Series, Form and Function: Comments on the Analytical Legacy of René Leibowitz and Apsects of Tonal Form in the Twelve-Tone Music of Schoenberg and Webern 1

John MacKay

Although the most important of Rend Leibowitzs writings on the series and thematic process, his "Treatise on Twelve-Tone Composition," remains unpublished, important indications of his analytical perspectives are to be be found in his discussion of the Webern Concerto op. 24 in Que'est-ce que c'est que la musique de douze sons? of 1948. 2 The present study will review portions of both of these writings and advance observations on tonal linearity and centering which, although extraneous to Leibowitz's analytical outlook, are clearly functional within the context of his structural and serial parsings. In many pieces a clear sense of tonal form and "argumentation" can be observed in much the manner that Schoenberg described in his Theory of Harmony,3 and his article "Problems in Modern Harmony"4 (1934) and this will be pursued at greater length in concluding analyses of the Minuet of the Suite op.25 for piano solo and the Klavierstücke op. 33b.

Qu'est-ce que c'est que la Musique de Douze Sons? and Classical Phrase Structure

The discussion of the structural function of the series in Qu'est-ce que c'est que la musique de douze sons? is embroidered with issues of esthetics, orchestration, twelve-tone technique and classicism. It is here that Leibowitz first juxtaposes thematic statements of Webern and Beethoven (the first movement theme of the Sonata op.2, #1 was also used by Webern in his The Path to the New Music 5) illustrating their respective "sentence" structures (see Example 1a) and making observations on Webern's micro-serialism (i.e. the intervallic retrograde, inversion and retrograde inversion of the three-note cellular motives of the series) throughout the thematic and motivic design. The model/response relationship of the Schoenbergian sentence is shown to articulate the functions of tonic/dominant in the Beethoven and prime form and retrograde serial structures, as well as the orchestrational contrast of the concertino and solo timbres in the Webem. The reductions of both thematic structures involve rhythmic intensifications leading to cadences on the dominant in Beethoven and an unfolding of the retrograde form in Webern which, as in the "response,u may indeed be his serial substitute for the dominant. Leibowitz observes therefore, that what Beethoven accomplishes in his preparation and cadence via a dynamic climax, ritardando and melodic descent onto the dominant, Webern's does in a similar dynamic intensification and a diminuendo, deceleration and increase in harmonic density.

Leibowitz's notion of the second theme in the Webern (he does not pursue any further parallelism with the Beethoven theme) is somewhat problematic in  Qu'est-ce que c'est que la musique de douze sons? since it does not correspond to the rhetoric of the tempo changes. Instead of admitting a transitional link to a new structure in measures 11 to 12 (see Example 1 b) Leibowitz sees the shift to more subdued dynamics and to the serial superpositions of these measures as the immediately distinguishing factors of the second theme. A developmental extension of the theme is seen in measures 13-22, and a third section consisting of a more cumulative elaboration is proposed in mm. 22-23 prior to the brief concluding phrase (mm. 24-25) which closes out the opening "A" section of the A-B-A sonata design of the movement.

Example 1a

Example 1b

Leibowitz's hyper-segmentation of these passages overlooks, however, a more contrapuntally complex model and repetition in measures 13-14 and 15-16 with distinctive recurrences of upper register G#s of m. 14 and 16 and mid-register E flats in measuresl 3 and 17. As in a sentence form, these provide a stable point of departure for the ensuing development of overlapping trichords of the following measures. Leibowitz does however make the important observation of the climactic resolution in m. 23 to a single sequential series which concludes the expository alternation between the successive serial dispositions of the primary theme and the harmonic relaxation into the superimposed series of the secondary ideas and in the structures which follow in the central section of the movement in measures 26-45.

Leibowitz similarly divides the central section into "phases": an initial phase (mm. 26 - 35) which has certain thematic qualities in its presentation and immediate varied repetition of the motivic cell followed by reductive shorter phrases (involving new grace-note figures) and a temporary cadence. A second phase (not shown) then appears to develop features of the first and second themes of the exposition to a registral and rhythmic climax in anticipation of the sixteenth-note figures of the return of the opening theme. The discussion of these measures relates more to orchestration - the role of the piano in the ensemble texture - and to the series, focusing pedagogically on the use of "pivot" tones between successive rows in Webern's serial technique. The central section however, also contributes certain features of tonal reference. The melodic trichords of Leibowitz's second theme in ms. 12 and 13 (A -C-B natural overlapping with E-E -G) are recalled in the opening thematic fragments of measures 26 and 27, and the cadential dyad E-E natural in the trumpet at the end of the exposition and at the end of the central section in the trombone.

Leibowitz comments on the condensations in the "reprise" (or recapitulation) of significant components of the exposition, particularly the elision of the "repetition" and in the abbreviated versions of the reductions and cadence in the first theme. Although the second theme is only slightly abbreviated, Leibowitz makes much out of the climactic extension which is expanded by the intercalation of measures 57 and 58 in the clarinet and oboe - recalling the grace-note figures of the central episode and providing a midregister preparation for the climax in quarter-note triplets and for the concluding, although still abbreviated, fortissimo statement of the first theme.

Leibowitz's structural insights can be pursued further in the resolution of the movement where the closing thematic iterations of the series (ms. 63-68) echo the presentation of the beginning of the recapitulation in ms. 45-49, saturating this.particular trichordal partitioning 6 of the series. In contrast, the final cadence in measure 69 reasserts the trichordal partitioning of the model and repetition of the very opening presentation of the first theme of the movement. This reflects further structural oppositions since the trichordal partitioning of the beginning of the secondary and central themes is also that of the model, repetition and reductions of the closing thematic statement, (ms. 63-68), and in the very opening theme itself the initial partitioning gives way to that of the secondary ideas at the beginning of the reductions (compare mm. 6-7 and 11-13 in the concertino.) The role of the cadential and pivotal E-E dyad is further echoed in measure 62 as the climactic development of the secondary theme unravels in anticipation of the final recapitulation of the opening thematic structure of the movement.

Leibowitz's discussions of the second and third movements of the Concerto op. 24 pursue the same sensitive treatments of formal structure, orchestration and serial disposition with which he began in the first movement and later continued in his Introduction a la Musique de Douze Tons. In these writings however, he does not pursue the same detailed attention to thematic structure and function in the disposition of the series. An opportunity in the Concerto is missed particularly in the second movement where, as observed by later analysts 7 , a very clear and elaborate period structure exists (see Example 2.)8 The symmetrical motivic structure of the antecedent plays with emphases on E , often in association with G natural (i.e. the motivic correspondence of measures land 5 and the cadential G-E of ms. 11 and. 12.) While the consequent moves away from these emphases, there is a clear return to the centering of a midregister E and accompanying G natural at the end of the central section (ms. 51-58, not shown) and at the close of the movement (ms. 74-75). E is similarly the goal of the interlocking descending thirds which begin in measure 65 D-B (ms.64-65), B-G (m.66-67), F#-D (ms. 69-70), E-13 in measure 72 before the final G-E in measures 74-75.

Example 2:

"A Treatise on Twelve-Tone Composition"

Leibowitz's "Treatise on Twelve-Tone Composition" was compiled as a teaching manual for his students in the early 1950's.9 It offers step-by-step pedagogical orientations in the use of the series in relation to Schoenbergian concepts of classical thematic structure, development, and formal process. In the first part of the treatise, he differentiates between "closed structures," the typical structures of prinicpal themes, "intermediary (i.e. transitional) structures" and "secondary" structures or the typical structures of secondary themes. He provides a wealth of examples from late Webern (Symphony op.21, Concerto op.24, Variations op.27, and Kantata op.29) and from Schoenberg's music of the 1930's and 1940's (Klavierstücke op. 33 a and b, the Fantasy for violin and piano, the Concerto for piano and orchestra, the Fourth String Quartet, Accompaniment to a Cinematographic Scene op.34, Chamber Concerto op.43 and Survivor from Warsaw .46.) A second portion of the treatise concerns the typical form of "re-expositional" structures (developments and recapitulations) as well as what he terms "autonomous structures" (introductions, episodes and cadenzas, codas, stretti and terminal cadential seqments) and a third part discusses "various thematic structures" (the "scherzo," slow movement and finale themes) "canons, fugatos and variations," "small forms" and "problems of dramatic forms and of forms in one movement." In addition to this skeleton of topics, the treatise, like Qu'est-ce que c'est que la Musique de Douze Sons? offers valuable asides on the classical elements of Schoenberg's compositional esthetics - variation process and musical thought, for example, and the "crisis of Romanticism" and its solution in the twelve-tone method etc. - which, unfortunately, cannot be treated in any specificity here. It is hoped nevertheless, that the following analytical survey will offer some tantalizing insight into the importance of Leibowitz's relationship to this music and his influence on the rising generation of mid-century composers.

Thematic Structure: The Sentence, Period and "Lied" Forms

Perhaps the easiest way to survey the "Treatise on Twelve-Tone Composition" is through its examples. Leibowitz continues his practice of juxtapposing thematic structures of Webern and Schoneberg and familiar themes from the classical literature, pairing in the first instance, Schoenberg's theme in Accompaniment to a Cinamatographic Scene op. 34 with the opening sentence from Beethoven's Sonata op.2 #2 (see Example 3). In this case, Leibowitz demonstrates the straightforward application of the sentence structure to the melodic segmentation of the series. The model articulates the first six notes of the series (subdivided, or "truncated" in Leibowitz's words, into groups of three and three) followed by the remaining six in the varied repetition of the model. The reductions and cadence are based on the inversion, beginning similarly on the E and culminating in a graceful arch on the emphatic G before receding to a midregister cadence. (Note the duplication and octave shift of the climactic F-G dyad.) In addition to Leibowitz's structural insights, it is also possible to notice a very deliberate chromatic linearity which is articulated in Schoenberg's sentence structure, slipping downward from the opening E to D natural and C# in the model and from A natural to A a d G in the repetition. The initial E -D is repeated and developed in the beginning of the reductions and the cadence ultimately recaptures the D of the opening model, but descends a step further to B natural instead of the C natural of the second measure.

Example 3: Schoenberg Accompaniment to a Cinematographic Scene op. 34 measures 1 - 8 Beethoven Sonata op. 2 #2, opening theme, first movement (Example 1 from "Treatise on Twelve-Tone Composition.")

Copyright @ 1930. Reproduced by permission of C.F. Peters Corporation on behalf of Hein richshofen's Verlag, Wilhelmshaven.

Leibowitz's example of a more expansive, developmental sentence structure in the first theme of the second movement of the Schoenberg Violin Concerto is appropriately paired with the opening theme of the Brahms Third Symphony (see Example 4). In both cases he points to a neutralization of the model (its reduction to a single repeated interval) in the sequences toward the cadence. The serial disposition in the Schoenberg also provides an example of the use of simultaneous row forms, putting the prime form in the violin melody against the inversion (at the interval of a perfect fifth below) in the orchestral accompaniment with a substantial amount of what might be called "pedalling" or 11 recycling" of the tones in the individual row segments row which are defined by the structural components of the theme. Leibowitz also notes the harmonic intensification of the accompaniment in relation to the melody beginning in the reductions where more and varied row forms are used toward the cadence as the solo melody echoes the A-C dyad through its different registers. 10

Example 4:

Once again, Leibowitz's analysis can be supplemented with observations on the linear and tonal development of the theme. In this example however, the linear emphases do not conform to the components of the sentence structure. The ascending chromatic line in the solo from F natural through G flat in measure 4 is completed in the G natural to A in measure 6 of the reductions which is a rhythmic variation of the descending quarter-note gesture of the opening model and unfolds downwards with strong harmonic implications through D natural-E flat and A-natural-C.

Leibowitz presents a further example of a thematic sentence from Schoenberg's Prelude op.46 "A Survivor from Warsaw" (see Example 5) to illustrate the partitioning of a single series between the melody and (in this instance,) a harmonic background in tremolo thirds. Leibowitz ascribes the function of a "hyphen" to the repetition of the G#-B between the last melodic interval of the model and the background sonority of the repetition. A similar process is observed in the tuba solo reductions, taking the two tones of the harmonic background as the initial tones of the melodic figures based on the first six tones of the prime form. However, as in the Leibowitz's previous example from the Schoenberg Concerto the reductions involve serial as well as a motivic development. Here rows become superimposed as the accompaniment involves the first six tones of the inversion, and the cadence unfolds the remaining hexachords of both of these forms. The complexity of the cadence, however, prompts Leibowitz to suggest that this particular sentence is a more thematically open structure which serves as a freestanding introduction to the work as a whole.

Example 5: Schoenberg Prelude "Survivor from Warsaw" op. 46, measures 1 -11. (Example 4 in "Treatise on Twelve-Tone Composition.")

Used by permission of Belmont Music Publisher, Pacific Palisades CA 90272.

Examples of simple period or antecedent/consequent structures which are less familiar in the Brahmsian spirit of Schoenberg's music are more common in Webern. Leibowitz makes an intriguing comparison of the structure of the theme from the third movement variations of the opus 27 and a well-known theme from Haydn (Sonata Hoboken #49, first movement, see Example 6). Here the structure is a type of developed period consisting of the parallel antecedent/consequent structures followed by a type of cadential conclusion. Leibowitz sees the disposition of the series in the Webern theme as, a melody and, accompaniment, the latter consisting of the initial graceful slurred sevenths and the isolated E and F natural in disparate registers. The pitch centricity of the theme however is striking in its framing of E as the initial tones of both antecedent and consequent structures and the final tone of the cadential extension, a fact perhaps recognized implicitly in Leibowitz's choice of a sonata theme in E flat as the model for his structural comparison.

Example 6:

Based largely on examples from Mozart, Leibowitz adapts a proposition/contrast model for what he sees as a typical structural opposition within antecedent and consequent phrases. This is borne out nicely in the Webern Kantata op.29 (Example 7) where four rows are unfolded simultaneously up to their ninth tones at the end of the antecedent (the "proposition" consisting of the first measure in the slow tempo and the "contrast" of the remainder of the antecedent to the cadence in measure 5) with the consequent via a two-note elision completing a subsequent row statement. However, while the Mozart excerpt may resemble a period, it is really a type of larger antecedent (in the form of a model plus repetition ending on a half-close) in an extended thematic structure which is not complete in Leibowitz's structural comparison. This example can also be taken as an illustration of Webern's typical incongruence of serial and thematic structure in contrast to their very strict correspondence in the music of Schoenberg.

Example 7:

Leibowitz cites the period structure of the Fantasy for Violin and Piano (Example 8) as an instance of a "proposition" which in the first measure presents the first six-tone truncation of the prime form of the row (the accompaniment' is based on the inversion) and a "contrast" in measures 2-3 which then presents the second six-note truncation of the inversion (the accompaniment taking over the completion of the series from the violin and beginning another inversion statement in measures three and four.) Leibowitz's consequent structure in measures 4-6 is reduced markedly by the virtual elision of the "contrast" segment and is based on a single hexachord of the retrograde against the other hexachord and the earlier form of the inversion. Although Leibowtiz's period structure (instead of a simple ternary sttucture) analysis may be somewhat forced, the function of the consequent phrase in the resolution of tonal tensions is quite clear i.e. the return to the G-F natural seventh in measure 5 from the emphatic and registrally disparate A -G in measures 2-3 and in the resolution in measure 6 of the mid-register B natural to the B of the opening measure.

Example 8:

All of the thematic period structures which Leibowitz observes in Schoenberg, are part of "Lied" (ABA') forms.11 So while they may be structurally analogous to periods (although even this was not the case with the opening structure of the Fantasy for Violin and Piano) they are dynamically incomplete pending the developments of the central section and the more conclusive cadences at the return of the opening ideas. The two most remarkable of these are the themes of the Variations op.31 and the Fourth String Quartet op.37. 12 The theme of the Variations is worth reviewing here since it is one of the few instances where Schoenberg actually gave a tonal harmonization of a twelve-tone theme (see Example 9a and b.13) Although there is no obvious "proposition/contrast" structure, the initial full close at the end of the antecedent and the cadential "modulation" at the end of the consequent are fairly clear. The melody is comprised serially of the unfolding of the prime form of the series against the retrograde of the accompaniment. The contrasting central section is defined serially by the unfolding of the retrograde form, and its hexachordal segmentation also contrasts with that of the outer sections which are more irregularly subdivided. The return of the A material is simply a retrograde of the consequent phrase (itself a form of the retrograde inversion.)

Example 9a:

Example 9b:

The chromatic harmonization in the radio interview realizes much of the implicit F major of the serial melody, playing about the dominant in the first three or four measures with an augmented sixth to tonic cadence at the close of the antecedent. The consequent phrase explores the heavier Neapolitan and F minor colorations but also closes with an augmented sixth cadence, this time on the dominant, albeit with evasive colorations in the dominant seventh E (m. 44) and in the motion in parallel tenths D -F, C-E at the cadence. Augmented sixth progressions persist in the descending chromatic bass lines of the contrasting section as well.

Schoenberg's serial harmonization reflects somewhat the tonality of F in the theme in its first cadence on an F major triad plus the Neapolitan shadows of G flat and B flat in the lower part which became important later in the tonal harmonization. The tonality of this sonority is much clearer in measure 46 but the remainder of the theme moves away from these initial polarities to a sequence in m. 48 and 49 of half-diminished sonorities and then a dominant seventh on G (m.51) before closing on more diffuse sonorities on C# (mm. 53-55,) and, with perhaps more than an innuendo of G in the final complex above B natural (mm. 56-57.) Like the earlier example from the Accompaniment to a Cinematographic Scene the theme of the Variations op.31 projects a considerabIe linear development, emanating from the initial F flat through E flat and then D natural, C# and finally the C natural of the first cadence. This is reflected in the closing phrase of the structure in the ascending line in the violins from C# to D (m.52) to E natural and eventually even resolving the E-B flat tritone to A and F natural (after the lengthy Neapolitan F#) at the final cadence.

One of the most dramatic thematic structures which Leibowitz considers is the three-part ("Lied") form of the opening theme of the Schoenberg Fourth String Quartet (see Example 10.) Leibowitz here makes another comparison with Beethoven (the Sonata op. 79) in which there seems to to be a similar reversal of "proposition" and "contrast" (the 'Y' and the "y" of the example diagram) and perhaps the same sense of formal imbalance and demand for subsequent development. The opening period of the Schoenberg unfolds the prime form of the series and the contrasting "b" section presents an immediate variation of a hexachordal motive derived from the initial period via the inversion of the series, as Leibowitz points out once more, at the perfect fifth below. The return unfolds the retrograde of the prime form of the opening A section which Leibowitz compares with the return to the tonic in the common-practice idioms. The tonal and linear implications of the Schoenberg theme are also very forceful in its cadential polarities.on D natural and B and its strong descending line from B beginning in the contrasting section (m.7) through B flat in m.8 which is recaptured in the lower octave in m.12 and pushed downward to the dominant A natural in m.13.

Example 10:

"Intermediate" Structures

Leibowitz is perhaps at his most meticulous in his discussion of the art and process of transition which he views as a mark of esthetic maturity, (frowning uncomprehendingly on the juxtappositional idioms of Stravinsky.) The opening up of the musical discourse which is the essential function of the transition involves, in particular, the opening up of the indially uclosed" structure (period, sentence, or "lied" forms) of the theme and the subsequent presentation of new "unclosed" and intervening ideas. This begins typically with a varied repetition of the opening structure itself from which various derivations emerge in a liquidation (i.e. elimination or dissolution 14) of the most prominent of the salient features of the theme, i.e. the replacement of the distinctive motives with motives of relatively "neutral" character such as the typical scalic figures or arpeggiations in the common-practice idiom. Having sufficiently "liquidated" and "neutralized" the opening ideas, a transitional moofel can then be introduced which bears little in common with the opening ideas but which, by its more insistent repetition, leads to a saturation or exhaustion of the thematic process in preparation for the entry of a secondary theme. One further term in Leibowtiz's discussion of transition is dissolution which, although not fully explained, seems to occur after the repetitive saturation as a preparatory link to the next idea.

In this light, Leibowitz offers a different discussion of the Webern Concerto op.24 first movement. What he sees in the "Treatise of Twelve-Tone Composition" as a varied (i.e. liquidated) repetition of the initial theme (i.e. measures 11-17, see Example 1b) is what was determined in Qu'est-ce que c'est que la musique de douze sons? to be a secondary theme in immediate succession to the primary theme. Beginning in measure 15 is a "second liquidation" and neutralization (to simple eighth-note values) of the original motives of the first theme.

Leibowitz presents more impressive examples of intermediate structures in the Schoenberg Fourth String Quartet and the Fantasy for Violin and, Piano. In the former, he notes the relatively abbreviated repetition and direct liquidation of the theme (into the triplet eighth-note motion based entirely on canonic imitation of the inversion of the series) since the theme itself already contains a repetition in its A-B-A structure (see Example 11.) The more extensive intermediate structure in the Fantasy for violin and piano involves the introduction of a new idea which is neutralized over 12 measures in a series of liquidations which culminate in a cadence and a dissolution signalled by a descending triplet figure in m. 4 (Example 12a.) This constitutes the intermediate structure of the movement and culminates the opening up of the initial thematic structure. What Leibowitz calls the "transition" follows where there is a similar saturation of a single model (see Example 12b); a near catalogue of the different techniques which are integral to Schoenberg's approach to serial development: different truncations of the series, a superimposition of an inversion of the series against the prime form however with a different ordering of the component truncations, and then a new transposition of the retrograde inversion against the original inversion with an exchange of the first and second truncation segments - all eventually opening the thematic process up into a constant sixteenth-note motion and a cadence (m. 24, see Example 12c,) and significantly in Leibowitz's view, interjecting a contracted summary of the opening theme within the space of a single serial statement.

Example 11:

Example 12a:

Examples 12b and c:


Secondary Structures

Secondary structures are said to be "open" in Leibowitz's terminology. Like intermediate structures, these are less symetrical and less cadentially closed, often involving new transpositions and segmentations of the series. Leibowitz raises the classical issues of the contasting character and tonality of secondary structures (typically the second theme in a sonata form) as well as their simpler but more continuous nature in comparison with primary structures. The second theme of the piano piece op. 33a (not shown) uses the same serial functions as the opening theme, a condition which Leibowitz ascribes to the brevity of the piece, in not introducing new serial functions. However, the second theme is distinguished by its partcular use of superimposed series as.opposed to the sequential presentations of the opening ideas. Leibowitz is particularly attracted to what he calls "flashbacks" or suggestions of the first theme in the accompaniment which suggest organic ties between the two thematic structures of a piece.

The secondary theme of the Fourth String Quartet has similar textural features to that of the piano piece op. 33a but does not unfold a complete series (see Example 13 - in this case a transposition of the prime form) at least in its first phrase where various tones are taken by the descending chromatic line of the accompaniment. As Leibowitz observes, however the second phrase presents a complete melodic statement of inversion with new truncations and once more, further use of "flashback" suggestions to the first theme in the accompaniment and the same transpositional relationship between the thematic and accompanimental series.

In contrast to the motivically saturated statements of its opening theme, and despite its complex accompaniment, Leibowitz notes that the secondary theme of the Prelude op.46 ("Survivor from Warsaw," see Example 14) is simple, and relates to the first theme in its use of the original serial complex and truncations, but with different relationships between the truncations. He notes the reversal of the superimposition between the two opening phrases of the theme, firstly truncation A (the first hexachord of the prime form) on B (the second hexachord of the prime form) and A' (the first hexachord of the inversion) on B' (the secod hexachord of the inversion), then A on A' and B on 13'. He also comments on the relative assymetry of the motivic alignment of the second phrase as an "open" structural property of the theme. This is enhanced by the use of a less predictable variety of the series tones in the melody rather than the opening two tones of the series which began the two motivic statements of the opening measure. The introduction of the new ideas in measure 22 further opens the theme motivically without the cadential structure which, might otherwise make a closed sentence with measures 19 and 20 as model and repetition.

Example 13:

Example 14:

Concluding Sections and Retransitions

Leibowitz reviews the typical thematic functions of recall and saturation involved in the classical models of concluding sections (of expositions) and retransitions or the reparatory and often anticipatory passages prior to the return of opening materials in the re-exposition. Based essentially on an example from a Mozart first movement (String Quartet K. 465, not shown) he notes the return in the codetta of the model of the primary theme and its manifold repetition on various harmonic degrees and its eventual liquidation and neutralization to a passage of equal (eighth-note) values. The retransition employs imitation to further saturate the process of thematic return and the anticipation of the return to the opening tonic via the repeated harmonic concentration of a dominant pedal.

Turning to twelve-tone music, Leibowitz observes in the initial retransition of the Prelude op.46 ("Survivor from Warsaw," example 15a and b) the return of the accompanimental thirds to the principal theme (he also notes their presence in the later sections of the secondary theme as well) as well as the concentration on the first six tones of the rows in the series complex (prime and retrograde forms) against a pedal on A-C# (Example 15a.) Indeed, the A major/minor sonority sustained in measure 24 behaves distinctly like a dominant to the obscure low-register'D natural of the beginnig of the initial theme (refer back to Example 5) and its cadence in m.10 resolves A-C# to D-F.

In the retransition of the Prelude (measures 51-54, Example 15b,) Leibowitz notes a more elaborate reference to the first theme followed by a more intensive saturation of the chordal model with two new series forms added to the end of the sequence, analaqous in function to the dominant. As in the concluding section, the retransition begins by isolating the thirds in the first two tones of the prime and inversions (the reference to the accompaniment to the principal theme) and then, for added harmonic weight, the same is done with retrograde forms, and we can note the emphasis of the high-register D-F third in anticipation of the return of the theme and in clear reference to the high D-F of m. 10 (compare this measure in Example 5 with measure 53 of Example 15b.)

Example 15a and b:

The concluding section of the Schoenberg Fourth Quartet is far more complex, involving an initial model derived from the accompaniment of the secondary theme (the sixteenth-note ideas in measure 78-79, see Example 16a) which is extended and subjected to imitative saturation (in the 'cello, viola and the two violins) and an eventual condensation in m. 82. What Leibowitz refers to as "another, more episodic motive" (the four-eighth-note figure in measures 79-81,) is developed at the same time against the reiteration of three-note chords which become the focus of the saturation in the remaining measures of the section. The aggregation of six tones in mm. 85-87 give way to another aggregate and a pedal in the 'cello and first violin in mm. 91-94. Leibowitz asserts an anticipatory reference to the primary theme in the three-note figures from measure 83 onward although the appearance of the series is obscured by its nonmelodic distribution before the return to the opening, apparently deriving a new model from the last four tones of the retrograde.

Example 16a:

In the retransition of the Fourth Quartet (measures 153-165, see Example 16b) Leibowitz notes the initial extraction and saturation of a melodic model from a trasposition of the prime form and then he notes the second, shorter model (mm. 157160) which is subjected to a process in which tones 12, 11 and 10 of the prime form are reiterated with a subsequent tone from the series being added in each reiteration. A similar poocess is carried out with tones 1 and 2 of an inversion, and the process continues using tones 7 and 8 of the inversion as the repeated elements of the model with a closer rhythmic interval of imitation in each stage of the process. The process f inally isolates the A -G head motive of the principal theme, appearing f irst in a recall of the first chords of the opening theme which is followed by a short link in the violin of the remaining tones of the series. In summation, Leibowitz proposes that the intensive motivic saturation of these passages compensates for their lesser harmonic weight, and perhaps this may be due to the strong harmonic weight in the imminent return of the opening theme of the movement. As we can observe ourselves, the harmonic direction of the passage is nevertheless very clear in the'cello line in measures 159-163 where the low E flat drops to C# and resolves to D in m. 162 prior to a cadential dominant in m. 164 which resolves into the return of the opening theme in the re-exposition.

Example 16b:

In summarizing his chapter on concluding sections and retransitions, Leibowitz ascribes qualities of less serial stability to these sections than to secondary structures but greater serial stability than their preceding intermediate structures. The series itself however may be concealed melodically and even harmonically in these sections and often with considerable freedom and asymmetry in its partitioning and disposition.

Re-expositional and Autonomous Structures

Leibowitz objects to the traditional distinction between exposition and development since development in some form or other is virtually present in all parts of the sonata form, and since typical development sections, which he prefers to call reexpositions, have structural qualities very much like expositions; they begin (typically) with thematic statements which are elaborated episodically in a new way, leading to another retransition. For Leibowitz's purposes, the German term Durchführung is more appropriate since, the central portions of development sections consist of a "leading through" different motivic transfomations and tonal regions.

The development of the first movement of the Webern Variations op.27 (the entire movement is in a simple ABA form) provides a very clear model, repetition and development (see Example 17) with very straightforward serial unfoldings. Although Leibowitz does not discuss this, an examination of the entire section would reveal a clear structural and quasi-thematic organization with a further repetition of the model in measures 15-19 and a reductive development of the 32nd note ideas to a cadence at the close of the section (not shown) in m. 36.

Example 17: Webem Variations op.27 first movement, measures 19 - 22. (Example 39 in "Treatise on Twelve-Tone Composition.")

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Leibowitz's discussion of the beginning of the recapitulation of the first movement from Schoenberg's Fourth Quartet is necessarily brief and incomplete but he focusses on the somewhat surprising recapitulation of the thematic model from the beginning of the development (mm. 116 - 121, see Example 18), noting its more thematic and "semi-closed" organization and the sharing of the series between the Haupt- and Nebenstimmen, against the inversion in the staccato chords, and its continued development as the model repeats in m. 122.

Example 18: Schoenberg Fourth String Quartet first movement measures 116 - 121.

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The summary remarks of the chapter comment on the symmetrical serial structuring of developmental models, however the disposition of the row in these structures is typically less trasparent than in the closed and secondary thematic structures. Re-expositional structures, Leibowitz continues, are based on the original occurences but with possible variations and extension, and typically new serial forms are applied to the more exact motivic re-expositions and more varied motivic re-expositions often entail row forms which are closer to the original occurrences.

Leibowitz goes on in the treatise to discuss an almost exhaustive array of formal contingencies: introductions, episodes and cadences, codas, stretti and terminal cadential segments, thematic structures in scherzi, slow movement and finales, serial structure in canons, fugatos and variations, and finally, the composition of small forms (songs with piano accompaniment). The preceding illustrations from Leibowitz's very thorough and influential analytical legacy can serve as a point of departure for a wider variety of interpretive considerations. My own analyses of the Minuet from the Suite op.25 and the Klavierstücke op. 33b are included here as examples of the application and extension Leibowitz's precepts to entire short and medium-length compositions.

The Minuet from the Suite for Piano op.25 and the Klavierstüke op.33b

The freedom in the ordering of the series in the Minuet is rationalized by the fact that it is a later movement of the work and the row must have become familiar enough by this point to withstand minor deviations in the order of tones. 15 In the case of the Minuet, the treatment of the row is entirely sequential (i.e. without superposition); many of the minor re-orderings may be motivated by tonal and harmonic clarifications. The prime row form is the following:

E F G G E A / D B C A D 13

As in the analysis of the second movement of the Concerto op. 24 of Webern, the initial tone of each row form is circled, but because of the ordering deviations, the placement of the row indications (R, P, Ri etc.) will indicate the actual beginning of individual row statements.


The sentence (model, repetition, reduction and cadence) structure of the theme is clearas is its initial tonal centering (E-D, D-E in mm.1-4 of Example 19) and the essential motivic importance of the tritone (D-A ) in the reductions as derived most likely from the opening measure. The insistence on B increases as the opening structure unfolds. In the initial cadence in m. 8, B flat slips to the lower neighbor A natural and we see the direct emphasis on B at the close of the extension - imitating the typical modulation of the classical minuet up to its cadence at the double bar. The emphasis on B as a pitch-class dominant is intensified through the bridge passage in measures 12 -16 in very close neighbor motion with C and A natural, perhaps with a slight recall of the E as D# in measure 14. The return of the thematic sentence begins much like a re-exposition via inversion of the upper and lower registers, and in the reductions, the reversal of the order and linear direction of the tritone quavers give the impression of a retrograde inversion. In the argumentation of tonal centers a climactic E pre-empts the resolution of the interpolated cadence (m. 24) but its emphasis disappears and it is the B 6 which ultimately prevails via a very deliberate chromatic linear ascent (the A of m. 32 is displaced registrally in the recall of the inital tritone D-A) to the closing sonority of the movement.

While Leibowitz discusses only the theme and transitional segments of the exposition of the Klavierstücke op.33b the piece is well worth studying in its entirety in terms of the functional perspectives reviewed here. The form of the piece, like the opus 33a is a sonata form with a first and second theme but a much longer development, and like the opus 33a, an abbreviated re-exposition. The row forms of the piece are the following:

prime (retrograde) : B C# F E A G# F# B G E C D

inversion (retrograde inversion) : EDBCF#G AFG#BD#C#

As Leibowtiz observed in many of in his analyses, the rows are combinatorial and related by inversion at the fifth. The close proximity of dominant seventh components on C# and C natural in the prime form as well as the half-diminished sonority (F-G#-B-D#) in the inversion all have distinctive tonal implications and powers of attraction which appear to function in the piece. Moreover, certain diatonic scale segments (C-D-E and B-C#-D#) contribute a melodic lucidity within contrapuntally and chromatically dense passages. Independent of these pre-compositional factors however, the thematic structure and processs give clear articulation to certain tonal centers and harmonic sonorities. The lower melodic tones in the opening motivic structure (the C# in m. 1, the B in m.2, the C natural in m.3 and the C in m.4 see example 20 16 ) are directed to B natural (m.4) to which the contrasting episode appends the B major melodic fragment (mm. 5-6). The contrasting fragment also establishes a certain harmonic clarity in the accompanimental dominant sevenths on C (mm.5-6) F (mm. 7-8) and D flat (mm. 9-10). Once again we see an example of the "pedalling" of a segment of the row (the D-C-E-G of mm. 5-7) with the discontinguous interjection of subsequent tones in the series (B flat and F#.)

The melodic orientation of the return in the opening 'lied" (ABA) structure is decidedly different, and based upon inversions of the series with dominant seventh sonorities emerging only at the beginning of the phrase (on C# in m.12) and at the end (on F# in m. 16.) It is here that we see the first manipulations in the order of the series in the piece - the reversal of the order of D and F natural in m. 14 and the displacement of the B flat of the second hexachord of the prime form to the very end of the phrase, to make the descending ninth with C natural and perhaps to give cadential emphasis to the leading tone of the dominant seventh sonority on F# at this point. The manipulation of the order of the series continues in the liquidation of the opening motive in the short transition in measures 17-18 with the clear re-articulation of the second hexachord of the retrograde inversion into trichords of F-A flat-A natural and B-C#-D#. The transition also has the function of preparing the 6/8 meter of the second theme.

Tonal innuendos become stratified in the contrapuntal complexity and the increased serial density (two row forms per measure) of the second theme. The implication of B major/minor however is clear in the upper line of the texture in mm. 21-22 as well as the continuation in the lower register in m. 23. Various tonalities are similarly implied as the ascending major scale-step motive climbs through the initial liquidation of the idea in the middle register in m. 24 and upper register in mm. 25 and 26. The subsequent diminutions of the motive similarly pass through a number of tonal implications including B major once more, A major and finally B major in m. 28. The concluding phrase of the exposition (Hetwas breiter" in mm. 29-31), following Leibowitz's prescription, resumes development of the thematic major second dyads of the opening theme, however over greater spans of the upper and mid-registers, and giving clear voice to the F major dominant in the ritardando of measure 31.

The tonal implications of the F major are realized at least temporarily in the F major-minor sonority of measure 32 (the G#-F-A-C natural in the second half of the measure) and the clear dominant thirteenth of the following measure. 17 Also consonant with Leibowitz's views of development sections as re-expositions departing from points of relatively closed thematic structure, measures 32-36 form a fairly clear model and repetition along the lines of the opening theme of the piece, with a cadence on D in measure 37. This structure however is preparatory to the return of the principal theme of measures 37-40 which is comprised solely of the motivic dyads as in reduction of the preceding model and repetition of measures 32 - 35. Incidental alterations in the order of the series allow the E of the E -D dyad to come after the B-A (note the B natural of the series appears incorrectly as B in the score in measure 37) and similar motivic adjustments of the serial order occur in the subsequent measures of the structure (mm.39-40). As in the original exposition, we have an intervening episode on the diatonic scale fragmnents placed in superposition to the scherzando ideas which occurred only fleetingly in the presentation of the principal theme in the exposition. Here however the ideas are expanded in parallel episodes with inversion of registral relationships. We also see the "pedalling" of row segments (dyads) A -F in the lower part of m.39 and G-F# and 136-C in the upper line in measures 41- 42 which is continued (at least in part, in the GF#) against a harmonic pedal involving A-G# and the low F natural in measures 43-44.

Example 20:

The cadence at measure 45-46 initiates an augmentation of the prime form against itself and the inversion. Formally, in terms of the development section (or 're-exposition") this parallels the return to the opening thematic dyads from the diatonic scalic fragment in the exposition of the principal theme. Here the episode continues in a harmonic pedaI18 and a cadence into the re-exposition and a return to the serial density of second theme. Some play with the serial ordering in the prime form is apparent in measures 49, 50 and 51 perhaps as a final dissolution of the theme (note the predominace of whole-tone fragments in the upper line of these measures) before the final re-exposition of the piece in m.52. The second theme in measure 52 is presented before the primary theme after being completely absent in the central re-exposition (development section) and undergoes the same, although abbreviated, liquidation both of the idea and the series as well which becomes virtually unrecognizable in measures 53-55. The primary theme recurs, in m. 57, now in a serial augmentation in which the prime form is once more superimposed upon itself and the inversion. The same process of augmentation generates the coda at this point in a progressive expansion and saturation of the motivic model in the upper parts of measure 61 which consists of superimposed hexachords of the prime-form. The retrograde inversion is placed beneath this in mm. 61-63 and is finally followed by the retrograde which "pedals" the F-A-A as in the close of the first re-exposition, leading to the ultimate B minor sonority plus C# which mirrors the first motive of the opening theme.

While the op. 33b is an admirable representation of the processes of closed structure, liquidation, neutralization, codetta and re-exposition which Leibowitz observes in Schoenberg's music, it is clear that it offers a somewhat different view of serial development than what Leibowitz studies in his "Treatise on Twelve-tone Composition." The restriction of the serial vocabulary to the four basic forms appears to prompt certain freedoms in ordering in addition to the derivations available through varied truncations and superimposition instead of those possible through transposition. Minor re-orderings which creep into the serial structure toward the close of the first theme appear to be motivated harmonically (mm.1 4 andl 5) and the transition in m. 18 presents re-truncations of the series which anticipate those of the second theme. While the initial statements of the second theme may follow the ordering confines of the series, further minor reorderings can be seen in the liquidations beginning in m.23 to privilege the diatonic trichords (the B-C#-D# of the retrograde inversion for example). The immediately subsequent measures are near-verticalized presentations of the series involving the diatonic trichords with little semblance of the original thematic order. Minor re-orderings persist in the closing of the exposition but by contrast the "etwas breiter" phrase of measures 29-31 represents a serial and thematic clarification in preparation for the ensuing re-exposition.

The "pedalling" of the series in the central section of the piece similarly creates opportunities for further diatonic manipulation and the augmentations (measures 46 - 48 and in the coda in measures 64 - 68) which seem also to have been anticipated by the minor motivic augmentation and reordering of measure 37 offer still further possibilities of harmonic implication. As seen in this analysis, serial manipulation is often closely linked to the emergence of tonal innuendo from the series, either transitionally as in the scalic link in the lower lines in measures 55-57, or at points of formal definition such as the quasi F major at the beginning of the re-exposition and the ultimate B minor resolution of the coda. In comparison with the techniques which Leibowitz explores in the Fourth String Quartet and the concertos, this seems to be part of a larger intuitive growth in Schoenberg's work toward the re-integration of harmonic tonality with the style of the "freedom of dissonance," much as he expresses in his own words in 1946 at the end of the second part of his article entitled "Composition with Twelve-Tones:"

In the course of about 10 years, some of the strictness of the rules concerning octave doubling and prominent appearances of fundamental chords of harmony have been loosened to some degree.

At first it became clear that such single events could not change the style of nontonality into tonality. There remained still those characteristic melodies, rhythms, phrasings and other formal devices which were bom simultaneously with the style of freedom of the dissonance.19

From the present study, it becomes clear that the significance of individual rows in Schoenberg lies not only in their tonal and serial implications but in the phraseology with which they are so closely linked. Unlike in Webern, Schoenberg's use of overlapping series is less common and there is a consistent relationship between series statements and the formal components which they define. Leibowitz's work therefore does much to elucidate developmental aspects of Schoenberg's serial language particularly in the demonstration of devices of liquidation, neutralization and retransition. What has become increasingly clear in applying these precepts analytically is the extent to which the essential factors of Schoenberg's tonal and serial language are integrated into the logic of thematic form and development: i.e. serial manipulation in the re-truncations of the original disposition of the series, new serial superpositions, the use of new transpositions or row forms; extra-serial manipulations in re-orderings of minor importance such as the inversion of the order a pair of notes, or more substantial re-orderings such as the inversion of the order of hexachord subgroups and finally, factors of harmonic and linear tonality, cadence and "modulation" in the interaction and inter-relation of tonal centers.



1 This paper has been given in different forms over the last couple of years at the West Coast Conference on Music Theory and Analysis at Mills College and in graduate studies colloquia at Yale University and the University of Pennsylvania. The author is grateful for these opportunities to present and discuss this work.

2 Qu'est-ce que c'est que la Musique de Douze Sons? Editions Dynamo, Pierre Aelberts ed. Liege, BeIgium, 1948 (no longer in print.)

3 Arnold Schoenberg Theory of Harmony, Roy Carter trans. Berkeley, UC Press, 1978. See in particular chapter 15 on modulation.

4 "Problems of Modem Harmony," in Style and Idea ed. Leonard Stein, Leo Black (London: Faber and Faber) 1975, 268 - 286. This article was written in 1934 at about the same time as many of the pieces which Leibowitz discusses in his "Treatise..." i.e. the opus 31 Variations, the opus 33 Piano Pieces, the opus 34 Accompaniment to a Cinematographic Scene, etc. Perhaps most significant is the attribution at the outset of the article of the essence of musical logic to fundamental factors of (harmonic) "tonality" and to the "art means" or means of form and motivic process of individual works. (p. 270.)

5 The Path to the New Music, Willi Reich ed. and trans., Bryn Mawr, PA: Theodore Presser 1963, pg. 35.

6 As has often been observed, the interval structure of the Webem's row allows a certain redundancy of pitch content at the level of the trichords of certain transpositions. For example the first trichord B-B-D) of the prime form is the same as the first trichord of the retrograde inversion starting on D (D-B -B) except for the different order of the three tones, and, as seen in Leibowitz's analysis in Example 1, all of the other trichords are identical in pitch content but different in their order of tones. In this way, a certain tonality emerges independently of the series in the combination of major thirds and semitones of trichords. Equally interesting are instances where one tone is different between two trichords such as in the last trichord of the opening prime form (C-C#-A) and the first trichord of the prime form starting on C# (C#-A-B ) which provide points of transition in the shifting tonal focus of certain passages.

7 See for example Leopold Spinner "Analysis of a Period," in die Reihe 2:46-50, and Christopher Wintle "Analysis and Performance: Webern's Concerto op. 24/11" in Music Analysis 1: 1 (1982) 78-99.

8 The first tone of each row are circled with references (P, 1, R, RI) to the particular row-form.

9 The authors contact with the "Treatise on Twelve-Tone Composition" arose via an English translation by Nancy François at UCSD received while the author was studying with Will Ogdon in the early 1980's. Copies of the original ms. in French were given to Will Ogdon and Jacques Monod (who actually worked with Leibowitz on the compilation of examples for the treatise) to have published in English and French, and Professor Ogdon recalls that Leibowitz had in mind to publish the treatise in many languages. Unfortunately, because of difficulties in obtaining a clear copyright to the document in the years following Leibowitz's death, it has remained unpublished although circulated considerably in xerox form. Only small portions of the first part of the treatise, on thematic, intermediate and secondary structures as well as closing, retransitional and re-expositional sections will be reviewed here. A fuller discussion of the treatise can be found in Professor Ogdon's "Concerning an Unpublished Treatise by Rene Leibowitz," Journal of the Amold Schoenberg Institute 2 (Oct 1977) 34-42, 4-25.

10 It is worth examining the complexity of the accompaniment of this passage which, as Leibowitz unravels it, is more hexachordal than serial. The inversion beginning on A natural unfolds and recycles its first six tones up to the end of measure 5, beginning the second half of the series with the F# of measure 6 up to the reappearance of the F- E natural motive in measure motive in measure 11.  At this point Leibowitz observes the first six tones of the retrograde inversion and then the remaining four tones of the series in measures 13, the missing of the missing tones of the second hexachord being taken by the taken in the A-C of the solo violin. Measures 15-17 similarly re-trace the prime form, harmonically echoing the second hexachord of the series.

11 The "Lied" or "short ternary" consists typically of an opening period structure followed by a motivically and harmonically contrasting fragment and a usually abbreviated return of the opening ideas. The "rounded binary" forms which are typical of the Minuet are similar in essence (the contrasting central section is commonly referred to as a "bridge" back to the return of the opening ideas) but with immediate repetitions of the initial period structure, and of the second part comprised of the bridge plus the abbreviated return of the initial period.

12 Leibowitz presents other examples: from the second movement of Webern's Symphonie op.21 and Schoenberg's theme in the beginning of the Klavierstücke op.33a.

13 "The Orchestral Variations op 31," Score 27, July 1960, p.33 - the score appeared as part of an interview which Schoenberg gave over the radio in which he played the "harmonized" version at the piano.

14 Compare this with Schoenberg's own view in "Connection of Musical Ideas," in Style and Idea where he says in the transition from an opening idea "One of the most important functions of the changing of features is the production of liquidation. By producing at least a preliminary end to a section it makes the appearance of a new idea a reasonable, if not necessary event. A liquidation is often carried out unto the entire elimination of all features. No wonder that in such a case the entrace of a terrifically strong contrast does note violate the feeling of balance, it is as if everything began anew." p. 288.

15 See Style and Idea ed. Leonard Stein, trans. by Leo Black (London: Faber and Faber) 1975 p.234 where he talks about the second movement Gavotte of the Suite and the subdivision of the row into separate groups of internally inviolate order although, in reality, the overlapping of the groups may violate the literal order of pitches in the series. In example 18 the initial tone of each series in the Minuet is indicated with and arrow. The indications of the series form P, I , R or RI are placed at the point where the pitches of the particular series begin to occur which is very often before the first tone of the series.

16 Once again arrows indicate the initial tone of each series, but because of the redundancy of the serial structure and for reasons of space, indications of prime, inversion, retrograde or retrograde inversion were not included. All forms starting on B natural are therefore prime forms; all forms starting on E are inversions; all forms starting on D are retrogrades and all forms starting on C#/D flat are retrograde inversions.

17 In this case, the F# needs to be interpreted as a neighbor-tone of an F major reading, or else as the root of a tritone-substitute dominant in which the C natural is a flatted fifth.

18 Note the missing treble clef in measure 50 of the score.

19 Style and Idea ed. Leonard Stein, Leo Black (London: Faber and Faber) 1975, p.244.


ex tempore
as published in Vol. VIII/1, Summer 1996