Yale University Department of MusicSome Intervallic Aspects of Pitch-Class Set Relations Author(s): Alan Chapman Source: Journal of Music Theory, Vol. 25, No. 2 (Autumn, 1981), pp. 275-290 Published by: Duke University Press on behalf of the Yale University Department of Music Stable URL: http://www.jstor.org/stable/843652 Accessed: 11-07-2015 20:58 UTC 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]. Duke University Press and Yale University Department of Music are collaborating with JSTOR to digitize, preserve and extend access to Journal of Music Theory. http://www.jstor.org This content downloaded from 128.59.222.12 on Sat, 11 Jul 2015 20:58:05 UTC All use subject to JSTOR Terms and Conditions They provide an organized means of classifying harmonic structures (or any other pitch combinations).222. the third chord.59. In Example 1a. pitch-classsets allow us to assign a name to every conceivablepitch combination.we run the risk of consideringonly the name of a pitch-class set at the expense of its contextual role. 11 Jul 2015 20:58:05 UTC All use subject to JSTOR Terms and Conditions . then. can we evaluate the structuralsignificanceof different pitch-classsets in a composition? Such evaluationsareinvariablybased 275 This content downloaded from 128. "triad"or "seventhchord") for the second chord. It too is linear in origin and to place the Roman numeral"iii" beneathit would be erroneous. How. the tonal analystwould recognizethe structuralconnection between the first and third chords and describe the second chord as a secondarypitch structurewhich is linearin origin.Everynon-tonalchord successionis comparable to Example ib. In Example ib. taken out of context.one which canmost easilybe exposed througha tonal analogy. In the analysis of non-tonal music. but becausewe lack the structuralcriteriaof functional harmony.Caseslike Examplel a.12 on Sat. therefore. but in context this is not its meaning.SOME INTERVALLICASPECTS OF PITCH-CLASS SET RELATIONS Alan Chapman Pitch-class sets are a necessary tool for the analysis of non-tonal music. do not arise. has a name ("minortriad"). but they also pose a problem.Thisjudgment is both facilitated and supportedby the lack of a tonal name (for example. an approachwhich is dependent upon the assumptionthat a composer chooses certainpitch combinations as musical materialsand presents them in varied forms throughouta composition. Ro. as distinct from a nonset.Criterion(2) is problematicin that non-Zhexachordsare their own complements.(3) if the set is a member of a Z-pair.they play only 276 This content downloaded from 128.1 Forte's first two criteriaare based upon set recurrence."3By criterion(2) above."2 A primary segment is "determinedby conventionalmeans.12 on Sat. In his analyses of significant sets.(4) the set is an "atonal" set. 11 Jul 2015 20:58:05 UTC All use subject to JSTOR Terms and Conditions . of course. The intervallic similarityrelations. is not always easy.upon the recurrenceof certain pitch-class sets.222. "the analytical procedureby which the significant musical units of a composition are determined. are abstractreflections of intervallicrelations.an eight-noteset. but no efficient analyst would identify a set by calculatingthe total intervalcontent of a collection of pitches and looking up the interval vector. of prime forms. A pitch-class set name is usually thought of as a designationfor a specific collection of pitches. Because occurrences of such large sets as primary segments are relatively infrequent.) If total interval content is relegated to a position of secondary importance.59."(2) the complement of the set occurs consistently throughout. the intervalvector."4 Employment of composite segments often obviates the necessity of dealingwith structuralrelations among individualverticalpitch combinations. Some informalguidesare: (1) the set occurs consistently throughout-it is not merely "local. analysis must rely to some extent on composite segments which are "formed by segments or subsegmentsthat are contiguousor linked in some way. a significanttetrachord would requirerecurrenceof its complement. Allen Forte writes: The determinationof a significantset. such as a melodic configuration[or a simultaneous vertical combination].thus criteria(1) and (2) are simultaneouslysatisfiedby multipleoccurrences of these sets. not a set that would occur in a tonal work. Forte emphasizessegmentation. then the intervallicpropertiesof pitch-classsets and their intervallic relations with other sets become tertiary. The association of set names with pitch classesis reinforcedby the use. in set identification. R1 and R2 (based upon comparisonof interval vectors). in which only a superficial representationof the set's total interval content is readily apparent. The present paper shows that it is often appropriateto think of the set name as a designation for a specific collection of intervallicproperties. (This method would not discriminatebetween Z-relatedsets in any case. There is.the other member also occurs. The AB (above bass) intervalset is the set of intervals(in semitones)relativeto the bass. A Tonal Analogy (a) (b) Example2. 277 This content downloaded from 128.(a) (b) Example 1.12 on Sat. Two Tetrachords 4-Z15 (soprano-bass): 1 (alto-bass): 4 (tenor-bass): 6 4-Z15 6 8 11 Example3.59. 11 Jul 2015 20:58:05 UTC All use subject to JSTOR Terms and Conditions .222. Example 4 shows the permutationsof AB: 1-4-6. 11 Jul 2015 20:58:05 UTC All use subject to JSTOR Terms and Conditions .) Let us considerthe measurementof the intervalsrepresentedin the two tetrachords of Example2.with the "extra"intervalreflecting the octave duplication of one of the upper voices. the interval sets representedin Example7 are VP:6/9/10 and VP:9/10/11.5These intervalsare displayedin Example3. 7) in descendingverticalorder below the staff. A given tetrachord may have up to twenty-four distinct forms based on pitch content. (The four voices will be identified as soprano. Interval sets: a new system. Because of the present focus on tetrachords. The intervalset thus obtainedwill be calledtheAB (for above bass) intervalset and will be written in the form AB:X-Y-Z. In this example. AB sets of trichordsare expressedas sets of three intervals. and so forth.59.6 278 This content downloaded from 128. in descendingvertical orderbelow the staff.trichords as model harmonic structures. where n is the cardinalityof the pitch-classset.It can be seen in Example 4 that permutation of the AB intervalsdoes not change the pitch-classset (4-19). Y and Z are the intervals in ascendingnumericalorder. where appropriate. For the purposesof this study. these permutationsare brought about by differentregistralplacementsof the uppervoices. tenor and bass. The numberof differentAB intervalsis equal to (n-1). where X. This intervalset will be called the VP(for voice pairs) intervalset and will be written in the form VP:X/Y/Z. This practice will allow us to explore certain intervallic relationships between trichordsand tetrachords. Investigationof intervallic properties and relations calls for consideration of pitch-classset intervalcontent as it is expressed in musical contexts.222. where X. but it may have a maximum of eight AB sets associatedwith it.a trichordwill have two different intervals. An AB set which consists of three different intervalsmay appearin six distinct permutations. Therefore.12 on Sat. as in Example 6.a minor role in most atonal analysis. We can first reckon intervals(in semitones)relativeto the bass. This study utilizes tetrachords and.It is evident that the propertiesand relationsexhibited are adaptableto sets of largercardinalities. Y and Z are the intervals in ascending numerical order. Each AB set which consists of three differentintervalsis unique to a specific tetrachord. Let us now return to the two tetrachordswe examinedin Examples 2 and 3 and measure the intervals (in semitones) between adjacent voices.this study will employ four-part voicings to demonstrate theoretical principles.usuallyappearingas supplemental observationsupon analysesbased on pitch-classset recurrence. These intervalsare displayed(Ex. alto. with the bass held constant.a five-note set will have four. The eight AB intervalsets of the pitch-classset 4-19 are shown in Example 5.the intervalsets representedin Example3 are AB:1-4-6 and AB:6-8-11. 4-Z15 I I 4 6 1 1 [AB:] 4 6 6 1 4 i A 4 1 6 1 6 4 6 4 1 Example4. 3-9 2 [AB:] 7 2 Example6. A given tetrachordmay have up to 24 distinct formsbased on pitch content. 4-19 1 [AB:] 4 8 1 5 9 3 4 3 4 4 4 4 7 7 8 5 8 8 11 8 8 11 9 Example5. An AB set which consists of three differentintervalsmay appearin six distinct permutations. This content downloaded from 128. (b) j (n) I 4-Z15 4-Z15 9 10 10 9 [VP:] 6 11 Example7. but it may have a maximum of 8 AB sets associated with it. 11 Jul 2015 20:58:05 UTC All use subject to JSTOR Terms and Conditions .12 on Sat. AB sets of trichords may be expressed as sets of three intervals.59. The extra interval reflects the octave duplication of one of the uppervoices.Each AB set which consists of three differentintervalsis uniqueto a specific tetrachord. The VP (voice pairs) intervalset is the set of intervals(in semitones)between adjacentvoices.222. The viola's A is emphasizedby its lower neighbor G?. from the third movement of Stravinsky's Three Pieces for String Quartet.occupyinga longer span. on the downbeat of the second measure.In addition. The relation shown in Example 9 is an importantextension. Pitch-classsets which have VP intervalsets in common will be called VP-relatedsets. as illustrated by Example 9. 4-25 and 4-24. Here VP:4/5/9 is transposed down one semitone as the interval exchangeoccurs. making a prolongedcommon-toneconnection between 4-12 and 4-13. Unlike AB sets. The viola's motion from F$ to A. Finally.59. it is possible for trichords to express some of the same VP intervalsets as tetrachords. In Example 13.The exchangewould occur at this point were it not for the displacementof D and DW(in violin I and 'cello) by their upper neighbors. In Example 11 it can be seen that a VP intervalset canbe transposedin the same sense as a pitch-class set.the anticipationof 4-18's A and F as the inner voices of 4-14 emphasizesthe new position of four semitonesas the middle VP interval. Another exchange involving the same VP intervalset occurs in the same movement between successivetetrachords(Ex. Example10a shows the entire two-measure refrain. Because we are constructing four-part voicings of trichords.D and D$ return in the outer voices and the exchangeis completed. 4-12.) Within3-5 and 4-14. (The formal statements of the voice-leading procedureswhich permute VP sets arebeyond the scope of this paper. on the last chord of the second measure.is shown in Example 12. one within 4-12 and one within 4-13. a givenVP intervalset may be expressed by specific voicingsof more than one pitch-classset. sets of different cardinalitieshave been connected solely by the subset relation. Thus. 11 Jul 2015 20:58:05 UTC All use subject to JSTOR Terms and Conditions . which will effect the exchange. 280 This content downloaded from 128. 4-13 and 4-Z15 are VP-related sets. 11).222.the VP exchange is framedby two identically-voicedtranspositionsof 4-18. a four-chord progression from 4-Z15 to 4-18. Hitherto.as illustrated in Example 10b.Example8 shows the permutations of VP:6/9/10.12 on Sat. Another symmetricalstructure. Here a symmetrical structurecan be observed. 3-11. Considerationof the interval sets involved reveals that the structureof the phraseis an embellished VPintervalexchange. 3-3. is made immediately. Framed by two rhythmically-stressedoccurrencesof the same form of 3-5 are two appearancesof VP:2/3/6.A VP interval set which consists of three different intervalsmay appearin six distinct permutations. which have VP:2/4/4 in common. the outer voices provide upper and lower neighbors to D and GM. Linearconsiderations:VP-relatedsets. in example 8. in example9. and 4-17 are VP-related. It is noteworthy that violin II retains C throughout the symmetrical structure.Eb and E. a givenVP set may be expressedby specific voicingsof more than one pitch-classset. Three Pieces for StringQuartet.59. A VP Interval Exchange: Stravinsky.222. However. 3-11 6 Id 3-11 4-17 4-17 3-3 3-3 iII 4 9 8 8 9 4 8 9 4 4 9 [VP:] 8 9 4 4 8 8 9 Example9.12 on Sat. Example 10. 15-16.4-13 4-13 4-Z15 4-Z15 4-12 4-12 6 10 6 10 9 9 9 10 10 6 6 9 [VP:] 10 6 6 9 9 10 Example8. 22-23 281 This content downloaded from 128. 11 Jul 2015 20:58:05 UTC All use subject to JSTOR Terms and Conditions . sets of different cardinalities may be VP-related.III. 4-Z15 4-18 VIo. mm.Unlike AB sets. A VP set which consists of three different intervalsmay appearin six distinct permutations. Hitherto sets of differentcardinalitieshave been connected solely by the subset relation. 8-9. A VP Interval Exchange: Stravinsky. mm.59. 3-5 4-12 2 [VP:] 6 4-13 3-5 2 3 Example 12.III.III.12 on Sat. Three Pieces for StringQuartet.4-18 5 [VP:] 6 9 4-18 5 4 9 4-Z15 4-18 3-5 5 9 0 6 5 4.Three Pieces for StringQuartet. 17-18 yiVi Vc I I . 19-21 282 This content downloaded from 128. mm.222. 5 6 9 Example 11. 11 Jul 2015 20:58:05 UTC All use subject to JSTOR Terms and Conditions . A VP Interval Exchange: Stravinsky. Example 14 shows that 4-19 is the AB source set of AB:4-8-9. bass II leaps down an octave and a distinctly differentintervalset is introduced. Structuralconsiderations: VP representation.after a VP intervalexchangewhich connects 3-11 and 3-4. Example 15 shows multiple instances of VP representationin a Bartok string quartet. A given three-intervalVP set may be expressedin the VP domainby one. a recurrentpitch-classset (in this case 4-Z15) is also involved in a system of intervallicrelations.3-3. In the second and third measures.12 on Sat. expressedby 4-18 and 4-Z15.4-5 and 4-6 function as VP representatives of the form of 4-Z15 (AB:2-5-6) which follows. is immediatelyprecededand immediately followed by two orderingsof VP:3/7/8. two or threedifferentpitch-classsets (dependingon the intervalcontent of the VP set). 4-Z29 and 4-16 introduce AB:1-6-10 and AB:5-6-10. functioning as VP representativesof 4-Z29 and 4-16.are connected by a "chromatic passing tone. Different verticalorderingsare. Then 4-6 and 4-Z29.which generates 4-27. Example 17 contains the same AB form of 4-19. expressedby differentpitch-classsets. contains two pitchclass sets which are each VP-representedby two other pitch-classsets. Example 16. will be called VPrepresentatives of the AB sourceset. respectively." DM. from Schoenberg'sGeorgeLieder.may be viewed as a variationon the succession of intervalsets which ends example 16. 14).recreate(in the VP domain) the succession of interval sets which occurred in the previousmeasure. In the third and fourth measures. Note that VP sets appear abovethe score and AB sets below.222. with the samevertical 283 This content downloaded from 128. in most cases. in the form AB:4-5-8. This pitch-classset is called the AB source set of that intervalset. 4-20 appearswith the set AB:3-7-8.3-11 and4-17 areseen to be VP representativesof 4-19 (Ex. 11 Jul 2015 20:58:05 UTC All use subject to JSTOR Terms and Conditions . The structureof the first measureis based not only upon the repetition of 4-18 and 4-12.can be expressedin the AB domain by one and only one pitch-classset. In the first measure.59. the intervalcontent of a givenAB set. 4-16 returnsat the beginningof the third measurewith a different ordering of AB:5-6-10. The following musical excerpts demonstratethe usefulness of interval sets in determiningstructuralrelations.as VP sets.4-18 and 4-Zl15 function as VP representativesof 4-20. Pitch-classsets which reproduce.As shown in Example 4. any three-intervalAB set. As in the Bartok example. In Example16. In the second measure. The sense of an end to this prolongationalunit comes with the motion to 4-5: the upper voices all move. 4-19. but also upon the function of the second form of 4-12 as a VP representativeof the first form of 4-18. Example 17. regardlessof ordering. from PierrotLunaire. In Example9. m.4-25 4-27 TenorI 4-5 4-24 IiI" r A j 4 [VP:] 2 4 2 4 4 T 4 6 1 Example 13. A VP IntervalExchange:Stravinsky.Zvezdoliki. In Example 9. 3-3. The AB source set of AB: 4-8-9 is 4-19. 11 4-19 4 [AB:] 8 9 Example 14.12 on Sat. 11 Jul 2015 20:58:05 UTC All use subject to JSTOR Terms and Conditions . 3-11 and 4-17 are seen to be VPrepresentativesof 4-19.59. 284 This content downloaded from 128.222. (third movement) 5 [VP:] 7 7 5 2 6 6 5 4-6 3 8 4-215 4-18 8 3 4-19 4-215 7 5 4 [AB:] 3 2 8 8 6 5 Example 16.tP'errotLunaire.4-9 4-2158 [VP:] 7 6 7 55 38 3 Example17. 15/4 5-8) 4-Z (mm. 8-9) 8 [AB:] 8 55 6 Example17.Pi GeorgeLieder. Schoenberg.6 10 1 6 10 5 11 [VP:] 3 4-18 AB:] 4-12 4-18 11 6 3 4-12 4-Z29 4-16 4-16 6 6 10 10 1 5 4-6 4-Z29 5 6 10 Example15. op.George 4-27 Lieder. Bartok.Second StringQuartet. 11 Jul 2015 20:58:05 UTC All use subject to JSTOR Terms and Conditions . 17. 8-9) This content downloaded from 128.59. Schoenberg. 4-19 op.222. Schoenberg. 21/17 (mm. 21/]17 (mm.op.op.12 on Sat. "4" means "4-4") between the staves. In example 19d.3-3. 4-9.ordering of intervals. 11 Jul 2015 20:58:05 UTC All use subject to JSTOR Terms and Conditions .59. 2. The first ordering.4-5. 4-12 expressesits recurrentAB set (AB:1-3-9) and introduces VP:1/2/6. 4.Example 19c groups 4-10 and 4-12 with their VP representatives. These voicings of 4-13 in turn express the set VP:1/9/9. Example 19b shows that two of these voicings of 4-11 are VP-relatedto 4-2 (VP:1/9/10). 4-10 occurs only once in the excerpt.12 on Sat. is includedin example 19d.but these tetrachordsare also involved in a numberof intervallicrelations. as in Example 16. Examples 19e-g show other significantintervalsets. but they are of comparableintervallic status. is expressedby 4-27 rather than 4-18. is expressedby 4-Z15. is associatedwith its VP representative. The second ordering. while 4-12 appearsfour times. which confirmthe structuralsimilarity. In example 19a. Examples 19a-d deal with interval sets which first appearin measure 7. not the pitch-classsets. differentintervals. the AB source set. Example 18 is a primarilytetrachordal excerpt from the second of Carl Ruggles'sEvocations (1943). The tetrachords are identified by ordinal number (for example.222. 3. Becausethis VP set containstwo. 4-12 is a VP representativeof 4-5 (AB:1-2-6).this trichord. not three. 286 This content downloaded from 128. which appearsonly once. 4-3.however. which in these two appearancesexpresses VP:1/8/10. The tetrachordsof example 19f are presentedin the orderin which they occur in the composition. itself a recurrentVP set (see Example 19a). 4-5 is a VP representativeof 4-20 (AB:1-5-8). 4-5 is a VP representative(VP:1/7/10) of four subsequentoccurrences of 4-13 (AB:1-7-10).Intervallicrelationsalso providea means of integrating the single occurrences of six other tetrachordsinto a structuralanalysisof the excerpt. Twelve of these tetrachordsmight be consideredstructurallysignificantsolely on the basis of recurrence. 4-20 is a VP representativeof 4-Z15 (AB:1-7-9). 4-4 and 4-Z15 are seen to be VP representatives(VP:1/8/10) of 4-11 (AB:1-8-10). In example 19e. It is the intervalsets. its AB source set is a trichord. also appears. Example 19 is a summaryof recurrentintervalsets in the excerpt. The table which accompaniesthe excerpt shows that eighteen tetrachordsappear at least once. Pitch-classsets whose names are outlined are those which occur only once. which introduces VP:1/7/9.Here 4-19 is also precededand followed by two orderings of VP:3/7/8. which introduces VP:1/5/8. Interval sets in a larger context. demonstratingan interestingsuccession of relationships: 1. a tempo 7ITea F. Ruggles. 19 1 11 4-Z29 2 18.59..222. Poio 9poco 4 1 13 3 Z29 9 . 4-19 1 1 -4-916417 4-200 4-10 201 odeato 2 .reqeni 2 1 4 5 5 4 2 3 1 10 12 oco r .4 -ab..12 on Sat.4 13 17 5 4- 13 8 5 14 7 12 4-8 5 13 i .. 11 Jul 2015 20:58:05 UTC All use subject to JSTOR Terms and Conditions . 287 This content downloaded from 128. II Example Evocations. . 8 10 8 [VP:I 1 8 10 (a) 1 1 10 I -.222. 11 Jul 2015 20:58:05 UTC All use subject to JSTOR Terms and Conditions .59." 4-4(7) . 4-Z15(14.15) V•I4-4(16) I _ 4-11(8) 4-11(11) 8 10 8 [AB:I 10 8 10 1 1 1 (b) 4-11(18) 9 10 10 [VP:1 1 9 9 10 1 1 4-2(7) 4-11(18) 4-11(8) 4-3(8) 9 [AB:110 1 [VP:] (C) 3 3 1 10 10 1 4-2(7) 4-1(16) 3 [AB:1 10 1 4 [VP:] 101 4-5(7) 4 10 1 4-2(16) 4-12(8) [AB:] 10 1 Example19.12 on Sat. Recurrent Interval Sets in Evocations. II (measurenumbers in parentheses) 288 This content downloaded from 128. 12 on Sat.222.[VP:] (d) 7 1 10 9 9 1 4-5(7) 4-13(10) [AB:] 9 9 1 9 9 1 4-13(14.59.15) 9 [AB:J 3 1 2 6 1 5 8 1 4 [VP:] 9 7 8 8 7 • 1 1 4-2(15) 1 [AB:] 10 8 98 1 7 9 1 1 Example 19 (continued) 289 This content downloaded from 128.15) 7 10 1 4-13(17) 3-3(14) 7 10 1 9 9 1 7 10 1 6 6 1 1 7 [VP:]7 (e) 4-5(11) 4-5(11) 4-9(10) k-[AB:] 6 2 8 5 4-12(9) 4-5(10) [VP:] (f) (g) 7 6 1 9 7 4-Z 15(14. 11 Jul 2015 20:58:05 UTC All use subject to JSTOR Terms and Conditions . 5. 209. For atonal sets see Allen Forte. 290 This content downloaded from 128. 210.Atonal Music. 2. but are not reducible to the same prime form. The sets of a Z-pair have the same total interval content. p. 45. NOTES 1.p. (In addition." Journal of Music Theory 17 (1973). in the "Basic Interval Patterns. with larger intervals reduced mod 12. p.222. respectively. Forte. The Structure of Atonal Music (New Haven: Yale Univ. interval sizes will range from 0 to 11 semitones inclusive. Allen Forte. 11:1 (1972). It is not to be inferredthat intervalset considerationsare intended to constitute a complete analysis. 4.Example 19g illustrates that three sets which occur once each (4-17. 210. The complement of a pitch-class set X is the set which contains all the elements not in X. 6. Forte. "Sets and Nonsets in Schoenberg's Atonal Music. refers to these interval sets as "basic interval patterns" (bips) and reduces the intervals therein to interval classes." Perspectives of New Music.12 on Sat. Thus the bips for Examples 7a and 7b would be 236 and 123. 3.It is my hope that this paperhas demonstratedthe potential for their incorporationinto a largermethodology. 4-19 and 4-8) are intervallicallyrelated. Forte. 4-17 and 4-19 are VP representatives of 4-8. Press. Once again we shall avoid abstract measurement and maintain absolute interval sizes.Atonal Music. Rather. 1973).59.p. 4-17 is VP-representedby 4-2. We shall not reduce intervals greater than a tritone and less than an octave to interval classes. 11 Jul 2015 20:58:05 UTC All use subject to JSTOR Terms and Conditions . the intervalset appearsthree times.) Each pitch-classset appearsonly once.