locanto - stravinsky's late technique

46
massimiliano locanto ‘Composing with Intervals’: Intervallic Syntax and Serial Technique in Late StravinskyTranslated by Chadwick Jenkins The use of intervals as the basic material of musical construction consistently served as an important and deeply personal characteristic of Stravinsky’s com- positional process. In his final serial compositions, however, this aspect assumed a more decisive role and underwent significant changes. Stravinsky’s late writ- ings, composed in collaboration with Robert Craft, reflect this renewed interest in intervallic construction. In them, the composer repeatedly describes the first stages of the creative process as ‘work with intervals’ and even projects this practice backwards to cover the entirety of his musical output. Stravinsky thus emphasises, perhaps excessively, the continuity of his thinking despite the evident changes that took place throughout his compositional career. 1 The new and more important role occupied by the intervallic component in Stravinsky’s serial compositions is, to a large extent, a consequence of the gradual abandonment of pitch collections (octatonic, whole-tone, diatonic, and so on), 2 which had played such a decisive role in his earlier music, in favour of a growing tendency to make systematic use of twelve-note aggregates – a ten- dency accentuated particularly in the compositions which follow Threni. 3 Although Stravinsky continued to prefer the same pc sets (particularly tetra- chords) which in his earlier compositions were derived from diatonic or octatonic collections, the sketches for his last compositions seem to indicate that the creative process has moved from specific intervals to larger combinations (and not vice versa), thereby producing harmonic environments which can appear variously diatonic, octatonic or chromatic. By freeing the treatment of intervals from a broader system of pitch organi- sation, 4 Stravinsky seems to have followed a path which presents some similari- ties to – but also some profound differences from – the predominantly motivic pathway followed by Schoenberg and his pupils in their gradual departure from traditional tonality. Although all of these composers treated the intervallic com- ponent in different ways, it became for them the foundational aspect of a ‘motivic’ technique – that is, one based on the use of a restricted number of intervallic configurations which serve a basic unifying function within a musical work.Yet the specific way in which Stravinsky used intervallic motives emerges through a study of his many unpublished versions of pieces, and these must be evaluated in relation to the peculiarities of his aesthetic. DOI: 10.1111/j.1468-2249.2011.00302.x Music Analysis, 28/ii-iii (2009) 221 © 2011 The Author. Music Analysis © 2011 Blackwell Publishing Ltd, 9600 Garsington Road, Oxford OX4 2DQ, UK and 350 Main Street, Malden, MA 02148, USA

Upload: 1111qwerasdf

Post on 21-Oct-2015

75 views

Category:

Documents


6 download

DESCRIPTION

Intervallic Syntax and SerialTechnique in Late Stravinsky

TRANSCRIPT

Page 1: Locanto - Stravinsky's Late Technique

massimiliano locanto

‘Composing with Intervals’: Intervallic Syntax and Serial

Technique in Late Stravinskymusa_302 221..266

Translated by Chadwick Jenkins

The use of intervals as the basic material of musical construction consistentlyserved as an important and deeply personal characteristic of Stravinsky’s com-positional process. In his final serial compositions, however, this aspect assumeda more decisive role and underwent significant changes. Stravinsky’s late writ-ings, composed in collaboration with Robert Craft, reflect this renewed interestin intervallic construction. In them, the composer repeatedly describes the firststages of the creative process as ‘work with intervals’ and even projects thispractice backwards to cover the entirety of his musical output. Stravinsky thusemphasises, perhaps excessively, the continuity of his thinking despite theevident changes that took place throughout his compositional career.1

The new and more important role occupied by the intervallic component inStravinsky’s serial compositions is, to a large extent, a consequence of thegradual abandonment of pitch collections (octatonic, whole-tone, diatonic, andso on),2 which had played such a decisive role in his earlier music, in favour ofa growing tendency to make systematic use of twelve-note aggregates – a ten-dency accentuated particularly in the compositions which follow Threni.3

Although Stravinsky continued to prefer the same pc sets (particularly tetra-chords) which in his earlier compositions were derived from diatonic or octatoniccollections, the sketches for his last compositions seem to indicate that thecreative process has moved from specific intervals to larger combinations (andnot vice versa), thereby producing harmonic environments which can appearvariously diatonic, octatonic or chromatic.

By freeing the treatment of intervals from a broader system of pitch organi-sation,4 Stravinsky seems to have followed a path which presents some similari-ties to – but also some profound differences from – the predominantly motivicpathway followed by Schoenberg and his pupils in their gradual departure fromtraditional tonality. Although all of these composers treated the intervallic com-ponent in different ways, it became for them the foundational aspect of a‘motivic’ technique – that is, one based on the use of a restricted number ofintervallic configurations which serve a basic unifying function within a musicalwork.Yet the specific way in which Stravinsky used intervallic motives emergesthrough a study of his many unpublished versions of pieces, and these must beevaluated in relation to the peculiarities of his aesthetic.

DOI: 10.1111/j.1468-2249.2011.00302.x

Music Analysis, 28/ii-iii (2009) 221© 2011 The Author.Music Analysis © 2011 Blackwell Publishing Ltd, 9600 Garsington Road, Oxford OX4 2DQ, UKand 350 Main Street, Malden, MA 02148, USA

Page 2: Locanto - Stravinsky's Late Technique

All of this raises an interesting issue: although in so-called atonal music the(presumable) absence of a hierarchy among the sounds allows the composer toemploy all twelve pitch classes as he pleases (considering them exclusively interms of their intervallic relationships),5 with the adoption of serial procedures anew constructive order is imposed. Now, while in the work of other composersone could say that this new constructive order encompasses and essentiallyidentifies with motivic-intervallic syntax, in Stravinsky the result of such identi-fication is instead rather problematic, because these two aspects operate accord-ing to slightly – but significantly – different criteria. In the following pages, Iwill attempt to demonstrate how, particularly in the compositions from Agononwards, this approach provided Stravinsky with a stimulus, rather than anobstacle, to composition. I will also attempt to interpret some characteristics ofStravinsky’s creative process and serial technique which are by now well knownbut whose deeper motivations still require further investigation.

Motivic-Intervallic Syntax: General Characteristics

Even if it is evident that the use of intervals as the basic material of thecompositional process constitutes a central feature of Stravinsky’s late musicalthought, the specific technical means employed lend themselves to beingdescribed in rather different ways.6 Stravinsky never specified exactly what hemeant by the expression ‘composing with intervals’, nor does the study of hissketches offer any definitive answers. A great deal of his work with pitch material,in fact, took place at the keyboard, in a phase prior to that documented in theearliest sketches, which in actuality record a stage in the creative process that isalready quite advanced.7 Given this modus operandi one can easily imagine thatStravinsky’s interval-based procedures were not codified according to any kindof systematic approach.This, however, does not exclude the possibility that suchprocedures can be described retrospectively in theoretical terms.

To this end, I will use as a brief first example the original twelve-note row ofA Sermon, a Narrative, and a Prayer (Ex. 1).The row can be subdivided into fourdistinct trichords, three of which – the first, second and fourth – belong to setclass [014]. Segments 2 and 4 are ordered as <0, 1, 4>, while segment 1 isordered as <1, 0, 4>.8 From an intervallic point of view, if we consider theintervals apart from their melodic direction (ascending or descending), that is, asunordered pitch-class intervals (interval classes), all three segments contain a

Ex. 1 Stravinsky, A Sermon, a Narrative, and a Prayer: subset structure of the originaltwelve-note row

[014] [014] [014][015]

1 1 1 334

3 4 4

222 massimiliano locanto

Music Analysis, 28/ii-iii (2009)© 2011 The Author.Music Analysis © 2011 Blackwell Publishing Ltd

Page 3: Locanto - Stravinsky's Late Technique

semitone (ic1), a minor third (ic3) and a major third (ic4).9 In consequence, wecould describe the three segments as three statements, differently ordered, of thesame group of three interval classes, or equally well as three statements, differ-ently ordered, of the same set class [014].

Considering the segments of the row as unordered sets corresponds to aconstructive logic which, far from being exclusive to Stravinsky, seems deeplyrooted in most twelve-note and serial music in general and is certainly veryfamiliar from the published literature on serialism. Many of the basic operationswhich concern the subset structure of the twelve-note row are based on thislogic. In Schoenberg, Berg and Webern, the rows are often organised in such away as to maximise certain segments which, if considered apart from the orderof the pitches, belong to the same set class. To mention only a couple ofexamples, one could cite the row of Schoenberg’s String Quartet No. 4, in whichone can identify four segments of three notes as belonging to set class [015](Ex. 2a),10 or the row employed in the twelve-note section of the third movementof Berg’s Lyric Suite, which contains four segments belonging to the class [0126](Ex. 2b).11

Furthermore, in many twelve-note compositions the idea of considering somesegments of the row as unordered sets constitutes an essential premise forestablishing various types of formal relationships. The internal structure of therow, in fact, allows some of its subsets to preserve the same global pitch contenteven after the typical transformational operations (transposition, retrograde andinversion) are applied.This gives rise to a network of relations among the variousforms of the row employed in a composition.12 The very notion of hexachordalcombinatoriality,13 which plays a fundamental role in Schoenberg’s twelve-notemusic, is based on the possibility of conceiving the hexachords as unorderedcollections.

The global intervallic content of these serial segments of course plays animportant role in enabling this type of relation from the moment that each

Ex. 2a Schoenberg, String Quartet No. 4: subset structure of the fundamentaltwelve-note row

[015] [015]

[015][015]

Ex. 2b Berg, Lyric Suite: subset structure of the twelve-note row of the thirdmovement

[0126] [0126]

[0126][0126]

‘Composing with Intervals’ 223

Music Analysis, 28/ii-iii (2009) © 2011 The Author.Music Analysis © 2011 Blackwell Publishing Ltd

Page 4: Locanto - Stravinsky's Late Technique

segment can be considered either as an unordered set of pitch classes or as anunordered set of interval classes. Particular attention to intervallic content formsthe basis of Webern’s practice of deriving a row from the reiteration of a singlebasic cell. To mention only one well-known example: in Cantata No. 1, Op. 29(Ex. 3),14 the four discrete trichords of the row are all members of the same setclass [014]. The importance of the global intervallic content of set class [014](semitone, minor third and major third) is underlined by the presence of thesemitone between the first and second segments and between the third andfourth segments, and by that of the minor third between the second and thirdsegments.

To summarise: the idea of globally considering the pitches and/or intervalscontained in some serial segments typically constitutes one of the basic con-structive criteria of twelve-note serialism. Nevertheless, this criterion corre-sponds only in part to the concept of intervallic motive which I hope toilluminate in the music of Stravinsky. Generally, the constituent pitches of amotive can be used in either a harmonic or a melodic sense,15 and compared inany order. Nevertheless, Stravinsky, working with the orientation of single inter-vals, radically modifies the physiognomy of his motives, which can therebyassume forms corresponding to different set classes. From this point of view,then, an intervallic motive no longer corresponds, in any sense, to a class ofunordered pc sets. Rather, Stravinsky’s operations act more on the level of singleintervals than on the level of the global configurations within which these singleintervals are included.

The difference becomes clear if we return to the third segment of the row ofA Sermon, a Narrative, and a Prayer (see again Ex. 1). This segment belongs notto set class [014], but rather to the class [015]. As a consequence, its globalintervallic content is different. However, this segment shares with the othersegments two of its three intervals (ic1 and ic4), which are merely arrangeddifferently (Ex. 4): in the third segment they are joined in the same direction,thereby producing an ic5; in the first, second and fourth segments they are joinedin opposed directions, thereby producing an ic3 (Ex. 4).16 In short, considered asunordered pc sets, only three of the four segments of the row of Ex. 1 turn outto belong to the same class; considered, however, as intervallic motives formedthrough the combination – in varying directions – of two intervals, they turn outto all be members of the same motive class (Ex. 5): the semitone and the majorthird conjoined, expressed symbolically as 1–4.17

Ex. 3 Webern, Cantata No. 1, Op. 29: subset and intervallic structure of the originaltwelve-note row

[014] [014] [014] [014]

O RI

1 3 1

224 massimiliano locanto

Music Analysis, 28/ii-iii (2009)© 2011 The Author.Music Analysis © 2011 Blackwell Publishing Ltd

Page 5: Locanto - Stravinsky's Late Technique

This simple alteration in intervallic direction constitutes a precious compo-sitional resource in Stravinsky’s hands. Joining various forms of a single (or atmost two) motivic class(es), he creates twelve-note rows – as in the previousexample – as well as smaller or larger successions of pitches to be employedeither melodically or harmonically in a musical passage. For instance, in Ex. 6 wecan observe the succession of pitches which serves as the basis for three episodes(bars 7–22) included in the first section (up to the prima volta) of the first of thefive Movements. This passage conceals a closely woven fabric of overlappingmotives of the semitone-tone (1–2) and semitone-tritone (1–6) types (indicatedby square brackets). Depending on the orientation assumed by the two intervals,the first motive (1–2) produces sets of three pitches belonging to set classes [012]and [013]. The second motive (1–6), on the other hand, produces collectionsbelonging to set class [016] regardless of the orientation assumed by the twointervals.18

It should be apparent that, in defining this type of intervallic syntax as motivic,the term ‘motive’ is being used with some degree of latitude. In the Formenlehretradition, for example, a motive is typically conceived as a structural nexus ofrhythm and intervals.19 In Stravinsky’s music, however, a motive is essentially anabstract configuration of intervals: pitch components and rhythm are treated asinitially distinct and separate dimensions which can subsequently be related.20 Assuggested above, in spite of some apparent similarities, this conception of inter-

Ex. 4 Intervallic motive class 1–4 in the two forms of set class [015] and [014]

5

1 1

00

ic4

(ic5)(ic4)

ic3

ic1ic1

4

1–4 motive

Ex. 5 Stravinsky, A Sermon, a Narrative, and a Prayer: motivic-intervallic structure ofthe original twelve-note row

[014] [014] [014][015]

1 1 1 334

3 4 44

5 1

E D B A

C B G

F

F

E D A4 4 4 4(3)(5)(3)(3)

1

1–4 motive

1 1 1

‘Composing with Intervals’ 225

Music Analysis, 28/ii-iii (2009) © 2011 The Author.Music Analysis © 2011 Blackwell Publishing Ltd

Page 6: Locanto - Stravinsky's Late Technique

vallic motive also differs radically from that familiarly applied to the post-tonalmusic of Schoenberg, Berg and Webern. For these composers, the concept ofmotivic elaboration which guaranteed coherence in tonal music was graduallyreplaced by a constructive principle based on the use of fundamental intervallicconstellations which operate at a more basic level. According to MartinaSichardt, this reduction of the various Gestalten within a passage to its mostelementary intervallic basis – a tendency Schoenberg himself had consciouslyput into practice in his own analytical formulations – represented a fundamentalpremise for the elaboration of the twelve-note method.21 In this respect, it isinteresting to note that most of the basic intervallic constellations which form theexpressive vocabulary of melodic gestures in Schoenberg’s compositions consistmerely of the union of two or three intervals – one of which is usually thesemitone – disposed in a particular arrangement.22

An interesting analogy with Stravinsky’s practice can nonetheless be glimpsedwherever Schoenberg subjects these basic combinations of intervals to a processof variation. Jack Boss, for example, has demonstrated that the majority of theintervallic motives in the first of Schoenberg’s Vier Lieder, Op. 22, could bederived by applying three types of modification to a motive formed from thecombination of one ic1 and one ic3.23 Boss considers all of the possible arrange-ments of these two intervals (that is, <+1, +3>, <+1, -3>, <-1, +3>, <-1, -3>, <+3,+1>, <+3, -1>, <-3, +1> and <-3, -1>) as variants belonging to the same motiviccategory. Moreover, each of these forms can undergo in its turn three funda-mental types of variation, two of which involve octave complementation andpitch reordering. All of this corresponds exactly to my definition of motiveclass 1–3. However, despite this similarity, a profound discrepancy remainsregarding the very concept of motive. Schoenberg’s procedures, as describedby Boss, ‘effectively identify motive as an entity which may be subjected to awide range of transformations while remaining largely recognisable’. Accordingto Boss, for example, the third basic category of variation employed by

Ex. 6 Stravinsky, Movements, i: motivic-intervallic structure of the succession ofpitches contained in each of the three solo episodes of bars 7–22

[012] [016] [016] [016] [016] [016] [016][012] [012][012] [016] [013] [013][016] [016] [016] [012][012]

[016] [016] [016] [016][012] [012] [013]

1–2 motive 1–6 motive

[013] [012] [016] [016]

1 1 1 1

2 6(3) [7] = 52

6(5)(1)

226 massimiliano locanto

Music Analysis, 28/ii-iii (2009)© 2011 The Author.Music Analysis © 2011 Blackwell Publishing Ltd

Page 7: Locanto - Stravinsky's Late Technique

Schoenberg involves the expansion of intervals.24 In this respect, the Schoenber-gian concept of variation implies a decisively greater quantity and varietyof forms derivable from a single motive than those which occur in Stravinsky.Still more important is the fact that the Schoenbergian concept of variationimplies a broader process, one which involves the entire plan of the musical form.Indeed, in Schoenberg, the variation of a motive cannot be dissociated fromthe concept, central to the Austro-German tradition, of motivic elaboration,understood as a means of conferring coherence and organic unity on a compo-sition.25 All of this is foreign to Stravinsky’s musical thought, in which themanipulation of intervallic motives is understood as a procedure for generatingprimary compositional material capable of being employed as a point of creativedeparture.

The idea – developed in particular through the work of George Perle26 – thata ‘basic cell’ or ‘referential sonority’ can be presented even in a vertical sensecould be considered another point of contact between Stravinsky’s motivic-intervallic syntax and the post-tonal harmonic language of Schoenberg, Bergand Webern. However, unlike the notion of the Stravinskian intervallic motive,the concept of the basic cell consists of a fixed configuration of intervals andcorresponds therefore to a single set class.27 The most decisive difference,however, concerns the contrasting aesthetic-musical aspects within which amotive unit is taken to function: in the music of Schoenberg, Berg and Webernan intervallic configuration disposed vertically always maintains a motivic char-acter – from which, in fact, the idea of ‘chord as motive’ arises – even in adynamic sense. The nature of this element is expressed by the Schoenbergianconcept of unrest:

What is a motive? A motive is something that gives rise to motion. A motion is thatchange in a state of rest, which turns it into its opposite. Thus, one can compare amotive with a driving force ... . What causes motion is a motor. One must distin-guish between motor and motive ... . A thing is termed a motive if it is already subjectto the effect of a driving force, has already received its impulse, and is on the verge ofreacting to it ... . The smallest musical event can become a motive if it is permittedto have an effect; even an individual tone can carry consequences. (Schoenberg1995, p. 386; emphases in original)

In the music of Schoenberg, the simultaneous presentation of pitches producedby an intervallic configuration can be considered the result of an extremeconcentration in time of an event whose essence is decidedly dynamic – tied,that is, to the movement of time. Therefore, if Schoenberg’s motivic conceptionis essentially temporal – the very idea of a suppression of temporality associatedwith the Schoenbergian ‘law’ of the unity of musical space implies in actualitythe concept of time – the Stravinskian approach conversely returns to a con-ception we may define as ‘spatial’ or ‘plastic-visual’: the motive is understood asa configuration of intervals which can be arranged in two dimensions, as invisual space.

‘Composing with Intervals’ 227

Music Analysis, 28/ii-iii (2009) © 2011 The Author.Music Analysis © 2011 Blackwell Publishing Ltd

Page 8: Locanto - Stravinsky's Late Technique

Theoretical Aspects

Before examining some specific examples, a number of purely theoretical con-siderations are worth reviewing in greater detail. As mentioned earlier, thepossibility of freely orienting any intervallic configuration ensures that a singlemotivic class corresponds to more than one set class.With respect to the motivesformed from two different intervals (the model Stravinsky most frequentlyemploys), each of the fifteen possible combinations generally produces twodifferent set classes, depending on the orientation of the two intervals conjoined.Only those motives containing the tritone produce a unique set class regardlessof intervallic orientation (see the left-hand side of Table 1).28

The right-hand side of Table 1, however, shows the motives which are capableof producing a particular set class of three pitches. For example: the set class[013] can be obtained by uniting one ic1 and one ic2 in the same direction(motive 1–2), or by uniting a single ic2 and a single ic3 in opposite directions(motive 2–3). As can be seen, according to the global intervallic content,29 eachset class can be produced by a variable number of up to three intervallic motives.Only set class [048], which contains three identical intervals (three ic4s), cannotbe produced by any motive formed from two different intervals.

The motives formed through the union of three different interval classes donot constitute an analytically relevant object since too many different set classesare generated. For example, a motive which combines a semitone, tone andminor third in whatever order and direction would produce ten different setclasses containing either three or four distinct pitch classes:30

Table 1 Motives formed by two different conjoined intervals

Motivic-intervallicclass

Set class(es)produced

Set class Associatedmotivic-intervallicclass(es)

1–2.............................................(012) (013) (012) ............................1–21–3............................................ (013) (014) (013) ............................1–2 1–3 2–31–4.............................................(014) (015) (014) ............................1–3 1–4 3–41–5.............................................(015) (016) (015) ............................1–4 1–5 4–51–6 .............................................(016) (016) ............................1–5 1–6 5–62–3.............................................(013) (025) (024) ............................2–42–4.............................................(024) (026) (025) ............................2–3 2–5 3–52–5.............................................(025) (027) (026) ............................2–4 2–6 4–62–6 .............................................(026) (027) ............................2–53–4.............................................(014) (037) (036) ............................3–63–5.............................................(025) (037) (037) ............................3–4 3–5 4–53–6............................................ (036) (048) ............................—4–5.............................................(015) (037)4–6 .............................................(026)5–6 .............................................(016)

228 massimiliano locanto

Music Analysis, 28/ii-iii (2009)© 2011 The Author.Music Analysis © 2011 Blackwell Publishing Ltd

Page 9: Locanto - Stravinsky's Late Technique

1–2–3: [013][023][0123][0124][0125][0134][0135][0136][0146][0236]

Moreover, the majority of these ten set classes can be derived from other motivescovering three different intervals. Thus every set class can be associated with anexcessively broad number of motives and vice versa.

However, motives that may be realised using only two interval classes, one ofwhich repeats once (for example, 1–2–1) to form a set of four pitches, arerelatively common. In Stravinsky’s case, motives of this type are typically thosewhich employ the semitone in conjunction with the whole tone or perfect fifth(1–2–1, 2–1–2, 1–5–1 and 5–1–5). As can be seen on the left-hand side ofTable 2, these motives produce only two or three different set classes, accordingto the orientation assumed by the intervals. The right-hand side of the sametable, on the other hand, show how two different motives of this type can, attimes, produce the same set class. For example, the set class [0123] is obtainedby both 1–2–1 and 2–1–2.

Intervallic Syntax and Serial Technique

The major discrepancy between the motivic-intervallic syntax described so farand serial technique – as conceived by Stravinsky himself – consists in the factthat while the first operates predominantly on the level of single intervals, thesecond acts essentially on the level of pitch-class sets, understood as the units ofprimary structural value.31 Different orientations of the single intervals of amotive can produce forms which belong to different set classes; by contrast,neither the retrograde, nor the inversion, nor the retrograde inversion, nor anytype of permutation of the order of a particular pc set is capable of generating adifferent set class. From the point of view of musical perception, one could evensay that motivic-intervallic syntax attributes to the quality of single intervals animportance superior to the globalising tendency of pc sets. Put simply, intervalliclogic tends towards disintegration, serial technique towards unification.

Table 2 Motives formed by two different intervals, with one of them repeated once

Motivic-intervallicclass

Set class(es)produced

Set class Associatedmotivic-intervallicclass(es)

1–2–1 ............................(012) (0123) (0134) (012)..............................1–2–12–1–2............................(0123) (0235) (0123)............................1–2–1 2–1–21–5–1 ............................(0145) (0156) (0167) (0134)............................1–2–15–1–5 ............................(0156) (0167) (0158) (0145)............................1–5–1

(0156)............................1–5–1 5–1–5(0158)............................5–1–5(0167)............................1–5–1 5–1–5(0235)............................2–1–2

‘Composing with Intervals’ 229

Music Analysis, 28/ii-iii (2009) © 2011 The Author.Music Analysis © 2011 Blackwell Publishing Ltd

Page 10: Locanto - Stravinsky's Late Technique

Despite these discrepancies, in Stravinsky’s compositional thought the twoaspects seem to aim towards the same end. In order to clarify how this occurs,consider Ex. 7, which reproduces the three choral statements at the beginning,in the middle and towards the end of the ‘Dies irae’ of the Requiem Canticles.These three fragments constitute an autonomous formal layer which interactswith the surrounding layers of contrasting musical material (omitted from theexample). The first choral statement (bars 82–83) divides into two parts: in thefirst part the chorus (doubled by the brass) intones the words ‘Dies irae’ on a fortechord repeated in a dotted rhythm; in the second, the single word ‘irae’ (chorusand horns con sordini) is repeated as an echo to a piano chord which bears acertain affinity with the preceding harmony. The second statement (bar 86) islimited instead to a repetition of the forte chord on ‘dies illa’, and without theecho response. The third statement (bar 97 onwards) is essentially a recapitula-tion of the first, although it presents a slight harmonic departure.Thus, the entirelayer assumes a type of ABA′ form.

My analytic symbols placed below the score in Ex. 7 show the motivic-intervallic construction of this layer. The two chords of the first statement (bars82–83) correspond to the two forms – [015] and [016] – of the motivic class 1–5.Moreover, notice that these forms share the pitches E� and A�, which togetherform ic5.The impression that the first chord is echoed by the second (come eco)therefore derives not only from the presence of two common tones, but also fromthe intrinsic motivic-intervallic affinity of the two harmonic simultaneities. Thesecond choral statement (bar 86) opens onto a symmetrical sonority, a memberof set class [0156] containing two ic1s and two ic5s. This sonority is obtainedthrough the sum of the two 1–5 motives appearing in the two chords of the firststatement: E�–F�–A� [015] + E�–A�–B [016] = E�–F�–A�–B [0156]. This time thechord is not simply repeated: in the middle of the bar, the lowest voice moves asemitone from B to B�, thereby giving rise to a sonority containing an ic1 (F–F�)and two conjoined ic5s (A�–E�–B�). The recapitulation (bar 97) is practicallyidentical to the first statement; nevertheless, in the first part (the forte chord on‘Dies irae’), the bass moves a semitone from F� to G. Finally, notice that theperfect fifth A�–E� is a constant presence throughout the layer in its entirety, thusforming a kind of operative tonal axis.

The sketches transcribed in Ex. 8a and 8b concern the composition of thesame formal layer. Ex. 8a reproduces a strip of paper containing an early versionof the first statement (bars 82–83), preceded by the pre-emptive instrumentalgesture which introduces it (bar 81).32 Note that in this version the harmony ofthe choral part includes a move of a semitone from F� to G in the bass voice, asolution which Stravinsky will subsequently adopt for the varied recapitulation(compare Ex. 8a with Ex. 7).33 Ex. 8b reproduces a little sheet containing twodifferent versions – in the upper and lower systems respectively – of the first andthe second statements, worked out as an unbroken succession. Here the secondstatement adumbrates the echo response, which in the final version Stravinskyuses only for the first statement (and for the recapitulation). In the first version

230 massimiliano locanto

Music Analysis, 28/ii-iii (2009)© 2011 The Author.Music Analysis © 2011 Blackwell Publishing Ltd

Page 11: Locanto - Stravinsky's Late Technique

Ex.

7S

trav

insk

y,R

equi

emC

antic

les,

‘Die

sir

ae’,

bars

81–8

3,86

and

97–9

8:m

otiv

ic-i

nter

valli

cco

nstr

ucti

on

Pia

no

Tim

pani

Di

esi

rae,

ira

e,{

come eco

CO

RO

T.

B.

A.

S.

Cor

.co

n so

rd.

III.

IV

II

{

Tr.

I. I

I

Trb

n. I

. II

ten.

DIE

S I

RA

E

= 1

36 (

= 6

8)

82

Di

esi

rae,

di

com

e ec

o

{

97

{86 dies

illa

,C

OR

O

S.

A.

B.

T.

Tr.

I. I

I

Trb

n. I

. II

ten.

, ,co

me

eco

{

Tr.

I. I

I

Trb

n. I

. II

ten.

Cor

.co

n so

rd.

II. I

V

I. I

II

CO

RO

S.

A.

B.

T.

[015

][0

16]

[015

6] =

[01

6] +

[01

5][0

16]

[015

]

E

F

A

EAB

AB E

F

B

EE

AA

FG

B

55

5

1

11

551

1

1

1

3

‘Composing with Intervals’ 231

Music Analysis, 28/ii-iii (2009) © 2011 The Author.Music Analysis © 2011 Blackwell Publishing Ltd

Page 12: Locanto - Stravinsky's Late Technique

of the passage (the upper system), the chord of the second statement is a seriesof three conjoined perfect fifths (B–F�–C�–G�). In the second version (the lowersystem) the B is modified to a B�. The alteration forms set class [0157], whichcontains two conjoined ic5s (F�–C�–G�) along with an ic1 between B� and C�. Inthe final version, Stravinsky preferred a sonority containing two ic1s and twoic5s, as we have already seen. In the event, all of the variants in the sketches, likethe final version, can be interpreted from the point of view of a systematic use ofcombinations of ic1 and ic5.

The only serial symbols discernible in the sketches are on the page transcribedin Ex. 8b and refer to the second of the two fundamental twelve-note rowsemployed in the Requiem Canticles,34 or, more precisely, to the two ‘rotationalarrays’ generated respectively by the first hexachord of series I (Ia) and the firsthexachord of series RI (RIa) of Ex. 8c.35Without going into detail on the variousproperties of this type of table and the ways of using it,36 I will briefly describeits construction. The pitches of the original hexachord are first made to rotatesystematically from right to the left: the first rotation begins with the secondpitch of the original hexachord, through which the first pitch moves into the finalposition; the second rotation begins with the second pitch of the first rotation(the third pitch of the original), and so on for five iterations (after which itreturns to the original form). The five rotated forms thus obtained are trans-posed successively so that they all begin on the same pitch as the originalhexachord (in the specific case of Ex. 8c, F for the forms generated by hexachordIa; G for the forms generated by hexachord RIa). Each of the five rotated-transposed forms thus obtained contains the same succession of intervals

Ex. 8a Stravinsky, sketches for Requiem Canticles, ‘Dies irae’ (compare bars 81–83 ofthe printed score) (Paul Sacher Foundation, Igor Stravinsky Collection)

BT

Di es i

3

rae3

i - [illegible]

SA

33

eco

3

3

TmpII Inv.

232 massimiliano locanto

Music Analysis, 28/ii-iii (2009)© 2011 The Author.Music Analysis © 2011 Blackwell Publishing Ltd

Page 13: Locanto - Stravinsky's Late Technique

globally – each time beginning at a different point within the succession – butdifferent pitch classes.37

The symbols on the sketch shown in Ex. 8b clearly indicate that the chords ofthe choral part result from the combination of dyads freely selected from the two

Ex. 8b–d (b) Stravinsky, sketches for Requiem Canticles, ‘Dies irae’ (cf. bars 81–83, 86and 97–98 of the printed score) (Paul Sacher Foundation, Igor Stravinsky Collec-tion); (c) rotational arrays of the hexachord a of the inversion (left-hand column)and retrograde inversion (right-hand column) of twelve-note row II, with encircledserial segments employed in the upper sketch (the circles and connecting arrows arenot part of Stravinsky’s original autograph); (d) motive 1–5, in the forms of sets[016] and [015]

TB

II1

2

inv. α

{ {

SA

Di es i rae, (irae) Di es illa

R inv. 1 (5 & 6)α

illa

BT

SA

Di es i rae (ir rae) di

R

es

inv. 1 (4 & 5)

illa

α

(ill a[sic])

1

2

3

4

5

6

1

2

3

4

5

6

[016] [015]

B

C

F

A

D

B5 51

1

Row II –hexachord Iα

Row II –hexachord RIα

st

nd

1

1

st

st

1 & 2

1 & 2 3 & 2 [T ]

1 & 3

2 & 311

(b)

(c) (d)

‘Composing with Intervals’ 233

Music Analysis, 28/ii-iii (2009) © 2011 The Author.Music Analysis © 2011 Blackwell Publishing Ltd

Page 14: Locanto - Stravinsky's Late Technique

rotational arrays (Ex. 8c).38 In general, the dyads derive from segments of twoconsecutive notes within a line of the tables. In one case, they even derive fromtwo non-consecutive notes (‘1st, 1 & 3’= the first and third note of the first line).In another case, the dyad derived from the third and second note of the secondline (‘2nd, 3 & 2’ = G�–F�) is transposed down a semitone (T11), so as to becomeG–E�. As can be seen, Stravinsky does not seem to have selected the dyads on thebasis of a pre-established criterion or precise order within the table. Rather, itseems that his only intention was to ensure the production of numerous ic1 andic5 relations.These intervals attain a certain importance within the original formof hexachord a, where they form two motives of class 1–5, in the forms [016] and[015], respectively (also shown in Ex. 8d). Moreover, given the structure of thetables, these intervallic motives also appear in the rotated(-transposed) forms.This justifies Stravinsky’s recourse to the rotational arrays, but it does notexplain his reason for extrapolating only dyadic segments, rather than completehexachordal units. Nevertheless, it is evident that, by operating in this manner,Stravinsky hoped to obtain a denser and more cohesive motivic constructionthan could be achieved using the hexachords in their entirety. Note, for example,that the two 1–5 motives interlaced to form the symmetrical set [0156] inthe second choral statement derive from neither hexachord a nor from itsrotated(-transposed) form. This demonstrates that from Stravinsky’s point ofview serial technique is not essential per se, but instead functions only as a meansto an end with regard to motivic-intervallic syntax. The use of complete serialforms does not, as a matter of fact, represent a restriction: if necessary, their usecan pass into the background in favour of a more immediate and direct engage-ment with single intervals.

From Intervallic Motives to Rows

The problem of the interaction between intervallic-motivic logic and serialtechnique becomes central in the compositions following Agon, which are sys-tematically based on the use of ordered pc sets (tetrachords, hexachords, twelve-note rows, and so on). This interaction can be observed in two distinctconceptual stages of the creative process: (1) the initial definition of a row ofpitches and (2) the transformation of the ‘abstract’ row into concrete musicalcontexts. In either of these stages, motivic-intervallic logic can take on a role ofgreater or lesser importance. In the first stage, such logic can determine thephysiognomy of the pc sets to be used as primary series. In the second stage, itcan determine the manner in which the rows are manifested musically. Theimplicit motivic-intervallic relations in the abstract formulation of the row, infact, can be revealed through a variety of devices.These devices can occasionallyelucidate different motivic-intervallic aspects latent in the row, which is thensubjected to a process of continual motivic interpretation.

In each composition, motivic-intervallic logic may be present in one ratherthan the other of these two conceptual stages of the creative process. In Agon, for

234 massimiliano locanto

Music Analysis, 28/ii-iii (2009)© 2011 The Author.Music Analysis © 2011 Blackwell Publishing Ltd

Page 15: Locanto - Stravinsky's Late Technique

example, this logic seems to have characterised the pre-compositional stage (thatis, the initial definition of a tone row). Indeed, the majority of the rows employedin the work demonstrate a very clear and definite motive-intervallic design.Elsewhere, I have tried to show how the overwhelming majority of rows (rangingfrom four to twelve pitches) which in recent studies of the compositional processhave been identified throughout the ballet,39 from the Triple Pas-de-Quatreforwards, can be traced back to motivic-intervallic combinations of ic1 and ic2.40

Consider, for example, the ordered tetrachord <0, 1, 4, 3>, which appears for thefirst time in the Pas-de-Deux: it can be generated by a 1–2–1 motive, with theintervals oriented in the same direction to form set class [0134], by simplyordering the sounds according to the succession <0, 1, 4, 3> in which the wholetone is found between the second and fourth notes and the two semitones oneither side. The central position is then occupied by a (non-structural) majorthird (Ex. 9).

The majority of the rows (from six to thirteen pitches) employed in thesucceeding movements of the ballet are obtained by combining different state-ments of this characteristic tetrachord.41 Consequently, the 1–2–1 motivebecomes the generating nucleus for the remainder of the work. Other rows notbased on the <0, 1, 4, 3> tetrachord can still be traced back to a particularcombination of ic1 and ic2.The first five notes of the hexachord stated in canonat the beginning of the Bransle Simple (Ex. 10a), for instance, are formed by asuccession of alternating tones and semitones, interpretable as two 1–2 motivesunited by a common pitch class. In this case, the particular design of the intervals– the first motive is in the form [013], with the intervals oriented in the samedirection; the second is in the form [012], with the intervals in opposite direc-tions – guarantees that between the highest pitch, D, and the lowest, G, an ic5 isformed, the same interval as that produced by the concluding B (the only notewhich lies outside the 1–2 pattern) and the preceding F�. This results in asymmetrical structure, with two ic5s (D–G and B–F�) at the distance of asemitone. The twelve-note row employed in the coda of the first Pas-de-Trois(presented for the first time in bars 185–189) is entirely formed from a chain of1–2 motives in the two forms [012] and [013] (Ex. 10b).

In Agon, the intervallic motives containing the tone and semitone, aside fromgenerating the majority of the fundamental rows, perform an important role evenin the movements not based on serial technique.42 Thus their presence imposescoherence on the work in its entirety, despite the different compositional tech-niques employed.43 The passage for the first violin (doubled by the cello) at bars

Ex. 9 Motive 1–2–1 in [0134] form, reordered as tetrachord <0, 1, 4, 3>

1–2–1 motive in <0134> form the same reordered as <0143>

1 2 1

‘Composing with Intervals’ 235

Music Analysis, 28/ii-iii (2009) © 2011 The Author.Music Analysis © 2011 Blackwell Publishing Ltd

Page 16: Locanto - Stravinsky's Late Technique

97–102 of the Triple Pas-de-Quatre, for example, derives from a dense chain ofoverlapping 2–1–2 motives (Ex. 11).The intervallic orientation of the motives isalmost always a zigzag, forming the chromatic set class [0123], but at times (seethe circled motives in Ex. 11) the intervals are oriented in the same direction,thus forming the set class [0235] (see again Table 2).44

By comparison with the rows used in Agon, the motivic structure of thetwelve-note rows employed in the compositions which succeed it chronologicallyappear to be less well defined. Beginning with Movements, Stravinsky seems tohave derived many of his twelve-note rows from a reading of a concrete musicalidea – most often a brief polyphonic passage.This procedure guarantees that theintervallic motives contained in the initial musical idea are less evident in therelated twelve-note row, in which the structural intervals can be found betweennon-adjacent pitches. This creates a sort of circularity between the two stagesinto which I have conceptually subdivided the creative process: from a concretemusical idea comes an abstract row of pitches, and on the basis of this row, new

Ex. 10a Stravinsky, Agon, Bransle Simple (opening): motivic construction

2 1

2 1

5

[012]

[013]

5

Ex. 10b Stravinsky, Agon, twelve-note row of the coda of the first Pas-de-Trois:motivic construction

2 2 2 2 2 11111

5

[012][012][013]

[013] [013]

Ex. 11 Stravinsky, Agon, Triple Pas-de-Quatre: motivic construction of bars 97–102

= 2–1–2 motive

[0123] [0123] [0123] [0123]

[0123][0123][0123][0123]

[0123] [0123] [0123]

[0123][0123]

(C. ni)

[0123] [0123][0123] [0123] [0123] [0123] [0123] [0123] [0123]

[0123][0123][0123][0123][0123][0123][0235][0235]

236 massimiliano locanto

Music Analysis, 28/ii-iii (2009)© 2011 The Author.Music Analysis © 2011 Blackwell Publishing Ltd

Page 17: Locanto - Stravinsky's Late Technique

and different concrete musical passages are realised. In effect, Stravinsky appearsto open up a wider field of compositional possibilities: if indeed the motivic-intervallic structure of the resultant row is more ambiguous, the row more easilypresents various musical realisations capable of illuminating different motivic-intervallic aspects that are implicit within it.

In his last works, Stravinsky hinted at different cases in which the formulationof a twelve-note row could derive from an initial concrete musical idea.45 Con-sider for example the following declaration concerning the composition ofEpitaphium:

I began the Epitaphium with flute-clarinet duet (which I had originally thought ofas a duet for two flutes, and which can be played by two flutes ... ). In the mannerI have described in our previous conversations, I heard and composed amelodic-harmonic phrase. I certainly did not (and never do) begin with a purelyserial idea, and, in fact, when I began I did not know, or care, whether all twelvenotes would be used. After I had written about half the first phrase I saw its serialpattern, however, and ... began to work toward that pattern. The constructiveproblem that first attracted me in the two-part counterpoint of the first phrase wasthe harmonic one of minor seconds. The flute-clarinet responses are mostlyseconds, and so are the harp responses, though the harp part is sometimescomplicated by the addition of third, fourth and fifth harmonic voices. (Stravinskyand Craft 1960, pp. 99–100)

Assertions of this sort often find confirmation in the sketches of the compo-sitions from Movements onwards, where one encounters some melodic or con-trapuntal annotations which probably served as the model for the formulation ofthe rows.46 A circumstance of this kind probably explains the origin of the twotwelve-note rows employed in the Requiem Canticles, which Stravinsky explicitlyattributed to ‘some intervallic designs which I expanded into contrapuntalforms’.47 The ‘intervallic designs’ to which this quotation alludes can be foundamong the sketches for the instrumental Interlude – which in actuality was thefirst movement in chronological order of composition.48

The amount of preparatory material which survives for the Interlude isuncommonly large, considering the Stravinskian standard: more than twentysmall sheets and strips of paper of different sizes, forms and typologies for apassage lasting only 67 bars.49 Some sketches contain only one or two briefmusical phrases, mostly corresponding to the exposition of a single form of rowI or row II (or of one of their constituent hexachords). Other pages assembletheir content from various earlier sketches. The final form of the passage thusresults from a sort of montage of single ideas which had been elaboratedindividually in the first stages of composition. Comparison of these versions,together with analysis of the various written materials employed and the auto-graph dates placed by Stravinsky on some pages, allows for chronological recon-struction of the sketches for the Interlude with a good degree of certainty.50 Oneof the very first ideas notated by Stravinsky is reproduced in Ex. 12a. It is formed

‘Composing with Intervals’ 237

Music Analysis, 28/ii-iii (2009) © 2011 The Author.Music Analysis © 2011 Blackwell Publishing Ltd

Page 18: Locanto - Stravinsky's Late Technique

from the union of two brief contrapuntal phrases based on the two original rowsemployed in the movement, as indicated by the autograph serial symbols.51 Thetwo phrases, initially notated separately on two small clippings of paper, werethen pasted onto a piece of cardboard (the continuity between the two phrasesis indicated by Stravinsky’s autograph arrow).52 These phrases correspondrespectively to bars 161–162 and 173–175 of the score, of which the clippingspreserve a very rudimentary version. In the following sketches, Stravinsky addednew musical material between the two phrases, which are at the same time thengradually reshaped. To this extended musical passage thus obtained (bars 163–172) Stravinsky subsequently added bars 176–192, thus creating the entireepisode for four flutes (bars 161–192), the largest and most important formalsection of the piece. In summary, it seems that the two musical ideas containedin the sketches transcribed in Ex. 12a were indeed the point of departure in thecomposition of the Interlude. If this is so, they may well feature the original‘intervallic designs’ to which Stravinsky alludes in the statement quoted above. Infact, the contrapuntal relations of the two musical phrases illuminate a very clear

Ex. 12a Stravinsky, sketches for Requiem Canticles, Interlude (Paul Sacher Founda-tion, Igor Stravinsky Collection)

II[ ]

0

3

I0

Ex. 12b Motivic analysis of the musical ideas contained in the sketches of Ex. 12a

3

F

C

B

D

A

A

D

F

GC

G

E

F

G

E

F

F

D

E

C

BC

D

C

B

A

GA

[016] [015] [013] [013] [013] [012] [013] [012]

122

1

122

11

1

1

2

12

15

5

51

55

1

238 massimiliano locanto

Music Analysis, 28/ii-iii (2009)© 2011 The Author.Music Analysis © 2011 Blackwell Publishing Ltd

Page 19: Locanto - Stravinsky's Late Technique

motivic-intervallic construction, based on the 1–2 and 1–5 motives (Ex. 12b).The triplet in the first crotchet of the second phrase, shown on the right inEx. 12b – probably composed first53 – presents within itself a sort of polyphony:the lower ‘part’, delineated by the pitches F–G–E placed in the same register,produces a 1–2 motive; the D� of the upper ‘part’ forms, however, a relation of asemitone with the lower E. By holding these two pitches firm and adding the F�,another 1–2 motive is obtained in the second crotchet of the phrase, this timevertically (D�–E–F�). Furthermore, the two motives (E–F–G and D�–E–F�) areseparated by the distance of a semitone. In the remaining part of the phrase,three overlapping 1–2 motives, in both [012] and [013] forms, are unfoldedhorizontally. In the first phrase (reproduced on the left in Ex. 12b), the first twovertical simultaneities of three pitches form two motives of class 1–5 – respec-tively in the forms [016] and [015] – while the following group of four pitchesdelineates a cycle of three ic5s (C�–D�–G�–F�), divided symmetrically into two(F�–C� in the bass; G�–D� in the upper parts). The last vertical sonority of threepitches (F�–E–G) forms a 1–2 motive which creates a strong link with thefollowing phrase, beginning with the motive F–G–E, another member of the 1–2motivic class. The link – illustrated also by Stravinsky’s cue in the upper right-hand corner of the first sheet of Ex. 12a – is reinforced by the presence of pitchclasses E and G in both of the motives.

The ‘intervallic designs’ contained in the two ideas thus become relativelyclear.We might ask at this point which came first, these two musical ideas or thetwo twelve-note rows – whether, in other words, the rows were obtained from themusical ideas or could instead have been fixed in advance as an abstractsequence of pitches on the basis of which the musical ideas were subsequentlyelaborated. The fact that the two musical ideas contain all twelve notes withoutrepetition does not mean we must prefer the second solution: generally speaking,in fact, we may suppose that Stravinsky initially elaborated his musical ideasfollowing a predominantly motivic-intervallic logic, and although even at thisstage – of which, however, hardly any written traces remain – he tended to exploitall twelve notes of the chromatic gamut, that did not prevent him from usingsome pitches more than once. Only in the final formulation of the idea were therepetitions eliminated until a fundamental twelve-note row was obtained.This isclearly demonstrated by an important document to which Joseph Straus hasdrawn attention:54 the photographs taken in 1967 by Arnold Newman in Stravin-sky’s Hollywood studio.55 Like the stills of a film, Newman’s photographs recordstep by step the creation of a musical idea – a brief instrumental passage – andits successive transformation into a twelve-note row.56 According to Straus, themusical passage reveals, above all else, some semitone-tone motives belonging toset class [013].57

From my point of view, conversely, the rethinking which took place in thecourse of elaboration was determined by a motivic idea of the semitone–majorthird type (1–4) in its two possible forms, [014] and [015]. In the very first bar,Stravinsky notated a portion of an Allintervallreihe (all-interval row; Ex. 13a) – as

‘Composing with Intervals’ 239

Music Analysis, 28/ii-iii (2009) © 2011 The Author.Music Analysis © 2011 Blackwell Publishing Ltd

Page 20: Locanto - Stravinsky's Late Technique

Straus also observes. In the next stage of composition (Ex. 13b), he abandonedthis initial idea to compose a brief counterpoint between two voices, withoutattempting to avoid the repetition of pitches (note the initial pitches B and C,repeated at the close of the passage). The third stage reveals the first significantstage of rethinking: comparing Ex. 13b with Ex. 13c, observe that Stravinskyreplaces the first of the two Cs with an E�, thus avoiding the presence ofrepetition. The choice of E� throws light on the motivic-intervallic logic whichguides the composition of the passage: the first three notes of the viola (B–E�–B�)now form a motive of the semitone–major third type (1–4). This motive – in itstwo forms, [014] and [015] – appears at numerous other points within thepassage: in the first five notes of the viola part (twice: B–E�–B� and B�–D�–A, withB� in common); in the last three notes of the cello (B�–G�–F), grouped togetheras a triplet; between the first two notes of the viola (B–E�) and the D of the cellowhich follows immediately afterwards (a semitone lower); and finally in thecontrapuntal relationship between the B�–D� of the viola and the D of the cello(Ex. 13f).

There is a second significant redrafting at the following stage (Ex. 13d), whereStravinsky replaces the first and the third notes of the viola (B and B�), bothrepeated, with G and A� respectively. By making this adjustment, Stravinsky notonly obtains all twelve notes without repetition, but also preserves intact the

Ex. 13a–g See Arnold Newman’s photos, reproduced in Craft (1967), pp. 14–15 and16–17. The sequence of photos goes across the volume’s two-page spreads of thesketches; photos 1 and 2 are on p. 14, photos 3 and 4 on p. 15, and so on

(a) stage 1 (cf. photos 1–10)

(b) stage 2 (cf. photos 11–13)

(c) stage 3 (cf. photos 14–17)

(d) stage 4 (cf. photos 18–20)

(e) analysis of stage 5(cf. photos 18–25)

(f) analysis of stage 3

(g) analysis of stage 4

3

3

[3]

3

[3]

inverted

[025] [027]

D

GE

A

D

E

5

5

22

3

[014] [015] [014] [014]

B B B A

DDE

E

BD D B41

1

4

4 4

1

1

[3]

[015]

F

G

B

4

1

3

[015] [015] [015]

D

E

G A

G

E A

D

D

1 14

144

[3]

[015]

F

G

B

4

1

240 massimiliano locanto

Music Analysis, 28/ii-iii (2009)© 2011 The Author.Music Analysis © 2011 Blackwell Publishing Ltd

Page 21: Locanto - Stravinsky's Late Technique

motivic construction of the passage, which remains based on the conspicuouspresence of a 1–4 motive – now in the form [015] (Ex. 13g).

The third and final phase of development occurs in the final stage (Ex. 13e),at the end of which Stravinsky obtains a twelve-note row, arranging in successionthe notes of the passage just composed.58 Here a small adjustment in the orderof the pitches suffices to mask the original motivic-intervallic aspect of thepassage while at the same time illuminating a new one. The second and sixthpitches (E� and E�) are inverted (see the circled notes in Ex. 13e).Thus, the firsttwo segments of three pitches (G–E–D and A�–D�–E�) become two motives of theclass 2–5. Moreover, the four segments of three pitches which form the rowdelineate a symmetrical structure: the combination of the even-numbered seg-ments forms a partial circle of fifths from G� to F, while the combination of theodd-numbered segments forms the remaining part of the circle (from C to B):59

1. E�–D�–A� (segment 2) + B�–G�–F (segment 4) = G�–D�–A�–E�–B�–F;2. G–D–E (segment 1) + A–B–C (segment 3) = C–G–D–A–E–B.

In the end, the simple exchange of E with E� in the final row creates, in this case,a marked estrangement of the motivic-intervallic construction of the originalmusical idea.60

From the Row to Intervallic Motives

At this point it is worth reflecting on the ways in which motivic-intervallic logicinfluences the musical concretisation of the row, once it has been definitivelyestablished. I will first consider a brief musical fragment drawn from the beginning(bars 46–48) of the second of the five Movements (Ex. 14a and b).61 The passageis based on the two discrete hexachords (labelled a and b) of the fundamental row

Ex. 14a Stravinsky, Movements, ii: hexachordal forms employed in the sketch shownin Ex. 14b

1

1

1

1

2

2

2

2 3

3

3

3 4

4

4

4 5

5

5

5 6

6

6

6

[016]

[016]

[012]

[012] [012]

[012]

[016]

[016]

Hexachord α

Hexachord β

RI (T )

R

6

‘Composing with Intervals’ 241

Music Analysis, 28/ii-iii (2009) © 2011 The Author.Music Analysis © 2011 Blackwell Publishing Ltd

Page 22: Locanto - Stravinsky's Late Technique

used throughout the entire composition (Ex. 14a, first and third lines). In theupper part of the sketch transcribed in Ex. 14b, Stravinsky wrote the RI (T6) formof hexachord a and the R form of hexachord b (Ex. 14a, second and fourth lines)– as indicated also by the autograph labels ‘Riv-Inv a’ and ‘Riv b’.62

As my analysis in Ex. 14a shows, hexachord b can be divided into twotrichords belonging to set class [012], which can in turn be related to motiveclass 1–2 (see again Table 1). Hexachord a is formed by two trichords of set class[016], which can be related to three different intervallic motives: 1–6, 1–5 or 5–6.In this case, Stravinsky clearly placed the intervals of a semitone and perfect fifth(motive 1–5) in relief. To this end, a particular permutation of the order of thepitches of hexachord a is carried out:63 besides placing the pitches in reverseorder (from the sixth to the first), he also reversed the order of the first twopitches of each trichord. In this manner the pitches which form ic5 (representedin bold in the schema below) are always adjacent:

6–5–4/3–2–1 becomes 5–6–4/3–1–2

(G–F�–C/D–D�–A� becomes F�–G–C/D–A�–D�).The purpose behind this particular reordering can be appreciated in the

musical passage outlined in the sketch transcribed in Ex. 14b, immediatelybelow the two hexachords: the pitches which form ic5 are arranged vertically asa perfect fifth; the pitches which form ic1 precede the fifth in a register at thedistance of an octave. The two fifths (C–G/D�–A�) are separated by a semitone,thereby producing a symmetrical configuration. All these choices are clearlyintended to throw the ic1 and ic5 into relief.

Ex. 14b Stravinsky, sketches for Movements, ii (cf. bars 46–48 of the printed score)(Paul Sacher Foundation, Igor Stravinsky Collection)

5 6 4 3 1 2

654321

6 43

12

5

2 3 4

651

Riv.

Riv β

Riv- Inv

Riv-Inv α

242 massimiliano locanto

Music Analysis, 28/ii-iii (2009)© 2011 The Author.Music Analysis © 2011 Blackwell Publishing Ltd

Page 23: Locanto - Stravinsky's Late Technique

This brief example shows how some operations which alter the physiognomyof the row were intended by Stravinsky to facilitate the transformation of anabstract row in a specific musical passage which highlights some particularmotivic-intervallic characteristics. One such operation is order permutation, aswe have just seen; another consists of extrapolating small segments, usually oftwo to four notes in length, and successively reconfiguring them.This technique,which I call serial fragmentation-recombination, was probably adopted for thefirst time by Stravinsky in Threni and was subsequently used in an increasinglysophisticated fashion.64 We have already observed one application, albeit a ratherlimited one, in the ‘Dies irae’ of the Requiem Canticles. To further illustrate itsfunction in relation to the motivic-intervallic syntax, I will now consider its usein the first part of the ‘Rex tremendae’ of the Requiem Canticles.

The serial construction of the passage is clearly illustrated on the first page ofthe autograph short score (containing bars 203–208), transcribed in Ex. 15a.65

The symbol ‘I Ra’ stands for row number I, retrograde form, hexachord a. The

Ex. 15a Stravinsky, first page of the short score for Requiem Canticles, ‘Rex tre-mendae’ (compare bars 203–207 of the printed score) (Paul Sacher Foundation, IgorStravinsky Collection)

B

6

T

8 Rex

4vlavccb

A

Rex

13 Fl

4 5 6

S

I R α

Rex3

Rex tre

β

men

Verticals

dae ma je sta tis

1 2 3

I

4

invα

5

3

6

rd

3 Fl

[ ]

thR α 5

[ ]I inv α 1 linest

vc

I inv α 4 lineth 3 linerd

I3 4 5

trmb

6 lineth

β 5th 54 6

456R α 1st

Strings vlc

3 linerd

‘Composing with Intervals’ 243

Music Analysis, 28/ii-iii (2009) © 2011 The Author.Music Analysis © 2011 Blackwell Publishing Ltd

Page 24: Locanto - Stravinsky's Late Technique

circled numbers at the beginning of each choral part in the short score indicatethe lines of the rotational arrays of the first hexachord. Only limited portions(three to five notes) of each line are used (shown circled in Ex. 15b).The zigzagline traced across the choral parts66 corresponds to the brass part (trumpet andtrombone) elaborated on the lower portion of the page. The autograph symbolsindicate that even this line was obtained by the combination of three serialsegments drawn from the two rotational arrays of the R hexachords. Forexample, the symbol ‘Ra 5th 3 4 5’ stands for retrograde, hexachord a, fifth lineof the rotational array, notes 3–4–5 (see also Ex. 15b, where the segments usedin the brass part are indicated within boxes). In sum: the first three bars67 of thechoral part and the brass parts are obtained exclusively through a combinationof serial segments extrapolated by two rotational arrays produced by hexachordsa and b of the retrograde form of row I. As in the case of the ‘Dies irae’, thesegments are selected in an apparently arbitrary manner. Nevertheless, theydemonstrate a significant presence of motives of class 1–2: four of the fivesegments of three-note segments (Ra 1st 4–6, Ra 4th 4–6, Ra 5th 3–5, Ra 6th4–6 and Rb 5th 4–6) directly correspond to this motive class; one of the twosegments of five notes (Ra 3rd 2–6) contains two overlapped 1–2 motives(C�–C�–C + C–B–A); and the other (Ra 1st 1–5) begins with a 1–2 motive. Thereason for this arrangement was Stravinsky’s desire to create an imitative texturebased on the motivic-intervallic element: at regular intervals of a minim thecontralto, the trombone, the sopranos and the tenors display motive 1–2; butbecause this motive can assume two different forms – [012] and [013] – andgiven that the three pitches can be combined in any order, the imitative responsesrepeat neither the same melodic profile (as in traditional imitative style) nor the

Ex. 15b Stravinsky, Requiem Canticles, twelve-note row I: rotational arrays of thehexachords a and b of the retrograde form, with encircled serial segments employedin the sketch of Ex. 15a

1

2

3

4

5

6

Row I – Retrogradeα β

244 massimiliano locanto

Music Analysis, 28/ii-iii (2009)© 2011 The Author.Music Analysis © 2011 Blackwell Publishing Ltd

Page 25: Locanto - Stravinsky's Late Technique

same set class. Therefore, motive 1–2 is repeated three times in the brass partwhich runs throughout the choral passage. Note also that the initial pitches ofeach imitative part (A�–G�–D�–F�) gradually take the form of a sequence ofperfect fifths (F�–C�–G�–D�–A�), which is completed in bar 3 with the addition ofthe pitches G� and C� in the bass part.

This homogenous motivic design is due largely to the structure of the rowitself. Indeed, it should be evident that numerous 1–2 motives are alreadycontained in row I of the Requiem Canticles (see again the right-hand section ofEx. 12b). Nevertheless, by segmenting the hexachords selectively, Stravinskycreated an imitative texture which was more coherent than would have beenpossible had he used complete hexachords. One might ask in what sense and towhat degree such a procedure could be defined as authentically serial. However,the fact that the composer had indicated the serial origin of the various segmentsin the short score demonstrates that he conceived of the row as a reservoir ofmotivic-intervallic material, and that he understood serial technique as a meansof managing this material systematically.

The beginning section of the first of the five Movements (see again Ex. 6)is likewise based on a systematic application of serial fragmentation-recombination. The passage divides into an initial introduction (bars 1–6) andthree solo episodes for, respectively, piano, first flute (accompanied by piano andclarinet) and piano again (accompanied by strings).The central flute solo is veryfamiliar to Stravinsky scholars; indeed, the composer himself drew attention toits complex serial construction, thereby instigating a long series of attempts atanalysis.68 As I will try to show, a motivic-intervallic approach can provide a newand logical key to reading the passage.

To this end, it is useful to begin with the subset structure of the two hexa-chords of the fundamental row (Ex. 16). The first hexachord is formed by twodisjunct trichords belonging to set class [016]. In the middle, starting with thethird note, one finds a trichord of set class [012].The second hexachord containstwo disjunct trichords of set class [012] and one [016] trichord at its centre, thuscomplementing the first hexachord. Other interesting properties of the twohexachords emerge when one takes into consideration their rotated forms.Ex. 17 cites one of the rotational arrays employed by Stravinsky for the compo-sition of Movements: columns a and b contain the rotated forms of the twohexachords of the original form of the row; columns g and d display the rotated-transposed forms. Ex. 18 reveals the subset structure of all these rotated(-transposed) forms. Eight of the twelve hexachords contain within them three

Ex. 16 Stravinsky, Movements, i: subset structure of the original twelve-note row

Orig.

[016] [016] [012] [012]

[016][012]

α β

‘Composing with Intervals’ 245

Music Analysis, 28/ii-iii (2009) © 2011 The Author.Music Analysis © 2011 Blackwell Publishing Ltd

Page 26: Locanto - Stravinsky's Late Technique

Ex. 17 Stravinsky, Movements, sketch showing the rotated (columns a and b) androtated-transposed (columns g and d) forms of the two hexachords of the originaltwelve-note row (Paul Sacher Foundation, Igor Stravinsky Collection)

I

II

III

IV

V

α β γ δ

Ex. 18 Stravinsky, Movements: subset structure of the rotated-transposed forms ofthe two hexachords of the fundamental row

Orig.

I

II

III

IV

V

[016] [016] [012] [012]

[016][012]

[016] [012]

[012] [016]

[016]

[012]

[012]

[016] [016]

[016] [016]

[012]

[012]

[016]

[012]

[016]

[012] [016]

[012]

[012] [012] [016] [016]

[016]

[012]

[012]

[016]

[016]

[012]

[016]

[016]

γ δ

246 massimiliano locanto

Music Analysis, 28/ii-iii (2009)© 2011 The Author.Music Analysis © 2011 Blackwell Publishing Ltd

Page 27: Locanto - Stravinsky's Late Technique

trichords belonging to set class [016] or [012]; two hexachords (IIg and IIIg)contain four which closely overlap with one another. The remaining two hexa-chords (Id and IVd) contain two each.

It is evident, therefore, that the internal structure of the two hexachords issuch that the rotated forms generate a large number of trichords belonging to setclasses [012] and [016]. Now, set [012] can be understood solely as a 1–2 motivewith the intervals arranged in opposite directions,69 while set [016] could beassociated with three different motives: 1–5, 1–6 or 5–6 (see again Table 1). Adeeper analysis will clarify which of these intervallic motives was the object ofStravinsky’s interest.

With the aid of the sketches, I have reconstructed the chronology and serialorigin of the three episodes which form the entire section, summarised inTable 3.70 As can be seen, all three episodes are based on a combination of serialsegments chosen from columns g and d of the rotational array. The serialsegments employed for the second episode, the flute solo (bars 13–17), areindicated in the sketch transcribed in Ex. 19a; the symbols here refer to thesketch with the rotational array transcribed in Ex. 17 (the Greek letters refer tocolumns g and d of the array and the roman numerals to the lines; the arabicnumeral indicate the selected segments). The 34 notes of the flute melody(Ex. 19a) are obtained by combining ten serial segments, freely chosen from thearray (Ex. 19b). In the final version (Ex. 19c), the melody thus obtained wascompletely transposed a major third higher (T4) to start on G instead of E�. In thethird episode (bars 18–22; see Ex. 20) the piano and string parts employ thesame serial segments as the flute solo, as is suggested by the serial symbols andthe indication ‘follow the flute solo before (same series)’ in the short score;however, this time it is not transposed (compare Ex. 20 with Ex. 19b).

The first episode (bars 7–12; Exs. 21 and 22) is essentially based upon thesame succession of serial segments, even if in the final version the resultingcorrespondence is obscured owing to some errors Stravinsky committed in the

Table 3 Chronology and serial origin of the three episodes in Stravinsky’s Movements,i, bars 7–22

Episodes (in chronological order) Twelve-note material employed

bb. 13–17: second episode(solo for flute I)

Serial segments drawn from columns g and d of therotational array of the hexachords. The entiresuccession is transposed T4 to G. The accompanimentuses complete hexachords.

bb. 7–12: first episode(first solo episode for piano)

The same serial segments from the flute solo buttransposed T2 (on F).

bb. 18–22: third episode(second solo episode for piano)

The same serial segments from the flute solo, nottransposed (on E�).

‘Composing with Intervals’ 247

Music Analysis, 28/ii-iii (2009) © 2011 The Author.Music Analysis © 2011 Blackwell Publishing Ltd

Page 28: Locanto - Stravinsky's Late Technique

preparation phase. On the first two lines on the page of sketches transcribed inEx. 21, up to the pitches B–C–D� before the notated clefs, the composer hadinitially outlined a first version of the passage transposed to begin on G, in amanner identical to the final version of the episode for flute. Ex. 22 presents thisfirst version together with the related serial symbols derived from his autographshort score. As is evident by comparing Ex. 22 with Ex. 19c, Stravinsky obtainedexactly the same pitches as the final version of the flute episode by utilising onlyslightly different serial fragments.71 Up until this point, therefore, the first pianoepisode corresponds exactly with the flute solo, with respect to pitch content. Ata later point, represented in lines 6–9 of the sketch in Ex. 21, Stravinsky elabo-rated a second version – different from the first in both rhythm and the octaveregistration of some pitches – transposed to begin on F instead of G. Later, after

Ex. 19a Stravinsky, sketches for Movements, i, bars 13–17 (Paul Sacher Foundation,Igor Stravinsky Collection)

1 2 3 4 5 6 1 12 2 23 3 3 3 34 4 4 4[4][4] 5 5 5 5 5 56 6 6 6 6 6

[ ] ][

I I I IIII IIIIIIIIIVVγ δ δ δ δγγγγγ

Ex. 19b Columns g and d of the array reproduced in Ex. 17 (hexachordal rotated-transposed forms), with encircled serial segments employed in the sketch shown inEx. 19a

Orig.

I

II

III

IV

V

γ δ

1

1

1

1

1

1 2

2

2

2

2

2 3

3

3

3

3

3 4

4

4

4

4

4 5

5

5

5

5

5 6

6

6

6

6

6 1

1

1

1

1

1 2

2

2

2

2

2 3

3

3

3

3

3 4

4

4

4

4

4

5

5

5

5

5 6

6

6

6

6

65

Ex. 19c Final version of the flute solo in bars 13–17 showing the serial segmentsemployed in it

Fl. I

γ I 1–3 (T )4 δ I 4–6 (T )4 γ I 4–6 (T ) γ I 3–6 (T )4 4δ V 1–3 (T )4 γ V 1–3 (T )4 δ III 3–6 (T )4 γ II 4–6 (T )4 γ III 4–6 (T )4 δ II 2–6 (T )4

248 massimiliano locanto

Music Analysis, 28/ii-iii (2009)© 2011 The Author.Music Analysis © 2011 Blackwell Publishing Ltd

Page 29: Locanto - Stravinsky's Late Technique

the clefs (where the elaboration of the first version was interrupted) on lines 1–2of the same page, Stravinsky completed the episode with the final missing part.Although neither the sketch nor the short score presents serial symbols at thispoint, it is obvious that the segments employed are the same as those used for thelast part of the flute episode (transposed by T2).72

To summarise, the second version of the piano passage stretches in part acrosslines 6–9, up to the dyad A–B� circled in the middle of the ninth line – whichcorresponds to the pitches B and C of the preceding version on G, as demon-strated by the vertical line drawn across the page – and in part (followingStravinsky’s arrow) on lines 1–2, to the right of the treble and bass clef.73 Insetting the entire passage in the short score, however, Stravinsky failed to recopypitches B and C, placed on the first line immediately after the treble clef, perhapsmistaking them for a repetition of the B and C immediately preceding it on thesecond line – which, however, belonged to the first version on G.Thus these twonotes do not appear in the final score. By restoring them, one will easily noticethat the entire sequence of pitches in the piano’s first episode correspondsexactly to that of the flute’s solo episode, transposed a tone lower.74

According to Stravinsky’s initial intent, then, all three episodes were to bebased on the same succession of pitches, transposed onto three different levels:G, F and E�. It was most likely in this manner that Stravinsky sought to obtainsomething similar to ‘tonal’ organisation on a broader formal plane.75 However,considerations of this type are beyond the scope of this study.What is importantfrom my point of view is to observe how the intricate combination of serialsegments on which the three episodes are based produces a homogeneous andcohesive motivic-intervallic texture.

Ex. 20 Stravinsky, Movements, i, bars 18–22

r.h.2

3

,

3

l.h.

18 19 20 21 3 2 13

Piano

γ I 1–3 δ I 4–6 δ V 1–3 γ V 1–3 δ III 3–6 γ I 4–6 γ II 4–6 γ III 4–6 γ I 3–6 δ II 2–6

22

3pizz.{ { sim.

3(non div.)

3

pizz.

Vle.

Vc.

C. B.

‘Composing with Intervals’ 249

Music Analysis, 28/ii-iii (2009) © 2011 The Author.Music Analysis © 2011 Blackwell Publishing Ltd

Page 30: Locanto - Stravinsky's Late Technique

Ex. 21 Stravinsky, page of sketches for Movements, i, bars 7–12 (Paul Sacher Foun-dation, Igor Stravinsky Collection)

[ ]

[ ]

[ ]

5

8

[ ]

][

[ ]

[ ]

5

va

vb

8va

[ ]

[

[ ]

]

[ ]

[ ][ ]

8

8vb

[ ]

250 massimiliano locanto

Music Analysis, 28/ii-iii (2009)© 2011 The Author.Music Analysis © 2011 Blackwell Publishing Ltd

Page 31: Locanto - Stravinsky's Late Technique

To this end, the question which arises is: according to what criterion didStravinsky select and combine these serial segments? Typically, if one takes intoconsideration the original hexachords from which the segments were extrapo-lated, it becomes apparent that a procedure of utilising pivotal pitches is ren-dered central. For example, the first segment of three pitches (gI 1–3) is followedby the pitch A� from the middle of its original hexachord gI (see again Ex. 19b).The second segment must therefore begin with this pitch. In its turn, the secondsegment (dI 4–6), also containing three pitches, is followed,76 in its originalhexachord, by the pitch C; thus the third segment must begin with that pitch.The passage consequently unfolds as follows:

γI E A G A1 3 4− −( ) = − − −( )� �δI A G D C4 6 1− −( ) = − − −( )� � �

δV C G G E1 3 4− −( ) = − − −( )� �Nevertheless, this criterion establishes only the first pitch of the following

segment and not the internal characteristics of that segment (as the serial tableshows, there are different segments which also begin with the same pitch).Moreover, the mechanism of pivotal pitches is employed only in a limitednumber of cases. A fuller rationale nevertheless emerges if the ten serial segmentsemployed across the three episodes are re-examined in relation to the concept ofintervallic motive. Given the structure of the hexachords (see again Exs. 16 and18), these segments are almost all members of set classes [012] and [016], asEx. 23 demonstrates (the only exceptions are the two segments indicated by theexclamation marks). All of the four-note segments (nos. 5, 9 and 10) containoverlapping sets [012] and [016], as Ex. 23 indicates.

What is intriguing, however, is the global result obtained from the combina-tion of the segments. Ex. 24 examines all of the consecutive three-note groups(bracketed above and below the musical stave) starting from each note of the

Ex. 22 Stravinsky, Movements, i (compare bars 7–10 of the printed score). Thesymbols above the staves represent serial segments drawn from the short score (PaulSacher Foundation, Igor Stravinsky Collection). Below them is an excerpt (first twostaves) from the sketch transcribed in Ex. 21

[ ]

γ V 3–6 (T ) δ I 5–6 (T ) δ V 1–3 (T )

5

δ III 4–6 (T ) γ I 4–6 (T ) γ II 4–6 (T )2 4 4 4 4 4 4γ V 1–4 (T )

‘Composing with Intervals’ 251

Music Analysis, 28/ii-iii (2009) © 2011 The Author.Music Analysis © 2011 Blackwell Publishing Ltd

Page 32: Locanto - Stravinsky's Late Technique

complete succession of pitches:77 with the sole exception of seven groups (indi-cated by the exclamation marks), all of the trichords belong to set class [012],[013] or [016]. These set classes, in turn, can be obtained from only twomotivic-intervallic classes; set classes [012] and [013] are, in fact, the two formsassumed, according to the orientation of the two intervals, by the characteristictone-semitone motive (1–2). It is important to note that the [013] form of thismotive is not included in the serial hexachord, which contains only the [012]form (see again Exs. 17 and 18). The presence of form [013] in the successionof pitches on which the three episodes are based is therefore the result of acollection of serial segments created ad hoc, which clearly demonstrates how thetechnique of serial fragmentation-recombination permitted the composer togenerate different motivic forms without limiting himself to those contained inthe row. Set class [016] could have been generated by three different motivic-intervallic classes: 1–5, 1–6 or 5–6 (see again Table 1). Nevertheless, the form[015] of motive 1–5 – obtained by orienting the two conjoined intervals inopposing directions – never appears in the entire succession of pitches, whichsuffices to exclude it as a possibility. Of the remaining two motives (5–6 and1–6), it is more logical and economical to think primarily in terms of motive 1–6since the entire passage can then be read from the point of view of only threeinterval classes – ic1, ic2 and ic6 – contained in motives 1–2 and 1–6.

It is evident, therefore, that the selection and arrangement of the serialsegments was done in such a way as to obtain a continuous interlocking of two

Ex. 23 Stravinsky, Movements, i: serial segments employed in the succession ofpitches contained in each of the three solo episodes of bars 7–22

[016] [016] [016] [016] [016][016] [016]

[012][012]

[012] [012]!!

γ I 1–3 δ I 4–6 δ V 1–3 γ V 1–3 δ III 3–6 γ I 4–6 γ II 4–6 γ III 4–6 γ I 3–6 δ II 2–6

1 2 3 4 5 6 7 8 9 10

Ex. 24 Stravinsky, Movements, i: intervallic motives in the succession of pitchescontained in each of the three solo episodes of bars 7–22

[012] [016] [016] [016] [016] [016] [016][012] [012][012] [016] [013] [013][016] [016] [016] [012][012]

[016] [016] [016] [016][012] [012] [013]

1–2 motive 1–6 motive

[013] [012] [016] [016]

1 1 1 1

2 6(3) [7] = 52

6(5)(1)

! ! ! !

!!!

252 massimiliano locanto

Music Analysis, 28/ii-iii (2009)© 2011 The Author.Music Analysis © 2011 Blackwell Publishing Ltd

Page 33: Locanto - Stravinsky's Late Technique

motivic classes (1–2 and 1–6),78 generated by just three interval classes (ic1, ic2and ic6). The use of complete serial forms (twelve-note rows or hexachords)would not produce so dense and coherent a fabric: hence the exigency offragmenting the hexachords into smaller units, which can then be recombined atwill.

Conclusions

In the evolution of compositional technique which runs from the melodic chainsof Agon to the complex combinations of serial segments in Movements or theRequiem Canticles, it is possible to discern a continuity of thought centred on theproblem of interaction between motivic-intervallic syntax and twelve-note tech-nique. In Agon, the motivic-intervallic logic determines, above all, the construc-tion of ordered sets, the combination of which gradually forms largeragglomerations, employed in their turn as fundamental rows. In later composi-tions, the row confronts a continuous motivic rereading, whether through dif-ferent means of musical realisation or through fragmentation into small groupsof pitches, which are then recombined into new configurations.

Examination of the creative process reveals that the tendency towards dis-integration resulting from the direct manipulation of single intervals is notincompatible with serial technique, providing the latter is understood in a cre-ative rather than a strictly procedural sense. Stravinsky’s adoption of serialismrelies upon an aesthetic vision which does not attribute to the row the valueof a fundamental Gestalt for the composition. The same can be said ofStravinskian motivic-intervallic syntax, at the base of which lies an aestheticconception foreign to the ideal of organic coherence which characterises theAustro-German tradition. The idea that intervallic configurations shouldprovide a unifying function for the general internal relations which govern anentire composition – a function comparable to that of traditional tonality – islargely alien to Stravinsky’s approach, which conceives of the intervallicmotives simply as starting materials for the act of musical construction. Thiscreative process consequently proceeds from the particular to the general,following an itinerary open to deviations and metamorphoses which arerealised through continuous motivic-intervallic rereading of both the pre-compositional row and its concrete musical correlates. The elementary mate-rials (the intervallic motives) from which this mode of construction primarilyderives evidently leave a mark, a recognisable impression on the final physi-ognomy of the musical edifice, without, however, assuming a determining rolewith regard to structural connection. To what degree and at what levelthis impression is discernible and perceptible is a question which evidentlyremains open. Whatever the answer, we must not lose sight of the essentialsignificance of working with intervals that characterised Stravinsky’s musicalthought.

‘Composing with Intervals’ 253

Music Analysis, 28/ii-iii (2009) © 2011 The Author.Music Analysis © 2011 Blackwell Publishing Ltd

Page 34: Locanto - Stravinsky's Late Technique

NOTES

This article derives, in large part, from my doctoral thesis (Locanto 2002). Alltranscriptions and facsimiles are published with the kind permission of the PaulSacher Foundation in Basel. The sketch transcriptions aim to reproduce theoriginals as faithfully as possible: all of the author’s annotations are either enclosedwithin square brackets or signalled in the accompanying captions. Since thenumbering of the microfilms in the Paul Sacher Foundation may change fromtime to time, items are identified here not by microfilm number, but by thecollection to which they belong and the type of material (sketch, short score, fullscore, etc.). In the case of the sketches for the Requiem Canticles, for which a micro-film copy has not yet been made, I refer only to the collection to which it belongs.Copyright clearance for musical examples was secured from the following sources:Stravinsky, Requiem Canticles © copyright 1967 by Boosey & Hawkes MusicPublishers Ltd., reproduced by permission of Boosey & Hawkes Music Pub-lishers Ltd.; Stravinsky, Movements © copyright 1960 by Hawkes & Son (London)Ltd., reproduced by permission of Boosey & Hawkes Music Publishers Ltd.

1. See for example Stravinsky and Craft (1959), p. 11: ‘I begin work by relatingintervals rhythmically’. (See also Stravinsky and Craft 1966, pp. 60–1, on theVariations, and Craft 1972, p. 98, on the Requiem Canticles.) In an interview with JayS. Harrison in the New York Herald Tribune, 21 December 1952 (cited in Tucker1992, vol. 2, p. 187), Stravinsky emphasised: ‘Always I have been interested inintervals. Not only horizontally in terms of melody, but also the vertical results thatarise from the combinations of intervals’.This point has been strongly endorsed byMilton Babbitt: ‘One of the remarkable things that Stravinsky said, when people feltthat he committed a treasonable act by starting to write pieces where you could finda succession of twelve [notes] at the beginning, was “There’s nothing to it; I’vealways composed with intervals”. Basically, of course, it was something of a witti-cism, but what it did show, much more than a witticism, was how profoundly thisis an interval kind of syntax and not just a pitch-class syntax – fundamentally andcentrally an interval syntax’ (Babbitt 1987b, p. 20; see also Babbitt 1968, p. 167).

2. On these topics, see especially Berger (1963), van den Toorn (1983) and Taruskin(1996), pp. 255–307. On the use of other non-diatonic collections in Stravinsky’smusic, see also Johnson (1987), Tymoczko (2002) and van den Toorn andTymoczko (2003).

3. After Threni, the systematic use of diatonic collections, which characterised evenStravinsky’s earlier serial compositions, is noticeably reduced in favour of moremarkedly chromatic situations, even if the latter contain a diatonic core (seeLocanto 2002, pp. 177–212). On the diatonic component in serial compositions upto Threni, see Neidhöfer (1999). Taruskin (1993) and (1996), pp. 1648–73, hashypothesised the persistence, up to the final serial compositions, of a similar routinebased on the use of octatonic collections. More recently this idea has been placedin a markedly different perspective, especially by Straus (2001), p. 39 and n. 79.

4. On the use of intervallic motives in Stravinsky’s pre-serial music, see for exampleStraus (1991).

254 massimiliano locanto

Music Analysis, 28/ii-iii (2009)© 2011 The Author.Music Analysis © 2011 Blackwell Publishing Ltd

Page 35: Locanto - Stravinsky's Late Technique

5. The underlying theoretical perspective is neatly summarised by Wason (1996), p.111: ‘In tonal music, motivic events are generally regarded as intervallic relation-ships, whose actual pitch-class representations change with reference to a fixed-pitch background – the tonal centre itself, or a related tonal region whichtemporarily comes to the fore. In so-called “atonal” music, that background ispresumably absent, leaving us with only the intervallic relationships – the immediateobject of most analytical investigations of this repertoire’.

6. For example, Straus, in attributing a basic role to the manipulation of single intervals,defines the concept of motive in a way which differs from the definition offered here.For him, a motive results from the combination of a limited number (usually two) of‘atomic’ intervals (one of which is generally a tone or semitone).Through the choiceof an appropriate transpositional level, these intervals produce particular sets (seeStraus 2001, pp. 82–92), which are then used as motives in melodic construction(Straus 2001, pp. 92–103). In this case, a motive, although initially obtained by aparticular combination of elementary intervals, corresponds to a set of pitch classeswhich could be represented in retrograde, inverted or reordered, but would continuenonetheless to belong to the same class. From my point of view, on the other hand,a motive is created by a variable configuration of intervals which can produce setsbelonging to different classes. Furthermore, Straus conceives of motives in anexclusively melodic sense: his analyses demonstrate the use of motives only in thehorizontal dimension. To my mind, however, the pitches produced by a particulararrangement of basic intervals can be situated either horizontally – in any order – orvertically. As an example of the different results to which these two approaches canlead, see n. 40.Yet another approach is adopted by Smyth (1997), pp. 21–3, whichconsiders interval types (not interval classes). See also Smyth (1999) and (2000).

7. As other documents also demonstrate: see for example Stravinsky and Craft (1959),pp. 11–12: ‘This exploration of possibilities is always conducted at the piano. Onlyafter I have established my melodic or harmonic relationships do I pass to compo-sition. Composition is a later expansion and organisation of material ... . I start tolook for this material, sometimes playing old masters (to put myself in motion),sometimes starting directly to improvise rhythmic units on a provisional row ofnotes (which can become a final row)’. Other sentiments of this type can be foundin Stravinsky and Craft (1962), p. 52, where the keyboard is described by Stravinskyas ‘the center of my life and the fulcrum of all my musical discoveries’; in Stravinskyand Craft (1966), pp. 23–4 and n. 8; and in Craft (1972), p. 131. See also thetestimony of Nicolas Nabokov (1949), p. 146, along with that of Stravinsky himselfin the documentary A Conversation with Igor Stravinsky, directed and produced byRobert D. Graff for the National Broadcasting Company in 1957 (the passage inquestion is transcribed with commentary in Tucker 1992, vol. 1, p. 23).

8. In this study, the prime forms (see Straus 1990, pp. 41–2) of unordered sets arerepresented by a sequence of numbers – each of which stands for a pitch class –between square brackets, arranged according to the conventional criterion dis-cussed in Straus (1990), pp. 41–2. Ordered sets are, however, represented by anumerical sequence in angle brackets which follows the actual order of the notes.For example: the succession of notes B�–A–C–B belongs to set class [0123]; con-sidered as an ordered set, however, it would be represented as <1, 0, 3, 2>.

9. For a definition of interval class see Straus (1990), pp. 6–8. In this study, intervalclasses are indicated in the orthodox manner by ‘ic’ followed by an arabic numeral

‘Composing with Intervals’ 255

Music Analysis, 28/ii-iii (2009) © 2011 The Author.Music Analysis © 2011 Blackwell Publishing Ltd

Page 36: Locanto - Stravinsky's Late Technique

indicating the interval class measured in semitones. The terms of traditional tonaltheory, when used, refer – unless otherwise indicated – to interval classes. Forreasons of space, in most of the musical examples the ‘ic’ is omitted and the intervalsare indicated by arabic numerals alone.

10. See Straus (1990), pp. 128–30.

11. This allows different row forms to project set class [0126] using the same globalpitch-class content, a property which Berg exploits to create a network of formalrelations.

12. This concerns, in essence, the phenomenon of pitch-class invariance, to whichMilton Babbitt drew attention in a fundamental collection of essays; see in particu-lar Babbitt (1960) and (1961). On invariance see also, for example, Lewin (1962)and Beach (1976).

13. On which there is an extensive body of literature; for an overview, see Perle (1991),pp. 98–9.

14. The example is based on Webern’s autograph row chart transcribed in Bailey(1996), p. 196.

15. It is instructive to compare this definition of intervallic motive to the followingaffirmation of Stravinsky’s: ‘Always I have been interested in intervals. Not onlyhorizontally, in terms of melody, but also the vertical results which arise from thecombinations of intervals’ (see again n. 1).

16. Since I am considering unordered pitch-class intervals, it may seem senseless tospeak of their ‘direction’ or ‘orientation’. However, the intervallic motives which arethe focus of my interest here always result from the union of two (or more)conjoined intervals, for which these terms refer simply to the orientation assumedby the intervals relative to each other. In fact, two conjoined intervals united in thesame direction produce a third interval corresponding to their sum (such as ic1 andic4, which together produce ic5 in Ex. 4) and, conversely, two conjoined intervalsunited in opposite directions produce an interval corresponding to their difference(such as ic1 and ic4, which together realise ic3 in Ex. 4). This is representedgraphically by the arrows in my examples, in which the pc sets are conventionallyarranged in their normal form from lowest to highest (for a definition of ‘normalform’ see Straus 1990, p. 27).

17. From here on, the intervallic motives will be represented by two or more arabicnumerals (corresponding to the interval classes) separated by a dash (–) andordered, only as a convention, from the smallest to the largest. For example: 1–2indicates a motivic class which includes the following possible configurations:<+1, +2>, <+1, -2>, <-1, +2>, <-1, -2>, <+2, +1>, <+2, -1>, <-2, +1>, <-2, -1>. Inwhat follows, I will often use, for the sake of simplicity, the term ‘motive’ in the moreprecise sense of ‘motive class’.

18. See in particular Table 1.

19. See for example the definition given in Schoenberg (1967), p. 8. According toDahlhaus (1986), p. 283, the concept of ‘motive’ which correlates with the Schoe-nbergian idea of ‘developing variation’ essentially concerns only the intervallicaspect. Nevertheless, in Schoenberg’s pedagogical writings, the motive is conceived

256 massimiliano locanto

Music Analysis, 28/ii-iii (2009)© 2011 The Author.Music Analysis © 2011 Blackwell Publishing Ltd

Page 37: Locanto - Stravinsky's Late Technique

always as a complex of different interconnected aspects (intervals, rhythm, metricalposition, dynamic level, and so on). See for example Schoenberg (1995), pp.168–71.

20. In Stravinsky’s own words: ‘I begin work by relating intervals rhythmically’ (seen. 1).

21. See Sichardt (1990), pp. 30–52.

22. See for example the motives identified by Sichardt (1990), pp. 50–2, in the com-positions and fragments dating from the years immediately preceding 1919.

23. See Boss (1992), pp. 125–50.

24. See Boss (1994), pp. 194–6.

25. It is more difficult, especially in twelve-note music, to establish whether (and, if so,to what extent) motivic elaboration, applied in the sense of the Schoenbergianconcept of developing variation, also confers a teleological orientation on themusical discourse, as maintained, for example, in Haimo (1997).

26. See for example Perle (1991), pp. 9–38, in particular his analyses of Schoenberg’sOp. 23 No. 1, based on a minor third–semitone cell. A ‘cell’ is defined by Perle (p.9) as a group of pitches which ‘may operate as a kind of microcosmic set of fixedintervallic content, statable either as a chord or as a melodic figure or as a combi-nation of both’.The difference between this and my definition of ‘intervallic motive’is obvious.

27. The same thing can be said of the cells which, according to different authors (seeespecially Perle 1955, Treitler 1959 and Antokoletz 1984, pp. 78–137), play adetermining role in the music of Béla Bartók. Nonetheless, Bartók’s use of inter-vallic cells presents some analogies with Stravinsky’s practice, especially in thepreference for symmetrical aggregates (see for example the cells labelled X,Y andZ in Treitler’s analysis). Different theoretical aspects concerning the use of inter-vallic cells in the music of Bartók are addressed in Antokoletz (1984), p. 16 n. 27and pp. 78–137.

28. This is because the tritone subdivides the octave into two equal parts.

29. The global intervallic content of a set class is represented synthetically by the‘interval vector’ defined in Forte (1972), p. 179.

30. In order to obtain all of the set classes produced by the union of three differentintervals, one can proceed as follows. For each of the six permutations of the threeintervals (1–2–3, 1–3–2, 2–1–3, 2–3–1, 3–1–2, 3–2–1) all of the possible arrange-ments of their directions (ascending or descending) are formed (four: <+ + +>, <+ +->, <+ - +>, <+ - ->). (Note: the four arrangements with initial descending intervalsproduce sets of the same classes as the four preceding arrangements,of which they aresimply inversions.) The number of set classes thus obtained will not necessarily be6 ¥ 4 = 24, because some arrangements will generate the same set class.

31. This latent discrepancy is also recognised by Straus (2001), p. 92, who notes ‘thebasic formal paradox of [Stravinsky’s] music, namely the centrifugal tendency ofthe musical units [intervals] toward isolation and the centripetal tendency of thetranspositions and inversions to link them together into larger wholes’. Neverthe-less, in Straus’s vision the discrepancy concerns ‘the very immediate level of

‘Composing with Intervals’ 257

Music Analysis, 28/ii-iii (2009) © 2011 The Author.Music Analysis © 2011 Blackwell Publishing Ltd

Page 38: Locanto - Stravinsky's Late Technique

structure’, that is, the structuring of the motives and ordered sets used in thecomposition. From my point of view, conversely, the necessity of finding a com-promise between the two different constructive logics conditions the compositionalprocess across multiple levels.

32. The serial symbol ‘II inv’ in the sketch transcribed in Ex. 8a indicates that thepassage is based on the inverse form of the second fundamental twelve-note row ofthe Requiem Canticles.

33. The version featuring the move of a semitone from F� to G is found also in the lowersystem of the sketch reproduced in Ex. 8b. In the upper system of this last sketchone finds a third version, with the F� which leaps directly to B, thus anticipating theB of the following chord.

34. The serial topography of this passage would be difficult to reconstruct without theassistance of the sketches: see for example the serial analyses of Powles (1995), p.288, which are theoretically plausible but in total contradiction of the documentaryevidence.

35. A clarification is needed with respect to the serial symbols O, R, I and RI employedin the sketches. Normally, Stravinsky does not obtain the fourth basic ordering ofthe row (RI) by reading the inverse form (I) backwards (as is typical of Schoenbergand his students), but by inverting the retrograde form (R) (on the respectiveconsequences, see for example Krenek 1940, p. 11). The two forms differ withrespect to the level of transposition: the inversion of the retrograde begins on thesame pitch as the retrograde; the retrograde of the inversion begins on the last pitchof the inversion. In order to avoid a disparity with the sketches which I havetranscribed, I will likewise employ the symbol RI to indicate the inversion of theretrograde. The factor of transposition (Tn) is computed in ascending semitonesfrom 0 to 11 (taking octave equivalence into account).

36. On which there already exists copious literature: see especially Spies (1965a),(1965b) and (1967); Rogers (1968); Hogan (1982); Van den Toorn (1983), pp.442–4; Babbitt (1986) and (1987a); Morris (1988); and Straus (2001), pp. 26–33.Stravinsky’s first composition to make use of rotational arrays is Movements (1958–9). In Locanto (2002), pp. 59–61, I interpret some characteristics of these tables inrelation to the serial procedures employed by Stravinsky in earlier serial composi-tion of the 1950s.

37. The two steps in which the procedure (rotation and transposition) is articulated areclearly illustrated in the autograph table transcribed in Ex. 17: columns a and bcontain the rotated forms; columns g and d contain the rotated-transposed forms.

38. The serial symbols of the sketch can be deciphered as follows: ‘II’ = second funda-mental row; ‘Inv’ = inversion (I); ‘R inv’ = retrograde inversion (RI); ‘a’ = firsthexachord; ‘1st’ = first line of the rotational array; ‘2nd’ = second line of the rota-tional array (etc.); ‘1st, 1 & 2’ = first and second notes of the first line of the array;‘2nd, 1 & 2’ = first and second notes of the second line of the array; and so on.

39. See in particular Tucker (1992), vol. 2, pp. 60–92; see also Pousseur (1971/i), pp.27–30, and Van den Toorn (1983), pp. 390–413.

40. See Locanto (2002), pp. 30–49. The use of motives formed by the combination oftones and semitones is a typical trait of many of Stravinsky’s serial compositions.

258 massimiliano locanto

Music Analysis, 28/ii-iii (2009)© 2011 The Author.Music Analysis © 2011 Blackwell Publishing Ltd

Page 39: Locanto - Stravinsky's Late Technique

Particularly common are the motives defined by Straus as ‘twist motives’, formed bythe tone and semitone conjoined in opposite directions to form set class [012] (seefor example Straus 2001, p. 91). These motives can be identified in many twelve-note rows, from Surge, Aquilo (Canticum sacrum) to The Owl and the Pussy-Cat (seethe statistic in Straus 2001, p. 90 n. 13, based on Jers 1986, pp. 33–5) and in severalpre-serial compositions (see for example Carter 1997). Although Straus (p. 90)observes that the semitone and tone often appear united in the same direction toform a set of class [013], in his analyses the two forms are considered to be distinctmotives. For example, his analysis of the serial melody of Fanfare for a New Theater(p. 91) notes only the presence of the motive [012] – in which the two intervals areoriented in opposing directions (see Schema 1). However, considering the set [013]as another form of the same motivic class 1–2, one will see that the row contains twomore motives between B and C�–D and between D–C and D�, thus appearing as acontinuous chain of seven motives (see Schema 2).

Schema 1A A B C D C D F E F G G

<–1, +2> <+1, –2> <+2, –1> <+2, –1>

<–1, +2>

Schema 2

41. As was demonstrated by Tucker (1992), vol. 2, p. 186, on the basis of a thoroughexamination of the sketches. For the sake of further clarity and completeness, I willbriefly summarise the way in which the principal rows employed in this part of theballet are obtained from the initial tetrachord (for further details I refer the readerto Tucker 1992, vol. 2, pp. 182–242; on the technique of tetrachordal linkage, seealso Van den Toorn 1983, pp. 409–14, and Locanto 2002, pp. 34–47). The serialheptachord G–A�–C�–B�–A–C–D� that appears in the coda of the Pas-de-Deux (bars495–496) results from the union through a common tone of a <0, 1, 4, 3> tetra-chord (G–A�–C�–B�) and its RI form (B�–A–C–D�). Then, by combining through acommon tone two forms of this heptachord placed a tritone apart (D–E�–G�–F–E–G–A� + A�–A–C–B–B�–D�–D), Stravinsky forms the thirteen-note row employed inthe second section (Adagio) of the Pas-de-Deux – a row containing all twelve pitchclasses with a single repetition. The twelve-note row used in the two final move-ments of the ballet (Four Duos and FourTrios) is obtained by uniting, through twocommon tones, the heptachord of the coda of the Pas-de-Deux with another row ofseven notes, arrived at through the union (again via a common tone) of thefundamental tetrachord <0, 1, 4, 3> and a slightly modified form of that tetrachord,ordered as <1, 4, 3, 0> instead of <0, 1, 4, 3>.

42. That is, from the Pas-de-Quatre to the Triple Pas-de-Quatre.

43. As Tucker (1992), vol. 2, pp. 76–80 (on the basis of the transcribed sketches, p. 13),demonstrates, a point of contact between the use of intervallic motives in the strictlymelodic sense, which characterises the non-serial movements of the ballet, and theserial technique employed in the following movements can be glimpsed in bars104–107 of theTriple Pas-de-Quatre, where the 2–1–2 motive, which first appearedin various configurations (see for example its use in the melody of Ex. 11), nowbecomes fixed in the form of the ordered tetrachord <1, 3, 2, 0>, which from thatmoment comes to be used as the referential row.

‘Composing with Intervals’ 259

Music Analysis, 28/ii-iii (2009) © 2011 The Author.Music Analysis © 2011 Blackwell Publishing Ltd

Page 40: Locanto - Stravinsky's Late Technique

44. See also the clarinet melody in bars 67–68 of the Double Pas-de-Quatre, analysedin Pousseur (1971/i), p. 42.

45. See for example Stravinsky and Craft (1966), p. 60, where the twelve-note row ofthe Variations is described as ‘a succession of notes that came to my mind as amelody’.

46. The genesis of the row for Epitaphium finds strong confirmation in a page ofsketches which was addressed recently by Straus (2001), pp. 61–3. Straus’s analysesfind within the row’s structure a certain number of segments belonging to set class[016], separated by the distance of T4 (see Straus 2001, pp. 61–3, 99–102 and130–1).The importance of the semitonal relation, which Stravinsky pointed out, isset in relief in Locanto (2002), pp. 128–31.

47. See Craft (1972), p. 98. As I have already mentioned, in the Requiem Canticles twodifferent fundamental rows are employed, indicated in the sketches by romannumerals I and II. Nevertheless, the two rows are employed simultaneously only inthe Interlude and the Postlude, while the remaining movements employ themalternatively.

48. According to the date of the sketches and the testimony of Robert Craft (seeStravinsky 1984, pp. 467–71), the Interlude was composed between the middle ofMarch and 17 October 1965.

49. For a complete inventory and a summary description of these sketches, see Locanto(2002), pp. 115–17.

50. See Locanto (2002), pp. 116–17.

51. Stravinsky (1984), p. 467, maintains that this sketch was preceded by a little sketch,dated March 1965, containing three annotations of row II, in each of which variousmetrical indications are visible, followed by a brief musical passage (also based onrow II) of which there is no trace in the final score. However, there is no proof thatthis sketch definitely preceded the sketch transcribed in my Ex. 12a, and in any caseit does not contain annotations of row I.

52. Currently the second clipping is detached from the piece of cardboard, on whichremain the traces of the adhesive tape which originally held them together.

53. Indeed, the row extracted from this phrase is numbered I.

54. See Straus (2001), pp. 49–52.

55. Reproduced in Craft (1967), pp. 13–17.

56. The row was then to have been employed in a symphonic composition, which wasnever brought to completion.

57. See Straus (2001), pp. 49–52. The divergence between my analysis and that ofStraus depends, even in this case, on our different conceptions of ‘intervallicmotive’, with respect to which see nn. 6 and 40. In this case, moreover, thedifference is accentuated by the fact that in my analysis, as opposed to Straus’s, thepresence of the motive also arises in the vertical dimension.

58. The row is clearly visible in Newman’s final photograph.

59. Note also that the enharmonic spelling of the row clarifies its basis in the circle offifths.

260 massimiliano locanto

Music Analysis, 28/ii-iii (2009)© 2011 The Author.Music Analysis © 2011 Blackwell Publishing Ltd

Page 41: Locanto - Stravinsky's Late Technique

60. One could perhaps glimpse an allusion to this substantial modification in Stravin-sky’s story, reported by Craft, commenting on Newman’s photographs: ‘During themorning of December 13, Stravinsky mentioned the “need to put an idea in order”,but when the sketch was completed, in about thirty-five minutes, he said that themusic he had actually written was something different and that it had not been inhis mind as long as an hour before. And he always seems to know exactly when hisimagination is at the starting line. Shortly after he had finished, Mrs. Stravinskyclapped her hands together in the hall below his studio, her signal that lunch isready, and Stravinsky applauded back, his signal that he is, too. At table, to showwhat he meant by putting in order, the composer placed three wine glasses inparade formation, then interchanged the first and third, saying, “It is a matter ofknowing that the notes must be this way and not the other” ’ (Craft 1967, p. 13, italicsadded; see also Craft 1972, pp. 303 and 311).

61. The dotted minim in the middle of the third line of the sketch shown in Ex. 14bclearly appears to be a B3 in the sketch. However, this is obviously an error derivingfrom the inconsistent inscription. The understood note is doubtless C4, as is seeneither in the serial hexachord employed in the passage (first line above in thesketch), or in the final score (bar 47, trombone I).

62. Which stand for RIa and Rb, respectively.

63. This demonstrates, once again, that from Stravinsky’s point of view the order ofsuccession of the notes was less important than their global intervallic content.

64. On the use of this technique in ‘De elegia tertia (Sensus spei)’ of Threni, see Tucker(1992), vol. 2, p. 251, where the author establishes, with the assistance of thesketches, the serial segments employed for the construction of the tenor melody of‘Eradicationem’ (bars 252–259). However, according toTucker, Stravinsky’s choiceof various segments ‘was determined by no compositional “system” ’ (p. 251). Onthe contrary, in Locanto (2002), pp. 134–7, I attempt to demonstrate that in all ofthe preliminary versions of the passage, as in the final version, the choice isdetermined by the desire to create a dense chain of overlapping 1–5 motives.

65. The complete autograph short score comprises four separate pages. Only the pagetranscribed in Ex. 15a – which, unlike the others, is merely drafted – contains serialsymbols. Its content corresponds to the printed score, with the sole exception of theE in place of D in the third bar of the tenor.The reading in the score (D) is probablyincorrect, inasmuch as the E of the autograph short score finds confirmation in theserial tables. However, even if this reading is treated as an intentional variantintroduced by Stravinsky (it appears for the first time in the clean copy of the score),that does not challenge the validity of my analysis, which concerns the creativeprocess from its earliest stages.

66. Note also the effect created by the occasional doubling of pitches in the vocal parts.

67. Starting in bar 4, the pitch organisation of the choral part is based on a differentserial technique, which involves reading the rotational arrays vertically. On thistechnique see especially Straus (2001), pp. 152–64.

68. See Stravinsky and Craft (1960), pp. 100–1. Analyses of the flute solo can be foundin White (1966), p. 612; Müller (1984); Babbitt (1986), p. 255; Tucker (1992), vol.2, p. 258; Rust (1994), pp. 64–71; and Straus (2001), pp. 65–8 (based on Rust) and125–30.

‘Composing with Intervals’ 261

Music Analysis, 28/ii-iii (2009) © 2011 The Author.Music Analysis © 2011 Blackwell Publishing Ltd

Page 42: Locanto - Stravinsky's Late Technique

69. Set class [012] could also be formed by uniting two semitones, but the motivesformed by two different intervals are far more typical of Stravinsky’s music.

70. The sketches which I consider (all in the Paul Sacher Foundation, Igor StravinskyCollection) include (1) the sketch for bars 13–17 transcribed in Ex. 19a; (2) thepage of sketches with two versions of bars 7–12 transcribed in Ex. 21; (3) a clippingwith a preliminary version of bars 18–21, without serial symbols (not reproducedhere); (4) a clipping with the definitive version of bars 18–21 and (on the other side)bar 22, without serial symbols (not reproduced here); and (5) the autograph shortscore, some of the serial symbols from which are transcribed in Ex. 22. If oneexcludes the serial analysis of the accompaniment in bars 13–17 in Rust (1994), p.70, based on the score (no. 5), sketches nos 2–5 of the list have never beenconsidered until now. In contrast, sketch no. 1 has been cited and discussednumerous times in the musicological literature. One of the first transcriptionsappeared in Neidhöfer (1991) (in the same year Joseph Straus presented thisdocument in a paper at the Annual Meeting of the Society for Music Theory inCincinnati, OH). Later the document was analysed by Tucker (1992), vol. 1, p. 66(transcription) and (1992), vol. 2, p. 258 (commentary); Rust (1994), pp. 63–4; andStraus (2001), p. 67, which finally established the serial construction of the flutesolo, without, however, comparing it with either the construction (actually quitesimilar) of the other two episodes, or with the motivic-intervallic construction towhich it is subjected.

71. The major discrepancy concerns the first two segments: in the short score of thepiano episode they are labelled segment 3–6 of hexachord g5 and segment 5–6 ofhexachord d1. In the sketch for the flute episode, however, they are labelled segment1–3 of hexachord g1 and segment 4–6 of hexachord d1. However, note that whenthe two piano segments are transposed by T2 (rather than T4, as in the flute), theresulting pitches are the same as in the flute episode.

72. In the first version, the serial segments were transposed a major third higher (T4);the second version, on the other hand, was obtained by lowering the first version bya tone (T2). Therefore, in the second version, the serial segments prove to betransposed a tone higher (T4 - T2 = T2). The short score does not contain serialsymbols for the final part of the passage, formed by the succession of thirteenpitches notated in the final part of the third line of Ex. 21. The same succession ofpitches – this time without the initial C� – is transcribed again lower on the page(lines 5–6) with a different choice of octave register for the single notes and usinga rhythm outlined for the final cadence.

73. The B that appears as the first note after the treble clef corresponds to the D� of thepreceding version, placed immediately before the treble clef. See also the line drawnby Stravinsky. A different version of the final part of the phrase alone (starting withthe dyad B�–A on line 13) is also notated by hand in the lower part of the page (lines12–14).

74. The error in the printed score has created some difficulties in the analyticalliterature. See for example Rust (1994), p. 68, where the omission of the notes Band C prevented the author from recognising the T2 relation between the last tennotes of the first piano episode and the corresponding notes of the flute solo.Moreover, Rust’s analysis omitted the last bar (bar 22) of the second episode for thepiano, so that he failed to notice the T8 relation with the flute solo in the last sixnotes.

262 massimiliano locanto

Music Analysis, 28/ii-iii (2009)© 2011 The Author.Music Analysis © 2011 Blackwell Publishing Ltd

Page 43: Locanto - Stravinsky's Late Technique

75. See Boykan (1963), p. 158; Walden (1979); Rust (1994), pp. 62–76; and Straus(2001), pp. 124–8.

76. On the basis of the principle of rotation on which the serial tables are based, onecould say that the pitch ‘following’ the final pitch of a hexachord is the first.

77. For the sake of convenience, in Ex. 24 the entire succession of pitches is transposedto F (as in the first piano episode).

78. Note that, in the definitive score, one of the very rare points at which this continu-ous motivic chain is interrupted corresponds to the juncture at which Stravinskyomitted the notes B and C in the first episode.

REFERENCES

Antokoletz, Elliott, 1984: The Music of Béla Bartók: a Study of Tonality andProgression in Twentieth-Century Music (Berkeley and Los Angeles, CA: Uni-versity of California Press).

Babbitt, Milton, 1960: ‘Twelve-Tone Invariants as Compositional Determi-nants’, Musical Quarterly, 46/ii, pp. 246–60.

______, 1961: ‘Set Structure as a Compositional Determinant’, Journal of MusicTheory, 5/ii, pp. 72–94.

______, 1968: ‘Remarks on the Recent Stravinsky’, in Benjamin Boretz andEdward T. Cone (eds.), Perspectives on Schoenberg and Stravinsky (Princeton,NJ: Princeton University Press), pp. 165–85.

______, 1986: ‘Order, Symmetry, and Centricity in Late Stravinsky’, in JannPasler (ed.), Confronting Stravinsky (Berkeley, CA: University of CaliforniaPress), pp. 247–61.

______, 1987a: ‘Stravinsky’s Verticals and Schoenberg’s Diagonals’, in EthanHaimo and Paul Johnson (eds.), Stravinsky Retrospectives (Lincoln, NE:University of Nebraska Press), pp. 15–35.

______, 1987b: Words about Music, ed. Stephen Dembski and Joseph N. Straus(Madison, WI: University of Wisconsin).

Bailey, Kathryn, 1996: ‘Webern’s Row Tables’, in Kathryn Bailey (ed.), WebernStudies (Cambridge: Cambridge University Press), pp. 170–228.

Beach, David, 1976: ‘Segmental Invariance and theTwelveTone System’, Journalof Music Theory, 20, pp. 157–84.

Berger, Arthur, 1963: ‘Problems of Pitch Organization in Stravinsky’s DiatonicMusic’, in Benjamin Boretz and Edward T. Cone (eds.), Perspectives onSchoenberg and Stravinsky (Princeton, NJ: Princeton University Press), pp.123–55.

Boss, Jack, 1992: ‘Schoenberg’s op. 22 Radio Talk and Developing Variation inAtonal Music’, Music Theory Spectrum, 14/ii, pp. 125–50.

______, 1994: ‘Schoenberg on Ornamentation and Structural Levels’, Journal ofMusic Theory, 38/ii, pp. 187–216.

Boykan, Martin, 1963, ‘Neoclassicism in Late Stravinsky’, Perspectives of NewMusic, 1/ii, pp. 155–69.

‘Composing with Intervals’ 263

Music Analysis, 28/ii-iii (2009) © 2011 The Author.Music Analysis © 2011 Blackwell Publishing Ltd

Page 44: Locanto - Stravinsky's Late Technique

Carter, Chandler, 1997: ‘Stravinsky’s “Special Sense”: the Rhetorical Use ofTonality in The Rake’s Progresss’, Music Theory Spectrum, 19/i, pp. 55–80.

Craft, Robert, 1967: Bravo Stravinsky. Photographs by Arnold Newman (Cleve-land, OH: World).

______, 1972: Stravinsky:Chronicle of a Friendship,1948–1971 (NewYork: Knopf).Dahlhaus, Carl, 1986: ‘Was heisst “entwickelnde Variation”?’ in Rudolph

Stephan and Sigrid Wiesmann (eds.), Bericht über den zweiten Kongress derInternationalen Schönberg-Gesellschaft (Wien: Universal), pp. 280–5.

Forte, Allen, 1972: The Structure of Atonal Music (New Haven, CT: Yale Uni-versity Press).

Haimo, Ethan, 1997: ‘Developing Variation and Schoenberg’s Serial Music’,Music Analysis, 16/iii, pp. 349–65.

Hogan, Catherine, 1982: ‘Treni: Stravinsky’s “Debt” to Krenek’, Tempo, 141, pp.22–9.

Jers, Norbert, 1986: Igor Stravinskys spate zwölf-tonWerke (1958–1966) (Regens-burg: Gustav Bosse).

Johnson, Paul, 1987: ‘Cross-CollectionalTechniques of Structure in Stravinsky’sCentric Music’, in Ethan Haimo and Paul Johnson (eds.), Stravinsky Retro-spectives (Lincoln, NE: University of Nebraska Press), pp. 55–75.

Krenek, Ernst, 1940: Studies in Counterpoint Based on the Twelve-Tone Technique(New York: G. Schirmer).

Lewin, David, 1962: ‘A Theory of Segmental Association in Twelve-ToneMusic’, Perspectives of New Music, 1/i, pp. 89–116.

Locanto, Massimiliano, 2002: ‘Pensiero musicale e procedimenti costruttivinell’ultimo Stravinsky’ (PhD diss., University of Pavia).

Morris, Robert, 1988: ‘Generalizing Rotational Arrays’, Journal of Music Theory,32/i, pp. 75–132.

Müller, Alfred, 1984: ‘Igor Strawinsky: Movements for Piano and Orchestra’,Melos, 46/ii, pp. 112–39.

Nabokov, Nicolas, 1949: ‘Christmas with Igor Stravinsky’, in Edwin Corle (ed.),Igor Stravinsky (New York: Sloan & Pearce), pp. 123–68.

Neidhöfer, Christoph, 1999: ‘An Approach to Interrelating Counterpoint andSerialism in the Music of Igor Stravinsky, Focusing on the Principal DiatonicWorks of His Transitional Period’ (PhD diss., Harvard University).

______, 1991: ‘Analysearbeit im Fach Komposition/Musiktheorie über dieMovements for Piano and Orchestra von Igor Strawinsky’ (master’s thesis,Musik Akademie der Stadt Basel).

Perle, George, 1955: ‘Symmetrical Formations in the String Quartets of BélaBartók’, Music Review, 16, pp. 300–12.

______, 1991: Serial Composition and Atonality: an Introduction to the Music ofSchoenberg,Berg,andWebern, 6th revised edn (Berkeley and Los Angeles, CA:University of California Press).

Pousseur, Henry, 1971: ‘Stravinsky selon Webern selon Stravinsky’, Musique enjeu, 3/i–ii, pp. 21–47 and 107–26.

264 massimiliano locanto

Music Analysis, 28/ii-iii (2009)© 2011 The Author.Music Analysis © 2011 Blackwell Publishing Ltd

Page 45: Locanto - Stravinsky's Late Technique

Powles, Jonathan C., 1995: ‘Continuity and Discontinuity in the Music ofStravinsky: Analysis, Theory and Meta-Theory’ (PhD diss., OxfordUniversity).

Rogers, John, 1968: ‘Some Properties of Non-duplicating Rotational Arrays’,Perspectives of New Music, 7/i, pp. 80–102.

Rust, Douglas, 1994: ‘Stravinsky’s Twelve-Note Loom: Composition and Pre-composition in Movements’, Music Theory Spectrum, 16/i, pp. 62–76.

Schoenberg, Arnold, 1967: ‘Fundamentals of Musical Composition’, ed. GeraldStrang and Leonard Stein (Boston and London: Faber and Faber).

______, 1995: The Musical Idea,and the Logic,Technique,and Art of Its Presentation,ed. and trans. Patricia Carpenter and Severine Neff (New York: ColumbiaUniversity Press).

Sichardt, Martina, 1990: Die Entstehung der Zwölftonmethode Arnold Schönbergs(Mainz: Schott).

Smyth, David H., 1997: ‘Stravinsky at the Threshold: a Sketch Leaf for Canti-cum sacrum’, Mitteilungen der Paul Sacher Stiftung, 10, pp. 21–6.

______, 1999: ‘Stravinsky’s Second Crisis: Reading the Early Serial Sketches’,Perspectives of New Music, 37/ii, pp. 117–46.

______, 2000: ‘Stravinsky as Serialist: the Sketches for Threni’, Music TheorySpectrum, 22/ii, pp. 205–24.

Spies, Claudio, 1965a: ‘Some Notes on Stravinsky’s Abraham and Isaac’, Per-spectives of New Music, 3/ii, pp. 186–209.

______, 1965b: ‘Some Notes on Stravinsky’s Variations’, Perspectives of NewMusic, 4/i, pp. 62–74.

______, 1967: ‘Some Notes on Stravinsky’s Requiem Settings’, Perspectives of NewMusic, 5/ii, pp. 98–123.

Straus, Joseph N., 1990: Introduction to Post-Tonal Theory (Englewood Cliffs, NJ:Prentice-Hall).

______, 1991: ‘The Progress of a Motive in Stravinsky’s The Rake’s Progress’,Journal of Musicology, 9/ii, pp. 165–85.

______, 2001: Stravinsky’s Late Music (Cambridge: Cambridge UniversityPress).

Stravinsky, Igor, 1984: Selected Correspondence, ed. Robert Craft, vol. 2 (Bostonand London: Faber and Faber).

Stravinsky, Igor and Craft, Robert, 1959: Conversations with Igor Stravinsky(Garden City, NY: Doubleday).

______, 1960: Memories and Commentaries (Garden City, NY: Doubleday).______, 1962: Expositions and Developments (Garden City, NY: Doubleday).______, 1966: Themes and Episodes (New York: Knopf).______, 1969: Retrospectives and Conclusions (New York: Knopf).Taruskin, Richard, 1993: ‘TheTradition Revisited: Stravinsky’s Requiem Canticles

as Russian Music’, in Christopher Hatch and David W. Bernstein (eds.),Music Theory and the Exploration of the Past (Chicago: University of ChicagoPress), pp. 525–50.

‘Composing with Intervals’ 265

Music Analysis, 28/ii-iii (2009) © 2011 The Author.Music Analysis © 2011 Blackwell Publishing Ltd

Page 46: Locanto - Stravinsky's Late Technique

______, 1996: Stravinsky and the RussianTradition:a Biography of theWorks through‘Mavra’ (Berkeley and Los Angeles, CA: University of California Press).

Treitler, Leo, 1959: ‘Harmonic Procedure in the Fourth Quartet of Béla Bartók’,Journal of Music Theory, 3/ii, pp. 292–98.

Tucker, Susannah, 1992: ‘Stravinsky and His Sketches: the Composition of Agonand Other SerialWorks of the 1950s’, 2 vols. (PhD diss., Oxford University).

Tymoczko, Dmitri, 2002: ‘Stravinsky and the Octatonic: a Reconsideration’,Music Theory Spectrum, 24/i, pp. 68–102.

Van den Toorn, Pieter C., 1983: The Music of Igor Stravinsky (New Haven, CT:Yale University Press).

Van den Toorn, Pieter and Tymoczko, Dmitri, 2003: Colloquy about ‘Stravinskyand the Octatonic: a Reconsideration’, Music Theory Spectrum, 25/i, pp.167–203.

Walden, William, 1979: ‘Stravinsky’s Movements for Piano and Orchestra: theRelationship of Formal Structure, Serial Technique, and Orchestration’,Journal of the Canadian Association of University Schools of Music, 9/i, pp.73–95.

Wason, Robert W., 1996: ‘A Pitch-Class Motive in Webern’s George Lieder, Op.3’, in Kathryn Bailey (ed.), Webern Studies (Cambridge: Cambridge Univer-sity Press), pp. 111–34.

White, Eric Walter, 1966: Stravinsky: the Composer and His Work (Berkeley, CA:University of California Press).

ABSTRACT

In Stravinsky’s final serial works, the intervallic component assumed a moredecisive role than in his in earlier compositions, becoming the foundationalaspect of a ‘motivic’ technique, the specific aspects of which are illustrated herethrough several sketch-based analyses. In Stravinsky’s case, motivic-intervallicsyntax and serial procedures operate according to slightly but significantly dif-ferent criteria: the first on the level of single intervals, the second on the level ofpitch-class sets. Using several specific examples drawn from compositionsranging from Agon to the Requiem Canticles, this article demonstrates that thediscrepancy provided Stravinsky with a stimulus, rather than an obstacle, tocomposition, and provides a guide to the interpretation of certain well-knowncharacteristics both of his creative process and of his serial technique. Themusic-theoretical aspects of Stravinsky’s intervallic syntax are illustrated, and itsinteraction with serial technique is observed from two conceptually differentdirections: from intervallic motives to rows (the initial definition of a row ofpitches), and from the row to intervallic motives (the transformation of the‘abstract’ row into concrete musical contexts).

266 massimiliano locanto

Music Analysis, 28/ii-iii (2009)© 2011 The Author.Music Analysis © 2011 Blackwell Publishing Ltd