FormalizedMusic THOUGHTANDMATHEMATICSINCOMPOSITION Revised Edition Iannis Xenakis Additional material compiled andedited by Sharon Kanach HARMONOLOGIASERIES No.6 PENDRAGON PRESS STUYVESANT NY Other Titles in the Harmonologia Series No.1 HeinrichSchenker:Index toanalY$isby Larry Laskowski(1978)ISBN 0-918728-06-1 1\'0.2 Marpurg'sThoroughbass and Composition Handbook: Anarrative trarulation and critical study by David A. SheLdon(1989) ISBN 0-918728-55-x No.3 Between Modesand Keys:Gennan Theory1592-1802 by Joel Lester (1990) ISBN0-918728-77-0 No.4 Music Theory fromZarlinotoSchenker: ABibliography and Guide by David Damschroder and David Russell Williams (1991) ISBN 0-918728-99-1 No.5 MusicalTime:TheScrueof Order by Barbara R.Barry (1990)ISBN 0-945193-01-7 Chapters I-VIII of this book were originally published in French. Portions of it appeared in Gravesann Blatter, nos. 1, 6,9,11/12,18-22, and 29 (195.'>-65). Chapters I-VI appeared originally as the book MusilJuesFormelles, copyright 1963, by Editions Richard-Masse, 7, place Saint-Sulpice, Paris.Chapter VII was first published in La Nef,no. 29 (1967);the English translation appeared in Tempo,no.93(1970).Chapter VII was originally published in Revue d'Esthtftiqul!, Tome XXI(1968). ChaptersIXandAppendicesIandIIWereaddedforthe edition by Indiana University Press, Blmington 1971. ChaptersX,XI,XII,XIV,andAppendi.xIII wereaddedfurthis edition, and all lists were updated to1991. Library of Congress Cataloging-Publication Data Xenakis, Iannis, 1922-Formalizedmusic: thought and mathematics in composition /hnnis Xenakis.. p.c.m.__(Harmonologia senes ; no. 6) "New expanded edition"--Pref. Includes bibliographical references and index. ISBN 0-945193-24-6 1. Music--20th ccntury--Philosophy and aesthetics.2. C ..(Msic)3.Music--Theory--20th century.4. omposluonu...I Music--20thcentury--History and cntlCIsm.1. Title.1. Series. ML3800.x41990 781.3--dc20 C .h1992'Pdragon Press opynglen1',.{I Contents Preface Preface to Musiques formelles Preface to the Pendragon Edition IFree Stochastic Music IIMarkovian Stochastic Music-Theory IIIMarkovian Stochastic Music-Applications IVMusical Strategy VFree Stochastic Music by Computer VISymbolic Music Conclusions and Extensions for Chapters I-VI VIITowards aMetamusic VIIITowards aPhilosophy of Music IXNew Proposals in Microsound Structure XConcerning Time, Space and Music XISieves XIISieves: A User's Guide XIIIDynamic Stochastic Synthesis XIVMore Thorough Stochastic Music Appendices I&IITwo Laws of Continuous Probability IIIThe New UPIC System Bibliography Discography Biography: Degrees and Honors Notes Index v vn ix xi 1 43 79 110 131 155 178 180 201 242 255 268 277 289 295 323,327 329 335 365 371 373 383 < Preface The formalizationthatIattemptedintryingtoreconstructpartofthe musicaledificecxnihilohasnotused,forwantof timeor of capacity,the mostadvancedaspectsof philosophicalandscientificthought.Butthe escalade is started and others will certainly enlarge and extend the new thesis. Thisbookisaddressedtoahybridpublic,but interdisciplinary hybridiza-tionfrequentlyproduces superb specimens. Icouldmmuptwentyyearsorpersonaleffortsbytheprogressive fillinginof thcfollowingTableor Cohercnccs.Mymusical,architectural, and visual works arcthe chips of thismosaic.It islikeanet whoscvariable latticescapturefugitivevirtualitiesandentwinctheminam u l t i t u ~ eof ways.Thistable,infact,sumsupthetruecoherencesor thesuccessive chronologicalchaptersof thisbook.Thechaptersstemmedfrommono-graphs,whichtriedasmuchaspossibletoavoidoverlapping. Buttheprofoundlessonofsuchatableofcohercnccsisthatany theoryorsolutiongivenononelevelcanbeassignedtothesolutionof problemson anotherlevel.Thusthe solutionsinmacrocompositiononthe Familieslevel(programmedstochasticmechanisms)canengender simpler and more powerfulnew perspectives inthe shaping of micro sounds thanthe usualtrigonometric(periodic)functionscan.Therefore,inconsidering cloudsof pointsandtheirdistribu tionoverapressure-timeplane,wccan bypasstheheavyharmonicanalysesandsynthesesandcreatesoundsthat have never beforeexisted.Only then willsound synthesis by computers and digital-to-analogueconvertersfinditstrueposition,freeof therootedbut ineffectualtraditionofelectronic,concrete,andinstrumentalmusicthat makesuseof Fouriersynthesisdespitethefailureof thistheory.Hence,in thisbook,questions havingtodomainly with ordlcstral sounds(whichare more diversified and more manageable)l1ndarich and immediate applica-tion assoonastheyare transferredtothe Microsound level in the pressure-timespace.Allmusicjsthusautomaticallyhomogenizedandunified. vii viii Prefaceto the Second Edition "Everythingiseverywhere"isthewordofthisbookanditsTableof Coherences;Herakleitoswouldsay that thc waysup and downare one. The French edition,Musiques Formelles,wasproduced thanks to Albert Richard,director of La RevueMusicale.The English edition, a corrected and completedversion,resultsfromtheinitiative of Mr.ChristopherButchers, whotranslatedthefirstsixchapters.MythanksalsogotoMr.G.W. Hopkins,andMr. andMrs. John Challifour, who translated Chapters VII and VIII,respectively; toMr. Michael Aronson andMr. Bernard Perry of IndianaUniversityPress,whodecidedtopublishit;andfinallytoMrs. NatalieWrubel,whoeditedthisdifficultbookwithinfinitepatience, correctingandrephrasing many obscure passages. 1.X. 1970 TABLE(MOSAIC)OFCOHERENCES Phil05tJ/>hy(intheetymological sense) \ \ \ Thrusttowardstruth,revelation.Questineverything.interrogation,harsh criticism, activethrough creativity. ChaplltJ(inthesenseof themethodsfollowed) Partially and experimental ARTS(VISUA1.,SONIClMIXED. ) Entirely inferentialand cxpcTimel'tal SCIENCES(OFMAN,NATURAL) PHYSICS.NATHE.MATJCS,LOGIC Other methods tocome ? This is why the arts are freer,and can therefore guide the sciences, which are entirely inferential and eyperimental. Categories()f QutstitmS(fragmentationof thedirectionsleadingto creative .knowledge,to philosophy) REALITY(nXISTENTIALITV);CAUSALITY;INFER.ENCE:CONNEXITYjCOMPACTNESS;TBMPORALANDSPATIALUBIQ.UtTY ASACONSKQUENCEOFNEWMENTALSTR.UCTURES jINDETERMINISM.-+- bi-pole -+.. DETERMINISM i .. Familiesof Solwionsorproced:"s(of theabovecategories)/.j. (examples of particularrealization) FREESTOClJASTtcMARKOVIANGAMESGROUPS materializedby acomputer pro-gram ACHORRIPSIS S1'/IO-I,090262 '1'/48-1,240162 ATRBES

ANALOOlQUEADUELAKRATA ANALOIHQ.UEBSTRATEGtENOMOSALPHA SYRMOSNOMOSGAMMA Classesof Sonic (sounds that areheardandrecognizedasawhole.and classifiedwith res.pecttotheir sources) ORCHESTRAL,ELECTRONiC(producedby analoguedevices) JCONCRETf,(microphonecollected),DIGITAL(realized withcomputersand digital-ta-analogueconverters),... A1icrosounds Formsandstnlctures inthepressure"time space,recognition of theclassesto whichmicrosoundsbelong or which microstructuresproduce. Microsoundtypesresultfromq\lestions andsolutions.thatwereadopte-dat CATECORtESJFAMILIES.and PICES levels. PrefacetoMusiquesFormelles This book isacollection of explorations in musical composition pursued in severaldirections. The effort toreducecertain soundsensations,to under-stand their logical causes, to dominate them, and then to use them in wanted constructions; the effort to materialize movements of thought through sounds, then to test themin compositions; the effort to understand better the pieces of the past,by searching foranunderlying unit whichwouldbeidentical withthatof thescientificthoughtof ourtime;theefforttomake"art" while" geometrizing,"that is, by giving it areasoned support less perishable thanthe impulseof themoment,and hencemore serious,moreworthy of the fiercefight which the human intelligence wages in all the other domains -all these efforts haveledto asort of abstraction and formalizationof the musicalcompositionalact.Thisabstractionandformalizationhasfound, ashave somany other sciences,anunexpected and,Ithink, fertilesupport incertainareasof mathematics.It isnotsomuchtheinevitableuseof mathematicsthatcharacterizestheattitudeof theseexperiments,asthe overriding needtoconsider sound and music asavast potential reservoir in whichaknowledgeof thelawsof thoughtandthe structuredcreations of thoughtmayfindacompletelynewmediumof materialization,i.e.,of communication. Forthispurposethequalification"beautiful"or"ugly"makesno senseforsound,norforthemusicthatderivesfromit;thequantityof intelligence carriedby the sounds must bethe true criterion of the validity of aparticular music. Thisdoesnotprevent theutilizationof soundsdefinedaspleasant or beautifulaccording tothe fashionof themoment,noreventheir study in theirownright,whichmayenrichsymbolization andalgebration.Efficacy isinitselfasignof intelligence.Wearesoconvincedof thehistorical necessityof thisstep,thatweshouldliketoseethevisualartstakean ix x PrefacetoMllsiquesFormel/es analogouspath-unless,thatis,"artists"of anewtypehavenotalready done it inlaboratories,shelteredfromnoisypublicity. These studies havealwaysbeen matchedby actualworks whichmark out the various stages.My compositionsconstitute the experimental dossier of thisundertaking.Inthebeginningmy compositionsandresearchwere recognizedand published,thankstotltefriendshipandmoral and material support of Pro!:Hermann Scherchen.Certain chapters inthepresent work reflecttheresultsof tlteteachingof certainmasters,suchasH.Scherchen andOlivier Messiaeninmusic,and Prof.G.Th.Guilbaud in mathematics, who,throughthevirtuosityandliberalityof histhought,hasgivenmea clearerviewofthealgebraswhichconstitutethefabricofthechapter devotedtoSymbolicMusic. I.X. 1962 Prefaceto thePendragon Edition Here isanew expanded edition of Formalized Music.Tt invites two fundamen-tal questions: Havethetheoreticalproposi60nswJlichIhavemadeoverthepast thirty-five years a) survived inmy music? b) been aesthetically efficient? To the first question, I will answer a general "yes." The theories which I havepresentedinthevariouschaptersprecedingthisneweditionhave alwaysbeenpresent inmy music,even if sometheorieshavebeen mingled withothersinasamework.Theexplorationof theconcepntalandsound worldinwhichIhavebeeninvolvednecessitatedanharmoniousoreven conflictingsynthesisofearliertheses.Itnecessitatedamoreglobal architecturalviewthanamerecomparativeconfrontationof thevarious procedures.But thesupremecriterionalwaysremained thevalidation,the aesthetic efficiency of the music which resulted. Naturally,itwasliptomeandtomealonetodeterminethe aesthetic criteria,consciously ornot,in virtueof thefirst principle which one cannot get around. The artist (man) has the duty and the privilege to decide, radically alone, hischoices and the value of the results.By no means should he choose any ot11er means; those of power, glory,money, ... Each time, hemllst throw himself andhischosen criteria into question allwhilestrivingtostanfromscratchyetnotforget.Weshouldnot "monkey"ourselvesby virtueof thehabitswesoeasilyacquireduetoour own "echolalic" properties.But to be reborn at each and every instant, like a child with a new and "independent" view of things. Allof thisispart of asecond principle:It isabsolutely necessary tofree oneself, asmuch as possible,from any and all contingencies. xi Xli Preface This may be consideredman'sdestinyinparticular,andtheuniverse's ingeneral.Indeed,theBeing'sconstant dislocations,betheycontinuousor not,deterministicor chaotic(orboth simultaneously)aremanifestationsof thevitalandincessantdrivetowardschange,towardsfreedomwithout return. An artist cannot remain isolatedintheuniversaloceanof formsand theirchanges.Hisinterestliesinembracingthemostvasthorizonof knowledgeandproblema tics,allinaccordancewiththetwoprinciples presented above.From hencecomesthenew chapter inthiseditionentitled "Concerning Time, Space and Music." Finally,tofinishwiththefirstquestion,Ihaveallalong continuedto developcertainthesesandtoopen up somenewones.Thenew chapter on "Sieves" isan example of this along with the computer program presented in AppendixIIIwhichrepresents alongaestheticandtheoreticalsearch.This researchwasdevelopedaswellasitsapplicationinsoundsynthesison UPIC.* Anotllerapproachtothemysteryof soundsistheuseofcellular automata which Ihaveemployed in several instrumental compositions tllese past fewyears. This can beexplained by anobservation which Imade: scales of pitch (sieves) automatically establish akind of global musical style,a sort of macroscopic"synthesis"ofmusicalworks,muchlikea"spectrumof frequencies,or iterations,"of the physicsof particles. Internal symmetries or theirdissymmetriesarethereasonbehindthis.Therefore,tJlrougha discerning logico-aestlleticchoiceof "non-octave" scales,wecan obtain very rich simultaneities (chords)or linear successions which reviveand generalize tonal,modalorserialaspects.Itisont11isbasi5ofsievesthatcellular automata canbe usefulin harmonic progressions whichcreatenew andrich timbric fusionswith orchestral instruments. Examples of thiscan befound in works of mine such asAta, Homs,etc. Today, there isawhole new fieldof investigation called "Experimental Mathematics,"that givesfascinating insights especiallyin automaticdynamic systems,bytlIeu!;eof math andcomputer graphics.Thus,many structures such ast1Iealready- mentioned cellular automata or those whichpossess self-*UPIC-Unite Polygogique Informatique duCEMAMu.A sort. of musicaldrawing boardwhich,throughthedigitalizationof adrawing,enablesonetocompose music,teachacoustics,engageinmusicalpedagogyat anyage.Thismachinewas developed at the Centre d'Etudes de Mathematiqucs et Automatiques Musicales de P ~ r i s . Prefacexiii similarities such as Julia or Mandelbrot set.s,are studied and visualized. These studiesleadoneright intothefrontiersof determinismandindeterminism. Chaos tosymmetry and thereverse orientation are once again being studied andareevenquitefashionable!Theyopenupnewhorizons,althoughfor me,theresultsarenovelaspectsof tlle equivalent compositionalproblemsI starteddealingwithabout thirty-fiveyearsago.Thethesespresentedintlle earlier editions of thisbook bear witnesstotllisfactalthoughthe dynamic of musicalworksdepends on severallcvcls simultaneously andnot only on tJIe caleu Iuslevel. Animportant taskof tlleresearchprogramat CEMAMuistodevelop synthesisthroughquantifiedsoundsbutwithup-to-datetoolscapableof involvingautosimilitudes,symmetriesordeterministicchaos,orstochastics within adynamic evolurjonof amplitudefrequencyframeswhereeach pixel correspondstoasoundquantumor"phonon,"asalreadyimaginedby Einsteininthe19105.Thisresearch,whichIstartedinE)58andwrongly attributedtoGabor,cannowbepursuedwithmuchmorepowerfuland modern means. Some surprises can be expected! InAppendixIVof thisedition,anew,morepreciseformulationof stochasrjcsoundsynthesis canbefoundasafollow-upof thelast chapter of the preceding edition of Formalized Music(presented here as Chapter IX). In theinterim,thisapproachhasbeentestedandusedinmywork La Ligerule d'Eerforseven-tracktape.Thisapproachwasdevelopedat the CEMAMu in Parisandworkedout at theWDR,theWest-GermanNationalRadiostudio inCologne.Thisworkwaspart of theDiatopewhichwasinstalledforthe inaugurationof thePompidou/BeaubourgCenterinParis.Theeventwas entirelyautomatedwithacompletelaserinstallationand1600electronic flashes.This synthesis is pan ofCEMAMu's permanent research program. Inthissamespirit,random walksorBrownianmovementshavebeen thebasisforseveralofmyworks,especiallyinstrumentalpiecessuchas N'Shima,whichmeans "breath"or "spirit"inHebrew;for2femalevoices,2 French Horns, 2trombones and1 'cello. This piece was written at the request of RechaFreier,founderofthe"Aliyamovement"andpremieredatthe Testimonium Festival in Jerusalem. Theanswertothesecondquestionposedatthebeginningof this Prefaceisnotuptome.Inabsoluteterms,theartisanmusician(nottosay creator)mustremaindoubtfulofthedecisionshehasmade,doubtful, howeversubtly,of theresult.Thepercentageof doubt shouldnot existin virtue ofthe principles elaborated above.But in relative terms, the public,or connoisseurs(either synchronicordiachronic),alonedecideuponawork's xivPreface efficiency. However, any culture's validationfollows"seasonal"rules, varying between periods of afewyears to centuries or even millennia. We must never forgetthenearly-totallackof considerationEgyptianart sufferedforover 2000 years,or Meso-American art. Onecanassimilateaworkof art,or,letussay,justawork,tothe information we can put on adocument,sealinabott1ewhichwe willthrow into the middle of the ocean. Will it ever befound?When and by whom and how will it be read, interpreted? MygratitudeandthanksgotoSharonRanach,whotranslatedand supervised the new material in this updated edition of Formalized Musicand to Robert Ressler, the courageous publisher. FormalizedMusic I PreliminarysketchAnalogiqueB(1959).SeeChapterIII, pp.l03-99. Preliminary sketch Analogique B(1959).SeeChapter III,pp. 103-09. .2 ChapterI FreeStochasticMusic Art,and aboveall,music hasafundamentalfunction,whichisto catalyze the sublimation that itcanbring about through allmeans of expression.It mustaimthroughfixationswhicharclandmarkstodrawtowardsatotal exaltationinwhichtheindividualmingles,losinghisconsciousnessina truthimmediate,rare,enormous,andperfect.If awork of art succeedsin thisundertaking evenforasinglemoment,itattains itsgoal. This tremen-dous truth isnot made or objects, emotions, or sensations; it isbeyond these, as Beethoven's Seventh Symphony isbeyond music. This is why art can lead to realmsthatreligionstilloccupies forsomepeople. But thistransmutation of every-day artistic material whichtransforms trivialproducts into meta-art isasecret.The;" possessed"reachit without knowing its" mechanisms." The others struggle in the ideological and tech-nical mainstream of their epoch whichconstitutes the perishable" climate" and the stylistic fashion.Keeping our eyes fixedon this supreme meta-artistic goal,we shall attempt to defineinamoremodest manner the paths which canleadto it fromourpoint of departure,whichisthe magmaof contra-dictions in present music. ThereexistsahistoricalparallelbetweenEuropeanmusicandthe successiveattempts to explain the worldby reason.The music of antiquity, causaland deterministic,wasalready strongly influencedby the schools of Pythagoras and Plato.Plato insistedonthe principle of causality,"for it is impossibleforanything,tocomeintobeingwithoutcause"(Timaeus). Strictcausalitylasteduntilthenineteenthcenturywhenitunderwenta The Englishtranslationof Chaps.I-VI isby A.Butchers. VI f III, , ;I 1., Fig.1-1.Score of Metastasis,1953/54,Bars309-17 3f9 I I 311 I I 312J{3 12lirst V I12 second violins V II 8 violas A -1-8 cellosVC6 double basses CB -,---e: '" ___ .-- "..,_ r .- -

- _"-_ ,_..,.,;- '1-- - -- .._'or.'fl,;;;:-- >7; .. tI' '-..-......:;",::-.-I I "- VI..... ____ .. - __ tJ,g Yl,- '"''.'"':::-. ___.r . .:. d'd':;....:;--.il ':'.'. cthcis!,!, ::===------:--- .,'/::-.:..- T . .. - .:--... ,...:::..I AI ../ "..rt, _/

co,-- '" ea ....-:. @f;; ./ 1_ /-.Il' "/ / j +-- +.!. 'J:& '" 1-_"" ...t .....Sl_ .;,

Fig. 1-2.StringGlissandi.Bars309-14 ofMetastasis 4FormalizedMusic brutal and fertile transformation asaresult of statisticaltheories in physics. Sinceantiquitytheconceptsof chance(tyche) ,disorder(ataxia),anddis-organization were considered asthe opposite and negation of reason(logos), order(taxis),and organization(systasis).It isonlyrecentlythat knowledge hasbeenabletopenetrate chanceandhasdiscovered howtoseparateits degrees-in other wordstorationalizeitprogressively,without,however, succeedinginadefinitiveandtotalexplanationof theproblemof "pure chance." Afteratimelagof severaldecades,atonalmusicbrokeupthetonal function and opened up anew path paralleltothat of thephysical sciences, but atthesametimeconstrictedbythevirtually absolutedeterminismof serial music. It is therefore not surprising that the presence or absence of the principle of causality,firstinphilosophyandtheninthe sciences,might influence musical composition.It caused it to followpaths that appeared tobe diver-gent,butwhich,infact,coalescedinprobabilitytheoryandfinallyin polyvalent logic,whichare kindsof generalization andenrichments of the principle of causality. The explanation of the world, and consequently of the sonicphenomena which surround usor which may be created, necessitated and profited fromthe enlargement of the principle of causality,the basisof which enlargement is formed by the law ofiarge numbers. This law implies anasymptoticevolutiontowardsastablestate,towardsakindof goal,of stochos,whencecomestheadjective"stochastic." Buteverythinginpuredeterminismor inlesspureindeterminismis subjectedtothe fundamentaloperational lawsof logic,whichweredisen-tangledbymathematicalthoughtunderthetitleofgeneralalgebra. These lawsoperate on isolated statesor onsetsof elements with the aid of operations,themostprimitiveof whicharctheunion,notatedU,the intersection,notatedfI,andthenegation.Equivalence,implication,and quantificationsareelementary relations fromwhichallcurrent science can beconstructed. Music,then,maybedefinedasan organization of theseelementary operationsandrelationsbetweensonicentitiesorbetweenfunctionsof sonicentities.Weunderstandthefirst-ratepositionwhichisoccupiedby set theory,not only forthe construction of new works,but also foranalysis andbettercomprehensionof theworksof thepast.Inthesamewaya stochasticconstructionoraninvestigationof historywiththehelpof stochastlcs cannot be carriedthroughwithout the heIp of logic-the queen of thesciences,andIwouldevenventuretosuggest,of thearts-orits mathematical formalgebra.For everything that issaid here onthe subject Free StochasticMusic5 isalsovalidforallformsofart(painting,sculpture,architecture,films, etc.). From this very general, fundamental point of view,fromwhich we wish to examine and makemusic,primary time appears as a wax or clay on which operations andrelations can be inscribed and engraved, first for the purposes of work,and then forcommunication with athird person.On this level,the asymmetric,noncommutativecharacter of timeisuse(Bafter A#- A after B,i.e.,lexicographicorder).Commutative,metrictime(symmetrical)is subjectedtothe same logical laws and can thereforealsoaidorganizational speculations.What isremarkable isthatthesefundamentalnotions,which are necessary forconstruction,are foundinman fromhistenderest age,and it isfascinatingtofollowtheirevolutionas J canPiaget1 has done. Arterthis shortpreamble ongeneralities we shall enter into thedetails of an approach tomusical composition which Ihave developed over several years.Icall it "stochastic," inhonor of probability theory, whieh has served asalogicalframeworkandasamethod of resolving the conflicts and knots encountered. The firsttask istoconstructanabstraction fromallinheritedconven-tionsandtoexerciseafundamentalcritiqueof actsof thoughtandtheir materialization.What,in fact,does amusicalcompositionofferstrictlyon theconstruction level?It offersacollection of sequences whichit wishesto be causal.When,forsimplification,themajor scaIcimplied its hierarchy of tonalfunctions-tonics,dominants,and subdominants-around whichthe othernotesgravitated,itconstructed,inahighlydeterministicmanner, linearprocesses,or melodieson the one hand,and simultaneousevents,or chords,ontheother.Thentheserialistsof the Vienna school,nothaving knownhowto master logicallytheindeterminismof atonality,returned to an organization which wasextremelycausalinthe strictest sense,more ab-stract thanthat of tonality;however,thisabstractionwastheir greatcon-tribution.Messiaengeneralizedthisprocessandtookagreatstepinsys-tematizing theabstractionof allthe variables of instrumental music. What is paradoxical is that he did this in the modal field.He created a multimodal musicwhichimmediatelyfoundimitatorsinserialmusic.Attheoutset Messiaen'sabstractsystematizationfounditsmost justifiableembodiment inamultiserialmusic.It isfromherethatthepostwar neo-serialistshave drawntheir inspiration.They could now,followingthe Vienna school and Messiaen,withsomeoccasionalborrowingfromStravinskyandDebussy, walk on with ears shut and proclaim atruth greater thantheothers.Other movementsweregrowingstronger;chief amongthemwasthesystematic exploration of sonic entities,new instruments, and "noises." Varese wasthe 1st Peak 'st Peak 7: ~ b A.Groundprolile 01theleft half 01 the"stomach:' The intentionwas to build a shell, composed ofasfew ruled surfaces aspossible, over the groundplan.A conoid(8)iscon-structed throughtheground profile curve: thiswall isboundedby two straight lines: thestraight directrix (rising lrom the left extremity 01the groundprofile),and the outermost generatrix(passingthrough the right extremity of theground profile). This produces thefirst"peak" of the pavilion. B.A ruledsurfaceconsistingof two co no ids, Bandd,islaid throughthe curvoboundingtheright half of the stomach." Thestraight directrix of d passesthroughthe firstpeak,and the outermost generatrix at this side forms a triangulnr exit with thegeneratrix01 e.Thestraight directrix of apasses througha secondpeak and is joined by anarc to thedirectrixof d. This basic form is theoneusedin thefirstdesign andwas retained, with somemodifications, inthefinal structure. Themainproblem of the designwas to establish anaesthetic balance betweenthetwo peaks. ~C.Atlemptto close the space between the two ruledsurfacesof thefirst designby flat surfaces(whichmight serve asprojection walls). Fig. 1-3. StagesintheDevelopmentoftheFirstDesignof the PhilipsPavilion D.Another attempt. Above the entrance channel a smalltriangular opening isformed. flankedby two hyperbolic paraboloids(g and k).and thewholeiscoveredwitha horizontal lop surface. E.Elaborationof D.Thethirdpeak begins totakeshape(shyly). 3rd Peak F.Thefirstdesigncompleted(see alsothe firstmodel.Fig.1-4). There arenolonger any nat surfaces.The thirdpeakis fully developedand creates.withitsopposingsWeep.a counterbalancefor thefirsttwo peaks. Theheights of the three peakshave been established.Thethird peakand thesmall arc connectingthe straight directrixes of conoid. a and d(seeB.) form. respectively. theapexandthe baseof a part of a cone I. ------ ~ ------ - ~ -----1st Peak 2ndPeak 1stPeak 2ndPeak 2ndPeak 8FormalizedMusic pioneerin thisfield,andelectromagneticmusichasbeenthebeneficiary (electronicmusicbeingabranchofinstrumentalmusic).However,in electromagnetic music, problems of construction and of morphology were not facedconscientiously.Multiserialmusic,afusionof themultimodalityof MessiaenandtheVienneseschool,remained,nevertheless,attheheart of the fundamentalproblem of music. But by 1954 it was already in the process of deflation, for the completely deterministic complexity of the operations of composition and of the works themselves produced an auditory andideological nonsense.Idescribedthe inevitable conclusionin"The Crisisof SerialMusic": Linear polyphony destroys itselfby its very complexity; what one hears is in reality nothing but a mass of notes in various registers. The enormous complexity prevents the audience fromfollowingthe inter-twiningof thelinesandhasasitsmacroscopiceffectanirrational and fortuitousdispersionof sounds over the wholeextent of the sonic spectrum.Thereisconsequentlyacontradictionbetweenthepoly-phoniclinear systemandtheheardresult,whichissurface or mass. Thiscontradictioninherentinpolyphonywilldisappearwhenthe independenceof soundsistotal.Infact,whenlinearcombinations andtheirpolyphonicsuperpositionsnolongeroperate,whatwill count will be the statistical mean of isolated states and of transform a-tions of sonic components at a given moment. The macroscopic effect canthenbecontrolledbythemeanof themovementsof e1ement.s which we select. The result isthe introduction ofthe notion ofproba-bility,whichimplies,inthisparticularcase,combinatorycalculus. Here,inafewwords,isthepossibleescaperoutefromthe"linear category"inmusicalthought.2 Thisarticle servedasabridgetomy introductionof mathematicsin music.For if,thanks to complexity,the strict, deterministic causality which the neo-serialistspostulatedwaslost,then it wasnecessarytoreplace it by a more general causality, by aprobabilistic logic which would contain strict serial causality as aparticular case. This isthe function of stochastic science. "Stochastics"studiesandformulatesthe law of largenumbers,whichhas alreadybeenmentioned,thelawsofrareevents,thedifferentaleatory procedures,etc.Asaresultof the impasseinserialmusic,aswellasother causes,Ioriginatedin1954amusicconstructedfromtheprincipleof indeterminism;twoyearslaterInamed it"StochasticMusic."Thelaws ofthecalculusofprobabilitiesenteredcompositionthroughmusical necessity. But other paths alsoledtothe same stochasticcrossroads-first of all, Free Stochastic Music9 natural evcnts such as the collisionof hailor rain with hard surfaces,orthe song of cicadas in a summer field.These sonic events arc made out of thou-sands of isolated sounds; this multitude of sounds, seen as atotality, isanew sonic event. This mass event isarticulated and formsaplastic mold of time, which itselffollows aleatory and stochastic laws.If one then wishes to forma largemassofpoint-notes,suchasstringpizzicati,onemustknowthese mathematical laws, which, in any case, are no more than atight and concise expressionof chainof logicalreasoning.Everyonehasobservedthesonic phenomenaof apoliticalcrowdof dozensorhundredsof thousandsof people. The human river shouts a slogan. in a uniform rhythm. Then another sloganspringsfromtheheadof thedemonstration;it spreadstowardsthe tail,replacing thc first.A wave of transition thus passes fromthe head to the tail.The clamor fillsthecity,andtheinhibiting forceof voiceandrhythm reachesaclimax.It isaneventof greatpowerandbeauty initsferocity. Thentheimpactbetweenthedemonstratorsandtheenemyoccurs.The perfectrhythmof thelastsloganbreaksupinahugeclusterof chaotic shouts,whichalsospreadstothetail.Imagine,inaddition,thereportsof dozens of machine guns and the whistle of bullets addipg their punctuations tothistotaldisorder.The crowdisthen rapidly dispersed,andaftersonic andvisualhellfollowsadetonating calm,fullof despair,dust,anddeath. The statisticallawsof theseevents,separated fromtheirpolitical or moral context, are the same asthose ofthc cicadas or the rain. They are the laws of thepassage fromcomplete order tototaldisorder inacontinuous or explo-sivemanner.They are stochasticlaws. Here wetouch on one of the great problemsthat have haunted human intelligencesinceantiquity:continuousordiscontinuoustransformation. The sophismsof movement(e.g.,Achillesand the tortoise)or of definition (e.g., baldness), especially the latter, are solved by statistical definition; that isto say, by stochastics. One may produce continuity with either continuous or discontinuous clements. A multitude of short glissandi on strings can give theimpressionof continuity,andsocanamultitudeof pizzicati.Passages fromadiscontinuousstatetoacontinuousstate arecontrollable withthc aid of probability theory.For sometime nowIhave beenconducting these fascinatingexperiments ininstrumental works;but the mathematical char-acterof thismusichasfrightcnedmusiciansandhasmadetheapproach especially difficul t. Here isanother directionthat converges onindeterminism.The study of the variation of rhythmposestheproblem of knowingwhat the limit of totalasymmetryis,and of theconsequent completedisruption of causality amongdurations.Thesoundsof aGeigercounterintheproximityof a lIiIII 10FormalizedMusic radioactivesourcegiveanimpressive ideaof this.Stochasticsprovidesthe necessarylaws. Beforeending thisshort inspectiontour of events rich inthe newlogic, whichwereclosedtotheunderstandinguntilrecently,1 wouldliketoin-cludeashortparenthesis.If glissandiarelongandsufficientlyinterlaced, weobtainsonicspacesof continuousevolution.Itispossibletoproduce ruledsurfacesbydrawingtheglissandiasstraightlines.Iperformedthis experiment withj\1etastasis[thiswork haditspremierein1955atDonau-esehingen).Severalyearslater,whenthearchitectLeCor busier,whose collaboratorIwas,askedme to suggestadesignforthearchitectureof the PhilipsPavilion in Brussels,my inspirationwaspin-pointedbythe experi-ment withMetastasis.ThusIbelievethat onthisoccasionmusicand archi-tecture found an intimate connection.3 Figs.1-1-5 indicate thecausalchain of ideaswhichledmetoformulatethearchiteetmeof thePhilipsPavilion fromthe scoreof Metastasis. Fig.1-4.FirstModelofPhilipsPavilion Free StochasticMusic 11 Fig,1-5,PhilipsPavilion,Brussels World'sFair,1958 12FormalizedMusic STOCHASTICLAWSANDINCARNATIONS Ishall give quickly some of the stochastic laws whichIintroduced into composition several years ago. We shall examine one by one the independent components of an instrumental sound. DURATIONS Time(metrical)isconsideredasastraightlineonwhichthepoints correspondingtothe variationsof theothercomponentsare marked.The intervalbetweentwopointsisidenticalwiththcduration.Amongallthe possible sequences of points,whichshallwechoose?Putthus,thequestion makesnosense. Ifa mean number of points isdesignated on a given lengththe question becomes: Given this mean, what isthe number of segments eq ual to alength fixedinadvance? The following formula,which derives from the principles of continuous probability,givestheprobabilities forall possiblelengths whenone knows themean number of points placed at random on astraight line. (SeeAppendixI.) in which(jisthe linear density of points,and xthelength of any segment. If wenowchoosesomepointsandcomparethemtoatheoretical distribution obeying the above law or any other distribution, we can deduce theamount of chance includedin our choice,orthemore or lessrigorous adaptationof ourchoicetothelawofdistribution,whichcanevenbe absolutelyfunctional.Thecomparisoncanbemadewiththeaidof tests, of whichthemostwidelyusedistheX2 criterionof Pearson.In ourcase, whereallthecomponents of sound canbemeasuredtoafirstapproxima-tion,we shalluseinaddition the correlation coefficient.It isknownthat if twopopulationsareinalinearfunctionalrelationship,thecorrelation coefficient isone.If thetwopopulations are independent,thecoefficient is zero.Allintermediate degreesof relationship arepossible. Clouds of Sounds Assumeagivendurationandasetof sound-pointsdefinedinthe intensity-pitch spacerealizedduringthisduration.Giventhemean super-ficialdensityof thistonecluster,whatistheprobabilityof aparticular densityoccurring inagivenregionof theintensity-pitchspace?Poisson's Law answersthis question: Free StochasticMusic13 where!LoisthemeandensityandfLisanyparticulardensity.Aswith durations,comparisons with other distributions of sound-points can fashion the law whichwewish our clustertoobey. INTERVALSOFINTENSITY,PITCH,ETC. For thesevariables the simplest law is 8(y)dy=~(I- ~ )dy, (SeeAppendix 1.) which gives the probability that a segment (interval of intensity, pitch, etc.) withinasegmentof lengtha,willhavealengthincludedwithinyand y+ dy,forO:=;;y:;;a. SPEEDS We have been speaking of sound-points, or granular sounds, which are in reality aparticular caseof sounds of continuousvariation.Among these let usconsider glissandi. Of all the possible formsthat aglissando sound can take, we shall choose the simplest-the uniformly continuous glissando. This glissando ean be assimilated sensorially and physically into the mathematical concept of speed.In aone-dimensionalvectorial representation,the scalar sizeof thevectorcanbe givenbythehypotenuseof therighttrianglein whichthedurationandthemelodicintervalcoveredformtheothertwo sides.Certain mathematical operations on the continuously variable sounds thus defined are then permitted. The traditional sounds of wind instruments are,forexample,particularcaseswherethespeediszero.Aglissando towardshigherfrequenciescanbedefinedaspositive,towardslowerfre-quenciesasnegative. Weshalldemonstratethesimplestlogicalhypotheseswhichleadus toamathematicalformulaforthedistributionof speeds.Thearguments which followare inreality oneof those"logical poems"whichthe human intelligence creates inorder to trap the superficial incoherencies of physical phenomena,andwhichcan serve,on the rebound,asapoint of departure forbuildingabstractentities,andthenincarnationsof theseentitiesin sound or light.It isforthesereasonsthat Ioffer themasexamples: Homogeneityhypotheses[11] * 1.The density of speed-animated sounds isconstant;i.e.,tworegions of equalextentonthepitchrangecontainthesameaveragenumberof mobile sounds(glissandi). * The numbers in brackets correspond tothe numbers in the Bibliography atthe end of thebook. 14FormalizedMusic 2.The absolutevalueof speeds(ascending or descending glissandi)is spreaduniformly;i.e.,themeanquadraticspeedof mobilesoundsisthe same in different registers. 3.Thereisisotropy;thatis,thereisnoprivilegeddirectionforthe movementsof mobilesounds inanyregister.There isan equalnumber of soundsascending and descending. Fromthesethree hypothesesof symmetry,wecandefinethefunction f(v)of the probability of the absolute speed v.(f(v)isthe relative frequency of occurrence of v.) Letn bethenumber of glissandiperunit of thepitchrange(density of mobile sounds), and r any portion taken fromthe range. Then the number of speed-animatedsoundsbetweenvandv+ dvandpositive,is,from hypotheses1 and3: n r!f(v} dv(the probability that the signis+ is!). Fromhypothesis2thenumberofanimatedsoundswithspeedof absolutevalueIvlisafunctionwhich depends on v2 only.Let thisfunction be g(v2).Wethen have the equation n r tf(v) dv=n r g(v2)dv. Moreover ifx=v,the probability functiong(v2)willbeequaltothe law of probability HoL'!:,when.ceg(v2)=H(x),or logg(v2)=h(x). Inorderthath(x)maydependonlyonx2 =v2,itisnecessaryand sufficient that the differentialsd log g(v2)=h'(x)dxandv dv=x dxhave a constant ratio: dlog g(v2)= h'(x)dx= constant=_2j, vdvxdx whence h'(x)=- 2jx,h(x)=- jx2 + c,and H(x)=ke-ix But H(x)is a function of elementary probabilities; therefore its integral from- Cf.)to+ 00mustbeequalto1.jispositiveandk=viI v7T.If j=lla2,it followsthat and tf(v)=g(v2)=H(x)=_1_ ,-V'fa2 aV7T f() - 2-v"laO V- --e aV7T forv=x, which isaGaussiandistribution. Free StochasticMusic15 Thischainof reasoningborrowedfromPaulLevywasestablishedafter Ivlaxwell,who,withBoltzmann,wasresponsibleforthekinetictheoryof gases.Therunctionf(u}givestheprobability of the speedv;the constant a definesthe" temperature"of thissonicatmosphere.Thearithmetic mean of v isequaltoa/ ,hr, andthestandarddeviationisa/ ,12. We oDer asan example severalbars fromthe work Pithoprakta for string orchestra(Fig.1-6), writtenin1955-56,and performedby Prof.Hermann SchcrchcninMunichinMarch1957.4 Thegraph(Fig.1-7)representsa setof speedsof temperatureproportionaltoa= 35. The abscissa represents timeinunitsof 5 em=26MM(MaIze!YIetronome).Thisunitissub-dividedintothree,four,andfiveequalparts,whichallowveryslight differencesof duration.Thepitchesaredrawnastheordinates,withthe unit 1 semitone=0.25 em.Iem on the vertical scale corresponds to a major third.There arc 46 stringed instruments,eachrepresentedby a jagged line. Eachof thelinesrepresentsaspeedtakenfromthetableof probabilities calculatedwiththeformula f() - _2_- v 2 ' a ~ V- e. aV7T A totalof 1148 speeds,distributed in 58distinct values according to Gauss's law,havebeen calculatedand traced [or this passage(measures 52-60, with aduration of 18.5sec.).The distributionbeing Gaussian,themacroscopic configuration isa plastic modulation of the sonic material. The same passage wastranscribedintotraditionalnotation.Tosumupwehaveasonic compoundinwhich: 1.The durationsdonot vary. 2.The massof pitchesisfreelymodulated. 3.Thedensity of soundsat eachmoment isconstant. 4.The dynamic is j jwithoutvariation. S.Thetimbreisconstant. 6.Thespeedsdeterminea"temperature"whichissubjecttolocal fluctuations.Their distribulionisGaussian. Aswehavealready had occasiontoremark,wecanestablishmoreor lessstrictrelationshipsbetweenthecomponent parts of sounds.s Themost usefulcoefficientwhichmeasuresthedegreeof correlationbetweentwo variablesxand y is r 2:(x- x) (y- y) 16FormalizedMusic where x and fj are the arithmetic means of the twovariables. Here then, isthe technical aspect of the starting point fora utilization of thetheoryandcalculusof probabilitiesinmusicalcomposition.With the above,wealready knowthat: 1.We can control continuous transformations oflarge sets of granular and/orcontinuoussounds.Infact,densities,durations,registers,speeds, etc.,canallbesubjectedto thelawof largenumberswiththenecessary approximations.Wecanthereforewiththeaidof meansanddeviations shapethesesetsandmakethemevolveindifferentdirections.Thebest known isthat which goesfromorder todisorder,or viceversa,and which introducestheconceptof entropy.Wecanconceiveof gj,Uk'For example:C3 played areoandforteonaviolin, one eighth noteinlength,atoneeighthnote=240MM,canberepresentedas CV!Ol.arco'h39(=e3),g4(= forte),U5 (=-l- sec.).Supposethatthesepoints .!v!are plotted on an axiswhich we shall call En and that through its origin we draw another axis t,at right anglesto axisErWe shallrepresent on this axis, called the axis of lexicographic time,the lexicographic-temporal succession of thepointsM.Thuswehavedefinedandconvenientlyrepresenteda two-dimensional space(Ent). This willallow ustopassto phase 3.,defini-tionof transformation,and4.,microcomposition,whichmust containthe answertotheproblemposed concerning theminimum of constraints. Tothis end, supposethat the points Mdefined above can appear with nonecessarycondition other than that of obeying an aleatory law without memory.This hypothesisisequivalent to saying that weadmit astochastic distributionof theeventsErinthespace(Ent).Admittingasufficiently weak superficialdistributionn,weenter aregion wherethe law of Poisson isapplicable: Incidentallywecanconsiderthisproblemasasynthesisof several conveniently chosen linear stochastic processes(law of radiation from radio-active bodies).(The secondmethod isperhaps more favorable foramecha-nizationof thetransformations.) A sufficiently long fragment of thisdistributionconstitutesthe musical work.The basie law defined above generates awhole family of compositions asafunctionof thesuperficialdensity.So wehaveaformalarchetypeof compositioninwhichthebasicaimistoattainthegreatestpossibleasym-metry(intheetymologicalsense)andtheminimumoj constraints,causalities, andrules.Wethinkthatfromthisarchetype,whichisperhapsthemost 24 FormalizedMusic general one,wecanredescendthe ladder of formsbyintroducing progres-sivelymorenumerous constraints,j,e"choices,restrictions,andnegations. Intheanalysisin severallinearprocesseswecanalsointroduce other pro-cesses:thoseofWiener-Levy,p,Levy'sinfinitelydivisibles,Markov chains,etc.,ormixturesofseveral.Itisthiswhichmakesthissecond method themore fertile, Theexplorationof thelimitsaandb of thisarchetypea:::;n:::;bis equally interesting, but on another level-that of the mutual comparison of samples.This implies,ineffect,agradationof theincrements of n inorder that the differences between the families111may berecognizable.Analogous remarksarc valid in thecaseof other linearprocesses. If weoptforaPoissonprocess,therearetwonecessaryhypotheses whichanswertbe_q ueslionof theminimum of constraints:1.there exists in agivenspacemusicalinstrumentsandmen;and2.thereexistmeansof contact between these men and these instruments which permit the emission of rare sonicevents. Thisistheonlyhypothesis(cf.theekklisisof Epicurus),Fromthese two constraints and withthe aid ofstochastics, Ibuilt an entire composition without admitting anyother restrictions.Achorripsis for21instrumentswas composed in 1956-57, and had its firstperformance in Buenos Aires in1958 under Prof,HermannScherchen,(SeeFig.1-8.) At thattime Iwrote:'" \\)\...."\l' TOyapaUTOVOHVfUTtVT8KateLVa! T6yapaUT6eiva!JUTtVTKaLOUKervatt ONTOLOGY In auniverse of nothingness.Abrief train of waves, so brief that its end and beginning coincide(negativetime)disengaging itself endlessly, Nothingnessresorbs,creates, It engendersbeing. Time,Causality. Theseraresoniceventscanbesomethingmorethanisolatedsounds. Theycanbemelodicfigures,cellstructures,oragglomerationswhose .. The following excerpt (through p. 37)is from" In Search of aStochastic Music," GravesanerBUill",no.11/12, t "For it isthe same tothink asto be" (Poemby Parmenidcs); and my paraphrase, "For itisthesametobeas nottobe," Free Stochastic Music25 characteristicsarealsoruledbythe lawsof chance,forexample,cloudsof sound-points or speed-temperatures.6 In each case they form asample of a successionof rare sonicevents. This sample may be represented by either a simple table of probabilities oradouble-entrytable,amatrix,inwhichthecellsarefilledbythefre-quenciesof events.Therowsrepresentthe particular qualificationsof the events, and the columns the dates (seeMatrix M, Fig. 1-9). The frequencies inthismatrix aredistributedaccording toPoisson'sformula,whichisthe law fortheappearances of rare random events. We should further define the sense of such a distribution and the manner inwhichwerealizeit.Thereisanadvantageindefiningchanceasan aesthetic law,asanormalphilosophy.Chance isthe limit of thenotionof evolvingsymmetry.Symmetrytendstoasymmetry,whichinthissenseis equivalent to the negationof traditionally inheritedbehavioral frameworks. Thisnegationnotonlyoperatesondetails,butmostimportantlyonthe composition of structures,hence tendencies in painting, sculpture,architec-ture, and other realms of thought. For example, in architecture, plans worked out withtheaidof regulatingdiagramsarerenderedmorecomplexand dynamic by exceptional events.Everything happens as if there were one-to-oneoscillationsbetweensymmetry,order,rationality,andasymmetry, disorder,irrationality in the reactionsbetweenthe epochs of civilizations. Atthebeginning of atransformationtowards asymmetry,exceptional eventsareintroducedintosymmetryandactasaestheticstimuli.When theseexceptional events multiply and become the general case,a jump to a higher leveloccurs.The levelisoneof disorder,which,at least in the arts and in the expressions of artists,proclaims itself as engenderedbythe com-plex,vast,andrichvisionof thebrutalencountersof modernlife.Forms suchasabstract and decorative art and action painting bear witnesstothis fact.Consequently chance,by whosesidewewalk inall our daily occupa-tions,isnothing but an extreme case of this controlled disorder(that which signifies the richness or poverty of the connections between events and which engendersthedependenceorindependenceof transformations);andby virtue of thenegation, it conversely enjoys all the benevolent characteristics of an artistic regulator. It is a regulator also ofsonic events, their appearance, and theirlife.But it isherethat the ironlogicof the lawsof chanceinter-venes;thischancecannotbecreatedwithouttotalsubmissiontoitsown laws.On thiscondition,chance checkedby its own forcebecomes ahydro-electric torrent. BafJkl B 'Xy'-11.-&1 gr:Tr. 1 YI.2 3 Vcl.2 3 1 Kb.2. 3

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VerifyifI relationyes-----------_________I I '-1/ I l'" Hasthisduration\...... r yes \.beenusedalready?;-no Store inmemory( no Isthe total12 7 yes Startnext cell Fig. 1-11. Composition for Double Orchestra,by Michel Philip pot,1959 FlowChartof theFirstMovement no r (Isthe total1Z? "-------..--yes Carryonthein-tervalaccording to cellI modulo 76 Choose duration aspreviously andverify no-Stopl 42FormalizedMusic it(aslongasthereisnoerrorof omission l)wastheanalysisof the composer,hismentalprocesses,andacertainliberationofthe imagination. Thebiggestdifficultyencounteredwasthatof aconsciousand voluntary splitinpersonality.On onehand,wasthecomposer who already had aclear idea and aprecise audition of the work he wished to obtain; and on the other was the experimenter who had to maintain aluciditywhichrapidlybecame burdensomeintheseconditions-a luciditywithrespecttohisowngesturesand decisions.We must not ignorethefactthatsuchexperimentsmustbeexaminedwiththe greatestprudence,foreveryoneknowsthatnoobservationofa phenomenonexistswhichdoesnot disturbthatphenomenon,andI fearthat the resulting disturbance might bcparticularly strong when itconcernssuchan ill-defineddomainandsuchadelicateactivity. Moreover,inthisparticular case,Ifearthat observationmightpro-vokeitsowndisturbance.If Iacceptedthisrisk,Ididnotunder-estimate its extent. At most, my ambition confined itself to the attempt toprojectonamarvelousunknown,thatof aestheticcreation,the timidlightof adarklantern.(The darklanternhadthereputation of being used especially by housebreakers. On several occasions Ihave been able to verify how much my thirst for investigation has made me appearintheeyesof themajorityasadangeroushousebreakerof inspiration. ) Chapter II MarkovianStochasticMusic-Theory Nowwecanrapidlygeneralizethestudy of musicalcomposition withthe aid of stochastics. Thefirstthesisisthat stochasticsisvaluablenotonlyin instrumental music,but alsoinelectromagneticmusic.We have demonstrated thiswith severalworks:Diamorphoses1957-58(B.A.M.Paris),ConcretPH(inthe Philips Pavilion at the Brussels Exhibition,1958);and Orient-Occident,music forthe filmof the same name by E.Fulchignoni, produced by UNESCO in 1960. Thesecondthesisisthatstochasticscanleadtothecreationof new sonic materials and to new forms.For thispurposewemust asapreamble put forward atemporary hypothesis which concerns the nature of sound,of all sound[19]. BASICTEMPORARYHYPOTHESIS(lemma)ANDDEFINITIONS Allsoundisanintegrationof grains,of elementary sonicparticles,of sonicquanta.Eachoftheseelementarygrainshasathreefoldnature: duration,frequency,andintensity.1Allsound,cvenallcontinuoussonic variation,isconceivedasanassemblageof alargenumberof elementary grains adequately disposed in time.So every sonic complex canbeanalyzed as aseriesof pure sinusoidal sounds even if the variations of these sinusoidal soundsareinfinitelyclose,short,andcomplex.Intheattack,body,and declineof acomplex sound,thousands of pure soundsappear inamore or lessshortintervalof time,/).t.Hecatombs of pure soundsarcnecessaryfor thecreationof acomplexsound.Acomplex soundmaybeimaginedasa multi-coloredfireworkinwhicheachpointof lightappearsandinstall-43 --44Formalized Music taneously disappearsagainstablack sky.But inthisfireworkthere would be such a quantity of points oflight organized in such a way that their rapid and teeming succession would create formsand spirals, slowly unfolding, or conversely> brief explosions setting the whole sky aflame. A line oflight would becreatedbyasufficientlylargemultitudeof pointsappearinganddis-appearing instantaneously. Ifwe consider the duration !:J.tof the grain as quite small but invariable, wecan ignore it in what followsand consider frequencyand intensity only. Thetwophysicalsubstancesof asoundarefrequencyandintensityin association. They constitute two sets, F and G,independent by their nature. They have a set product F x G,which is the elementary grain ofsound. SetF canbeput inanykindof correspondencewithG:many-valued,single-valued,one-to-onemapping,.... The correspondence can begivenby an extensiverepresentation,amatrix representation,or acanonical represen-tation. EXAMPLESOFREPRESENTATIONS Extensive(termby term): Frequencies 111 1213 Intensities g3 g" g3 Matrix(inthe formof atable): .l- 111213141516 gl + 0 + 000 g2 0 + 000 + g3 000 ++ 0 Canonical(inthe formof afunction): V'1=Kg 1 =frequency g=intensity K=coefficient. 14 gil. 17 + 0 0 The correspondencemay also be indeterminate(stochastic),andhere themostconvenientrepresentationisthematricalone,whichgivesthe transitionprobabilities. Markovian StochasticMusic-Theory45 Example: ~II f2f3f4 gl 0.500.20 g2 00.30.3 ga 0.50.70.50 The table shouldbeinterpretedasfollows:foreachvalue froff thereare one orseveralcorrespondingintensityvalues gbdefinedbyaprobability. For example,thetwointensities g2andgacorrespondtothefrequency f2' with 30% and70%chance of occurrence,respectively.On the other hand, eachof thetwosetsFandG canbe furnishedwithastructure-that isto say,internal relationsand lawsof composition. Time t isconsidered asatotallyordered setmappedonto For G in a lexicographic form. Examples: a.flf2fa t=1,2, c. h.fa.513f';l1fx t=0.5,3,.yll, x, flflf2flf2hfnfa.. .. t=ABCDE.. .. .. .. .. ... ~ t~ tbtbtbtatbt Example c.isthcmost general sincecontinuous evolution issectioned into slicesof asinglethickness~ t ,whichtransforms it in discontinuity;this makes it mucheasier toisolate and examineunder the magnifying glass. GRAPHICALREPRESENTATIONS We can plot the values of pure frequenciesinunits of octaves or semi-tones on the abscissa axis, and the intensity values in decibels on the ordinate axis,usinglogarithmicscales(seeFig.II-I). Thiscloudof pointsisthe cylindricalprojectionontheplane(FG)of thegrainscontainedinathin slice~ t(seeFig.II-2). The graphicalrepresentationsFigs.II-2 andII-3 make moretangible the abstract possibilitiesraisedup tothispoint. Psychophysiology Weareconfrontedwithacloudof evolvingpoints.This cloudisthe product of the two sels F and G in the slice oftime bt. What are the possible 46 G (dB) . . .' . FormalizedMusic Elementarygrain consideredasan instantaneous associ-ation of anintensity 9anda freque ncy f Frequenciesinlogarithmic units(e.g ..semitones) Fig.11-1L....___________________=_ F G G \ ~ ~F F;gI I ~~ ' l , - - ~ - ~ - ~ - - - - - - - - - - - F F Fig.11-3 t - '" II I)M Regions 2 " 5 ,; 1 8910II,/r1& (h) Fig.111-16 2.Twogroupsofjntensityregionsgo,gl'asinFig.III-l7.The protocolsofthisgroupwillagainobeythesame(MTP),swiththeir parameters(y)and(e): '"I go .Ii' t go g, go I 0.20.8 ,t:o 0.850.4 (y) 0.80.2 (e) 0.150,6 glgl GG 00 '" '" v 0 'II "' .... L 00

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Fig.111-17 F Markovian StochasticMusic-Applications105 3.Two groups of density regions do,d},as in Fig. III-18. The protocols of thisgroup willhave the same(MTP)'s with parameters("-)and(p.): Fig. 111-18 0.8 0.2 (p.) 0.85 0.15 0.4 0.6 Thischoicegivesusthe principalscreensA,B,C,D,E,F,G,H,as showninFig.111-19.Thedurationilt of eachscreenisabout0.5sec. The period of exposition of aperturbationor of astationary state isabout 15sec. Weshallchoosethesameprotocolof exchangesbetweenperturba-tionsand stationary states of (MTPZ),that of AnalogiqueA. ~E _ ~ _ ~ _ E _ ~ _ ~ _ ~ _ ~ _ E _ ~ The screensof AnalogiqueBcalculateduptonowconstituteaspecial choice.Laterinthecourseof thiscompositionother screenswillbeused moreparticularly,buttheywillalwaysobeythesamerulesof coupling andthe same(MTPZ).In fact,if weconsider the combinations of regions of thevariable fj of ascreen,wenoticethatwithouttamperingwiththe name of the variable); itsstructure may be changed. -Screen A (lofodo) ScreenC (1.. #., d.) Screen D ai, ti,) ScreenF (I,/.d,) ScreenG (f."., d.) Fig.111-19 t; 4 3 .; I t; 4 j .2 f Ii 3 :J. .( ./2J"S"6'1- II>CDIt) NOTE: If two stages areused. eachorchestrais arrangedinthe classic manner. 0 :!!:'U !. ~ 3r:: g~ . 'C0 ::T" o " '" [f.J ,... ., ~ @ t""' 5' n ~ ., "0 ., 0 ~ l>l S S 5' ~ I :J 0.. ~ .: en (;' e-0 0 S '1;;10 ~ , ::t. 0 :J -t-.:) VI 126 CO/'v COlo -Formalized Music e.Actuallyallthese waysconstitutewhat onemay call"degenerate" competitive situations. The only worthwhilesetup,whichaddssomething newinthecaseof morethanoneorchestra,isoncthatintroducesdual conflictbetweentheconductors.Inthiscasethepairsof tacticsareper-formedsimultaneouslywithoutinterruptionfromonechoicetothenext (secFig.IV-4), and the decisionsmade by the conductors are conditioned by the winnings or lossescontainedinthegame matrix. Of?X GAINJ78724&.:s& XIVXVPleT/CSIX"-VIII VII Ol?y IfAiNJ 524048:28/dJ -----rAcrlCSVII'XIXXVV Fig.IV-4 2.Limiting the game.The game may be limitedin severalways:a.The conductors agree to play to a certain numbcrofpoints, and the first to reach it isthe winner.b.The conductors agrec in advance toplay n engagements. The one with more points at theend of thenthengagement isthewinner. c.Theconductorsdecideonthedurationforthegame,mseconds(or minutes),forinstance.Theonewithmorepointsat theendof themth second (or minute) isthe winner. 3.Awarding points. a.One method istohaveoneor tworefereescountingthepointsin two columns, one for conductor X and one for conductor Y,both in positive numbers.The refereesstopthegameaftertheagreedlimit andannounce the result to thepublic. b.Another method has noreferees,but usesan automatic system that consistsof anindividualboardforcachconductor.Theboardhasthe nxn cells of the game matrix used. Each ccll has the corresponding partial score and apush button. Supposethat thegame matrix isthe large one of 19x19cells.Ifconductor X chooses tactic XV againstY'sIV, he presses the button at the intersection of row XV and column IV.Corresponding to this intersection is thc cell containing thc partial score of 28 points for X and thebuttonthat Xmust push.Each button isconnectedtoasmalladding machine which totals up the results on an electric panel sothat they can be seen by the public asthe game proceeds, just: like the panels in the football stadium, but on asmaller scale. 4.Assigning of rowsor columns ismade by the conductors tossing acoin. 5.Dedding who startsthe game isdeterminedby a secondtoss. Strategy, Linear Programming, andMusical Composition127 6.Readingthetactics.The orchestras performthe tacticscyclically on a closed loop. Thus the cessationof atactic ismade instantaneously at abar line, at the discretion of the conductor. The subsequent eventual resumption of thistacticcanbe made either by: Q.reckoning from the bar line defined above, or h.reckoning fromabar line identified by a particular letter. The conductor willusuallyindicatetheletterhewishesbydisplayingalarge card tothe orchestra. If he has a pile of cards bearing the letters A through U,hehasavailable 22differentpointsof entry foreachone of the tactics. In the scorethetacticshaveaduration of at least twominutes. When the conductor reaches the end of a tactic he starts again at the beginning, hence the"da capo" writtenonthe score. 7.Durationoj theengagements.Thedurationof eachengagementis optional. It is a good idea, however,to fix alower limit of about 10 seconds; i.e.,if aconductorengagesinatactichemustkeepit up forat least10 seconds.This limit may vary fromconcert toconcert. It constitutes awish on the part of the composer rather than an obligation, and the c.onductors havetherighttodecidethelowerlimit of durationforeachengagement beforethegame.There isnoupperlimit,forthegameitself conditions whether tomaintain or to change thetactic. 8.Result afthe contest.To demonstrate the dual structure of this compo-sitionandtohonortheconductorwhomorefaithfullyfollowedthecon-ditionsimposedbythecomposerinthegamematrix,attheendof the combat onemightQ.proclaimavictor,or h.awardaprize,bouquetof flowers,cup,ormedal,whatevertheconcertimpresariomightcareto donate. 9.Choiceojmatrix.In Strategie there exist three matrices. The large one, 19rowsx19columns(Fig.IV-5), contains allthe partial scores forpairs of the fundamental tactics Ito VI and their combinations. The two smaller matrices, 3x3, also contain these but in the following manner: RowI and column 1 contain the fundamental tactics fromIto VI without discrimina-tion; row 2 and column 2 contain the two-by-two compatible combinations of the fundamentaltactics;androw3 and column 3 contain the three-by-threecompatiblecombinationso[ thesetactics.Thechoicebetweenthe large19x19matrix and oneof the3x3matricesdependsonthe ease with which the conductors can read amatrix. The cellswith positive scores meanagainforconductorXandautomaticallyasymmetricallossfor conductorY.Conversely,thecellswithnegativescoresmeanalossfor conductorXandautomaticallyasymmetricalgainforconductorY. Thetwosimpler,3x3matriceswithdifferentstrategiesareshownin Fig. IV-6. 128Formalized Music MATRIXOFTHEGAME ..,. ConductorY (columns) "1"-=-". .. .I:rIT Jjl.... "Jf&__ ...uIN.--r"I 2' :r-.1 -/10 " *:r -110 .It fit -,a ""-32 11-l6 x-Itl tJ " 4& I" /1 :w"-J/) fJ88 til3 .. ., -36 fI.-$2 /0 ,&

-.to H -r" -6 II) It 1+ '9' -lD -/+ , 88 92--14 -S2 ", -*8 -!of-u 9'9' JIt8+ -8& " l.t -/11 to ,I. -.r1.-6 -1&-3,t -Ii " 11: -+ -lIB -JJI -u-tt> -u '" " 18 ID9-28 r2-J8 I -66 " .. Fig. IV-5. Strategy II -s2 -20S.I .. " -u. I"" -II -6 &+ 3&J#

" ,t " -" a.t " -18 -3.-tot -1'2-f-v -311-.rt -11 I, .to,& " I} 0 -2 Ii-G ,." -6D-0. -#-8 -J' .II -.r. -f.1.1.18 (; -t8-" " -t&-J/-u4" -tt.-98-I,,, IA-!ll " -8 -1+-.0 i' #20-II " B-8 -M-8JIt " -:II(

-n-w 4-Sit-1111 D -4 '" :J 0 140 FormalizedMusic 5isexpressedasthedifferencebetweenthehighest andlowestpitchesthat can be played onthe instrument. 7.Attribution qfa glissando speed if class r is characterized as n glissando.The homogeneityhypothesesin Chap.Iledustothe formula I()=_2_-v'la' vaY1Tc, and by thetransformationvia=u toitshomologue: 2(U T(u)=Y1T Joe-u2 du, for which there are tables.f(v)isthe probability of oct:urrence of the speed v (whichisexpressedinsemitones/sec.);it hasaparametera,whichispro-portional tothe standard deviation s (a=s-yl2). a isdefinedasafunctionof the logarithm of the density of sequenceat by:aninverselyproportional function or adirectlyproportional function or afunctionindependent of density a=17.7+ 35k, wherek isarandom number between0and1. The constants of thepreceding formulaederive fromthelimitsof the speedsthat string glissandimaytake. Thus for(DA)j=145sounds/sec. a=53.2semi tones/sec. 25=7Ssemitones/sec., and for(DAyj=0.13sounds/sec. a=17.7semitones/sec. 25=25semitones/sec. 8.Attribution qf a durationxtothesoundsemitted.To simplifyweestablish amean duration for eachinstrument, which isindependent oftessitura and Free StochasticMusicbyComputer141 nuance.Consequently wereservethe right tomodifyit whentranscribing into traditional notation. The following isthe list of constraints that wetake into account fortheestablishment of duration x: G,themaximumlengthof respiration or desiredduration (DA)i,thedensityof the sequence qr,the probabilityof class r pn,theprobability of theinstrument n Thenifwedefinezasaparameterofasound'sduration, zcouldbe inverselyproportionaltotheprobabilityoftheoccurrenceofthe instrument,sothat z willbe at its maximum when (DA),PnqT is at its minimum, and in this case wecouldchoosez",ax=C. Instead of lettingZmax=G,weshallestablishalogarithmic lawsoas to freezethe growthof z.This law applies forany given value of z. z'= G In z/ln zmax Since we admit atotal independence,the distribution of thedurations x willbeGaussian: 1 J(x)=---e-(x-m)2/Zs> , sV27r wherem isthe arithmeticmean of thedurations, s the standard deviation. and m- 4.25s=0 m+ 4.25s=z' the linear system which furnishes us with the constants m and s.By assuming u=(x- m)/sy2wefindthefunctionT(u),forwhichweconsultthe tables. Finally,thedurationx of the soundwillbe givenbytherelation x=usy2 + m. We do not take into account incompatibilities between instruments, for this wouldneedlesslyburden the machine's program andcalculation. 9.Attributionof dynamic formstothesoundsemitted.Wedefinefour zones of meanintensities: ppp,p, J, if.Taken threeat atimethey yield43 = 64 142 FormalizedMusic permutations, of which 44 are different (an urn with 44 colors); for example, pppp. 10.Thesameoperationsarehegunagainfor eachsound of theclusterNa,. 11.Recalculations of the same sort aremade for theother sequences. An extract from the sequential statement wasreproduced in Fig. V-I. NowwemustproceedtothetranscriptionintoFortranIV,alanguage "understood" by the machine(seeFig.V-3). It isnot our purposetodescribethetransformationof the flowchart intoFortran.However,it wouldbeinterestingtoshowan example of the adaptation of amathematical expressiontomachine methods. Let usconsider the elementary law of probability(density function) f(x)dx=ere,. dx.[20] How shallweproceedin order forthecomputer to giveuslengths xwith theprobability f(x)dx?The machinecanonlydrawrandomnumbers Yo withequiprobabilitybetween0and1.Weshall"modulate"thisproba-bility:Assume some length Xo;then wehave prob.(0SxSxo)=L%O f(x)dx=1- rexo=F(xo) where F(xo)isthe distribution functionof x.But F(xo)=prob.(0sysYo)=Yo then I- e-C:Jeo= Yo and xo= In(1- Yo) c forallXo;;::O. Oncetheprogramistranscribedintolanguagethatthemachine's internal organization can assimilate, aprocess that can take several months, wecanproceedtopunching thecardsand settingup certaintests.Short sectionsare run onthemachineto detect errorsof logicand orthography and to determine the values of the entry parameters,which are introduced in the formof variables. This is a very important phase, for it permits us to exploreallparts of theprogram and determine the modalities of its opera-60n. The final phase is the decoding of the results into traditional notation, unlessan automatic transcriber isavailable. 143 Free Stochastic Music by Computer Table of the44 Intens'\tYFormsDerived from4Mean Intensity Values,ppp,p,f, ff iff - -tPf til --==:::f-::::==-!I'I' f !I'I' f ?II-=c:/ ==-j)! I----->!/; III--=--=-- # /il if!II I!i --===:::f :==--1 I=--=-I!!-==-} } ===-----I::::::==- III I =-==- IW-==-I /I! -====- ..;-==- /' JL- /'/'1-=1 f::::==- Iii---> If j> If;::::==-- /I! J----.-! J --==-)11-==/ t ---==--- 1;::::==-/6 /'I! 1/ =------== -ill-===== f I111-==:::# J-=====- If.::::::=:=-III I J-=::::::::# I-======j>-=====- #:==-I // - /' ---===::::./ 1------1 I II) -=:::1 /'-====/

#==---=======I ;/ II

-144Formalized Music Conclusions Alargenumber of compositions of the samekindasST/lO-l,080262 ispossible for a large number of orchestral combinations. Other works have alreadybeenwritten:ST/48-1,240162,forlarge orchestra,commissioned by RTF (France III); Atdes for ten soloists; and Morisma-Amorisima,for four soloists. AlthoughthisprogramgivesasatisfactorysolutiontotheminiInal structure, it is, however,necessary to jump to the stage of pure composition by coupling a digital-to-analogue converter to the computer. The numerical calculations would then be changed into sound, whose internal organization hadbeenconceivedbeforehand.Atthispoint one,ould bring to fruition and generalizetheconcepts described in the preceding chapters. The following are several of theadvantages of using electronic compu-ters in musicalcomposition: 1.The long laborious calculation made by hand isreduced to nothing. The speed of a machine sueh asthe IBM-7090 istremendous-of the order of 500,000 elementary operations/sec. 2.Freed from tedious calculations the composer isable to devote him-self to the general problems that the new musical form poses and to explore the nooks and crannies of this form while modifying the values of the input data.For example,hemay testall instrumental combinations fromsoloists to chamber orchestras,to large orchestras.Withtheaid of electronic com-putersthecomposer becomes asort of pilot:he pressesthebuttons,intro-ducescoordinates,and supervisesthecontrolsof acosmicvesselsailing in the spaceof sound,acrosssonicconstellationsandgalaxiesthathecould formerly glimpse only asadistant dream.Now he can explorethem at his ease,seated in an armchair. 3.Theprogram,i.e.,thelistof sequentialoperationsthat constitute thenewmusicalform,isanobjectivemanifestationof thisform.The programmay consequentlybedispatchedtoanypoint ontheearththat possessescomputersof the appropriatetype,and maybe exploitedby any composerpilot. 4.Becauseof certainuncertaintiesintroducedintheprogram.the composer-pilot can instill his own personality in the sonic result he obtains. Free Stochastic Musicby Computer Fig. V-3. Stochastic Music RewritteninFortranIV c c C C C C C C C C C C C C C C C C C C C C c C C C C C C C C C C C C C C C C C C C C C C c c c PROGRAMFREESTOCHASTICMUSIC(FORTRANIV) GLOSSARVOFTHEPR[NC1PALABBREVIATIONS A- DURATIONOFEACHSEOUENCEINSECONOS XEN XEN XEN XEN - NUMBERSFORGLISSANDOCALCULATIONXEN ALEA- PARAMETERUSEDTOALTERTHERESULTOFASECONDRUN THEXEN SAMEINPUTDATAXEN ALFAt31- THREEEXPRESSIONSENTERINGINTOTHETHREESPEEDVALUESXEN OFTHESLIDINGTONESCGLISSANDIIXEN ALIM- MAXIMUMLIMITOFSEQUENCEDURATIONAXEN TABLeOFANEXPRESSIONENTERINGINTOTHEXEN CALCULATIONOFTHENOTELENGTHINPARTBXEN BF- DYNAMICFORMNUMBER.THELISTISESTABLISHEDINDEPENDENTLVXEN OFTHtSPROGRAMANDlSSUBJECTTOMODIFlCATIONXEN DELTA- THERECIPROCALOFTHEMEANDENSITYOFSOUNDEVENTSDURINGXEN ASEDUENCEOFDURATIONAXEN CEtl.J).I=I,KTR.J=I,KTEI- PROBABILITIESOFTHEKTRTIMBRECLASSESXEN INTRODUCEDASINPUTDATA.DEPENOINGONTHECLASSNUMBERI=KRANOXEN ONTHE"POWER.J=1IORTAINE"DFROMV3*EXPFCUI=DAXEN EPsr- FORACCURACVINCALCULATINGPNANDECI,J),WHICH ITISADVISABLETORETAIN.XEN rGNtl,J}.1=1.KTRtJ=1,KTSJ- TA5LEOFTHEGIVEN OFBREATHXEN FOREACHINSTRUMENT,DEPENDINGONCLASS1ANDINSTRUMENT GTNA-NUMBEROFNOTESIN OF AXEN GTNS- OFNOTESINKWLOOPSXEN CHAMIN(l.J),HAMAXfl.J).HBMIN(I.J),HBMAXCt.Jl.t=I.KTR.J=l.KTS)XEN TABLEOFINSTRUMENTCOMPASSLIMITS.DEPENDINGONTIMBRECLASSXEN ANDINSTRUMENT.J.TESTINSTRUCTION460INPART6DETERMINESXEN WHETHERTHEHAORTHEHBTABLEISTHENUMBER1IS ARBtTRARV. JW- ORDINALNUMBEROFTHESEQuENCECOMPUTEO. KNL- NUMBEROFLINESPAGEOFTHEPRINTEDRESULT.KNL=50 - NUMBERINTHECLASSKR:lUSEDFORPERCUSSIONORINSTRUMENTS WITHOUTADEFINITEPITCH. KTF- POWEROFTHEEXPONENTIALCOEFF1CIENTESUCHTHAT OAfMAX1=V3*CE*4fKTE-ll' KTR- NUMBEROFTIMBRECLASSES - MAXIMUMNUMBEROFJW KTEST1,TAVl.ETC- EXPRESSIONSUSEFULINCALCULATINGHawLONGTHE VARIOUSPARTSOFTHEPROGRAMWILLRUN. KTI- ZEROIFTHEPROGRAMISBEINGRUN.NONZERODURINGDEBUGGING KT?- NUMBEROFLOOPStEQUALTO15BVARBITRARYDEFINITION. (M001CIXS).rXS=7.1)AUXILIARYFUNCTIONTOVALUESIN THF.TETAC256)TAALE 80 81 82 8:3 8 .. 85 86 87 88 89 90 91 92 93 XEN94 PRINT4(\,TETA,Zt.22XEN95 40rORMAT( THETETATABLE= -./.2tI12FI0.6.;,.4FIO.6./////.XN96 **THEZlTABLE.*,'.7F6.2.E12.3.;/;.*THE22TABLE /.SF14.B,/XEN97 IHIIXEN98 REAO50.DELTA.V3.AtO,A20,A17.A30.A35,BF,SQPI.EPSI.VITLtM.ALEA.AXEN99 +LtMXN100 50XENlt4 REAO60,KTI.KT2.KW,KNL,KTR.KTE.KRl.GTNA.GTNS,(NTI11.tcl,KTR)XEN115 60FORMATtSJ3.212.2F6.0.1212)XEN126 PRINT70,OELTA,V3.A10,A20,A17.A30,A35,BF.SQPt.EPSI.VITLJM.ALEA.AXEN127 itLIMtKTI ,KT2.t'W,KNL,KTR,ICTE.t0,then wehaveachordcomposedof aninfinityof vectorsof duratione"(thick constant glissando).(SeeFig.VI.:...6.) -t, LL Fig. VI-4 Ch - --- ---- --.1-1 - - - -cL4,t 1-1c. Fig.VI-5 Fig.VI-B t,5 168 Formalized Music For dH/dt= Ch,H= cht+ k,and dU/dt= cu,U= cut+ r,we have a chordofaninfinityof vectorsof variabledurationsandpitches.(See Fig.VI-7.) Fig. VI-7 14=e ... t ,He U.C",.t.k For dH/dt= Ch,H= Cht+ k,and dU/dt= J(t),U= F(t), we have a chord of an infinity of vectors.(See Fig.VI -8.) k lJF (t) Fig. VI-8 t, Ii FordH/dt= JCt),H= F(t),anddU/dt= 0,U= cu,ifCu 0,thenwehavea chord of an infinity of vectors of duration Cv.(thick variable glissando).(See Fig.VI-9.) H. F(t) u..c .... Fig. VI-9 f,u Symbolic Music 169 For dR/dt=J(t},R=F(t),anddUtdt= s(t),U= S (t),wehavea chord of an infinity of vectors.(SeeFig.VI-10.) u=Slt) Fig. VI-10 i,a In the example drawn fromBeethoven, set Aof the vectors XI isnot a continuous functionof t.The correspondencemay bewritten 1 XoXlX2XaX4Xs ~4~~4~ Becauseof this correspondence the vectors are not commutable. Set B is analogous to set A. The fundamental difference lies in the change of basein spaceEarelativetothebaseof A.Butweshallnot pursuethe analysis. Remark Ifour musical space has two dimensions, e.g., pitch-time, pitch-intensity, pressure-time, etc., it isinteresting tointroduce complex variables.Let x be the time and y the pitch, plotted on the i axis. Then z=x+ yi is a sound of pitch ywiththeattackattheinstantx.Lettherebeaplaneuvwiththe followingequalities: u=u(x,y), v = v(x,g), and w=u+ vi.They define amappingwhichestablishesacorrespondencebetweenpointsintheuv and xgplanes.In general anyw isatransformationof z. Thefourformsof amelodicline(orof atwelve-tonerow)canbe representedby the following complex mappings: w=z, with u=x and v= y,which corresponds to identity (original form) w=IzI2/z,with u= xand v=-g, whiehcorrespondstoinversion w=IzI2/- z with u=-x and v= y,which corresponds to retrogradation w= -Z, with u=-x and v=-g, whichcorrespondsto invertedretro-gradation. 170 FormalizedMusic These transformations formU1CKlcin g r o u p . ~ Other transformations,as yet unknown, eventopresent-day musicians, couldbeenvisaged.Theycouldbeappliedtoanyproductof twosetsof soundcharacteristics.Forexamplc,w=(A Z2 + Bz+ C)/(DZ2+ Ez+F), whichcanbe consideredas acombinationor twobilineartransformations separatedbyatransformationofthetypep=02Furthermore,fora musical space of more than two dimensions wecan introduce hypercomplex systemssuchasthesystemof quaternions. EXTENSIONOFTHETHREEALGEBRASTOSETSOF SONICEVENTS(anapplication) We havenotedin the abovethree kindsof algebras: 1.Thealgebraofthecomponentsof asonicevent,withitsvector language,independentoftheprocessionoftime,thereforeanalgehra outside-time. 2.A temporal algebra,which the sonic events create on the axis of metric time,and which isindependent of the vector space. 3.Analgebrain-lime,issuingfromthecorrespondencesand functional relations between thedements of the set of vectorsX and of thc set of metric time,T,independent of the set of X. All that has been said about sonic events themselves,their components, and about time can be generalized for setsof sonic events X and for setsT. Inthischapter wehaveassumedthatthereader isfamiliarwiththe conceptof theset,andinparticularwiththcconceptof theclassasitis interpretedinBooleanalgebra.Weshalladoptthisspecificalgebra, which isisomorphic withthetheory of sets. Tosimplifytheexposition,weshallfirsttakeaconcreteexampleby consideringthereferentialor universalsetR,consisting of allthesoundsof a piano.We shallconsider only the pitches;timbres, attacks, intensities,and durationswillbeutilizedinordertoclarifytheexpositionof thelogical operationsandrelations whichwe shallimposeonthe set of pitches. Suppose,then,asetAofkeysthathaveacharacteristicproperty. ThiswillbesetA,asubsetof setR,whichconsistsof allthekeysof the piano.This subset ischosenaprioriandthe characteristic propertyisthe particular choice of acertainnumber of keys. For theamnesicobserverthisclassmaybepresentedbyplayingthe keys one after the other, with aperiod ofsilenee in between. He willdeduce fromthisthat hehasheardacollectionof sounds,or alistingof elements. Symbolic Music171 Anotherclass,B,consistingof acertainnumber of keys,ischoseninthe same way.It isstatedafterclassAby causingtheelements of Btosound. The observer hearingthe twoclasses,Aand B,will note thetemporal fact:AbeforeB;ATB,(T=before).Next hebeginstonoticerelation-ships between the elements of the two classes.If certain elements or keys are commontobothclassestheclassesintersect.If noneare common,theyare disjoint.Hall the elements of B are common to one part of A he deduces that Bisaclassincluded inA.If all the elements of Bare foundin A,and all the elements of Aare foundinB,he deducesthat thetwoclassesare indistin-guishable,that they are equal. Let uschooseA and Bin suchaway that they have some clements in common.Let theobserver hear first A, then B,then thecommon part.He will deduce that: 1. there was a choice of keys , A; 2. there was a second choice of keys,B; and 3.the part common to Aand B was considered.The opera-tion of intersection(conjunction)hasthereforebeen used: ABorBA. This operation has therefore engendered anew class, which was symbolized by the sonic enumeration of thepart commonto Aand B. If the observer,having heard Aand B,hearsamixture of allthe ele-ments of A and B,he willdeduce that anew classisbeing considered,and that alogical summation hasbeen performed onthe firsttwo classes.This operation isthe union(disjunction)andiswritten A+ BorB+ A. If class A has been symbolized or played tohim and he ismade to hear allthe sounds of R except those of A,he willdeducethat the complement of AwithrespecttoRhasbeenchosen.Thisisanewoperation,negation, whichiswritten A. Hithertowehaveshownbyanimaginaryexperimentthatwecan defineandstateclassesof sonicevents(whiletakingprecautionsforclarity inthesymbolization);andeffectthreeoperationsof fundamentalimpor-tance:intersection,union,andnegation. Ontheotherhand,anobservermustundertakeanintellectualtask in order todeducefromthisboth classesandoperations.On our planc of immediate comprehension,wereplacedgraphic signsby sonicevents.We considerthesesoniceventsassymbolsof abstractentitiesfurnishedwith abstract logicalrelationson whichwemay effectat leastthefundamental operations of thelogicof classes.Wehavenot allowedspecialsymbols for thestatementof theclasses;onlythesonicenumerationof thegcneric -172 FormalizedMusic elementswasallowed(thoughincertaincases,if theclassesarealready knownandif thereisnoambiguity,shortcutsmaybetakeninthestate-menttoadmitasortofmnemotechnicalorevenpsychophysiological stenosym bolization). Wehavenotallowedspecialsonicsymbolsforthethreeoperations which are expressed graphically by"+,- ; onlythe classes resulting from these operations arc expressed, and the operations are consequently deduced mentallybytheobserver.Inthesamewaytheobservermustdeducethe relation of equality of thctwoclasses,andthe relation of implicationbased on the concept of inclusion. The empty class,however,may be symbolized by a duly presented silence.In sum,then, wecan only state classes,not the operations.Thefollowingisalistof correspond epeesbetweenthesonic symbolization and the graphical symbolization aswehave just definedit: Graphic symbols ClassesA,B,C,... Intersection(.) Union(+) Negation(-) Implication(--+) Membership(E) A AB A+B A:::lB A=B Sonic symbols Sonicenumerationofthegeneric elementshavingthepropertiesA,B, C,. (with possible shortcuts) Sonic enumeration of the elementsof R not included inA Sonic enumeration of the elementsof AB Sonicenumeration of the elementsof A+B This table shows that wecan reason by pinning down our thoughts by meansof sound.Thisistrueeveninthepresentcasewhere,becauseof a concern for economy of means, and in order to remain close to that immedi-ate intuition from which all sciences are built, we do not yet wish to propose sonicconventionssymbolizingtheoperations"+,-, andtherelations =, --+.Thuspropositions of the formA,E,I,0may not besymbolized by sounds, nor may theorems. Syllogisms and demonstrations of theorems may only be inferred. SymbolicMusic173 Besidestheselogicalrelationsandoperationsoutside-time,wehave seenthat wemay obtain temporal classes(T classes)issuing fromthe sonic symbolization that defines distances or intervals on the axis of time. The role of time isagain defined in a new way. I t serves primarily as a crucible, mold, or space in which are inscribed the classes whose relations one must decipher. Time isin some waysequivalent tothe area of asheet of paper or ablack-board. It isonly inasecondary sense that it maybeconsidered ascarrying genericelements(temporal distances)andrelationsor operationsbetween theseelements(temporalalgebra). Relationsandcorrespondencesmaybeestablishedbetweenthese temporal classes and the outside-time classes, and wemay recognizein-time operations and relations ontheclasslevel. After these general considerations, we shall givean example of musical compositionconstructedwith theaid of the algebraof classes.For thiswe must search out anecessity,aknot of interest. Construction EveryBooleanexpressionor function F(A,B,C),forexample,of the three classesA, B, C canbeexpressedinthe formcalleddisjunctivecanonic: where at=0;Iand kl=A .BC, A.B.C, A.B.C, A.B.C, ;r.B.C, ;r.BC, XBC, ;rBC. A Boolean function with n variables can always be written in such a way astobring in amaximumof operations+,"-, equalto3n 211-2 - 1. For n=3 thisnumber is17,and isfoundinthefunction F=ABC +AEC +A.BC +ABC.(1) For three classes,each of whichintersectswiththe othertwo,function(1) can be representedbythe Venn diagramin Fig.VI-ll. The flowchart of the operations isshownin Fig.VI-12. This same function Fcan beobtained withonly ten operations: F=(AB+ A.E).C + (A.B+ AB) C.(2) Its flowchart isgivenin Fig.VI-I3. Ifwe compare the two expressions ofF, each of which defines a different procedureinthecomposition of classesA,B,C,wenoticeamoreelegant 1 Formalized Music 174 Fig.VI-11 Fig. VI-12 Symbolic Music175 Fig.VI-13 in(1)than in(2). On the other hand(2)ismore economical(ten operationsasagainst seventeen).It isthiscomparisonthat waschosenfor the realization of Herma,awork forpiano.Fig.VI-14 showsthe flowchart that directstheoperationsof (1)and(2)on twoparallelplanes,andFig. VI-l5 showsthe preciseplan of theconstruction of Her-rna. The three classes A, B, C result in an appropriate set of keys ofthe piano. There exists a stochastic correspondence betwccn the pitch components and themomentsof occurrence insetT,whichthemselvesfollowastochastic law.Theintensitiesanddensities(numberof vectors/sec.),aswellasthc helpclarifythe levelsof the composition.This work was composed in1960-61,andwasfirstperformedby the extraordinary Japanese pianist YujiTakahashiinTokyo in February1962. 176 FormalizedMusic Inconclusionwecansaythat our argumentsarebasedonrelatively simplegenericelements.Withmuchmorecomplexgenericelementswe couldstillhavedescribedthesamelogicalrelationsandoperations.We would simplyhavechanged the level.An algebra on severalparallellevels isthereforepossiblewithtransverseoperationsandrelationsbetweenthe variouslevels. Fig.VI-14 L SymbolicMusic I ;:;I . I '", - I ".. D (llx24). Suprastructures One can apply a stricter structure to acompound sieve or simply leave the choice of elements to a stochastic function.We shall obtaina statistical Towards aMetamusic 199 colorationof thechromatictotalwhichhasahigherlevelof complexity. Usingmetabolae.Wcknowthat at every cyclic combination of the sieve indices(transpositions)and at every change inthemodule or moduli ofihe sieve(modulation)weobtainametabola.Asexamplesof metabolictrans-formationsletustakethe5mallestresiduesthatareprimetoapositive numberr.TheywillformanAbelian(commutative)groupwhenthe compositionlawfortheseresiduesisdefinedasmultiplicationwithreduc-tiontothe leastpositiveresiduewithregardtor.For anumerical example let r=18;the resid uesI, 5, 7,II,13,17are primes to it, and their afterreductionmoduloISwillremainwithinthisgroup(closure).The finitecommutativegrouptheyformcanbeexemplifiedbythefollowing fragment: 5x7=35;35- 18=17; 11xII=121;121- (6x18)=13;etc. ModulesI,7,13formacyclic sub-group of order 3. The following isa logical expressionof the two sieveshavingmodules5and13: L(5,13)=(13n+4 V13"+5V13n+7 V13n +9 ) A5n+1 V(5n + 2V5n +4 )A13n +9V13n + 6 Onecanimagineatransformationof modulesinpairs,startingfromthe Abelian group defined above. Thus thc cinematic diagram(in-time) will be L(5,13) 17)11)I)->-L(j,5)... 13) soastoreturntothe initialterm(closure).25 Thissievetheory canbeput intomanykindsof architecture,soasto create included or successivelyintersecting classes,thus stagesof increasing complexity;inotherwords,orientationstowardsincreaseddeterminisms inselection,and intopologicaltexturesof neighborhood. Subsequently wecanput intoin-timepracticethisveritahlehistology of outside-time music by means of temporal functions,forinstance by giving functionsof change-ofindiees,moduli,or unitary displacement-in other words,encasedlogicalfunctionsparametric withtime. Sieve theory isvery general and consequently isapplicable to any other sound characteristics that may be provided with atotally ordered structure, suchasintensity,instants,density,degreesof order,speed,etc.Ihaveal-ready saidthis elsewhere,as in the axiomatics of sieves.But this method can beappliedequallyto visual scalesandtothe optical arts of the future. Moreover,inthe immediate futureweshallwitness

'I, [/" .. '.:(f' Unl\.'f- !. llII 200 FormalizedMusic thistheoryanditswidespreadusewiththehelpof computers,foritis entirelymechanizable.Then,inasubsequentstage,therewillbeastudy of partially ordered structures, suchasaretobefoundintheclassification of timbres,forexample,bymeans of lattice or graphtechniques. Conclusion Ibelieve that music today could surpass itsclfby research into the out-side-timecategory,whichhasbeenatrophiedanddominatedbythe temporalcategory.Moreoverthismethodcanunifytheexpressionof fundamentalstructures of allAsian, African, and Europeanmusic.It has a considerableadvantage:itsmechanization-hencetestsandmodelsof allsortscanbefedintocomputers,whichwilleffectgreatprogressinthe musical sciences. In fact,what wearewitnessingisanindustrialization of musicwhich hasalready started,whether welikeit or not.It alreadyfloodsour ears in manypublicplaces,shops,radio,TV,andairlines,theworldover.It permits a consumption of music on a fantastic scale, never before approached. Butthismusicisof thelowestkind,madefromacollectionof outdated cliches fromthe dregs of the musical mind. Now it isnot a matter of stopping this invasion,which, afterall,increasesparticipation inmusic,even if only passively.It is rather aquestion of effecting a qualitative conversion of this music by exercising a radicalbut constructive critique of our ways of think-ing and of making music.Only in thisway,asIhavetriedtoshow in the present study,willthemusiciansucceedindominatingandtransforming thispoisonthatisdischargedintoourears,andonlyif hesetsaboutit without furtherado.But onemustalsoenvisage,andinthe sameway,a radicalconversionof musicaleducation,fromprimarystudiesonwards, throughouttheentireworld(allnationalcouncilsformusictakenote). Non-decimalsystemsand thelogicof classesarealreadytaught incertain countries,sowhy not their applicationtoanewmusicaltheory,suchasis sketched out here? Chapter VIII TowardsaPhilosophyof Music PRELIMINARIES Wearegoingtoattemptbriefly:1.an"unveilingof thehistorical tradition"of music,l and 2.toconstruct amusic. "Reasoning" about phenomena and their explanation was the greatest step accomplishedby man in the courseof hisliberation and growth.This iswhytheIonian pioneers-Thales,Anaximander,Anaximenes-must be .consideredasthestartingpointof ourtruestculture,thatof "reason." When I say" reason," it is not in the sense of a logical sequence of arguments, syllogisms,orlogieo-technicalmechanisms,butthatveryextraordinary quality of feelinganuneasiness,acuriosity,then of applying the question, It is,infact,impossibletoimaginethis advance,which,inIonia, created cosmology from nothing, in spite of religions and powerful mystiques, whichwereearlyformsof "reasoning."Forexample,Orphism,whichso influencedPythagorism,taughtthatthehumansoulisafallengod,that onlyek-stasis,thedeparturefromself,canrevealitstruenature,andthat with the aid of purifications (Ka,8apflot)and sacraments (oP'Y,a,)it can regain itslostpositionand escape the Wheel of Birth(TpOX0!> bhavachakra) that is to say, the fate of reincarnations as an animal or vegetable. I am citing thismystiquebecauseitseemstobeaveryoldandwidespreadformof thought, which existedindependently about the same time in the Hinduism of India.2 Aboveall,wemustnotethattheopeningtakenbytheIonianshas finallysurpassedallmystiquesandallreligions,includingChristianity. Englishtranslationof Chapter VIn by John andAmberChallifour. 201 .. 202Formalized Music Neverhasthespiritof thisphilosophybeenasuniversalastoday:The U.S.,China,U.S.S.R.,andEurope,thepresentprincipalprotagonists, restateit withahomogeneityandauniformitythat Iwouldevendare to qualify asdisturbing. Having beenestablished,thequestion(EAYXOS')embodiedaWheel of Birthsuigeneris,andthevariouspre-Socraticschoolsflourishedbycon-ditioning all further development of philosophyuntil our time.Two are in myopinionthehighpointsof thisperiod:thePythagoreanconceptof numbers and theParmenideandialectics-bothuniqueexpressionsof the same preoccupation. Asit wentthroughitsphasesof adaptation,uptothefourthcentury B.C.,the Pythagorean concept of numbers affirmed that things are numbers, or that allthingsarefurnishedwithnumbers,or thatthings are similar to numbers. This thesis developed (and this in particular interests the musician) fromthe study of musicalintervalsinorder to obtaintheorphic catharsis, foraccordingtoAristoxenos,thePythagoreansusedmusictocleanse, the soulastheyusedmedicinetocleansethebody.Thismethodisfoundin other orgia,likethat of Korybantes,asconfirmedby Plato inthe Laws.In everyway,Pythagorismhaspermeatedalloccidentalthought,firstof all, G