Having the Last Word: Schoenberg and the Ultimate Cadenza

brian alegant and andrew mead

In this essay we examine the ultimate cadenzas of Arnold Schoenberg’s Violin Concerto, Op. 36, andPiano Concerto, Op. 42, and summarize the main partitioning strategies in these compositions,including two constructs we refer to as dyadic and trichordal complexes. The analysis details thestructure and rhetoric of the cadenzas and concludes with a brief reconsideration of the universe ofdyadic complexes.*

Keywords: cadenza, dyadic complex, Op. 36, Op. 42, Schoenberg, triadic complex

This study analyzes the ultimate cadenzas of ArnoldSchoenberg’s ‘‘American’’-period concertos. The ini-tial cadenzas of the Violin Concerto, Op. 36, and

Piano Concerto, Op. 42, have been the focus of several analyticalstudies. David Lewin and Andrew Mead have each shown thatthe cadenza of the first movement of the Violin Concerto servesas a focal point, or perhaps even nexus, for the movement’sprimary motives and partitioning strategies. Brian Alegant hasargued that the solo cadenza in the Adagio (movement III) ofthe Piano Concerto is the most extended and sophisticatedtrichordal passage in Schoenberg’s oeuvre.1 But the initial ca-denzas in these serve merely as previews for these cadenzas,which are even more expansive and elaborate in terms of com-positional technique, motivic development, and rhetoricalflourish. With this in mind, it is surprising that scholars havevirtually ignored these final cadenzas.

This essay attempts to remedy the oversight. We argue thatthe final cadenza of the Violin Concerto serves as a summary forthe entire work, just as the opening cadenza encapsulates theevents of the first movement. We also argue that the secondcadenza of the Piano Concerto functions in a similar manner:the orchestral tutti recasts the events of the solo piano cadenzaand brings to a culmination several other partitioning strategiesthat pervade the work. We begin by reviewing some basicprocedures of Schoenberg’s mature twelve-tone praxis. We thenexamine the final cadenzas of these concertos in turn, payingparticular attention to the partitioning strategies and aggregaterealizations on the surface, and the larger-scale associationsamong the constituent elements of different harmonic regions.We conclude with a few thoughts on the articulation of ‘‘cadenzaspace,’’ which are followed by an appendix that reconsidersa construct we refer to as the dyadic complex.

initial considerations

The foundations of Schoenberg’s harmonic praxis are wellknown and long understood.2 Example 1(a) illustrates theprocedure of hexachordal inversional combinatoriality with the‘‘home’’ rows of the Violin Concerto, P9 and I2.3 These rowsform a combinatorial array containing two aggregates, eachaggregate comprising six dyads sharing the same order positionin the rows: f2,9g, f1,Ag, f3,8g, fB,0g, f4,7g, and f6,5g. (Thedyads are the cycles of index number 11, or B.) The array can bepartitioned in various ways to generate ‘‘row harmonies.’’ Oneharmonization combines the non-overlapping dyads of the rowsinto tetrachords, with the first one containing the elements atorder positions 0 and 1, the second having the elements at orderpositions 2 and 3, and the remaining tetrachords combining theelements at order positions 4 and 5, 6 and 7, 8 and 9, and A andB. Example 1(b) shows that the resulting tetrachords belong toset-classes 4-1[0123], 4-7[0145], and 4-17[0347]; note that, inthis array, tetrachords of the same set class are related by T6. Wecall this set of six tetrachords the dyadic complex, and we label itstetrachords [1] through [6].4

Example 2 shows the combinatorial array of the PianoConcerto’s home rows, P3 and I8. Example 2(a) shows thedyadic complex. The complex of Op. 42 shares two tetrachordal

* An earlier version of this essay was delivered at the annual Music TheorySociety of New York State Meeting, held at Ithaca College, Ithaca, NewYork, in April 2008.

1 Analyses of Op. 36 include Lewin (1962) and Mead (1985) and (1993);analyses of Op. 42 include Rothstein (1980); Mead (1989); Alegant andMcLean (2001); and Alegant (2002–3). A sampling of other twelve-toneanalyses that invoke partitional approaches includes Kurth (1992); Mead(1985) and (1989); Morris and Alegant (1988); and Peles (1983–84) and(2000).

2 Babbitt (1960) and (1961) and Lewin (1962), (1967), and (1968) areamong the earliest analyses to address the use of inversion inSchoenberg’s twelve-tone music.

3 Some conventions: We use C¼0, C�¼1, . . . , B�¼‘‘A,’’ and B�¼‘‘B.’’ Weidentify P and I rows by their initial pitch class and R and RI rows by theirfinal pitch class. Thus, any P9 opens with A natural; any RI2 ends with D.Set classes are identified with labels from Forte (1973) and Rahn (1980).Ordered collections are indicated with angled brackets, h i, and unorderedcollections are designated by accolades, f g. Order numbers are denoted bythe italics 0 through B.

4 Babbitt (in Dembski and Straus [1987]) discusses the pairing ofinversionally related dyads in Schoenberg’s Opus 33a; his combinatorialarray closely resembles our Example 1. Babbitt remarks: ‘‘The essentialquestion in Opus 33a is how you regard these two sets when they’retaken in dyads and when those dyads are regarded as making tetrachords,not linearly but in groups of twos’’ (77–79). Peles (2000, 127–29) showshow the pitch symmetry of the opening violin tune of the Fourth Quartetprefigures its array of dyads.

107

example 1. Characteristics of the row of Op. 36.

example 2. Characteristics of the row of Op. 42.

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set classes with that of Op. 36 (4-7[0145] and 4-17[0347]), butits profile is different. Specifically, the Piano Concerto’s arrayyields tetrachords from five set classes; additionally, its third andsixth sonorities are identical. Example 2(b) shows anotherpartitioning strategy for generating row harmonies. Here, therows’ non-overlapping trichords, which belong to set-classes3-4[015], 3-9[027], and 3-6[024], are grouped into four hexa-chords. The first one combines the elements at order positions0–2; the second combines them at order positions 3–5. Theseyield the Z-related pair 6-Z6[012567] and 6-Z38[012378].The third and fourth hexachords combine the elements at orderpositions 6–8 and 9–B; these conjoin to form all-combinatorial,diatonic collections that belong to set class 6-32[023457]. Werefer to the collection of these hexachords as the trichordalcomplex.5

From a labeling standpoint, the trichordal complex containsfour partitions that belong to three distinct mosaics.6 A partitionis a collection of non-overlapping, unordered pitch-class setsthat form an aggregate; a mosaic is the set of partitions that arerelated by transposition and inversion. One mosaic includes thesegmental trichords of P3 and I8; this mosaic houses two par-titions, which we designate fp1,p2,p3,p4g and fi1,i2,i3,i4g.The trichords of these partitions generate a pair of 6-9[012357]hexachords, a pair of 6-16[014568] hexachords, and a Z-relatedpair, 6-Z26[013578] and 6-Z48[012579]. The first half of thecomplex contains another trichordal partition from anothermosaic; we represent this partition as fp1,p2,i1,i2g. Partitions ofthis mosaic yield pairs of 6-9 and 6-20[014589] hexachords andanother Z-pair, 6-Z6[012567] and 6-Z38[012378]. Finally, thesecond half of the complex contains another partition that be-longs to a third mosaic, fp3,p4,i3,i4g. The constituent trichordsof this partition create pairs of 6-9, 6-22[012468], and 6-32[024579] hexachords. This latter set class, the diatonic all-combinatorial hexachord, plays a significant role in the ultimatecadenza.

The dyadic and trichordal complexes in Schoenberg’s twelve-tone compositions yield a unique set of tetrachords and hexa-chords. (By definition, all of these collections are inversionallysymmetrical under an odd index number.) Depending on theproperties of the source row, the tetrachords and hexachords inthe complexes may or may not match those that are found as rowsegments. In practice, Schoenberg tends to project the elementsof the dyadic complexes in a linear fashion, moving incremen-tally from [1] through [6] or from [6] through [1]; moreover,he takes pains to exploit—or avoid—their characteristic

properties.7 Through the technique of varied repetition, thesecollections come to function as referential sonorities.

A cross partition is another strategy commonly found inSchoenberg’s twelve-tone music.8 In simplest terms, a crosspartition arranges the pitch classes of an aggregate (or row) ina two-dimensional rectangular design. Typically, the verticalcolumns of a cross partition are derived from a row’s segments,while the horizontal rows of a cross partition contain non-adjacent elements of the row. The vertical pitch classes of a crosspartition are often realized as chords, while the horizontalpitch classes are differentiated from one another by registral,timbral, or other means. Below we show the ‘‘even’’ crosspartitions, which we designate 62, 43, 34, and 26, respectively.(The first integer specifies the number of vertical elements;the exponent specifies the number of horizontal elements.Thus, a 62 cross partition is essentially a presentation of twosimultaneous hexachords.)

62 43 34 26

* * * * * * * * * * * * * * *

* * * * * * * * * * * * * * *

* * * * * * * * *

* * * * *

* *

* *

Although many other irregular configurations are possible,twelve-tone composers (notably Schoenberg and Dallapiccola)tend to gravitate to the four even configurations. Once con-structed, the pitch classes in the columns of the cross partitioncan be permuted—as if subjected to ‘‘slot-machine’’ transforma-tions. These permutations preserve the elements in the verticaldimension while varying the material in the horizontal dimension,in a sense, maintaining the original row’s ‘‘harmonies’’ and gen-erating different ‘‘melodies.’’ To illustrate, the configurationsbelow are some of the many slot-machine variations of a single 34

cross partition.9

0 3 6 9 0 4 7 B 0 5 6 9 2 3 8 A

1 4 7 A 1 3 8 A 1 4 8 A 1 4 7 B

2 5 8 B 2 5 6 9 2 3 7 B 0 5 6 9

5 Martino (1961) discusses at length the phenomena of dyadic and trichordalcomplexes in the twelve-tone system, and categorizes the harmonies thatarise in all two- and four-part arrays. Starr (1984) further extends thetheoretical implications of Martino’s study.

6 See Morris and Alegant (1988). The relation between a partition anda mosaic is analogous to that between a pitch-class set and a set class. Mead(1988) and Kurth (1992) use mosaic and mosaic class instead of partition andmosaic.

7 For instance, the trichordal complex of the Violin Concerto yields twopairs of Z-related hexachords: 6-Z42[012369] and 6-Z13[013467], and6-Z6[012567] and 6-Z38[012378]. The latter pair also appears as the fronthalf of the complex of the Piano Concerto. The Violin Concerto’s trichordalcomplex appears only on rare occasions; it is not a feature of the cadenzaunder discussion.

8 This construct is discussed in Martino (1961) and explored further inAlegant (1993), (2001), and (2010). The cross-partition is a remarkablyflexible device. A familiar example is the opening of Op. 33a, in which thedisjunct tetrachords of the P and RI rows are presented as chords; over timethe tetrachords are unraveled and the row as a string is revealed.

9 Such slot-machine transformations can generate many variations of theoriginal design. The number of distinct configurations (those unrelatedby Tn or TnI) is determined by the properties of the pitch-class sets inthe columns.

having the last word: schoenberg and the ultimate cadenza 109

Like the dyadic and trichordal complexes, cross partitionsoffer an easy and organic way to generate non-segmentalmaterial. However, it is important to remember that cross par-titions are derived from a single row, whereas dyadic and tri-chordal complexes are derived from pairs of rows.

the violin concerto

Schoenberg’s Violin Concerto is in three large movementsfollowing the traditional model for the genre. While eachmovement has its own form and character, several strands ofmotivic continuity run through the entire work. These strandsare summarized in an extended notated cadenza toward the endof the last movement, in which the soloist revisits the primarymaterials of all three movements. Unlike the cadenza (alsonotated) in the first movement, the final cadenza includesinterjections from the orchestra. The following discussion tracessome of the connections between the final cadenza and the restof the work, paying particular attention to motivic associationsand the return of combinatorial row pairs that appear atimportant junctures.

One significant feature of the Violin Concerto is the pre-ponderance of tetrachordal motives containing discrete pairs of

half steps. There are five such tetrachords; three invert ontothemselves under an odd index number while the other twoinvert into themselves under an even index number. Of these, setclasses 4-1[0123], 4-7[0145], and 4-17[0347] are found in thedyadic complex, while 4-8[0156] and 4-9[0167] are segments ofthe row. In addition, a sixth tetrachord, 4-3[0134], the emblematicmotive of the concerto, combines the first dyads of the firsthexachords of all P and I rows (and the final dyads of the secondhexachords in all R and RI rows). Each tetrachordal set class isprominently realized in the opening, shown in Example 3.

These tetrachords can be derived in a variety of ways—fromsingle rows or pairs of rows, and from adjacent or non-adjacentorder positions. Throughout the concerto Schoenberg consis-tently evokes the same handful of partitioning schemes, each ofwhich is characterized by specific sonorities. The ensuing dis-cussion traces the history of one such scheme.

Example 4(a) shows a passage from the opening of the firstmovement. Its material is based on the P9/I2 complex, whichappears at the bottom of the example. Of particular significanceare sonorities [2] and [6], which belong to set-class 4-17[0347];note that these (unordered) tetrachords are related by T6.Example 4(b) shows a recapitulation of this passage later inthe movement; the pitch-class material for this return is basedon the P3/I8 complex, which is a tritone removed from the

example 3. Prominent tetrachords in the opening of Op. 36.

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example 4. Invariant [0347] sonorities among T6-related complexes.

having the last word: schoenberg and the ultimate cadenza 111

previous one. Applying the same partitioning scheme to T6-related complexes exchanges the tetrachordal collections of thecombinatorial arrays. At the same time, while the row pairs inthe passages are related by transposition, the surface realiza-tion highlights the inversional relationships between the rowpairs. In this way the passage in Example 4(b) is able to evokethe initial gesture of the concerto through the flute’s fA,B�gdyad.

The handling of these [0347] motives is vivid and memo-rable. The passage in Example 4(a) is marked by the first use of

double stops and octave shifts within aggregates. The oboe’s halfstep echoes the rhythm of the movement’s opening motive,while at the same time employing a new dyad for that motive;nonetheless, the fG,A�g dyad is not entirely unfamiliar: in anearlier passage it was played by the winds, in the same register,and with dynamic emphasis. The return of the violin figurationin Example 4(b) is noteworthy not only for its octave leaps(which are reserved for these two passages), but also because thispassage immediately follows a varied recapitulation of themeasures leading to its first occurrence. Transforming the oboe’s

example 4. [Continued]

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familiar half step motive into the opening fA,B�g dyad furtherunderlines the sense of return.

Example 5 highlights another facet of the dyadic complex.Each passage in this example unfolds the tetrachords of

a complex from [1] through [6] or vice versa. The excerpts inExamples 5(a) and 5(b) occur a mere seven measures apart,toward the end of the first main section of the first movement. InExample 5(a) the solo violin leapfrogs through [6] and [3], while

example 5. Another scheme involving the tetrachords of the complex.

having the last word: schoenberg and the ultimate cadenza 113

the orchestra combines [5] and [4], and [2] and [1].10 The entirepassage unfolds thus:

R6: h B,2 5,4 A,9 3,1 8,0 7,6 iRIB: h 6,3 0,1 7,8 2,4 5,9 A,B i

[6] [5] [4] [3] [2] [1]

[0347] [0145] [0123] [0123] [0347] [0145]

Examples 5(b) and 5(c) show the results when the parti-tioning scheme in 5(a) is applied to different regions. Despitevariations in the violin’s contour and content, the partitionalcorrespondences between these events are quite recognizable.The passage in Example 5(d) anticipates the complex that ap-pears near the end of the first movement’s cadenza. The violinmoves through the complex in rapid-fire succession, squeezingall six of its tetrachords into the span previously reserved for [6]and [3]. As a result, Example 5(d) can be heard as a summary orreflection of the passages in (a), (b), and (c).

We now turn to the vast cadenza that concludes the thirdmovement of the work and recapitulates, both by motive and byrow pair, music from the entire concerto. Example 6 shows thelead-in and the first section of the cadenza (mm. 640–60). Thelead-in employs the home rows that open the work, articulatedin the rhythmic and motivic contours of this last-movementrondo’s refrain. The cadenza brings a significant rhetoricalchange: the orchestral accompaniment seems to dissolve, leav-ing the violin to play a very broad and dramatic figure. Thefollowing passage (not shown) features a shift to the lowestregister, a change of dynamic and tempo, and a kind of musicalregrouping—a pulling back from the driving march that hasdominated the last movement.

The cadenza brings back in order the row pairs from theopening of the first movement (P9 and I2, m. 647), the openingof that movement’s recapitulation (P0 and I5, m. 661), and there-entrance of the violin in the recapitulation (P7 and I0, on beatthree of m. 671). (These are discussed below.) At the beginningof the cadenza, the soloist returns to the atmosphere of the veryopening of the first movement, accompanied by figures foundin the winds. At the same time, the upper strings shape thecombinatorial counterpart of the row of the opening so as tobring back a motive introduced in the third movement (see, forexample, mm. 492ff. in the example). The passage reprises manyof the collections we have been discussing, including severalidentified in the complex of Example 1(b).

Example 7 shows the cadenza’s second subsection, whichrecalls motives from several earlier passages. Measures 661–64recapture a melody from the middle of the third movement (seemm. 572ff.). In m. 666, the orchestral strings rejoin the soloist,thus recalling the opening of the second movement, which itselfrecalls the rhythmic figuration of mm. 6–7 of the first movement(shown in Example 3). Measures 666–68 are based upon the

latter half of the dyadic complex. The solo violin combines ic-3dyads to form sonority [6], while the orchestral strings project[5] and [4] in imitation. (This complex is a T6-transposition ofthe design shown in Example 5[a].) The remainder of thissubsection recalls the opening of the second movement,including an extensive portion of its lyrical opening melody.

The third subsection deals primarily with references tomaterial from the third movement; at the same time, it relatesto the motive that had opened the cadenza of the first move-ment. (See Example 8.) This same material, realized in a vari-ety of ways, also concludes the cadenza. An interesting featureof mm. 676ff. is the way in which the variations of the primarymotive retain its rhythmic profile and its pairing of discrete halfsteps. The overall effect from m. 677 through the winds’presentation of the motive in m. 681 is a chromatic ascent fromD to F�. Measures 686ff. also include chromatic strings ofpitch classes, here derived from a partitioning scheme that isintroduced in the first movement.11 This partition yields theemblematic motive of the entire work, h9,B,0,1i; this tetra-chord is projected in the upper orchestral strings. Measure 692brings yet another allusion to the opening motive. Here,however, the half steps fA�,Gg and fD,C�g are related bytransposition (not inversion), and combine to form a memberof 4-9[0167].

It is intriguing that this passage articulates precisely the tri-tone transposition linking the complexes of two regions, whilebringing forth two specific pitch-class dyads associated with theopening passages of the first movement. (See, for instance, thefA�,Gg dyad in Example 4 and the fD,C�g dyad in the soloviolin’s opening line in Example 3.)

Schoenberg takes a different approach to row presentationand articulation in the next subsection, which is illustrated inExample 9. Here, a sequence of single rows, presented withouttheir combinatorial counterparts, projects the segmental dyadsfound by pairing order numbers with an odd number first,leaving the ends of the rows as singletons. (This procedurecontrasts with the manufacturing of segmental dyads in thedyadic complex, which pairs order numbers with an evennumber first.) Several interesting features emerge in the reali-zation. First, this partitioning strategy emphasizes the singleelements at the beginnings and ends of the rows. Looking at thefirst two parallel gestures in the passage, we can see that the fournotes marked with asterisks, hD,B�,C�,Ai, replicate the firsttetrachord of the opening dyadic complex (shown in Example 6).Second, we can read the sets of segmental dyads framed by therows’ end-notes as three tritones and two perfect fifths, andobserve the wealth of invariance patterns shared among the rowsin this passage, some partial and some complete. Some of thesecommonalities are noted in the pitch-class representations atthe bottom of the example. Lastly, we can appreciate howSchoenberg combines a desire for segmental invariance witha desire to bring out the open-string dyads of the violin fG,Dgand fA,Eg, articulated here by left-hand pizzicati. These dyads

10 The content of the solo violin is based entirely on [3] and [6], whichtogether form pitch-class set fB,1,2,3,4,6g, a member of the all-combinatorial 6-8[023457]. 11 See Mead (1985).

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example 6. The lead-in and Part 1 of the cadenza (mm. 640–60).

having the last word: schoenberg and the ultimate cadenza 115

example 6. [Continued]

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example 7. Part 2 of the cadenza (mm. 661–75).

having the last word: schoenberg and the ultimate cadenza 117

example 8. Part 3 of the cadenza (mm. 676–94).

118 music theory spectrum 34 (2012)

example 8. [Continued]

having the last word: schoenberg and the ultimate cadenza 119

are found in virtually every measure of Example 9, and are easilyidentified with the traditional ‘‘þ’’ symbols that indicate thismode of articulation.

The cadenza then makes one more overt reference to earliermaterial. As Example 10 shows, the pickup to m. 705 recalls theopening of the first movement, derived from the member of therow class used at that point. Two specific details deserve men-tion: the accompanimental figures in the violin articulate thesegmental [0156] and [0167] tetrachords featured in the pre-vious example, while the clarinet sustains the notes that form theopening dyad of the solo part in the second movement. Whatfollows is a return to the motive from the third movement thataccompanies the opening of the cadenza.

Example 11 shows the close of the cadenza and the return ofthe full orchestra. Measure 715 is based on a remarkablycompact realization of a dyadic complex, one that articulatesthe set of tetrachords both by timbre and by rhythm. It is alsoworth pointing out the material played in the lowest register bythree trombones and tuba in unison: the dyads fA,B�g andfE�,Eg (boxed on the lower winds staff of the second systemof the example). These dyads fulfill two functions: they refer

to a motive in the third movement and they recall the precisedyads found in the very first measures, as part of a call andresponse between the soloist and the orchestra (shown inExample 3).

Obviously, the above analysis only scratches the surface of theViolin Concerto, which entails a wide variety of invariance re-lations based not only on symmetrical tetrachords but also oncollections of many sorts derived in a variety of ways. But themotivic presence of repertories of symmetrical tetrachords, andtheir tendency to engage inversionally related rows, invites thisparticular perspective on the work. As a final gloss on theconcerto, Example 12 summarizes the cadenza’s motivic andharmonic structure, noting some of the connections with otherparts of the work.

the piano concerto

Schoenberg’s Piano Concerto is in many respects tradi-tional and nostalgic. Its cantabile melodies, lilting rhythms,and tonal ‘‘artifacts’’ (such as trills and perfect-fifth bassmotions at points of arrival) have much in common with its

example 9. Part 4 of the cadenza (mm. 694–704).

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example 10. Part 5 of the cadenza (mm. 705–12).

having the last word: schoenberg and the ultimate cadenza 121

example 11. Part 6 of the cadenza (conclusion, mm. 713–19).

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eighteenth- and nineteenth-century Viennese predecessors.At the same time, several aspects of the work are atypical ofSchoenberg’s praxis. The first, and most conspicuous, is thewealth of octave doubling (and even tripling); the thickorchestration is much closer in spirit to turn-of-the-centuryworks such as Gurrelieder or Verklarte Nacht than to the ViolinConcerto.12 A second innovation in the work is the organizationof hexachordal regions in the first movement. As WilliamRothstein, Andrew Mead, and others have noted, the sequenceof hexachordal regions is based upon the succession of pitchclasses in the source row, P3. One way to understand theprogression is to consider each note of the row as a tokenrepresentative of a specific region. Thus, pitch class 3 can beinterpreted as a token for the quartet that includes rows P3 and I8,and their retrogrades, R3 and RI8. The first four notes of P3 areE�, B�, D, and F; and the first four regions include rows drawnfrom the regions that include rows P3, PA, P2, and P5. The arrivalof the row’s last note (which is commensurate with the twelfthharmonic region) heralds a dazzling and intricate recapitulationin which the opening P3 row is nested on two temporal levels.13

A third defining aspect of the concerto is its program; the fol-lowing brief descriptions of the movements are written clearly ona manuscript page: Andante, ‘‘life was so easy’’; Molto Allegro,‘‘suddenly hatred broke out’’; Adagio, ‘‘a grave situation wascreated’’; Giocoso, ‘‘but life goes on.’’14

The cadenzas play vital roles in this ‘‘grave situation.’’Structurally, the first cadenza for solo piano is the saturationpoint of trichordal structuring not only in this work, but inSchoenberg’s oeuvre: its pitch-class material is generated entirelyfrom trichordal complexes. The ultimate cadenza, which com-bines the piano and full orchestra, serves as the culmination firstof the trichordal complex and then of the dyadic complex.Significantly, neither of these appears in the fourth movement,save for a flashback of the dyadic complex in the final measures.

It will be helpful to trace the history of the dyadic complex. Itappears in each of the first three movements; through sheerrepetition, its tetrachords come to serve as referential sonorities.The first hint of the complex occurs near the end of the first-movement exposition, which presents each row of the homeregion, in the order P3, RI8, R3, and I8. Example 13(a) showsthe concluding measures of the realization of row I8. The final

Measure Region Reference

647 P9/I2 (home) I, 1; III, 492

661 P0/I5 III, 572 666 I, 6; II, 266 669 II, 266 672 P7/I0 II, 270 677 P2/I7 III, 492 681 P5/IA

685 PB/I4

689 P3/I8

692.5 I 695.5 P2

696.5 P1

697.5 RI9

698.5 RIB

699.75 P6

701 IB

702.5 I3 I 704.75 P9/I2 (home) I, 1 708 PA/I3 III, 492 710 P1/I6

712 P8/I1

715 P4/I9

716 P9/I2 (h2 of home)

example 12. Summary of the cadenza.

12 The preceding work, the Ode to Napoleon, Op. 41, also makes heavy use ofoctave doublings.

13 Rothstein (1980), Mead (1989), and Alegant and McLean (2001) discussthe relationship between the row’s intervallic profile and the succession ofregions in the first movement; Alegant and McLean discuss the

enlargement schemes in the movement; Mead offers a close reading ofthe exposition and the nestings in the recapitulation.

14 The String Trio is thought to have a program as well, one with references toSchoenberg’s near-fatal heart attack.

having the last word: schoenberg and the ultimate cadenza 123

dyad, {C,E}, is accompanied by three tetrachords, which welabel [1], [2], and [3]. As Example 13(b) shows, these sametetrachords return in the final measures, fortissimo and resonant.Here, the piano projects the initial (h1) hexachords of P3 and I8

in strict inversion, combining their final dyads in an instance ofset class 4–20[0158]. The hexachords are punctuated byorchestral realizations of [1] and [2] that are enhanced by octavedoublings. Midway through the first movement, the dyadiccomplex is realized with a specific rhythmic motive with a short-medium, short-long pattern, which we call motive x. The manyrepetitions of x throughout the first three movements lend it thestatus of a leitrhythm. Example 13(c) shows an early manifes-tation of x; Example 13(d) shows x harmonized by the tetra-chords in the second half of the PA/I3 complex; and Example13(e) shows x applied to the P6/IB and P1/I6 complexes in thepassage immediately prior to the orchestral cadenza. As was thecase in the Violin Concerto, Schoenberg typically moves ina linear fashion through either a half or a whole dyadic complex.The passage in Example 13(e) realizes a palindromic design,punctuating statements of [1], [3], and [5] with tremolos of [2],[4], and [6]. With the re-articulation of [6] the progression isreversed, as follows:

[2] [4] [6]–[6] [3] [1]

[1] [3] [5]-[5] [4] [2]

The second half of m. 315 is a variation of material in m.313. Schoenberg modulates to another complex, inverts themelodic presentation of x, and repositions the tremolos in thebass register. Observe the subtle variations of like sonorities[3] and [6]: the realization of [3] articulates two ic 4s, B�–Dand F–A, and then crescendoes, whereas the realization of[6] highlights ic 1 (B�–A) and ic 3 (D–F) and fades awayto ppp.

The trichordal complex follows a trajectory similar to itsdyadic counterpart: it is introduced in the first movement,developed in the second, and brought to a culmination in thethird. It also assumes a wide variety of configurations. Example14(a) shows an early formation of the trichordal complex:a condensed realization, pitting the trichords of IA in the pianoagainst the trichords of P5 in the orchestra. Example 14(b)shows a later episode in the same movement. Here, the pianojourneys through the entire complex in a sentential design, ex-ploiting the rhythms of x. A significant aspect of this passage isthe presence of 6-32[024579] hexachords. These diatonic, all-combinatorial sonorities are produced by the conflation of p4and i4 in the right-hand ‘‘trills’’ and the pairing of p3 and i3 inthe left-hand statements of x. Example 15(c) skips ahead toa codetta passage in the second movement (Molto Allegro). Thepitch-class material of this excerpt incorporates the latter halvesof two complexes. Measures 259–61 unfold the elements of thePA/I3 complex, with p4 and i4 in the piano and p3 and i3 in theorchestra. A textural change in m. 262 announces a modulationto the P4/I9 complex, which lies a tritone away from the previous

complex. Here, too, the diatonic hexachords are clearly exposed.Example 14(c) highlights the invariant 6-32[024579] hexa-chords of the T6-related complexes.

Example 15 traces the histories of x and the complexes in thefirst three movements (there is little to mention about them inthe fourth movement, Giocoso, save for a final presentation ofthe dyadic complex in the final measures). The example(though not drawn to scale) is designed to show the frequentpairings of motive x with the elements of the dyadic and tri-chordal complexes, and the two-part design of the Adagio,each part comprising an interlude, an intensification, anda cadenza.

There is one more passage to consider before turning to theorchestral cadenza. Example 16 shows the opening of the solopiano cadenza (mm. 286–87). The pitch-class material for thisexplosive outburst is based on the trichords of the P2/I7 complex.Three features of the piano’s cadenza are directly imported intothe orchestral cadenza. The first of these is the tempo markingPiu largo (quarter¼44); this tempo is reserved exclusively forthe two cadenzas. The second feature is the trichordal andhexachordal structuring on the surface, courtesy of the trichordalcomplex. The third feature is the melody projected by the highestpitches of the trichordal cross partition: hA�

6,F6,A5,B5,B�5, . . . i.

The orchestral cadenza divides into two sections on the basisof partitioning schemes, dynamics, texture, and orchestration.We examine each section in turn, paying particular attention tothe set-class and pitch-class configurations of the aggregatepartitions. Example 17 provides an annotated reduction of theinitial two measures, which are based on the P7/I0 complex. Inthe lower staff, the cellos and basses restate the initial tune of thepiano’s cadenza, transposing it down three octaves and a perfectfifth. In terms of derivation, the tune takes one note from each ofthe segmental trichords of I0; the remaining notes of the tri-chords are distributed to the bassoons. The pitch classes of themelody are highlighted in the reduction at the bottom of theexample. In the upper system, the trombones and lower stringsproject the segmental trichords of P7 as simultaneities; the addedpercussion suggests a military topic and a sense of the macabre.

The next passage, shown in Example 18, brings a change incharacter, texture, and hexachordal region. The soundscapebecomes introverted and dolce, despite the increased rhythmicactivity and the octave doublings. The pitch-class material forthis subsection is derived from the P1/I6 complex, which is re-produced at the bottom of the example. At first the horn de-velops the Hauptstimme by taking single notes from differenttrichords; then it articulates segments of the complex, projectingi4 and p4 in retrograde inversion. The flute, clarinet, horn, andone set of strings combine i1 and i2, and i3 and i4, while thepiano, whose left hand is reinforced by the other group ofstrings, alternates p1 and p2, and p3 and p4.

Example 19 shows the third part of the first section. Theenergy builds dramatically: the harmonic rhythm further accel-erates, the texture becomes thicker still, and the diatonic [027]trichords and [024579] hexachords come to the fore. The sus-tained chord before the fermata is the climax of the entire

124 music theory spectrum 34 (2012)

example 13. A brief history of dyadic complexes in Op. 42.

having the last word: schoenberg and the ultimate cadenza 125

example 13. [Continued]

126 music theory spectrum 34 (2012)

example 14. A brief history of trichordal complexes.

having the last word: schoenberg and the ultimate cadenza 127

Andante (mm. 1−175) Recapitulation 36−39 62 86 92 97 101 117 126 133 155 160 163 166

DC * * x * * * * * * * * * * * * * * ********** * * * *TC * * * * * * * * * * * * * * * ***** * * * *

Molto Allegro (mm. 176−263) 176 182 186 190 193 212 218 223 235 → 245 249 →263

DC * * * * * * ******** x * * * * * * * * * * * * * * * * * * * * *** * TC * * * * * * * * * * * *

Adagio (mm. 264−329) Interlude Build-up Piano cadenza Interlude Catastrophe Orchestral cadenza

264 284 286 → 301 302 313 319 325−29 DC * * * * * * * * * * x * * * * * * * *TC * * *********** * * * *

example 15. Summary of the dyadic and trichordal complexes and motive x.

example 16. Opening of the solo piano cadenza (mm. 286–87).

128 music theory spectrum 34 (2012)

example 17. Section 1, Part 1 of the orchestral cadenza (mm. 319–20).

example 18. Section 1, Part 2 (mm. 321–22).

having the last word: schoenberg and the ultimate cadenza 129

concerto. A closer look at the pitch-class and set-class structurereveals that each quarter note projects the 6-32[024579]sonorities from the second halves of different trichordal com-plexes. The piano, brass, and strings combine p3 and i3 whilethe woodwinds combine p4 and i4. As we can see from thebottom of the example, the pairs of complexes are related by T6,which preserves the (unordered) hexachords. To designate thisinvariance, we use the labels h1 and h2 to denote the [024579]collections of the P1/I6 and P7/I0 complexes, and the labels h3 andh4 to denote the collections of the P2/I7 and P8/I1 complexes.

The surface changes dramatically after the diatonic outburst.Example 20 shows the first part of the second section of thecadenza. Having exhausted the materials of the trichordal com-plex, Schoenberg now fashions a mini-cadenza for solo piano. Butunlike the earlier solo cadenza, which was extroverted, angular,

and trichordal, this mini-cadenza is introverted, grazioso, anddyadic. It is an escape from reality, and a respite from the angstsaturating the Adagio. The soft dynamics and thin texturessuggest an air of fragility and delicacy; the rapid shifts in tempoand free-flowing, unmeasured rhythms have the feeling ofimprovisatory fantasy. Tremolos alternate with written-out burstsof figuration that often exhibit horizontal or vertical symmetry(palindromic pitch structures or axial inversion). But, uncharac-teristically, the elements of the dyadic complex do not unfold ina linear fashion; rather, the almost haphazard progression ofsonorities partially obscures the complex’s tetrachordal harmonies.It is only in the final section of the cadenza that the dyads arerealigned and a sense of linearity is restored.

Example 21 gives the conclusion of the orchestral cadenza.As before, the example provides the pitch-class contents of the

example 19. Section 1, Part 3 (mm. 323–25).

130 music theory spectrum 34 (2012)

example 20. Section 2, Part 1 (mm. 325–26).

example 21. Section 2, Part 2 (conclusion, mm. 327–29).

having the last word: schoenberg and the ultimate cadenza 131

complex; a dotted line separates the discrete complexes. Thereappearance of motive x heralds a return to reality (see the firsttwo gestures in m. 327), recapitulating music heard immediatelybefore the cadenza. At the same time, the dyadic complex re-surfaces, with note-against-note presentations of [6] and [5], and[4] and [3] in strict pitch inversion. The inversional symmetrydissolves at the fB,D�g dyad in the left hand, and the dyads aresubsequently staggered. The fC,Eg dyad, which belongs to theP8/I1 and P3/I8 complexes, functions as a kind of pivot. Belowrepetitions of this fC,Eg dyad in the right hand, the left handleisurely unfolds the elements of row RI8. In what is admittedlya subtle instance of hidden repetition, the tail of the left-handmelody is retrograded to begin the Giocoso.15

By way of summary, Example 22 models the formal structureof the ultimate cadenza. It shows the T6-related regions of tri-chordal complexes and cross partitions in the first part, thedyadic complexes and note-against-note presentations in thesecond part, and the pervasive influence of x. The outer-right

column traces the history of several partitioning strategies thatare integrated into the cadenza.

final considerations

It should come as no surprise that both of Schoenberg’stwelve-tone concertos have strong ties to the Romantic andClassical traditions, despite their serial language. The ViolinConcerto clearly echoes the models of the past, with its three-movement structure, while the Piano Concerto bears a closeresemblance to Liszt’s Piano Concerto No. 2, with its cyclicthemes that transform from a lyrical presentation to a marchover the span of the work. Orchestral interjections in cadenzas,especially ultimate cadenzas, are found in the last movement ofSchumann’s Cello Concerto and Brahms’s Violin Concerto.Perhaps the most vivid precursor to Schoenberg’s final cadenzain Op. 36 is the finale to Elgar’s Violin Concerto, in which anextended summary and review of the entire work is presented asa cadenza with orchestral accompaniment. A striking feature ofboth compositions is the way in which the action of the lastmovement grinds to a halt in order to recall the more contem-plative moments of previous movements.

First section: trichordal complexes

Measures Region Material Precedent Culmination

319−20 P7/I0 Tune of the piano cadenza III, 286−87 Cross partitions I, 126−31; 133−36

I, 142−43; 155−57; 163−64 II, 176−79; 184−85 III, 284−85 III, 285−88; 295−96 321−22 P1/I7 Tune of the piano cadenza III, 288−89 323−24.5 P1/I6, P7/I0 [027] and [024579] III, 251−52; 255−63 Trichords P2/I7, P8/I1 [024579]

Second section: dyadic complexes

Measures Region Material Precedent Culmination

324.5−26 P8/I1, P6/IB Fantasy (new material) (none) 327−28.5 P8/I1 Rhythmic motive x I, 86−90; 97−100; 117−31; 160−62; 164−67 II, 176−83; 192; 215−19 III, 311−12

x, harmonized III, 313−18 Dyads, xComplex as chorale I, 36−39; 62; 132

II, 182−83; 191−92; 235−42 IV, 490−92 (end) 328.5− 29 P3/I8 (home) Dénouement, retransition

example 22. Formal overview of the cadenza.

15 Those concerned with ‘‘twelve counting’’ the surface will detect anirregularity in the ultimate dyad of the cadenza: the D should be an A.Regardless of whether this is a ledger-line error, we can offer no compellingreason as to why D is paired with F� rather than A.

132 music theory spectrum 34 (2012)

example 23. Dyadic complexes in selected works.

Pitch classes (unordered) Prime forms (ordered) Forte/Rahn Invariance

1) 0312 4e5t 6978 0123 0123 0167 1 1 9 T6

2) 0312 4e78 5t69 0123 0145 0347 1 7 17 3) 0312 4e69 5t78 0123 0235 0257 1 10 23 M, OZ 4) 034e 1278 5t69 0145 0145 0167 7 7 9 T65) 034e 1269 5t78 0235 0145 0158 7 10 20 6) 0369 1278 4e5t 0167 0167 0369 9 9 28 T6, M, OZ 7) 0378 1269 4e5t 0167 0158 0158 9 20 20 T68) 035t 1278 4e78 0167 0257 0257 9 23 23 T69) 0369 124e 5t78 0235 0235 0369 10 10 28 T610) 0369 125t 4e78 0347 0347 0369 17 17 28 T611) 035t 1269 4e78 0347 0158 0257 17 20 23 M

4B5A4B78 5A694B69 5A78

034B 5A69034B 5A78

4B5A4B5A

035A 4B78124B 5A78125A 4B78

035A 4B78

example 24. The universe of the dyadic complex.

having the last word: schoenberg and the ultimate cadenza 133

Concertos offer particularly vivid ground for studying therelationship between our immediate experience of the music andour sense of its dramatic structure. Our appreciation of thedrama in a concerto involves many factors, including theinteraction of the soloist and orchestra, the handling of soloentrances and departures, the placement of orchestral tuttis, andthe incidence and content of cadenzas, both solo and accom-panied.16 Larger questions regarding dramatic structure arisewhen we consider the ways in which motives emerge and recurin solo or orchestral passages. In this light it is interesting toconsider the aftermath of the cadenzas.

The final cadenza of Op. 36 is vast—and necessarily so,because it integrates the main tunes of each movement witha large cast of tetrachordal motives.17 But the dyadic complexand its 4-1[0123], 4-5[0145], and 4-17[0347] tetrachords playonly subsidiary roles. In contrast, the final cadenza of Op. 42 isa mere eleven measures long, yet still manages to place a spot-light on the dyadic and trichordal complexes. In Op. 36, thepassages immediately following the final cadenzas of the firstand third movements are quite brief; thus they function ascodas—to the first movement and to the work as a whole. In Op.42, the orchestral cadenza occurs at the juncture between thethird, slow movement of the work and the finale. As a result, thepiano concerto’s concluding movement does not carry the samekind of musical or dramatic weight as the earlier movements;it serves as a coda to the entire work rather than as a newdeparture.18 The point is that in both of these compositions, theact of summing up is essentially done. All that is left is rhetoricalflourish.

appendix: reconsidering the dyadic complex

The Violin and Piano Concertos are not the only American-period compositions to evoke dyadic complexes; we can also findthem in the Fourth Quartet, Op. 37, and the Phantasy for Violinand Piano, Op. 47. Example 23 compares the dyadic complexesof these works. The complexes in Opp. 36 and 42 appear in (a)and (b); the complexes of Opp. 37 and 47 appear in (c) and(d).19 Note that each complex generates a unique collection oftetrachordal set classes, all of which are distinct from the discretesegments of the source rows. Additionally, note that the com-plexes in (a) and (c) generate three set classes and are invariantunder T6, while the complexes in (b) and (d) produce five dis-tinct set classes.

It is an easy task to enumerate and classify the universe of dyadiccomplexes. Let us take as a point of departure the dyads of indexnumber 3. Every odd index number has six pairs of odd intervalclasses; we represent these with the letters a through f:

f0,3g f1,2g f4,Bg f5,Ag f6,9g f7,8ga b c d e f

Any inversional setting that is based on an odd index numberwill (by definition) project certain pairs of these dyads. Toillustrate, the pitch-class representation below shows a designfrom the Violin Concerto using these dyads. The first threetetrachords combine dyads a–c, b–d, and e–f ; the second threetetrachords combine a–b, d–e, and c–f.

PB: h B,0 5,1 6,8 2,3 9,A 4,7 iI4: h 4,3 A,2 9,7 1,0 6,7 B,8 i

aþc bþd eþf aþb dþe cþf

But if we reorder these rows we obtain different tetra-chords—which leads us to ask: How many distinct tetrachordal

Set class ICV Degree of symmetry M-invariant 4-1[0123] 2 4-7[0145] 2 4-9[0167] 4 * 4-10[0235] 2 * 4-17[0347] 2 * 4-20[0158] 2 4-23[0257] 2 4-28[0369] 8 *

321000

201210

200022

122010

102210

101220

021030

004002

example 25. A closer look at the tetrachords in the dyadic complex.

16 They further involve considerations of the presentational role of the soloistin a given passage: a soloist may lead, comment, add figuration, oraccompany. Timbre and register are other factors: the violin onlyoccupies the music’s upper register, but can claim timbral membership inthe ensemble; conversely, the piano can compete with the orchestrathroughout its full range, although it remains a timbral outsider.

17 By ‘‘vast’’ we mean that the cadenza is sixty-one measures in length:precisely one-twelfth of the whole.

18 This makes sense in light of the programmatic inscription: ‘‘but life goeson.’’

19 Most of the Phantasy is based on the presentation of unordered hexachordsand trichordal complexes. Measures 110–16 represent the only clearassertion of the dyadic complex in the work. The analyses by Lewin(1967) and Lester (2000) mention neither this passage nor the influenceof the dyadic complex.

134 music theory spectrum 34 (2012)

arrangements are possible? Below are listed the fifteen ways todivide six elements into groups of three.20

ab cd ef ab ce df ab cf de

ac bd ef ac be df ac bf de

ad bc ef ad be cf ad bf cd

ae bc df ae bd cf ae bf cd

af bc de af bd ce af be cd

As it turns out, four of these arrangements relate to each otherunder Tn/TnI, which leaves eleven distinct formations. Example24 lists these configurations along with their (unordered) pitch-class collections, set classes, and invariance properties.21 From thisperspective, the complex in the Violin Concerto contains twoversions of #2, the complex in the Piano Concerto contains #5and #11, the complex in the Fourth Quartet combines #1 and #8(both of which are invariant under T6), and the complex in thePhantasy combines #1 and #11.

We can generalize further and consider the universe ofsymmetrical tetrachords arising in complexes that are nothexachordally combinatorial. In any design that is based on anodd index number, only eight symmetrical tetrachords canensue—whether the underlying harmonic context is twelve-tone, atonal, or octatonic. Example 25 provides a closer look atthese tetrachordal set classes. Each tetrachord contains pairs ofthe same interval class, and can map into itself under at least oneodd index number. (The Degree of Symmetry reflects thenumber of T and I operations that map a collection into oronto itself; M invariance indicates the ability of a collection tomap onto itself under the multiplicative operators, M5 or M7.M-invariance occurs when a collection has the same number ofinterval-class 1 and 5 in its ICV.) Along with the five sym-metrical trichords ([012],[024],[027],[036],[048]), and the sixdyads ([01], [02], . . . ,[06]), these collections define the har-monic universe of odd index numbers.22

works cited

Alegant, Brian. 1993. ‘‘The 77 Partitions of the Aggregate:Analytical and Theoretical Implications.’’ Ph.D. diss.,University of Rochester.

———. 1996. ‘‘Unveiling Schoenberg’s Op. 33b.’’ Music TheorySpectrum 18 (2): 143–66.

———. 1999. ‘‘When Even Becomes Odd: A PartitionalApproach to Inversion.’’ Journal of Music Theory 43 (2):190–230.

———. 2001. ‘‘Cross Partitions as Harmony and Voice Leadingin Twelve-Tone Music.’’ Music Theory Spectrum 23 (1): 1–40.

———. 2002–3. ‘‘Inside the Cadenza of Schoenberg’s PianoConcerto.’’ Integral 16–17: 67–102.

———. 2010. The Twelve-Tone Music of Luigi Dallapiccola.Rochester: University of Rochester Press.

Alegant, Brian, and Don McLean. 2001. ‘‘On the Nature ofEnlargement.’’ Journal of Music Theory 45 (1): 31–71.

Babbitt, Milton. 1960. ‘‘Twelve-Tone Invariants asCompositional Determinants.’’ The Musical Quarterly46(2): 246–59. Repr. in The Collected Essays of MiltonBabbitt. Ed. Stephen Peles with Stephen Dembski,Andrew Mead, and Joseph N. Straus. 55–69. Princeton:Princeton University Press, 2003.

———. 1961. ‘‘Set Structure as a Compositional Determinant.’’Journal of Music Theory 5 (1): 72–94. Repr. in The CollectedEssays of Milton Babbitt. Ed. Stephen Peles with StephenDembski, Andrew Mead, and Joseph N. Straus. 86–108.Princeton: Princeton University Press, 2003.

Dembski, Stephen, and Joseph N. Straus, eds. 1987. MiltonBabbitt: Words about Music. The Madison Lectures.Madison: University of Wisconsin Press.

Forte, Allen. 1973. The Structure of Atonal Music. New Haven:Yale University Press.

Kurth, Richard. 1992. ‘‘Mosaic Polyphony: Formal Balance,Imbalance, and Phrase Formation in the Prelude ofSchoenberg’s Suite, Op. 25.’’ Music Theory Spectrum 14 (2):188–208.

Lester, Joel. 2000. ‘‘Analysis and Performance in Schoenberg’sPhantasy, Op. 47.’’ In Pianist, Scholar, Connoisseur: Essays inHonor of Jacob Lateiner. Ed. Bruce Brubaker and JaneGottlieb. 151–74. Stuyvesant [NY]: Pendragon Press.

Lewin, David. 1962. ‘‘A Theory of Segmental Association inTwelve-Tone Music.’’ Perspectives of New Music 1 (1):89–116.

———. 1967. ‘‘A Study of Hexachord Levels in Schoenberg’sViolin Phantasy.’’ Perspectives of New Music 6 (1): 18–32.

———. 1968. ‘‘Inversional Balance as an Organizing Force inSchoenberg’s Music and Thought.’’ Perspectives of New Music6 (2): 1–21.

Martino, Donald. 1961. ‘‘The Source Set and Its AggregateFormations.’’ Journal of Music Theory 5 (2): 224–73.

Mead, Andrew. 1983–84. ‘‘Pitch Structure in Elliott Carter’sString Quartet No. 3.’’ Perspectives of New Music 22 (1–2):31–60.

———. 1985. ‘‘Large-Scale Strategy in Arnold Schoenberg’sTwelve-Tone Music.’’ Perspectives of New Music 24 (1):120–57.

———. 1988. ‘‘Some Implications of the Pitch Class/OrderNumber Isomorphism Inherent in the Twelve-Tone

20 Alegant (1999) enumerates the inventories of even and odd index numbersin a similar ‘‘bottom up’’ fashion.

21 Two points: first, ‘‘OZ,’’ discussed in Mead (1988), rotates the pitch classesof a collection through either the even or the odd whole-tone scale, leavingthe pitch classes of the other scale untouched. To illustrate, an ‘‘odd’’application of OZ on f0,1,2,3g yields f0,2,3,5g: 1 moves to 3, 3 movesto 5, and 0 and 2 are unchanged. Second, this view of the dyadic complexhighlights the ways in which a whole-tone collection can be partitioned intodiscrete ic 2, ic 4, and ic 6 dyads: we can have two ic 2s with an ic 6; two ic 4swith an ic 6; three ic 6s; three ic 2s, and two ic 4s with an ic 2.

22 These set classes are a subset of a larger class of tetrachords generated bypairs of interval classes. These in turn are themselves a non-trivial subset ofall those types whose constituent tetrachords can map onto themselvesunder inversion by some index number, even or odd—but such aninquiry is beyond the scope of this essay.

having the last word: schoenberg and the ultimate cadenza 135

System: Part One.’’ Perspectives of New Music 26 (2):96–163.

———. 1989. ‘‘Twelve-Tone Organizational Strategies: AnAnalytical Sampler.’’ Integral 3: 93–169.

———. 1993. ‘‘ ‘The Key to the Treasure . . . ’ .’’ Theory andPractice 18: 29–56.

Morris, Robert D., and Brian Alegant. 1988. ‘‘The EvenPartitions in Twelve-Tone Music.’’ Music Theory Spectrum10: 74–101.

Peles, Stephen. 1983–84. ‘‘Interpretations of Sets in MultipleDimensions: Notes on the Second Movement of ArnoldSchoenberg’s String Quartet #3.’’ Perspectives of New Music22 (1–2): 303–52.

———. 2000. ‘‘Schoenberg and the Tradition of ImitativeCounterpoint.’’ In Music of My Future: The SchoenbergQuartets and Trio. Ed. Reinhold Brinkmann and ChristophWolff. 117–38. Cambridge [MA]: Harvard University Press.

Rahn, John. 1980. Basic Atonal Theory. New York: Longman.Rothstein, William. 1980. ‘‘Linear Structure in the Twelve-

Tone System: An Analysis of Donald Martino’sPianississimo.’’ Journal of Music Theory 24 (2): 129–65.

Samet, Sydney Bruce. 1985. ‘‘Hearing Aggregates.’’ Ph.D. diss.,Princeton University.

Starr, Daniel. 1984. ‘‘Derivation and Polyphony.’’ Perspectives ofNew Music 23 (1): 180–257.

Music Theory Spectrum, Vol. 34, Issue 2, pp. 107–136, ISSN 0195-6167,electronic ISSN 1533-8339. © 2012 by The Society for Music Theory. Allrights reserved. Please direct all requests for permission to photocopy orreproduce article content through the University of California Press’sRights and Permissions website, at http://www.ucpressjournals.com/reprintinfo.asp. DOI: 10.1525/mts.2012.34.2.107

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