Koenigrsquos ProjectOne and Project Tworsquorsquo Computer Music Journal5(4)54ndash65

Lorrain D 1980 lsquolsquoA Panoply of Stochastic CanonsrsquorsquoComputer Music Journal 4(1)53ndash81

Lyon D 1995 lsquolsquoUsing Stochastic Petri Nets for Real-time Nth-Order Stochastic Compositionrsquorsquo ComputerMusic Journal 19(4)13ndash22

Maia A P Do Valle J Manzolli and L N S Pereira1999 lsquolsquoA Computer Environment for Polymodal Mu-sicrsquorsquo Organised Sound 5(2)111ndash114

McAlpine K E R Miranda and H Gerard 1999 lsquolsquoMak-ing Music with Algorithms A Case-Study SystemrsquorsquoComputer Music Journal 23(2)19ndash30

Nattiez J J 1990 Music and Discourse Towards a Se-miology of Music trans C Abbate Princeton NewJersey Princeton University Press pp 102ndash129

Pressing J 1988 lsquolsquoNonlinear Maps as Generator of Musi-cal Designrsquorsquo Computer Music Journal 12(2)35ndash46

Ross B J 1995 lsquolsquoA Process Algebra for Stochastic MusicCompositionrsquorsquo Proceedings of the1995 InternationalComputer Music Conference San Francisco Interna-tional Computer Music Association pp 448ndash451

Schillinger J 1948 The Mathematical Basis of the ArtsNew York The Philosophical Library (Reprinted 1976Cambridge Massachussetts Da Capo Press)

Smalley D 1986 lsquolsquoSpectro-Morphology and StructuringProcessesrsquorsquo In S Emmerson ed The Language of Elec-

troacoustic Music Basingstoke UK Macmillanpp 61ndash93

Smalley D 1997 lsquolsquoSpectromorphology ExplainingSound-Shapesrsquorsquo Organised Sound 2(2)107ndash120

Sturm B L 2001 lsquolsquoComposing for an Ensemble of At-oms The Metamorphosis of Scienti c Experiment IntoMusicrsquorsquo Organised Sound 6(2)131ndash145

Truax B 1982 lsquolsquoTimbral Construction in Arras as a Sto-chastic Processrsquorsquo Computer Music Journal 6(3)72ndash77

Truax B 1990a lsquolsquoChaotic or Non-Linear Systems andDigital Synthesis An Exploratory Studyrsquorsquo Proceedingsof the 1990 International Computer Music Confer-ence San Francisco International Computer MusicAssociation pp 100ndash103

Truax B 1990b lsquolsquoComposing with Real-Time GranularSoundrsquorsquo Perspectives of New Music 28(2)120ndash134

Vaggione H 2001 lsquolsquoSome Ontological Remarks aboutMusic Composition Processesrsquorsquo Computer Music Jour-nal 25(1)54ndash61

Vaughan M 1994 lsquolsquoThe Human-Machine Interface inElectroacoustic Music Compositionrsquorsquo ContemporaryMusic Review 10111ndash128

Xenakis I 1992 Formalized Music revised editionStuyvesant New York Pendragon Press

Appendix

Schroedingerrsquos Equation for an Atomic Potentialwith Radial Symmetry

]Y21 sup1 Y ` V(r)Y 4 i (A1)2m ]t

Using spherical coordinates

W(rhft) 4 R(r) H(h) W(f) T(t) (A2)

where

r is the distance of a point to the center of thepotential

h is the elevation anglef is the horizontal angleand t is the time R H W and T are given below

n 1 l 1 1 k2Zrl Zrnr0R(r) 4 (Zr) e b (A3)o k3 1 2 4nrk 4 0 0

mH(h) 4 P (cosh) (A4)l

65Fischman

5 imW(f) 4 e (A5)

iE`T(t) 4 e t (A6)n

where

n is the principal quantum number (n 4 1 2 3 )

l is the angular momentum quantum number (l4 0 1 2 n 1 1)

m is the magnetic quantum number (1 l m l)r0 is the minimum possible distance from the po-

tential centerZ is the atomic number denoting the charge that

originates the radial potential as a multiple ofthe charge of a proton (equal in value and ofopposite sign to that of the electron)

is the ionization energy (Rydberg en-4m qe eE 4` 22

ergy) or the energy required in order to pushthe electron outside the reach of the potential

qe and me are the charge and mass of an electronrespectively andis a family of polynomials known as the asso-mPl

ciated Legendre functions

The principal quantum number determines theoverall energy of the electron according to the fol-lowing equation

1E 4 1 E (A7)n `2n

The energy required by the electron to movefrom state a given by the number na to state bgiven by nb is found by subtracting the energy ofthe former from that of the latter as follows

1 1E 4 1 2 E (A8)a U b `3 42 2n na b

The square of the absolute value of Y (|W(r h ft)|2) is the density function associated to the proba-bility of nding the electron at a particular point inspace and time Thus the probability of nding anelectron in a volume element dV within a time in-terval dt is |Y(r h f t)|2 dVdt Using spherical co-ordinates we have

2 2probability 4 |Y(rhf)| r sin(h)drdhdfdt (A9)dVdt

As a result of radial symmetry (|U(f)|2 4 1) and auniform temporal distribution (|T(t)|2 4 1) it isconvenient to work only with the distributionfunctions r2|R(r)|2 and sin(h)|H(h)|2 (Physically thismeans that there is the same probability of ndingan electron at any time interval dt and that there isno favored horizontal angle) Substituting thesevalues in equation A9 and rearranging terms weobtain

probabilitydV

2 2 24 (r |R(r)| ) (sin(h)|H(h)| ) drdhdf (A10)

It is also worth noting that R is only dependenton Z n and l and that H is dependent only on land m

Cloud Density

To convert time from source duration to cloud du-ration we use the relation

cloud_durationt 4 t 2 (A11)c e envelope_duration

where

tc is the timing of the cloud density breakpointte is the timing of the envelope breakpointenvelope_duration is extracted from the envelope

of the sourceand cloud_duration is given by the composer

Calculation of Density from the Input Envelope

The source envelope drives the radial and angulardistributions

density(t ) 4 densityc min

` (density 1 density ) (A12)max min

2 2 22 eumlr |R(r(t )) |ucirc 2 eumlsin(h(t ))|H(h(t ))| ucirci c c c

where

densitymin and densitymax are the values of theminimum and maximum densities provided bythe composer

ri is the ith entry in the envelope table andtc is the time from the beginning of the cloud

66 Computer Music Journal

In equation A12 r and h may be determined ac-cording to equations A13 or A14 below at the com-poserrsquos discretion If r and h are both directlyproportional to the envelope then we have

amp(t )er(t ) 4 r ` (r 1 r ) 2c m in m ax min ampm ax (A13)(h 1 h ) amp(t )max min e

h(t ) 4 2c 2 ampmax

If r is directly proportional to the envelope and h isdirectly proportional to its slope we have

amp(t )er(t ) 4 r ` (r 1 r ) 2c min max min ampmax (A14)h(t ) 4 arctan(amp 8 (t ))c e

where

rmin rmax are the minimum and maximum radialvalues given by the composer

hmin hm ax are the minimum and maximum eleva-tion angle values (set to 0 and p )

amp(te) is the corresponding envelope amplitudevalue at time te

ampm ax is the maximum amplitude extractedfrom the envelope and

amp 8 (te) is the derivative of the envelope attime te

Alternatively we can require direct proportional-ity between the source envelope and the density

density(t ) 4 densityc m in

amp(t )e` (density 1 density ) 2 (A15)max min ampm ax

Cloud Amplitude Scaling

We can scale the cloud amplitudes according to thefollowing equation

gain(t ) 4 gain ` (gain 1 gain )c min max min

(amp(t ) 1 amp )e min2 (A16)

(amp 1 amp )max m in

where gainm in and gainmax are the values of the min-imum and maximum gain (provided by the com-poser) and ampmin and ampm ax are the minimum

and maximum amplitudes extracted from the enve-lope

Cloud Motion

Width and Depth

2 (1 1 x)A 4 L

4 22 (1 ` y) ` x

2 (1 ` x)A 4 1 1 x 1R

4 22 (1 ` y) ` x (A17)

2A 4 A 4 0 x 1 1L R

4 4(1 ` y) ` x

2A 4 0 A 4 x 1L R

4 4(1 ` y) ` xwhere

x is the widthy is the depth (see Figure 2) andAL and AR are amplitude factors that multiply

the left and right channels respectively

Trajectory Creation

rx(t ) 4 mWidth 2 sinh cosfc m ax r 1 rm ax m in

ry(t ) 4 mDepth 2 sinh abs(sinf)c m ax r 1 rm ax m in

z(t ) 4 mSemitones ` (mSemitones 1 mSemitones )c m in m ax m in

r2 cos h

r 1 rm ax min (A18)

where r h and f are generated from distributionsR(r) H(h) and U(f) The other parameters are givenby the composer rm in rm ax are the minimum andmaximum radial values mWidthm ax and mDepthmax

are the maximum absolute value of the width andthe maximum depth and mSemitonesmin and

67Fischman

mSemitonesmax are the low and high transpositionboundaries

Cloud Volume (Overall Grain Scatter)

rxScatter(t ) 4 sWidth 2 abs(sin h cos f)c max r 1 rmax min

ryScatter(t ) 4 sDepth 2 abs(sin h sin f)c max r 1 rmax min

rzScatter(t ) 4 sSemitones 2 cos hc max r 1 rmax min

(A19)

where r h and f are generated randomly from dis-tributions R(r) H(h) and U(f) The other parame-ters are given by the composer rmin rmax are theminimum and maximum radial values andsWidthmax sDepthmax and sSemitonesm ax are thewidth depth and height (in semitones) of thescatter

Grain Amplitude Scaling

Scaling proportional to Source Envelope

gGain(t ) 4 gGainc min

amp(t )e` (gGain 1 gGain ) 2 (A20)max m in ampm ax

where

gGainm in and gGainm ax are the minimum andmaximum gain (given by the composer)

amp(te) is the corresponding envelope amplitudevalue at time te and

ampmax is the maximum amplitude extractedfrom the envelope

Scaling Proportional to Schrodingerrsquos Distribution

gGain(t ) 4 gGainc min

` (gGain 1 gGain ) 2 F(t ) (A21)max m in c

where F(tc) is given by

2r(t )c 2 2F(t ) 4 sin(h(t ))|R(r(t ))| |H(h(t ))| (A22)c c c c3 4rmax

and

tcr(t ) 4 r ` (r 1 r ) 2c min max min cloudDuration (A23)

tch(t ) 4 h ` (h 1 h ) 2c min max min cloudDuration

Here rm in and rmax are the minimum and maximumradial values (given by the composer) hm in and hmax

are the minimum and maximum elevation angles

Scaling Proportional to the Complement ofSchrodingerrsquos Distribution

Replace the following de nition of F(tc) in equationA22

2r(t )c 2 2F(t ) 4 1 1 sin(h(t )) |R(r(t ))| |H(ht ))| (A24)c c c c3 4rmax

Scaling Proportional to Schrodingerrsquos CumulativeProbability

Replace the following de nition of F(tc) in equationA22

(t ) h(t ) r(t )f c c c2r(t )cF(t ) 4 (A25)c 3 4rm ax

h rmin min min

2 2sin(h(t ))|R(r(t ))| |H(h(t ))| drdhdfc c c

Scaling Proportional to the Complement ofSchrodingerrsquos Cumulative Probability

Replace the following de nition of F(tc) in equationA22

(t ) h(t ) r(t )f c c c2r(t )cF(t ) 4 1 1 (A26)c 3 4rm ax

h rmin min min

2 2sin(h(t ))|R(r(t ))| |H(h(t ))| drdhdfc c c

68 Computer Music Journal

Scaling generated randomly from variousdistributions

Replace F(tc) in equation A22 with values generatedby one of the following distributions

2 2r |R(r)| (A27)

2sin(h)|H(h)| (A28)

2|W(f)| (A29)

2 2 2r sin(h)|R(r)| |H(h)| (A30)

2 2 2r sin(h)|R(r)| |W(f)| (A31)

2 2sin(h)|H(h)| |W(f)| (A32)

2 2 2 2r sin(h)|R(r)| |H(h)| |W(f)| (A33)

Grain Duration

Duration Proportional to Source Envelope

gDuration(t ) 4 gDurationc m in

` (gDuration 1 gDuration ) (A34)m ax m in

amp(t )e2ampmax

where gDurationmin and gDurationmax are the mini-mum and maximum durations (given by the com-poser)

Duration Proportional to SchrodingerrsquosDistributions

gDuration(t ) 4 gDurationc min

` (gDuration 1 gDuration ) 2 F(t ) (A35)max min c

The various functions used are analogous to thoseused for grain amplitude with F(tc) given in equa-tions A22 A24 A25 and A26

Duration Generated from Various Distributions

The distributions used are the same as those inequations A27-A33

Grain Envelope

Distributions Used as Envelopes

2 2r |R(r)| (A36)

2 2Reverse of r |R(r)| (A37)

2sin(h)|H(h)| (A38)

2 2r sin(h)|Y(rhft)| (A39)

2 2Reverse of r sin(h)|Y(rhft)| (A40)

Grain Scatter Path

gxScatter4 gsWidth ` (gsWidth 1 gsWidth )min m in

gyScatter4 gsDepth ` (gsDepth 1 gsDepth )min m in

gzScatter4 gsSemitonesmax

` (gsSemitones 1 gsSemitones )min

(A41)

where

gsWidthm in gsWidthmax are the minimum andmaximum width scatter

gsDepthm in gsDepthm ax are the minimum andmaximum depth scatter and

gsSemitonesmin gsSemitonesm ax are the minimumand maximum semitone scatter

All of these values are given by the composerThe values gsWidth gsDepth and gsSemitones arederived from the mean and standard deviation ofthe distribution functions

gxScatter(t )4 rScatter 2 sin(hScatter)cos(fScatter)c

gyScatter(t )4 rScatter 2 sin(hScatter)sin(fScatter)c

gzScatter(t )4 rScatter 2 cos(hScatter)c (A42)

Here

r 1 rmeanrScatter 4rstdDeviation

h 1 hm eanhScatter 4 (A43)

hstdDeviation

f 1 fm eanfScatter 4

fstd Deviation

where

69Fischman

r h and f are generated randomly from the dis-tributions R(r) H(f) and U(f)

rmean and rstd Deviation are the mean and standarddeviation of R(r)

hm ean and hstdDeviation are the mean and standarddeviation of H(h) and

fm ean and fstd Deviation are the mean and standarddeviation of U(f)

Grain Overall Motion

x (t ) 4 gxScatter(t ) ` xScatter ` x(t )g c c c (A44)y (t ) 4 gyScatter(t ) ` yScatter ` y(t )g c c c

The values gxScatter(tc) and gyScatter(tc) are de- ned in equation A42 xScatter and yScatter are de- ned in equation A19 and x(tc) and y(tc) are de nedin equation A18

Grain Doppler Shift

The pitch shift relative to the listener is invariantunder rotation Therefore it is possible to simplifythe calculation of the Doppler Effect if we rotatethe coordinate system by an angle a(tc) so that theline joining the loudspeakers is kept parallel to theinstantaneous velocity of the grain as shown inFigure 5 The angle a(tc) is then

dyga(t ) 4 arctg (A45)c 1 2dxg

Also to maintain consistency a(tc) must be posi-tive if it produces a counterclockwise rotation

The Doppler Shift may be expressed as a time-varying frequency ratio calculated according to thefollowing equation

1frequencyRatio(t ) 4 (A46)c V (t )coshg c1 1

Vs

where

Vg(tc) is the instantaneous velocity of the grainVs is the velocity of sound in air andh is the angle of the grainrsquos velocity relative to

the listener (see Figure 5)

In the rotated system Vg and a(tc) can be calcu-lated from the rotated coordinates xag and yag which can be obtained from the standard equationsfor a rotation

x (t ) 4 x (t )cosa ` (y (t ) ` 1)sinaag c g c g c

y (t ) 4 1 x (t )sina ` (y (t ) ` 1)cosa 1 1ag c g c g c

(A47)

Thus we have

dxagV 4 (A48)s dt

and

xagcosh 4 1 (A49)2 2(1 ` y ) ` x ag ag

66 Computer Music Journal

In equation A12 r and h may be determined ac-cording to equations A13 or A14 below at the com-poserrsquos discretion If r and h are both directlyproportional to the envelope then we have

amp(t )er(t ) 4 r ` (r 1 r ) 2c m in m ax min ampm ax (A13)(h 1 h ) amp(t )max min e

h(t ) 4 2c 2 ampmax

If r is directly proportional to the envelope and h isdirectly proportional to its slope we have

amp(t )er(t ) 4 r ` (r 1 r ) 2c min max min ampmax (A14)h(t ) 4 arctan(amp 8 (t ))c e

where

rmin rmax are the minimum and maximum radialvalues given by the composer

hmin hm ax are the minimum and maximum eleva-tion angle values (set to 0 and p )

amp(te) is the corresponding envelope amplitudevalue at time te

ampm ax is the maximum amplitude extractedfrom the envelope and

amp 8 (te) is the derivative of the envelope attime te

Alternatively we can require direct proportional-ity between the source envelope and the density

density(t ) 4 densityc m in

amp(t )e` (density 1 density ) 2 (A15)max min ampm ax

Cloud Amplitude Scaling

We can scale the cloud amplitudes according to thefollowing equation

gain(t ) 4 gain ` (gain 1 gain )c min max min

(amp(t ) 1 amp )e min2 (A16)

(amp 1 amp )max m in

where gainm in and gainmax are the values of the min-imum and maximum gain (provided by the com-poser) and ampmin and ampm ax are the minimum

and maximum amplitudes extracted from the enve-lope

Cloud Motion

Width and Depth

2 (1 1 x)A 4 L

4 22 (1 ` y) ` x

2 (1 ` x)A 4 1 1 x 1R

4 22 (1 ` y) ` x (A17)

2A 4 A 4 0 x 1 1L R

4 4(1 ` y) ` x

2A 4 0 A 4 x 1L R

4 4(1 ` y) ` xwhere

x is the widthy is the depth (see Figure 2) andAL and AR are amplitude factors that multiply

the left and right channels respectively

Trajectory Creation

rx(t ) 4 mWidth 2 sinh cosfc m ax r 1 rm ax m in

ry(t ) 4 mDepth 2 sinh abs(sinf)c m ax r 1 rm ax m in

z(t ) 4 mSemitones ` (mSemitones 1 mSemitones )c m in m ax m in

r2 cos h

r 1 rm ax min (A18)

where r h and f are generated from distributionsR(r) H(h) and U(f) The other parameters are givenby the composer rm in rm ax are the minimum andmaximum radial values mWidthm ax and mDepthmax

are the maximum absolute value of the width andthe maximum depth and mSemitonesmin and

67Fischman

mSemitonesmax are the low and high transpositionboundaries

Cloud Volume (Overall Grain Scatter)

rxScatter(t ) 4 sWidth 2 abs(sin h cos f)c max r 1 rmax min

ryScatter(t ) 4 sDepth 2 abs(sin h sin f)c max r 1 rmax min

rzScatter(t ) 4 sSemitones 2 cos hc max r 1 rmax min

(A19)

where r h and f are generated randomly from dis-tributions R(r) H(h) and U(f) The other parame-ters are given by the composer rmin rmax are theminimum and maximum radial values andsWidthmax sDepthmax and sSemitonesm ax are thewidth depth and height (in semitones) of thescatter

Grain Amplitude Scaling

Scaling proportional to Source Envelope

gGain(t ) 4 gGainc min

amp(t )e` (gGain 1 gGain ) 2 (A20)max m in ampm ax

where

gGainm in and gGainm ax are the minimum andmaximum gain (given by the composer)

amp(te) is the corresponding envelope amplitudevalue at time te and

ampmax is the maximum amplitude extractedfrom the envelope

Scaling Proportional to Schrodingerrsquos Distribution

gGain(t ) 4 gGainc min

` (gGain 1 gGain ) 2 F(t ) (A21)max m in c

where F(tc) is given by

2r(t )c 2 2F(t ) 4 sin(h(t ))|R(r(t ))| |H(h(t ))| (A22)c c c c3 4rmax

and

tcr(t ) 4 r ` (r 1 r ) 2c min max min cloudDuration (A23)

tch(t ) 4 h ` (h 1 h ) 2c min max min cloudDuration

Here rm in and rmax are the minimum and maximumradial values (given by the composer) hm in and hmax

are the minimum and maximum elevation angles

Scaling Proportional to the Complement ofSchrodingerrsquos Distribution

Replace the following de nition of F(tc) in equationA22

2r(t )c 2 2F(t ) 4 1 1 sin(h(t )) |R(r(t ))| |H(ht ))| (A24)c c c c3 4rmax

Scaling Proportional to Schrodingerrsquos CumulativeProbability

Replace the following de nition of F(tc) in equationA22

(t ) h(t ) r(t )f c c c2r(t )cF(t ) 4 (A25)c 3 4rm ax

h rmin min min

2 2sin(h(t ))|R(r(t ))| |H(h(t ))| drdhdfc c c

Scaling Proportional to the Complement ofSchrodingerrsquos Cumulative Probability

Replace the following de nition of F(tc) in equationA22

(t ) h(t ) r(t )f c c c2r(t )cF(t ) 4 1 1 (A26)c 3 4rm ax

h rmin min min

2 2sin(h(t ))|R(r(t ))| |H(h(t ))| drdhdfc c c

68 Computer Music Journal

Scaling generated randomly from variousdistributions

Replace F(tc) in equation A22 with values generatedby one of the following distributions

2 2r |R(r)| (A27)

2sin(h)|H(h)| (A28)

2|W(f)| (A29)

2 2 2r sin(h)|R(r)| |H(h)| (A30)

2 2 2r sin(h)|R(r)| |W(f)| (A31)

2 2sin(h)|H(h)| |W(f)| (A32)

2 2 2 2r sin(h)|R(r)| |H(h)| |W(f)| (A33)

Grain Duration

Duration Proportional to Source Envelope

gDuration(t ) 4 gDurationc m in

` (gDuration 1 gDuration ) (A34)m ax m in

amp(t )e2ampmax

where gDurationmin and gDurationmax are the mini-mum and maximum durations (given by the com-poser)

Duration Proportional to SchrodingerrsquosDistributions

gDuration(t ) 4 gDurationc min

` (gDuration 1 gDuration ) 2 F(t ) (A35)max min c

The various functions used are analogous to thoseused for grain amplitude with F(tc) given in equa-tions A22 A24 A25 and A26

Duration Generated from Various Distributions

The distributions used are the same as those inequations A27-A33

Grain Envelope

Distributions Used as Envelopes

2 2r |R(r)| (A36)

2 2Reverse of r |R(r)| (A37)

2sin(h)|H(h)| (A38)

2 2r sin(h)|Y(rhft)| (A39)

2 2Reverse of r sin(h)|Y(rhft)| (A40)

Grain Scatter Path

gxScatter4 gsWidth ` (gsWidth 1 gsWidth )min m in

gyScatter4 gsDepth ` (gsDepth 1 gsDepth )min m in

gzScatter4 gsSemitonesmax

` (gsSemitones 1 gsSemitones )min

(A41)

where

gsWidthm in gsWidthmax are the minimum andmaximum width scatter

gsDepthm in gsDepthm ax are the minimum andmaximum depth scatter and

gsSemitonesmin gsSemitonesm ax are the minimumand maximum semitone scatter

All of these values are given by the composerThe values gsWidth gsDepth and gsSemitones arederived from the mean and standard deviation ofthe distribution functions

gxScatter(t )4 rScatter 2 sin(hScatter)cos(fScatter)c

gyScatter(t )4 rScatter 2 sin(hScatter)sin(fScatter)c

gzScatter(t )4 rScatter 2 cos(hScatter)c (A42)

Here

r 1 rmeanrScatter 4rstdDeviation

h 1 hm eanhScatter 4 (A43)

hstdDeviation

f 1 fm eanfScatter 4

fstd Deviation

where

69Fischman

r h and f are generated randomly from the dis-tributions R(r) H(f) and U(f)

rmean and rstd Deviation are the mean and standarddeviation of R(r)

hm ean and hstdDeviation are the mean and standarddeviation of H(h) and

fm ean and fstd Deviation are the mean and standarddeviation of U(f)

Grain Overall Motion

x (t ) 4 gxScatter(t ) ` xScatter ` x(t )g c c c (A44)y (t ) 4 gyScatter(t ) ` yScatter ` y(t )g c c c

The values gxScatter(tc) and gyScatter(tc) are de- ned in equation A42 xScatter and yScatter are de- ned in equation A19 and x(tc) and y(tc) are de nedin equation A18

Grain Doppler Shift

The pitch shift relative to the listener is invariantunder rotation Therefore it is possible to simplifythe calculation of the Doppler Effect if we rotatethe coordinate system by an angle a(tc) so that theline joining the loudspeakers is kept parallel to theinstantaneous velocity of the grain as shown inFigure 5 The angle a(tc) is then

dyga(t ) 4 arctg (A45)c 1 2dxg

Also to maintain consistency a(tc) must be posi-tive if it produces a counterclockwise rotation

The Doppler Shift may be expressed as a time-varying frequency ratio calculated according to thefollowing equation

1frequencyRatio(t ) 4 (A46)c V (t )coshg c1 1

Vs

where

Vg(tc) is the instantaneous velocity of the grainVs is the velocity of sound in air andh is the angle of the grainrsquos velocity relative to

the listener (see Figure 5)

In the rotated system Vg and a(tc) can be calcu-lated from the rotated coordinates xag and yag which can be obtained from the standard equationsfor a rotation

x (t ) 4 x (t )cosa ` (y (t ) ` 1)sinaag c g c g c

y (t ) 4 1 x (t )sina ` (y (t ) ` 1)cosa 1 1ag c g c g c

(A47)

Thus we have

dxagV 4 (A48)s dt

and

xagcosh 4 1 (A49)2 2(1 ` y ) ` x ag ag

67Fischman

mSemitonesmax are the low and high transpositionboundaries

Cloud Volume (Overall Grain Scatter)

rxScatter(t ) 4 sWidth 2 abs(sin h cos f)c max r 1 rmax min

ryScatter(t ) 4 sDepth 2 abs(sin h sin f)c max r 1 rmax min

rzScatter(t ) 4 sSemitones 2 cos hc max r 1 rmax min

(A19)

where r h and f are generated randomly from dis-tributions R(r) H(h) and U(f) The other parame-ters are given by the composer rmin rmax are theminimum and maximum radial values andsWidthmax sDepthmax and sSemitonesm ax are thewidth depth and height (in semitones) of thescatter

Grain Amplitude Scaling

Scaling proportional to Source Envelope

gGain(t ) 4 gGainc min

amp(t )e` (gGain 1 gGain ) 2 (A20)max m in ampm ax

where

gGainm in and gGainm ax are the minimum andmaximum gain (given by the composer)

amp(te) is the corresponding envelope amplitudevalue at time te and

ampmax is the maximum amplitude extractedfrom the envelope

Scaling Proportional to Schrodingerrsquos Distribution

gGain(t ) 4 gGainc min

` (gGain 1 gGain ) 2 F(t ) (A21)max m in c

where F(tc) is given by

2r(t )c 2 2F(t ) 4 sin(h(t ))|R(r(t ))| |H(h(t ))| (A22)c c c c3 4rmax

and

tcr(t ) 4 r ` (r 1 r ) 2c min max min cloudDuration (A23)

tch(t ) 4 h ` (h 1 h ) 2c min max min cloudDuration

Here rm in and rmax are the minimum and maximumradial values (given by the composer) hm in and hmax

are the minimum and maximum elevation angles

Scaling Proportional to the Complement ofSchrodingerrsquos Distribution

Replace the following de nition of F(tc) in equationA22

2r(t )c 2 2F(t ) 4 1 1 sin(h(t )) |R(r(t ))| |H(ht ))| (A24)c c c c3 4rmax

Scaling Proportional to Schrodingerrsquos CumulativeProbability

Replace the following de nition of F(tc) in equationA22

(t ) h(t ) r(t )f c c c2r(t )cF(t ) 4 (A25)c 3 4rm ax

h rmin min min

2 2sin(h(t ))|R(r(t ))| |H(h(t ))| drdhdfc c c

Scaling Proportional to the Complement ofSchrodingerrsquos Cumulative Probability

Replace the following de nition of F(tc) in equationA22

(t ) h(t ) r(t )f c c c2r(t )cF(t ) 4 1 1 (A26)c 3 4rm ax

h rmin min min

2 2sin(h(t ))|R(r(t ))| |H(h(t ))| drdhdfc c c

68 Computer Music Journal

Scaling generated randomly from variousdistributions

Replace F(tc) in equation A22 with values generatedby one of the following distributions

2 2r |R(r)| (A27)

2sin(h)|H(h)| (A28)

2|W(f)| (A29)

2 2 2r sin(h)|R(r)| |H(h)| (A30)

2 2 2r sin(h)|R(r)| |W(f)| (A31)

2 2sin(h)|H(h)| |W(f)| (A32)

2 2 2 2r sin(h)|R(r)| |H(h)| |W(f)| (A33)

Grain Duration

Duration Proportional to Source Envelope

gDuration(t ) 4 gDurationc m in

` (gDuration 1 gDuration ) (A34)m ax m in

amp(t )e2ampmax

where gDurationmin and gDurationmax are the mini-mum and maximum durations (given by the com-poser)

Duration Proportional to SchrodingerrsquosDistributions

gDuration(t ) 4 gDurationc min

` (gDuration 1 gDuration ) 2 F(t ) (A35)max min c

The various functions used are analogous to thoseused for grain amplitude with F(tc) given in equa-tions A22 A24 A25 and A26

Duration Generated from Various Distributions

The distributions used are the same as those inequations A27-A33

Grain Envelope

Distributions Used as Envelopes

2 2r |R(r)| (A36)

2 2Reverse of r |R(r)| (A37)

2sin(h)|H(h)| (A38)

2 2r sin(h)|Y(rhft)| (A39)

2 2Reverse of r sin(h)|Y(rhft)| (A40)

Grain Scatter Path

gxScatter4 gsWidth ` (gsWidth 1 gsWidth )min m in

gyScatter4 gsDepth ` (gsDepth 1 gsDepth )min m in

gzScatter4 gsSemitonesmax

` (gsSemitones 1 gsSemitones )min

(A41)

where

gsWidthm in gsWidthmax are the minimum andmaximum width scatter

gsDepthm in gsDepthm ax are the minimum andmaximum depth scatter and

gsSemitonesmin gsSemitonesm ax are the minimumand maximum semitone scatter

All of these values are given by the composerThe values gsWidth gsDepth and gsSemitones arederived from the mean and standard deviation ofthe distribution functions

gxScatter(t )4 rScatter 2 sin(hScatter)cos(fScatter)c

gyScatter(t )4 rScatter 2 sin(hScatter)sin(fScatter)c

gzScatter(t )4 rScatter 2 cos(hScatter)c (A42)

Here

r 1 rmeanrScatter 4rstdDeviation

h 1 hm eanhScatter 4 (A43)

hstdDeviation

f 1 fm eanfScatter 4

fstd Deviation

where

69Fischman

r h and f are generated randomly from the dis-tributions R(r) H(f) and U(f)

rmean and rstd Deviation are the mean and standarddeviation of R(r)

hm ean and hstdDeviation are the mean and standarddeviation of H(h) and

fm ean and fstd Deviation are the mean and standarddeviation of U(f)

Grain Overall Motion

x (t ) 4 gxScatter(t ) ` xScatter ` x(t )g c c c (A44)y (t ) 4 gyScatter(t ) ` yScatter ` y(t )g c c c

The values gxScatter(tc) and gyScatter(tc) are de- ned in equation A42 xScatter and yScatter are de- ned in equation A19 and x(tc) and y(tc) are de nedin equation A18

Grain Doppler Shift

The pitch shift relative to the listener is invariantunder rotation Therefore it is possible to simplifythe calculation of the Doppler Effect if we rotatethe coordinate system by an angle a(tc) so that theline joining the loudspeakers is kept parallel to theinstantaneous velocity of the grain as shown inFigure 5 The angle a(tc) is then

dyga(t ) 4 arctg (A45)c 1 2dxg

Also to maintain consistency a(tc) must be posi-tive if it produces a counterclockwise rotation

The Doppler Shift may be expressed as a time-varying frequency ratio calculated according to thefollowing equation

1frequencyRatio(t ) 4 (A46)c V (t )coshg c1 1

Vs

where

Vg(tc) is the instantaneous velocity of the grainVs is the velocity of sound in air andh is the angle of the grainrsquos velocity relative to

the listener (see Figure 5)

In the rotated system Vg and a(tc) can be calcu-lated from the rotated coordinates xag and yag which can be obtained from the standard equationsfor a rotation

x (t ) 4 x (t )cosa ` (y (t ) ` 1)sinaag c g c g c

y (t ) 4 1 x (t )sina ` (y (t ) ` 1)cosa 1 1ag c g c g c

(A47)

Thus we have

dxagV 4 (A48)s dt

and

xagcosh 4 1 (A49)2 2(1 ` y ) ` x ag ag

68 Computer Music Journal

Scaling generated randomly from variousdistributions

Replace F(tc) in equation A22 with values generatedby one of the following distributions

2 2r |R(r)| (A27)

2sin(h)|H(h)| (A28)

2|W(f)| (A29)

2 2 2r sin(h)|R(r)| |H(h)| (A30)

2 2 2r sin(h)|R(r)| |W(f)| (A31)

2 2sin(h)|H(h)| |W(f)| (A32)

2 2 2 2r sin(h)|R(r)| |H(h)| |W(f)| (A33)

Grain Duration

Duration Proportional to Source Envelope

gDuration(t ) 4 gDurationc m in

` (gDuration 1 gDuration ) (A34)m ax m in

amp(t )e2ampmax

where gDurationmin and gDurationmax are the mini-mum and maximum durations (given by the com-poser)

Duration Proportional to SchrodingerrsquosDistributions

gDuration(t ) 4 gDurationc min

` (gDuration 1 gDuration ) 2 F(t ) (A35)max min c

The various functions used are analogous to thoseused for grain amplitude with F(tc) given in equa-tions A22 A24 A25 and A26

Duration Generated from Various Distributions

The distributions used are the same as those inequations A27-A33

Grain Envelope

Distributions Used as Envelopes

2 2r |R(r)| (A36)

2 2Reverse of r |R(r)| (A37)

2sin(h)|H(h)| (A38)

2 2r sin(h)|Y(rhft)| (A39)

2 2Reverse of r sin(h)|Y(rhft)| (A40)

Grain Scatter Path

gxScatter4 gsWidth ` (gsWidth 1 gsWidth )min m in

gyScatter4 gsDepth ` (gsDepth 1 gsDepth )min m in

gzScatter4 gsSemitonesmax

` (gsSemitones 1 gsSemitones )min

(A41)

where

gsWidthm in gsWidthmax are the minimum andmaximum width scatter

gsDepthm in gsDepthm ax are the minimum andmaximum depth scatter and

gsSemitonesmin gsSemitonesm ax are the minimumand maximum semitone scatter

All of these values are given by the composerThe values gsWidth gsDepth and gsSemitones arederived from the mean and standard deviation ofthe distribution functions

gxScatter(t )4 rScatter 2 sin(hScatter)cos(fScatter)c

gyScatter(t )4 rScatter 2 sin(hScatter)sin(fScatter)c

gzScatter(t )4 rScatter 2 cos(hScatter)c (A42)

Here

r 1 rmeanrScatter 4rstdDeviation

h 1 hm eanhScatter 4 (A43)

hstdDeviation

f 1 fm eanfScatter 4

fstd Deviation

where

69Fischman

r h and f are generated randomly from the dis-tributions R(r) H(f) and U(f)

rmean and rstd Deviation are the mean and standarddeviation of R(r)

hm ean and hstdDeviation are the mean and standarddeviation of H(h) and

fm ean and fstd Deviation are the mean and standarddeviation of U(f)

Grain Overall Motion

x (t ) 4 gxScatter(t ) ` xScatter ` x(t )g c c c (A44)y (t ) 4 gyScatter(t ) ` yScatter ` y(t )g c c c

The values gxScatter(tc) and gyScatter(tc) are de- ned in equation A42 xScatter and yScatter are de- ned in equation A19 and x(tc) and y(tc) are de nedin equation A18

Grain Doppler Shift

The pitch shift relative to the listener is invariantunder rotation Therefore it is possible to simplifythe calculation of the Doppler Effect if we rotatethe coordinate system by an angle a(tc) so that theline joining the loudspeakers is kept parallel to theinstantaneous velocity of the grain as shown inFigure 5 The angle a(tc) is then

dyga(t ) 4 arctg (A45)c 1 2dxg

Also to maintain consistency a(tc) must be posi-tive if it produces a counterclockwise rotation

The Doppler Shift may be expressed as a time-varying frequency ratio calculated according to thefollowing equation

1frequencyRatio(t ) 4 (A46)c V (t )coshg c1 1

Vs

where

Vg(tc) is the instantaneous velocity of the grainVs is the velocity of sound in air andh is the angle of the grainrsquos velocity relative to

the listener (see Figure 5)

In the rotated system Vg and a(tc) can be calcu-lated from the rotated coordinates xag and yag which can be obtained from the standard equationsfor a rotation

x (t ) 4 x (t )cosa ` (y (t ) ` 1)sinaag c g c g c

y (t ) 4 1 x (t )sina ` (y (t ) ` 1)cosa 1 1ag c g c g c

(A47)

Thus we have

dxagV 4 (A48)s dt

and

xagcosh 4 1 (A49)2 2(1 ` y ) ` x ag ag

69Fischman

r h and f are generated randomly from the dis-tributions R(r) H(f) and U(f)

rmean and rstd Deviation are the mean and standarddeviation of R(r)

hm ean and hstdDeviation are the mean and standarddeviation of H(h) and

fm ean and fstd Deviation are the mean and standarddeviation of U(f)

Grain Overall Motion

x (t ) 4 gxScatter(t ) ` xScatter ` x(t )g c c c (A44)y (t ) 4 gyScatter(t ) ` yScatter ` y(t )g c c c

The values gxScatter(tc) and gyScatter(tc) are de- ned in equation A42 xScatter and yScatter are de- ned in equation A19 and x(tc) and y(tc) are de nedin equation A18

Grain Doppler Shift

The pitch shift relative to the listener is invariantunder rotation Therefore it is possible to simplifythe calculation of the Doppler Effect if we rotatethe coordinate system by an angle a(tc) so that theline joining the loudspeakers is kept parallel to theinstantaneous velocity of the grain as shown inFigure 5 The angle a(tc) is then

dyga(t ) 4 arctg (A45)c 1 2dxg

Also to maintain consistency a(tc) must be posi-tive if it produces a counterclockwise rotation

The Doppler Shift may be expressed as a time-varying frequency ratio calculated according to thefollowing equation

1frequencyRatio(t ) 4 (A46)c V (t )coshg c1 1

Vs

where

Vg(tc) is the instantaneous velocity of the grainVs is the velocity of sound in air andh is the angle of the grainrsquos velocity relative to

the listener (see Figure 5)

In the rotated system Vg and a(tc) can be calcu-lated from the rotated coordinates xag and yag which can be obtained from the standard equationsfor a rotation

x (t ) 4 x (t )cosa ` (y (t ) ` 1)sinaag c g c g c

y (t ) 4 1 x (t )sina ` (y (t ) ` 1)cosa 1 1ag c g c g c

(A47)

Thus we have

dxagV 4 (A48)s dt

and

xagcosh 4 1 (A49)2 2(1 ` y ) ` x ag ag