Wallace Bartok

8/12/2019 Wallace Bartok

1/94

Symmetrical Interval Sets and Derivative Pitch Materials in Bartok's String Quartet No. 3Author(s): Wallace BerrySource: Perspectives of New Music, Vol. 18, No. 1/2 (Autumn, 1979 - Summer, 1980), pp. 287-379Published by: Perspectives of New MusicStable URL: http://www.jstor.org/stable/832989.

Accessed: 19/07/2013 05:02

Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at.http://www.jstor.org/page/info/about/policies/terms.jsp

.JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of

content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms

of scholarship. For more information about JSTOR, please contact [email protected].

.

Perspectives of New Musicis collaborating with JSTOR to digitize, preserve and extend access to Perspectives

of New Music.

http://www.jstor.org

This content downloaded from 147.162.119.135 on Fri, 19 Jul 2013 05:02:43 AMAll use subject to JSTOR Terms and Conditions
http://www.jstor.org/action/showPublisher?publisherCode=pnmhttp://www.jstor.org/stable/832989?origin=JSTOR-pdfhttp://www.jstor.org/page/info/about/policies/terms.jsphttp://www.jstor.org/page/info/about/policies/terms.jsphttp://www.jstor.org/page/info/about/policies/terms.jsphttp://www.jstor.org/page/info/about/policies/terms.jsphttp://www.jstor.org/page/info/about/policies/terms.jsphttp://www.jstor.org/stable/832989?origin=JSTOR-pdfhttp://www.jstor.org/action/showPublisher?publisherCode=pnm


2/94


3/94

1 1 t2Z I Z 1 2

1 2 1

-L. I .. I-J21 2.2 1 2 Z2 2 1 2 2

_ . r\1L-

. - )a_ - _ ,I3 z 2 3 [ V

The talic apitaletters used or eferenceopitch-classPC); staffotation,or owercaseetterswith uperscriptumeralsor he ctavesscendingrommiddle -c1), for pecific itchreferences.pecific itchesn thesecondoctave elow ' aredenotedyuppercaseetters.

O-*

' -LC

w

http://www.jstor.org/page/info/about/policies/terms.jsphttp://www.jstor.org/page/info/about/policies/terms.jsphttp://www.jstor.org/page/info/about/policies/terms.jsp


4/94

SYMMETRICAL INTERVAL SETSAND DERIVATIVE PITCH MATERIALSIN BARTOK'S STRING QUARTET NO. 3WALLACEBERRY

While its explicitfunctions re oftenproblematic,onalityan bedemonstratedo be a potentorganizing actor hrough irtuallyll ofBart6k'smusic. n thethird uartet, s in itsfive ompanions,he tonal'implicationsf ines ndverticalitch ffiliationsreofprimarymportance.Thus,while onalitys as such not a central oncern fthis tudy,tseemsnecessaryogiveexplicitcknowledgement,nprefatoryomment,nd nrepresentativenalyticalllustration,oa fewmanifestationsfquasi-tonalfunction.fsuch functionsmoreproblematicnthe third uartet han nthe firstwo, t is neverthelessfvital mportance, hatever herange fplausible nterpretationsf its particular pplications. hus, the broadreferentialignificancefC#1 s immediatelyvidentn itsappearancesthe root' fcadential onoritiesoncludinghe rima arte nd thework sa whole.



5/94

2The termfunction'smeant o refer o the entral ole f particularC,and at times n harmonic tructureaving hatPC as 'root', s thefocal,orientingoint f inearmotions;n this ense, onalfunction's seenhere sdistinctlynalogous o thatof the tonicand itssubsidiaryreparationsnconventionalonality, hileof coursedepartingn importantarticularsfapplication.

N,1



6/94

291Somethingf the nature fquasi-tonalunctionnthequartet anbeillustrated,s a perspectiven which otreat nderlyingotivicndgenerative

intervalets, n a few nterpretiveketches.At the sametime, tcan besuggestedhatfactors fquasi-tonalentralityre identifiableotonly spitch-classes,ut as specific itches: or xample, : andThis principle,ne of quasi-tonal entralityf pitch-registralpecificityandnotmerelyfpitch-class,ears xplorationnanyfull onsiderationfthework's tonality';t is a principle t leasttentativelyvident n thefollowingxamplesof quasi-tonal unction2nd successiontoward hepitch-classPC) C#, nd thepitchC#as cited bove.

The initiatingpedal' omplex onsistingf a chromatic etrachordnC# s ofpalpable ignificance;essimmediatelypparents thedescendinglinear equence nderlyingheforegroundf theopening irst iolin hrase,a sequencedirected owardG#,theclarifyingthabove the sustained'tonic'root Ex. 1).

There s thus persuasivexpressionftonal rientationt thework'sinception,n bothcomponentsfthe texture fthe ntroductoryhrase,themoving oice ontrolledna stronglyirectedfundamentallyonjunct)linear uccessionwhich s onlyone ofmany xamples f itskind n thequartet.Consider, or urtherxample, hefirst iolin inewhich ollows,mm. 7-21,or thecellophrasebeginningt m. 83-in its own terms fforceful escent towardC#, and in interaction ithconcurrent oices(concludingnA#, thenB#). Wemay onsider fewother xamples frelativelyxplicitonal xpression ithinarticular,ifferingircumstances,and of other evelsof structure.



7/94

-- nihatin iiiheAfor mitativentril?mm., $'

II7 C1__$t-

__ y- -- 1 . n .-,,- it-, I

etumrallymolatedieheTt U-

3The abbreviation. p. denotesrima arte,with omparableesignationsfor econdaparte s. p.) and coda (c.). Thereare no quotationsrom hericapitulazione.

0.3 t ..)-, , - '' (a) * (-

10PC6,Zackingzial Cj,

? nr . I I -- - II L.. . 1 -I . _ _ - - ' _,

\) i.. ^ ^.T-T- ^^tlX


8/94

293The sectionbeginning t m. 66, and extending o the relativelyunequivocal ello arrival t m. 87, towhichreferences madeabove, s

strikingn its ambivalence etween endencies f tonalambiguityndrelativelarityi.e.,transitoryresolution')uring process fexploitationofoneofthe work's ellularmotives,o be a subject ftreatmentater nthispaper. t is notpossible otreatnadequatedetail hedevices ftonalemphasis ccordedC#,F#,and G#in thisdevelopmentalrea,as root-prominencesn 'chordal' tructuress well as in relations f imitativeentries.A glimpse fthese endencies an be seen nthefollowingketch,inwhichBart6k's urposefulppositionfrelativelyneutral'omplexesinwhich hetritonesmuch nvolved)with elativelyfocal'pointsC#,F#,inisolated,ompositionallyingled-outccurrences,nd ntherelations fimitativentries tm. 84) is vivid. See Ex. 23.)

(Compare hechords fmm.53-64,andtheorientationsfthispassagetoward, ndon, C andF#, and their nharmonicquivalents.)

The fugato fthe econda artehas A as its tonic', ndhasas itsbasisan interval et scale)offundamentalmportancenthequartet. hatset,which asextensivereatmentithinhis aper'smajor oncern,s llustratedinEx. 3, together ith tstranspositionp seven emitonest7),thebasisfor he ubject's nswersnthefugato xposition.

This collection s symmetricaln a sense to be defined ater[theintervalequence s (2)-2-1-2-1-2-(2)],ts nferablexialcenter # nd thatof itsduplication 7G#,PCs ofprimaryonalcentralityn thequartet'sprimaparte.Moreover, he set and its transpositionmbrace en PCs,lacking nlyC#andG# nthese, tsfundamentalorms.



9/94


10/94

295The 'tonicization'f A in theinitiatingubject tatements. p., m.242) is the result f:itsemphaticssertions thesubject's irstttack,ts

insistent epetitions the subject'sconcluding itch, nd of courseitsreturn fter he answer nE (m. 246, cello),notto mentionheelaboratepreparationf thefugato,ndofA, at least as farback as m. 194,whichinauguratesbroadly acedharmonic uccessionfquasi-dominants,uasi-subdominants,nd quasi-tonics,he latter n 'firstnversion',with theprima arte'sC#as bass.With ll ofthis sprecedent,he nitiatingubjectstatementeveals till nother,ndmore ubtle, ctof tonicization'.

Thisstatement,uoted nEx. 4, isaccompaniedy tsown reduction',a derivativeorm ywhicht senhancedn modifieddoubling',n imitation'of restrictedontrapuntalnteractionfchangingangles' nddirectionsfrelation. hese relationsrerepresentedn Ex. 4, in which anbeseenthetonally ritical eparturesftheviola'sreiterationn m. 245, where hecomplementaryCs C#andG#appear na tonally larifying,ffirmativealterationfthesubject's wopenultimate itches-theG# leading-tone'especiallyotentn tsresolutive,onicizingunction.xample also ndicatescontrapuntalmotivations f voice-leadingmplicit n the viola's part,significant actors n the functional uccessionbywhichthe alreadyestablished isenhanced s tonic'.As a final eferenceoquasi-tonalunction,ereof a higherevel, nextremelyroad, elescopic iewof theessential asspitches ftheprimapartewould how hebroad entral unctionfC# especiallyn tsregistralspecificitys C#,two leger inesbelow the bass clef) in a number fprolongingecurrences. oreover,n underlyingrinciplewhichwill becharacterizeds black-white'oppositionits ourcentwo entatonicntervalsets ivennEx. 6i persistsntheuxtapositionfC#andGatthe onclusionof the rima arte.For all of the relative larity f theseexamples nd referencesbywhich llustrations affordedfmeansof nquiryntoBart6k's uasi-tonalfunctions),muchof the work's itch tructures predictablyroblematicwith especto tonality',lthoughhe hapingppositionftonal mbiguitywithrelativeoften adential) laritys in itself telling spectof tonal'structure.



11/94

4,a b

4A soleexception,hepentachordf Ex. 6d,may e seen s issuing romgeneric ource, hesymmetricalmajor'hexachord,epresentedn Ex. 6c.Thus,3-2-2-3would eregardeds 2-(2+ 1)-2-2,nverted.



12/94

297A second ndvery mportantrinciplefpitch elationsn theBartokhas already een implied: hebasis of pitchstructuresn certain calar

formulationsiketheheptachordited nEx. 3,or thetetrachordiven sEx. 5a. Such collections an ofcourse be-and havebeen-viewed,andreferredo, nvariousways.Thus,for xample, he major' etrachordC-D-E-F, f t s onC) can bereferredo as to tsconstituentCs,C-D-E-F, r0-2-4-5, rderedn a unique, implest rrangement,uiteapartfrom nytonicmplicationsfC, inwhich he mallest ossiblentervalseparatetselements. r such setmight e dentifiedy nassigned ymbol,umericalorother,odistinguishtfromther ollections.n suchmodes freferencethe set'stranspositions seenas a discrete orm: or xample, he et 4-6-8-9 (E-F#-G#-A)s a duplicationreducible'otheprototype-2-4-5, ne inwhichPC numbers an be seen as derived ytheaddition f a constant(here, , the ntervalftransposition).Inversionndretrogradeorms anofcoursebe viewed s derived hrough omplementationfPC numbersandordernumbersnwelldefined rocesseswhichneednotbe reviewedfurthernthepresentontext.)A further odeofreferenceosuch a set swith especto ts nterval(or interval-class,C) content iewed s to thesequenceof intervalsasdistinct rom heset'svector ftotal C content) ccurringrom egree odegreewhenthe set is disposed n its 'simplest' orm-i.e., when it isordered o that he ntervallicpaces nthat ineardispositionre ntheirsmallest orms,rso that he ntervallicistancencompassedy he et sreduced o ts mallest otentialange.nthis ense,Ex. 5a is simpler'hanEx. 5b; Ex. 5a representshemajortetrachordn itssimplest, efinitiveform.

(Again, he ssueoftonalfunction,fthetonicpotentialfC orF, isbesidethepresent oint.)Tetrachordalnd othergerminativeollectionscited s basic ntheBart6k recited ntheir implest ormsnthis ense.4

Referenceo sucha collection s to its ntervalequence n simplestformuggests modeof dentificationna sensemore bstract han hat fPC namesor numbers-one in whichthe 'major'tetrachordmightbereferredo as 2-2-1, he ntegersenotinghe uccession f ntervalsnteriortothe etdisposedn ts implestorm.A modeofreferencenwhich ucha collectionssimplydentifiedy hosen ymbol--e.g.,s No. 1-would ofcoursebe comparablyabstract',utwould nnowaybe suggestivef theset'scontent.) eyond hedirect pplicabilityftheabstract esignation-



13/94

5Orderingfthe ntervalsnthesequenceinterior'o thescalar et s ofcourseproblematichere ucha setspans 7th thedistance f 10 or 11semitones)ndthen s so structureds torepeattselfn a subsequentctave:i.e.,where heptachordike hat fEx.3 begins'sproblematic.nsuch ases,the rderingf he ntervalsinterior'othe et n tsdefinitive'orms seen sto demonstrablepplicationsnthework,s stemmingromonalfactorsaswith he conventional ajor cale),and/ors to compositionalpplicationswhich irectlyeflectnherentymmetries.Allthree riteriaffirmhe nitiatingfunctionsfA andE in the twosetsofEx. 3.) Most oftheunderlyingetsdiscussed ere re, rare reducibleo, ymmetricalrichordsndtetrachords.

6Examplesof calar etsnot tatablensymmetricalorm re? , . 0 and .

7Transposedne semitonep,orby compoundrcomplementaryorm fthat istance.



14/94

2992-1 to the entirerangeof potential ransposition,uch a nomenclaturefurther akes mmediatelyisible uch a propertys the set's symmetry',or asymmetry'.oreover,onnectionsetweenhebasic ina sensesource')set andsuchpermutationss the mirrorin theforegoingase, 1-2-2, r2-2-1) re mmediatelyisible. itchmaterials ithiniscrete ormalegmentsin theBart6kquartet re attributableo certain nterval ets,variously'explained' s to relations ogerminativer derivativeets,which an beviewedas fundamentalnits, hesourcesof lines, motives, nd verticalcomplexesnmuchof thework.These collections re as a rule symmetrical'n the sense that thesequences f nteriorntervallicuantities5avethe ommon haracteristicthat nesideofa central xis s themirrorfthe ther. hus,a symmetricalset e.g.,C-D-F-G)s identicaln ntervaltructurei.e.,2-3-2)viewed romeither irection. he setgiven n Ex. 3 is symmetricalnthis ense:2-1-2-1-2,andit is exceedinglymportantn thequartet.The 'pentatonic'etissymmetricalf rdered -2-2-3,lthough ot n ts implestorm sdefined,encompassingmajor6th.6

[Sucha sethas,ofcourse, hepotentialor ombinatorialssociationwith tstranspositionst certain evels, ssumingt is notlarger han ahexachord:.g.,C-D-F-G nd tsfive ranspositionsy Cs, 1, 4,and 6 (i.e.,onC#orB, E orG#, ndF#). twillbesuggestedhatBart6k anbe seenin thethird uartet o adopt n certaindegrees he (oftenblack-white')oppositionfsuchcombinatorialairs consider he3-2-2-3 etofEx. 6iand thetransposition,l,7 shownwith t). The 'pentatonic'et 3-2-2-3salsoapplied noppositiono tscombinatoriallyelated ranspositiont thetritone,t t6, with in view ofdistinct extural ifferentiations)itonalimplicationsobenoted.]

The balance f this tudysdevoted o thediscussion fextracts romthequartetndemonstrationf theapplicabilityfcertain ntervalets as'sources' fpitch tructures.uchapplicationsre, ften, ot s toultimatepotentialntervallicontentinthevector ense),butmaybewith especttointervalrderingnthedesignatedet nitssimplestorm. hus,itwillbesuggestedhatwhileBartok'structuresanincertainnstances e seenasgroupingsf nterval-classesnherentnthebasic et nquestion,withoutrespect o simplestrderingEx. 42b, andothers), heapplicabilityf theset is often f motivic' idelityo its interiornterval rdering. hat is,Bart6k ommonlyerives romhebasic ollectionsines hat un long hecontiguousntervalequence fthegerminativeet n ts implestorm,sseen nmany ftheexampleswhich ollow.



15/94

2---2 (2-2-0-z-2)

2-0~-2

z

$ff o



16/94

301The ordered,nteriorntervalequenceoftheset-e.g., 2-2-1for he'major' etrachordwill beadopteds a basisfordentificationndreference.

PC duplicationna second ctavemayberepresentedarentheticallyas inEx. 3) where t is inferablen the set'sextensionnpractice; arenthesesmay lso be usedtosuggestuchextensionr ts ncipiencey ymmetricalrepetitions.oints faxis, nvolvingometimes neand sometimeswoPCelements,reoftenncircledntherepresentationsf ets.Where principleof xtensionscarriedutonly artially,ts heoreticalontinuationotentialis suggested yellipsesfollowinghe integer enoting he final ntervalactually epresentedi.e., -2 . . =-2, etcetera). tems nd beams areusedonlyto draw attentiono significantecurrentr germinativeubsets rother ignificantubunits. he choiceof one transpositionr anothernthenotation,he actualPC content hown, s often rbitrary,r mayberelevanto a particularpplication.nderscoredxamplesreof mportancein thequartet; hosefollowed y ?' areofuncertainignificance. hereprinciplefpotentialxtensionr derivations notactuallyarried ut inthe pecimensfEx. 6,which ollows,t sbecause t sadjudged o havenosignificancentheworke.g.,the xtensionfthe et1-2-1 s 1-2-1-2-1,r1-2-1-1-2-1).While thefollowinget of 'sources'forpitchmaterialsstreated s a distinctoncern part romheforegoing,imited onsiderationofquasi-tonal unctions,twillbe clearthat hetwoapproachesrebynomeans exclusive: lreadythas beennotedthat heset 2-1-2-1-2Ex. 3)has, in its applicationsn the secondaparte, strong onalimplications.Indeed, onal onsiderationsre attimes hebasisforudgments to how aset begins'. inally,umbers rethose ommonlysedtorepresentnterval-classes:1for he emitone,for hemajor ndor ts nharmonicquivalents,etc., through forthe tritonen all its forms.As a rule,the setsarerestrictedothe mallerntervals,2,andoccasionally. Insomethingfa'chicken-and-egg'ircumstance,hefollowingiscussionutlines plausible'path' fgenerativerocesseswhile ttimes uggestinglternateiews.

It ispossible o view he ymmetrical,major'hexachords theoverallprogenitormaterial, ontainings it does the major'tetrachord,-2-1(overlapped y wodegreeswith tsownmirrornversion,-2-2), s well s,in its middlefourdegrees or two intervals), he symmetricalminor'tetrachord.



17/94

alZo2-3-2-2... t *2 - 3 - 2 - -(3 - 2)

8Applicationsf mirrornversionsfconjecturederminativentervaletsare of ignificanceolelywith especto themajor' etrachord,-2-1, ince llothersntheoutline resymmetrical,hus uplicatedxactlyn nversion.

ybb6 Y(2z)-- -)--2-(z)

ortequivalenfnPCconfent,ot n Tinttrrat trucurc):

2


18/94

303As tothemajor'etrachord,-2-1,nd tsmirror,-2-2,verlappedby nedegreeo form he ymmetricalexachord-2-1-2-2,heresthefurtherossibilityf xtensionf he etrachordyuxtaposition,efinedssuperpositionf tstransposeduplicationr nversion8ith he inkingintervalneof hose stablishednthe nitiatingnit--inhis ase2 or1.While uxtapositiony nterval is ofno apparentignificancen thequartet,uxtapositionyntervalcanberegardedsthe ourcef he eryimportantactorlreadydentifiednEx.3. (SeeEx.6b.)Thehexachord-2-1-2-2,egardedsanoverlappingy wo egreesfthemajor'etrachordnd tsmirrorEx.6a),has heminor'etrachord,-

1-2, sitsmiddleubset;tcanthus lsoberegardeds originating'nthe'minor'etrachordxtendedyoverlappingfonedegreeinterval) ineachdirectionEx.6c).Thisprocess, ith tsbasisnthe2-1-2et,wouldthus lsobe seen s source' or hemportantentachord-2-2-3Ex.6d).The atters, s notedarlier,therwiseieweds themajor' exachord-2-1-2-2ackingts hirdegree2- 3 -2-2); eeExx.6a and6d.2+1

There repracticaleasonsor egardingheminor'etrachord,hesymmetrical-1-2,sgerminativeaterial.otonly oes tgeneratehesymmetricalmajor' exachord,-2-1-2-2,y he ule llustratednEx.6c,but tcontains,romitherxtremity,roceedingy nterval inonedirection,nd nterval or 3 intheother5 - 2 or31), he xtremelyimportantotivicdea llustratednEx.6e.

Animportantpplicationndvariationfthe2-1-2 etrachordsitsextensiony ranspositionsverlappedy nedegreeEx. 6f), ormingnintervallicesourcevident s thebasisofcertainines n thequartet.These xtensionsppearttimesobeconsiderable;othe xtenthat t scarriedut nEx.6f,t an lsoberegardedsembodying,n symmetricalcomplex,he -2-1-2-2exachordof x.6c)togetherithts ransposition(att5)overlappedy hreeegrees-twontervals-asllustrated;hus:2-2-12 122-2 1-2-2.



19/94

2-2-1-z-z)o1-ZO,z -2- 1 (~ - I --2 2 2 1 : :i" 2 ){DA2


20/94

305Because of its relevance o certainpitchstructuresn thequartet(illustrated ater),the symmetricalnterval et 2-3-2 mightbe noted

parentheticallytthispoint,na senseanexpansion f the2 - 3 formu-(1+2)lationnotednEx. 6e,and a subset f thepentachordiven s Ex. 6d. Thatintervalet 2-3-2)has twooccurrencesnthemajor'hexachord-2-1-2-2,onethetranspositiont2) oftheother, s seen nEx. 6g.

Further,nincipientstring'asedon the2-1-2 et cf.Ex. 6f)containsthesymmetricalpentatonic'et 3-2-2-3 Ex. 6d) togetherwith tstrans-positiont5). (See Ex. 6h.)

Note should also be takenof thesymmetricalormulationEx. 6i)which ncludes hepentachord-2-2-3 ogether ith ts 'combinatorially'related) ranspositiontl): it s importantnthequartet s a manifestationof bitonal' ppositionsfpitch tructuresfparallel nterval ontent utexclusive C contentn black-white'uxtapositions.

Overlappingf thegerminativeminor' etrachord-1-2by wodegrees(one interval) roduces ne ofthequartet'smost mportant aterials,hehexachord-1-2-1-2,iscussednconnection ith x. 3. Example j showsthehexachordalorm, -1-2-1-2,s well as itsextension,2)-2-1-2-1-2-(2),as applied n thequartet's econdaparte. See Ex. 4.) The latter an beregarded,ther han s anextension f the2-1-2-1-2 exachordmanatingfrom he2-1-2 etrachordverlapped ytwodegrees, s derivingrom he'major'tetrachord-2-1 uxtaposedwith ts nversion, -2-2,yieldinghesame mportantet, s noted arlierseeEx. 6b).



21/94


22/94

307The et -1-2-1-2,n urn,ieldshe xceedinglymportantymmetricaltetrachord-2-1, recurrentlypplicableource fpitchtructurenthe

quartet.n itsrepresentationsEx.6k, heres includedarentheticallybeamedelineationf he et'somponentnterval-classotive-1, ccurringinboth irections.

The2-1-2 tring, ith verlappingy nedegreeseeEx.6f),yieldsthe ymmetricalentachord-2-2-1,s shownnEx. 61.Thatpentachordcan lsoberegardedsderivingromhemajor'etrachord-2-1verlappedwithts nversiony hree egreestwontervals)-alsohownnEx. 61.

The set1-2-2-1-2-2-1Ex.6m) sderiveds the verlappingy wodegreesftheforegoing-2-2-1. nd heuxtapositionfthe et1-2-2-1withts ranspositiont8)producesnine-PCet, -2-2-1-2-1-2-2-1,hichcanberegardeds a furtherossibleourcef hemportantexachord-1-2-1-2cf.Exx.6b, ) togetherithts xpandedorms appliednthequartet'seconda arte Ex. 6n).

Athesisould lsobedevelopedhathemportantexachord-1-2-1-2 [orheptachord2)-2-1-2-1-2-(2),s tunderlieshe ugatof he uartet'ssecondaarte]s itself basisfor erivative,ndfurtherenerative,ets(Ex. 7).



23/94

ft

,uLll

0q~1

Itx~

m

If a



24/94

309

A word houldbe interposedtthispointregardingheprinciples,rrules,bywhich xtended orms fthegenerativeetsare derivednEx. 6,aswellas somefurtherbservationsoncerninguchgenerativeets oftentetrachords)hemselves.Of the ight ossible ermutationsnvolvingetrachordalrderingsfthree ontiguousCs 1 and2,byfar hemost ignificantntheBartok rethe opposite' orms -2-1 nd 2-1-2 Ex. 8). The anomalous1-1-2 nd2-1-1 do not, so faras it has been possibleto determine, ave significantapplication.hechromaticetrachord,-1-1,ndthewhole-toneetrachord,2-2-2,have imitedignificance,lbeitvery xplicit ndappreciablempactat certain imes, ndmaybe regardeds derivativeatherhangenerativeformswhere hey rebrieflypplicable,s may he major' etrachordndits nversion,-2-1 -> 1-2-2,which an also beregardedsgenerative,snoted arlier.

One rule of extension foreffecting erivative orms) s that ofsuperimpositionfa tetrachordnd tstranspositiono form symmetricalunit nwhich he nterval f uncture, suallyC2, is oneofthoseofthegerminativeet.Therelation ithinhe ompositesoneofmirrornversion,or ifthegerminativeet is itself ymmetrical,ach componentmirrorsitselfis itsown nversion)tthe ametime hat hispropertysrelevant otheextended,erivative nit See Ex. 9a).

A secondrule s that foverlappingyone, two,or three egrees oform erivative,ymmetricaleptachords,exachords,rpentachords,sillustratedn Ex. 9b and in segments fEx. 6. In this ense,the naturalminor' cale s a derivativefthe minor' etrachord-1-2.

[Twosuchmodes fextensionmayproduce ariantsquivalentnPCcontent.For example,givena symmetricalerivative ysuperposition



25/94

4.10- z - R- --

iJ, $r2[7 2 P I

d ~ _L j -1- Z

I T 2 "zII,L -mV

(in P2 1 2''inPCon)bnt, I I I2 1 2

1 2 2

not nintrvaljuccesoion)

(in sametmnr)

._ - l



26/94

311extendingothereturnf the nitial C, it willbeequivalentn PC contentto a form erived y overlappingyone degree--thatormn which heduplicated C is axis (Ex. 10). For thegreater art, ets are seen in thepresent nalysis s generatingines n which C succession s, as distinctfromPC content r broad C vector, f significance,o thatone or theother f suchPC equivalentsmay ppear obe a 'better' ource na givenapplication. hus, as to thegeneratingf lines as opposedto verticalcomplexes,ntervaluccession,nwhich wo uchforms renot quivalent,commonlyurpassesn mportancehefactorfequivalencefPC content.]

The issue of relations f PC content risesonlylimitedlyn thepresent nalysis,s inthesuggestionhat point s madebyBart6k f thedeliberate pposition f certaincombinatorial'lignments-setstotallyopposed n PC contentwhileduplicativen interval tructure.See thepotentialorblack-white'ppositionllustratednEx. 6i.)Moreover,while ommonly unctioningnlines,maintainingpecificintervaluccession, ndwithreadily iscerniblemotivic ignificance,hecitedmaterialsespeciallyhetetrachordalet 1-2-1) ccasionallyunctionas IC and PC 'source' ets nthework, eneratingntervallicndpitch-classrelationsmultidirectionallyhroughhetexture.

We maybegin seriesof analyticalitationsn thequartetwith insomedegree rbitrary)eferenceo thesymmetricalminor' etrachord,-1-2, nd ts xtensions,s well s its xtremelymportant otivic erivative(oneofrelativelypecific p-down onfigurations)ndicatednEx. 6e. Asthis eries f xamples egins,t mustbeemphasizedhat heyre ustthat:specimensfderivative aterialsased n thevarious ources ited, electedpassages aken or llustrationfsuggested rinciples.



27/94

l.C(, o.17)(pp.,mm.6)

yA y^ ^[) .?,;5jl., ^ J

9As oftenn Bart6k,pparentnomalies f enharmonic otationre ac-countableodifferingevels ffunction:he nitial b/a#lsa case npoint;tsimmediateelationstoa', whilethasa longer-rangeunctionalelationob1,from hich he ineultimatelyescendsog# (Ex. 1)10Most ignificants that he set 2-1-2 s stated ytheviolin's irst ourpitches,swell s inthenext hreewithhebasspedal. tcanalsobe seen-notsurprisinglyn view of the six measures' ggregateontent-onsix other'roots'. hus, djacentndproximateitchesormhe2-1-2 etoneight f hesoundedggregatef welve Cs.e^.12 P.., mm,-39) (G4, O oppoiion)

V2n.,mm.1-13;ln. , 1-38;vla.,22-25; vla,, S-38;vc., 5(TrillPC ourroundingc' aXi)

Z/n ^ - via., 8-34 nd3B-39

...2- 1-() -1-Z...( f,F oppoition)9: 9 * 9 ' V/la.,0-11a., 1-Z1,-9. * ' vc.,5- VC,16-21,yC.,639



28/94

313Reference asbeenmadeto thequartet'sntroductoryhrasemm.1-6) as toquasi-tonal unction,nd these pening arswill be thesubject ffurthereferencenconnection ithmotivicnd ellularntervallicontent.Ingeneral,tcanbe notedhere hat heopening hrase s anexpositionfthe fullaggregate f twelvePCs: a chromatic etrachordn the pedal'voices on C#),with heviolinmelody nfoldingheremaining,omple-mentaryet of ninePCs. Giventhat ggregateesourceandaswithmanycollections f eightto twelvePCs noted n subsequent xamples) t isobvious hat very richordal,etrachordal,r other ubset an bededuced;observationsfmeaningfulntervallicontentn subsetgroupingsmustrefer hereforeoadjacent rconcurrentccurrencesfrelatedC attacks.Content s to thederivative -3 (Ex. 6e) is especiallytriking,s noted nEx. 17a, buta case can bemade npassingwithrespect o the fullminortetrachord-1-2,whichhas important anifestations,speciallyndmostobviouslynthe nterrelationf thefirstour Cs ofthemovingineofthefirst iolin Ex. 11)9 (In one of thefollowingnstances citedPC, G#,occursonlyproximately.)10therviewsof nterval-classelationsn mm.1-6 aregiven nlater xamples.

The extended erivativetring f 2-1-2tetrachordsllustratednEx.12 can beseenas thebasis for he nitial tatementfthese nthe econdaparte.PC contentsthetotal ggregate,utoccurrencesfthegerminative2-1-2tetrachordnd its extensions y overlapingf one degree re dem-onstrateds to adjacencies n the inear treams f ndividual oices,'ntheexampleparticularegmentsfthecomplex re dentifieds tovoicesin the Bart6k xpositionowhich hey elate s intervallic ases.For themoment,eferences tothefirsthirty-nineeasures fthe econda arte.



29/94

4^ 13 .'2., m,40,49)



30/94

315A strikingeaturefthe bove sthedelineationfmaterialselativelyequidistantround he xialcenter, l, a particularlyivid eaturefwhich

is theemploymentf thePCs D, Eb, andA, Bb inthetrills. ignificantnthe upper' egments theG, Gbambivalence,s isthatbetween andF#in the lower'.Comparably,noverlappingtringf 2-1-2 etrachordsanbe seen asa modelfor the area followingm. 40. Again,discrete egments f thequartet'sndividual oicesrunalong segmentsfthemodel, nd arethebasisfor pposed istributionsfPCs,oftenowardblack-white'ollections.In thepresentxample heres considerableppositionfthesegmentn

C# withthaton A, effecting kindof pitch-classounterpoint'here,involvingertainduplications)which s prevalentn muchof the work.Thisoppositionfthe etfrom #toB with hat rom toG issymbolizedin the insistence n theG#,A dyadwith which thepassageis muchpreoccupiedmm.40-44and52-54):moreover,hese re the xialPCs forthecitedmodel. n therepresentationfthe tringEx. 13) someportionsofthe2-1-2 omponentsreomitted s irrelevantothe xtractnquestion.A strikingeaturehatmight e noted s Bart6k's recipitatemodulation'toward hestring'sower egmentsn mm.47-48,then oward hemiddlesegmentn48-49, n viola and cello quoted npart nEx. 13).



31/94

.4.14 (4.P.,mm.277-305)1V.,90-99f,

1lt.,vc., 94-97VlU., c.,2994; vn.1, 98-901 \

40l(i.,S, mm.334-42)

Afloicou,77 1; vln.2, 81-lbf,'1"""" ::::--- , rIt, -I1 .

2-i1-cjz(-V

Vln. , la,334S3S

vlln.2, c., 35-36

,ln., via.,vc., 40-42

4*.1 (c., mm. 4, 99ff.)

O.O.v fr E z

~~r~ ~$ Ar-*

't)v 't. him9pt*-"_ . .LYin, | , Inm9ff,

te.,00/



32/94

317As a furtherxamplefcomparable aterialEx. 14) referencesmade o mm. 77-300citednconnectionith differentntervalet, -

1-2-1-2,ta later oint).uggestionsf malleregmentsf ine elatedosmall egmentsf hemodelre racednly artiallynthe uotationromm. 295. SuchPC oppositionss F, F# ndB, Bb, inherentnoctave-spanningegmentsfthe tring,reexploitedntheworknd, ndeed,palpablyelt.While theconclusionf thequoted xampleand thecontinuationom.316of he econdaarte) anbe seen sembodyingnemergencef hewhole-toneetrachord-2-2,heyan lsoberegardeds'leaping' ithinhe2-1-2 tring,s,for xample,nthe uccessionsb-C-D-EandF-G-A-B.

The 2-1-2 etrachordith singleverlappingy nedegreeanbeseen s thebasis fmm.329ff., hichppearncontrasto a precedingdenselyhromaticassage. hemotives: (mm. 34-35, 40-41) rediscretelyased ndiatonicegmentss indicated,hile C contentsotherwisextendedoembracehe ymmetricallyisposed andE# Ex.15).Resultantontentfnine Csexcludes, C,andB7,the atterwo finsistentmportancenthebars ollowing342ff.).

A 2-1-2 tring, ith verlappingyonedegree,ffive omponentsegments,anbeseen sbasis or he irstiolin eginningtm.94of hecoda nd he econd iolin romm.99 (Ex.16).



33/94

4.17a. B.?,Xmm.6)(cf.4^11)

'-. :_.r- - - -"-,.17b

2t3 tt3



34/94

319The precedingarsoftheviolins, nd theviolaand celloparts nthissegment f thecoda,are also relevant o thisresource-again,notmerelyas to PC content, ut as toderivation n the basisofsuccessionswithincomponent arts fthe model.The first iolin'sDn (see Ex. 16) posesaquestionwhich an onlybe touched npassing, lso relevanto theviolaand cellolines-of-chordsetweenmm.84 and 105 (comparem. 42 ofthecodaand,ultimately,hecello themewithwhich he econda artebegins).The resource or heviolaand cellopartsnthis ection an be interpreted(1) as to an expandedymmetricaletofnine PCs: (A#), C#,D#,E, F#,G#,A, B, (D); (2) as having riginnthe2-1-2 tringombinedwith hechromatic1-1-1)tetrachordsxpressed ytheviolins n 84ff.; r (3) asconformantothe -1-2 tring, ith xploitationf uch nherenthromaticambivalences s theD#,D relation,nd theA#,A pair.Throughouthispassagethe 2-1-2-2-1-2esource s, in general, een in one of its mostexplicitontexts.

Onlyveryimited ote s takenhereofthe 2-1-2 etrachords sourceofthe mportantormulation-3, nd ts onsequentmotiviconfigurations5 3 (or reverse) nd5 2 (orreverse); ee Ex. 6e. Buta few xamples fmanifestationsf thisderivative-especiallyessobvious nes-are given.[Alsorelevants the set2-2-1-2-2nd thederivative-3-2, epresentednEx. 6ganddiscussedater.] hecompositionfthe2-1-2 etrachordeen stooccurrencesfthe2-3motivic ormulationnbothdirectionsagain, x.6e) isone ofthemost ersuasiveactorsnderscoringhebroadmportanceofthat etrachordn thequartet.The critical,nsomeways vasive nddifficult,pening hrasemustagainbe cited s tothegenesis fthis mportantonstruct. he followingspeculationsre relevant. he initiatingwoappearancesfthe3-2factoroverlaprelatively nequivocallyo form he 2-1-2tetrachord,s alreadynoted.Elements n Ex. 17a are representeds PCs, notconsistentlyspitches.Again it should be emphasized hatdiscussion n the presentconnection, s in others, s of IC proximities-usually djacencies ndsimultaneities-andot foverallontent,ere mbracinghe otal hromaticaggregate.[Thefirst iolin's tartingotes analsobeinterpretedsbringinghemotive ut n a kind f evolving'Ex. 17b)throughn interim'2-1-)forminthe onfigurationt34 orreverse),ppearinghree imes, reliminaryothe mmediatelyollowingexpanded', efinitiveorm5t 3p)on F/E# nmm.4-5. n thisregard,ee also mm.20-21.]



35/94

44. 1 ip, .33)

* ev r

V V9:,It0a ;t

Jr Jrj irr rjLrJrJ: I

(ij-*. 19a , if. ,.00-5

4. 19b2., mm.871)

0f19c(p.p. ru.974109f)"

g$\)0

1 n Ex.19c, henotationsputnto lignmento how imultaneitiesxpressingthe3-2,2-3relation.he violadoubles he econd iolin.



36/94

321Suchexplicittatementss those f mm.7ff., 7ff.,nd35ff.,nadditiononumerousthersnthe rimaartes well s notherections,

arenot itedn thesexamples;utnotem.33,where ourranspositionsof he -2 richordre xplicitithinhe esourcef he -2-2-3entachordonA (A-C-D-E-G)na complexf nsistentirectnessEx. 18).

The chordsollowing.54ofthe rimaartere lsorelevant,s istheir reparationnmm.50-53 Ex. 19a).Examples 9bandc providefurthereferencesnwhich he3-2derivativeanbeseen.

A comparableine ccursnviola nd econd iolinowardhe nd fthe rima arte,wherehepredominancef4ths nd 5thsIC5) extendsinto ccompanyingoices-a cluetothe nfluencefthe3-2,2-3factorthroughouthe exture.ome f hese elationsre ndicatednEx. 19c.1



37/94

'4 20(u) (~~~~~~~~~~~~~~-e-)...2-Z... (a) J ~

;z-(0zThis content downloaded from 147.162.119.135 on Fri, 19 Jul 2013 05:02:43 AM

All use subject to JSTOR Terms and Conditions


38/94

323The motivic ormulation hichsthebasisfor heprecedingxamplesis, indeed,one of the quartet'smostomnipresentesources, nd many

furthernstances f its applicationsmightbe cited: it is important,fcourse, nthericapitulazionee.g.,at itsbeginning,t m. 32ff.) nd nthecoda (m. 34, m. 43ff.and m.118). Its presence s felt and is clearlyappreciable, otmerely educible)wherever,n fact, itch tructuresrebuilt n 4ths, 5ths,2nds, (ICs 5, 2), such structuresegarded ere asexpressionsfthe3-2 IC setand itsparent -1-2;or ofthe combinativeform,he2-3-2 C setdiscussedn thefollowingection s derived rom(embeddedndduplicatedn) thehexachord -2-1-2-2seeEx. 6g); orthe'pentatonic'-2-2-3seeEx. 6h). In allofthesefundamental aterials,he3-2 formulations a distinctive,nddistinguishing,actor; heforegoingexamplesdemonstratehat its applicationsn Bartok'spitchstructuresreachfarbeyondts mportant elodic onfigurationC5t3 (or 2\ .

A number of the exampleswhich follow show the symmetricalhexachord 2-2-1-2-2)and/orpentachord3-2-2-3) in 'combinatorial'association,ftennwhathasbeen characterized ere s deliberateblack-white'opposition; ubsequently,he derivative -3-2,a basisforfurthermotivic ormulationsf the kinddiscussed n theprecedingection, sillustrated.xample20 notes, npassing, heextremelylose relations ywhich ny fthesematerials anbe construed ogenerateheothers.



39/94

^. 21e., mm.7-33)

5- Z-2-3.721 /

f'Ks^.22a(ZE?,mrm.5-42)

m.35 ^i3i

3-2-2-3



40/94

325Byfar hemosttrikingxamplef he ppositionnd extural-rhythmicseparationftwo ombinatoriallyssociatedentachords-oneblack',he

otherwhite'-occursarlyn theprima arte, ollowingherelativelydense,hromaticirstage ftheprintedcore,nd nsevere ontrastothe irst6measures.xample1 s a representationf he assageollowingm.26, nwhich hecounterpoint'f harplypposedC contentsevidentatonce, s isthe xplicit otivicontent:t31,7$51,etc.

Followingirectlysa passagenwhich, hile he pperoices nfolda fullyhromaticet theoctachord#,G,G#,A, A#,B, C, C#), helowerviola, ello)expressherelationhownn Ex. 21 (seeEx. 22a).These wo oices lsoprojecthemotiveiscussednthe oregoingectioninoneof tsmost aked,urelyterativeontexts:C 513N-3+4-2f5t,sillustratedn Ex. 22b.



41/94

7 -ljZ-i) aI/ 2 -1-(0-1 -

m.57A .

aw .* V f f -7ffE/ a * 4 A '-

roAV*-r "t%' *- -Wm,.S ,

--- o increaida,dmifure,teiniip in tm.5J biparlionf..._ _- -11

? . *: U~W M VPc1 f: W -Jr- ^. .W9= i9: V~lS. . p & 9:K -

over-allaggregateresource

3- E~-z)-

4.325

2 (),4

4 23bE ., me.-.58)

m. 4r'A

Iv

Itl} '- I1- -

A_ _ ~ . I

.r Sw ,_*s

- 'W Mf -



42/94

327Measures 4-64 re nelaborateevelopmentf hemotivend f heoppositionrincipleotednthe bovexamples:hilehemotivendergoesbroad ntervallicxpansionEx.23a)-extremenmm. 0-61, he black-white' pposition,he white' actor owextendednto hefull iatonicheptachord,s maintainedonsiderablyEx. 23b).As thisdevelopmentunfolds,he ppositionfdiscreteets scompromisedittley ittle, iththe wo pposedomponentsncreasinglyntermingled-i.e.,ithn volvingaggregatef ncreasinglyomplexinthis ne ense) exture.

The closing ection f theprimaparte see mm.87-109) brings hisprinciplegain ntofocus, lthoughot withouteviance. he interiorsegmentf hatlosingectionmm. 3-102) ringsther, orehromaticmaterialsnto perationincludinghe1-2-1 et obediscussedater); et,therescontinuedombinatorialppositionf ssentiallyentachordalets,persistingven o he inalonoritynwhich# ndG# re pposedoG.



43/94

4O12 (c., nnm. 6-21)

im.110 m.1204.A" A-i - - a

* k. "-

U-- 4-,I--

~~~~~~~~~~~~~~~ Lif I?1 I;-* ~ ti ~ .

.-- Jt ( -r

2-2-C-2-2

- A

.ZJL . . a 4 R

F3 2>

A lg12'1I3 2

h

Im.117-?



44/94

329The quartet's onclusion s also relevant o the sameprincipleEx.24). Interpretationsf black-white'pposed ets reshownn thefollowing

summationfPC contentnthecoda's final ars. n the black'pentachordonD#,thePC F# somittedaswasthe ase ntheprima arte'somparablepassageat mm. 87-109); it is easyto speculate hat ts omission s forreasons f tonalclarity. hus, thestrong epresentationf G# (togetherwith heabsenceofF#)greatlyortifiesheultimatemergencefC#asunequivocaltonic'.

Example greferso anintervalet2-3-2, iscussednthat ontextnrelationo the major'hexachord.t can alsobe regardeds a combinativederivativef themotivic -3,3-2 Ex. 25a) oras,like herelated -3 or3-2) trichord,segmentf the pentatonic'cale Ex. 25b).



45/94

4-.b (., mm.7-53)2 ,- Two12-tonerrqa~t:2--2 2 - 3 -I N I 2-3-2

-- I L I--- ,r -' b P.-cL- -



46/94

331The most trikingpplicationfthe set2-3-2, n a highlyhromaticcomplex, s at mm. 47-53 of theprimaparte. n thispassage,2-3-2 s

manifest erymuch as an interval-classet,takinghowever consistentform: ; 2f 1. t s mitatedna series f diagonal' elations t2J, hen t1land 3J, with tsoccurrencest t levelswhichbring boutcombinatorialassociationsoften f black-white'pposition)ffectingwo otal hromaticaggregates,s pointedut nEx. 26. Moreover,hePC levels freiterationprepareunfold,ay ut)thePC contentfthe hordswhich ollow,iscussedearliernconnectionwithEx. 23. The viola'sBb, C, Eb (mm.49-50) arebestviewed s continuing'hefirst iolin's (sustained 2) nan extensionofthe2-3-2resource,s indicated.

Other xampleswillnot be citedwith pecific eferenceo the2-3-2formulation,ut can be seen in relation o the associated ets (whichinclude rcanbe seenas otherwise elated o the2-3-2)discussed arlier.See,for xample,m. 54 or m. 87ff. ftheprima arte,mm.408-14 (cello)ofthe econdaparte, r thequartet'soncludingars;or seeEx. 19bor Ex.23, above.

Weturn tthispoint othe mportantymmetricalexachord-1-2-1-2 [or, 2)-2-1-2-1-2-(2)];xx. 3, 4, 6j and7 are relevant o thisresource,manifestarticularlyntheseconda arte.Moreover, xx. 6b and 7 showthe hexachords the nternalixdegrees f the setformed ythe major'tetrachord-2-1 nd ts nversion -2-2 uperposed ith asthe nterval fjuncture. xample6j representstas the2-1-2 etrachordverlapped ithitself ytwodegrees. herefollows series fexamplesnwhich omeofitsvarious pplicationsrecited nanalysisf extracts rom hequartet.



47/94

peIal... 27 + =(p -(pE=-= 2- 1- ( --

.r 28f(I., mm.33-4; cf. %32)in.233

13t" .1

3 -' 11-

Compnltmntaryierachord:/ wndafrf - / --t (-)--i M234

Jibdfuintihration

z -1- 2- - - -2- I-2

^27



48/94

333Althoughemonstrablencertainxposedonfigurationse.g., ntheimmediatelyirrored-3at the ummitf theviolin ine ndthe ame

line's adentialrichord;ee Ex. 27) itseems oubtfulhat he2-1-2-1-2resource assignificantanifestationnthequartet'srucial,ndprob-lematic,peninghrase.hemost efinitivexpositionfmaterialserivingexplicitlyrom he 2-1-2-1-2et occurs n thefugato fthe econda arte,asnotednExx.3and .Commentsmade elowoncerninganifestations(andemergence)fthe et napproacheso thefugato, herensistent'pedal'pplicationsf he onallyriticalCsA andE are tthe ame imemuchn vidence,othnpitchterationndn terationf -rootedhordsandA-rootedscales' f nalogoustructure.Inthe ugatoxposition,hemost xpliciterivativefthe2-1-2-1-2etappears:n A-rootedubjectansweredy n E-rootedubject.)

It iswell onote npassinghederivativetructurefthefew arsimmediatelyrecedinghe nitiatingubjecttatement:hese onsist fiterationsf he itch and nE-rooteduasi-dominanthord; ut tmm.233-35movingines,uicklyborted,ive dumbrationf he pproachingsubjectnconfigurationshatncludehe edalA (a1, ustainedremolandosulponticellontheviola) nadditionocoursingpanddownwithinhemodelllustratednEx.28.

Thefugatoubjecttselfm.242ff.,econdiolin)xpresseshe riticaldistinguishingeatures f the 2-1-2-1-2et: its two tritonesnd twodiminishedriadsn the nitial,mirroredegments;nditssymmetry,expressednthenotedmirroringnd lso nthebroadonvergencenthecentralnterval2) andonthe mpliedxialC#.That onallyritical#(later,nthe nswer'sranspositionf he et,G ) s mportantnthe ocaltonal ctions fthe ccompanyingountersubject'smpliedounterpoint,as treatednEx. 4. Significantly,hree f ourCsof he wo ritonesf hetransposednswerm.246ff.,ello) re bsentromhe nitialCcollection.Some of the mportantualitiesfthe2-1-2-1-2et,and of its nitial



49/94

2.29

L_.4-FmirroredrifonienddiminishedfriadjnherentnVheetvector 34122)

abieni C^: 3 later$), and hetrifoneniialing heranopoiedanlutrtofollow

the ubjecl'diitinctivezpreiyonofJ e et' inherentymmeery

r 2 - . tZheJubljet' urther,roaderi- *- L (' eeipreiionf hatynmmtryoward~--r^-- r- p focalconvergencen heaial C}- 2 I a v la,, .'5he10PCdf he et nd ti$- transpositiont IC7excludehea%ial %nd # thetonallycentralCgnthequarttao a whole

12 That the 2-1-2-1-2set can be regarded s progenitor orthe whole-tonesuccessionsprominentncertain fthe ines of thepassagesfollowingm. 80 isevident n itspotential,nthe form2)-2-1-2-1-2-(2), or2-2-2-2 uccession-asin the PC successionEb-F-G-A-B fthe set is on A, as in thefugato ubject.Since, in thefollowing iscussion, s inpreceding nes,references attimes osets of considerable C scope, t s importanto noteagainthat nsuch instancesthe quartet'smaterials re seen as derived uccessionsof specific ntervals-actual segmentsof scales and scalar derivatives seen as (often extended)models-not merely s to IC contentdispersed nd unordered. till, smallersets, ike the derivative -2-1-e.g., B-C-D-Eb-will be discussed as thebasisfor uchdispersalsfderivativematerials fconsistentnterval-classormulations,as was the2-3-2set,for xample, n connectionwithEx. 26.

iJTWUJ

1niL



50/94

335applicationnsubject tatement,resummarizedraphicallynEx. 29. Thereaderwillsee, on evencursoryxaminationf thedevelopment hichfollows hefugato xpositionm. 256ff.),manyderivative ormulations,somefragmentary.ertain f thesehave specific nalysisn connectionwith ater xamples.

The areaofthe seconda artefollowingm. 80 is (within hepresentpurview) ne ofthequartet'smost ntriguingndproblematic: develop-mental lay n all'of hework's undamentalcalar esourcesndderivatives,including hedirectlyhromatic1-1-1. .) as well as whole-tone2-2-2... ), and the tetrachord-2-1 which ies at the center fthe2-1-2-1-2formulation,ndwhichsdecisively revalentna numberf arge reas)-of all ofwhichmore ater.The 2-1-2tetrachordnditsstrings,iscussedearlier,re relevant eretoo:see,for xample, heviolins tmm.131-38.Itmaywell be that he 2-1-2-1-2 esource,which mergesnexplicitpredominancen thefugato,s incipient ere, lternating ith,oinedto,opposed ocontrastingrderivativematerials,na constant luctuationfreference o intervallic nd motivicformulations avingmore stableexpositionnother,motivicallyess fluctuantegmentsfthework.12



51/94

IN

It4I

Ikl

IIt-

?

I

F-

-I-I-

F-

tI

-I-A-A-I

J

-AiN

-I

I

t0.E]-I

#I~\z-1

,".

(

I



52/94

337The 2-1-2-1-2 et can be seenas emergent,or xample,ntheviolinfollowingm. 95, compromisedyreferenceso (derivative?) hole-tone

andsemitonal uccessions. he lineoftheviolins ontinuesnformulationssuggestivef the 2-1-2-1-2 nG#; thenonconformant# at itsregistralsummit s a link,perhaps,with he2-1-2 tring2-1-2-2-1-2,entered nC# oftheconjoined iola andcello,or as evidence fthe nterposition,ttimes revalent,fdirect emitonaluccession-here, imited o 1-1at theapproachothehighest itch. hroughouthis reaoffluctuant,ariegatedscalarand motivic eferences,he formulations-1-2-1-e.g.,C#-D#-E-F#-G ntheviola t mm.112-14)and1-2-1e.g.,D#-E-F#-Gfthat amesuccession) revividly epresentativefthe parent' -1-2-1-2.The 2-1-2-1-2et s in furthervidence t m. 127ff.cello,viola,onG#)and 130ff.cello,viola,on C#).Such deviants s theB#/CPC of m.133 express he1-2-1motivic-cellularerivative hich akes na tentative,quasi-independentxistencen this ection, ssuming iscrete unctionnother reastobe seen. Notetoo thesuccession1-2-1E#-F#-G#A ofcello and viola at 135-38.)OthergroupingsG#-A-B-C1-2-1

are traceable o thefamiliar-1-2 tring, ery xplicit t m. 131ff.,nthefirst iolin, nE.Example30 traces omparable evelopmentsnmm.149-60,where'pure'2-1-2-1-2manifestationsre bracketed: hey ccuron A andF#.Chromatic,deviant' roupingsreencircled. fcourse, he equenceA-B-C-D is attributableo either ransposition.Chromaticomponentsan be viewed a) as symmetricallyisposedextensions f 2-1-2-1-2 etson A andF#-an interpretationortifiednthatAb, G, andF# ppear nassociationwith he set onA, andF, E. andD# withthaton F# (Ex. 30b); (b) as to an overall 10-PC resource fsuperposed -2-2-1 entachordsntwopairs ontaininghe 2-1-2-1-2ndcenteredn it (Ex. 30c); or (c) as 'interchanged'actors etween hetwosets:2 PCs ofeachoccurringnthe linear nvironments'ftheother two(2)-2-1-2-1-2-(2)ets, n A andF#,themselvesmbrace 0PCs,excludingonlyBb andC# . As to the atter iew, ee Ex. 30d.The 2-1-2-1-2et,onB, G, E, andA, canbe seen as thebasisofmm.187-93 again, econda arte) n a context fgeneral revalencefthe2-1-2 stringonBb andEb . Furtheregmentsf inesnotrepresentednEx.31 arerelatednexpressionf he ubset -2-1,reatedt engthnsubsequentexamples.



53/94

< .31 .p.m,mm.16-93)

Im,lIb~~~~~~~~~~

onr

m.lJ6o, .n. ? 19 .

on

o. 32a .p, mm 16-29)22S

n2916 229 -onL- - . * - --.. ,.. - , . 4- c_ cr * - e- .'

222?- 2241onC- L L . 219 -on ;

2~~~~~,Lt Pd -W t-07 (0



54/94


55/94

4.S32b(cf. ^.46af)

.32c d, mm 22-29)m. 2Z , iL- . . AL- - .---.&~~~~irFr~~~~~~~~J~~~TLT~ ~~~~-~J a O

4y33 , , mm 70-74)

m,70Cs-h6r h?rTb~

44'3(fU mm.31-35)

m. 12 - -1--1-

40.3S'(c.,m.S4)

r 4 Is" 5,4 p4zZ.113 0)i~pu' (13 V. -

2 -2-1-3The comparable ectionfromm. 77, criticallymportants tothe 2-1-2-1-2set-witness the firstnotes of either violin-is treated ateras to the 1-2-1subset.

m.274+

(cf.ytn ,Za.)

,q-~j'f-ftx I rY' . ^ . rI~b& 0



56/94

341motiveanotherpplicationfthework's nderlyingf 1configuration)na series ffluctuant-1-2-1-2llusions lternatingith thermaterialsnmm. 275-94 (see especially he secondviolin at 289-95). This motive,occurringt a number ftranspositionalevels nd nimitationt inthecitedpassage, s indicatednEx. 33.Althoughhe coda is as such thesubject f ater eference,ertain fitsevidences f the 2-1-2-1-2etcanbe notedbriefly.he cello ine fromm. 31 (from #') is a simple ase inpoint Ex. 34). So is the owest ellovoice fromm. 43, viewedbeyond he immediate oregrounds to its'structural'itches: #-G#-A-B-c.orfurther,elevantllustration,x. 35quotesfrom heviola t mm.53-54.The intervaluccession -1-2-1-,r-1-2-1-2, s invariablyuggestivef thisresource.Also pertinentothe 2-1-2-1-2s themobileyet,n a broader ense,static) assage tm. 55ff. fthe oda. 3Here, nalternation ithpure'1-2-1 statements ot representedn the followingxample, nd occurringlargelynB, accumulativentervallicormulationsre nterpretables 2-1-2-1-2-based,s constitutedfa modelofthe2-1-2-1-2verlapped yonedegreewith ts transpositiont IC8 (Ex. 36). This model can also beexpressed s extracted rom n extended2-1-2string n A, and otherinterpretationsould be adduced.Example36 includes arentheticallyhepunctuatinghords,nwhichD andE, lackingnthe10-PCcited esourcefor heactivepassages, re ntroduced.

A number freferencesavealready een made topitch tructurestraceable othe et 1-2-1, elated o source f?derivativef?)the 2-1-2-1-2; itsrelation o2-1-2-1-2maybedescribedwith hestatementhat t s asubsetyingwithin 'radius' f twodegrees f the xis. See Exx.6j,k,and7.)TheIC-set1-2-1ssignificantnallfour f he uartet major ivisions.It is, in a sense, opposite' o theimportant-1-2; ike2-1-2, tyieldsmotivicntitymanifestnbothdirections: -1-2 -> 2-3,or 3-2; 1-2-1 >1-3,or 3-1 see Ex. 37). The derivative -3 orits nversion)s relevant san interval-classet yielding tructures-horizontal,ertical-ofgreatflexibilityf ntervalize,direction,ndregistral lacement.



57/94

4e. 36(c.,mm.-70)

2-1 -2-1-2bl j ' ~"T T ( fOC,Zacking ,)

2 - 1- 2 2 - - 2 ymmcrically itpoud omponen*trachordK - 1 z 2-2 1 - 2f ,T- JbJb-- 72-. - I'...- - ,..,

s.357(2-1-2) J0 3

2-0- 2 Zt31t 2

1- -1 31 1t 3i

3 1 1$U Y (I- -)31 5t35 5i 3t etc.?1 3 31t 11; tol ctc./

1-2-12-1-2 A00000~~~~~~~0~



58/94

343Among he ections ftheBart6kmost ividlyelevanto this esourceare:primaparte,mm.43-46, 72-87 (andpreceding),nd93-96; secondaparte,242-69 (and beyond), nd 342-400; ricapitulazione,5-end;andcoda, 55-77; theseare notedas perhaps hemostexplicit nd extensivemanifestations,lthoughhere remany thers f esssingularndexplicit,orpassing, mplications,omeofthem lready iscussed.It is extremelynterestingo speculate oncerninghe 1-2-1set inemergencenthequartet's arliest hrases.tsgenesismaywellbeviewedas occurringnmm.3-4.To see that ritical pening hrase n a furtherlight, t can be noted that a fundamentalmotivic hape- and-evolves (perhapsnprogressivelyiderntervallicnits, trend

thatcan be seen as continuinghroughmost oftheprimaparte). fwenumber hefirstwelve itches,mittinghe nitial uplicationfa#',butincluding epetitionsfpitch ther han hat inglemmediatene,wecanmake hefollowing,elevantbservations: -2 -3 (2t3); 3 -4 -5 (2\3t);4 - 5 -6 (31 2); 5 -6- 7 (2j3f). [The subsequent - 9- 10- 11 - 12 havealready eencited s tothe succession t3 332(the motivic t31, tc.).]And within he firstevenattacks,gainexcludinghe nitial uplication,the1-2-1 et s inemergence: ote speciallyhesuccession - 5 -6 -7 asthe et1-2-1i.e.,reordered , G#,A#,B). Example 8 isa representationofthesefactors.

Relativelycovert'manifestationsfthe 1-2-1 et attimes ettereenas 3-1, 1-3) in thesection mmediatelyollowinghe ntroductoryhrasearerepresenteds specimensn Ex. 39, inBart6k's hythmiclignments.



59/94

c0c~.99-, 1-2- 3- 11 ~-2-1 I -I r-

4.40? Zinmn,-3-46)1-i- 1



60/94

345

Meastres43-46ofthe rima arte re thefirstnequivocal xpressionofthe1-2-1 et.While hromatic1-1-1)tetrachordsnd octachordsertaininthemirrored,irectlyertical elationse.g.,m. 43: A, Bb,B, C, D, F,F#,G,'or m. 45: G#,A, A#,B, C,Db, D, Eb)-a situationwhichopensout' in the 2-3-2collectionswhichfollow see Ex. 26), the 1-2-1 etofrelationssclearlyvidentnhorizontalnddiagonal eadingsfthedoublyimitativeines imitativetcontractingime ntervalsfJ, J,J). ollowingthe nitial,mostexplicit xposition fthe 1-2-1, hat et continues obemanifestn interactions'fthe ines, na relativedispersion'Ex. 40).

In the ello atmm.64 and71, the1-2-1 ssumes particularmotivic'form; hiscan be seen in thePC successionA# -A -F#,a specificCcomplexwhich merges opredominatena numberfareas nthe uartet.Example 1a quotes heviolaandcello at mm.68-71,with ttention rawntothe celloofm. 69 as an important otivic onfiguration--311 . Noexhaustiveecountingfthismaterial,nymore han fothers,sattemptedhere, nd cited xamples reagainto beread s specimens.



61/94

4v41a., mnm.8-71)

^ ,lb (.,, mm.273)^/ 72

.%

71?3tv

=A

13'' )*'-1~tl,tt-o?1CV.10 a~~~~~~i

mm. 0-71 7Z (72) 72-73 73 75 ?7677f zevtd:

7S-77( 72)7-77 ( 11PCi,( r >Z. - lin(* 'r Df)

13 1/3$X$w-j: /:3-19: so -

t - 3 j^^y^^ 41Jr (31 1f,),?/fa,--



62/94

347

The 1-2-1 nd tsderivative -3,3-1undergoesreatmentnBart6k'sdevelopmentsn importantwaysparallelto that of the 2-1-2 (or the'pentatonic'et)andthederivative -3,3-2:themotivic esources -3,3-1and2-3,3-2-the attererhapsnterpretables an expansion'f he ormer-areofprimaryignificancenallparts f the uartet,ach of them ssumingcharacteristic,istinctiveonfigurations.The 3t11 motivic orm oted boveprevailsn m.72ff.andagain tm. 84ff.-notquotedhere). n thenext xample41b) only fewbars regiven s basis or n nteresting,upplementary,elatedoint: he onsiderableextent o which ordered ranspositionalevelsof the motivicunitformcontiguous, igher-level,arallel etswhich re furthermanifestationsfthe1-2-1 nd tsderivative -3,3-1.

It is interestingo notethatEx. 40 (p. p., m.43) representshe1-2-1set n mitationtthe ame t evels s are topertainwhen he et ppearsnits-perhaps-contextof origin'within he 2-1-2-1-2nthefugato fthesecondaparte.There,the countersubjects'viola,m. 242ff. r violin1,246ff.; nd violin2, 247ff.)havebeen discussedEx 4 andaccompanyingcomment)s to 1) tonallyonditionedmplied oice-leadingnwhich woPCs external otheunderlyinget reofdecisive unction:#,G#;and 2)its thematic unction s a slowermoving, educedform f thesubject,similarlyxpressinghe et's ymmetry,na sense imitating'tatchangingdiagonal elations.Within hepresentoncern,t shouldbeconsideredooas to itsexpressionsfthe isolated'andinsome of ts forms ersuasivelyexposed)1-2-1: hefirsts a natural onsequence f tsmimicryf the2-1-2-1-2 ubject, he econd ntroducing ore overt, etnsistent, anifesta-tions Ex. 42a).



63/94

4*i4OfC2 mm. 42-416,46-49)m.2242 ?

m,24b

Jt f

orWIComparabZe,dditionalmplication,f tHto'count'riubject'n aiociation:

e4,4,z b , mm,,5-59)



64/94

349Ofcourse,wherever hesubject ppearsn thefugato xpositionndsubsequent evelopmentalpisodes he1-2-1 s in evidence s thecentral

subset f the 2-1-2-1-2. utit is importantosee its argelyindependent'life ogether ith hederivative, otivic -3,3-1nd other -1-2-1-2-derivedmaterialsnthe nitial pisode, artiallyepresentednEx. 42b. Hereagain,the haracteristic otivic orm t11I smuch nevidence longwith ther,related orms.

The anomalous oncludingreas oftheseconda arte fromm. 329)are ttimesxceedinglyuggestivef he1-2-1 et, lternatingithhromaticmaterialsi.e., 1-1-1 . .) and at timescombinedntodensely hromaticcollections fas many s 10 PCs (see Ex. 43) nonethelessppliednwaysthat, yvirtue fproximatessociations oth inear ndvertical,trikeheearwithrecurrentat times onstant,multidirectional)xplicit emindersofthe1-2-1 and 1-3,3-1)underlyingesource. he concludingection fthe secondapartecan be characterizeds one of progressiveendencytowardncreasinglyhromaticontent'chromatic'een, gain, s a matterofdegree elatingothefrequencyndconsistencyfoccurrence ithinset ofthe uccession -1... ).In the area followingm. 361, the 1-2-1 s increasinglyominant,increasinglyeadily erceptible,etalternating ithchromaticmaterials(e.g., D#- E - E# - F$ - G of mm. 363-64, or comparable lissando'motives'). f significant,ndpotent,mportnthis onnection retheff,ruvido, own-bowttacksnC,C#, recurrenttteranceecoming -2-1-based nan evolutionary'evelopmentnmm.374-77, nwhich he1-2-1is finallyxtended o embrace ncreasinglyhromaticuccessions.Eventhen, s at mm.378-83, 1-2-1 etsonB, C, C#,andD arekept exturallydiscrete.)Example 3 quotesfrom he core s. p.,mm.383-95)to show ertainof theseexplicitmanifestationsf the 1-2-1resource nd its 3-1, 1-3derivatives.



65/94

4,43 (4,y"mm?383-95),m,.383$ 0 :

5)i iJ_I -

| t< 2 (col_____We ___

4%4(c., mmI. 0) -1-2Zn, I^^^^^I /

\ _ __ \-\

= ,",,*. L_..6" o ?-A ._7*

J f) ? l

tC+ r t ~~~~~~~~~~~~~~I

v \ \\ / \ / \ / _I ,11-2-1

4%. (c., mM 8-8t4)-"? -._r_. 1e . . ..-------------------I 2-1-2-1...i) - ( ii)l4 I.7,

fm.5S, 1 6J 80 III I II ,I Ii, i i -

ilv 1

I

/1 I \ I

-



66/94

351

Passages n the coda to whichreferenceouldbe made n the1-2-1connection re mm. 55-57 (wherethe 1-2-1 set is constant,nd at itsfamiliarC levelonB); mm.57-72 wheret sintermittent);ndmm.72-77 (constant),n additiono areasof 2-1-2-1-2pplication.wo lessovertmanifestationsan be foundn mm.24-33 and43-55 ofthe coda. In the'bitonal' rea,mm.24-35,the cello is tonally pposed ouppervoices, swell s insomedegreenthe black-white'ense, nd n ts oncentrationnthe2-1-2stringon G#,C#,F#) intowhich heset 1-2-1 s interposed,marked yoccurrence f the PCs B# and G, external o the otherwisepredominant-1-2 tring.

Example36 presents viewof a segmentf thecodaas, nsignificantpart,modeled n a 2-1-2-1-2tringnwhich, fcourse, he1-2-1 etrachordis prominent,aving ecurrent,ppreciable pplicationnthat ontext.nthat connection wo additional ointscan be made: Bartok's xtremelyactive inescanbeseenas alternatingetweenhe etrachords-1-2 nd 1-2-1 (at times xclusively,s inthefirst iolin, or xample, tmm.66-68;at timesncombination ith hemajor etrachordnd ts nversion:-2-1,1-2-2). xample 4 indicateshis etrachordalontent;nd a second, elatedpointsuggestspossibleapplicabilityf the 1-2-1tetrachordt another,higherevelofstructureEx. 45). That is, ifthehighpointsnpitch retraced hroughhis ctive assage, heyrefound oconfigurehe1-2-1, ran overlappedombinationf 1-1-1 . (Bartok'sinesrise ndfall,with1-2-1each ascent o a pitchhigherhan ndconjunctlyelated o,thepreceding,up to the d#3prolongation-withts semitonalneighbors3 and d3-applicable o theconcluding 8 measures f thequartet).



67/94

2-1--1-2- 1-2-0-2- -2 - --1 - 2

i -aalio: - I I, . (.- a ) a, 4a

in 76 t21_

"'? i',w,^ __f~ ~ L. I

4. 46a (c, mnm7-84)

vtn.

ytn. ^^ a

4X. b(cf. .32)

4%.4bC

4^, 6d



68/94

353Measures 71-84 are,as is apparentn themost ursory iew, vitalarea of concentrationn these materials.Measures 71-76 manifestn

exclusive reoccupation ith heset1-2-1 t thePC levelsB, F# thetwolevels ssociatedwith ts definitive'ugato ccurrencen the econda arte),andCo. As to mm.77-84, t swellto consider heuppervoicesfirst,ndthe owervoices ignificantlynrelationotheupper oices.A summaryfpitchesn theviolins'ctive arts eveals,na number f ights, developmenton the ets1-2-1 nd2-1-2, swell s anexplicit pplicationfthe nclusive2-1-2-1-2 s the synthesis'fthese.The secondviolinduplicateshefirst,at tl 1, in a canonat 1, ; the structurefthis anonicvoice,given nEx.46a, showsa symmetricalomposite ompoundedfsymmetricalubsetstowhichwe havereferred.

Example 6 'interprets'nternal ontentfthe anon s to a number fways fregardingtscomponentubsets.Itcanbe seen,for xample, s aself-duplicatingoctatonic' uccession, s in Ex. 46a, or (Ex. 46b) as theoverlappedbytwoor four egrees) tringf2-1-2-1-2ets.)Itcan be viewed s the et2-1-2 verlapped y wo egrees,r uperposedwith ts transpositiont IC6, with C1 as the interval f junctureEx.46c). It is alsotheset1-2-1 verlapped ytwodegrees, rsuperposedt t6withC2 as the ntervalf unctureEx. 46d). Or, t s the lternationfthefundamentallymportantetrachords-2-1and 2-1-2 (Ex. 46e). Thesecomponentelationsmaybesuggestive fthe et2-1-2-1-2sfunctionalnthegenesisofits components -2-1 nd 2-1-2, ndependentlyreated yBart6k s generatorsorderivative itch tructuresnvarious reas ofthequartet,rconverselyuggestivef 2-1-2-1-2s the ompositemalgamoftwo, ymmetrical,erminativeetrachords.n any vent, hese mportant,functionalnterrelationsrepersuasivelyxposed, ndsummarized,nthecoda.It is possible hattheunequivocal, onsistentmphasis n the 1-2-1tetrachordn the mmediatelyreceding m.72-76conditionshe ognitionof thesubsequentines s to thecentralityf that esource. he set1-2-1,in thesection s a whole c., mm.72-84) appears,nterestingly,t all PClevels xceptA, C, andDo. These three Cs are 1) a partial, ritical asisfor the viola and cello parts, s discussedbelow; and (2) thePCs givenprominentxposure yvirtue ftheirunctionss cadential igh oints npitchin theascending,nd reascending,iolin ines ofmm. 77-84. Wehave alreadynoted that the last of these,d#3, undergoes rolongationextendingo theendofthe work.Wehave alsonoted Ex. 45) that hesehighpoints rethemselvesart fa higher-level anifestationf the1-2-1formulation.



69/94

4. b/f 1-&vlnI. ^ r; :a

2- 1 - 2...t. 4b9 ? mm.6-83)

Cf, h two4ucctlion'encircling', then $-ontt&he iminution the ther.14Inthismportantunctionf onalresolution'owardhework'sonclusionthe encirclement'fC#byC, D#, and D is of vital ffect,s is the voice-leading epresentedynoptically,ndwithoutomprehensivereatment,nEx.46g.

f-2-1-2-1- 2- 1....') 1



70/94

355During hispassage, heglissando iguresf theviolaand cello select'and doublecertain rucial, ritonallyelated Cs of the violin ines. To amarkedxtent hey oublepitches s wellas PCs.) Itseems hat his sonefunctionfthe ower oices-the extractionfandemphasisnkey ritonesreflectinghe ntervalsftranspositionftheviolins' omponent-1-2 nd1-2-1 ets.These tritonesEx. 46f)haveagogic mphasisntheviolinsandcanbeseenbeyondhe mmediateoregrounds a furtherxpositionfthe1-2-1-2-1.. set, s seen n Ex. 46f);thetwotritones oublednthe owervoices onstitute furtherymmetryuggestivef whole-toneasis.A second critical unctionf the lowervoices shouldbe notedpar-enthetically:hat f tonal-harmonicuccesstionoward # (Ex. 46g).14

Otherkinds fintervallicollections aving ignificancentheworkin varying egrees houldbe considered riefly. ostof thesehave beennoted npassing.The 'whole-tone'ollection2-2-2... ) can be defined sexplicit o the extent he succession2-2...pertains,althought is likelythattrue whole-toneharacter equires hatsuccession o the extent fthreeinto hefourthegree,nto C6). That succession2-2-2.. ) can bebroughtbout,givena particular elevantC set,by anyofa number f'rules'of overlap, uperpositionwitha transposed orm), xpansion rcontraction,xtension,eletion,tc.Butwe have noted hat t s incipientinthe2-1-2-1-2etextendedo theoctave, s appliednthe econda arte fthequartet: 2)-2-1-2-1-2-(2),s thesetrepeatstselfn a secondoctave.Some whole-toneccurrencesntheworkseeExx. 52, 53) aresuggestiveofthat ource.A setmay edefineds chromaticothe xtentf onsistentuccession1-1... and a setmaybe characterizeds totallyhromaticwhenno otherIC successionpertai;ns,r relativelyhromaticor nonchromatic)o theextent f thenumber fcontiguousC pairs1-1. Again, anynumber f'rules' ould be adduced s tothegenesis fchromatic etsfrom thers, rothers romhromaticets:e.g.,the fillingn'of the mportant-2-1withitsaxisproduces hesuccession -1-1-1.)In the2-1-2string,o whichconsiderable ttention as beengivenwithrespect o itsunderlyingunctionsn thequartet, ther etrachordal



71/94


72/94

357and arger uccessions anofcoursebenoted-e.g., thestringan be seen(Ex. 47) as one of major' etrachords,-2-1, rofthepentachordalet1-2-2-1overlapped ytwodegreesby nterval ). It seems ikelyhatbothofthesehavesubsidiaryunctionn thequartet.

The stringfthree -1-2 omponentsasthe mportantharacteristicofincorporating single hromatic elation:.e., therelation -Bb ifthestring eginson A, as illustratedn Ex. 47. Within hat cope ofthreecomponents)here ccurtwotritones--ofignificanceor heBart6k: -F,E-Bb.Such distinctionsf effect's 2-1-2, -1-2, -1-2 sopposed o2-2-1,2-2-1,2-2-1depend n compositionalpplicationswhere hestring,unc-tionings a model', roducesinear egments): hetherhe ppliedegmentisinitiated-1or 2-2 nmelodic xtracts; revalencef uchcharacteristicsas thetritonee.g., delimitinghe set 1-2-2-1) nd suchfactors ertainingtothestructuresfdiscreteegmentsf themodel ccurringncomposedlines.More simply,hesuccessionbasedon thecomponent -1-2has a'minor' ound, hatbasedon the2-2-1 'major' ound, ndthat xposingthe1-2-2-1 tritonal'ound,where uchdistinctionsre n fact ppreciable.There follow ertain xamples nd referencesowhich hese bservationsarerelevant.The 'simplest' ollections-simplestn the sense thateach ofthemcontainsbut a single C-are considered irst:1-1-1..., 2-2-2..., the'chromatic'nd whole-tone'ets.We havealreadyeen that tetrachord-1-1 sexposednthepedalvoices fthe ntroductoryhrase,mm. 1-6.Thetetrachords 'fully' hromaticno IC succession ther han1), and it iscomplemented,n thestrict ense,bythemoving ine ofthefirst iolin,which nfolds hefullyhromaticctachord #-B#-A -G -G#-B-E#-F#, nthat rder,o effect ithinhese peningixbars nexpositionf hetotal hromaticggregate.Followinghe ntroductoryhrase heresan interestingcounterpoint'between ndividual ines of relatively onchromaticharacter gainstchromaticontenthroughouthetextureassuming collection obe,thelargertscomponent,ontiguousntervals,natleastone importantensemore distant'from he chromatic tate within he diatonic-chromaticspectrum].I5here s total hromaticontent rom heupbeat f m. 7 totheD of m. 9, from hatpoint o B ofm. 14,or from hebeginningfm.21 to the G and G# of m. 22-suggestingan acceleratingrhythm'fchromaticaccrual'.



73/94

4*S (.., inm.b39)v1n. m. 9

S'\_ t- $ - 1 -...4%.9 .B, mm.1-41)doublingf C1: doublingfIC2:

i $ + $'-l . tiVIAn.+W, 1 (i) -1 1-1-...



74/94


75/94


76/94


77/94


78/94

363thecello and viola have semitonal escentfrom #' to g (viola) and e(cello)-preparing hePC basisof the sectionmmediatelyollowing;nd,against his, he violins scendchromaticallyrom he samepitch, # toC#3 (violin 2) and d#3 violin 1). (Example30 is also relevant o theconsiderationf 1-1... formulations.)

Explicit, nambiguous pplicationfsets sbynomeans ninvariablerule in the quartet; yet,a numberof apparent erivations re seen asrelativelynequivocalexpositions'ffunctionalntervallicollectionsuchas thefugato f thesecondaparte,or pentachordaletssimply tated ncertainportions f theprimaparte,and thelike. In the secondaparte,following . 80 (uptom. 149oftheviolin arts, rm. 151 ofthe elloandviola) thereoccurs a provocativedevelopment'f numerousntervallicvariants ftheunderlyingontour fthematiconfigurationtated ythecelloat thebeginningftheseconda arte. The fugato ubject s one ofthese.) n this hangeable, luctuantituationertain ets statednratherstraightforward,scalar'dispositions)re in evidence, ftenwithoutasilyidentifiableonsistency.ertainlyhis rea sworthyf ubstantialnvestiga-tionbeyondhe ummaryommentaryossiblehere;theres,for xample,evidence fapplicationsf mportantets lreadyreatede.g.,2-1-2-1-2nthe ello t mm.127-30, tc.).Onefactorhat an benotedsthe mergenceof 2-2-2... successions t m. 80ff., ften ompromisedr aborted. he-followingine Ex. 52), statedbythelowervoices, s suggestivefsuchwhole-tonemergence;heviolins t the ame ime, ot uoted, ignificantlyexpress hecollectionGb -Ab -Bb -C -D -E.


8/12/2019 Wallace Ba

Documents

Wallace Bartok