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A Primer for Atonal Set Theory Author(s): Joseph N. Straus Reviewed work(s): Source: College Music Symposium, Vol. 31 (1991), pp. 1-26 Published by: College Music Society Stable URL: http://www.jstor.org/stable/40374122 . Accessed: 22/01/2013 07:30 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]. . College Music Society is collaborating with JSTOR to digitize, preserve and extend access to College Music Symposium. http://www.jstor.org This content downloaded on Tue, 22 Jan 2013 07:30:12 AM All use subject to JSTOR Terms and Conditions Pedagogyof Music Theory A Primer for AtonalSetTheory1 JosephN. Straus SET THEORY HAS A BAD REPUTATION. LlKE SCHENKERIAN ANALYSIS IN its earlier days, set theoryhas had an air of the secret society about it, withadmissiongrantedonly to those who possess the magic password,a forbidding technical vocabulary bristlingwith expressions like "6-Z44" and "intervalvector." It has thus oftenappeared to the uninitiatedas the sterile application of arcane, mathematicalconcepts to inaudible and uninteresting musical relationships.This situationhas created understandablefrustration has grownas discussions of twentiethamong musicians,and the frustration literaturehave come to be in theoretical the music professional century in this unfamiliar almost language. entirely expressed Where did this theorycome fromand how has it managed to become so dominant?Set theoryemergedin response to the motivicand contextualnatureof post-tonalmusic. Tonal music uses only a small numberof referential sonorities(triads and seventhchords); post-tonalmusic presentsan extraorTonal music sharesa commonpractice dinaryvarietyof musicalconfigurations. of harmonyand voice leading; post-tonalmusic is more highlyself-referential- each work defines anew its basic shapes and modes of progression.In tonal music, motivicrelationshipsare constrainedby the normsof tonal syntax; in post-tonalmusic, motivesbecome independentand functionas primary In this situation,a new music theorywas needed, free structural determinants. of traditionaltonal relationshipsand flexibleenoughto describe a wide range of new musical constructions.The structuralnovelties of post-tonalmusic caused theoriststo rethinkeven the most basic elementsof music, pitch and interval.On this new foundation,a new theoryarose, flexible and powerful enoughto describe a new musical world. This new theoryhas its roots in work by Milton Babbitt,Allen Forte, David Lewin, RobertMorris,GeorgePerle, and JohnRahn,but it has flowered 1 This article is adapted from my Introductionto Post-Tonal Theory, an undergraduatetextbook recentlypublished by Prentice-Hall. This content downloaded on Tue, 22 Jan 2013 07:30:12 AM All use subject to JSTOR Terms and Conditions 2 COLLEGE MUSIC SYMPOSIUM in many differentdirections.2 Set theoryis not a single language, but a communityof local dialects and subcultures.It is best understoodnot as a rigidlyprescribedpractice,but as an arrayof flexible tools for discovering and interpreting musical relationships.It should be emphasized that these can and shouldbe enjoyablyaudible.If set theoryhas occasionally relationships veered into the inaudiblyarcane, that should be blamed on the excesses of individualpractitioners, not on the theoryitself.At its best, the theorymakes possible rich and interestinghearings of the finest musical works of the twentiethcentury. Despite its occasionally forbiddingappearance, atonal set theoryis not particularlycomplicated,at least in its basic applications.No high-powered computersor advanced degrees in mathematicsare needed- just a commitmentto twentieth-century music,the abilityto add and subtractsmall integers, and some good will. What follows is a primerof basic concepts of atonal set theory. OctaveEquivalence There is somethingspecial about the octave. Pitches separated by one or more octaves are usually perceived as in some sense equivalent. Our musical notation reflects that equivalence by giving the same name to octave-related pitches.Since equivalencerelationshipsunderpinmuchof atonal set theory,it should be emphasized at the outset that equivalence does not mean identity.Example 1 shows two melodic lines fromSchoenberg'sString QuartetNo. 4, one fromthe beginningof the firstmovementand one a few measuresfromthe end.3 2A complete, annotated bibliographyof post-tonal theory,compiled by Martha Hyde and Andrew Mead, may be found in Music TheorySpectrum11/1 (1989): 44-48. The field as a whole originates with Milton Babbitt's influentialarticles and teaching, particularly"Some Aspects of TwelveTone Composition," The Score and I. M. A. Magazine 12 (1955): 53-61; "Twelve-Tone Invariantsas Musical Quarterly 46 (1960): 246-59; "Set Structureas a ComposiCompositional Determinants,** tional Determinant,'*Journal of Music Theory 5/2 (1961): 72-94. Although these articles focus on twelve-tone music, their theoretical categories have broad application in post-tonal music. Some of Babbitt's central concerns are presentedmore informallyin Milton Babbitt: Words About Music, eds. Stephen Dembski and Joseph N. Straus (Madison: Universityof Wisconsin Press, 1987). The basic concepts presented in this article are drawn from Babbitt*s work, and also from three widely used books: Allen Forte, The Structureof Atonal Music (New Haven: Yale UniversityPress, 1973); John Rahn, Basic Atonal Theory (New York: Longman, 1980); and George Perle, Serial Composition and Atonality,5th ed. (Berkeley: Universityof California Press, 1981). Two importantrecent books offer profoundnew perspectives on this basic material,and much else besides: David Lewin, Generalized Musical Intervals and Transformations(New Haven: Yale UniversityPress, 1987); Robert Morris, Composition withPitch Classes (New Haven: Yale UniversityPress, 1987). 3 Although atonal set theory is most closely associated with the "free atonal** music of Schoenberg, Webern, and Berg, it has much more general applications throughoutthe range of posttonal music, including particularlytwelve-toneserial music. This content downloaded on Tue, 22 Jan 2013 07:30:12 AM All use subject to JSTOR Terms and Conditions A PRIMER FOR ATONAL SET THEORY 3 Example 1. Two equivalentmelodies (Schoenberg,StringQuartetNo. 4, first movement). i liLj I 'i I viol™ \\ ! » _ ii A ^ I iJiJIlJJJ-i^ i J. JiuJ*io // c* 1^1 IJLJ 1 M 1 ^ i i i 1 i/' I 1 Althoughthe two lines differin many ways, particularlyin theirrange and versionsof one single unthey are still understoodas two different rhythm, are In other idea. words, equivalent. they derlying In Example 2, the openingof Schoenberg'sPiano Piece, Op. 11, No. 1, comparethe firstthreenotes of the melodywiththe sustainednotes in measures 4-5. Example 2. Two equivalentmusical ideas (Schoenberg,Piano Piece, Op. 11, No. 1). Moderate / ^~ -^-^^ I Til 1'i| I'^^^J There are manydifferencesbetweenthe two collectionsof notes (register,articulation,rhythm, etc.), but theysound equivalentbecause theyboth contain only a B, a Gl, and a G. As with otherequivalences,the purpose of invoking octave equivalenceis not to smoothout the varietyof the musical surface, but to reveal the relationshipsthatlie beneaththe surface,lendingunityand coherenceto musical works. This content downloaded on Tue, 22 Jan 2013 07:30:12 AM All use subject to JSTOR Terms and Conditions 4 COLLEGE MUSIC SYMPOSIUM Pitch Class It is usefulto distinguish betweena pitch (a tone witha certainfrequency) and a pitch class (a group of pitches one or more octaves apart fromone another).When we say thatthe lowest note on the cello is C, we are referring to a specific pitch. We can notatethat pitch on the second ledger line beneaththe bass staff.When we say thatthe tonic of Beethoven's FifthSymnot to some particularpitch C, but to pitch-class phonyis C, we are referring C. Pitch-classC is an abstractionand cannotbe accuratelynotatedon musical staves. Sometimes,for convenience,a pitch class will be representedin musical notation,but, in reality,a pitch class is not a single thing; it is a class of things.A pitch and all the otherpitches one or more octaves away fromit are membersof the same pitchclass. EnharmonicEquivalence In common-practice tonal music, Bl>is not equivalentto A*. Even on an like the piano, the tonal systemgives Bt and At equal-temperedinstrument In forexample, At is t2 whereasBt is lo, and different G-major, functions. have differentmusical roles. This distinctionis 2 3 and very scale-degrees in where notes that are enharmonically abandoned music, largely post-tonal are also equivalent functionallyequivalent. Composers may occasionally notate pitches in what seems like a functionalway (sharps for ascending and flats for descending,for example). For the most part,however,the notation is functionallymeaningless,determinedprimarilyby simple convenienceand legibility. IntegerNotation Tonal music uses seven scale degrees or step classes. In C-maJor,for example,At, A*i,and At, in all octaves, are membersof scale-degreeS. Posttonal music uses twelve pitch classes. All Bts, Os, and Dl4>sare membersof a single pitch class, as are all the Cts and Dts, all the CXs, Ds, and Etts, and so on. Our theoreticaland analyticalpurposes are best served by cutting throughthe notationaldiversityand assigningintegersfrom0 through11 to the twelve pitch classes. Set theorycustomarilyuses a "fixed do" notation: The pitch class containingC, Bt, and DW»is arbitrarilyassigned the integer 0; Ct and Dt are membersof pitch-class 1; D is a memberof pitch-class2; and so on. Integersare simple to grasp and to manipulate.They are traditionalin music (as in figuredbass numbers,for example) and useful for representing certainmusical relationships.As long as we avoid committingthe "numeromathematicaloperationswithoutregardto their logical fallacy"- performing This content downloaded on Tue, 22 Jan 2013 07:30:12 AM All use subject to JSTOR Terms and Conditions A PRIMER FOR ATONAL SET THEORY 5 - we will findthema greathelp.4 Set theoryuses nummusical implications and rewardingto hear. bers to model musical relationshipsthatare interesting "mathematical"any The theoryand the music it describes are not, therefore, more than our lives are "mathematical"just because we count our ages in integers. Mod 12 Every pitchbelongs to one of the twelvepitchclasses. If you begin with a memberof some pitchclass, thengo up or down one or more octaves, you arriveat anothermemberof the same pitch class. In arithmeticalterms,add12 (or a multipleof 12) does not changepitch-classidentity. ing or subtracting reliesupon arithmetic modulo Thus, 12 is a modulus,and set theoryfrequently 12, for which mod 12 is an abbreviation.In a mod 12 system,-12 = 0 = 12 = 24. Similarly,-13, -1, 23, and 35 are all equivalentto 11 (and to each other) because they are related to 11 (and to each other) by adding or subtracting12 (or multiplesof 12). It's just like tellingtime- twelve hours after fouro'clock it's fouro'clock again. Set theorysometimesuses negativenumbers (for example, to suggest the idea of descending) and sometimesuses numbers largerthan 11 (forexample,to representthe distancebetweentwo widely separatedpitches)but,in general,such numbersare discussedin termsof their mod 12 equivalents,from0 to 11 inclusive. PitchIntervals names to inA theoryof post-tonalmusic has no need to give different and fourths diminished as such absolute tervalswiththe same size, majorthirds. A for crucial. are third, example, In tonal music, such distinctions functionally is an intervalthat spans threesteps of a diatonic scale, while a fourthspans four steps. A major thirdis consonantwhile a diminishedfourthis dissonant. In music that doesn't use diatonic scales and doesn't systematically betweenconsonanceand dissonance,it seems cumbersomeand even distinguish misleadingto use traditionalintervalnames. It is easier and more accurate musicallyjust to name intervalsaccordingto the numberof semitonesthey contain.The intervalsbetween C and E and between C and Ft both contain foursemitonesand are both instancesof interval4, as are Bt-Ft,C-DX, and so on. A pitch intervalis simply the distance between two pitches,measured numberof semitonesbetweenthem.If we are concernedabout the dithe by rectionof the interval,whetherit is ascendingor descending,we can precede the numberof semitoneswith either a plus sign (to indicate an ascending interval)or a minus sign (to indicatea descendinginterval).Intervalswith a 4See Rahn,Basic AtonalTheory,19, fora relateddiscussionof the numerological fallacy. This content downloaded on Tue, 22 Jan 2013 07:30:12 AM All use subject to JSTOR Terms and Conditions 6 COLLEGE MUSIC SYMPOSIUM plus or minus sign are called directed or ordered intervals.If we are concernedonlywiththe absolutespace betweentwo pitches(an unorderedinterval), we need only the numberitself. Whetheran intervalis consideredorderedor unordereddepends on our particularanalyticalinterestsat the time.Considerthe melodyby Weberngiven in Example 3. Example 3. Motivic developmentvia orderedpitch intervals(Webern,"Wie bin ich froh!"fromThree Songs, Op. 25). I^J,l'p'hiHinjlyj'flnr H'r^l^] Wie bin ich froh! noch ein-mal wird mir al - les griin und leuch- tet so! The intervallicideas of its openingnotes echo throughoutthe line. The orderedpitch intervalsformedby its firstthreepitches occur again in two otherplaces, in measure2 (D-B-Bt) and measure3 (C-A-Gt).The second fragment spans a rest and helps to link two lines of the poem. The thirdone leaps up to a high G>,the highpointof the melody and the last of the twelve pitch classes to be heard. These intervallicechoes, based on orderedpitch intervalsare easy to hear. The threemelodynotes at the beginningof the second measure,Cl-F-D, also relate to the opening three-notefigure,but in a more subtle way: they have the same unorderedpitchintervals(see Example 4). Example 4. Motivic developmentvia unorderedpitch intervals. * LsJLnJ L8JL3J * * The relationshipis not as obvious as the one based on orderedpitch intervals, but it is still not hard to hear. The opening figure can thus be understoodsimultaneouslyin termsof its orderedand unorderedpitch intervals. The orderedpitch intervalsfocus attentionon the contourof the line, its balance of risingand fallingmotion.The unorderedpitch intervalsignore contourand concentrateentirelyon the spaces betweenthe pitches. Pitch-ClassIntervals A pitch-classintervalis the distancebetweentwo pitchclasses. Like pitch intervals,pitch-classintervalscan be thoughtof eitheras orderedor unordered. To calculate an orderedpitch-classinterval,envisionthe pitch classes This content downloaded on Tue, 22 Jan 2013 07:30:12 AM All use subject to JSTOR Terms and Conditions A PRIMER FOR ATONAL SET THEORY 7 arrangedaround a circularclockface and count clockwise (or envision a piano keyboardand countupward) fromthe firstpitchclass to the second. The orderedpitch-classintervalfromQ to A, for example, is 8. Notice thatthe orderedpitch-classintervalbetween A and Q (4) is differentfromthatbeordered tweenCf and A (8). Excluding the unison,thereare eleven different pitch-classintervals. In the Webern melody discussed above, the firstfourpitch classes are the same, in order,as the last four:G-E-Dt-Ft(see Example 5). Example 5. Differentcontour,but the same orderedpitch-classintervals. orderedpitch intervals: orderedpitch-class intervals: +9 -3+11+3 Wie bin ich froh! 9 11 3 -13 3 +3 und leuch- tet 9 11 so! 3 The contoursof the two fragments(theirorderedpitch intervals)are different,but the orderedpitch-classintervalsare the same: 9-11-3. This similarity the rhyme is a nice way of roundingoffthe melodicphraseand of reinforcing in the text: "Wie bin ich froh!. . . und leuchtetso!" For unorderedpitch-classintervals,directionno longer matters.All we care about now is the space betweentwo pitchclasses. Unorderedpitch-class intervalsare calculatedfromone pitchclass to the otherby the shortestavailable route,eitherup or down. The unorderedpitch-classintervalbetweenCf and A is 4, because it is only 4 semitonesfromany Cf to the nearestavailable A. Notice that the unorderedpitch-classintervalbetween Ct and A is the same as the unorderedpitch-classintervalbetweenA and Ct- 4 in both unorderedpitch-class cases. Excludingthe unison,thereare only six different intervalsbecause, to get fromone pitch class to any otherpitch class, one neverhas to travelfartherthan six semitones. Let's returnone more time to our Webern melody. We noted that the last fourpitch classes were the same as the firstfour,but given a different contour.This change of contourmakes somethinginteresting happen: it puts the E up in a high registerand groups the G, Df, and Ft togetherin a low register.That registrallydefinedthree-notecollection (G-Dt-Ff)containsunorderedpitch-classintervals1 (G-Ft),3 (Dt-Ft),and 4 (G-Di). These are exactly the same as those formedby the firstthree notes of the figure (G-E-Dt): E-Dt is 1, G-E is 3, and G-Dt is 4. The entire melody develops musical ideas fromits opening figure,sometimesby imitatingits orderedpitch intervals,sometimesby imitatingits unorderedpitch intervals,and sometimes, more subtly,by imitatingits orderedor unorderedpitch-classintervals(see Example 6). This content downloaded on Tue, 22 Jan 2013 07:30:12 AM All use subject to JSTOR Terms and Conditions 8 COLLEGE MUSIC SYMPOSIUM Example 6. Developmentof the initialmelodic figure. intervals orderedpitch-class nnr>rHf»iwrftttr4vJnt- orderedpitcfrmteixals -"" . Ultuuirn1/! U^sitrfrrvals [iJUill . " ****-* ■ """"""' ^S^ ^^^"^^ An unorderedpitch-classintervalis also called an intervalclass. Justas each pitch-classcontains many individualpitches, so each intervalclass contains many individualpitch intervals.A given pitch interval,its compounds,and its inversionswith respect to the octave are membersof the same interval class. Thus, for example, pitch intervals23 (compound major seventh), 13 (minorninth),11 (major seventh),and 1 (semitone) are all membersof interval-class1. We thushave fourdifferent ways of talkingabout intervals:orderedpitch unordered interval, pitch interval,orderedpitch-classinterval,and unordered interval. If, for example, we want to describe the intervalfrom pitch-class the lowest note in the Webern melody, B3, to the highest,GI5, we can do so in fourdifferent ways. If we call it a +21, we have describedit veryspecifically,conveyingboth the size of the intervaland its direction.If we call it a 21, we express only its size. If we call it a 9, we have reduced a compound intervalto its simple equivalent.If we call it a 3, we have expressed the intervalin its simplest,most abstractform.None of these labels is better or more correctthanthe others.It's just thatsome are more concreteand specificwhile othersare more generaland abstract. It's like describingany object in the world- what you see depends upon where you stand. If you stand a few inches away froma painting,for example, you may be aware of the subtlestdetails, rightdown to the individual If you standback a bit, you will be betterable to see the larger brushstrokes. and overall design. There is no single "right"place to stand; to the shapes the appreciate painting,you should be willing to move fromplace to place. One of the nice thingsabout music is thatyou can hear a single object like an intervalin manydifferent ways at once, as thoughit were possible to stand in several different places at the same time. Interval-ClassContent The sound of a sonorityis determinedto a significantdegree by its interval-classcontent.This can be summarizedin Scoreboardfashionby indicating,in the appropriatecolumn,the numberof occurrencesof each of the six intervalclasses. Example 7 refersagain to the three-notesonorityfrom Schoenberg'sPiano Piece, Op. 11, No. 1. This content downloaded on Tue, 22 Jan 2013 07:30:12 AM All use subject to JSTOR Terms and Conditions A PRIMER FOR ATONAL SET THEORY 9 Example 7. Interval-class content of a three-notesonority (Schoenberg, Piano Piece, Op. 11, No. 1). Moderate "T^^ "7^-^-^^ I"1"TTTi 1'n \^^§-j A &\ - -^ rJmmmmm^4r * 1 1U ff" "^h» ^ class Interval I 2 3 of number 1 0 110 occurrences 11111 4 5 6 0 - in this Like any three-notesonority,it necessarilycontains three intervals case one occurrenceeach of interval-classes1, 3, and 4 (and no 2, 5, or 6). this is fromthe sonoritiespreferredby Stravinskyin the pasHow different from his opera The Rake's Progress,labeled 1 and 2 in Example 8! These sage sonoritiescontainonly 2s and 5s. Example 8. Interval-class content of a contrasting three-note sonority (Stravinsky,The Rake's Progress, Act I, Scene 1). j 7Q 35:= ^^sS^f^^E^^^ls^^^^ 5 class Interval numberof occurrences 0 | 0 2 0 10 6 5 4 3 12 | \ This content downloaded on Tue, 22 Jan 2013 07:30:12 AM All use subject to JSTOR Terms and Conditions COLLEGE MUSIC SYMPOSIUM 10 The interval-classcontentis usually presentedas a stringof six numbers with no spaces intervening.This is called an intervalvector.The first numberin an intervalvectorgives the numberof occurrencesof interval-class 1; the second numbergives the numberof occurrencesof interval-class2; and so on. The intervalvectorforthe sonorityin Example7, G-Gt-B,is 101100 and the intervalvector for the sonoritiesin Example 8 (A-B-E and D-E-A) is 010020. Their differencein sound is clearly suggestedby the difference in theirintervalvectors. Intervalvectors can be constructedfor sonoritiesof any size or shape. Intervalvectorsare notreallynecessaryfortalkingabouttraditional tonalmusic. - four kinds of triads and five kinds of There, only a few basic sonorities seventhchords- are regularlyin use. Post-tonalmusic, however,confrontsus witha huge varietyof musical sonorities.The intervalvectoris a convenient way of summarizingtheirbasic sound. Even thoughthe intervalvectoris not as necessarya tool for tonal music as forpost-tonalmusic,it can offeran interesting perspectiveon traditional formations.Example 9 calculates the intervalvectorforthe major scale. Example 9. IntervalVector forthe major scale. class: 12 Interval | tr j^g^J; ^n. - ^ j iL. ■ i&.. 4 5 i 2 2 l 2 I l • ' ■ • *•' totalnumber 4 6 11110 ii^ ifi7Tr^m ^ 3 l ' __! <; 9 4 l^l^l4!^!0'1 ofoccurrences: ' ' ^ This content downloaded on Tue, 22 Jan 2013 07:30:12 AM All use subject to JSTOR Terms and Conditions fi 1 A PRIMER FOR ATONAL SET THEORY 11 Notice the methodicalprocess of extractingeach intervalclass. First,the intervalsformedwiththe firstnote are extracted,thenthose formedwiththe second note, and so on. As with any seven-notecollection,thereare 21 intervalsin all. Certain intervallicpropertiesof the major scale are immediatelyapparent fromits intervalvector. It has only one tritone(fewer than any other interval)and 6 occurrences(more than any otherinterval)of interval-class5 whichcontainsthe perfectfourthand fifth.This probablyonly confirmswhat we alreadyknew about this scale, but the intervalvectormakes the same kind of informationavailable about less familiarcollections. The intervalvector - it contains a different of the major scale has anotherinterestingproperty numberof occurrencesof each of the intervalclasses. This is an extremely and rareproperty(only threeothercollectionshave it), and it makes important traditionaltonal hierarchyof closely and distantlyrelatedkeys. the possible of intervalclasses in the major scale, In comparisonto the unique multiplicity the intervalvectorforthe whole-tonescale- 060603- presentsa starkeither/ or. The contrastingpropertiesof the two scales are suggestedby theircontrastingintervalvectors. Pitch-ClassSets Pitch-classsets are the basic buildingblocks of much post-tonalmusic. A pitch-classset is simplyan unorderedcollectionof pitch-classes.It is like - register, characteristics a motive withoutmany of its customaryidentifying occurrencesof a single and order.Example 10 shows five different rhythm, Suite forPiin from the Gavotte F, E, G], set, Schoenberg's [Dt, pitch-class ano, Op. 25. This content downloaded on Tue, 22 Jan 2013 07:30:12 AM All use subject to JSTOR Terms and Conditions 12 COLLEGE MUSIC SYMPOSIUM Example 10. A single pitch-class set expressed in five differentways (Schoenberg,Gavotte,fromSuite forPiano, Op. 25). (\A f (ifo /fi^K ffn \ l ^ 26 *tf~^"\ ^ /i r ===== i»;4 \k\jk * rit. - ^ ' • - s^ a It is expressed in a varietyof ways, but always retainsits basic pitch-class and interval-classidentity.A balance of compositionalunityand diversitycan be achievedby selectinga pitch-classset (or a small numberof different pitchclass sets) as a basic structuralunit,and thentransforming thatbasic unit in various ways. When we listen to or analyze music, we search for coherence. In a great deal of post-tonalmusic, thatcoherence is determinedby the use of pitch-classsets. NormalForm A pitch-classset can be presentedmusicallyin a varietyof ways. Conmusical figurescan representthe same pitch-classset. versely,manydifferent To aid in recognizinga pitch-classset no matterhow it is presentedin the music,it is oftenhelpfulto put it into a simple,compact,easily graspedform This content downloaded on Tue, 22 Jan 2013 07:30:12 AM All use subject to JSTOR Terms and Conditions A PRIMER FOR ATONAL SET THEORY 13 - themostcompressedway of writing called the normalform.5The normalform a pitch-classset makes it easy to see the essential attributesof a sonority and to compare it to othersonorities. The normalformof a pitch-classset is similarto the root positionof a sonoritiesthat triad,in thatboth are simple,compressedways of representing can occur in many positions and spacings. There are importantdifferences however. In traditionaltonal theory,the inversionsof a triad are generated fromthe relativelymore stable root position. The normalform,in contrast, has no particularstabilityor priority.It is just a convenientway of writing sets so thattheycan be more easily studiedand compared. To put a set in normal form,its pitch classes must be arrangedin ascending order,withinan octave, with the smallestpossible intervalbetween the firstand last notes, and withthe smallerintervalspacked towardthe bottom.6 For most sets, the normalformis evidentfromsimple inspection,but here is the step-by-stepprocedure: 1. Excluding doublings,writethe pitch classes ascending withinan octave. There will be as many differentways of doing this as there are pitch classes in the set, since an orderingcan begin on any of the pitch classes in the set. (The set fromSchoenberg'sGavotte,forexample,can be writtenout as E-F-G-Dt,F-G-Dt-E,G-Dt-E-F,and Dt-E-F-G). 2. Choose the orderingwhich has the smallest intervalfromthe lowest to thehighestnote.(The normalformof theset fromSchoenberg'sGavotte is [Dt, E, F, G]- the intervalfromDt to G is only 6). 3. If two or more orderingsshare the same smallest intervalfromthe lowest to the highestnote, choose the orderingthat is most packed to the left.To determinewhich is most packed to the left,comparethe intervals between the firstand second-to-lastnotes. If the determination is stillnot clear,comparethe intervalsbetweenfirstand third-to-last notes, and so on. (For example,the famous "changingchord" fromSchoenberg's Five OrchestralPieces, Op. 16, No. 3, can be writtenin two ways with only an 8 fromlowest to highest:E-Gi-A-B-C and Gi-A-B-C-E.Of these, [G#,A, B, C, E] is the normal form,since it has the smaller interval fromfirstto second-to-last). 4. If all of the comparisonsof Rule 3 still produceno clear choice, then arbitrarilypick the orderingbeginningwith the pitch class represented by the smallest integer.(For example, A-Q-F, Q-F-A, and F-A-Q are 5 The concept of "normal form" is original with Milton Babbitt. See "Set Structureas a Compositional Determinant." 6Allen Forte (The Structureof Atonal Music) and JohnRahn (Basic Atonal Theory)differslightly in their definitionof normal form, but this results in only a small number of discrepancies. This article adopts Rahn's formulation.Unfortunately,many differentnotationalconventions are currently in use. In this article, normal formswill be given in square brackets. This content downloaded on Tue, 22 Jan 2013 07:30:12 AM All use subject to JSTOR Terms and Conditions 14 COLLEGE MUSIC SYMPOSIUM in a three-waytie accordingto Rule 3. [Ci, F, A] is designatedas the normalformsince its firstpitchclass is 1, which is lower than5 or 9.) Transposition The termtranspositiontraditionallyrefersto lines of pitches. When we transposea tune fromC major to G major, we transposeeach pitch, in order, by some pitch interval. This operation preserves the ordered pitch intervalsin the line and thus its contour.Because contouris such a basic musical feature,it is easy to recognizewhen two lines of pitches are related by transposition. Transposinga set (not a line) of pitch classes (not pitches) is somewhat To transposea pitch-classset by some intervaln, an operationrepdifferent. resentedby the expressionTn, simply add n to each pitch class in the set. For example, to transpose[5, 7, 8, 11] by pitch-classinterval8, simplyadd 8 (mod 12) to each elementin the set to create a new set [1, 3, 4, 7]. If the firstset is in normalform,its transpositionwill be also (with a small numnormalform). ber of exceptionsrelatedto the fourthrule fordetermining A pitch-classset is a collection with no specified order or contour.As a result,transpositionof a set preservesneitherordernor contour.The four pitch-classsets circled in Example 11 are all transpositionsof one another. This content downloaded on Tue, 22 Jan 2013 07:30:12 AM All use subject to JSTOR Terms and Conditions A PRIMER FOR ATONAL SET THEORY 15 Example 11. Transpositionallyequivalentpitch-classsets (Webern,Concerto forNine Instruments, Op. 24, second movement). Sehr Langsamo - ca40 " ========^=^ F1-mi obis==EE^^m=^=^==== - fill 1 ^ l (TijnimermitDmpf. Trp( Ak"(i|J\ I \ " «* r~fl *hi ' I / " \, V ^ 1/ I / \ / /mil / Dmpf. VI -s- \ - i ■ ===== --mivpmpfX Iffhr \ '/ ^EE^=^=^ ==H" I .■ I M ¥r>i 1 ■ l^ -• l^^^^Jr,) iD I I . " I i I" 1 [I This content downloaded on Tue, 22 Jan 2013 07:30:12 AM All use subject to JSTOR Terms and Conditions ■ 16 COLLEGE MUSIC SYMPOSIUM These sets are dissimilarin manyobvious ways,includingtheirpitchand pitchclass content,theirmannerof presentation, theircontour,and theirorder. Withall of thesedifferences, thingsin comtheystill have two important mon.First,theycontainone-to-onecorrespondences amongtheirelements.This is particularlyclear when the sets are writtenin normal form,as they are beneaththe score in Example 11. For example, the Df has the same position in the firstset that the D has in the second set, the B in the thirdset, and the At in the fourthset. Second, theyall have the same intervalvector;each of themcontainsinterval-classes1, 3, and 4, and no others.This gives them a similarsound.Transposition of a set of pitchclasses may changemanythings, but it preservesinterval-classcontent.Along with inversion,transpositionis comone of only two operationsthatdo so, and consequently,is an important positionalmeans of creatinga deeperunitybeneatha complexmusical surface. to one of the songs from Considerthe beginningof the piano introduction Schoenberg'sBook of the Hanging Gardens, Op. 15, shown in Example 12. Example 12. Transpositionally equivalentpitch-classsets, and a transpositional path (Schoenberg,Book of the Hanging Gardens, Op. 15, No. 11). Sehrruhig(* = 4£> Gesang (fc(« -JsL-i* /' Klavier Jj, «P [d.E.F.G»] T3 l . i I -^-'^ % */ PP ■ = == * - h - IAYttJ* - * I " | [BKI>, D, F] To = \ n^!^- ■ " E \ ' I ■== [F.G».A.C] pocorit.T7- ppp The openingfour-note melodyin the righthand of the piano describesa certain pitch-classset, [Bt, Dt, D, F] in normalform.When the music continuesin measure 2, the righthand of the piano transposes[Bt, Dt, D, F] to T3, resultingin [O, E, F, G#].Then, in measure4, it transposesagain, this time to T7, resultingin [F, G#,A, C]. The music thus takes an initial musical idea and projects it througha transpositionalpath: TO-T3-T7.That succession of This content downloaded on Tue, 22 Jan 2013 07:30:12 AM All use subject to JSTOR Terms and Conditions A PRIMER FOR ATONAL SET THEORY 17 transpositionlevels, 0-3-7, exactly mirrorsthe intervallicsuccession of the firstthreenotes of the melody,Bt-Dt-F.The passage is thus audiblyunified, not only by the transpositionalequivalence of threepitch class sets, but by its large-scalereproductionof a small-scalemotivicidea.7 Inversion inversionis an operationtraditionallyapplied to lines Like transposition, of pitches. In invertinga line of pitches,order is preservedand contouris - each ascendingpitch intervalis replaced by a descendingone and reversed traditionaltonal practicerequiresonly that interval vice versa. Furthermore, sizes be maintained,not intervalqualities (major can become minor,and vice versa). It is best understoodas Inversionof a pitch-classset is a bit different. a compoundoperationexpressedas TnI,where "I" means "invert" and "Tn" means "transposeby some intervaln." By convention,we will always invert firstand thentranspose.The inversionof pitch-classn is 12-n. Pitch-class 1 invertsto -1, or 11. 2 invertsto 10; 3 invertsto 9; and so on. Afterwe invert,we will transposein the usual way. Example 13 shows again the opening of Schoenberg'sPiano Piece, Op. 11, No. 1. Example 13. Inversionallyequivalentpitch-classsets. Moderate) QX-.- ^ fi\ } V InlMl tfHf^ 13 3 1 3 1 Comparethe firsttwo sets circled in the example. They have the same interval-class content,but theirintervalsare arrangedin reverseorder.The second 7The eleventh song fromSchoenberg's Book of the Hanging Gardens has been analyzed briefly by Tom Demske ("Registral Centers of Balance in Atonal Works by Schoenberg and Webern," In TheoryOnly 9/2-3 (1986): 60-76), and, in great and compelling detail by David Lewin ("Toward the Analysis of a Schoenberg Song (Op. 15, No. 11)," Perspectives of New Music 12/1-2 (1973-4): 4386). My brief discussion is indebted to the latter. This content downloaded on Tue, 22 Jan 2013 07:30:12 AM All use subject to JSTOR Terms and Conditions 18 COLLEGE MUSIC SYMPOSIUM set has the same intervalsreadingfromthe top down as the firstdoes reading fromthe bottomup. Sets related by inversioncan always be writtenin this way. Now comparethe firstand thirdsets. Again, these two sets are related by inversion.They have the same interval-classcontent,but the intervals are in reverseorder. To inverta set, simplyinverteach memberof the set in turn.For example, to apply the operationT5I to the set [1, 3, 4, 7], just apply T5I to each integerin turn.Rememberingto invertbeforetransposing,we get ((12l)+5, (12-3)+5, (12-4)+5, (12-7)+5) = (4, 2, 1, 10). Notice that if we write this new set in reverseorder [10, 1, 2, 41 it will be in normal form.There will be some exceptions,but generallywhen you inverta set in normalform, the resultingset will be in normalformwrittenbackwards. The conceptof index numberoffersa simplerway both of invertingsets and of telling if two sets are inversionallyrelated.8 When we compared related sets, we subtractedcorrespondingelementsin each transpositionally set and called that differencethe "transpositionnumber."When comparing inversionallyrelated sets, we will add correspondingelementsand call that sum an "index number."When two sets are relatedby transpositionand both sets are in normalform,the firstelementin one set correspondsto the first elementin the other,the second to the second, and so on. When two sets are related by inversionand both are in normal form,the firstelement in one set will usually correspondto the last elementin the other,the second to the second-to-last,and so on. This is because inversionallyrelated sets are mirrorimages of each other. Here again are the firstand thirdsets fromExample 13, writtenin integer notation:[7, 8, 11] and [6, 9, 10]. If we add the pairs of corresponding elements(firstto last, second to second-to-last,and last to first),we get 5 (mod 12) in each case. These two sets are related at T5I. Any two sets in which the correspondingelementsall have the same sum are related by inversionand thatsum is the index number.To findthe index numberfortwo inversionallyrelated pitch classes, simplyadd themtogether.Conversely,to performthe operationTnI on some pitch class, simplysubtractit fromn. To performthe operationT4I on [10, 1, 2, 6], for example, subtracteach elementin turnfrom4: (4-10, 4-1, 4-2, 4-6) = (6, 3, 2, 10). As before,inverting a set in normal formproduces the normal formof a new set writtenbackwards. The normalformof (6, 3, 2, 10) is [10, 2, 3, 6]. • The concept of "index number"was firstdiscussed by Milton Babbitt in "Twelve-Tone Rhythmic Structureand the Electronic Medium/*Perspectives of New Music 1/1 (1962): 49-79; reprinted in Perspectives on ContemporaryMusic Theory, eds. Bo retz and Cone (New York: Norton, 1972), 148-79. He developed this concept in many of his articles, including "ContemporaryMusic Composition and Music Theory as ContemporaryIntellectualHistory,"Perspectives in Musicology, eds. Brook, Downes, and Van Solkema (New York: Norton, 1971), 151-84. This content downloaded on Tue, 22 Jan 2013 07:30:12 AM All use subject to JSTOR Terms and Conditions A PRIMER FOR ATONAL SET THEORY 19 Set Class Consider the followingcollection of pitch-classsets, all given in normal form. [2, 5, 6] [3, 6,7] [4, 7, 8] [5, 8, 9] [6, 9, 10] [7, 10, 11] [8, 11,0] [9, 0, 1] [10, 1, 2] [11,2, 3] [0, 3, 4] [1, 4, 5] [6, 7, 10] [7, 8, 11] [8, 9, 0] [9, 10, 1] [10, 11, 2] [11, 0, 3] [0, 1,4] [1, 2, 5] [2, 3, 6] [3,4,7] [4, 5, 8] [5, 6, 9] chosen set which is In the firstcolumn,the firstentryis an arbitrarily Each of the twelve sets levels. twelve of the each at transposition transposed is related to the remainingeleven by transposition.The second column begins with an inversionof the set, which is similarlytransposedat each of the twelve levels. In the second column,as in the first,each pitch-classset is relatedby transpositionto the othereleven. Now consider all twenty-four is relatedto all of the others of these sets together.Each of the twenty-four The inversion. or either pitch-classsets thuscomtwenty-four transposition by class. Each pitch-class a set sets called of related a family closely single, prise set is a memberof the set-class. Justas a pitch class contains many equivalent pitches and an interval class contains many equivalent intervals,a set class contains many equivamembers.A set class lent sets. Normally,a set class contains twenty-four class of the diminThe set fewer. has however, sets, containingsymmetrical ished-seventhchord,for example, containsonly threedistinctmembers:[Q, E, G, Bt], [D, F, Al>,B], and [C, Eb, F#,A]. Few sets are as redundantas this one (althoughone set, the whole-tonescale, is even more so). Most set classes containtwenty-four members;the rest have betweentwo and twentyfour. Set-class membershipis an importantpart of post-tonalmusical structure.Everypitch-classset belongs to a single set class. The sets in a set class are all related to each otherby eitherTn or TnL As a result,they all have the same interval-classcontent.By using membersof a single set class in a composition,a composer can create underlyingcoherencewhile varyingthe musical surface. Let's look yet again at the openingsectionof Schoenberg'sPiano Piece, Op. 11, No. 1 to see how set-class membershipcan lend musical coherence to a varied surface.The passage (omittingthe briefvaried repetitionsin measures 5-8) is shown in Example 14 with a numberof pitch-classsets circled or joined by a beam. All of these pitch-classsets are membersof the same set class. This content downloaded on Tue, 22 Jan 2013 07:30:12 AM All use subject to JSTOR Terms and Conditions COLLEGE MUSIC SYMPOSIUM 20 Example 14. Varied presentationof membersof a single-setclass. L_ 1 >> > 1 1 In the firstthreemeasures, a single continuousmelody descends from its highpointon B. In measures 4-5, the melody is reduced to a two-note fragmentthat reaches up to G (two immediaterepetitionsof this figureare omittedfromthe example). In measures9-11, the opening melody returnsin a varied formwith a highpointof Gt. These threenotes, B-G-Gi, are separated in time but associated as contourhighpoints.These are the same pitch classes as the firstthreenotes of the piece and the sustainednotes in measures 4-5. Each note in this large-scalestatementis also partof at least one smallscale statementof anothermemberof the same set class. The B in measure 1 is part of the collection [G, Gf, B]. In measures 4-5, the highpointG is part of not only the sustainedchord [G, Gi, B] but also part of the registral grouping[G, Bt, B]. An additionalmemberof the set class, [Bt, B, D], is formedin the middle of the textureand still another,[Ft, A, Ai], is formed by the movingpart in the tenorvoice. In measure 10, the G#is part of the collection[G#,A, C]. Additionalmembersof the set class are sprinkledthroughout the passage. In measure 3, forexample,both the left-handchord and the highestthreenotes in the measure are membersof the set class. The passage as a whole is also spannedby a large-scalestatementin the bass. The chords in measures 2-3 presentGt-Btin the bass. When the chords returnin measures 10-11, the Gt is restated,thenG arrivesto complete the set-statement, [Gk G, Bt]. The passage is virtuallysaturatedwith occurrencesof this set class. It occurs as a melodic fragment,as a chord, and as a combinationof melody and chord.It is articulatedby registerand, over a large span, by contourand mode of attack.An entirenetworkof musical associations radiates fromthe melodic figure.Some of the laterstatements have the same openingthree-note some the same content. Some are related content, pitch pitch-class by transsome All inversion. are members of the same set As in class. position, by This content downloaded on Tue, 22 Jan 2013 07:30:12 AM All use subject to JSTOR Terms and Conditions A PRIMER FOR ATONAL SET THEORY 21 an initialmusical idea grows and tonal music,but witheven greaterintensity, develops as the music proceeds.9 Prime Form There are two standardways of naming set classes. First, Allen Forte, has compiled a well-knownlist of set classes. He identifieseach with a pair of numbersseparatedby a dash (for example, 3-4). The numberbefore the dash tells the numberof pitch classes in the set. The numberafterthe dash gives its position on Forte's list. Set-class 3-4, for example, is the fourthon Forte's list of three-noteset classes. Forte's names are widely used. The second common way of identifyingset classes is to look at all of the membersof the set class, select the one with the "most normal"of normal forms,and use thatto name the set-class as a whole. This optimalform, called the primeform, begins with 0 and is most packed to the left. Of the pitch-classsets listed at the beginningof the previous section of twenty-four this article,two begin with0: 034 and 014. Of these,(014) is the mostpacked sets are all membersof to the left and is the primeform.Those twenty-four the set class with prime form(014). More familiarly,we say that each of those sets "is a (014)." In the rest of this article,set classes will be identified by both theirname accordingto Forte and, in parentheses,theirprime form.10 Here is the procedurefor findingthe prime formof a set class, a process usually referredto as puttinga set in primeform: 1. Put the set into normalform.(Take [1, 5, 6, 7] as an example.) 2. Transpose the set so that the firstelementis 0. (If we transpose [1, 5, 6, 7] by Tn, we get [0, 4, 5, 6].) 3. Invertthe set and repeatsteps 1 and 2. ([1, 5, 6, 7] invertsto (11, 7, 6, 5). The normalformof that set is [5, 6, 7, 11]. If that set is transposedat T7, the resultis [0, 1, 2, 6].) 4. Comparethe resultsof step 2 and step 3; whicheveris morepacked to the left is the primeform.([0, 1, 2, 6] is more packed to the leftthan [0, 4, 5, 6], so (0126) is the primeformof the set class of which [1, 5, 6, 7], our example,is a member.) Lists of set classes are available in manypublishedworks,includingthe books by Forte, Rahn, and Morris cited in the notes to this article. Considfew set classes. eringthe largenumberof pitchclass sets,thereare surprisingly 9 Schoenberg*s Piano Piece, Op. 11, No. 1, has been widely analyzed. George Perle discusses its intensive use of this same three-notemotive (which he calls a "basic cell") in Serial Composition and Atonality.See also Allen Forte, "The Magical Kaleidoscope: Schoenberg*s First Atonal Masterwork, Opus 11, No. 1," Journal of the Arnold Schoenberg Institute 5 (1981): 127-68; and Gary Wittlich, "Intervallic Set Structurein Schoenberg's Op. 11, No. 1," Perspectives of New Music 13 (1972): 41-55. 10As with normal form,conventionsfor notatingprime formsvary fromsource to source. This content downloaded on Tue, 22 Jan 2013 07:30:12 AM All use subject to JSTOR Terms and Conditions 22 COLLEGE MUSIC SYMPOSIUM For example, thereare 220 distincttrichords,but these can be grouped into trichord-classes.There are also only twelve nonachordjust twelve different and octachord-classes, classes. Similarly,thereare only 29 tetrachord-classes 38 pentachord-classesand septachord-classes),and 50 hexachord-classes. At this very basic level, set theoryinvolves namingthings,findingprecise, accurate,and consistentways of describingmusicalobjects. While naming thingsmay be a low-level analyticalact, it is an indispensablefirststep toward musical understanding.Like Adam in the Garden of Eden, we find ourselvesin a new world and,just as his firsttask was to name all the beasts of the field and fowl of the air, ours has been to develop a flexible,accurate nomenclature fornotes,intervals,and collections.A reliable music theory mustnot stop there,but it mustcertainlystartthere. Z-relation Any two sets related by transpositionor inversionmust have the same interval-classcontent.The converse,however,is not true. There are pairs of sets (one pair of tetrachords, threepairs of pentachords,and fifteenpairs of that have the same interval-classcontent,but are not relatedto hexachords) each otherby eithertranspositionor inversionand thus are not membersof the same set class. Sets thathave the same intervalcontentbut are not transbetween positionsor inversionsof each otherare Z-related and the relationship themis the Z-relation(the Z doesn't stand for anythingin particular).11Sets in the Z-relationwill sound similarbecause theyhave the same interval-class content,but they won't be as closely related to each other as sets that are membersof the same set class. If the membersof a set class are like siblings withina tightlyknit nuclear family,then Z-related sets are like first cousins. In Example 15, an excerptfromthe thirdof Stravinsky'sPieces for StringQuartet,the occurrencesof 4-Z15 (0146) in measures24-26 are strongly linkedto the similaroccurrencesof 4-Z29 (0137) in measures27-28. 11The Z-relation was firstdescribed David Lewin in "The IntervallicContentof a Colship by lectionof Notes,Intervallic Relationsbetweena Collectionof Notesand Its Complement: An Applicationto Schoenberg'sHexachordalPieces,'*Journalof Music Theory4 (1960): 98-101. The use of thelabel "Z" to referto thisrelationship is a coinageof Allen Forte's (see The Structure of Atonal Music,21-24). This content downloaded on Tue, 22 Jan 2013 07:30:12 AM All use subject to JSTOR Terms and Conditions A PRIMER FOR ATONAL SET THEORY 23 Example 15. The Z-relation(Stravinsky,Pieces for StringQuartet,No. 3). 1\itti glissezavec Varcheten toutesa longueur . ifi .1 jfc.i ri|/iiiij jY i i Tr ' I r rir rr ' ^ 4-Z15(014 6) tj^g, [ MEiz=a=±jl sonsi«els\yV Jll 1 J | ±== ^L^ 4-Z29 (013 7) Any set with a Z in its name has a "Z-correspondent,"anotherset with a different prime formbut the same intervalvector. On most set lists, the Zrelated hexachordsare listed across fromone another,but you will have to look throughthe list forthe Z-relatedsets of othersizes. Relation Complement For any set, the pitch classes it excludes constituteits complement.The complementof the set [3, 6, 7], forexample, is [8, 9, 10, 11, 0, 1, 2, 4, 5]. will containall twelvepitchclasses. takentogether, Anyset and its complement, For any set containingn elements,its complementwill contain 12-n elements. There is an importantintervallicsimilaritybetweena set and its complement. It mightseem logical to suppose that whateverintervalsa set has in abundance,its complementwill have few of, and vice versa. It turnsout, however, that a set and its complementalways have a similar distributionof intervals.For complementary sets, the differencein the numberof occurrences of each intervalis equal to the differencein the size of the sets (except for This content downloaded on Tue, 22 Jan 2013 07:30:12 AM All use subject to JSTOR Terms and Conditions 24 COLLEGE MUSIC SYMPOSIUM the tritone,where it will be half that difference).12If a tetrachordhas the intervalvector021030, its eight-notecomplementwill have the vector465472. The eight-noteset has fourmoreof everything (except forthe tritoneof which it has two more). The larger set is like an expanded version of its smaller complement. This intervallicrelationshipholds even if the two sets are not literally complementsof each other,so long as they are membersof complement-related set classes. For example, [3, 4, 5] and [5, 6, 7, 8, 9, 10, 11, 0, 1] are not literal complementsof each other,since they share pitch-class5. Howset classes. In otherwords,each ever,theyare membersof complement-related set is related by transpositionor inversionto the literal complementof the of intervals.Complementother,and therefore theyhave a similardistribution relatedsets do not have as muchin commonas transpositionally or inversionally related sets, but they do have a similar sound because of the similarityof theirintervalcontent. The complementrelationis particularlyimportantin any music in which the twelve pitch classes are circulatingrelativelyfreelyand in which the aggregate (a collection containing all twelve pitch classes) is an important structuralunit. Wheneverthe aggregateis divided into two parts,each part will have a similardistribution of intervals.If, forexample, five of the pitch classes are played as a melody and the remainingseven are played in accompanyingchords,the melody and the accompanimentwill have a similar sound because theycontaina similardistribution of intervals. The finalfour-note chordof Schoenberg'sLittlePiano Piece, Op. 19, No. 2 is a formof 4-19 (0148), a set class prominentthroughoutthatpiece and one commonin much of Schoenberg'smusic (see Example 16). Example 16. The complementrelation (Schoenberg,Little Piano Piece, Op. 19, No. 2). exactlyin time v 4-19 (O148)( , J | * PP u r*i * * H i i 1lfB FF pocorit. I*** n 'im 8-19 ■= ^ (01245689) 12The intervallic relationshipof complementarysets was firstdiscovered by Milton Babbitt with regard to hexachords. Generalizing this relationshipto sets of other sizes, was the work of Babbitt and David Lewin (see Lewin's "The IntervallicContent of a Collection of Notes"). Babbitt discusses the development of his theorem about hexachords and its subsequent generalization in Milton Babbitt: Words About Music, 104-106. This content downloaded on Tue, 22 Jan 2013 07:30:12 AM All use subject to JSTOR Terms and Conditions A PRIMER FOR ATONAL SET THEORY 25 The last eight notes of the piece (which, of course, include that final fournote chord), are a formof 8-19 (01245689), the complementaryset class. Comparethe intervalvectorsof these two sets: the vectorfor4-19 is 101310 and the vectorfor 8-19 is 545752. Both sets are particularlyrich in intervalclass 4. In fact, no four-or eight-noteset contains more 4s than these do. the 4s are featuredin the music. Because of the And notice how prominently complementrelation,the final four-notechord sounds similar to the larger eight-notecollection of which it is a part. Most lists of sets place complesets across fromone another.In Forte's set names,complementary ment-related sets always have the same numberfollowingthe dash. Thus, 4-19 and 8-19 are complementsof each otheras are 3-6 and 9-6, 5-Z12 and 7-Z12, and so on. The complementrelationshipholds particularinterestfor hexachords. - theyand theircomplementsare Some hexachordsare "self-complementary" membersof the same set class. For a simple example,considerthe hexachord [2, 3, 4, 5, 6, 7]. Its complementis [8, 9, 10, 11, 0, 1]. But both of these sets are membersof set-class6-1 (0123456). In otherwords,self-complementary hexachords are those that are related to their complementsby either T, or T.I. then it must be Z-related to If a hexachordis not self-complementary, sets, the differencein its complement.Rememberthat with complementary the numberof occurrencesof any intervalis equal to the differencein the size of the two sets. But a hexachordis exactlythe same size as its complement.As a result,a hexachordalways has exactly the same intervalcontent as its complement.If it is also relatedto its complementby Tn or TnI, then If not, then it is Z-relatedto its complement.This it is self-complementary. hexachordsis particularlyimintervallicrelationshipbetweencomplementary music. twelve-tone much for portant In additionto the basic nomenclatureand relationshipsdescribedabove, atonal set theory has developed a relatively sophisticated vocabulary for discussing common tones under transpositionand inversion,13the similarity suband axes of symmetry,15 of non-equivalentsets,14inversionalsymmetry 13Commontones undertransposition and inversionare discussedin Forte,The Structureof AtonalMusic,29-46 and Rahn,Basic AtonalTheory,97-115. For Morris'sapproachto the question, usuallythroughthe use of matrices,see his reviewof Rahn,Basic AtonalTheoryin Music Theory withPitchClasses, 70-78. Lewin describesthe 4 (1982): 138-55and his own Composition Spectrum samephenomenain termsof his Embeddingand InjectionFunctions.See GeneralizedMusical Intervals, 88-156. 14On similarity of AtonalMusic,46-59; CharlesLord, relations,see Allen Forte,The Structure "IntervallicSimilarityRelationsin AtonalSet Analysis,"Journalof Music Theory25 (1981): 91IndexforPitch-Class Sets/*Perspectives ofNewMusic 18/1-2(1979111; RobertMorris,"A Similarity 1980), 445-60; JohnRahn,"RelatingSets," Perspectivesof New Music 18/1-2(1979-80): 483-98; and David Lewin,"A Responseto a Response:On PC Set Relatedness," of New Music Perspectives 18/1-2(1979-80): 498-502. 15On inversionalsymmetry t 49and its musicalconsequences,see Rahn,Basic AtonalTheory 51 and 91-95; Lewin, "InversionalBalance as an OrganizingForce in Schoenberg'sMusic and Tonality(Berkeley: Thought,"Perspectivesof New Music 6/2 (1968): 1-21; and Perle,Twelve-Tone of CaliforniaPress,1977). University This content downloaded on Tue, 22 Jan 2013 07:30:12 AM All use subject to JSTOR Terms and Conditions 26 COLLEGE MUSIC SYMPOSIUM set and superset relations,16and ways of groupingset classes into interrelated complexes.17Despite this sophistication,post-tonaltheoryis still in its relativeinfancy,particularlycomparedto tonal theory,now enteringat least its fourthcenturyof development.That offersboth bad news and good news to those eager to understandpost-tonalmusic. The bad news is that we are still at a relativelyprimitiveanalytical stage, with considerable effortstill devoted to comparativelyrudimentary analytical acts, like taking an invenor of harmonic describing motivic structure.Of course, tory vocabulary there is more to music (and should be more to music analysis) than harmonic and motivic structure.The good news is that in recent years, set contour,20 theoryhas begun to branch out into voice leading,18rhythm,19 and timbre.21As crucial as these topics are to a complete understandingof post-tonalmusic, they have had to await the prior constructionof a secure theoryof pitch organization.Now that a reliable theoryof pitch organization is in place, a rapid expansion into othermusical areas has become not only possible, but an excitingfact of currentpost-tonaltheory. 16Subset and superset relations are discussed in Forte, The Structureof Atonal Music, 24-29 and Rahn, Basic Atonal Theory, 115-117. 17Forte's K and Kh relations are the best known models of set complexes. He has recently evolved a new model in "Pitch-Class Set Genera and the Origin of Modern Harmonic Species/*Journal of Music Theory 32/2 (1988): 187-270. 18See Alan Chapman, "Some IntervallicAspects of Pitch-Class Set Relations/*Journal of Music Theory25 (1981): 275-90; Allen Forte, "New Approaches to the Linear Analysis of Music,**Journal of the American Musicological Society 41 (1988): 315-48; ChristopherHasty, "On the Problem of Succession and Continuityin Twentieth-CenturyMusic,**Music Theory Spectrum 8 (1986): 58-74. See also two importantrecent dissertations:John Roeder, "A Theory of Voice-Leading for Atonal Music'* (Yale University,1984) and Henry Klumpenhouwer,"A Generalized Model of Voice-Leading for Atonal Music**(Harvard University,1991). 19See David Lewin, Generalized Musical Intervals, particularly22-30 and 60-87; Allen Forte, "Aspects of Rhythmin Webern*s Atonal Music,**Music Theory Spectrum 2 (1980): 90-109; Allen Forte, "Foreground Rhythmin Early Twentieth-Century Music,'* Music Analysis 2/3 (1983): 239-68; Martha Hyde, "A Theory of Twelve-Tone Meter,**Music Theory Spectrum 6 (1984): 63-78; Christopher Hasty, "Rhythmin Post-Tonal Music: PreliminaryQuestions of Duration and Motion,**Journal of Music Theory 25 (1981): 183-216. 20See Morris, Composition with Pitch Classes, 23-33; Michael Friedman, "A Methodology for the Discussion of Contour: Its Application to Schoenberg's Music,**Journal of Music Theory 29/2 (1985): 223-48; Elizabeth West Marvin and Paul A. Laprade, "Relating Musical Contours: Extensions of a Theory for Contour,** Journal of Music Theory 31/2 (1987): 225-67. 21See Wayne Slawson, "The Color of Sound: A Theoretical Study in Musical Timbre,**Music TheorySpectrum3 (1981): 132-41. This content downloaded on Tue, 22 Jan 2013 07:30:12 AM All use subject to JSTOR Terms and Conditions