The languages of western tonality PDF
Preview The languages of western tonality
Computational Music Science Eytan Agmon The Languages of Western Tonality Computational Music Science Series Editors Guerino Mazzola Moreno Andreatta For furthervolumes: http://www.springer.com/series/8349 ThiSisaFMBlankPage Eytan Agmon The Languages of Western Tonality EytanAgmon DepartmentofMusic Bar-IlanUniversity Ramat-Gan Israel ISSN1868-0305 ISSN1868-0313(electronic) ComputationalMusicScience ISBN978-3-642-39586-4 ISBN978-3-642-39587-1(eBook) DOI10.1007/978-3-642-39587-1 SpringerHeidelbergNewYorkDordrechtLondon LibraryofCongressControlNumber:2013951826 ©Springer-VerlagBerlinHeidelberg2013 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpart of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation,broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionor informationstorageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilar methodologynowknownorhereafterdeveloped.Exemptedfromthislegalreservationarebriefexcerpts inconnectionwithreviewsorscholarlyanalysisormaterialsuppliedspecificallyforthepurposeofbeing enteredandexecutedonacomputersystem,forexclusiveusebythepurchaserofthework.Duplication ofthispublicationorpartsthereofispermittedonlyundertheprovisionsoftheCopyrightLawofthe Publisher’s location, in its current version, and permission for use must always be obtained from Springer.PermissionsforusemaybeobtainedthroughRightsLinkattheCopyrightClearanceCenter. ViolationsareliabletoprosecutionundertherespectiveCopyrightLaw. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publicationdoesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexempt fromtherelevantprotectivelawsandregulationsandthereforefreeforgeneraluse. While the advice and information in this book are believed to be true and accurate at the date of publication,neithertheauthorsnortheeditorsnorthepublishercanacceptanylegalresponsibilityfor anyerrorsoromissionsthatmaybemade.Thepublishermakesnowarranty,expressorimplied,with respecttothematerialcontainedherein. Printedonacid-freepaper SpringerispartofSpringerScience+BusinessMedia(www.springer.com) Preface ... Each of these four notes governs, as its subjects, a pair of tropes.... Thus every melody... is necessarily led back to one of these same four [notes]. Therefore they are called“finals,”becauseanythingthatissungfindsitsending(finem)in[oneof]them. —HucbaldofSaint-Amand,lateNinthCentury1 As an object of inquiry, “tonal music” is far from homogenous. The music of the ninthcenturywithwhichHucbaldwasfamiliarwasverydifferent,wemayassume, from (say) the music of the seventeenth century. Nonetheless, there are striking points of contact. Most notably, a “background system” of exactly seven notes, orderablewithintheoctaveasacyclicpermutationofthesequenceT-T-S-T-T-T-S oftonesandsemitones(alternatively,asasequenceofperfectfifths—thefifthbeing themostprivilegedintervalfollowingtheperfectoctave),is“governed”byoneof its members—the final (and “co-governed,” one might add, by another—the cofinal,theperfectfifthabovethefinal).Howcanoneaccountfortheremarkable stabilityofsuchbasicfeaturesof“Tonality,”and,atthesametime,dojusticetothe equallyremarkablevarietyofstyles—nay,languages—thatthehistoryofWestern tonalmusichastaughtusexist?Thisbookisanattempttoanswerthesequestions. The book is divided into two main parts. Part I, Proto-tonality, studies the background system of notes prior to the selection of a final. The “proto-tonal system”ultimatelypositedisharmonicandcontainsa“harmonicmessage.”How- ever,theharmonicmessagemaybeempty,inwhichcasethesystemreducestoits diatonic component. In other words, a harmonic system is diatonic, but not vice versa(adiatonicsystemisobliviousofsuchconstructsas“chord,”“chordprogres- sion,”and“voiceleading”).Animportantcomponentofeverydiatonicsystemisits “core”:alength-7segmentofthe“lineoffifths,”forexample,F,C,...,B. After some preliminaries that concern consonance and chromaticism, Part II, The Languages ofWestern Tonality (also the title ofthe book as awhole), begins withthenotion“mode.”Amodeisassumedtocontaina“nucleus.”Anucleusisa subset of the core that is consonant while containing a maximal number of 1Trans.fromCohen(2002),p.322. v vi Preface elements.Themode’sfinalistheuniquenucleuselementthatisarootrelativeto every other nucleus element. For example, the final would be the lower member (ratherthantheupper)ofaperfect-fifthdyadicsubsetofthenucleus. Crucially, the notion of “consonance” is itself dependent on the proto-tonal system’s status as diatonic or harmonic. In particular, in the harmonic case the major and minor third and sixth are deemed consonant in addition to the diatonic perfectoctave,fifth,andfourth.Asaresult,themodalnucleusisaperfect-fifthdyad inthediatoniccaseandatriad(majororminor)intheharmoniccase.EarlyinPart II, in other words, a distinction is established between two types of mode, dyadic and triadic. Moreover, within each type, the degree to which the nucleus (or a privilegedsubsetthereof)isconsonantinrelationtothenon-nucleuscoreelements defines“semi-key”asaspecialtypeofmodeand“key”asaspecialtypeofsemi- key. On the basis of these distinctions, seventeenth-century music (for example) presents itself as a (tonal) language of triadic semi-keys; ninth-century music, by contrast,isa(tonal)languageofdyadicmodes. Thus,inanutshell,thetheoryaccountsfortonalvariety. Thequestionoftonal stabilityisaddressedmainlyintheproto-tonalPartI,thoughitcontinuestoinform ideas put forth in Part II as well. In a nutshell again, tonality is seen as a highly successful “communication system.” Communication, indeed, is the most impor- tanthigh-levelprinciplethatguidesthetheoryofferedinthisbook. Theory; History; Cognition Fromtheprecedingremarksitisclearthatthebookdrawsuponthreedistinctfields ofstudy,namelymusictheory,musichistory,andmusiccognition.Likethethree edges of an equilateral triangle, the contribution of each field to the project as a wholeisinconceivablewithouttheothertwo. Musictheoryistheoldestandmostestablishedofthethree.Musictheoryhasnot only handed us, early in the nineteenth century, a valuable though elusive concept—Tonality—but is on record for centuries if not millennia for attempting todemystifythatextraordinarygiftofmankindtoitself:music.Thestoryofmusic theory is fascinating in its own right, replete as it is with turns and twists, progressions and regressions. Be that as it may, the present project is unthinkable intheabsenceoftherichandcomplexheritageofideasthatconstitutethemusic- theoreticendeavor. It is all too easy to absorb oneself in the familiar and the readily accessible, forgettingnotonlythatthepastmayhavebeendifferentfromthepresent,butalso thatthepresentmaybeverydifferentelsewhere.Historicalmusicologyandethno- musicologyhavetaughtustorespectthechronologicalandculturalOther. AsshouldbeclearfromthePreface’sopeningremarks,thisstudytakesseriously the historical challenge, offering a theory that, while not explicitly diachronic, nonethelessrendersconceivableahistoricalprocessofthesortthatseemstohave takenplaceinWesternculture,namelyfromdyadictotriadictonality,andmoreor Preface vii lessconcurrently,frommodes,throughsemi-keys,tokeys.Self-consciouslyfocusing on the West, the study is obviously less responsive to the cultural challenge. Nonetheless, a reference to the Javanese pe´log scale may be found in Part I, Sect.8.1;areferencetonorthIndianmusicmaybefoundinPartII,Sect.14.1. Finally,musiccognitionhasmadeuskeenlyawarethatmusicisareflectionof thehumanmind.Incurrentmusic-cognitivediscoursemuchemphasisisplacedon perception. The present book, by contrast, places equal emphasis on conception. The difference reflects the book’s communicative bias, coupled with the observa- tionthatcommunicationtakes placewhere perceptionmeets conception. Inmany ways, we shall see, the languages of Western tonality reflect the logical and cognitiveconstraintsthatmakemusicalcommunicationpossible. Even for the professional music theorist, the book is no easy reading. This is especially true of Part I, Proto-tonality, where abstraction and formal rigor reign supreme. Formal rigor in the book, however, is no ornament. As explained in Chap.2,thebookstrivesforthehighestpossiblestandardofscientificacceptability, namelyexplanatoryadequacy.Tothisend,itwasnecessarytostriptonalitydown to its barest elements, generalizing parameters whenever possible. That this approachpaysoffbecomesapparentalreadyinChap.8,whereanalternativetheory iscomparedtotheproposedtheorypreciselyintermsofexplanatoryadequacy. Anumberofstrategiesmayhelptheinterestedreaderovercomethedifficulties of Part I. The tried-and-true strategy of nonsequential reading may not only help combatfatiguebutmayofferalarger(ifnotcompletelycoherent)perspectivefrom which the intricacies of a given phase in the theory may be easier to digest. In particular,Chap.4,“TheConventionalNomenclaturesforNotesandIntervals,”isa relativelyaccessibleexercisethatmaybefruitfullystudiedbeforeorconcurrently with Chap. 3. In general, examples of formal definitions are presented after the definitionsthemselves.Easier accessintoanabstruse definitionmaybegainedby skippingaheadtoanaccompanyingexample. The reader may feel overwhelmed by the sheer number of definitions and notationsintroduced.Partialhelpinthismatterisprovidedbythreeindicesfound in the back of the book: a General Index, where formal definitions are easily identified by the corresponding page number’s formatting in bold; a List of Definitions; and a List of Notation. Two mathematical appendices provide the basicmathematicalbackground. The origins of this book go back to my 1986 PhD dissertation, “Diatonicism, Chromaticism, and Enharmonicism: A Study in Cognition and Perception.” The bookhasthusacquiredmanydebtsovertheyears. Carl Schachter, the dissertation’s supervisor, with whom I spent a couple of yearsasagraduatestudentattheCityUniversityofNewYork,hashadaprofound influenceonmyidentityasamusictheorist.Schachter’simmenseknowledgeand deep understanding of music and music theory, his astounding eloquence in verbalizing his ideas and insights, his generosity towards students and colleagues alike, and his down-to-earth, unpretentious human warmth have enriched and nourishedmeforlife. viii Preface I was very lucky to have had Joel Lester as a reader of my dissertation. Lester showed interest in my work early on and has always been extremely generous in lending support and guidance. Lester provided important feedback on early versionsofChaps.12and13. JohnRinkandWilliamRothsteincommentedinsightfullyonanearlyversionof Chap.15.Section15.2infactdevelopedfromabrilliantsuggestionofRothstein’s. Anumberofindividualsprovidedinvaluablehelpwiththemathematicalaspects ofthebook. Ifeelveryluckytohavetworenownedmathematiciansinmyimmediatefamily: my father, Shmuel Agmon, and my brother, Ehud de-Shalit. Both have been involved in my work since the dissertation. Without their patient guidance over theyears,thisbookcouldnothavecomeintobeing. In2010(cid:1)2011NoriJacobyandIgaveacourseatBar-IlanUniversity,“Music, Mathematics, and Cognition,” based on Part I of the book. Nori contributed numerous improvements to the mathematics, substantive as well as stylistic. ThanksmustalsogotoThomasNoll,whoreadPartIandofferedvaluableinsights andsuggestions,toAvinoam Bravermanforcommenting insightfullyonChap.3, andtoReuvenNavehforcommenting onaveryearlyversionofChap.4. Although it has become a cliche´ to thank one’s spouse and children, in the present case mywife, Lea, andtwo wonderfuldaughters, Einatand Orly, are true partnersintheendeavor.Mywife,anaccomplishedmusicianinherownright,has been partofthisprojectfromthe verystart,lending her ear,heart,andmindwith uncompromising patience and devotion at every turn. My daughters have grown accustomedtomusic-theoreticdiscussionsatfamilydinners.Theyhavefoundearly onthatthebestwaytogetDaddygenuinelyupsetistoarguewithhimthatthereis nodifferencebetweenG andA. IdedicatethisbooktoLea,Einat,andOrly,withlove. Reference Cohen,D.(2002).Notes,scales,andmodesintheearliermiddleages.InT.Christensen(Ed.),The CambridgehistoryofWesternmusictheory(pp.307–363).Cambridge:CambridgeUniversity Press. Contents 1 Proto-tonalTheory:TappingintoNinth-CenturyInsights. . . . . . . 1 References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 PartI Proto-tonality 2 Preliminaries. .. . . .. . . .. . . .. . . .. . . .. . .. . . .. . . .. . . .. . . .. 19 2.1 DescriptiveandExplanatoryProto-tonalAdequacy: ALessonfromLinguistics. . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 2.2 TheCommunicationPrinciple. . . . . . . . . . . . . . . . . . . . . . . . . . 24 2.3 ThreeAdditionalGuidingIdeas. . . . . . . . . . .. . . . . . . . . . . .. . 26 2.3.1 TheEconomicalPrinciple. . . . . . . . . . . . . . . . . . . . . . . 26 2.3.2 TheCategoricalPrinciple. . . . . . . . . . . . . . . . . . . . . . . . 26 2.3.3 TheMaximalistPrinciple. . . . . . . . . . . . . . . . . . . . . . . . 27 2.4 EventSequences. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 3 CommunicatingPitchesandTransmittingNotes. . . . . . . . . . . . . . 31 3.1 Octave-EndowedNoteSystems. . . . . . . . . . . . . . . . . . . . . . . . . 31 3.2 BasesoftheIntervalSpace. . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 3.3 Pitch-CommunicationSystems. . . . . . . . . . . . . . . . . . . . . . . . . 36 3.4 Absolute,Relative,andReflexivePitchCommunication. . . . . . . 45 3.4.1 TwoPostscripts. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 3.5 CompositeToneSystems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 4 TheConventionalNomenclaturesforNotesandIntervals. . . . . . . 59 4.1 TheConventionalNomenclaturesforNotesandIntervals. . . . . . 60 4.2 StaffNotationandItsIdiosyncrasies. . . . . . . . . . . . . . . . . . . . . 63 ix
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