allen forte - context and continuity in an atonal work

Upload: peter-osim

Post on 15-Oct-2015

17 views

Category:

Documents


2 download

DESCRIPTION

Context and Continuity in an Atonal Work

TRANSCRIPT

  • 5/25/2018 Allen Forte - Context and Continuity in an Atonal Work

    http:///reader/full/allen-forte-context-and-continuity-in-an-atonal-work-5

    Context and Continuity in an Atonal Work: A Set-Theoretic ApproachAuthor(s): Allen ForteSource: Perspectives of New Music, Vol. 1, No. 2 (Spring, 1963), pp. 72-82Published by: Perspectives of New MusicStable URL: http://www.jstor.org/stable/832105

    Accessed: 12/01/2009 10:39

    Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available at

    http://www.jstor.org/page/info/about/policies/terms.jsp. JSTOR's Terms and Conditions of Use provides, in part, that unless

    you have obtained prior permission, you may not download an entire issue of a journal or multiple copies of articles, and you

    may use content in the JSTOR archive only for your personal, non-commercial use.

    Please contact the publisher regarding any further use of this work. Publisher contact information may be obtained at

    http://www.jstor.org/action/showPublisher?publisherCode=pnm.

    Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed

    page of such transmission.

    JSTOR is a not-for-profit organization founded in 1995 to build trusted digital archives for scholarship. We work with thescholarly community to preserve their work and the materials they rely upon, and to build a common research platform that

    promotes the discovery and use of these resources. For more information about JSTOR, please contact [email protected].

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

    http://www.jstor.org

    http://www.jstor.org/stable/832105?origin=JSTOR-pdfhttp://www.jstor.org/page/info/about/policies/terms.jsphttp://www.jstor.org/action/showPublisher?publisherCode=pnmhttp://www.jstor.org/action/showPublisher?publisherCode=pnmhttp://www.jstor.org/page/info/about/policies/terms.jsphttp://www.jstor.org/stable/832105?origin=JSTOR-pdf
  • 5/25/2018 Allen Forte - Context and Continuity in an Atonal Work

    http:///reader/full/allen-forte-context-and-continuity-in-an-atonal-work-5

    CONTEXT AND CONTINUITYIN AN ATONAL WORKA SET-THEORETIC APPROACH

    ALLEN FORTE

    WHEN Schoenberg abandoned triadic tonality in 1908 and beganto compose so-called atonal music he introduced into the historicalflux certainproblemsthat remainunsolved to this day. His own evi-dent concern as to whether composing was a logical or a sensoryprocess may have obstructed any personal efforts to find solutions.Whatever the case, we know that he did not explain adequatelythestructural bases of his atonalworks, but seemed to regard the dis-covery and developmentof the method of composing with twelvetones as a satisfactorydenouement.It now appearsthat althoughwemay have by-passedthe problemsof the atonalperiod quite success-fully we are still left with a large body of problematicmusic fromthat period-some of which is performed regularly, little of whichis understood.In aneffort to attackthe problemof atonal music morestrategicallythe following discussiondeparts from the familiar terms context andcontinuity. One need only reflect for a moment upon the degree towhich both context and continuitywere determinedby triadic tonal-ity-indeed, were essential conditions-in order to realize that theabandonment f that musicalsystemendangeredwhat many regardedas fundamentalpropertiesof musical composition.Certainobserversrespondedto the threatby denying the musicality f atonal works.More recently others have attemptedto show that the logic of suchatonal works still resides in triadic tonality, so that one is to under-stand atonalconfigurationsn terms of an implicit triadic norm. Bothresponseshave only furtherobscuredthe problems.Both have failedto recognizeSchoenberg's ruly revolutionary pirit, on the one hand,and his remarkable ntuitivegrasp of musical essentials on the other.But perhaps most misguided of all is the condescending observerwhose stock-in-trades that precious commodity style, for Schoen-berg has indicatedthe fruitlessnessof such an approach:* 72 -

  • 5/25/2018 Allen Forte - Context and Continuity in an Atonal Work

    http:///reader/full/allen-forte-context-and-continuity-in-an-atonal-work-5

    CONTEXT AND CONTINUITY IN AN ATONAL WORKIn my music there has neverbeen a concernfor style, but rathera constantconcernfor content [Inhalt] and its most precise repre-sentation[Wiedergabe].Therefore my youthfulworks prepareforthe understandingof my musical thought, and one would do wellto familiarize himself with those works [written]before my meansof expressionbecame so concise-as in my latest works. (From aletter to Leo Kestenberg, 1939)The above quotationwill also serve to introduce the compositionwhich is used here as an example of atonal music: Schoenberg'sSechs kleine Klavierstiicke,Op. 19. With the exception of the sixth

    piece this work is dated February 19, 1911, at which time thecomposer was thirty-six. The six famous miniatures form a singlework. They belong together, for they are all compositionalprojectionsof the same relationalsystem. The propertiesof that system are setforth below, together with several sample passages from Opus 19.As indicated n the title of the presentarticle,mathematicalset theoryunderlies the analytical approach.The set-theoreticformulation,inturn, reflectsthe general viewpoint that the analysis of a structuralsystem begins with the determinationof a set of elements and thecombinationalrelationswhich they exhibit.DESCRIPTIONOF THE SYSTEM

    We assume the usual partition of the available equal-temperedpitches,such that there is an equivalencerelationbasedon the intervalof the octave. The universalcollectionor set of our system, then, isthe chromaticscale, and its elements are placed in one-to-onecorre-spondencewith zero and the positive integers from 1 to 11 as shownin Ex. 1.x , o -t o o ? #.. ? I.0 1 2 3 4 5 6 7 8 9 10 1

    Ex. 1This universalset containsa set of five subsetsdesignatedA whosemembers are selected in the following way. First, a subset of twoelements x1 and x2 is definedas the interval-of-reference. his sub-set, designated X in Ex. 2, consists of the pitches G, B, which intraditionalparlance form the interval of a major third and in thenumericallanguage used herethe interval4. The meaningof the terminterval-of-referencebecomes clearer when the selection of all theelements of set A [the collection of subsets A1 . . . A5] has been ex-

    *73 -

  • 5/25/2018 Allen Forte - Context and Continuity in an Atonal Work

    http:///reader/full/allen-forte-context-and-continuity-in-an-atonal-work-5

    PERSPECTIVES OF NEW MUSICplained. Each subset of set A contains the pitch elements of a singleinterval relation on X. Thus A1 contains all the half-step relations(1-relations), A2 contains all the whole-step relations (2-relations),. . Each element forms a pair (not ordered) either with G or withB or in a few cases with both. The basis of association will be evi-dent in a moment. First it is important to state two limitations uponthis relational process. 1) As a result of the principle of octave equiva-lence the process does not continue beyond the 6-relation. 2) Theprocess observes what may be called the condition of proximity: Anypitch associated with xi must be either nearer to xl than to x2 orequidistant from both. The same condition holds for pitches associatedwith x2. As a result of this condition one otherwise possible subset isexcluded, the subset [2, 4] which is based upon the 5-relation. In thisway both xl and x2 are deprived of possible dominant associates.

    A =A1 . A5]

    XJ,X1 ^2X = Interval-of-Reference

    j$J-

    3fiL uLJ

    A1 = set of all 1-relatlons on X

    A2 = set of all 2-relations on X

    A3.= set of all 3-relations on X

    A4 = set of all 4-relations on X

    A5 = set of all 6-relations on X

    [7,11]

    0o, ,8,10]

    15,s,9]

    2, 4]

    [3]

    [1,5]Ex. 2

    * 74 ?

    A

  • 5/25/2018 Allen Forte - Context and Continuity in an Atonal Work

    http:///reader/full/allen-forte-context-and-continuity-in-an-atonal-work-5

    CONTEXT AND CONTINUITY IN AN ATONAL WORKEx. 2 displays these subsets in musical and in correspondingnumerical notation.The musical notationwill facilitate study of theexcerpts from the score, while the numerical notationenables us tosee certainpropertiesmorereadily.These structurally mportantprop-erties are most efficientlydiscussed if we first extractanotherset B ofsubsets assumedto havespecialinterest.This set is displayed n Ex. 3.

    B =B1 . . . Bs

    B1 = set of all elements associated [4, 6,8]~-' ^' *' 1=?:~~?;:,?~?~???-with G only

    i Lf f ~2 = set of all elements associated [0, 2,10]with B onlyJ J1 J J -==B3 = set of all elements associated [0,2,4,6,8,10]g - J~.,^-V ~either with G or B

    ^O- ; i J JB4 = set of all elements associated [1, 3, 5,9]with both G and B* ' 'bltj B5 = set with only one number [3]

    J| |-| J>tW xj; A X = complement of X [0,1,2,3,4,5,6.8,9,10]1 J} ~J'-tf J (all pitch-classes not contained in X)Ex. 3

    Set B may be of interestwhen we come to examinethe composition,since it sorts out in various ways the relations on x1 and x2, wherexl and x2 are regardedas discreteelements,whereas set A shows theclasses of interval relationson X without emphasizingthe individualroles of its members. It should be remarkedhere that in the composi-tion the interactionof elements associatedexclusively with pitch Gor with pitch B, contrastedwith the interactionof elementsassociatedwith both, is a significantaspect of continuityand of the controlledtransformationof context. Both set A and set B are essential to anunderstandingof the relational system, and the explanation of im-portant propertiesthat follows will consider them individually andin combination.* 75 -

  • 5/25/2018 Allen Forte - Context and Continuity in an Atonal Work

    http:///reader/full/allen-forte-context-and-continuity-in-an-atonal-work-5

    PERSPECTIVES OF NEW MUSICEquivalenceSet A containsno two equivalent(identical) subsets. Set B likewisecontains no two equivalentsubsets. This suggests that each subsethas in some way a special structure with respect to X. From theanalytic standpoint his is advantageous, or we can deal with specificand differentiated relations rather than with an undifferentiatedcollection of pitch elements.Equivalencerelations on A and B are as follows:1) A =B2) A4 = B5 (That is to say, the set with only onemember[Eb]

    is the same as the set of all 4-relations on X. Asmight be expected, this unique property is ex-ploitedin the composition.)3) A=X4) B=XInclusionTaking sets A and B separatelywe see that1) A5 c A2 (i.e. A5 is included in A2)2) B2 c B33) B1 CBsInclusioninvolvingboth A and B yields4) Al cB35) A2 C B46) A3 c B7) A4 C B48) A5 cB4.Intersection

    The operation ntersectionperhaps s of morecompositional nterestsince it suggests possibilitiesfor connectingone subset to anotheror,conversely,detachingcertainelementswithoutendangeringcontinuity.Intersection follows naturally from the list of subsets immediatelyabove, for in all cases the set resulting from intersection s the sameas the smallersubset of each of the pairs listed. That is, A2 n A5 =[1, 5], that is [Db, F], andA2 n B4 = [1, 5, 9], that is [C#, F, A],...One might raise a question at this point regarding intersection ofmore than two subsets of A and B, since presumablythis occurrencewouldtendto makecertainpitchelementsmoreredundant hanothers.It happensthat every element occurs in three differentsubsets, withthe exceptionof the pitch-elementA, which occursonly in A2 and B4.76-

  • 5/25/2018 Allen Forte - Context and Continuity in an Atonal Work

    http:///reader/full/allen-forte-context-and-continuity-in-an-atonal-work-5

    CONTEXT AND CONTINUITY IN AN ATONAL WORKEvery elementoccursin one subset of A andtwo subsetsof B, with theexceptionof the pitch-elementF which occursin A2, A5, andB4. Bothpitches F and A are prominentin Opus 19.ComplementationThe complementof X has alreadybeendisplayed(Ex. 3) and it hasbeen remarked that X = A. Another kind of partition, one that ismore important in the work, involves two whole-tone hexachordswhich arise within the system. This is explainedin the next section.Union

    Unions are easily read from the numericalrepresentation n Exam-ples 2 and 3 since a subset containsnumbersdrawn either from theset 0 and the even numbers, or it contains only odd numbers. Thus1) A UA3= B3 (The union of the set of all 1-relationsandthe set of all 2-relationsis the same as theset of all elements associated with eitherG or B.)2) A2UA4= B4 (The union of the set of all 2-relationsandthe set of all 4-relations is the same as theset of all elements associated with bothG andB.)3) B1UB2= B3, and4) A1UA3 B1UB2

    5) B4U X B3The last subset listed, B3, is so important n the compositionthat itwill be designatedY. In terms of pitch elements Y is the whole-tonehexachordwhichtogetherwith B3formsapartitionof theuniversalset.It shouldnow be apparent hat the relationalsystem offersa multi-tude of compositionalpossibilities, suggesting ways of combiningelementsinto largerunits as well as ways of extractingsmallerunitsfrom larger. In addition, it provideseffectiveanalogues to harmonicprogressionandvoice-leading n the tonal system. For demonstrationsof some of these attributes let us turn now to the compositionalprojections n Opus 19.

    COMPOSITIONAL PROJECTIONSAs we approach he composition n terms of the relationalsystemitis importantto regard the system as a formalizedstatementor sum-maryof the structuralbasis of the composition.It is not an analysisofthe work. In orderto emphasizethis distinctionand at the same time77-

  • 5/25/2018 Allen Forte - Context and Continuity in an Atonal Work

    http:///reader/full/allen-forte-context-and-continuity-in-an-atonal-work-5

    PERSPECTIVES OF NEW MUSICrelate the musical events to the abstractsets of the system the termcompositional set is here introduced.It is also importantto realize that there are no independent le-ments in this composition,just as there are none in a tonal work.There are, however,elementswhich areless determinativehanothers,as may be inferred from the foregoing explanation of the system.Such elements standforth in a varietyof contextsand thus makecleara certain structuralhierarchy, which, in turn, makes it possible todescribethe structuralmeaning of a compositional et in termsof theelements which dominate it. For instance, at the beginning of thefifth piece (Ex. 4) we discoverthat the first phrase is controlledbyF, A, and D#, while the lower (accompanimental)part of the phraseis controlledby Db and G. These strategicallyplacedpitch elementsare the union of X and B4, the special subset Y.

    Etwasrasch (b)

    -zart, abetvoll p

    Ex. 4To bring these preliminaryremarksto an end, a question:What,then, determines he structuralmeaningof a given note in this compo-sition? Response: Its context as interpretedby the underlying rela-tional system shown in Exx. 2 and 3.The first excerptto be discussed is the opening phrase of the firstpiece.

    Leicht, zart ( ) 1

    2 3 4Ex. 5

    For the sake of convenient reference the compositional sets areenclosed and numberedto correspondto the remarkswhich follow.1. The first threepitchesof this melodic et belong to the specialsubsetY. The last element,F#, is associatedonly with G (as in B1),* 78 ?

  • 5/25/2018 Allen Forte - Context and Continuity in an Atonal Work

    http:///reader/full/allen-forte-context-and-continuity-in-an-atonal-work-5

    CONTEXT AND CONTINUITY IN AN ATONAL WORKso that over the span of the phrasethe directionof the melodyis fromB toward G. The concludingmelodic successionF-F# is one of threesimilarpairsprominent n the work. These semitonesuccessionsmaybe interpretedas unions of two elements, one drawn from each ofthe whole-tone hexachordsB3 and Y.2. This set provides a context for B of the melody, which wecan take to be the controllingelement here. It departsfromA, a pitchelement from subset Y. Of the remaining elements of the set thepitch C associatesexclusivelywith the pitch B (as in B2), G belongsto X, andG# associatesexclusivelywith G (as in B1). Taken togetherwith B in the treble staff this compositional et exhibitsa characteristicfeature which pervades the work: equal distribution or balance ofpitch elements with respect to the interval-of-reference.n this caseeach element of the interval-of-referenceas its own exclusive associ-ate, while the pitch A associateswith both. Balancedsets of this kind,which occurat crucialstructuralpoints, are analogousto fundamentalharmonies n the triadicsystem.3. In terms of the relationalsystem the fundamental pitch B iscombinedwith two pitches from set B4 (the set of all elementsassoci-atedwith both G andB). The remainingelement E is one of the threepitches associatedonly with G; in this context it servesto preparetheprogressionto the finalcompositional et, which centers on G. In thisconnection it is importantto recognize that, as a continuity factor,the repetitionof the melodic successionB-D# (in set 1) by the har-monic pair B-D# is secondary n significanceto the transformationofcontext which takes place by virtue of the presence of the two addi-tionalelementsE and F.

    4. The finalsimultaneityof the phraseconsistsof the fundamentalpitch G together with one exclusive pitch associate and one pitchexclusively associated with B, the other fundamentalpitch. Unlikethe balancedsonorityat the beginning of the phrase this sonorityisweighted toward G.In the second piece the interval-of-referenceerves as a kind ofostinatoagainst which various compositionalsets are projected.The

    closing part of the piece is shown in Ex. 6.1. Here again the elements of this initial compositionalset aredistributed equally with respect to the interval-of-reference.Onlyregisterandrhythmicaccentdifferentiate.The emphasizedpair C-Eb,in particular, s importanthere (and throughoutthe first part of thepiece). The pitch C associatesexclusively with B, while Eb is theonly set with only one member and associateseither with G or with79

  • 5/25/2018 Allen Forte - Context and Continuity in an Atonal Work

    http:///reader/full/allen-forte-context-and-continuity-in-an-atonal-work-5

    PERSPECTIVES OF NEW MUSIC

    Ex. 6B depending upon context. Here it associates with both since botharepresent.2. At this point the progressionculminates in a harmonywhich isweighted in favor of the pitch B. Observe that although the pairC-Eb descends to the pair D-B in the upper part, it is retained(notated enharmonically) in the lower part of the harmony.3. The restatementof the interval-of-reference t the beginning ofthis phrase representsthe final resolution f the pair C-Eb. Thisentireset is a complete projectionof Y.4. The final set consists of an almost complete statement of B3,the complementof Y. Only Gj is lacking. The complementation sexpressedinsteadby the juxtapositionof two trichords,a distributionconsistentwith the texture of thirds that characterizes his piece.In the thirdpiecewe encountera different extureaswell asdifferenttechniquesof projection.Ex. 7 summarizesthe controllingcomposi-tional sets.

    i I-opp v -a b d

    Ex. 7Both in Ex. 7 and in Ex. 8 the different elements of the controllingsets aredistinguishedby notationalmeans.The interval-of-referencesnotated in half notes and the pitches associated with it are shown as8th notes. The other controlling set F-A is given in quarter notes,while its pitch associates are shown as 16ths. The complementof the

    80 -

  • 5/25/2018 Allen Forte - Context and Continuity in an Atonal Work

    http:///reader/full/allen-forte-context-and-continuity-in-an-atonal-work-5

    CONTEXT AND CONTINUITY IN AN ATONAL WORKunion of sets a and b is designatedd. Let us now see how these setsinteract in the composition.Example 8 is a sketch of the first fourbars. Secondaryharmonicnotes (from subset d) have been omittedin order to show the controllingsets more clearly.

    b:r ^tj J4 -^-J-,(-d i : - bg^ tB ^Ex. 8

    Observethat the same elements of the subsets areconsistentlycounter-pointed (vertical arrows), guaranteeing continuity over the longerspan. Observe also that the interval-of-reference,which controlsthefirst part of the passage, gradually yields to the interval F-A, in amannerreminiscentof tonal modulation.To extend the analogy, thismight be called a naturalmodulationsince F-A belongs to the set B4,the set of allelementsassociatedwith both G andB. The entirepassageterminatesin a whole-tone context consisting of the trichord F-A-Dbplayed by the right hand and a disjuncttetrachord(partial statementof the complement) in the left hand. (The separatenessof the twosubsets is emphasized n this instanceby an idiosyncraticperformanceinstruction: Throughout he first four measuresthe right hand is toplay f, the left hand pp. ) This whole-tonecontextpreparesthe firstpitchesof the followingphrase (not shown), the interval-of-reference.

    i 'jp - -4. -ipp23 _^==> ' ^- '7 7_ p r 1U-1* _ ___, ^

    Ex. 9The closing passage of the third piece, shown in Ex. 9, beginswith another balanced harmony, exploits the unique pitch set B5(Eb) and ends with melodic emphasis placed upon the pairs Eb-Gand F-A. The final harmony is weighted toward G, yet two of thefour elements,D and Bb, associateonly with B.As a final illustrationEx. 10 shows the closing set of the last piece.Not only are the elements of the completecompositionalset perfectlybalanced with respect to the interval-of-referencebut each of the

    81

  • 5/25/2018 Allen Forte - Context and Continuity in an Atonal Work

    http:///reader/full/allen-forte-context-and-continuity-in-an-atonal-work-5

    PERSPECTIVES OF NEW MUSIC

    Ex. 10subsets is also distributedin that way. Thus the first subset, whichcontains the pitch B, containsas counterbalancean associate of G andan element of B4. The elements of the second subset are similarlybalancedoff against G. The third subset contains one element of B2and one element of B1. Had this piece been composed some fifteenyears later this final gesture would have been followed by the whole-tone dyads E-D and Eb-Db, the complement of the final set, sostronglysuggested by the concludingpair.

    It is hopedthat the foregoing has indicatedthat with the assistanceof new techniquesatonalmusic can be studiedeffectively.To exploreother aspects of the particular system unfolded here or to drawconclusionsregardingthe historicaldevelopmentof the artbaseduponthe materialpresentedwould exceed both the physicalbounds as wellas the intent of the article.

    * 82 ?