Music, Mathematics and Language: The New Horizon of Computational Musicology Opened by Information Science PDF
Preview Music, Mathematics and Language: The New Horizon of Computational Musicology Opened by Information Science
Keiji Hirata Satoshi Tojo Masatoshi Hamanaka Music, Mathematics and Language The New Horizon of Computational Musicology Opened by Information Science Music, Mathematics and Language · · Keiji Hirata Satoshi Tojo Masatoshi Hamanaka Music, Mathematics and Language The New Horizon of Computational Musicology Opened by Information Science KeijiHirata SatoshiTojo DepartmentofComplexandIntelligent GraduateSchoolofInformationScience Systems JapanAdvancedInstituteofScience FutureUniversityHakodate andTechnology Hakodate,Hokkaido,Japan Ishikawa,Japan MasatoshiHamanaka MusicInformationIntelligenceTeam RIKENCenterforAdvancedIntelligence Project Chuo,Tokyo,Japan ISBN 978-981-19-5165-7 ISBN 978-981-19-5166-4 (eBook) https://doi.org/10.1007/978-981-19-5166-4 ©TheEditor(s)(ifapplicable)andTheAuthor(s),underexclusivelicensetoSpringerNature SingaporePteLtd.2022 Thisworkissubjecttocopyright.AllrightsaresolelyandexclusivelylicensedbythePublisher,whether thewholeorpartofthematerialisconcerned,specificallytherightsofreprinting,reuseofillustrations, recitation,broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionor informationstorageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilar methodologynowknownorhereafterdeveloped. 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The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721, Singapore Preface Wearelivinginaveryrichcultureofmusic.Nowadays,suchgenredistinctionsas jazz,rock,pop,andclassicalaremeaningless.Fusionhasoccurredin1970s;Miles Davis introduced electric instruments, as well as church modes used in medieval times,andhastriedtoopenanewhorizonforthefuturejazz.Evenaprofessional musiciantrainedinonegenrelookenjoyingothergenres.ToruTakemitsu,oneofthe best-knowncomposersinexperimentalmusic,issaidtohavehadastrongcompassion withSergeantPepper’sLonelyHeartsClubBandoftheBeatles.Suchatendencyof mixturewouldproceedfurtherinfuture. Atthesametime,theauthenticclassicalmusicseemslosingitspopularity.Those faithfulguardiansoftraditionmaystillthinkthattheclassicalmusicisabranchof liberalartsfortheeducatedclass,andthustheymayconsiderthatthefaithfulrepli- cationofwrittenscoresissignificant.However,archivingtheclassicsandenjoying music are two different things. Classical music was written when there were no recordingtechnology.Fromthecurrentviewpoint,therepriseofthesamephrasesis nomoreneeded,andthuswecouldshortentheirplayingtimetoavoidbeingbored. In the beginning of twentieth century, many composers tried to break an old style and to innovate music. One such representative is Arnold Schoenberg who triedtodestroy thetonalitywhich had governed thetraditionalmusic,introducing dodecaphony.However,thistrialwasnotsuccessfulatacquiringthepopularity.On thecontrary,so-calledpopularmusicseemsthemostfaithfulsuccessoroftherather oldstyleoftonalmusic,consistingofamelodyaccompaniedbyharmonicchords, andthestylehasfunctionedtoevokeouremotions. Still,wewantnewmusicpieces.Thereappearnewpiecesonthewebeveryday, and we try to catch them by digital methods like Apple Music, Amazon Prime, YouTube, and so on. We say a new hit song is new; surely, there may be a new constituentinachord,ormaybeanewprogressionofchordswhichhasnotbeen heardbefore,andsuchaninventionwouldappearfurthermoreaslongasthereare anexplosivenumberoftheircombinations.However,pleasedonotforgetthatthey are still written in do-re-mi-fa-sol-la-ti-do, with chords of do-mi-sol or sol-ti-re. Whatarealternatives?Thetemperamentofscaleshasalonghistoryinselectinga diatonic set of notes and in fixing the pitches of these notes in an octave. We feel v vi Preface harmonizedwhentheratiooftwofrequenciesisasimplerationalnumber,butanytwo notesarenotrigidlyharmonicinthecurrentlyprevalent12-toneequaltemperament, whichisaproductofmanycompromises.Yet,weappearsufficientlyadaptedtothis temperament. Do we need to abandon the temperament for exploring the new possibilities in music? Then, where is music headed? Who wants the change? Or, will songs are createdcontinuouslyinthesameformalism?Willmusicforeverprogressandevolve, orwillitsaturateatsomepoint?Ifmusicisgoingtosaturatesomewhere,istherea limittohumancreativity?Whatismusicinthefirstplace?Composers,performers, andmusicologistsofthepastmusthavebeenraisingthesamequestionsandinves- tigating music. They must have taken different approaches and set different goals, wonderingwhattechniques,skills,andknowledgetheywouldneedtoacquireinorder tocreatenewmusic.Theresultsincludeactualmusicalmasterpieces,performances, andmusictheories. Inthelasthalf-century,weobtainedanapparatustomirrorourcognitiveprocesses andintelligentfunctions—acomputer.Nowadays,weareabletosimulatetosome extentourbrainfunctionsandintelligentactivitiesinsidethecomputersinisolated environments from the real world, and to even conduct as many experiments that would not be allowed in reality as required. We call a virtual person who owns itsindependentandsubjectivecognitionanagent.Thenotionofcognitionmaybe different among various psychologists; some requires a perfect ego, that is, being conscious of ‘self,’ and others are rather tolerant of the definition and may call a robot with a simple inference mechanism with independent knowledge base an agent. Naturally, implementing a cognitive and emotional ego agent in computer with a rigorous description in formal language is difficult. In the early days of an electroniccomputer,suchanoptimisticpredictionthathumanintelligencecouldbe realizedinacomputerwithin10or20yearswasprevalent.Itwasin1948whenthe precursoroftheideafortheTuringtestwaspublished.Sincethen,therehavebeen severalcyclesofenthusiasmanddisappointmentalongwithexcessiveexpectations andtirelesstrialsanderrors.WearenowinthemidstofanotherAIboom,ineffect a deep learning boom; these ebbs and flows can be also regarded as the history of developmenttodescribetheintelligentactivitiesandfunctions. If music is one of the intellectual activities of human beings, we believe that tackling the seemingly unanswered question “what is intelligence in music?” or “what is beauty in music?” using the powerful tool of a computer can provide us withsomeusefulnovelinsights.Itwouldbemeaningfultoconstructacomputational musictheorybasedontraditionalmusictheory,justascomputationallinguisticshas beenconstructedandhasproducedvaluableresultsbasedontraditionallinguistics. Oneofthepurposesofwritingthisbookwastodemonstrateanapproachtorealizing thiscomputationalmusictheory. Now,tofindthekeytorealizingcomputationalmusictheory,letusgobackand thinkabouttheclaimthattheoriginofmusicandlanguageisoneandthesame.As wecannotobtainfossilsofoldlanguages,wecannotprovethisintheevolutionary linguistics. However, observing various facts, we can convince ourselves that the claimseemssomewhatplausible.Apartfromsuchlinguisticaspect,thelanguageis Preface vii themostusefulmethodtoexternalizethenotionofbeauty.Toexpressthebeautyin language means that we symbolize and find structure of music as language. Jean- JacquesNattiez,followinghismasterJeanMolino,consideredthatmusiccouldbe structured and analyzed with regard to semiology, and thus the meaning of music could be externalized. However, in Nattiez’s book, he confessed his own remark includedacontradiction1.Inhisinterviewwithvirtualhimself,hehonestlyadmitted thatitwasimpossibletofullyexpressthemusicbeautyinsymbols.AsNattiezfinally claimedthatthereshouldbeanadequatetheoryoflinguisticstoconstructameaning ofmusic,webelievethatclarifyingthegrammarstructureofmusicisanessential andmoststeadywaytoacomputationalmusictheory. Inthisbook,wefirstconsiderthemeaningofmusic,regardingcognitivereality. Tothispurpose,weintroducethetheoryofHeinrichSchenker2,whereheclaimed that there was a fundamental skeleton common in every music piece, and we can obtain this via the procedure called reduction and how these skeleton worked on our cognition. Next, we consider why the 12-tone equal temperament came to be prevalent in Chap. 2 titled ‘The mathematics of ebony and ivory keys.’ The equal temperament,aswehavetold,isaresultofcompromises;however,wecaneasilyget accustomedtoitandcanenjoymusicwithit.Thischapterfunctionsasanintroduction tomathematicaldefinitionofharmonythatconcernstheratiosoffrequencyintonic waves.Chapter3‘Musicandlanguage’explainsthefundamentalsofgrammartheory aswellasthecompositionalityprinciple,thatis,thesemanticsofasentencecanbe composedinparalleltosyntacticstructure.Wewillcontendthatmusicalsopossesses thesamecontext-freegrammarasnaturallanguagethoughsomewhatinaweaksense. In Chap. 4, we explain the most prevalent score notation called Berklee method, which was initiated from Berklee School of Music in New York. Nowadays, we mostlyemploychordnamesaccordingtothemethodinpopularmusicandconsider howefficientthismethodis.InChaps.5and6,wegivetwotheoriesoncomputational musicology.WefirstintroduceImplication-Realizationmodel(IRmodel),thatisa theorytoreflectour human intuitiononmelody progression.Next,weexplain the GenerativeTheoryofTonalMusic(GTTM)withitscomplementarytheoryofTonal PitchSpace(TPS).Chapters7,8,and9aredevotedtotheapplicationsofGTTM.In Chap.7,wediscussalgebraicoperationsontrees.InChap.8,wereportoureffortsto implementGTTManalyzerwhichretrievesatreefromagivenpiece.InChap.9,we illustrateourworkonmusicmorph-ing,togenerateanewmusicpiecefromgiven two other pieces. Finally, Epilogue, we discuss the future of music with regard to computationandAI,posingHilbert-stylequestions. Thisbookaimsatageneralreaderofmusic,inadditiontoprofessionalscientists of music information processing. Therefore, we try to make each chapter be as 1Jean-Jacques Nattiez. “Je vous prends en flagrant délit de contradiction!” in La Musique, La Recherche,etLaVie,LeméacEditeurInc.,Montréal,(1999/2004). 2HeinrichSchenker,DerFreieSatz.NeuemusikalischeTheorienundPhantasien,Margada,Liège, Belgium(1935). viii Preface independentofotherchaptersaspossible.Thus,wewouldliketoshareacommon interestindiscussing‘musicologyincomputerage’inbroadercommunity. Hakodate,Japan KeijiHirata Ishikawa,Japan SatoshiTojo Tokyo,Japan MasatoshiHamanaka March2022 Acknowledgements Asauthors,wewouldliketothankeveryonewhohasbeeninvolvedinourresearch; research collaborations, discussions and chats at conferences, laboratory visits, hostingresidencies,emailexchanges,events,andsoonhaveallcontributedtothe resultsofourresearch. We would like to thank Tory Honor for English rewriting, corrections, and translation. WewouldliketothankMioSuginoofSpringer.Infact,wewouldnothavebeen able to publish this book without the patient but kind encouragement and careful supportofMioSuginothroughoutthisproject. MuchoftheworkpresentedinthisbookwassupportedbyJSPSKAKENHIGrant Numbers20300035,23500145,25330434,26280089,16H01744,and21H03572. ix Contents 1 MachinethatComputestheMeaningofMusic .................... 1 1.1 MusicTheorythatEnjoystheBenefitsofComputers ........... 1 1.2 AShortHistoryofProgramMusic ........................... 3 1.3 UnderstandingMusicfromaGestaltPointofView ............. 5 1.4 AttributionofMeaningtoMelodybySemiotics ............... 8 1.5 CategorizationofMusicalMeaning .......................... 11 1.6 ReductionHypothesisandSchenkerianAnalysis ............... 13 1.7 AnalysisofRetrograde ..................................... 15 1.8 UnderstandingMusicthatImitatesUnderstandingLanguage ..... 19 1.9 CognitiveReality .......................................... 23 References ..................................................... 25 2 TheMathematicsofEbonyandIvoryKeys ....................... 27 2.1 Scalewith2and3 ......................................... 27 2.2 RevisedPythagorasScale ................................... 31 2.3 JustIntonation—Introductionof5 ........................... 33 2.4 Eitz’sNotation ............................................ 34 2.5 WhatIsOvertone? ......................................... 36 2.6 Meantone ................................................ 37 2.7 IssueonModulation ....................................... 38 2.8 EqualTemperament ....................................... 39 2.9 Scale,Mode,andKey ...................................... 43 2.10 Historyof12-ToneEqualTemperament ...................... 44 2.11 Symmetry ................................................ 46 References ..................................................... 49 3 MusicasFormalLanguage ...................................... 51 3.1 ChomskyandGenerativeGrammar .......................... 52 3.2 FormalLanguageandAutomata ............................. 55 3.3 HumanLanguageintheHierarchy ........................... 61 3.4 LanguageClassofChordProgression ........................ 63 3.4.1 Context-FreeSyntaxofCadence ...................... 63 xi
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