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Conditional and spontaneous asymmetry of harmonicprogressions in madrigal cycles from Verdelot to

MonteverdiChristophe Guillotel-Nothmann

To cite this version:Christophe Guillotel-Nothmann. Conditional and spontaneous asymmetry of harmonic progressionsin madrigal cycles from Verdelot to Monteverdi. 2017. �halshs-01617805�

ChristopheGuillotel-NothmannCNRS-IReMusUMR8223,chargéderecherchecontractuel

christophe.guillotel-nothmann@cnrs.fr

Conditional and spontaneous asymmetry of harmonicprogressionsinmadrigalcyclesfromVerdelottoMonteverdi.

[1]Thetheoryofharmonicvectors(THV)postulatesthatofthesixpossiblerootprogressionsinagiventonalitythoseupafourth,downathirdandupasecond(+4,-3,+2)arepresentinsignificantlygreaternumbersthanthecomplementaryrootmotionsdownafourth,upathirdanddownasecond(-4,+3,-2)(Meeùs1988,1989,2000).Thisimbalancebetweenbothrootprogression categories is considered to shed light on a specific aspectof the tonal system(Meeùs2001,63).

[2]PreviousstudieshaveconfirmedthetendencyfortheprimarygrouptodominateingenresrangingfromRenaissancepolyphonyto20thcenturypopularmusic(Desbordes2001,Meyer2009,O'Donnell2011,Cathé2012,Guillotel-Nothmann2013).Butwhilerootprogressionsofthefirsttypedobecomemoreprevalentintonality,theincreaseisnotasdramaticasmighthavebeenexpected(Tymoczko2003,43;Hedges&Rohrmeier2011).Thiscouldsuggestthattheprevalenceislesscrucialforharmonictonalitythaninitiallythought.However,thenatureoftheseprogressionsalsoneedstobeconsidered.

[3]Iwillarguethatitislessthechangeinfrequencyofthesepreferredprogressionsthatiscriticalforharmonictonality,thanthechangeintheirquality.Theprogressions+4,-3,+2arisealmostaccidentallyinpre-tonalpolyphonythroughtheconstraintsofcontrapuntalrules.Onthecontrary, in later repertoires, theybecomeadecisivesyntactical featurewhichactivelyconstrainstonality.

[4]Totestthishypothesis,amodelthatcombinesvoice-leadingandharmonicprogressionswillbeconsideredagainstabodyofmadrigalcyclesbyVerdelot,Arcadelt,Lassus,Rore,WertandMonteverdi.Thesecycles,publishedbetweenc.1530and1638,containabout50000chordprogressions.Theempiricalresultsinconjunctionwiththemodelwillallowforacloseexaminationofhowandwhyoneparticulargroupofrootprogressionsdominates.Theywillshowthephenomenathatreflectthechangingstatusoftheprevalentrootprogressionsandthe technical aspects which may have fostered it. Finally, they will help to identify thecompositionalpossibilitieswhichresultfromtheevolutionoutlined.

1.Vectors,Voice-leadingandAsymmetryofrootprogressions

1.1.TheTheoryofHarmonicVectors

[5]TheTHVisbasedonasystematicclassificationofrootprogressionsandprovidesrulesofsyntaxthatconstitutetheembryoofatonalgrammar(Meeùs2003,8).TheTheorycategorizesharmonicprogressionsintotwodistinctgroupsofdominantandsubdominantfunctions.Eachgroupincludesonemainprogressionthatmovesbyfourthandtwosubstituteprogressionswhichmovebyathirdorasecond(example1).

Dominantvectors Subdominantvectors

Mainprogression +4 -4

Substitutions -3 +3

+2 -2

Example1.ClassificationofharmonicprogressionsintheTHV.

[6]Theprogressionupafourth(+4)andthesubstitutionsdownathird(-3)andupasecond(+2)areclassifiedasdominantvectors.Thecomplementarychordprogressions (-4,+3, -2)belongtoacategoryofsubdominantvectors.Whilethemainprogressionsarenamedwithreference to the dominant and subdominant progressions in the perfect (V-I) and plagalcadence (IV-I), the substitutions are inferred – in the theory’s initial formulation – fromRameau’sdoubleemploi (Rameau1737)andRiemannianfunctionalequivalences(Riemann19097).

[7]Studiesofcorporaofsixteenthandseventeenthcenturymusichaveshownthatdominantvectorsarealwaysinthemajority.Thisimbalanceincreases,andgraduallystabilizeshoweverfromthe17thcenturyonwards(Cathé2012).

[8] The hegemony of dominant vectors can be deduced most effectively from the tonalcadence I-IV-V-I-IV-I (example 2), where all progressions, with the exception of the plagalclosure IV-I, correspond to dominant vectors. Following Tymoczko (2003, 38), I term thisimbalance as the ‘asymmetry of root progressions’. This asymmetry corresponds to thedifferenceinfrequencybetweendominantvectors(DV)andsubdominantvectors(SV):

Asymmetry=DV–SV

[9]IntheTHVthevectorcategoriesareassignedopposingdirections.Thesearevisualisedwitharrows to the right, for dominant progressions, and arrows to the left for subdominant

progressions(seeexample2).Thepredominanceoftherightwardarrowsmakesexplicitthecadentialteleologywhichplaysakeyroleintonality.Thiscadentialdirectionwillbereferredto as the ‘privileged direction’ of chord progressions. The THV does not claim that tonalharmonycanbeexclusivelyreducedtodominantprogressions1.Itdoeshighlighthoweverthecrucialroleoftheseprogressionsintonalsyntax.

Example2.Paradigmaticcadencewithharmonicvectors.

1.2.Model

[10]Asshallbedemonstrated,theasymmetryofrootprogressionshasitsrootsincadencepatternswhichgobacktothelateMiddleAgesandtheearlyRenaissance.

[11]Theearliestmadrigalsconsideredinthisstudydatefromtheearlysixteenthcentury,atime when the cadence becomes a locus of theoretical thought. Through the concept ofclausula formalis, and taking as point of departure the idiomatic melodic and harmonicformulae associated with the cantus and the tenor, theorists of that time describe theadditionalbassusandaltus lines,thevoice’spermutation,the interminglingofdissonancesandtheexceptionofthemi-cadence.

[12] The intervallicprogression sixth tooctavebetween thepenultimateand the finalisofexample 3 has remarkable qualities. It combines a gradual transition from the relativeimperfection (imperfect consonance) to the relative perfection (perfect consonance) withstepwiseupward(cantus)anddownward(tenor)motion.Thischaracteristicprogression(anditscomplementaryprogressionthirdtounison)becomesestablishedinthe14thcenturyasthecanonicalcadentialformulaofthecantus-tenorframework(Eberlein1992,34)2.Fromthe15thcenturyonwards,becausetriadicharmonyweakensthedistinctionbetween imperfectandperfect consonances, this formula is regularly preceded by a dissonant suspension. Thisdissonanceontheantepenultimate(D4-C5inexample3)expandsandreinforcesthecadencebylaunchingtheteleologicaldriveearlier(Dahlhaus1990a).

1 Tymoczko(2003,46-47)drawsattentiontotheparticularstatusofsubdominantprogressions.Occurringonspecificscaledegrees,theycanplayacriticalroleinestablishingtonality,asforexamplethesubdominantprogressionsincludedinthea-b-apatternsI-V-IandI-IV-I.2 Theoristsofthe14thcenturyawardaspecialstatustotwoparticularprogressions:1.majorsixth–octave2.minor third – unison. They thus systematically apply the principle of voice-leading proximity to intervallicclasses.

Example3.Cantus-tenorframework.

[13]Themodelofexample4presentsthetriadiccontextsthatmayfollowfromthecantus-tenorframeworkin3a.Itshowsthepossibleharmonizationsandcompatiblerootsthroughwhichthecantusandtenorlinesmaypass.TheinitialconsonanceE4-C5whichpreparesthesuspensionmaybeharmonizedbyatriadoneitherCorA(inblack).ThedissonanceD4-C5thatfollowscanbeharmonizedbytriadsonD,BorG(indarkgrey).TheresolutionontotheD4-B4 imperfectconsonance that follows isharmonizedby triadsonBorG (in lightgrey).Finally,theresolutionontotheperfectconsonanceC4-C5istiedtotherootsC,AandF(inblack).Theseharmonizationsleadtoseveralobservations.

Example4a.Model:Harmonicprogressionsimpliedbythecantus-tenorframework.

[14] The cadence-pattern as awhole implies a fall through a cycle of thirds between thedifferent intervals of the cadential chain. The arrows in example 4a represent the rootprogressionsthatmayoccurbetweenthepreparation,theimpactandtheresolutionofthedissonance. They indicate that in the harmonic progressions generated, dominant vectors(56%) occur more frequently than subdominant vectors (44%). This confirms that underspecific conditions the cantus-tenor framework is a potential source of asymmetry. Theimbalancebetweenbothvectorcategoriesisyetnotablylow.

[15]Somerootsarehoweverexcludedfromthecadencepattern.TheharmonizationofthefinalconsonanceC4-C5usingatriadwithrootF,althoughtheoreticallypossible,wasnotincludedhere.WiththeexceptionofthePhrygiancadence(theharmonizationofafinalEwithatriadonA),composerstendtoavoidtheharmonizationofthefinalbythelowerfifth.IfoneexcludethechordonrootFasfinalchord,theasymmetrybetweendominant(64%)andsubdominant(35%)vectorssignificantlyincreases,asshownintheanimationofexample4b.

Example4b.Model:Harmonicprogressionsimpliedbythecantus-tenorframeworkwithrestrictedfinalchord.

[16]Furthermore,thethirdtriadinthecycle(alsoonF)cannotbeinvolvedinthecadencepatternasthetenorlineexcludesitsuse.ThesameappliestothetriadonE,theseventhtriadinthecycle.

[17]Thetenorlineishowevernotentirelystableintheoryandinpractice.Itcanbereplacedbythemovement^1-^2-^1orbeabsent(seeEberlein1992,56-62and2.1).Inthefirstcase(^1-^2-^1),harmonizationwithatriadonFbecomespossibleatthebeginningandleadstothree additional dominant progressions (F-D, F-B and F-G) thus reinforcing the asymmetrybetween dominant (71%) and subdominant vectors (29%), as shown by the animation inexample4c.

Example4c.Model:Harmonicprogressionsimpliedbythemodifiedtenorline^1-^2-^1inthecantus-tenorframework.

[18]Inthesecondcase(absenceofthetenorline),thepossibilitiesevenincrease:atriadwithrootEbecomespossibleattheresolutiononB.Thisleadsagaintointensifyingtheasymmetrybetweendominantvectors (76%)andsubdoinantvectors (36%)byallowing fiveadditionaldominantprogressions,asinexample5(D-E,B-E,G-E,E-C,E-A)3,asshownbytheanimationofexample4d.

Example4d.Model:Harmonicprogressionsimpliedbythecantusline.

3TherelationshipbetweentheevolutionofcadentiallinesandharmonicprogressionshasbeenstudiedindetailinGuillotel-Nothmann&Meyer2013andGuillotel-Nothmann2015,459-469.

[19]Interestingly,thismodelhasaffinitieswithTymoczko’sthird-based-grammarofelementarytonalharmony(Tymoczko,2011,226-30).Bothfavorthedownwarddirectioninthecycleofthirdswiththeupwardmotionlimitedtoarestrictednumberofprogressions.Theyalsobothrestricttheuseofthechordonscaledegreeiii(Einexample3)butallowsubdominantprogressions,especiallyinI-V-I,orI-VII-I(C-G-C,C-B-Cinexample2).[20]Themodeldoesnotclaim,however,tobeanaccuraterepresentationoftonalharmony.Its contrapuntal constraintsdonotequate to constraints that govern tonalorganization. ItneverthelessallowsseveralimportantobservationsaboutasymmetryinWesternpolyphonyanditslinkswithothercharacteristicsofharmonictonality.

[21]1.Themodelsupportsthehypothesisthatdominantprogressionsarenottheresultoftonalitybut,onthecontrary,helptocreatesomeofitscharacteristics,bothontologicallyandhistorically.Itsuggeststhatrulesofroot-motionthatconstraintonalitymighthaveevolvedatanearlierstagethroughcontrapuntalconstraintsatthecadence.ThisindirectlycorroboratesanintuitionLowinsky(1962,4)hadwhenhequalifiedthecadenceasthe“cradleoftonality”.

[22]2.Themodelalsoshedsanotherlightontheconceptofsubstitution.TheoriesofchordprogressionfromRameau(1721)toDeJong&Noll(2008)andtheoriesofharmonicfunctionsuch as Riemann’s Funktionstheorie assume (at least implicitly) a hierarchy betweenmainrepresentativesandsubstitutions4.Ontheotherhand,thecontrapuntalperspectivepresentedin thismodel conceives of the alternative harmonizations as equivalent. This equivalenceresults fromtheharmonicaffinitiesbetweenthirds inadistinctlydiatoniccontext,andthepositionoftheharmonizationsinthecadence.Theconceptofsubstitutionthen,bothinitstransformational and in its functional interpretation, is closely linked to the asymmetry ofdominant versus subdominant root progressions. It also appears as a hierarchicalreinterpretationofamoregeneralprincipleinaspecifictonalcontext.

[23]3.Themodelshowsthatasymmetry,substitutionandtonicisationareinter-related.Theconfirmation of the tonic through the cadential teleology both elicits dominant chordprogressionsandconcludesthem.Withoutthisgravitationalforce,theharmonywouldmoveforwardperpetuallyinanunlimitedharmonicspace(Meeùs2003).Directionaltendencythuscannotbe theonlycriterion foranadvanced theoryofharmonic tonalitybecause it isnotrestrictiveenough(Noll2008,87).It isnonethelessthiscriterion,directionaltendency,thatfacilitatestheutteranceofthetonicandconstrainstonality,andnottheconverse.

[24]4.Finally,itshouldbenotedthatthemodeldoesnotallowustoinferacausalrelationshipbetween contrapuntal constraints and asymmetry. On the one hand, contrapuntal rulesactively affect root progressions and encourage asymmetry.On the other hand, dominantvectors always allow thepreparation anddownward resolutionof thedissonance and theupwardmotionoftheleadingtone.Therefore,Iarguethatcontrapuntalconstraintsandthe

4ThesequestionsofthehierarchybetweendifferentrepresentativesofthesametonalfunctionandofapossibledistinctionbetweenamainrepresentativeandsubstituteshavebeendiscussedindetailinDahlhaus1966,1975.

hegemonyofdominantvectorsinfluenceeachothermutually.However,thislinkevolvesandweseetheircausalrelationshipchangethroughcenturiesofpolyphony.Thisevolutioniskeytomyargumentandtotheconceptsofconditionalandspontaneousasymmetrypresentedbelow.

1.3.Changingasymmetries

1.3.1.Spontaneousasymmetry

[25]NicolasMeeùs (1992a)arguesthat in tonality thechord’smodeandthecharacteristicdissonances–i.e.thesubdominant6/5orthedominant7th–aresuperficialelements.Thesefeatures are compared to what the linguist Sechehaye (1926, 86) calls “rection”, i.e. acharacterizingelementthatconfirmsagrammaticalconstructionwithoutbeingdecisiveforit.ThisspecificsituationisillustratedintheharmonizationofHerzlichliebhabichDich,oHerrbyJ.S.Bach(example5).Thecadential6/5andthepassingdominantseventhintheexample’slastbarreinforcethecadentialteleology.Thesedissonancesarehowevernotdecisiveforthecadenceitself.Ascorroboratedbythemodelcadence(example2)whichusesnodissonanceson IVandV, it ishow theharmonicunitsarearrivedatandare left– i.e. inbothcasesbydominantvectorsbetweenI-IVandIV-V–thatiscrucialforthegrammaticalconstruction.Thedissonances reinforce the cadential teleology and contribute to characterize the chord’sfunctionaspredominantanddominant,butareneitherdecisiveforthesefunctionsnorforthe prevalent dominant direction. In both cases, the dominant direction, critical for thesyntacticalorganizationofthecadencepattern,isestablishedforitsownwill,independentlyofcontrapuntalconstraintssuchasthepreparationorresolutionofdissonances.ThisiswhatIcallspontaneousasymmetry.

Example5.J.S.Bach,HerzlichliebhabichDich,oHerr,fromCantata149MansingetmitfreudenvomSieg,BWV149.

[26]Theconceptofspontaneousasymmetrysupposestheassimilationbythelisteneroftheprevalent dominant direction, which thus becomes a key syntactical element. Dahlhaus(1990b, 133) alludes to this when he argues that chordal dissonances are the result of a“reciprocal relationship between root progression and the resolution of dissonance”. Thismeansthatthechordaldissonancereliesonadynamicinterpretationofchordprogressions,onebasedontheexpectationofspecificchordprogressions(i.e.dominantvectors+4or+2)thatcoincidewiththedissonance’sresolution.

[27]Spontaneousasymmetrythusinteractswithvoice-leading:theupwardsdirectionoftheleadingtoneorthedownwardresolutionofthedissonancesinexample5aretheconsequenceofaspecifickindoflisteningwhichhearsrootmotionasanessentialrelationshipbetweenthechords. Or put in other words: “The ‘dynamic’ conceptions of root progressions and theresolutionofdissonancearetwosidesofthesamecoin”(Dahlhaus2014,134).

[28]Thistonallyorientedunderstandingofmusicalsyntaxalsomeansthatthegrammaticalconsistencyofthepolyphonycanbepreserveddespiteirregularitiesintheforeground,asinexample6.Here,thecadentialeffectismaintaineddespitetheextensiveelaborationofthetonic(bars131-132),andtheincompletedominantandtonicchords,whicharedeprivedoftheupwardmotionoftheleading-tone(bars.133-134)andreducedtothecharacteristicbassmovement down a fifth. Accordingly, spontaneous asymmetry, based as it is on a chordalbackgroundthat impliesdirectional tendencies,alsocarriesnewcompositionalpossibilitiessuchasregistertransfer,diminution,elision,elaborationorirregularvoice-leading,whichcanbeexploitedinfreecomposition.

Example6.Beethoven,Sonataop.14.1,Rondo,131-134.

[29] This organization around spontaneous asymmetry is the result of a specific tonallyorientedtypeof listening.Butharmonicsyntaxandthestatusofasymmetryhavechangedduring thehistoryofWesternpolyphony.This iswhy it isnecessary to take intoaccountapossibleshiftbetween,ontheonehand,constitutiveelementsthatarecriticalforsyntacticmeaningand,on theother, characterizingelements,which reinforce thismeaningwithoutbeingdecisive.

1.3.2Conditionalasymmetry

[30] InDufay'sMissa Se la faceaypale theprogression from thepenultimate to the finalharmonyattheendoftheKyrie(example7)cannotbedistinguishedfromatonaldominant-tonicprogression,asinthemodelcadenceinexample2.However,inthemiddleofthe15thcentury,the+4progressionbetweenpenultimateandfinalisisnottheresultofanemergingawarenessofdominant-tonicrelationshipsasBesseler(1950)suggests.Itisinsteadtheresultofastrictobservationofcompositionalrulesinfour-voicemodalcounterpoint(Eberlein1992,39-41).

Example7.Dufay,MissaSelafaceaypale,Kyrie,bars75-76.

[31] The impact of writing constraints is also evident earlier in this cadence. In the usualconfigurationof theclausula formalis, theharmonicprogressionbetweenantepenultimateand penultimate usually consists of a subdominant vector -4 (example 7). However, theintroductionofadissonantseventhinthepenultimateharmony,asseeninexample8withthe dissonance C3-Bb3 in bar 24, means the last two progressions now both move in adominantdirection,withadominantvector+2betweenantepenultimateandpenultimateharmony. It is the preparation of the dissonance 7th in example 8 that necessitates aharmonisation on the antepenultimate which generates the dominant vector that waspreviouslyabsent.

[32]Inthesecases,thedominantdirectionisinfactconditionedbythecontrapuntalrulesofpreparationandresolution.Thisspecifictypeofasymmetry,wherethedominantprogressionisinducedbycontrapuntalconstraints,Idescribeasconditionalasymmetry.

Example8.FrottolelibroPrimo(1504),FrottolaXXIX,Tromboncino,Ahpartialeecrudamorte.

[33] In this type of writing, the syntactic consistency, and more precisely the cadentialmeaning,resultsfrommelodicfluidity–i.e.parsimoniousvoice-leading–andthechangeofconsonant quality, particularly the alternation between dissonance, imperfect and perfectconsonancewhichplaysacriticalroleforcadentialteleology.Thepreferenceforthedominantdirectionofchordprogressionsarisesalmostaccidentallyfromcontrapuntalconstraintsandplaysonlyasecondaryroleasafactorofsyntacticalcoherence.

[34]Correspondingly,compositionaltechniquesarenotaffectedbythistypeofasymmetry.Although polyphony becomes inherently triadic from the beginning of the 16th centuryonwards (Lowinsky 1962, 3), the compositional possibilities that result from the triadicbackground–i.e.registertransfers,harmonicelaborations,irregularvoice-leading(see1.3.1and4)–arenotfullydeveloped.Thepreferreddirection isnotanelementactingonvoiceleadingbutisaphenomenonwhichresultsfromcontrapuntalconstraints.

[35]Thesedifferentfeatureswhichcharacterizeconditionalandspontaneousasymmetryaresummarizedinthetableofexample9:

Conditionalasymmetry Spontaneousasymmetry

• Consequence of contrapuntalconstraints.

• Interactswithvoice-leading.

• Resultofmediationbetweenmelodicfluidityandchangeofconsonantquality.

• Consequence of a dynamicinterpretationofrootprogressions.

• Secondary criterion for syntacticcoherence.

• Essential criterion of syntacticcoherence.

• Does not affect compositionaltechniques.

• Affectscompositionaltechniques.

Example9.Conditionalasymmetryvs.spontaneousasymmetry.

1.3.2.Empiricalverification

[36]Howdoestherelationshipbetweenconditionalandspontaneousasymmetryevolveinthecorpus?Toanswerthisquestion,thelineAsyTinexample10illustratesthevariationofthetotalasymmetryencounteredinthemadrigalcycles5.Contrarytoexpectations,theprivilegeddirectiondoesnotincreaseatthesameratefromthebeginningtotheendofthecorpus6.Theasymmetry issignificantlyhigh inArcadelt'sandVerdelot'sbooks. It is low in thecyclesby

5Thetotalasymmetry(AsyT)correspondstothedifferencebetweenalldominant(DV)andsubdominant(SV)vectorsencounteredinthemadrigalcycles(DV-SV=AsyT).Forexample,thetotalasymmetryinArcadeltIequatesto62,72%–32,28%=35,44%.ThelineAsyAinexample11representsthedifferencebetweenalldominant(DVA)andsubdominant(SVA)vectorsassociatedwiththepreparationandtheresolutionofthedissonance.Seebelowandnote2.6ComparetheseresultswiththoseobtainedbyCathe2012,25,129,159.

Lassus and Rore and then increases with a remarkable regularity over fifty years untilMonteverdi'sbookV,peakinginhisbooksVIIandVIII.

[37]Theconstantimbalanceinfavorofthedominantvectorsconfirmsthatasymmetryisnotspecifictoharmonictonality.ItisageneralfeatureofWesternPolyphony,althoughthisfeaturetendstointensifyand–asthisarticleargues–changeinquality.Furthermore,thewaytheasymmetryvarieswithinthecorpusimpliesthatthephenomenonissuperficialanddoesnotbelong to the deep structure of the harmonic language. It must depend instead oncompositional techniques (homorhythmic textures, imitative counterpoint, thorough bassetc.)andstylisticcriteria(strictcounterpoint,freecounterpoint,stilerecitativoetc.).

[38]Fromthisperspective,itisworthnotingthatthelevelofasymmetryfoundinVerdelot’sandArcadelt’scycles–whichstillshareimportantcharacteristicswiththeearlierfrottola–arealmostidenticalwiththelevelofasymmetryfoundmorethanacenturylaterinMonteverdi’sbooks VII and VIII, in which tonality partly crystallizes. Einstein (1949, 865) had alreadyobserved“howcloselytheextremesapproachoneanother–thebeginning[ofthemadrigal],about1500,andtheend,about1620”.These“strangebonds”,werealsoindirectlygraspedfromamoreabstractperspectivebyLowinsky(1962,14),whoclaimedthatsomeaspectsoftonality–especiallyofthemajormode–areanticipatedinthelighterpolyphonicgenresoftheearly16thcentury.

[39] The statistical results here clearly confirm an affinity between the earlier and laterrepertoiresofthecorpus,fromthepointofviewoftheasymmetryofrootprogressions.Theyalsoshowthatthemadrigalcyclesmorerepresentativeofmodalpolyphony–suchasLassus’bookIof1555–standoutthroughtheirsyntacticproperties.

[40]Atthesametime,acloserexaminationofthenatureoftheasymmetrysuggeststhatitsconceptual background changed dramatically between the beginning and the end of thiscorpus.Example10showshowthevariationinoverallasymmetryinthecomposers'work(lineAsyT)relatestocontrapuntalconstraints.Tothisend,thelineAsyAshowstheasymmetrythatrelatesspecificallytochordprogressionswhichinvolvedissonances.Ittakesintoaccountthoserootprogressionsthatareassociatedwithsuspensionsandnoteagainstnotedissonances7.Acomparisonofbothlinesconfirmsthatthetotalasymmetryandasymmetryassociatedwithdissonance are strongly correlated from Verdelot to Monteverdi's book III. However, thiscorrelation weakens in Monteverdi’s books IV to VI and the significant increase in theasymmetrybetweenbooksVIandVIIoccursindependentlyofthedissonance.

[41]Itisnotsomuchtheaccentuationoftheasymmetrywhichseemscriticalhere,butthefactthatthisincreaseisnotconditionedbyasymmetryassociatedwiththedissonance,which

7 In thecaseof thesuspensiondissonance, thechordprogressionsassociatedwith thepreparation,and thesuccessiveresolutionofthedissonanceontotheimperfectandperfectconsonancehavebeentakenintoaccount(see themodel in example 2). In the case of the dissonance note against note, only the chord progressionsassociatedwiththedissonance’simpactandimmediateresolutionhavebeenselected.

decreasesbetweenVIandVII.Wherepreviouslytheasymmetrywasinducedbycontrapuntalconstraints, it now occurs largely independently and spontaneously. Considering thetheoretical implicationsofasymmetry(1.2)anditspossiblerepercussionsoncompositionaltechniques (4), I argue that this emancipation is an important hint of the progressivecrystallizationoftonality.

Example10.Fluctuationoftotalasymmetryandasymmetryassociatedwiththedissonance.

[42] This move from conditional to spontaneous asymmetry is modeled in example 11.Contrapuntal constraints, represented by the arrows at the top, give rise to intervallicprogressions thatmediatebetween thechangeofharmonicquality (dissonance, imperfectconsonance and perfect consonance) and parsimonious voice-leading. Root progression,indicatedbythearrowbelow,constitutes,ontheotherhand,anabstractrepresentationoftheseintervallicpatterns.

[43]Thecontrapuntalandharmonicperspectivesareneveropposed:harmonicqualityandparsimoniousvoice-leadingontheonehandandrootprogressionsontheotheraremutuallydependent.However,thecausalrelationshipbetweenthesecriteriaevolves.Thisissuggestedbytheverticalarrowsandbythegradient fromredtowhite (andconversely). Initially, thecontrapuntalconstraintsaffectrootprogressionandgiverisetoconditionalasymmetry.Attheend of this development it is the root progression and its constraints on tonality whichbecomes the main vector of harmonic meaning and actively determines the polyphonicstream.

36,48% 35,44%

24,32%

17,82%13,80%

21,60%20,55%

26,40% 26,76%30,33% 31,52%

26,50%

39,61%42,05%

21,42%

23,95%

6,26%

6,34%

2,82%

13,12%

8,05%14,09%

18,01%

14,55%

14,20%

17,43%

11,86%

15,72%

0,00%

5,00%

10,00%

15,00%

20,00%

25,00%

30,00%

35,00%

40,00%

45,00%

Ver

delo

t

Arc

adel

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Lass

us I

Rore

II

Rore

V

Wer

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Mon

teve

rdi I

Mon

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rdi I

I

Mon

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rdi I

II

Mon

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rdi I

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Mon

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rdi V

Mon

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AstT AsyA

Example11.Evolutionfromconditionalasymmetrytospontaneousasymmetry.

[44]Tymoczko(2011,232)haspointedoutthatthecircleofthirds,asmodeledinexample3,might indirectly have “influenced the developing conventions of functional harmony”. Thescenarioofa shift fromconditional to spontaneousasymmetryputs thishypothesis intoatangiblehistoricalcontextandhelps,toconfirmitthroughempiricalevidence.

[45]Thismovefromconditionaltospontaneousasymmetrycontinuesbeyondtheendofthecorpus. It must also be borne in mind that both extremes – a completely conditionedasymmetry and a fully spontaneous one – never appear in music; neither in pre-tonalpolyphonynorincommonpracticeharmony.

[46]Thisiscorroboratedbytheempiricalresults.Ontheonehand,inallthemadrigalcyclesconsidered,thereisaresidualasymmetry,seeninthegapbetweenbothlinesofthehistogramin example 10. This residual asymmetry, which by definition cannot be explained bycontrapuntalconstraints,showsthattriadicprogressionsarenevercompletelyfreeasregardstheirdirection,andtheyareincreasinglyinvolvedindominantmovement.Ontheotherhand,even in the last twomadrigal cycles, characterizedby thehighest residual asymmetry, theoverallasymmetryremainspartlycorrelatedtotheasymmetrythatistiedtothedissonance:betweenbookVIIandVIIItheincreaseofthetotalasymmetry(AsyT)goeshandinhandwiththeincreaseoftheasymmetryassociatedwiththedissonance(AsyA).Thus,theresultsandthemodel do not imply a radical reversal but rather a gradual exchange in the hierarchybetweenbothkindsofasymmetries.

2.Signsoftheevolutionfromconditionaltospontaneousasymmetry

[47]Theshiftfromconditionaltospontaneousasymmetry isconceptual innature. It isnotinherent in the harmonic syntax itself but in how the syntax is interpreted. This shift cannevertheless be inferred from the evolution of compositional techniques. From themanycriteria which might reflect this evolution, three will be examined in detail here: 1. therealization of the cantus-tenor-framework, 2. the usemajor andminor chords and 3. themorphologyofdissonantchords.

2.1.Cantus-tenorframework

[48]Thecantus-tenorprogressionoftheclausulaformalishasremarkableproperties.Asthetenorandcantusshowinexample128,itcombinesparsimoniousvoice-leadingwithagradualchangeofharmonicquality.Thiscanbeseeninthemovefromthesecondquarternoteofm.23tothefirstofm.24inthetenorandcantusparts:thechangefromthedissonance(C4-D4)to the imperfect consonance (B3-D4) and the perfect consonance (C4-C4). However, thisframeworkisnotalwayscomplete.Thedownwardstepwisemotionofthetenorline(D4-C4atthecantusinexample12)isfrequentlyreplacedbyanupwardmovementtothethirdofthefinalchord.Thisisthecaseinexample13wherethistenorlineissungbythealtus(D4-E4),andwherethecantuslineissungbythetenor(B3-C4).

Example 12. Arcadelt (1539),Benedettiimartiri,25-26.

Example13.Verdelot (1530),Amorquantopiùlieto,6-7.

[49]Asshowninexample14,therealizationofthecantus-tenorframeworkshowshighlevelsofvariation,whileexhibitingagradualdecrease.Thisevolutionislinkedtoquestionsofvoice-leadingandtothechangingstatusoftriadicharmony.Infour-partwriting,theregularcantus-tenorframeworkpreventsaconclusiononafulltriadandonlypermitsanendingonathirdoron an empty fifth, as in example 12. The upward movement of the modified tenor line,however,allowsafulltriadasisthecaseinexample13(iftheGinthecantusisprolonged).

[50]Whatmakes a cadence in themiddle of the 16th century is no longer the intervallicprogression from the imperfect to the perfect consonance since this change of harmonicqualityiscompletelydissolvedinthegeneralimperfectsonorityofthetriads.Itisatthesametime the passage from the dissonance to the consonance and the characteristic melodic

8Inthisandthefollowingexamples,arrowspointingtotherightandtotheleftrespectivelyindicatedominantandsubdominantvectors.Therootprogressionisidentifiedbythepositive(upwardprogression)andnegative(downwardprogression)numbersoverthearrows.

movementsofthecantus-bassusframework(Eberlein&Fricke1992,57).Thisleads,ontheonehand,toanincreaseduseofdissonantstructureswhich,ashasbeenshown,conditiontheasymmetryofchordprogressions.Ontheotherhand,thischangeinwhatconstitutesacadence also brings increased attention to the outer voices. From this point of view, theidiomatic bass movements become, in the long term, a prominent factor of syntacticcoherence. The empirical results thus corroborate the hypothesis that the logic behindintervalliccadencepatternscoexistswithonebasedonanincreasingroleforthelowestvoice.Thelattertendshowevertoincreaseduringtheperiodunderconsideration.

Example14.Realizationofthecantus-tenorframeworkinthemadrigalcyclesexamined.

2.2.Majorversusminortriads

[51]Asshowninthediagraminexample15,therelationshipbetweenmajortriads(M3p5)andminortriads(m3p5)evolvesinaninterestingmanner.WhileVerdelotandArcadeltuseboth chord types in equal measure, the use of the major triad increases significantly infrequencywithLassusand thenagainat theendof thecorpus fromMonteverdi’sBook IIIonwards.

[52]TheFrenchmusicologistSergeGuthadalreadynoticedasimilarevolutionwhen,inhisstudyoftheharmonicthird,heexaminedthefinalchordincadencesfromlargepolyphoniccorporaoftheMiddleAgesandtheRenaissance.Gut(1969,191)suggeststwoexplanationsforthisphenomenon.Thefirstproposesapossiblelinkbetweena)thepredominanceofthemajormode,b)thesystematicuseoftheleadingtoneintheminormodeandc)thehegemonyof the major final chord. The second argues that an awareness of harmonic overtones,togetherwithanevolution froman intervallicunderstandingtoaharmonicunderstanding,contributed to the increased use of major chords. Both explanations however arequestionable, partly because thehypothesis of a gradual integrationof overtones remainsdebatableandpartlybecausethetwoargumentsfailtoexplainwhytheincreaseintheuseofthemajortriadhappensatthisparticularperiodinthehistoryofWesternmusic.

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[53]Iwouldliketosuggestinsteadthattheincreaseduseofmajortriadsmaypartlybelinkedtoanincreasinglytonallyorientedtypeoflistening.Oncethetriadisunderstoodasawhole,oncethebassisperceivedasthetriad’srootandoncerootprogressionbecomesalientfroma cognitive point of view, then theuse of themajor triad reinforces the link between thesyntactical units. This is bothbecauseof theharmonicqualityof the triadanddue to theupwardmelodicmovement,bysemitone,ofthemajorthird.

[54] In this explanation,which characterises the chord’smode as “rection” (see 1.2.), theincreaseduseof themajor triad isalso linked toan increasinglydynamic interpretationofchordprogressions.Thus,thisexplanationalsomaintainsthatthegreateruseofthemajorthirdindirectlyreflectstheincreaseofspontaneousasymmetry.

Example15.Majorandminortriadsinthemadrigalcyclesexamined.

2.3.Morphologyofseventhchords

[55]Thepossibleshiftfromconditionaltospontaneousasymmetrycanultimatelybededucedbyconsideringthemake-upofdissonantchords,especiallyseventhchords.Thechordappears,bothwithandwithoutthefifthfromthebeginningofthecorpus,asinthemadrigalFrapiùbeifioribyArcadelt(example16),wherethedescendingchainofsuspensionsinbars20-21givesrisetothreedissonantchords:twoseventhsharmonizedwithonlythethird(A-C-GandG-B-F),andoneseventhtowhichthethirdandthefifthhavebeenadded(F-A-C-E).

Example16.Arcadelt(1539),Frapiùbeifiori,19-23.

[56]Thehistograminexample17showsthattheuseofbothchordtypesiscorrelatedintheearlymadrigalcyclesandthatbothincreaseinusebetweenRoreandMonteverdi’sbookIII.However,fromMonteverdi’sBookIIIon,theuseofthetwokindsofseventhchordsdiverges:thefullseventhchordincreasesandpeaksinMonteverdi’sbooksVandVIII,whiletheseventhchordwithoutthefifthisusedmuchlessfrequently.

[57]Thistendencyisinterestingsince,inthe19thcentury,theBelgiantheoristFétisusesthepresenceofthefifth,alongwiththemajorthirdandminorseventh,todistinguishaseventhasnon-harmonictonefromatruedominantseventhchord.ForFétis,thelatterappearsfromMonteverdi’sbookIIIonwards,andmarksthebeginningofthe“tonalitémoderne”(Fétis1840,36).

[58]Theevidencefromthecorporasuggestsarealchangeintheuseofthechordbothwithandwithoutitsfifthatthebeginningofthe17thcentury.Thisevolutionhowevercanbynowwayexplainthedramaticchangespolyphonicsyntaxundergoesatthattime.

[59]AsDahlhaushasshown,theveryconceptofchordaldissonancedependsonadynamicinterpretationofchordprogression(see1.3.1andDahlhaus1990b,133-135).Hisdeductionreflectsthemoregeneralconvictionthatthedistinctionbetweenessentialandnon-essentialchordsmadebyalistenerisnotanaturalphenomenonbutdependsonharmonicschemasassimilated in a tonally oriented type of listening. The traditional distinction betweensuspensionsasnon-harmoniceventsandas“real”chordaldissonances,as impliedbyFétisandstilllargelyusedinrudimentarymusictheory,isthusirrelevantinsyntaxwhichessentiallyreliesonconditionalasymmetryi.e.syntaxpriortoharmonictonality.Ontheotherhand,thegradualriseofthecompleteseventhchordmaybeindicativeofatransformationofthewayharmonicsyntax is implicitlyunderstood.But itcanhoweverneither justify the increaseof

spontaneousasymmetrynorbeconsideredasadecisivefeaturethatactivelycontributestotheoriginof“tonalitémoderne”.

Example17.Morphologyof7thchordsinthemadrigalcyclesexamined.

3. Factors contributing to an intensification of spontaneousasymmetry

[60] Several factors may have fostered the evolution from conditional to spontaneousasymmetry. I alreadymentioned the hegemony of triadic harmony. The practice of bassocontinuomayalsohavecontributedtothisevolution.Thereishoweveranotherfactorthatappearscrucial:theirregulardissonancesatthestartofthe17thcentury.Thesystematicuseofcontrapuntal licenceundermines, inthelongterm,therelationshipbetweentheverticalandhorizontaldimensionsofpolyphonyandpartlyworksagainstsyntacticalcoherence.Sincethis coherence isno longerderivedexclusively fromparsimonyand the smooth changeofharmonic quality, characteristic of intervallic writing, movements of the real bass andthereafterabstractrootprogressionbecomeaprominentfactorofharmonicmeaning.

[61]Signsofthisshiftcanbeseeninbars22-26ofMonteverdi’smadrigalCh’iot’ami(example18)where theeightnote-against-notedissonances infringe radicallyoncontrapuntal rules.Unquestionablytheparsimoniousvoice-leadingandthecontrarymotionoftheoutervoiceshelp to soften the harshness of these irregular structures. However, it seems at least assignificant from the point of view of syntactic consistency that, despite the contrapuntalirregularityandwithonlyoneexception,allharmonicunitsarearrivedatandleftbydominantvectors.

Example18.Monteverdi(1605),Ch’iot’ami,et’amipiúdellamiavita,bars22-28.

[62] These observations lead to the following hypothesis: that opposition to regularcounterpoint at the beginning of the 17th century provoked a search for new criteria ofsyntacticcoherence.Asaresult,rootmotionbecomesthefocalpointandtherebythemainvectorofcadentialmeaning.Contrapuntallicencesthatarecompatiblewithdominantvectorsaremaintained,whereasthosethatareincompatiblearegraduallyexcluded.Thisleadstothestrengtheningofspontaneousasymmetry.

[63] The table in example 19, showing the rates of dominant and subdominant vectorsassociatedwithirregulardissonancesinbooksI-VIandVII-VIIIbyMonteverdi,supportsthishypothesis. Irregular dissonancesmainly occurwith dominant vectors in both sub-corpora(booksI-VIandVII-VIII).Onaveragehowever,thepercentageofirregularstructureslinkedtodominantvectorsincreasesby6%inthelastcycles,regardlessofanycontrapuntalconstraint.These results suggest that irregular structures were gradually integrated in line with thedominantdirectionofharmonicvectorsoccurringspontaneously.

[64]Becausetheseirregulardissonancesarecompatiblewiththedominantdirectionofchordprogression(whichalwaysallowsthepreparationandthedownwardstepwiseresolutionofdissonances, see 1.2.), they can be reduced at a deeper level (as per Schenker’s ‘middleground’)tostrictcounterpoint.However,thisisnottheresultofaslavishobediencetothestrictcounterpointofthepast.Itisinsteadtheindirectconsequenceoftheprivilegeddirectionofrootprogressionsoccurringspontaneously.

Contrapuntallicence

MonteverdiI-VI MonteverdiVII-VIII

DV SV DV SV

Syncopatuttacattiva 84.44% 15.56% 88.00% 12.00%

Dissonantpreparation 69.86% 30.14% 87.10% 12.90%

Noteagainstnote 64.44% 35.56% 68.75% 31.25%

Absenceofresolution 80.33% 19.67% 78.57% 21.43%

Resolutiononasilence 66.67% 33.33% 75.00% 25.00%

Resolutiononadissonance 58.43% 41.57% 65.00% 35.00%

Average 70.69% 29.31% 77.07% 22.93%

Example19.RatesofdominantandsubdominantvectorsassociatedwithirregulardissonancesinMonteverdi’sbooksI-VIandVII-VIII.

4.Spontaneousasymmetryandcompositionaltechniques

[65]Ascanbeinferredfromthesebondsbetweenasymmetryandthetreatmentofirregulardissonance,theemancipationoftheprivilegeddirectionfromcontrapuntalconstraintsgrantsnewcompositionalpossibilities.Thislargefield,worthyofaseparatestudy,willbediscussedhereexclusivelyfromthepointofviewofpossiblelinksbetweenspontaneousasymmetryandelaborationtechniques.

[66] In her study on the transition from modal to tonal organization in the works ofMonteverdi,SusanMcClaryarguesthatthehierarchicalshiftbetweenthestructurallineandthemelodicforegroundisdecisivefortheemergenceoftonality(1976,179).Accordingtothisview,inpre-tonalcounterpointtheharmony-generatingstructurallinecorrespondsinaone-to-one relationship to the foregroundmelody.Both the structural lineand the foregroundmelodybelongtothesamehierarchicallevel.Intonalityontheotherhand,eachpitchofthestructural linegivesrisetofurtherelaborations: thestructural linecorrespondstoahigherhierarchical level,distinctfromthesubordinatedmelodicforeground.Thisshiftalsoaffectsharmony. On the one hand, each structural pitch is able to generate what McClary calls‘harmoniccollections’.Ontheother,harmonyactivelyprojectsthestructuralline,articulatesit and becomes decisive both for freer voice-leading and for increased elaboration in theforeground.

[67]Theincreasedelaborationandornamentationoftheforeground,whichgoeshandinhandwithadecreaseintheharmonicrhythm,supposesthatthechordshavetobeunderstoodasimmediateharmonicunitsandthattheirprogressionhastobeinterpretedfromthedynamicpointofview,asoutlinedabove.TheseconditionscanalsobeinferredfromSchenker's(1954,p.155)explanationoffreecomposition:

[68]"Inrealityhowever,thetacticsofvoice-leadingbecomeeverfreertotheextenttowhich,infreecomposition,thereeruptssuddenlytheforceofthescale-step,underwhosecovertheindividualpartsmaymanoeuvreinalessinhibitedwayeventhaninstrictcomposition.Thescale-stepsthenresemblepowerfulprojectorlights: intheirilluminatedspherethepartsgothroughtheirevolutioninahigherandfreercontrapuntalsense,unitinginharmonicchords,which,however,neverbecomeendinthemselvesbutalwaysresultfromthefreemovement.[…]"

[69] The projector-lights-metaphor reflects the importance of the harmonic degree as aconceptual unit that enhances freer voice-leading. If we accept that the scale degree'smeaning depends on how it is arrived at and left, it can be argued that at least someelaborationtechniquesarecloselylinkedtotheinincreaseinspontaneousasymmetry.Twoparticular cases of elaboration, closely linked to the dynamic interpretation of chordprogressions,willbediscussedhere:theelaborationofthedominant itselfandofthepre-dominant.

4.1.Elaborationofthedominant

Example20.Monteverdi(1585),Senelpartirdavoi,24-30.

[70]Atbar28ofthemadrigalSenelpartirdavoifromMonteverdifirstbook(example20),thesyncopatedseventhF-Ebbetweenbassusandquintus,resolvescorrectlyontothemajorsixthF-D.However,theharmonicunitasawholecontainsadissonantfourthF-Bbbetweenbassusandquintus,which shows that thedominant chordon root F is conceivedandelaboratedacrossthetwobarsthatprecedethefinalchord.

[71] This stagnation has two consequences. On the one hand, it suspends the distinctionbetween the antepenultimate and the penultimate in the cadence. More precisely, thisdistinctionisnolongerestablishedfromacontrapuntalpointofview,butthroughharmonicprogression.Asisshownbythereductioninexample21,harmonicallytheantepenultimateispulledbackwards,andnowcorrespondstothechordonrootCinbar27.

[72]Ontheotherhand,thisstagnationprolongsthepenultimateonFbystretchingitintime.Inparticular,theelaborationoftheseventhEb,firsttreatedasasuspensionandthenasapassing note, reinforces the cadential effect of the following bass leap down a fifth. Thereinterpretation of the cadence and the elaboration of the dominant are compositionalpossibilitiesthatareultimatelygrantedbyadynamicinterpretationofchordprogressions.

Example21.Monteverdi(1585),Senelpartirdavoi,27-30,reduction.

4.2.Elaborationofthepre-dominant

Example22.Monteverdi(1592),Poich’ellainsétornò,36-40.

[73] At bar 38 of themadrigalPoi ch'ella in sé torno fromMonteverdi’s book III, the pre-dominantisreachedbyanote-against-notedissonanceD-C(example22).However,atalocallevel,thedissonantpitchisimplicitlypreparedbytherootCofthepreviouschordandcouldbeexplainedbyaregistertransfer.Thisisshownbythereduction(example23)andconfirmsthatfreecompositioncanalwaysbereducedtostrictparsimonyandregularcounterpointifatriadicbackgroundispresumedandifthetriadsmoveinadominantdirection.Whatismore,thepre-dominantDisinvolvedinrelativelyimportantelaborationsfrombar36onwards.Thus,fromabroaderperspective,C5atbar38couldbeinterpretedasamerepassingnotewhichresolvesontoB4,thefifthofthedominant.Thefactthatthesubdominantinvolvesanote-against-notedissonanceandthatitiselaboratedintensivelyoverthreebarsgrantsitrelativeautonomy.Thissuggeststhatitderivesitslegitimacyandmeaningfromhowitisarrivedatandleft,harmonically.

Example23.Monteverdi(1592),Poich’ellainsétornò,36-40,reduction.

[74]Theelaborationtechniquesusedinbothexamplesdonotleadtoalargedistancebetweenforeground and background. However, they show a real link between asymmetry andelaboration,whichallowsustomaketwoobservations.Firstly,theelaborationshappeninatonal context.They focus specificallyon thedominantandpre-dominant toenhance tonalcoherence.Secondly,spontaneousasymmetryiscrucialtotheintegrationofthesesuperficialphenomena.Itisthewaytheharmonicunitsarereachedandmovedfromthatallowsfortheintegrationofthecontrapuntallicensesandthatgrantsthemaparticularsyntacticalmeaning.From this perspective, the above elaborations result indirectly from the integration ofspontaneousasymmetry.

[75]Inhisattempttoformalizeagenerativesyntaxoftonalharmony,MartinRohrmeier(2011)tries to reconcile Riemannian tradition with recursive and prolongational approaches. Hisresultstendtoconfirmthatacomparativelysimplesetofrulessufficesfortheexplanationofalargerangeofexamplesbecause“tonalharmonyisfundamentallygroundedinelaborationsofcadentialharmony”(2011,48).

[76] The empirical results and theoretical reflection outlined here partly corroborate thishypothesis.Atthesametime,theysuggestthatboththecrystallizationofstatictonalfunctions

and the hierarchical articulation of tonal syntaxmay be historically linked to the changingstatusoftheasymmetryofrootprogression.Asystematicconsiderationoftheasymmetryofrootprogressionscouldthusprovideabetterunderstandingofthehierarchicalandfunctionalcharacteristics of tonality, historically9. What is more, it could also help to integratecontrapuntalstructures,thatlieoutsidethetonalframework,intothissameunderstandingoftonalharmony.

5.Conclusion

[77]Toconclude,theTheoryofHarmonicVectorsarguesthatthedominantdirectionofchordprogressionsisacharacteristicfeatureoftonality.Thispaperinturnsuggeststhatitisnotsomuchthemereprevalentdirectionthatiscrucialbutthefactthatthisdirectionaltendencyemancipates itself from contrapuntal constraints and acts on them. This evolution from aconditional toaspontaneousasymmetry, though immaterial, canbededuced fromseveralphenomena:thedissolutionofthecantus-tenorframework,theincreaseduseofmajortriads,themorphologyoftheseventhchordorthefactthatirregulardissonancesareincreasinglycompatiblewiththepreferreddominantdirection.

[78]Therecurrentuseofcadentialprogressionshelpedassimilatethispreferreddirectionoverdecades.Furthermore,irregulardissonances,temporarilysuspendingcontrapuntallogic,mayhaveplayedadecisiveroleintheincreaseofspontaneousasymmetry.Finally,thisstudyhasshown that the assimilation of spontaneous asymmetry provides new compositionalpossibilitiesandcontributestoagreaterdistancebetweenforegroundandbackground.

[79]Thisevolutiondoesnotcometoanendat the turnof the17thcenturybutcontinuesthroughthefollowingcenturies.However,thecorpusexaminedhereevolvesdrasticallyandsuggests that the shift fromconditional to spontaneousasymmetry,withall its theoreticalimplications, has already taken place, at least in part, in the lastmadrigal cycles analyzedabove.

[80]Thispaperdoesnotclaimthatallaspectsoftonalitycanbereducedtoasymmetry.Itdoesnot even suggest that asymmetry is a characteristic feature of tonality. However, becauseasymmetryistightlyrelatedtootherimportanttonalfeaturesandbecauseitinteractswiththese features, this analytical and theoretical criterion has a significant heuristic value forunderstandingtonality.

9 Tymoczko 2003, 42 has argued that if the historian could confirm that functional tonality appearedwhencomposersgraduallybegantofavordominantprogressionsoversubdominantprogressions,itwouldconstituteadecisivesteptowardsanexplanationoftonalharmony.Theresultsobtainedheresuggestthatitisnotsomuchthe accentuation of asymmetry that seems decisive from a historical point of view for the crystallization offunctionalharmonybutthechangingstatusofasymmetry.

6.Acknowledgments

[81] Iwould like to thankAidanO’Donnell for his endless patience and invaluable help inediting and proofreading this paper. Thank you also to the anonymous reviewers for thehelpfulsuggestionsduringtherevisionofthisarticle.

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