Musical Analysis by Computer Following Cognitive Model ofInduction of Analogies

Olivier LartillotIrcam- CentreGeorges-Pompidouemail:Olivier.Lartillot@ircam.fr

Abstract

Theobjectiveis thedesignof a softwarecapableof car-rying outananalysisof a musicalscore,not in keepingwith thepreceptsof an a priori musicaltheory, but onthe contrary as autonomouslyand neutrally as possi-ble. For this purpose,inductivemechanismshavetobe integrated to the system. A cognitivemetaphor–in particular the procedural vision of induction pro-posedby Holland et al. (Holland, Holyoak,Nisbeth,andThagard 1989),with mentalmodelas a semanticnetworkfeaturingmulti-weightedhypothesesconflict-ing and corrobating each other –, overriding logicalor probabilisticinconsistencies,efficientlyanswersthisproblem. Theoretical inquiries about induction haveshownthe fundamentalrelationshipsbetweeninduc-tion and analogy. This is even more pertinentin ourcontext, sinceanalogy, as has beenshownby eithercognitiveor musicologicstudies,is the core mecha-nismfor musicalentitiesemergence. Our visionof mu-sical analysisthrough systematicinduction of analo-giesintegratesmelody, harmonyand form into a uni-fiedframework andsuggestsnew kindsof analysisthatcould graspthosemusic– non-occidental,contempo-rary, electro-acoustic,improvised– not understoodbytraditionalanalyses.

1 Introduction

With music,humanmay simulatethe whole mys-tery of nature:its order, its beautybut alsotheunlim-ited complexity of its expression.Indeed,whenlisten-ing to music,our perceptionbeingcontinuouslybesetby ahugeflow of orderedstimuli,wefeelasif weexpe-riencean idol-like representationof natureitself. Oneimportantaimof musicanalysiswouldbeto explicit indetail the organizationin music that inducesso mucheffect in our consciousness.We proposeto focus onthisperceptivepointof view of musicanalysis– or nat-ural, again,becauseit considersthewaymusicis reallyperceived.Becausethistaskis socomplex, computerishereof greatinterest.For this reason,andalsobecauseit may be preferableto analyzemusicobjectively, theinner mechanismsthat enableus to understandmusichave to be modellized. Here also, we will show the

importanceof anaturalpointof view, thatis,adescrip-tion of cognitive processes.We will discover, by theway, theessentialcognitivemechanismof inferenceofanalogy. Throughthis investigation,we proposea newanalyticaltool thatwouldbeableto analyzeany kindsof music,including electro-acousticandimprovised–”real-time” – ones.

2 Induction: A natural approachof music analysis

Somesaythattoday’stechniquesof musicanalysisaresufficient for understandingthe essenceof music.HenceNicholasCook”seeno intrinsicmerit in thede-velopmentof ever morerigorousandsophisticatedan-alyticalmethods:thoughthereareareaswhichareana-lytically under-developed(earlymusicis an importantone), [...] our presentanalyticaltechniquesarerathersuccessful.”1 For him, theideawouldbemoreto com-bine pointsof view conveyed by differenttechniquesthaninventingnew ones.

The troubleis, thoseareas,concededby NicholasCook,in whichmostof thosetraditionalanalyticaltoolsarefairly lost,consistin factof any styleof musicthatwas not explicitly taken into accountby thesetoolswhen conceived. As a matterof fact, theseanalytictools, asthey implicitly describethecharacteristicsofa certainstyleof music,maythemselvesbeconsideredmoreasananalyticalresult– amethodto retrievesomestyleaspectsin every piece– thanasa pureanalyticalprocess.Moreover, evenfor themorecanonicalmusi-cal works, thesekind of analysisreducetheir contentinsteadof explicating theirspecificity.

This ideawasalreadyformalized,in avery generalepistemologicalpoint of view, in theseventeenthcen-tury, by thephilosopherFrancisBacon.”Thereare,andcanbeonly two waysof searchinginto anddiscoveringtruth”2:

� ”The oneflies from thesensesandparticularstothemostgeneralaxioms,andfrom theseprinci-ples, the truth of which it takesfor settledand

1(Cook1987),p. 3.2(Bacon1620),book1, aphorism18.

immovable,proceedsto judgmentandto thedis-covery of middle axioms. And this way is nowin fashion.” In musicalcontext, thesegeneralax-ioms are thosetraditional music theories,con-structed,like scientifictheories,hypothetically.

� ”The otherderivesaxiomsfrom the sensesandparticulars,rising by a gradualandunbrokenas-cent,sothatit arrivesat themostgeneralaxiomslast of all. This is the true way, but as yet un-tried.” This may take into accountthe particu-larity of any phenomenon,andproducegeneralknowledgeinsteadof simplyneedingit. Thisap-proach,vigorouslydefendedby FrancisBacon,is calledinductive, becauseit aimsat producingknowledgefrom phenomena.

The inductive paradigmhasbeenrelevantly criti-cized by modernepistemology, becausetherecannotbe a reductionof scientificknowledgein termsof ob-served phenomena.Theremustbe hypotheticaxiomssomewhere. This objection,however, is not valid intherealmof communicationprocesses,or semioticsys-tems(Nattiez 1990) if you prefer. The music itself,concretizedin a scoreor a signal – the neutral level–, is the result of a poietic process– the act of com-position–, andis now subjectto theesthesicgraspofeither the reader(musicianor analyst)or the listener.As mostof theabstractionof musicallanguageseemsto be – potentially, andmainly implicitly – reducibledirectly to our mereperceptionof it3, musicanalysis,asa kind of perception,may profit from an inductiveapproach.

This ”true way”, nowadays,hasbeentried in a mu-sical context. RudolphReti hasexperiencedan ana-lytical methodologythat studiesthe scorevery minu-tiously, trying to understandeachnote in its context.He then proceedto a ”gradual andunbrokenascent”from microscopic(motivic) to macroscopic(formal)level. ”And the true structuraldynamismof a com-position, its form in the fullest meaningof the term,can be conceived only by comprehendingas a con-certedstreamboth the groupsand proportionsof itsoutershapingandthethematicevolutionbeneath.”4

Thisinductiveapproachis theonly waytoachieveasatisfyingunderstandingof musicallanguage.But thistaskis socomplex andimpliessuchan overwhelmingcombinatorythat ”Moti vic analysiseasilydegeneratesinto a purelymechanicalexercicein which thescoreisanalyzedwithout ever really beingreadproperly[...].The whole tendency of motivic analysisis to suggestthatmusicis somekind of complicatedcipher, andthatthe way to breakthe codeis to stareat the scoreforlong enough. It doesnot encouragesensitive listen-ing.”5 Hopefully, theuseof computer, alleviatingusof

3It is true that somekinds of music– serialismof the 1950sinparticular– featurepoieticknowledgethatcannotbeinducedby thelistener.

4(Reti1951),p. 114.5(Cook1987),p. 114.

themechanicalexercice,mayanswerto this objection.But we needthento implement– and,beforethat, tomodel– theseinductivemechanisms.

Reti himselfwasblaimedfor not proceedingto re-ally objective analyses.He wasindeedinclined to ex-pressimplicitly his subjective estheticof music. Forthis reasontoo, inductive mechanismshave to be ex-plicited objectively.

3 Cognition: A natural modellingof inductive mechanisms

A longphilosophicalinquiry hastried,sinceAntiq-uity, to understandthephenomenonof induction.Aris-tote, when trying to definethe conceptof induction,integratesit in a logical framework, by consideringitasa kind of reverseof syllogism.This logical point ofview hasbeenfairly developed,especiallyduring theXXth century, in particularwith theinductive logic ofRudolf Carnap.It hasbeena failure, though,becauseinduction, contrary to deduction,cannotbe artifiallyreducedto someelementaryandabstractaxioms,andalsobecausewecannotproceedto inductionif wecon-siderknowledgein theform of predicatesor linguisticpropositions. In a word, induction is not an abstractcalculus,but a pragmaticprocess.

AlthoughAristoteformalizedinductioninsidealog-ical framework, hekeptin mind theimportantfact thatinductionis a naturalandpsychologicprocessthaten-ablesusto catchageneralideaoutof phenomena.Thispsychologicdimensionof induction has beendevel-opedespeciallybyDavid Hume(Hume1748).Hechar-acterizesit asakind of habit,and,moreprecisely, demon-stratesits foundationon imagination.Thisdescription,however, is only partial becauseinduction has to berootedin a priori mechanisms,assaidImmanuelKant(Kant 1781).CharlesPeirce(Peirce1992)managedtoformalizedefficiently theseideasof imagination,andhenceinduction,with thehelpof graphlogic, or, moregenerally, a network. Indeed,the connexionnismof anetworkof concepts– or semanticnetwork– may beconsideredasageneralizationof logic.

Theinductivelogicof RudolfCarnapnotonly failedbecauseof theobsolescenceof logic, but alsobecauseof its foundationon a relatedparadigm,namely, prob-ability. Leibniz inventedthe conceptof probability inorder to explicate the degreeof certitudeof uncertainknowledgein a mathematicalframework. But, in anyway we considerprobability, eithersubjectively – byconsideringa universeof possible– or objectively –throughstatisticalmeasurements–, it is a unidimen-sionalquantitythathasto be fixed for any hypothese.In the connexionnistvision of knowledgeasa seman-tic network,theideaof probability is givenup andre-placedby a setof several distinct quantitiesthat takeinto accountdifferent aspectsof knowledgerelation-ships:degreeof match,pastexperienceaward,support

from otherhypotheses,etc.The ideaby Peirceof a networkof conceptstakes

placein his pragmatistprogram,alleviatingconceptualframework of uselessparadigmandintegratingnew ideasdescribingpsychologicalandeffectiverealities.Today,suchpragmaticideasareechoed,in away, by cognitivesciences.The cognitive point of view is of epistemo-logical importance,becauseit explainsourunderstand-ing by describingits nature.Cognitiveapproachof in-duction,especiallyby the collectif of AI researchers,experimentalpsychologistsandphilosopherHollandetal. (Holland, Holyoak, Nisbeth,andThagard1989),takesbenefitfrom the conclusionsof all thesephilo-sophicalinquiries. They emphasizethe needto con-siderknowledgeasa semanticnetwork,wherethefir-mity of hypotheses,in conflict andcorroborationeachother, dependson thoseof parentconceptsin the net-work. Hollandet al. addtheessentialideathat induc-tion is a temporalprocess,wherehypothesesarecon-stantlytrying to explain thenew observedphenomena.

4 Analogy: A natural mechanismof music perception

JohnStuartMill (Mill 1866)hasshown thata lot ofknowledgeof particularfacts,insteadof beingdeducedfrom generalconcepts,are basedupon the degreeofressemblancebetweentheconsideredphenomenonanda setof referencephenomena.This meansthat anal-ogy is anessentialcognitive mechanism.And we cansupposethat even whenwe takeinto accountgeneralconcepts,we alsohave to find theadequateconceptbyanalogybetweenthe consideredphenomenonandthegeneralconcept. In any way, therefore,inferenceofknowledgeabouta phenomenonfatally needsananal-ogywith otherphenomena– eitherothersamplesor anabstractedone–. We maysupposethat thechoicebe-tweenthesetwo alternativesgenerallydependson thequantityof known analogs.

This is in fact what LeonardMeyer meansin histheoryof expectation.Indeed,he envisionsmusiclis-teningasadynamicprocess.At any time,”musicarousesexpectations,someconsciousandother unconscious,whichmayor maynotbedirectlyandimmediatelysat-isfied”.6 Theactualcontinuation,if not whatwaspre-dicted, triggersemotion,because”emotion is evokedwhen a tendency to respondis inhibited”.7 And thecoreideais thattheseexpectationsarelearned,becausethey rely in fact on thememoryof pastmusicalexam-ples.Meyer, becausehethinksthat”embodiedmusicalmeaningis [...] a productof expectations”8, implicitlyappliesMill’ spointof view in a musicalcontext.

The idea of analogyhas beeneven more explic-itly consideredin theparadigmaticanalysismethodol-

6(Meyer1956),p. 25.7(Meyer1956),p. 22.8(Meyer1956),p. 35.

ogy appliedin musicby NicolasRuwet(Ruwet1972).Inspiredby linguistic, Ruwetproposesan analysisofmusicwhich, througha researchof repetitions,detectsthedifferentmotives,their innerstructurationandtheirglobalorganization.However, Ruwet’sapproachis farfrom achieving Reti’sideal. Indeed,thelinguisticmetaphor,thoughproductive, doesnot take into accountthe in-trinsic specificityof music. Indeed,suchan approachconsidersmusicasamonodicflow – or asuperpositionof monodicflows – but never asa polyphonicnetworkof intricatedflows. Moreover, thestaticlinguistic ideaof paradigmis totally contradictoryto thedynamicmu-sical ideaof development. And this methodologycan-not beimplementedon computerunlesscriteriaof de-tectionof similarity bedefinedexplicitly anda priori.

LerdahlandJackendoff ’sanalyticmethodologyshareswith Ruwet’stheideaof ahierarchicalmusicrepresen-tation. They are sensitive to the ideaof analogyandrepetition– which they call parallelism– but recognizenot to be”preparedto gobeyondthis”, andto ”feel that[their] failure to fleshout thenotionof parallelismis aseriousgap in [this] attemptto formulatea fully ex-plicit theoryof musicalunderstanding”.9 This is duetothatfact thatthey rely onastaticgrammar, insteadof apragmaticstudyof inductiveprocess.Moreover, a per-tinent modellingof ”parallelism” would expressmorefreely throughanassociative networkthana hierarchi-cal tree.

Analogyhasbeenimplementedin artificial intelli-genceapplications,in particularby DouglasHofstadter(Hofstadter1995). He agreesin a way with Hollandandal.’s framework, in particularwith theideaof anet-work of activatedconcepts.Throughadrawing of mul-tiple possibleanalogiesbetweenthedifferentelementsof thestructure,Hofstadter’ssoftwareCopycat, whoseaim is to analyzeshort sequencesof letters,builds asemanticnetworkof relations. In this analogyframe-work, we would like to add the otherpivotal ideaofHollandetal.’smodellingof induction,namelythepro-ceduralapproach.Indeedmusicis atemporalobject,aswould sayHusserl,anda naturalway of explaining itis by consideringits temporaleffect in conscious,asinLeonardMeyer’s expectative approach.

5 kanthume theory of analogy

5.1 Principles

We have shown why a cognitiveapproachof musi-cal analysisis of greatimportance,andwhy, alongthetemporalprogressionof musicperception,it will con-sist mainly of a researchof analogybetweencurrentinstantandpastones.Our software,calledkanthume,is a tentativeof simulatingthispointof view. WeshareRuwet’s idea that motives– of notes,but alsoof setsof notes,of motives,etc. – have to be found through

9(LerdahlandJackendoff 1983),p. 53.

the detectionof their repetition,variedor not. But ifthis researchhasto be efficient enoughsuchas to beableto detectmotiveshiddenin a polyphonicflux, it isnecessaryto go beyondReti’shierarchicalframework.

First we proposeto formalize any musicalstruc-turesimply in termsof a motive, or a sequenceof ele-ments,which themselves may recursively be motivestoo. That is, every musical structureis ordered. Itseemsthat relationshipsbetweenmusicalentitiesaredeterminedalong two dimensions,whosebasicrela-tionsare:

� relationsof analogy, betweentwo analogs.� relationsof concatenation,betweenlateral ele-

mentswithin a motive.

Now if we considerthat motives emerge becauseoftheir repetition– even whenthey vary – thenthis canbepossibleonly if:

1. Somethingin thebeginningof therepeatedmo-tive triggersthe idea of analogywith first mo-tive: eithera samevalueof a parameterfor thesameelementof the two motives (pitch, dura-tion, etc.),or a similarity of an interval betweentwo elementsin thetwo motives(herealsopitchinterval,onsetinterval, etc.).

2. The successive next musical items sharesomesimilaritywith thecorrespondingonesin thefirstmotive: mostlybecauseof similarity of interval,but, why not, of similarity of an absolutepa-rameterof one particularnote. Eachnew ele-mentwill beconsideredasthecontinuationof arepeatedmotive if it canbe linked (particularlyby an interval) to an elementof its beginningandif this link hasits analogin thefirst motive.This elementto which it is linked canbecalled,metaphorically, theanchor of thenew element.

5.2 Description

Duringanalysis,musicis consideredincrementallyin a chronologicalsense. At eachstep,new note ofthe scoreis considered.From currentnote � , a seriesof intervals aredrawn to all precedentnoteswithin ashorttermarea.For eachof theseintervals

����� ��� , thesystemfinds the setof similar previous pastintervals.For eachof thesesimilar intervals

���� � :1. if

is the right noteof an interval

����� � whichconcludesa motive ����� ����� ��� , andif � is alsotheright noteof an interval

������� � which concludesanothermotive ������� ������� ��� , andif bothsequencesare analoguous,then there is an analogy, or asequencing, betweenthe two extendedmotives����� ����� � ���� ��������������� ������� � ���� ����� ;

2. elseif thetwonotesand

�areanaloguous,then

thereis ananalogybetweenthetwo new motives

� ����� ��� ���!� ������� ����� , where � �����"� ��� is the mo-tive constitutedby thesimplenote

andthe in-

terval���"� � ;

3. if no analogiesat all can be inferred betweenand

�, then only a analogymay be drawn –

if necessary– betweenthe two simple motives� ����"� ���#���$� ���� ����� .

In this way, a networkof analogiesis drawn fromthenotesof thescore.Theseanalogiesalsoemphasizethenotesandintervalsthatbelongto them. That is tosay, themorea note,or an interval, belongsto numer-ous or big sequences,the more probablywill it be acandidateanalog.

Now eachanalogyis itself an interval whosetwoelementsareits twoanalogs(thatis, thetwo wholemo-tives).The(multi-dimensional)valueof this interval iscalledtheanalog-interval. Whentwo analogiesof thesamekind have similar analog-interval, new analogiesare triggeredin the sameway as for the similarity ofpreviousintervals

���� � and ���� ��� .Ourhypotheticclaimwouldbethatthewholesetof

links automaticallyinferredby this theoryis sufficientto retrieve all theconceptinducedby traditionalmusi-cal analysis,and,muchbetter, all theunderstandingofmusic implicitly experiencedby a simple listeningofmusic.

Thearchitectureof kanthume, thatwewill now de-scribe,hasbeendeterminedin orderto fulfill this re-searchof analogies.

6 kanthume architecture

6.1 The relationship network

The note object. Eachnoteof the original scoreisrepresentedasa noteobjectinsidetherelationshipnet-work. Thenoteobjectcontainsthevalueof thenotepa-rameters:basically, pitch,dateandduration.Thenotesareinsertedinsidetherelationshipnetwork,incremen-tally andin a chronologicorder. The relationshipnet-work progressively digeststhe notesof eachnew in-stantof the score: that is, eachtime a new chord isadded,new relationshipspropagatealongthenetwork,which entersa stablestatebeforeconsideringnew mu-sicalevents.

Thenoteobjectalsofeaturespointersontothedif-ferentsequencesandanalogiesto which it belongs,ei-therasananaloguousnote,or asanelementof anana-loguousinterval.

The note parameter hash-tables. For eachpossiblenoteparameter, a hash-tableassociateseachparametervaluewith the setof the noteswherethis valueholds(seeFigure1). The pitch hash-tableis consideredintwo ways: in anabsolutepoint of view asa correspon-dancebetweenany pitchvalueandits occurrences,and

in a chromaticpoint of view asa correspondancebe-tweenany pitch of the chromaticscaleandits occur-rences.Thesecondpoint of view consistsof consider-ing the setof all the pitch valueequalto the absolutepitchvaluemodulo12.

The interval network. Eachtimeanew pitchor datevalueis addedto thehash-table,thenew valueis linkedto eachpossibleold value(thevalues,not theevents).For eachof thesesetsof links, forming two intervalnetworks(a pitch-intervalnetwork,anda time-intervalnetwork),is associatedthevalueof theinterval.

The interval hash-tables. As for noteobjects,twoadditionalhash-tables,onefor onsetandonefor pitch(seeFigure2),associateeachinterval valuewith thesetof thelinks insidetheintervalnetworkwherethisvalueholds. Onceagainthepitch interval hash-tablecanbeconsideredfollowing two pointsof view : theabsoluteoneandthechromaticone.

Note relationships. Eachnew note is linked to itscorrespondingnoteparameterhash-tables.Thesehash-tables,by definition, automaticallydetectsthe equal-ity of the currentnote parameterswith old ones. Inour framework, we proposenot to considerthesehash-tablesin a binary point of view. We prefer insteadadding a similarity-distancethat enablesto considernotonly equality, but alsosimilarity of values,for eachpossibleparameter. For eachcandidatenote is asso-ciated an activation degree that consistsof all thesesimilarity-distances,plus the note supportparameter(seeparagraph6.2). Whenactivationexceedsacertainactivation-threshold,ananalogyis inferredbetweenthecurrentnoteandtheactivatedone.

Interval relationships. The sameis true for inter-vals. The trouble is, the comparisonof every possi-ble interval from currentnotewith everypossibleotherinterval is of coursea taskthatmayexplodefor a longmusicalsequence.It is necessary, therefore,to limit thescopeof thestudyof interval relationships.Concerningthechoiceof interval from thecurrentnote,two factorsaretakeninto consideration:the time interval andthesupport(seeparagraph6.2)of thenoteat theotherex-tremity of this interval, that inducetwo new distances,namelytime-distanceandsupport-distance.Oncetheseintervalsarechosen,theactivationof relatedintervals(using two support-distancesfor eachextremity) andthe triggeringof analogyfollow themodelof notere-lationshipactivation,this time usingtheinterval hash-tables.For thecomparisonof absoluteintervals,a newdistanceis added:the pitch-distancebetweenthe twohighnotes(or two low notes).

6.2 The analogy network

The analogy object. Any analogyrelationshipmaybe representedin generalasa coupleof two analogs.The analogyobject also lists the parametersthat arecommonto theanalogs,andtheamountof correspond-ing similitude. Finally, as for noteobject, it containsa list of pointersto higher-orderanalogies,in which itbelongsasananalog.

Support. To any noteor analogyis associateda dy-namic parametercalledsupport, equalto the numberof analogiesof any orderthatareconstructedfrom it.This measureindicatesthe importanceof presentnoteor analogy, andplaysa role in detectionof new analo-gies:themoreanoteor analogyis supported,themoreit will beusedasananalogfor a new analogy.

Analogy inference. Whenconsideringtriggeringnewanalogies,several parametersaretakeninto account:the distanceof eachanaloguousinterval, the multidi-mensionaldegreeof similitudebetweeneachcandidateanalogue,thesupportof theanchor(seepoint 2 of sec-tion 5.2). Thesedifferent parametersare consideredin parallel,that is, analogiesaretriggeredif onepara-meter– or any collaborationbetweenseveral parame-ters – is particularlysignificative. To this frameworkis addedtheconstraintof a limited numberof trigger-ings: in caseof competition,only themostfavourablecandidateswill bechosen.Thedifferentfunctionsandthresholdthat control all thesecompetitions– whichexactly correspondto the competitive model of Hol-landet al. – canbeeditedby theuser.

7 An example of analysis

In orderto appreciatethemusicologicalinterestofsucha framework, hereis how kanthumeanalyzestheninefirst barsof thefifth symphony of Beethoven,re-ducedfor pianoasin figure3. Throughoutthis analy-sis, noteswill bedenotatedby thenumberof their in-stantand their rank within the instant– from high tolow – by a letter( % , & , ' ).7.1 Instant #1

Thevalueof theparameters(chromaticandabsolutepitche,duration)of thesethreefirst notesareregisteredin their respective hash-tables.Interval parameters(in-terpitchandinteronset)areregisteredtoo. Both threenoteshave samechromaticpitch (G), but since theyaresynchronized,they arenot consideredasanalogu-ous(becauseweonly consideranalogywith old notes).Idemfor thetwo octave intervals.

7.2 Instant #2

Similarity is detectedbetweeneachof thetwo cur-rent octave intervalsandeachof the two previous oc-tave intervals.Following point 3 of paragraph5.2,this

may triggerfour possibleanalogiesbetweenintervals.But point 2 is alsotrue: in particular, asthe two highnotesof both instantsareequal(sameabsolutepitch,sameduration),thenthereis ananalogybetweenthesetwo high notes,idem for the two extremenotes. Seescores1 and2 of figure3.

��()% � ()% � (*&+�������$�-,.% � ,/% � ,.&0��� (1)��()% � ()% � (*'+�������$�-,.% � ,/% � ,.'0��� (2)

Concerningnow the two low intervals, eachhighnotesis includedinsideeachanalogintervalsof anal-ogy 1. Hencetheanalogybetweenthesetwo intervalsfollows point 1 of paragraph5.2, that is, a motive iscreated.Seescore3 of figure3.

��()% � (*% � (*&+� � ()& � ()'+���1���$�-,.% � ,/% � ,.&0� � ,/& � ,.'+��� (3)Are also registeredthe intervals betweencurrent

andprecedentinstants,in particularthe unissoninter-vals: between1a and2a, between1b and2b andbe-tween1cand2c.

7.3 Instant #3

Theintervallic similaritybetween2aand3aandbe-tween1a and2a triggersnew analogy. Seescore4 offigure3.

��(*% � ()% � ,.%��������$�-,.% � ,/% ��2 %���� (4)Theintervallic similaritybetween2aand2bandbe-

tween3aand3b triggersananalogy.

�-,.% � ,.% � ,/&+�������$� 2 % �2 % ��2 &0��� (5)Similarity betweentheanalog-interval of analogies

1 and5 leads(seescore5) :

����(*% � ()% � ()&+����� ,/% � ,3% � ,/&������1���$��� ,/% � ,.% � ,.&+����� 2 % ��2 % ��2 &0�����(6)

We will not list all thepossibleanalogies,andwillpreferfocusinghereon importantones. In particular,asfor analogy3:

�-,.% � ,/% � ,/&+� � ,.& � ,.'+���1���$� 2 % �2 % ��2 &0� �2 & ��2 '+��� (7)Similarity betweentheanalog-interval of analogies

3 and7 leads(seescore6):

����(*% � ()% � ()&+� � ()& � (*'+����� ,/% � ,.% � ,/&+� � ,.& � ,.'+��������4���-,/% � ,.% � ,.&+� � ,.& � ,/'+����� 2 % �2 % ��2 &+� ��2 & ��2 '+����� (8)

7.4 Instant #4

Similarity of time interval between2a and3a andbetween3aand4a, inducesanextensionof analogy4(seescore7):

��(*% � ()% � ,.%�� � ,.% ��2 %����1���$�-,.% � ,/% ��2 %5� �2 % ��6 %���� (9)Similar asanalogies3 and7:

� 2 % ��2 % ��2 &+� ��2 & ��2 '0���1���$� 6 % ��6 % ��6 &+� ��6 & ��6 '+��� (10)Similarity betweentheanalog-intervalof analogies

7 and10 leads:

� �-,.% � ,.% � ,/&+� � ,.& � ,.'0����� 2 % ��2 % ��2 &+� ��2 & ��2 '+������� � � 2 % ��2 % ��2 &+� ��2 & ��2 '0����� 6 % ��6 % ��6 &+� ��6 & ��6 '+����(11)

Analogies8 and11 inducea sequencing(seescore8):

� � ��(*% � ()%7()&0� � (*&3()'0�����-,.% � ,/%5,/&+� � ,.&�,.'0����� � ,/% � ,.%�,/&+� � ,.&+,/'+����� 2 % �2 % 2 &+� �2 & 2 '+���8������4� � �-,/% � ,.%�,.&0� � ,/&+,.'0����� 2 % �2 % 2 &+� ��2 & 2 '0����� � 2 % �2 % 2 &+� ��2 & 2 '+����� 6 % ��6 % 6 &+� ��6 & 6 '+���8��� (12)

7.5 Instant #6

Now relative intervalssimilaritiesbetweenthe be-ginning of this motive and the previous one are de-tected(seescore9).

� � (*% � ()&0� � (*& � ()'+���9����� ��: % ��: &+� �: & ��: '+��� (13)7.6 Instant #7

Therepetitionis detected(seescore10):

� : % �: % ��: &+��������� ;/% � ;.% � ;.&+��� (14)andis comparedwith thefirst repetition(seescore

11 and12):

� (*% � ,.%��

7.8 Instant #14

Following similar way, after several sequencings,thenew motive is plainly detected(seescore14):

��()% � ()% � ,/%�� � ,/% ��2 %5� �2 % ��6 %�������?��(.( � (/( � (),.� � (*, � ( 2 � � ( 2@� ( 6 %5��� (18)� � >/% ��A %5���9���$� � ( 6 % � ( 6 &0��� (19)

Moreover, avery interestingrelationshipof intervalis inducedby similarity betweenanalog-intervals (seescore15):

� ����2 % ��6 %�� � >/% ��A %5����������� ��� >.% ��A %�� � ( 6 % � ( 6 &+����� (20)7.9 Instant #18

The threeoccurrencesof main motive (seescore16) is detectedis asimilarway. And alsotherepetitionof a samechord(seescore17).

7.10 Instant #34

Finally, the equivalencebetweenthe first andsec-ond half of the whole exampleis detected(seescore18).

8 Discussion

It canberemarkedthatwe shareMeyer’s ideacon-cerningmusicallistening,of a constantrelationof thepresentinstantwith known similar context (eitherex-periencedin thepastof theworkor learnedasanaspectof musicalstyle),but not theotherpart– themostim-portantone,accordingto him – of his theory, namelytheexpectationof learnedcontinuation.In a phenom-enologicterminology, thismeansthatwetakeinto con-siderationthe retentionalaspectof perception,but notthe protentionalone. It would be possible– andalsonecessary, if we would want to prolongethecognitivemetaphor– to implementtheprotentionalpart,but wewould like to know if it is possiblefor a cognitivesys-temsuchasacomputersimulation,to avoid protention.Wewouldtendto think that,whenfacingwith complexenvironment,protentionis necessary, becauseproten-tional capacityhassomegoodevolutionist reasonstoexist.

Althoughour systemgetsinspiredby cognitivere-searcheson inductive mechanismsandanalogy, its ar-chitecturehasbeenestablishedfollowingpragmaticcon-siderationsandphenomenologicintuitions.Thecogni-tivemodellingis usedhereonly asa kind of metaphor,in abiomimicdemarche.It wouldbeof agreatestinter-estto build a cognitively foundedmodel,by a collab-orationwith experimentalcognitivepsychologyandinparticularby measuringthe parametersof this modelthroughexperimentalmeasurements.

kanthumeis implementedasa library of Ircammu-sical representationsoftwareOpenMusic. Thepresentversion(OMkanthumeO.1) displaysthe resultsof itsanalysisthroughlist of texts,asshownin previouspara-graph.Addedto theproblematicof conceptionof cog-nitive modelling,arisesthen the questionof interfaceandergonomy. Theresultof theanalysishasto bedis-playedgraphically in a kind of network of relations,above thescoreitself. Becauseof its complexity – notgraphicallyrepresentableandin factnotcatchyfor hu-man – this networkshouldnot be entirely displayed,but only a partof it. Theusershouldbe ableto navi-gateinsidethis network,by choosingtemporalobjectsandhierarchicallevel of the network. Finally, in ourfirst version, following standardalgorithmic, the dif-ferenthypothesesareconsideredsequentially. In orderto follow carefully the cognitive metaphor, we wouldundoubtedlyneedto considera parallelmodel,for ex-ampleby implementinga multithreadedversion.

Thosearethekindsof questionsthatareconsideredin my currentPhD,directedby EmmanuelSaint-James(LIP6, ParisVI) andGérardAssayag(MusicalRepre-sentationTeam,Ircam).10

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10See website for up-to-date developments:http://www.ircam.fr/equipes/repmus/lartillot.

Figure1: Two noteobjectslinked to onsetandpitch hash-tables.

Figure2: Theintervalsbetweenpitch valuesD andF andbetweenE andG, bothworth3 semitones,arelinked totheinterval hash-table.

Figure3: kanthumeanalysisof pianoreductionof theninefirst barsof Beethoven’sfifth symphony.