Wei-Chieh Lin "In Search of Time Musical Time and Form in Dérive 2 by Pierre Boulez by Wei-Chieh Lin

IN SEARCH OF TIME: MUSICAL TIME AND FORM IN DÉRIVE 2 BY PIERRE BOULEZ

BY WEI-CHIEH LIN

SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DOCTOR OF MUSICAL ARTS DEGREE, May 2012.

ABSTRACT Dérive 2 (1988-2011) for 11 instruments by Pierre Boulez is based on the concept and properties of periodicity. Aspects of pitch, rhythm, duration, harmonic rhythm, phrasing, instrumentation, dynamic, tempo, and large-scale formal structure are all derived from an architectural design of the underlying multi-layered periodicity. This document will focus on two important aspects in Dérive 2: Musical Time and Formal Structure. The first chapter discusses Boulez’s conception of musical time in general, and more specifically traces some of the conceptual developments and practices with regard to rhythmic construction, smooth and striated time, meter, grace-note, tempo, and periodicity from the earlier works up to and including Dérive 2. The second chapter addresses Boulez’s approach to form and the idea of trajectory of large-scale organization in the previous works, and in relation to Dérive 2. The third and last chapter examines Dérive 2 in detail, and in addition traces the compositional development and the revisions of the work from its initial conception in 1988 to its latest modification in 2011. An appendix is provided with a list of performances of Dérive 2.

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AUTOBIOGRAPHICAL NOTE Born in Taichung, Taiwan, Wei-Chieh Lin’s music has been performed at venues including the Gaudeamus Muziekweek, Centre Pompidou in Paris, Avery Fisher Hall and Alice Tully Hall, Peter Jay Sharp Theater at Lincoln Center, Weill Recital Hall at Carnegie Hall, Jordan Hall in Boston, Yallow Barn, Nasher Sculpture Center, and the National Concert Halls in Taiwan. Among the ensembles that have performed or commissioned his works are the, Ensemble Intercontemporain, Orchestre Philharmonique de Radio France, National Taiwan Symphony Orchestra, Taipei Philharmonic Orchestra, Le Nouvel Ensemble Moderne, Insomnio Ensemble, Alarm Will Sound, Cadillac moon Ensemble, Formosa Quartet, members of Klangforum Wien, Xasax Ensemble, Makrokosmos Ensemble, The New Juilliard Ensemble, the Juilliard Orchestra, members of eighth blackbird ensemble, DZ4 Wind Quartet, the Formosa Performing Arts Association, the Hudson Symphony Orchetra, the New York Asian Symphony Orchestra, the New Asia Chamber Music Society, and the New York Classical Players Ensemble. Mr. Lin also has an upcoming performance with the Asko/Schoenberg Ensemble. Mr. Lin’s compositions have received awards including the Honorable Mention of the Gaudeamus Muziek Prize of 2011, selection for the 2012 International Composer Pyramid Competition, two ASCAP Morton Gould Young Composer Awards, the 2009 and 2010 National Symphony Orchestra of Taiwan Composition Competitions, the 2010, 2011, and 2012 National Taiwan Symphony Commissioning Award, and the Palmer Dixon Award. Mr. Lin has participated in music festivals including the Aspen Music Festival, The Wellesley Composers Conference and Chamber Music Center, MusicX Festival, Académie musicale de Villecroze, Domain Forget Asian Composers League Music Festival (Taiwan), active participant in the Manifeste/Acanthes@Ircam, Voix Nouvelles Royaumont, and the 2013 Mizzou International Festival. Mr. Lin completed his BM, MM, and DMA with scholarship at The Juilliard School under the tutelage of Milton Babbitt.

ACKNOWLEDGEMENT I would like to thank the Paul Sacher Foundation in Basel, Switzerland for allowing me to access and reproduce the sketches and drafts of Dérive 2, plus all the relevant research materials in the Pierre Boulez Collection. I also would like to thank the staff of the Foundation for their support, especially to Dr. Robert Piencikowski for his generosity and assistance in understanding Boulez’s sketches, as well as his invaluable knowledge on Boulez’s music in general. I also would like to thank the Universal Edition for granting me permission to use passages from Dérive 2 in my examples, as well as providing information regarding the publications and the performances of Dérive 2. A special gratitude also goes to Yu-Ting Hung for assisting the preparation of all the musical examples and her encouragement throughout the whole process of completing this document. I must thank the Doctoral Committee and the faculty members at Juilliard for their guidance over the years. I particularly like to thank and acknowledge my advisor, Jonathan Dawe, for his invaluable advice and countless hours of support not only through the process of this document, but also for the years I have known him. A special tribute must be paid to my teacher, Milton Babbitt, from whom I have learned so much over the years, and there is no word for me to express my deepest gratitude for everything he had given me. Finally, to my parents, whose support, love, and all they have done for me is beyond anything I could have asked for. I owe an enormous debt to them.

CONTENTS ABSTRACT AUTOBIOGRAPHICAL NOTE ACKNOWLEDGEMENT LIST OF FIGURES AND MUSIC EXAMPLES PROLOGUE

iii iv v vii 1

CHAPTER I. MUSICAL TIME 1.1. Rhythmic and Temporal Conception in Boulez’s Early Works 1.2. Smooth Time and Striated Time 1.3. Tempo 1.4. Periodicity

11 16 23 39

CHAPTER II. FORMAL STRUCTURE 2.1. Formal Organization in Boulez’s Early Works

CHAPTER III. DÉRIVE 2 3.1. Pitch Construction in Dérive 2 3.2. Periodicity in Dérive 2 3.3. Formal Structural in Dérive 2 3.4. Revisions in Dérive 2

74 122 166 196

EPILOGUE

232

Appendix A. Periodicity Chart Appendix B. Performance History of Dérive 2

238 250

BIBLIOGRAPHY

254

LIST OF FIGURES AND MUSIC EXAMPLES Chapter 1.2 Ex. 1.2.1. Morse Code of the name SACHER translated into rhythmic patterns Chapter 1.3 Fig. 1.3.1. Tempo manipulation and fluctuation from R66 to R83 Ex. 1.3.2. Written-in accelerando from 3-R26 to 1-R26 Ex. 1.3.3. Two measures before R10 to 5 measures after R10 Ex. 1.3.4. Tempo modulation at R10 Fig. 1.3.5. Tempo markings from R167 to R221 Fig. 1.3.6. Tempo markings from R26 to R129

22 27 29 31-32 34 36-37 38

Chapter 1.4 Ex. 1.4.1. Pulsation Ex. 1.4.2. Periodicity of pulsation and event Ex. 1.4.3. Periodicity of multiple lines of equal length cycle Ex. 1.4.4. Periodicity of multiple lines of unequal-length cycles Ex. 1.4.5. Periodicity of multiple lines of unequal-length cycles Ex. 1.4.6. Polyrhythm ratio 9:10 between the two pulsations Ex. 1.4.7. Multi-layered Polyrhythms as periodicities Ex. 1.4.8. Point of convergences in 9:10 polyrhythm Ex. 1.4.9. Symmetrical structure of the polyrhythm Ex. 1.4.10. Out-of-phase periodicities where the pulses do not converge Ex. 1.4.11. In-phase periodicity Ex. 1.4.12. A three-layer out-of-phase periodicity Ex. 1.4.13. A three-layer periodicity of rhythmic ratio of 15:10:6 Ex. 1.4.14. Ligeti Piano Concerto – Piano solo and percussion part Ex. 1.4.15. Ligeti Piano Concerto – String pizzicatos Ex. 1.4.16. Two rhythmic patterns superimposed Ex. 1.4.17. The polyrhythmic ratio between the two periodicities Ex. 1.4.18. 56:45 polyrhythm in A Celebration of Some 100x150 Notes Ex. 1.4.19. A reduction of the opening measures of sur Incises

40 41 41 42 43 43 44 45 45 46 46 47 48 55 56 57 58 63 66

Chapter 2.1 Fig. 2.1.1. Formal and instrumental outline of Le Marteau san maître

Chapter 3.1 Ex. 3.1.1. The opening viola line up to the first double barline Ex. 3.1.2. The opening viola line partitioned into 24 trichords, labeled with set-classes Ex. 3.1.3. Trichords in the woodwinds and strings in the opening measures up to the first double barline (boxed numbers refer to trichords in ex. 3.3.2) Ex. 3.1.4. Opening horn part, along with cello, cor anglais, and bassoon Ex. 3.1.5. Trichords are combined to form 6-element collection Ex. 3.1.6. Trichords are combined to form 9-element collections Ex. 3.1.7. Trichords are combined to form 12-element collections Ex. 3.1.8. Ascending order for the combined 6, 9, and 12-element collections

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74 75 78-80 81 82 83 83 84

Ex. 3.1.9. Horn part starting from R5 to R13 Ex. 3.1.10. A realization of the 6-element collections from R5 to R6-3 Ex. 3.1.11. A realization of the 6-element collections from R6-3 to R8 Ex. 3.1.12. A realization of the 6-element collections from R8 to R9-3 Ex. 3.1.13. A realization of the 6-element collections from R9-3 to R9-4 Ex. 3.1.14. A realization of the 6-element collections from 2-R11 to R13 Ex. 3.1.15. A realization of the 9-element collections from R14 to R16 Ex. 3.1.16. The opening all-trichordal hexachord Ex. 3.1.17. A reduction of the simultaneities in the opening measures Chapter 3.2 Ex. 3.1.1. The opening viola line up to the first double barline Ex. 3.1.2. The opening viola line partitioned into 24 trichords, labeled with set-classes Ex. 3.1.3. Trichords in the woodwinds and strings in the opening measures up to the first double barline (boxed numbers refer to trichords in ex. 3.3.2) Ex. 3.1.4. Opening horn part, along with cello, cor anglais, and bassoon Ex. 3.1.5. Trichords are combined to form 6-element collection Ex. 3.1.6. Trichords are combined to form 9-element collections Ex. 3.1.7. Trichords are combined to form 12-element collections Ex. 3.1.8. Ascending order for the combined 6, 9, and 12-element collections Ex. 3.1.9. Horn part starting from R5 to R13 Ex. 3.1.10. A realization of the 6-element collections from R5 to R6-3 Ex. 3.1.11. A realization of the 6-element collections from R6-3 to R8 Ex. 3.1.12. A realization of the 6-element collections from R8 to R9-3 Ex. 3.1.13. A realization of the 6-element collections from R9-3 to R9-4 Ex. 3.1.14. A realization of the 6-element collections from 2-R11 to R13 Ex. 3.1.15. A realization of the 9-element collections from R14 to R16 Ex. 3.1.16. The opening all-trichordal hexachord Ex. 3.1.17. A reduction of the simultaneities in the opening measures Chapter 3.2 Ex. 3.2.1. Rhythmic pattern of 2-1-3-1-1-2-1 Ex. 3.2.2. The opening viola line re-barred under the pattern of 2-1-3-1-1-2-1 Ex. 3.2.3. The opening instrumental parts re-barred under the rhythmic pattern of 2-1-3-1-1-2-1 Ex. 3.2.4. Rhythmic cycle of 2-1-3-1-1-2-1 articulated by horn and other instruments Ex. 3.2.5. Partitioning the rhythmic pattern of 2-1-3-1-1-2-1 with E-n-n-n-E-E-n-E-E-n-n Fig. 3.2.6. Instrumental assignments according to the E-n-n-n-E-E-n-E-E-n-n partitioning Ex. 3.2.7. Opening measures re-barred according to the instrumental partitioning of E-n-n-n-E-E-n-E-E-n-n Ex. 3.2.8. Periodicities of the resonant instruments in alignment with the rhythmic pattern of 2-1-3-1-1-2-1 and the pattern of E-n-n-n-E-E-n-E-E-n-n Fig. 3.2.9. Periodicities of Section A Fig. 3.2.10. Periodicities of Section B

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87 89-90 93-94 96-97 98 99-103 106-109 111 111-117 74 75 78-80 81 82 83 83 84 87 89-90 93-94 96-97 98 99-103 106-109 111 111-117 122 122-123 123-126 127-128 129 129 131-133 135-138 140 143

Fig. 3.2.11. Instrumental and Periodicity assignment in Section B Ex. 3.2.12. The first 3 cycles of EN with annotated rhythmic patterns in Section B Ex. 3.2.13. The next 2.5 cycles of EN with annotated rhythmic patterns in Section B Ex. 3.2.14. The next 2 cycles of EN with annotated rhythmic patterns in Section B Ex. 3.2.15. The next 1.5 cycles of EN with annotated rhythmic patterns in Section B Fig. 3.2.16. Dynamic scheme of Section B Fig. 3.2.17. Periodicities of Section C Ex. 3.2.18. Rhythmic transformation in Section C Ex. 3.2.19. Horn part with annotated periodicities in Section C Ex. 3.2.20. Annotated periodicities in Section C Ex. 3.2.21. Periodicities of Section L Fig. 3.2.22. Instrumental assignment in Section L Ex. 3.2.23. Annotated periodicities in Section L

150 151-152 153 153 154 155-159 161 161 162-165

Chapter 3.3 Fig. 3.3.1. Formal Structure of Dérive 2 Fig. 3.3.2. The periodicity cycle of 3 x 4 x 7 Fig. 3.3.3. The periodicity cycle of 3 x 4 x 7 Ex. 3.3.4. The form of Dérive 2, with the 3 x 4 x 7 periodicity and the interruptions Ex. 3.3.5. Instrumental assignment in Coda I Fig. 3.3.6. Dynamic Scheme in the first half of Coda I from R225 to R231 Fig. 3.3.7. Dynamic Scheme in the second half of Coda I from R231 to R234 Fig. 3.3.8. Duration of phrases in Coda II Ex. 3.3.9. The concluding measures of Dérive 2. Ex. 3.3.10. Opening horn solo Ex. 3.3.11. The first interruption Ex. 3.3.12. The second interruption Ex. 3.3.13. The fourth interruption Ex. 3.3.14 SACHER pitch sequence Ex. 3.3.15. Symmetrical pitch structure in the interrupting sections

168-171 172 173 174-175 179 179 179 180 182-183 185 186 187 188 190 191-195

Chapter 3.4 Ex. 3.4.1. Ending for the 1993 version Ex. 3.4.2. Ending for the 2001 version Ex. 3.4.3. Ending for the second 2001 version Ex. 3.4.4. Ending for the 2002 version Ex. 3.4.5. Ending for the 2004 and 2005 version Ex. 3.4.6. Concluding measures in an earlier version of 2006 and 2007 (UE31940) Ex. 3.4.7. Ending for the most current version Ex. 3.4.8. Earlier version at R129 Ex. 3.4.9. Chronological history of Dérive 2

143 144-145 146-147 148-149

204-205 206-207 208-209 210-213 214-218 219-224 225-230 231 232-3

PROLOGUE Dérive 2 (1988-2009) for 11 instruments by Pierre Boulez (1925-) is one of his longest works, second only to Pli selon pli: Portrait de Mallarmé (1957-62/83/89/2000).1 The current version of Dérive 2 lasts approximately 45-50 minutes, while Pli selon pli lasts about 65 minutes.2 Dérive 2 is written for the second largest chamber ensemble that Boulez has ever scored, rivaled only by Éclat (1965).3 Dérive 2 marks an important departure from Boulez’s later works which are categorized into three genres. The first is works involving electronics (Dialogue de l’ombre double (1982/84/85) for clarinet and tape, Répons for 6 soloists, ensemble, computer sounds and live electronics, …explosantefixe…(1991/…), …explosante-fixe…Transitoire V (1991/1993), …explosantefixe…Transitoire VII (1991-93) for flute, various ensembles, and live electronics, and Anthèmes 2 (1997) for violin and live electronics). The second group consists of works for large orchestral forces (Répons, Notations I-IV, Notations VII for orchestra, revisions of Pli selon Pli for soprano and orchestra, and Boulez’s orchestration of Ravel’s Frontispice for orchestra). The third comprises solo instrumental works (Anthèmes (1991) for solo violin, Incises (1993-94/2001) for solo piano, and Une page d’éphéméride (2005) for solo piano). Boulez’s return to a purely instrumental medium of mixed chamber ensemble in his late

This does not include the “unfinished” or “work-in-progress” pieces such as the Third Piano Sonata (1955Like most of Boulez’s works, it is uncertain whether Boulez plans to revise Dérive 2 in the future. The other two major late works, Répons and sur Incises (1996/98), last about 43 minutes and 38 minutes, respectively, in their latest versions. 3 Éclat is scored for 15 instruments: piano, celesta, harp, glockenspiel, vibraphone, mandolin, guitar, cimbalom, tubular bells, alto flute, cor anglais, trumpet, trombone, viola, and violoncello; though in the subtitle Boulez regards Éclat as “pour orchestra.” 2

years,4 a genre that had interested him throughout much of his earlier career, is significant, if not refreshing.5 Dérive 2 is scored for 11 instruments: cor anglais, clarinet in A, bassoon, violin, viola, violoncello, horn in F, vibraphone, marimba, harp, and piano.6 Its mixed instrumentation is quite rare in the chamber ensemble genre; however, it is typical of Boulez to employ a dichotomy in contrasting resonances between natural and artificial resonant instruments, exemplified in works such as Pli selon pli, Éclat, sur Incises, Répons, or even Le Marteau sans maître.7 Moreover, the combination of and the fascination for the sound of piano, harp, vibraphone, and marimba can be found in many works throughout his œuvre.8 A characteristic in this scoring is that the four woodwind instruments are the only monophonic instruments in the whole ensemble (Boulez does not employ any 4

The only other mixed chamber ensemble work in recent years is sur Incises, scored for three pianos, three harps, and three percussionists. This instrumentation should not be considered as a mixed ensemble since Boulez typically groups these instruments together as a single unified force. There are also two lesser-known relatively recent chamber works, Initiale (1987/92) for brass ensemble and Petite Dérive – en echo (1998) for alto flute, clarinet, vibraphone, marimba, violin, and viola; though both works are quite short and have not been widely performed. 5 Some of the major mixed chamber ensemble works before 1988 are Le Marteau san maître (1953/57), Éclat, Pour le Dr. Kalmus (1969), …explosante-fixe…(1972-74), Dérive 1 (1984/86) for 6 instruments, and Mémoriale (…explosante-fixe…Originel) (1985) for 9 instruments. 6 This is the first time where Boulez prominently used cor anglais and bassoon in his chamber works—two instruments that are less common in the chamber music repertoire, similar to his use of alto flute in Marteau. As for the other instruments in the ensemble, Boulez has either written solo works or works that prominently feature them. For example, clarinet in Domain (1961/68) and Dialogues de l’ombre double, violin in Anthèmes 1 & 2, viola in Marteau, violoncello in Messagesquisse (1976), and horn in Tombeau (1959/62/83) are works that all feature these instruments prominently. The scoring of Dérive 2 would normally be considered a large chamber ensemble, though it could arguably be categorized as a small chamber orchestra or a sinfonietta force, since the subtitle is particularly vague, as Boulez simply writes as “for 11 instruments.” 7 For Boulez, the natural resonant instruments are instruments where the resonance after the initial attack is not manufactured by the performer—instruments such as piano, harp, guitar, vibraphone, marimba, and various percussion instruments. The artificial resonant instruments are ones where the performer needs to maintain the sustained sound after the initial attack—such as all the string, woodwind, and brass instruments. Boulez will utilize this contrast in timbre, resonance, and sound production effectively in Dérive 2. 8 Works such as Pli selon pli (especially in Improvisation sur Mallarmé no. 1, 2, and 3), Éclat, Éclat/Multiples (1966-70), Répons, and especially sur Incises where these instruments are frequently grouped as a unified ensemble, and are featured prominently in these works as well.

multiphonics). This is comparable to his approach to the three string instruments, despite the fact that the strings are not limited to monody.9 Thus, the whole group is divided into seven melodic instruments (woodwinds and strings) against four harmonic and resonant instruments (piano, harp, and percussion). This is an important characteristic that Boulez utilizes in Dérive 2. In comparison to Boulez’s other works, or even to most contemporary music, Dérive 2 is quite traditional in its approach to instrumental timbre: it does not employ any indeterminate-pitch percussion instruments, nor does Boulez explore indeterminatepitches from these traditional instruments. In addition, the three string instruments draw solely on conventional extended techniques (such as pizzicatos, snap pizzicatos, harmonics, jéte/thrown bow, con sordino, and a few instances of col legno and sul ponticello). In comparison to the rest of the ensemble, the string instruments produce the widest range of timbre differences; while the instrumental writing for the others is straightforward, as the rest of the ensemble does not employ extended techniques such as slap tongue, flutter-tongue, multiphonics, quarter-tones (as used in the earlier versions of Le Visage Nuptial (1947/52/89), or the quarter-toned harps in Improvisation III sur Mallarmé), harmonics in the piano (compare to the opening of Dérive 1 and sur Incises where the piano uses the sostenuto pedal with pressed keys in the lower register to create sympathetic resonances and overtones)10, or any nonconventional modes of playing. The instrumentation of Dérive 2 naturally suggests several possible pairings based on timbre: a trio of strings is paired with a trio of woodwinds, a duet of piano and harp is 9

Excepting the Coda (rehearsal number 225) and a few other instances, the strings rarely play double-stops. This of course is not the case for the use of the piano resonance (pedal effects) seen in the very opening measures of Dérive 2, where the pedal “catches” the resonances after the attack of the keys. 10

coupled with a duet of vibraphone and marimba, while the horn acts as a mediator that blends all the timbres together.11 An alternative arrangement based on range offers another possible combination, with bassoon, violoncello, viola, and horn covering the lower register; clarinet, cor anglais, violin, and vibraphone covering the upper register; and piano, harp, and marimba covering the whole range from low to high. This registral distribution is significant since the range of these instruments (especially the woodwind section) tends to have a richer sonority in the mid-low register, specifically with the use of clarinet in A and cor anglais instead of oboe; while piano, violin and harp are relatively speaking the only instruments that can cover the upper end of the registral spectrum. Thus with these registral and timbral constraints, the majority of the work is fixed within the mid-low register. As the work unfolds and approaches the midpoint, the register of the full ensemble expands to its fullest range, reaching the highest possible register at rehearsal numbers 114 and 129, and the lowest possible register at the end of rehearsal number 166 (where the midpoint section also concludes.) As the work retraces its steps and moves toward the Coda at rehearsal number 226, the registral span likewise slowly retracts back to the mid-low register.12 With this set of instruments, Dérive 2 offers a unique sonority of colors and timbres, and the intentional mixture of instrumental forces will serve Boulez to create a wide variety of textural variations, sonic explorations, as well as unconventional instrumental mixtures—all of which become vital aspects of Dérive 2.

This is obviously Boulez’s intension, since his sketches, drafts, and the engraved score all list the instruments in these particular combinations (woodwinds, strings, horn, percussion, harp, and piano) instead of the customary orchestral ordering. 12 This registral expansion and retraction reinforces the symmetrical structure of the work, an essential feature which will be discussed in chapter 3. The Coda (starting with rehearsal number 225) does not fall into this registral outline.

Dérive 2 is dedicated to Elliott Carter for his 80th birthday anniversary in 1988.13 It was premièred on June 21, 1990 at the Grande Salle, Conservatorio di musica Giuseppe Verdi di Milano, conducted by the composer with the Ensemble Intercontemporain.14 Dérive 2, similarly to most of Boulez’s works, has gone through numerous revisions and extensions throughout the intervening years.15 Boulez began the work in 1988, and the latest primary version was completed in 2006, published by Universal Edition (UE 32528), while Boulez continued to make minor adjustments and revisions to that edition until 2011.16 The two available commercial recordings of the piece, one is of an earlier edition and was made by Boulez conducting the Ensemble Intercontemporain, released in 2005, while the other was recorded in 2011, and released in 2012.17 The former recording runs about twenty-four and half minutes, while the latter runs about fifty-one minutes. One of the more recent performances, also given by the composer and the Ensemble 13

The majority of Boulez’s works have dedicatees: Le Visage Nuptial was originally dedicated to Messiaen, the First Piano Sonata (1946) was originally dedicated to René Leibowitz; Le Marteau sans maître is dedicated to Hans Rosbaud; Tombeau (the last movement of Pli selon pli) was a tribute to Prince Max Egon zu Fürstenberg; the initial version of …explosante-fixe… (1971) was written as a memorial to Stravinsky; Mémoriale (…explosante-fixe… Originel) was dedicated to flautist Lawrence Beauregard; Rituel in memoriam Bruno Maderna (1975) is dedicated to Bruno Maderna; Messagesquisse (1976) and sur Incises are dedicated to Paul Sacher; Dérive 1 (1984) is dedicated to Sir William Glock; Dialogues de l’ombre double was written for Luciano Berio’s 60th birthday; Répons and Anthèmes are dedicated to Alfred Schlee; Fanfare (1992) was written for Sir Georg Solti’s 80th birthday; Derive 3 is dedicated to Michael Gielen, and Petit Dérive – en echo was dedicated to Elliott Carter’s 90th birthday. Consequently, Carter has also dedicated works to Boulez: Esprit Rude/Esprit Doux (1985) was written for Boulez’s 60th birthday, Esprit Rude/Esprit Doux II (1995) was written for Boulez’s 70th birthday, Retrouvailles (2000) was written for Boulez’s 75th birthday, and Réflexions (2005) was written for Boulez’s 80th birthday. 14 On the same program, Boulez also conducted Octandre by Edgard Varèse, Boulez’s own Mémoriale (…explosante-fix e…originel) and Dérive 1, Tema by Franco Donatoni, Jalons by Iannis Xenakis, and Oiseaux exotiques by Olivier Messiaen. See Susanne Galaise, “Les écrits et la carrière de Pierre Boulez: catalogue et chronologie” (PhD diss., Université de Montréal, 2001). 15 The history of revisions and extensions of Dérive 2 will be documented in chapter 3. 16 Pierre Boulez, Dérive 2 pour 11 instruments 1988/2006 (Vienna: Universal Edition (UE 32528), 2006). 17 Pierre Boulez, Le Marteau sans maître, Dérive 1 & 2, Ensemble InterContemporain dir. Pierre Boulez, Deutsche Grammophon CD DDD 0289 477 5327 8 GH (20/21 Series), 2005. This recording was recorded at Ircam in Paris in September of 2002, but was not published until February of 2005. This recording used the 2002 edition of Dérive 2. A more recent recording by the Ensemble Orchestral Contemporain, conducted by Daniel Kawka, was released by Naïve Records in March of 2012.

Intercontemporain in 2010, is about 45 minutes, almost double the length of the first commercially available recording.18 Curiously, both performances that took place in 1998 and 2001 only lasted about 7 minutes.19 Between 1988 and 2011 not only has Boulez drastically expanded the work, he has written and revised several other pieces as well.20 In this regard, Dérive 2 represents a significant work, a magnum opus in Boulez’s catalogue as he continuously expanded it for more than 20 years, longer than any of his previous revisions of works.21 Dérive 2 bears little or no relation to Dérive 1.22 The only connection between the two is that both works derived materials from Répons.23 Yet, unlike Dérive 1 where the

This performance was given by Pierre Boulez with the Ensemble Intercontemporain at the Cité de la musique in Paris on December 3rd, 2010. 19 Both performances were given by Pierre Boulez with the Ensemble Intercontemporain at the Cité de la musique in Paris on October 28th, 1998 and March 17th, 2001 respectively. 20 Between 1988 and 2010, Boulez had written and revised several works: in 1988, revisions were made to Livre pour cordes and Figures, Doubles, Prismses; in 1989, a third version was made to Don (from Pli selon pli); from 1991 to 1992, Anthèmes was composed; in 1992, Fanfare was composed; from 1993 and 1994, …explosante-fixe… was composed and revised in various versions and extensions, and the first version of Incises was written; in 1995, Dialogue di l’ombre double for bassoon and electronics was transcribed; in 1996, the first version of sur Incises was composed; in 1997, Anthèmes 2 and Notation VII were composed; in 1998, revision was made to sur Incises; in 2001, revision was made to Incises and Messagesquisse was transcribed for violas; and in 2005, Une page d’éphéméride was composed. This list does not include the continuous revisions he has made on Dérive 2 during this time span. This chronology is important since these works share a close connection to Dérive 2, and many ideas that were formed in the later stages of Dérive 2 are indirectly informed by and seeded within these works. Moreover, what makes Dérive 2 unique among the late works is that it is the only work where the pitch and temporal material is constructed anew individually instead of reusing materials from an earlier work. The later works are predominately based on either the 7-note series taken from …explosante-fixe…, or based on the SACHER series. See Galaise for a detailed chronology of Boulez’s works during this time span. Galaise, “Les écrits et la carrière de Pierre Boulez: catalogue et chronologie.” 21 One can argue that the intervening number of years Boulez has worked on Notations is far greater, since he wrote the piano version of Notations in 1945, and the orchestra version starting in 1978 up to the present day. Another case can be argued for Pli selon pli, since all the movements were written or revised sporadically from 1957 to 2000. 22 Dérive 1 (1984) was written for the occasion of Sir William Glock’s retirement, and was premiered on January 31st, 1985, conducted by Oliver Knussen with the London Sinfonietta. 23 “It’s a tree which gives another tree which is another tree. Therefore, the title Dérive. Derive 1, for instance, is from Répons. Part of the material for Répons I did not use, actually, and that became the material for Dérive 1. And Dérive 2 is from studies I did for the part of Répons which is still not written.” Andrew Carvin and Joshua Cody, “Pierre Boulez: An Interview,” Paris New Music Review (November 1993) http://www.paristranstlantic.com/magazine/interviews/boulez.html (accessed October 18, 2011).

principal pitch material (the SACHER series based on name of its dedicatee: Eb, A, C, B, E, D) is shared with Répons (Dérive 1 is built exclusively on the SACHER chord and its transposed rotational array), Dérive 2 is constructed on completely new and independent material, a rarity for Boulez.24 The two works also do not share any rhythmic or temporal scheme. Moreover, the two works are not considered or performed as a cycle, unlike several other works that share the same title in Boulez's catalogue.25 The instrumentation between the two works is also entirely different: Dérive 1 is scored only for a Pierrot ensemble of six instruments (flute, clarinet in A, violin, violoncello, vibraphone and piano), lasting about 8 minutes. On the other hand, another sibling to these two works, Dérive 3, is an incomplete piece that originated as Initiale (1987) for brass ensemble.26 It was then expanded with an added percussion section to become Fanfare in 199227, which was then expanded again and retitled as Dérive 3 in 1997.28 Dérive 3 offers very little resemblance to 24

Boulez has a history of reusing materials from work to work, or using the same series or collection in a number of works. A good example is the 7-note series that first appeared in …explosante-fixe… (1971), which is used again in Rituel, …explosante-fixe for MIDI flute, ensemble and live electronics (1991/…), Mémoriale (…explosante-fixe…Originel), Anthèmes 1, and Anthèmes 2. The already mentioned SACHER series is employed not only in Répons and Dérive 1, but in fact began in Messagesquisse (1976) where the series is derived from the last name of the dedicatee (Paul Sacher), and subsequently, used again in Incises and sur Incises, the latter piece also dedicated to Paul Sacher. Though there are instances of the SACHER chord in Dérive 2, it is not as importance and visible as in the other works, and the underlying pitch construction in Dérive 2 has very little to do with the SACHER series. From this perspective, Boulez is radically different from Schoenberg, Webern, and Stravinsky who use a new series or collection for each piece. Boulez’s method of working, especially reusing the same series or collection in a number of works, is much more similar to the methods of Carter and Babbitt. Carter typically uses the two all-interval tetrachords and the all-trichordal hexachord in almost all of his works, while Babbitt tends to use the same all-partitioned array in a number of his works. 25 The cycle of pieces refers to works that share the same title, and are closely related with one work inspiring the other: Éclat/Multiples is the expanded version of Éclat, Anthèmes 2 is the expansion of Anthèmes 1; sur Incises is based on Incises; Notations I-IV, VII for orchestra are based on the short piano pieces with the same title; and the …explosante-fixe… series all came from the same source. Dérive 1, Dérive 2, and Dérive 3 do not share such a connection; however, earlier performances of Dérive 2 were often paired with Dérive 1, though such pairing is no longer in practice in the recent years of performances. 26 Initiale was premiered on June 4th, 1987 in Houston, Texas. 27 Fanfare was premiered as a birthday present for Georg Solti’s 80th birthday in Chicago. 28 Boulez, however, claims that Dérive 3 is derived from Le Visage Nuptial. See Carvin and Cody, “Pierre Boulez: An Interview.”

Dérive 2 in terms of pitch material, rhythmic structure, and instrumentation. In contrast with the other two works, Dérive 2 has a relatively individual and distinct compositional history, since it does not relate or rely on any previous work, but its history after the premiere makes it difficult to decipher the exact creative path due to the voluminous revisions Boulez has made. Despite these trivial differences, the three Dérives nonetheless share many similar approaches to harmony, harmonic rhythm, texture, rhythm, meter/pulsation, and musical time in general. The program note to Dérive 2 provided by the publisher states, “Dérive 2, dedicated to Elliott Carter on his 80th birthday, was the result of ‘research into periodicity’. ‘When I reflected on some of Ligeti’s compositions,’ wrote Boulez, ‘I felt the desire to dedicate myself to some almost theoretical research into periodicity in order to systematically examine its overlays, its shifts and its exchange.’ “29 In a more in-depth program note, Boulez wrote: The sustained contact with certain works from Ligeti brought me to think about the rhythmic life of a musical composition. Starting from a narrowly focused perspective, Dérive 2 has evolved in compositional time well beyond my initial point, for meanwhile it has endured interferences from other completed works. In short, the word Derive evokes the meandering that this work has progressed through in its realization. Thus it became a kind of journal that reflects not only the evolution of the musical ideas themselves, but also the ways of organizing them in a kind of narrative.30 29

Universal Edition, “Pierre Boulez, Dérive 2 for 11 instruments – Work Introduction,” Universal Edition, http://www.universaledition.com/Pierre-Boulez/composers-and works/composer/88/work/9777/ work_introduction (accessed October 18, 2011). 30 “Le contact suivi avec certaines oeuvres de Ligeti m’a amené à réfléchir sur la vie rythmique de l’oeuvre musicale. Partant d’un point de vue étroitement ciblé, Dérive 2 s’est développée dans un temps de composition bien au-delà du point de départ, subissant les interférences d’autres oeuvres achevées entretemps. Le mot dérive peut, en somme, s’appliquer aux nombreux méandres que cette oeuvre a décrits le long de sa réalisation. Elle est devenue ainsi une sorte de journal reflétant l’évolution des idées musicales proprement dites, mais également la façon de les organiser dans une sorte de mosaïque narrative.” Pierre Boulez, “Program note to Dérive 2,” November 7th, 2006 at the Cité de la musique in Paris. Translation provided by the author.

Furthermore, during a BBC interview that took place at the Barbican in 2004, Boulez clarified his thoughts: Dérive means derived from, but in the same time when a boat as you know [with] no motor, and no sail, and just lets the currents push him, that’s also derive … it drifts, … there are these two meanings and I like these two meanings, … I like these types of pun. … When I write a piece generally I have a lot of sketches which are not always used. You know that’s like pages of a diary which I take and then amplify, and expand. Dérive 2 is not derived from a work specifically. But I was interested by the periodicity in the works in the recent works by Ligeti, in the works by Carter, the notion of time in Carter, and also Carter gave me a clue to that, to the works of Nancarrow, the American who was living in Mexico. What I think [of] my periodicity that’s when you have you know phenomena which come[s] back at a regular pace, or according to certain developments, and when you have something which is periodic and something which is not periodic you find that the fights between the two is interesting. And therefore, I gave a series of lectures31 on the periodicity in these three composers, with not only examples from these composers but examples from myself, just sketches you know, [to] articulate examples let’s say. And it gave me the incentive to write a piece according to these datas.32 And the secondary reason in this direction: what I did in this piece that is to try to have a tempo quick, and tempo rather slow but with full of activity. By tempo I mean the change of harmony, but I don’t take that from Nancarrow, or from Carter, or from Ligeti, I take that from Beethoven. Simply the system of variations of Beethoven, the slower it is, the more notes you have; you have even 64th and 128th-notes in some variations by Beethoven, but the harmony is very slow, but the number of events is very big. And then I tried you know to have this kind of dialectic with quick tempo with few events, relatively, and slow tempo with lots of it. … To simplify; quick, many, slow, few.33

Boulez is referring to a series of lectures he gave at the Collège de France between 1976 to 1995 on various musical topics. These have been subsequently published as Points de repére III: Leçons de musique: Deux décennies d’enseignement au Collège de France, ed. Jean-Jacques Nattiez (Paris: Christian Bourgois Éditeur, 2005). 32 According to Wolfgang Fink, these periodicities and examples “were initially worked out on a computer so that they could be handled more easily, but in the process they developed a life of their own.” See Wolfgang Fink, linear notes for Le marteau sans maître, Dérive 1 & 2 by Pierre Boulez, Deutsche Grammophon CD DDD 0289 477 5327 8 GH (20/21 Series), 2005. 33 This interview along with a performance of Dérive 2 took place on October 13th, 2004 at the Barbican in London, with Boulez conducting members of the London Symphony Orchestra. This performance lasts approximately 25 and half minutes and it is already different from the commercial recording of 2002 and the 2003 edition of the score. The final horn solo (which imitates the opening) was heard here for the first time. This Boulez interview is transcribed by the author.

This document, then, will focus on two important aspects in Dérive 2: Musical Time and Formal Structure. The first chapter presents Boulez’s conception of musical time in general, and more specifically traces some of the conceptual developments and practices with regard to rhythmic construction (chapter 1.1), smooth and striated time (chapter 1.2), meter, grace-note, tempo (chapter 1.3), and periodicity (chapter 1.4) from the earlier works up to and including Dérive 2. The second chapter addresses Boulez’s conception of form and the idea of trajectory of large-scale organization in the previous works in relation to Dérive 2. The third and last chapter examines Dérive 2 in detail, and the process of realizing these concepts of musical time and formal structure that were discussed in the previous two chapters, in addition to their transformations and manipulations by Boulez in the discourse of Dérive 2.

CHAPTER 1. MUSICAL TIME Chapter 1.1. Rhythmic and Temporal Conception in Boulez’s Early Works Influenced by the rhythmic innovations of Bartok, Stravinsky, and Messiaen1, Boulez set out to develop his own approach to rhythm.2 From Messiaen, he learned the importance of the independence of rhythm. Rhythm should no longer be considered as a secondary parameter to pitch; rather, rhythmic construction can and should have its own internal structure.3 Unlike Messiaen, Bartok, and Stravinsky, Boulez was not interested in transporting another culture’s rhythmic constructions to his own music, nor was he interested in imitating another culture’s rhythmic devices.4 In fact, Boulez strongly criticized Messiaen’s use of Greek or Indian rhythms in his music.5 Instead, one sees in early Boulez the use of automatism in rhythmic construction through mathematical or 1

From Messiaen, Boulez not only had gained an interest in rhythm, but also in harmonic thinking, voicing, and a keen sensitivity to color and timbre, not to mention Messiaen’s interest in the music of Debussy, and in non-Western music. All these ideas continued to influence Boulez for many years to come. See Peter O’Hagen, “Pierre Boulez and the Foundation of Ircam,” in French Music since Berlioz, eds. Richard Langham Smith and Caroline Potter (Aldershot: Ashgate, 2006), 303-307. 2 See Pierre Boulez, “Proposals,” in Relevés d’apprenti, ed. Pale Thévenin, trans. Stephen Walsh as Stocktakings from an Apprenticeship (New York: Oxford University Press), 47-54. 3 “… I already considered that rhythmic writing ought to be something worked on for its own sake, and I think this is the lesson I learnt from Messiaen, particularly from his classes on music from Stravinsky onwards. After having analyzed The Rite of Spring with him, or even his own works, I was convinced for the necessity of working at purely rhythmic invention.” Pierre Boulez, Par volonté et par hasard: entretiens avc Célestin Deliége, trans. Robert Wangermé as Pierre Boulez: Conversations with Célestin Deliége (London: Eulenburg Books, 1976), 13-14. 4 “You could say I was very influenced when I was young by cultures of Asia and Africa. But I saw these cultures from a very different point of view than that of the original cultures. I could not pretend that I understood them. I understood them not from their point of view; I understood them from my point of view, which is completely different. And that I called a misunderstanding—a misunderstanding because, fundamentally, what you pick up is the kind of impression you have of the music you hear.” Rocco Di Pietro, Dialogues with Boulez (Lanham: Scarecrow Press, 2001), 47-48. 5 “As a matter of fact, that side of Messiaen has never much interested me. As you know, he very often makes use of rhythms drawn from either Greek or Indian music, and to my way of thinking this poses a problem. It is very difficult to introduce fragments of another civilization into a work. This is what I believe now, but I also believed it then: we have to invent our own rhythmic vocabulary in accordance with our own norms. In this sense, even in my earliest works, there is what one might call a contrast between free forms (sometimes there are, for instance, extremely free rhythms – almost improvised, or written down as they are thought up) and on the other hand extremely strict sections. This is something I still practice; it is one of my main ideas.” Boulez, Par volonté et par hasard: entretiens avc Célestin Deliége, 13.

serial formulas, and any sense of traditional rhythm and meter in addition to any cultural references or connotations is completely abandoned if not destroyed.6 In the Sonatine for Flute and Piano (1946/49), for example, Boulez had constructed a rhythmic scheme based on two principle motives that are then extended to become the rhythmic basis for the whole piece.7 Besides the serial technique and its realization of the 12-tone row in the pitch domain, what is innovative in the Sonatine is that the rhythmic device is constructed entirely in isolation from pitch.8 The rhythmic complexity (the abandonment of meter, the use of polyrhythm, and the layering of superimposed rhythms) is unmatched by any piece Boulez had written up to this point. A few years later, in Polyphonie X (1949/51) for ensemble, Structures I (1951-52) for two pianos, and Le Marteau san Maître for voice and ensemble, Boulez conceived of an analogy of the serial method in the pitch domain applied to the temporal domain with the durational series9—a technique that provides Boulez complete control of the rhythmic

Boulez had gotten to know the principals of the 12-tone technique (and the music of Schoenberg and Webern as well) from René Leibowitz, who was a student of Schoenberg. The 12-tone system, for Boulez, posits a completely new solution to musical coherence. For Schoenberg and Berg, the 12-tone system was a renewed grammar and syntax to re-create the tonal system and language, and it is quite evident in their use of form, meter/rhythm, and the thematic treatment of the series in their serial works (for example, Schoenberg’s works after op. 25 show a great detail of imitation of the classical forms, which are drastically different from his innovative use of form and rhythm in the post-tonal works such as Pierrot Lunair, Erwartung, and the piano pieces of op. 11 and op. 19. Similarly, the use of classical forms in Berg’s Wozzeck also poses a problem for Boulez). See Boulez, Par volonté et par hasard: entretiens avc Célestin Deliége, 17-18. Boulez was not interested in renewing the past, instead he uses the 12-tone system to create a new path. 7 By this time Boulez had already been studying with Messiaen for more than a year at the Conservatoire, while studying counterpoint privately with Andreé Vaurabourg (the wife of Honegger) since April 1944, and also privately with René Leibowitz starting in 1945. Though most people usually associate Boulez’s musical influences from Webern, however, at this point Boulez hardly knew any late Webern works (with the exception of Symphony Op. 21), he was still under the influence of Messiaen and Honegger. See Gerald Bennett, “The early works,” in Pierre Boulez: a Symposium, edited by William Glock (London: Eulenburg, 1986), 41-84; and O’Hagan, “Pierre Boulez and the Foundation of IRCAM,” 303-330. 8 See Bennett, “The early works,” and Boulez’s own analysis of the use of the “rational” and “irrational” rhythmic cells in the Sonatine. Boulez, “Proposals,” in Relevés d’apprenti, 47-54. 9 Boulez is not the only one who had experimented with the durational series, Messiaen, Babbitt, Berio, Nono, Stockhausen, Wuroinen, Ferneyhough, and even Grisey had come up with similar processes in their

domain and insures its independence.10 A durational series subsequently allowed him to apply the same kind of transformation and manipulation that takes place in the pitch domain to rhythmic construction.11 The rhythmic structure is thus freed from meter, pulse, or temporal hierarchies, and it breaks away from any tradition or cultural implications, in addition to eliminating any personal freedom and choice to a certain extent. Rhythms, along with pitch, dynamics, and articulations (modes of attack) thus have a parallel and equal weight in the structure of the music, and these four parameters are completely independent of each other. Most important, all four parameters are governed and generated from a single 12-tone series (hence Total-Serialism), thus unifying all aspects of music to and from the same structure.12

music with varying approaches. Babbitt also developed a “time-point” system in which a measure or a meter is analogous to the octave in the pitch domain, and a measure divided into 12 equal time-points, with each point corresponding to a number in the series. See Milton Babbitt, “Twelve-Tone Rhythmic Structures and the Electronic Medium.” Perspectives of New Music 1, no. 1 (1962): 42-79. On the contrary, Stockhausen experimented with the Fibonacci and the overtone series to determine rhythmic structure and construction. 10 “Looking at Schoenberg’s method, which was preoccupied only with pitches, I thought (especially given the influence of Messiaen and his rhythmic procedures) “Why not try to make everything under the same control and order?” At this moment, it was called pointillistic, because we were dealing with point after point, and the reunion of point. After a while, I was bored, because you can’t only work with separated notes, you can’t always only work with number one. It wasn’t enough for me, and once I had taken the consequences of serialism as far as that, I was aware that anarchy produced practically the same results. It proved the absurdity of the extreme logic, which is equivalent to the absurdity of no logic at all.” Paul Steenhuisen, “Interview with Pierre Boulez,” in Sonic Mosaics: Conversations with Composers (Edmonton: University of Alberta Press, 2009), 66. 11 With the durational series, Boulez permutes the series instead of transposing it as in the case of the pitch domain. 12 “The series is—in very general terms—the germ of a developing hierarchy based on certain psychophysiological acoustical properties, and endowed with a greater or lesser selectivity, with a view to organizing a FINITE ensemble of creative possibilities connected by predominant affinities in relation to a given character; this ensemble of possibilities is deduced from an initial series by a FUNCTIONAL generative process (not simply the consecutive exposition of a certain number of objects, permutated according to restrictive numerical data). Consequently, all that is needed to set up this hierarchy is a necessary and sufficient premise which will ensure the total cohesion of the whole and the relationships between its successive parts.” Pierre Boulez, Penser la musique aujourd’hui, trans, Susan Bradshaw and Richard Rodney Bennett as Boulez on Music Today (London: Faber and Faber, 1971), 35-36.

The serial method in the temporal domain (mainly the durational series) provided a short experiment for Boulez, and he immediately knew the limits of pure TotalSerialism: It is very easy to see how the situation has developed in just this way. When the serial principle was first applied to all the components of sound, we were thrown bodily, or rather headlong, into a cauldron of figures, recklessly mixing mathematics and elementary arithmetic; the theory of permutations used in serial music is not a very complex scientific concept … rhythmic organization [of the serial method] disregarded realizable metric relationships, structures of timbres scorned the registers and dynamics of instruments, dynamic principles paid no heed to balance, groups of pitches were unrelated to harmonic considerations or to the limits of tessitura. Each system, carefully worked out in its own terms, could only cohabit with the others through a miraculous coincidence. The works of this period also show an extreme inflexibility in all their aspects.13 From another perspective, Boulez’s total-serial works led to surprising comparisons to Cage’s chance or aleatoric works, since arguably, both methods could inevitably lead to similar results: If you are only obsessed with organization, then practically you arrive at chaos, because an excess of organization in physics bring chaos. Chaos alone does not bring any order. Therefore, I have to accept the stream, which is an order, and within this stream I must accept the unforeseeable elements, which you can not control. But I have to make the best use of them that is possible. I think that in life that’s exactly the same. You have opportunities which come which will never come back, for instance, and you have to see-to judge, even-if this opportunity will serve your purpose or not. If you see that it is of no value for you, or it takes away more than it contributes to your purpose, then you have to let the accident go.14 Boulez quickly moved away from the rigidity and automatism of Structures Ia, and starting with Marteau and Pli selon pli, Boulez allowed personal freedom, creativity, spontaneity, and “accidents” into the process of music making, while greatly extending the

13 14

Boulez, Penser la musique aujourd’hui, 25. Di Pietro, Dialogues with Boulez, 98.

possibilities of the series through many layers of manipulation and transformation in all parameters.15 Even though Marteau is still composed with the durational series, its surface rhythmic construction and the temporal organization are much more varied and are manipulated on a detailed level. The works that followed Marteau, Pli selon pli and Troisième Sonate, showed even greater freedom and inventiveness on the realization of the rhythmic construction in relation to the multiplication technique.16 One no longer perceives the “point” based rhythmic values, but instead, a fluidity of rhythms and a smoother surface. However, after Marteau, Boulez knew he had to reconstruct once again a new syntax for his rhythmic construction and conception.

“A progressive loosening of the vice-like grip of strict writing will finally lead to complete freedom – freedom, of course, within general structural principles. … The play of structures implicitly suggests a scale of relationships going from the chance of automatism to the chance of choice.” Boulez, Penser la musique aujourd’hui, 106. 16 See also Catherine Losada, “Isography and structure in the music of Boulez,” Journal of Mathematics and Music 2, no. 3 (November 2008): 135-155.

Chapter 1.2. Smooth Time and Striated Time French philosopher Gilles Deleuze (1925-1995) proposed two readings of time: Chronos time and Aion time, where “each one of which is complete and excludes the other: on one hand, the always limited present, which measures the action of bodies as causes and the state of their mixtures in depth (Chronos); on the other, the essentially unlimited past and future, which gather incorporeal events, at the surface, as effects (Aion).”17 Deleuze further defines Chronos time as: … only the present exists in time. Past, present, and future are not three dimensions of time; only the present fills time, whereas past and future are two dimensions relative to the present in time. In another words, whatever is future or past in relation to a certain present (a certain extension or duration) belongs to a more vast present which has a greater extension or duration. Thus, the relativity of past and future with respect to the present entails a relativity of presents themselves, in relation to each other.18 On the contrary, Aion is defined as: … only the past and future inhere or subsist in time. Instead of a present which absorbs the past and future, a future and past divide the present at every instant and subdivide it ad infinitum into past and future, in both directions at once. Or rather, it is the instant without thickness and without extension, which subdivides each present into past and future, rather than vast and thick present which comprehend both future and past in relation to one another.19 Aion is, then, “infinitely subdivisible”20 and it is “no longer the future and past which subvert the existing present; it is the instant which perverts the present into inhering future and past.”21 For Deleuze, the duality between Chronos and Aion time is:

Gilles Deleuze, The Logic of Sense, eds. Constantin V. Boundas, trans. Mark Lester and Charles Stivalep (NY: Columbia University Press, 1990), 61. 18 Ibid., 162. 19 Ibid., 164. 20 Ibid., 61. 21 Ibid., 165.

[Chronos is] limited and infinite, Aion is unlimited, the way that future and past are unlimited, and finite like the instant. Whereas Chronos was inseparable from circularity and its accidents—such as blockages or precipitations, explosions, disconnections, and indurations—Aion stretches out in a straight line, limitless in either direction. Always already passed and eternally yet to come, Aion is the eternal truth of time: pure empty from of time which has freed itself of its present corporal content and has thereby unwound its own circle, stretching itself out in a straight line.22 Deleuze then relates his concept of Aion time and Chronos time to Boulez’s concept of Smooth time—temps lisse—and Striated time—temps strié, respectively,23 where smooth time refers to music that is non-pulsed, while striated time refers to music that is pulsed. According to Boulez, In Pulsed time, the structures of duration will be related to chronometric time as landmarks, or, one might say, systematically placed regular or irregular beacones: these constitute a pulsation, either of the smallest unit … or of a simple multiple of this unit (the smallest common multiple of all the values used), or of a simple multiple of this unit (two or three times its value). … Amorphous [non-pulsed] time is only related to chronometric time in a global sense; duration, whether with defined proportions (not values) or having no indication of proportion, appear in a field of time. Only pulsed time is susceptible to speed, acceleration or deceleration: the regular or irregular referential system on which it is based is a function of a chronometric time of greater or lesser delimitation, breadth or variability. The relationship of chronometric time to the number of pulsations will be the index of speed. Amorphous time can vary only in density according to the statistical number of events which take place during a chronometric global timespan; the relationship of this density or an amorphous timespan will be the index of content. … Beneath a line of reference, place a completely smooth surface and a striated surface … I will call these two categories smooth time and striated time.24 22

Ibid., 165. “Aeon [Aion]: the indefinite time of the event, the floating line that knows only speed and continually divides that which transpires into an already-there that is at the same time not-yet-here, a simultaneous toolate and too-early, a something that is both going to happen and has just happened. Chronos: the time of measure that situates things and persons, develops a form, and determines a subject. Boulez distinguishes tempo and nontempo in music: the “pulsed time” of a formal and functional music based on values versus the “nonpulsed time” of a floating music, both floating and machinic, which has nothing but speeds or differences in dynamic.” Gilles Deleuze and Félix Guattari, A Thousand Plateaus: Capitalism and Schizophrenia, trans. Brian Massumi (Minneapolis: University of Minnesota Press, 1988), 262; see also in the same book, “The Smooth and the Striated,” 474-500, for detailed explanations. 24 Boulez, Penser la musique aujourd’hui, 88-89. 23

Deleuze proposes that smooth and non-pulsed time is a “kind of floating time that more or less corresponds to what Proust called ‘a bit of pure time’,” and that is duration “freed from measure, be it a regular or irregular, simple or complex measure.”25 Additionally, nonpulsed time “puts us first and foremost in the presence of a multiplicity of heterochronous, qualitative, non-coincident, non-communicating duration,”26 and that the “non-pulsed space-time detaches itself in turn from the striated one. It only refers to chronometry in a global way: the cuts are indeterminate of an irrational type, and the measures are replaced by distances and proximities that cannot be broken down and that express the density or rareness of what appears there (statistical distribution of events).”27 In comparing the two, as Deleuze puts it most vividly, that smooth time and non-pulsed time “occupies [time] without counting instead of [striated time and pulsed time] counting to occupy.”28 Boulez has scrutinized extensively his concept of smooth and striated time as first defined in Penser la musique aujourd’hui in various forms throughout his own writing, and it is clear that such concept is important to him and to his own music.29 Boulez even goes as far as to state that “I regard the two categories – smooth and striated time – as capable of reciprocal interaction, since time cannot be only smooth or only striated. But I can say that 25

Gilles Deleuze, “Making Inaudible Forces Audible”, in Two Regimes of Madness: Text and Interviews 19751995, rev. ed., ed. David Lapoujade, trans. Ames Hodges and Mike Taormina (New York: Semiotet(e), 2006), 157. 26 Ibid. 27 Deleuze, “Occupy Without Counting: Boulez, Proust and Time”, in Two Regimes of Madness, 294. In this article Deleuze corresponds Boulez’s conception of time to Proust’s temporal qualities in À la recherché du temps perdu. 28 Ibid. 29 Gérard Grisey dismissed these two temporal concepts by Boulez, since “the notion of smooth (unmeasured) and striated (measured) time … is merely the invention of a conductor bereft of any phenomenological awareness. Who perceives the differences between time divided up periodically by a meter … [or] by a virtual pulse maintained by the composer/musicians, and smooth time, without a pulse, if the rhythms which overlay it are there precisely to destroy all feeling of periodicity?” Gérard Grisey, “Tempus ex Machina: A Composer’s Reflections on Musical Time,” Contemporary Music Review 2 (1987): 240.

whole formal time system is based on these two categories and on them alone.”30 Boulez’s concept of smooth and striated time is not new; in fact, it can be traced to a number of sources. The most obvious similarity to smooth time is static time–time that is nondirectional, immobile, and is a “temporal continuum determined by progression toward unpredictable goals.”31 Static time can be found in the music of Stravinsky and Messiaen, or even earlier in the music of Wagner (The Ring cycle and Parsifal) and Debussy (Pelléas et Mélisande). Boulez, however, also associates static time with the gagaku Japanese court music: If I hear the gagaku Japanese court music (as I will remember hearing it in 1945 or ’46), the first thing which is very impressive is the expansion of time, because the sustained pace of this music is very slow for a long time. And that’s completely different from our conceptions of time. A related phenomenon is the use of small intervals in gagaku-intervals which do not seem to move. This is kind of destruction of the intervals. And that, for me, was very important because I speak in my own terms when I speak of the category of time. That’s comparing my perception of gagku time with that we are doing here, generally, in Western civilization-namely, to go from a point to another point, which is moving and developing. On the contrary, here in gagaku, I had the impression that the music is completely static.32 These two seemingly opposed temporal conception “cannot be separated in watertight compartments, as antithetical but must be regarded as two poles of a continuum.”33 They can in fact co-exist simultaneously, or juxtaposed intermittently and in succession in time. The distinction between the two concepts is a matter of perception

Pierre Boulez, Orientations: Collected Writings by Pierre Boulez, ed. Jean-Jacques Nattiez, trans. Martin Cooper (London: Faber and Faber, 1986), 87. 31 Jonathan D, Kramer, The Time of Music; New Meanings, New Temporalities, New Listening Strategies (New York: Schirmer, 1988), 452 32 Di Pietro, Dialogues with Boulez, 48-49. 33 Jean-Jacques Nattiez, “On Reading Boulez,” in Orientations, 22.

and perspective, as each can be easily infused with the other.34 While giving a lecture at IRCAM in February 1978,35 Deleuze discussed the mixture of pulsed time and nonpulsed time, where the two concepts are closely related: We saw a kind of non-pulsed time emerging from a pulsed time, even though this non-pulsed could become a new form of pulsation. The first work (Ligeti) [Ligeti’s Kammerkonzert] showed how a non-pulsed time rose from a certain pulsation; the second [Messiaen’s Mode de valeurs et d’intensités], third [Boulez’s Êclat] and fourth [Stockhausen’s Zeitmasse] works developed or showed different aspects of this non-pulsed time; the fifth and last work by Carter [Carter’s A Mirror on which to Dwell] showed how a non-pulsed time could lead to a new form of original pulsation, a very particular, very new pulsation.36 As early as Pli selon pli, Boulez was already composing music in non-pulsed and pulsed time (for example, the senza tempo sections). This is a significant departure from the earlier serial works Structures, Polyphonie X, and Marteau where the rhythmic and temporal conceptions are completely detached from the idea of pulsation. However, it is not until Éclat that Boulez had refined these ideas of smooth and striated time in the most practical and audible way. In Éclat, the resonances of the ensemble (piano in particular) and their acoustics are composed into the score, and the duration of these resonances is left to be determined by the conductor. These sections correspond to the smooth time conception, since it is not possible to sense any internal pulsation or rhythmic pattern 34

“[Deleuze] is surely correct in suggesting that Boulez’s distinction of the smooth and the striated is of less value as a division than as a continuum, since they can be alternated or superposed, a phenomenon we will meet, for example, in sections of Répons.” Edward Campbell, Boulez, Music and Philosophy (New York: Cambridge University Press, 2010), 132. 35 Boulez organized and gave a series of lectures at Ircam, titled “Le Temps Musical,” between 17th to 21th of February in 1978. Boulez presented and analyzed the following five works: Kammerkonzert by György Ligeti; Mode de valeurs et d’intensités by Olivier Messiaen; Éclat by Boulez, Zetimasse by Karlheinz Stockhausen; and A Mirror on which to Dwell by Elliott Carter. Deleuze lectured on the last day of these lectures. 36 Deleuze, “Making Inaudible Forces Audible,” in Two Regimes of Madness: Text and Interviews 1975-1995, 157.

from these undetermined resonances and their unpredictable acoustics (for example, rehearsal numbers from 3 to 7 and 14 to 21). Rhythms in these sections are usually grouped as gestures without any metric clarity. This is emphasized through long passages of static harmony of stretched durations. The middle “Vif” section (rehearsal numbers 8 to 13) and the last section (rehearsal 27 to the end) of Éclat corresponds to the striated time concept, despite the fact that these rhythms do not form any clear pattern or meter, the underlying constant 16th-note pulsation is clearly audible, and the entire rhythmic construction is based completely on the 16th-note and its multiples. In Rituel, the two contrasting sections – static harmonic long held chords by the full ensemble and the multilayered texture of heterophony of various instrumental groups, with each group accompanied separately by a steady pulsation played by the percussion, are perhaps the clearest demonstration of the contrast between pulsed and non-pulsed time in Boulez’s music.37 Similarly, in Messagesquisse, the running 16th-note and 8th-note sections (rehearsal numbers 4 to 7, and 10 to 12, respectively) are pulsed by a permutation of a rhythmic pattern, and every single beat is filled in with at least one event. On the other hand, the solo sections (rehearsal numbers 1 to 3, 8 to 9) are semi-improvisatory in nature, the music purposely avoids any clear sense of pulsation or meter, and the discrete rhythmic patterns in the accompaniment are Morse codes derived from the dedicatee’s last name, with permutations in each cello part (ex. 1.2.1).38 These striated sections are also characteristically filled with long pauses, rests, and unpredictable durations.

See Jonathan Goldman, The Musical Language of Pierre Boulez: Writings and Compositions (New York: Cambridge University Press, 2011), 100-115. 38 Antoine Bonnet, “Ecriture and perception: on Messagesquisse by Pierre Boulez,” Contemporary Music Review 2 (1987): 174.

Ex. 1.2.1. Morse Code of the name SACHER translated into rhythmic patterns S A C H E R

B B B

s s s s Ò Òs Òs s s s s s s Òs

bœ. œ-.

bœ. œ. œ. œ. œ. œ. œ. œ. œ. œ-. R bœ-. œ. œ-. œ. œ. œ. œ. œ. œ. œ. œ. œ. œ-. bœ-. œ. œ-. œ. œ. œ. œ. œ. œ. R -œ. œ. œ-. œ. œ. œ. œ. bœ. œ-. œ-. œ. œ-. œ. œ. œ. œ. œ. œ. œ. œ. bœ. œ-. œ-. œ. œ-. œ. œ. œ.

. . . . -. . . . . -. -. . -. B bœ œ œ œ œ bœ œ œ œ œ œ œ œ

œ. bœ. œ. œ. œ. œ-. œ-.

œ. œ-.

œ. œ. œ. œ. œ. bœ. œ. œ. œ. œ-.

œ-.

œ. œ-.

œ-.

œ. œ-.

œ.

œ. œ. œ. œ. œ-. œ. œ. œ. œ. œ. œ. R œ. œ. œ. œ. œ. œ. R

Boulez’s next large work, Répons, takes these two temporal conceptions to the next level in the latter half of the work, where these two opposing ideas are juxtaposed simultaneously – the notated rhythmic and pulsed orchestra parts against the ornamented unmetered music played by the resonant instruments (rehearsal numbers 42 to 47). Boulez continues to use this opposition in temporal conception in his more recent works such as Dérive 1, Incises, sur Incises, and Dérive 2.39 The unpredictable resonances of the instruments become an important method for Boulez to have musical time that is flexible, a time that is not restricted by pulsation. Time in these works is therefore guided by sound, not by arbitrary serialized durations or integer based pulsed rhythms. The striated and metered sections that involve complex subdivisions or the constantly varied pulsations and rhythms, still confine musical time into subdivided compartments. Dérive 2 will utilize these two concepts to challenge our perception of musical time.

On the employment of Smooth and Striated time in Incises, see Goldman, The Musical Language of Pierre Boulez, 174-185.

Chapter 1.3. Tempo As an experienced conductor, Boulez knows intimately the effect and the capability of tempo manipulation on both local and structural levels. Tempo, more precisely, the fluctuation of tempi becomes another method for Boulez to engage in more levels of temporal manipulation beyond the typical operations of rhythm and meter.40 Tempo, for both Carter and Boulez,41 is different from rhythm and meter: If I distinguish between tempo and rhythm as such, it is because I regard these two aspects of musical time as essentially different. The latter is a function of the unit of duration, a unit which can, for the purpose of analysis, be abstracted from the speed at which it happens. The former is ultimately a rate of unfolding of musical text, and by its nature pragmatic. … The crucial factor in the choice of different tempos … [is] a hierarchy of tempo relationships, which comes back to a relation between speeds of musical unfolding.42 Boulez’s interest in utilizing tempo as another temporal manipulation is much more evident in the more recent works. These works employ tempo in various ways. In some works, tempo manipulations are used to fluctuate the surface level of speed. These 40

Boulez’s interest in the manipulation of tempi, much more prevalent in the later works, might be due to the fact that Boulez was becoming an experienced conductor. Not only has he conducted music which requires constant tempo fluctuations, both interpretatively and specifically notated in metronomic markings, such as in the music of Wagner, Mahler, Schoenberg, Bartok, Berg, Debussy, Stravinsky, and Carter; Boulez has also conducted almost all the premieres and the subsequent performances of his own music. Such experience allows him to manipulate tempo in a practical manner, as well as be confident that these tempo manipulations and interpretations can be altered and decided when he conducts his own music. Furthermore, Boulez often leaves these temporal decisions open in the score, so that the interpretative quality of tempi and the specific temporal modifications directed by the conductor would allow him freedom to determine these features during the performances. 41 See Elliott Carter, Collected Essays and Lectures, 1937-1995, ed. Jonathan W. Bernard (Rochester, New York: University of Rochester Press, 1997), 224-228; and 262-280. One typically associates “metric modulation” with Carter’s music. However, it is a misleading term since Carter generally works with tempi, not meter in his music. Meter implies a hierarchy of downbeats and upbeats, a pattern within pulsations. Carter’s pulsations do not imply a hierarchy of order or pattern. When Carter “modulates” from one tempo to the other, or changes from one meter to the other, the underlying pulsation changes its speed while keeping its internal rhythmic subdivisions the same (but certainly the notation will obviously be different). Therefore, it is the change of tempo that interests Carter, not the change of meter. It is much more appropriate to use the term “tempo modulation” or “temporal modulation” to refer to Carter’s technique of temporal transformation. However, in 90+, the long-range pulse remains fixed while the surface tempo changes. 42 Boulez, Relevés d’apprenti, 132.

manipulations are placed between important structural points or tempi, but their speeds and internal organizations do not form part of an overall scheme (for example, see in the 1986 version of Cummings ist der dichter or in Mémoriale (…explosante-fixe) where almost every single measure has surface tempo manipulation). In some other works, tempo serves as a structural device to differentiate the various sections or their internal divisions. In Dérive 1, for example, the whole work can be divided by its temporal design into two halves.43 In the first half, measures 1 to 27, the tempo marking “Très lent, immutable (q = maximum 40)” is maintained throughout without a single fluctuation (with the exception of the last measure). The meter is also fixed in 4/4 throughout, with only 16thnote rhythmic value and its multiples, though the meter is obscured by the use of varying length of grace-notes in all the instruments. The second half, measures 27 to 46, goes through multiple tempo changes. As the larger pulse/speed becomes slower, the subdivisions become larger and more complex for the first part; and after the midpoint, as the larger pulse/speed becomes faster, the rhythmic subdivisions become less complicated for the second part. This whole second half starts with “Très lent (q = maximum 40)” tempo, with only quarter-notes, 8th-notes, and 8th-note triplets (measure 27). Then the tempo changes to “Elargir le tempo (e = 72)” in measure 36, and the rhythmic value grow to include 16th-note triplets and 16th-note quintuplets. Then the tempo changes to “Elagir le tempo encore plus (e = 63)” in measure 38, and the rhythmic values adds 32nd-notes. In measure 41, the tempo changes to “Elargir le tempo toujours davantage (e = 60)”, while the rhythmic values further include 32nd-sextuplet notes as the music reaches its slowest 43

Jonathan Goldman also makes a similar point with regard to Dérive 1’s symmetrical structure based on its temporal organization of striated time. See Jonathan Goldman, The Musical Language of Pierre Boulez, 116127.

speed and the highest degree of subdivision. Starting from measure 42, the tempo markings are reversed. At first the tempo marking changes to “Resserrer le tempo vers le tempo initial (e = 66)”, while the rhythmic value again add 16th-note septuplets. In measure 44, the tempo changes to “Resserrer encore plus vers le tempo initial (e = 69)”, while rhythmic values are limited to 16th-notes, 8th-notes, 16th-note triplets, and 8th-note triplets. In measure 45, the tempo changes to “Resserrer advantage (e = 75)”, while rhythmic values are limited to only 16th-notes, 8th-notes, and 8th-note triplets. At the end of measure 45, the tempo changes back to the initial tempo of (e = 80), and rhythmic values are only left with 8th-notes and 8th-note triplets. In the second half of the piece, grace-notes are used sporadically, and each time with only a few notes. In comparison, the first half is rhythmically simple, but metrically ambiguous with the use of many gracenotes while keeping the underlying speed/pulse the same. The second half is rhythmically complex, but metrically simple while varying the underlying speed/pulse. The whole musical time in the entirety of Dérive 1 relies fundamentally based on these tempo changes and their rhythmic subdivisions to create temporal fluctuations.44

Goldman analyses these oppositions in musical time in terms of Boulez’s concept of smooth and striated time. Ibid., 116.

In Dérive 2, tempo or metronomic speed has two principal purposes: one functions at the local level and does not have an overall organization, and the basic pulse or speed remains undisturbed; the other functions at the structural level and manifests on a larger time scale, and at times, alters the underlying pulse and speed. The first function of local fluctuation (accelerando and ritardando) often bridges adjacent sub-sections or phrases that are in different tempi. These manipulations of tempi stretch or compress the duration of events; and more specifically, slow down or speed up the rate of unfolding of musical materials while keeping the internal activities (rhythm, meter, or pulsation) unaltered. Boulez’s constant fluctuation of tempi is directly influenced by Mahler and Debussy, whose music constantly require suspension of time and temporal modifications on the local level.45 In a way, these local tempo fluctuations are similar to rubatos in Chopin’s music, whose moments of tempo fluctuation suspend meter and the flow of time. In Dérive 2, the various local accelerations and retardations tend to happen in the interruption sections – sections that are outside of the overall periodicity structure that governs the whole piece.46 These surface level tempo modifications usually help to bridge sections that are in different tempi. Between R6647 and R83, one of the longest interruptions in the piece, the tempo goes through numerous subtle and drastic changes (fig. 1.3.1). Within this section, Boulez has maintained an equal subdivision of 32nd-notes throughout (with the exception of from rehearsal numbers 81 to 83 where the subdivision goes to a sextuplet within an 8th-note), and all the rhythmic values are limited to 32nd-notes 45

Boulez has conducted and recorded almost all the orchestral works of Mahler and Debussy. In works such as Mahler’s symphonies, and Debussy’s La Mer and Jeux, one sees frequent changes of tempo. 46 The interrupting sections and their function in relation to periodicity will be discussed in chapter 3. 47 Throughout this document, Ry will denote rehearsal number y, x-Ry will denote x measures before rehearsal y, while Ry-x will denote x measures after rehearsal y.

and their multiples. With the constant 32nd-notes and 8th-notes as the constant pulsation in the background, the tempo changes are immediately audible. Nevertheless, these changes do not alter the basic pulsation - an overall idea that Boulez was very much influenced by Carter: I was immediately fascinated by his rhythmic writing. I think the principle of rhythmic modulations is particularly interesting. The shifts in rhythmic values are perfectly clear where they occur in the writing; the rhythmic relationships reach their fullest expression, as it were, when they can be heard. In some cases, however, you have to rely on the metronomic relationships. You realize this when conducting, because if you’re not in control of what you hear, the rhythmic modulations are executed more loosely. This idea is found in a simpler form in Stravinsky’s work. More often than not, he used relationships of 3 to 2 or 4 to 3, as in the Symphonies for Wind Instruments, a work which is entirely based on rhythmic relationships of 2, 3, and 4.48 R66 2-R67 R67 R67-2 3-R68 2-R68 R68 R68-2 R68-3 R68-4 … R72 R72-4 R73 R74 R74-2 1-R75 R75 R75-3 R75-4 1-R76 R76

Assez vif (e = 120) en reserrant le tempo Plus vif (e = 132) revenir au Assez vif (e = 120) Plus vif (e = 132/138) revenir au Assez vif (e = 120) acelérer brusquement Plus vif (e = 132/138) … Sub. Tempo (e = 104) Sub. Tempo (e = 96) Sub. Tempo (e = 90) Sub. Assez vif (e = 120) Plus vif ralentir à Modéré (e = 84) poco accel. À peine plus animé (e = 88) poco accel. Un peu plus animé (e = 92)

R76-4 R76-5 1-R77 R77 R77-4 R77-5 R77-6 R78 R78-5 R78-7 R79

poco rall. Plus calme (e = 88) poco rall. Encore plus calme, mais san trainer (e = 84) accélérer Plus vif ravenir rapidement à Très calme, hesitant (e = 78) accélérer progressivement à Animé (e = 98), poco rall. Sub. piécipité (e = 132) revenir à Assez vif (e = 120) ralentir rapidement à Très modéré (e = 98), ralentir Plus calme (e = 90), ralentir Encore plus calme, presque lent (e = 82) accélérer brusquement et beaucoup Assez vif (e = 120) aussi rapide que possible sans changer le tempo Plus modéré (e = 98)

R79-3 R80 R80-1 R80-2 R80-3 R81 R81-2 R81-4 R83

Fig. 1.3.1. Tempo manipulation and fluctuation from R66 to R83 48

Philippe Albéra, and Pierre Boulez, “Pierre Boulez in Interview (2): On Elliott Carter, ‘A Composer Who Spurs Me On’,” Tempo 217 (July 2001): 2.

Example 1.3.2 demonstrates an instance where the acceleration is not only written in the note values, but also indicated with a tempo modifier (the accel. marking). Writtenin acceleration and retardation are often found in the music of Carter, whose practice Boulez admires.49 In this example, the alternation of chords between monophonic instruments and resonant instruments occur from two per measure, to three per measure, and finally to seven per measure, as the harmonic rhythm speeds up (ex. 1.3.2).

“[My] point of interest in Carter’s music was the principle of the written-out accelerandi and ritardandi which are often found in his music and which are extremely effective. One might say that these are primarily gestures; but, in Carter’s work, they form part of the structures along with various intricately overlaid values.” Philippe Albéra, and Pierre Boulez, “Pierre Boulez in Interview (2): On Elliott Carter, ‘A Composer Who Spurs Me On’,” Tempo 217 (July 2001): 2-3.

Ex. 1.3.2. Written-in accelerando from rehearsal 3-R26 to 1-R26 Cor anglais

° &

¸ ≈ œ

‰

¸ ≈ œ

‰

Clarinette en la

Basson

Violon

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≈

‰

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¸ œ ≈

nœ

nœfi¸ œ

nœ

° Vibraphone &

¸ ≈ œ. >

n>œ ˝

≈

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& Piano

b>œ

œ ˝ ≈ 3

> bnœœ

> nbœœ mf

‰

nn>œœ

‰

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‰

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n-œ

^ nnbœœœ nœ

bœ

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nœ

> nœ > ff

^ n#œœ ff

n œ^ #œ ^ nbn#œœœœ

^ #bnœœœœ

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¸ œœœœ ≈ ‰ b##œœœ > >œ n n œ œœœœ #œ ˝ ≈ ‰ * °

≈ nœ nœ f

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> ≈ nbœœ ˝ > nO ≈ nœ ˝

≈ nœ #œ f

>. œœ nnœœ

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¸ nnnœœ ™™™ œ>

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accélérer beaucoup - - - - - - - - - - - - jusqu'à - - - - - -

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nœ >

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accélérer assez peu - - - - - - - - - - - - - - - - - - - - - -

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nœ >

>œ f

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> #œ nœ >

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¸ #œfi

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Structural tempo, a second approach, distinguishes sections by employing two different tempi or speeds, or uses the same tempo to draw a connection between two or

more non-adjacent sections, even if the subdivision within each section is different. Tempo, for both Carter and Boulez, is not limited to surface level modifications of speed, but it is capable of serving a structural role in a large-scale composition, as it can have an organization of its own. Structural tempo in Dérive 2 essentially functions as an important divider for all the sections throughout the entire work. For example, the metronome marking at R5 is q = 138, and the smallest subdivision is the 16th-note, which divides the quarter-note into 4 equal parts. At R10, the metronome marking slows down to q = 92. Yet because the change of tempo, it is immediately clear that the basic pulsation has shifted to a slower rate, and signaling the beginning of a different section (ex. 1.3.3). Four measures later, the tempo goes back to q = 138 (2 measures before R11) and with the same 16th-notes as in R5. There is no doubt that one immediately draws a connection to the previous section just right before the tempo change at R10. In retrospect, it is evident that the music between R10 to two measures before R11 (a total of 4 measures) is an interruption or an “insertion” of some sort, since it interrupts the quarter-note pulse of q = 138. By the virtue of disrupting the pulse, this interruption section sounds “out of time.” Additionally, Boulez emphasized the change of time in these 4 measures by eliminating all the 16th-notes in the first two measures of R10, so that the shift of pulse is clearly articulated (ex. 1.3.4).

Ex. 1.3.3. Two measures before R10 to 5 measures after R10 Cor anglais

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nœ ‰ Œ #>œ > ff

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> B Œ nœ #>œ ‰ Œ ¸ bœfi#œ¸nœ p

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> ‰ nœ ˝

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> B Œ nœ #>œ ‰ Œ

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B bœ ¢ œ ‰ & œ ‰

¸ ¸#œ n œfi# œ

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> # œfi¸n œ œ ˝ ‰

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nœfi¸

¸ nœfin˙

œn œfi¸œ ˙ œ™

¸ nœfin˝œ ˙

¸nœ nœfi

œ nw ¸ nœfi

‰bœ¸ >

sffp

fi¸ ¸ . #nœœ #œ ‰ nœ

# œ. ˝

¸ ¸ ¸ ¢& ‰ œ ‰nœfi n œ¸‰ #œ # œ¸ ‰ nœ # œ

¸ nœfi#œnœ nœ p

∑

∑ >Æ nnœbœœ

‰&

n#nœœœ >'

‰ Œ >Æ ? bnœ œ ‰ Œ

ralentir très brusquement - - - (q = 92)

∑

Ó™

∑

‰ #n#nœœœœ >'

∑

‰

∑

¸ n œ.

¸ ‰ Œ nœ

∑

‰ #nœœ >'

‰ ‰

& Œ nnœœ # œ ‰ Œ >' >' f >Æ > ? Œ n œ n#œœÆ ‰ Œ

#>œ

#œ ‰nœfi¸#œ ˝ ‰ ‰

° fi¸ ¸ Vibraphone & nœ ‰ ‰ nbœ œ ¸#œ. # œ Œ . # œ. .

Piano

‰ bœ nœ œ >

¸ ‰ Ó™ œ

nœ

∑

(q = 138)

Harpe

¸ nœ

nœ œ n˝œ ‰ ˝ bœ

Marimba

nœnœfi¸nœbœ Œ

nœfi¸ p

¸ #œfi

> > > ‰ #œ nœ nœ

Violoncelle

ff sub.

Basson

Ÿ~~~~~~~~~~~~~~~~~~~~~~~~~~ & Œ œfi¸ ¸#œ nœ nœ #œ œ œ œ

Clarinette en la

‰ œfi¸#œ ¸‰ ‰ nœ œ

¸ nœ

Mib Fa# Sol# La§ Réb Si§

Œ f

bœ nn œœ >'

Ex. 1.3.3. (cont’d) Cor anglais

° &

˙™ <#>˙ ™

Clarinette en la

cresc.

poco

cresc.

Basson

Violon

Cor en fa

˙™

poco

¸ #œfi

n œ œ. œ. œ. œ. œ. >'

poco

¸ nœfiœ œ œ œ

˙™ poco

.. >' .

poco

¸ bœfi f

˙ ™

cresc.

poco

˙™

‰ fi¸#œ nœ nœ bœ nœn œ p > mp

¢&

œ™

∑

nœ

‰nœfi¸ #œ nœ n œ bœn œ bœn œ#œ

nœ œ >

∑ ¸ #œfinœ

#œ n œ nœbœ

Sib

nœ

nœ nœ œ >

¸nœ œ bœfi œ œ

¸ nœfi n œ bn œ œ

nœ

nœ b œ

¸ #œfi f

Æ nnœœ ‰ Œ ˝

¸ bœfinœ¸

¸‰ Œ nœ

¸ ¸ b œfin œ

bœ ff

¸‰ Œ nœ f

‰

° *

nœ nœ

‰

> >> ‰ bœ nœb>œ nœnœ‰#œfi¸nœ#œ #œ nœ bœ #œ nœ f

bœ

*°*

¸ #œfi

¸‰ nœ

nœ#œ ‰

Œ f

Si§

nœfi¸n œ

¸‰ nœ '

bœ œ

nœbœ #œ Œ

‰ #˝œÆ

b œ n>œ >b>œ fi¸n œ#œ #œ nœ #œ nœ ‰ bœ n>œ nœ ‰#œ

nœ ‰nnœœfi¸bœ œ Œ #>œ n>œ ' ˝ ¸ nœfinœ

¸ nœ '

¸ n ‰ Œ n œœ '

bœ Œ ¸ bœfi bœ#œ f

n œfi¸ bœ .

¸ bœfi

nœb œ #œ Œ

bœÆ n>œb>œ#>œ‰#œfi¸#˝œÆ ‰ ˝ n>œ

bœfi¸

‰

‰ bœ nœ #œ‰#œfi¸ œ#œ #œ nœ bœ b>œ > #œ nœ nœ >>

¸nœ n œfi n œn œ

œbœ #œ Œ

bœ Œ ¸ ‰nœfi¸ n œ. n œ bn>œœ n œ ' >

poco

∑

Harpe

¸ bœfi

revenir peu à peu au - - - - - - - - - - - - - - - - - - - - - - - - - -

nœ

¸bœ. nœfi ˝

‰ bœÆ nœ bœ#œ‰#œfi¸n˝œÆ ˝ nœ > > >>

œ nœfi¸œ œ n œ œ œ œ œ œ œ œ œ œ œ œ . . . . . . . . . .... >' cresc. mp f cresc. cresc.

¸ ‰ n œ.

cresc.

Piano

œ™

¸ bœfi

> ¸. . . . . . . . . . . . . . . . . . nœfinœ œ œ œ œ œ nœ œ œ œ œ œ œ œ œ œ œ œ

˙™

° Vibraphone &

Marimba

œ™

° &

Alto

Violoncelle

˙™

‰#œfi¸nœ#œ œ ‰ bœ

nœ bœnœnœbœ nœ

nœ. ‰nnœœ bnœœ ˝ #œ nœ

nœ

nœ‰ Œ

¸ nœfi bœ nœ

#œ‰

Œ ?

From another perspective, the metronome marking (q = 92) at R10 is actually a tempo modulation from the previous metronome marking (q = 138), where the relationship between the two tempi is 2 to 3. A dotted half-note in q = 138 is equivalent to

a quarter pulse in q = 46, while a half-note in q = 92 is also equivalent to a quarter pulse in q = 46. This modulation is emphasized through the use of grace-notes. One measure before R10 the rhythmic grouping articulates a duration of a dotted half-note (starting with the grace-notes in cor anglais, bassoon, horn, and marimba), while at R10 all the grace-notes articulate an equal subdivision of the half-note in the new tempo. The larger pulse that takes place in both parts, the dotted half-note in the old tempo, and the halfnote is the new tempo, is actually the same; but their subdivisions are different. The effect is that as if a constant pulse is being divided different from one division to the other.

Ex. 1.3.4. Tempo modulation at R10 Tempo modulation

& Œ

˙™

subdivision

& Œ

¸ ¸ œ œ œ ‰ ‰ œ

œ œ œ œ

œ œ

œ œ œ

Cor anglais

¸ œfi #œ

& œ¸ ‰

Basson

Violon

B œ ‰ ˝

¸ ‰ œ

nœ

Ó™

∑

#œ ˝ ‰ ‰ nœ¸

∑

¸ nœfi #œ p

¸ œ ‰

Ó™

¸n œ > nœfi nœfi¸nœ

∑

Cor en fa

B & Œ

° Marimba & Œ ¢

‰

∑ ¸ #œfi #œ

¸ œ ‰ ‰

œ ˝

> œ ˝

# œfi¸n œ

f sfpp

sfpp

nœ

fi¸ ‰ nœ nœ¸ œ > f sfpp

Ó™

Violoncelle

¸ nœfi#˙ ™

f sfpp

Alto

œ œ œ œ

œ nœ

f sfpp

∑

œ ¸ #œfi

¸ n œfin ˙

Clarinette en la

¸ ‰ ‰ #œfi¸ ¸ nœ œ nœ

¸ nœfi nœ ™

n œfi¸

œ nœ ˝

¸n ˙ nœfi

¸ ¸ n œfi nœ

œn œfi¸œ

œ™

nœ nœfi¸

sfpp

¸ nœfi nœ

¸ nœfi n˙

˙ ¸ nœfi n˝œ

¸nœ nœfi

¸ nœfi

Ó™

‰ bœ¸ >

sffp

(q = 138)

¸ nœfi #œ nœ

nœ Ó mp

Ó™

ralentir très brusquement - - - - - - - - - - - - - - - (q = 92)

¸ ‰ Œ nœ

∑

Consequently, if the fastest speed at R5 and 2 measures before R11 is the 16thnote MM = 582 (where q = 138, 4 x 138 = 582), Boulez maintains the same relative speed of the smallest subdivision in the subsequent sections, despite the change of the metronomic tempo. For instance, at R14, the smallest subdivision is the sextuplet within a quarter-note, which makes the speed of a single sextuplet MM = 540 (where q = 90, 90 x 6 = 540). At R26, the smallest subdivision within a quarter-note is the 32nd-notes, which makes the speed of a single 32nd-note MM = 568/608 (where e = 142/152, 142/152 x 4 34

= 568/608). At R83, the smallest subdivision is the sextuplet within an eighth-note, which makes the speed of a single sextuple MM = 588 (where e = 98, 98 x 6 = 588). The speed of the fastest note in each of these sections, MM = 582, 540, 568/608, and 588, respectively, are relatively similar within a very small margin, which makes the association among these discrete sections clearly audible. The effect is that all the fast notes will sound fairly equal in speed, while the larger pulse shifts to a faster or a slower rate.50 Structural tempo has another role that spans through a longer duration. Starting at R167, the subsequent tempo markings clearly distinguish two contrasting and alternating sections. One section is marked “Rythmique, énergique” while the other is marked “Très calme” (fig. 1.3.5). These two opposing characters, one clearly rhythmic and polyphonic, the other less metrically orientated and homophonic, are juxtaposed in alternation. The tempi in the beginning for both sections are relatively close, q = 74 and q = 60, respectively. As both sections unfold, the “Rythmique énergique” tempo gradually increases from q = 74 to q = 156, while the “Très calme” tempo gradually decreases from q = 60 to e = 96 (q = 48). Furthermore, the durations of each section are systematically shortened towards the end of this large section. At R220 the two contrasting tempi are juxtaposed down to measure to measure with the greatest difference in tempo, further highlighting the “opposite” characters between the two.

Curiously, this method is opposite from Carter, whose tempo changes typically maintains the same basic pulse intact while varying the surface rhythm and the subdivisions within this overall pulsation.

Rehearsal Numbers R167 R175 R176 R180 R181 R185 R188 R189 R190 R193 R194 R196 R197 R199 R200 R202 R203 R205 R206 R208 R209 R210 R211 1-R213 R213 R213-4 R214 R214-5 R215 R215-4 R216 R216-3 R217 R217-3 R217-4 R217-5 R218 R218-2 R218-5 R218-6 R219 R219-3 R219-4

Tempo Marking Rythmique, énergique (q = 74, e = 148) Très calme, suspend, régulier (q = 60) Rythmique, énergique (e = 150) Très calme, suspend, régulier (q = 60) Rythmique, énergique (e = 156) Sub. Très calme, suspend, régulier (q = 60) Rythmique, énergique (e = 160) Très calme, suspend, régulier (q = 60) Rythmique, énergique (e = 166) Très calme, suspend, régulier (q = 58) Rythmique, énergique (e = 180) Très calme, suspend, régulier (q = 58) Rythmique, énergique (e = 180) Très calme, suspend, régulier (q = 56) Rythmique, énergique (e = 184) Très calme, suspend, régulier (q = 56) Rythmique, énergique (e = 184) Très calme, suspend, régulier (q = 56) Rythmique, énergique (tempo rigide) (e = 188) Très calme, suspend, régulier (q = 54) Rythmique, énergique (e = 188) Très calme, suspend, régulier (q = 54) Rythmique, énergique (e = 192) Très calme, suspend, régulier mais tendu (q = 52) Rythmique, énergique (e = 192) Très calme, suspend, régulier mais tendu (q = 52) Rythmique, énergique (e = 192) Très calme, suspend, régulier mais tendu (q = 52) Rythmique, énergique (e = 196) Très calme, suspend, régulier (étirer le tempo) (q = 52) Rythmique, énergique (q = 108, e = 216) Très calme, suspend, régulier mais tendu (q = 52) Rythmique, énergique (q = 114) Sub. Très calme, plus étiré (q = 52) Rythmique, énergique (q = 120) Sub. Très calme, toujours plus étiré (q = 52) Rythmique, vif (q = 126) Sub. Très calme, tendu (q = 50) Rythmique, vif (q = 132) Très calme, tendu (q = 50, e = 100) Rythmique, vif (q = 138) Très calme, tendu (e = 100) Rythmique, vif (q = 144)

R219-5 R220 R220-3 R220-5 R220-6 R221

Très calme, tendu (e = 100) Rythmique, vif (q = 150) Très calme, tendu (e = 96) Rythmique, vif (q = 156) Très calme, tendu (e = 96) Très rapide (q = 164)

Fig. 1.3.5. Tempo markings from R167 to R221 Between R26 to R66, the music goes through several interruptions and tempo changes. It is not immediately clear what relationships there are between these sections, or their underlying organization. Boulez, however, maintains the same tempo marking for some of these sections while varying the tempi for the others, prominently highlighting the alternation of these contrasting sections. In the “Vif” sections, Boulez also rhythmically preserves the same subdivision of 32nd-notes and sextuplet throughout, so that these sections are perceived as a unified whole, with intermittently interruptions that use different rhythmic subdivisions (fig. 1.3.6). Similarly, the next section from R83 to R129 goes through the same process of maintaining the same tempo marking for one group of sections while varying the other, further highlighting these alternations and interruptions.

Rehearsal numbers R26 R35 R37 R45 R49 R56 R63 R66

R83 R85 R87 R89 R91 R94 R102 R105 R111 R114 R126 R129

Tempo Markings Vif (e = 142/152) (e = 108) Vif, comme précédemment (e = 142/152) (e = 96 – 108 – 102 – 96 – 92 - 108) Vif, comme précédemment (e e = 142/152) (e = 108 – 138 – 118 – 114 – 110 – 106 – 102) o T Vif (e = 142/152) Assez vif (e = 120 – 132 – 138 – 132 – 120 – 132 – 112 – 104 – 96 – 90 – 84 – 88 – 92 – 88 – 84 – 78 – 98 – 132 – 120 – 98 – 90 – 82 – 120) Plus modéré (e = 98) Plus calme (e = 76) Plus modéré (e = 98) Plus calme (e = 74) Plus modéré (e = 98) (e = 84 – 92/88) Plus modéré (e = 98) Tempo détendu (e = 74), très régulier Plus modéré (e = 98) Large (e = 62), très régulier Plus modéré (e = 98) Assez modéré (e = 76 – 74 – 72 – 70 – 68 - 66)

Fig. 1.3.6. Tempo markings from R26 to R129

Chapter 1.4. Periodicity We do not consider periodicity as either basic material nor as the unit of rhythmic structure, but the most simple, most probable phenomenon; it is tempting to see it as an ideal point of reference for the perception of time, as is a sinusoidal sound for the perception of pitches, but not at all the a priori foundation of a hierarchical system. …Periodicity is irreplaceable; it allows a pause in the music’s unfolding, the suspension of time and, sometimes, a redundancy helpful to our powers of comprehension. When the musical structure demands it, we use it for its intrinsic qualities, avoiding both rejection and obsession.51 The word periodicity, according to the Oxford English Dictionary, is defined as “the quality or character of being periodic; the quality of regular recurrence; tendency to recur especially at regular intervals.”52 Periodicity in music refers to an uninterrupted temporal phenomenon in which an event or a series of events of a fixed duration is repeated at a regular interval.53 Periodicity, then, can be understood as comprised of two essential elements: pulsation and event. Pulsation is the underlying pulse that occurs at the beginning of each cycle (or period), and it is an “uninterrupted sequence of reference points with respect to which rhythmic flow is organized.”54 Event is the object that is reiterated in each cycle.55 Example 1.4.1 demonstrates a pulsation, which in fact can be

Gérard Grisey, “Tempus ex Machina: A Composer’s Reflections on Musical Time,” Contemporary Music Review 2, (1987): 245-247. 52 Oxford English Dictionary, 2nd ed., s.v. “periodicity.” 53 "All forms of music in the Central Aferican Republic are constructed according to a principle of periodicity." Simha Arom, African Polyphony and Polyrhythm: Musical Structure and Methodology, translated by Martin Thom, Barbara Tuckett and Raymond Boyd (Cambridge: Cambridge University Press, 1991): 230. 54 Simha Arom, African Polyphony and Polyrhythm, 202. 55 "Metrically speaking, the period can thus be broken down on two lower levels, into the pulsation and the operational values it contains. We must remember tat, characteristically, this organisation involves no intermediate level between the period itself and the pulsation, consisting of a regular accentual system, i.e., the 'measure' with its characteristic strong beat, as found in Western music. Consequently, the 'beats' comprising the period all have equal status." Simha Arom, African Polyphony and Polyrhythm, 231.

considered as a hybrid form of periodicity, since each pulse is equivalent to an event.56 However, pulsation cannot be considered as rhythm, since each pulse functions primarily as a reference point; it is an implied temporal grid rather than a sounding event. Pulsation can be best understood as a "unit of measurement; its infrastructure is chronometric time, e.g. the second [in a clock]."57

Ex. 1.4.1. Pulsation Pulsation

/ œ

Example 1.4.2 demonstrates two illustrations of a single-line periodicity. The event, which is composed of a rhythmic pattern, repeats for each successive measure at a fixed interval, while the pulsation marks the beginning of each cycle.58 In the example, the underlying pulsation is an implied temporal grid of this rhythmic pattern. Similarly, ostinato is also a form of periodicity, since ostinato refers to "the regular and uninterrupted repetition of a rhythmic or melodic-rhythmic figure, with an unchanging periodicity underlying it."59

"The pulsation is an isochronous reference unit used by a given culture for the measurement of time. It consists of a regular sequence of points in relation to which rhythmic events are ordered. Moreover, in polyrhythmic music, the pulsation is the common denominator, from the standpoint of temporal organization, for all the parts in a piece. It is therefore the basic unit of time with respect to which all durations are defined." Simha Arom, African Polyphony and Polyrhythm, 230. 57 Gérard Grisey, “Tempus ex Machina: A Composer’s Reflections on Musical Time,” 239. 58 A cycle can also be described as a period, which "provides a temporal framework for rhythmic events." Simha Arom, African Polyphony and Polyrhythm, 230. 59 Simha Arom, African Polyphony and Polyrhythm, 40.

Ex. 1.4.2. Periodicity of pulsation and event

° / 43 œ œ œ œ œj ‰ œ œ œ œ œj ‰ œ œ œ œ œj ‰ œ œ œ œ œj ‰ event

Pulsation

3 ¢/ 4 œ

cycle/period

> > > > > > > > > > > > ° 44 œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ / event

Pulsation

4 ¢/ 4 œ Œ Œ Œ

œ Œ Œ Œ

cycle/period

Example 1.4.3 demonstrates a periodicity consisting of multiple lines of equal-length cycles unfolding simultaneously.60 Despite the differences in the rhythmic patterns within each line, both cycles and pulsations align throughout since the duration of the rhythmic pattern for each cycle is the same. Ex. 1.4.3. Periodicity of multiple lines of equal length cycle > > > > > > > > > > > > > > > > ° / 46 œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ Pulsation

6 ¢/ 4 œ

Ó™

cycle/period

Ó™

>> > > > >> > > > >> > > > >> > > > ° / 46 œj œ œ œ œ œ œ œ œ œ œ œ œj œ œ œ œ œ œ œ œ œ œ œ œj œ œ œ œ œ œ œ œ œ œ œ œj œ œ œ œ œ œ œ œ œ œ œ Pulsation

Ó™

cycle/period

Ó™

Arom defines such periodicity, regardless of length, as macroperiod: "a macroperiod is the cycle obtained when periods of different lengths are superposed, and each individually is shorter. This happens, for example, when two or more periods stand in a ratio of 2:3 and/or 3:4. The macroperiod then provides the only point at which all the period will coincide." Simha Arom, African Polyphony and Polyrhythm, 231.

Ex. 1.4.4. Periodicity of multiple lines of unequal-length cycles > > > > > > > > > > > > > > > > > > > > ° / 45 œ œ œ œ œ œ œj œ œ œ œ œ œ œ œ œ œj œ œ œ œ œ œ œ œ œ œj œ œ œ œ œ œ œ œ œ œj œ œ œ œ œ œ œ œ œ œj œ œ œ Pulsation

5 ¢/ 4 œ

cycle/period

> > > > > > > > > > > > > > > > > ° / 45 œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ Pulsation

5 ¢/ 4 œ

cycle/period

j ‰ œ Œ

cycle/period

‰ œ

‰ ‰ Œ

‰ œ

‰

NEW CYCLE BEGINS

> > > > > > > > > > > > > > > > > > > > ° / œ œ œ œ œ œ œj œ œ œ œ œ œ œ œ œ œj œ œ œ œ œ œ œ œ œ œj œ œ œ œ œ œ œ œ œ œj œ œ œ œ œ œ œ œ œ œj œ œ œ Pulsation

¢/ œ

>j > > > > > > > > > > > > > > > > / œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ j Œ ‰ œ ‰ Ó Œ œ Œ Ó Œ Œ œ Ó ‰ œ Œ ‰ Ó œ Œ Œ Œ ‰ œ Pulsation ¢ / °

Example 1.4.4 demonstrates a periodicity of multiple lines of unequal-length cycles unfolding simultaneously. In this example, each rhythmic pattern in each line is not only different from the other in terms of rhythmic construction, but the total duration as well: the rhythmic pattern in the top line consists of ten 8th-notes, while the rhythmic pattern of the bottom line consists of nine 8th-notes. Because the two rhythmic patterns are unequal in length, the pulsations will not align throughout; it is not until measure 10 that the two pulsations align once again. Each line is considered as a periodicity on its own, while the whole cycle between the two layers is another periodicity at a larger-scale, which takes nine pulsations for the top line and ten pulsations for the bottom line to complete a cycle, where the next cycle begins by the convergence of the two pulsations at measure 10. Example 1.4.5 is another representation of example 1.4.4. The ratio between the two pulsations as a cycle can be best described and expressed as a polyrhythm 9:10 (ex. 1.4.6).

Ex. 1.4.5. Periodicity of multiple lines of unequal-length cycles

Ex. 1.4.6. Polyrhythm ratio 9:10 between the two pulsations

° ¢/ œ ° ¢/ œ

œ œ

The rhythmic ratio between the two pulsations can be understood as a large-scale polyrhythm 9:10. In fact, any polyrhythm where each individual part remains consistent or equally-spaced temporally can be regarded as a periodicity between two or more streams of pulses (ex. 1.4.7).61 More specifically, "linear repetition create a rhythm of durations between such repetitions, so that there is also the sense in which repetitions of different periodicities in simultaneous instrumental statements create 'polyrhythms,' and in which the individual rhythmic lines constitute a partitioning of time units 61

In this context, a polyrhythm is limited to “a system of two or more streams of periodic pulsations,” although technically a polyrhythm can refer to a combination of two or more rhythms of any kind that is not necessary periodic or equally-spaced. See John Link, “Long-Range Polyrhythms in Elliott Carter’s Recent Music” (PhD diss., City University of New York, 1994): 8.

corresponding to the partitioning of smaller units by pitch repetition in the individual line, or by repetition of simultaneities in the ensemble."62 In this context, a polyrhythm refers to a system of two or more streams of periodic pulsations, though the term technically refers to a combination of two or more independent rhythms of any kind. Since the entire duration of the polyrhythm from one convergent point to the next is a cycle between the pulsations, the point of convergence becomes an important structural pillar for indicating the starting point of each successive cycle (ex. 1.4.8).63

Ex. 1.4.7. Multi-layered Polyrhythms as periodicities

° ¢/ œ œ œ œ œ œ œ œ œ 9

° ¢/ œ œ œ œ œ œ œ œ œ œ 10

° ¢/ œ ° œ ¢/

œ 2

œ œ

° ¢/ œ

° ¢/ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ 17

° ¢/ œ œ œ œ œ œ œ œ œ œ œ 11

° ¢/ œ œ œ œ œ œ œ 7

° œ œ œ œ ¢/

° œ œ œ œ œ œ ¢/

Milton Babbitt, "Edgard Varèse: A Few Observations of His Music." Perspectives of New Music 4, no. 2 (1966): 19. 63 “Periodicity is an extremely potent device with different effects and effectiveness at various levels. The musical material which establishes regularity encourages focus on other, while simultaneously contributing a specific character to the music. Many musical figures involve some periodicity at a pulse or sub-pulse level, and often these exist within a hierarchical structure. As periodicities under three or four seconds are grasped directly, their incorporation into musical configurations helps recognition of future recurrences of those configurations. By maintaining a contrast in the periods of different coexisting layers of material, the composer can clarify the definition of each layer, thus making the interplay of the various layers more audible.” Rosemary Mountain, “Time and Texture in Lutoslawski’s Concerto for Orchestra and Ligeti’s Chamber Concerto,” Ex-tempore 7, no. 1 (Summer 1994).

Ex. 1.4.8. Point of convergences in 9:10 polyrhythm ° ¢/ œ œ œ œ œ œ œ œ œ 9

œ œ œ œ œ œ œ œ œ

° ¢/ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ 10

"point of convergence"

Any polyrhythm that is derived from the pulsations of the periodicities, regardless of its duration, the number of streams within, or the complexity between rational or irrational subdivisions, is always symmetrical. The midpoint of any polyrhythm is exactly halfway between the two convergence points, and the cross-rhythm that takes place after the midpoint is exactly the retrograde inversion of the first half (ex. 1.4.9). However, such symmetrical characteristic is only true to in-phase polyrhythms, it does not apply to out-of-phase polyrhythms, since the streams of pulses do not converge at all (ex. 1.4.10).

Ex. 1.4.9. Symmetrical structure of the polyrhythm

° ¢/ œ

°/ œ œ œ œ œ œ œ œ œ œ ¢ 10

"point of convergence"

œ œ œ œ œ œ œ œ œ œ

"midpoint"

Ex. 1.4.10. Out-of-phase periodicities where the pulses do not converge

° Pulsation

¢/

3 3 j j3 œ œ œœ œ œ œ

œ œ œ œ

°/ œ œ œ œ œ œ œ œ œ 5

Pulsation

¢/ œ Œ

3 3 j j3 œ œ œœ œ œ œ

œ 5

œ 3

œ œ œ œ œ œœœœ œ

3 j œ œ

œ 5

œ 3

œ œ œ œ œ œœœœ œ

Example 1.4.11 demonstrates two instances of in-phase periodicity, where streams of pulses converge at the beginning of each cycle. Example 1.4.12 demonstrates a three-layer out-of-phase periodicity, where the three pulsations do not align.

Ex. 1.4.11. In-phase periodicity

Ex. 1.4.12. A three-layer out-of-phase periodicity

° / 49 œ œ œ œ œ œœœœœœœœ œ œ œ œ œ œ œ œ œ œ œ œ œ œœœœœœœœ œ œ œ œ œ œ œ œ œœœœœ 5

Pulsation

9 ¢/ 4 œ

œ œ™

9 ¢/ 4 œ ° / 49

Pulsation

9 ¢/ 4

j œ™

° / 49 œ œ œ œ Pulsation

œ œ™

œœ œ œ

œ œ œœ

œ 3

œ œ™

œ 5

œ œ œ œ œ œ œ œœ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ

Example 1.4.13 illustrates an example of a three-layer in-phase periodicity. The ratio among the three streams of pulses become much more complex, as it is a polyrhythmic ratio of 15:10:6. The rhythmic pattern for the first line lasts two quarternotes, the rhythmic pattern for the second line lasts three quarter-notes, while the rhythmic pattern for the third line lasts five quarter-notes. The total duration of the entire periodicity lasts 30 quarter-notes, since 2 x 3 x 5 = 30. The midpoint of this largescale periodicity occurs after 15 quarter-notes, where coincidentally the bottom two lines also converge for the second time.

Ex. 1.4.13. A three-layer periodicity of rhythmic ratio of 15:10:6 NEW CYCLE BEGINS

° 44 œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ / Pulsation

4 ¢/ 4 œ Œ œ Œ

œ Œ

° 44 œ œ œ œ œ œ / Pulsation

4 ¢/ 4 œ Œ Œ œ ° 44 œ œ œ œ œ œ œ /

Pulsation

4 ¢/ 4 œ Œ Ó ° ¢/ œ ° ¢/ œ ° ¢/ œ

œ Œ

œ Œ œ Œ

œ Œ

œœœœ œ œœ

Œ Œ

œ Œ

Œ œ Œ Œ

œ Œ Œ œ

œ œœ œœœœ œ œ œœœœ œœœ œ œœ œœœœ œ œ

œœœœœœœ

œ œœœœœœ œ œ

œœœœ

œ Œ Œ Œ

Œ œ Œ Œ

œ Œ

œ œ

œ Œ

œœœœ œ œœ œœœ œœœœ œ œœœœœ

œ Œ œ Œ

œœœœ œ œœ œœœ œœœœ œ œœœœœ œ Œ

Œ œ

œ Œ œ Œ

Œ Œ

œ œ

Œ œ Œ Œ

Œ Œ œ Œ 15

œ œ

œ œ œ

œ Œ Œ œ

Œ Œ Œ œ œ

œ œ

∑ œ

Œ Œ

œ Œ

œ œ

Periodicity is a relatively recent temporal concept in Western music, developed mostly in the latter half of the 20th-century, especially in the works of Oliver Messiaen, Edgard Varèse, Elliott Carter, Colon Nancarrow, John Cage, Henri Pousseur, Iannis Xenakis, György Ligeti, Luciano Berio, Harrison Birtwistle, Morton Feldman, Steve Reich, Philip Glass, Brian Ferneyhough, Gérard Grisey, Tristan Murail, John Adams, Beat Furrer, Sciarrino, Michael Jarrell, Yan Marsez, and Bruno Mantovani, just to name a few.64 The concept of periodicity does not originate from Western music, as it is often found in African, Indian, and Balinese music,65 as well as in most Asian music (especially in the Japanese Gagaku). This non-Western temporal conception of periodicity has

14th-century isorhythmic motets can be considered as a form of periodicity between pitch and rhythm. “All forms of music in the Central African Republic are constructed according to a principle of periodicity.” Simha Arom, African Polyphony and Polyrhythm, 230; and Simha Arom, “Time Structure in the Music of Central Africa: Periodicity, Meter, Rhythm and Polyrhythmics.” Leonard 22, no. 1 (1989): 91-99. 65

greatly influenced composers in the last sixty years. Regarding Messiaen’s use of periodicity, Boulez writes: As in the Indian tradition … [a] feature of these sounds – their duration or rhythmic relations – is governed by laws that Messiaen was gradually to formulate with increasing precision and eventually to build into a logically coherent system. … The idea of regular metre gradually disappears from his music and is replaced by two fundamental principles. The first of these is the inequality of basic note values starting from the smallest pulse (a point of which he was greatly influenced by Stravinsky); and the second is the rhythmic sequence based on a type series in which the initial cell determines the rhythmic construction and is no longer obliged either to take its place in, or to counter, a regular rhythm. Detached from any obligatory and limited periodicity, the idea of duration becomes increasingly supple and subtle. Periodicity becomes variable and can be long term as well as short term. (Here Indian rhythmic pattern played an important part).66 The core concept of periodicity is radically different from much of the temporal conception in the traditional Western music, which mainly focuses on meter and rhythm.67 Meter implies and constitutes a hierarchy of patterns – circular and repetitive patterns. Even if the meter is ambiguous, changes irregularly, or has an uneven and unusual number of beats, the sense of downbeat and upbeat remains an important characteristic of meter, and these strong and weak beats are crucial indicators to the circular structure of meter. The strong and weak beats also form a circular formation, where strong beats are followed by weak beats, and weak beats are served as preparations 66

Boulez, Orientations, 409. “A basic and traditional structural function of periodicity is the establishing of a pulse framework. Traditionally, this framework was manifest as a metric hierarchy. In additive forms, increasingly in contemporary works, a single pulse layer maybe the only unifying force. In either type, the regularity is likely to provide a background grid for more irregular and expressive rhythms. In the case of metric-style hierarchical frameworks, the irregularities are frequently at the foreground level, resulting from a variety of subdivisions of the pulse or ‘super-pulse.’ (the term employed here for bar-length periods). In a additive framework, however, irregularities are typically at higher grouping levels, as a result of addition or multiplication of pulse or sub-pulse.” Rosemary Mountain, “Time and Texture in Lutoslawski’s Concerto for Orchestra and Ligeti’s Chamber Concerto.” 67

for strong beats. Rhythm, on the other hand, solely comprises of patterns of long and short durations. These patterns would interact with meter to create highly complex relationships of predictabilities of patterns or unexpectations. Periodicity, on the contrary, is free from downbeats and upbeats, and it is not circular. Periodicity constitutes points on a straight line that extends into infinity; it has neither beginning nor end. Each pulse in a periodicity is no more important than the other; there are no internal patterns or hierarchies within a periodicity. Pulsation is not analogous to the "beat" in the traditional metrical structure, and the regular succession of equal durations of pulsations do no suggest any internal or large-scale groupings. The essential difference between periodicity and meter is that meter is structured within a closed system, measured and subdivided by a specific rhythmic grouping, while periodicity does not rely on any such internal structure. However, the large-scale pulsation that arises from meter (hypermeter for example) can be argued as a form of periodicity; in a way, meter can be understood as a micro-level structure imposed on periodicity. Furthermore, periodicity/meter establishes a grid of reference points for various durations to be perceived and understood as rhythms, since, rhythm can be identified "by relating it to a given pulse, the meter, in the form of a periodic reference point. . . .Each rhythm is perceived in its qualitative relationship to meter (on the beat, off the beat) but also in its quantitative relationship to meter (longer or shorter than the beat). Without a reference pulse we are no longer talking of rhythm but of durations. Each duration is perceived quantitatively by its relationship to preceding and successive durations."68

GĂŠrard Grisey, "Tempus ex Machina: A Composerâ&#x20AC;&#x2122;s Reflections on Musical Time,â&#x20AC;? 240-241.

Another major difference between meter and periodicity is that meter inescapably creates expectations of repetition and arrival. Metrical innovations and unexpectedness are rooted in play on the listener’s expectations. Periodicity lacks a dramatic device for the listener to expect an arrival of any kind, and “absolute, mechanical periodicity tires the listener as much as a ceiling or wall composed of perfectly equidistant tiles. We have all noticed how the periodicity of the synthesizer or computer in its perfect redundancy merely induces boredom and inattention.”69 Therefore composers began to explore various ways to incorporate periodicity within the musical fabric with dramatic effects, either through various methods of intensification, modification, multi-layering of periodicities, synchronization, or superimposition with other rhythmic patterns, interruptions, or progressive rhythmic phasings so that the underlying periodicities can have motion and direction. Periodicity then becomes goal-oriented, and it is capable of creating expectations and hierarchy of structures for the listener. The progressive variations and transformations that take place in a periodicity become an intriguing technique and process for the composer to manipulate such simple temporal concept. Both Carter and Ligeti, the two composers that Boulez credited as the main influences for the periodicity concept in Dérive 2, often employ periodicities in their own music.70 What is common between these two composers is that their temporal conception in music is not limited only to the manipulation of meter or rhythmic groupings; rather, the emphasis of their music tends to focus on polyrhythm, periodicity, and pulsation – all of which rely on linear temporal 69

Grisey, “Tempus ex Machina,” 245. Boulez has mentioned these two composers, along with Nancarrow, as the influences for the concept of periodicity in Dérive 2 in many different occasions. See prologue for detailed documentations on these occasions. Boulez knows these two composers works intimately, as he has conducted, recorded, and commissioned their works numerous times. 70

conception in multiple levels of time unfolding simultaneously, rather than the traditional circular approach of meter.

György Ligeti Periodicity in Ligeti can be found in a number of works such as Poème Symphonique für 100 metronomes (1962), Ramifications (1968-9), Kammerkonzert (196970), Clocks and Clouds (1972-73), Horn Trio (1982), Piano Concerto (1985-88), Études pour piano (1985-2001), Violin Concerto (1990/92), and Hamburgisches Konzert (1998/99, 2002). In these works, one often finds multiple periodicities unfolding simultaneously, and these multiple layers of periodicities of complex polyrhythms and temporalities of dissimilar pulsations often create rhythmic and metric tension and conflict in both the structure and the surface level of the music.71 Ligeti was very much influenced by the music from the Central African Republic in the use of polyrhythms: The first African recording hit me like a bomb...It was in the fall of '82 that I heard this recording...[and] the music of the Banda tribe; theirs is a very special kind of music. It had a very profound effect on me... There is a twenty-piece orchestra on this recording of the Banda... Every instrument has its own rhythmic pattern. This music is very reminiscent of the technique of Machaut and Philippe de Vitry, but it is much, much more complicated rhythmically. This kind of polyrhythm does not exist in the European musical tradition.72 Furthermore: That which is eminently new in my piano études is the possibility of a single interpreter being able to produce the illusion of several simultaneous layers of different tempi. That is to say, our perception can be outwitted by imposing a 'European' accent pattern onto the non-accentuated 'African' 71

"What attracts me is the idea of superimposing several levels, several different time-grids moving at different speeds, and so very subtly achieving rhythmic deviations." György Ligeti, interview by Josef Häusler, Györgt Ligeti in Conversation, trans. Sarah E., Soulsby (London: Eulenberg Books, 1983): 108; moreover, "here what you perceive as rhythm is not rhythm coming from the succession of notes your fingers play. The actual rhythm of the piece is a pulsation that emerges from the distribution of the notes, from the frequency of their repetitions...the accelerando of the rhythm is therefore the result of an increased frequency of a note, it is realized through a modified note distribution." György Ligeti, interview by Péter Várnai, György Ligeti in Conversation , trans. Gabor J. Schabert (London: Eulenberg Books: 1983), 61. 72 Stephen Satory, "Colloquy: An Interview with György Ligeti in Hamburg," Canadian Music Review 10, no. 1 (1990): 111.

pulsation...I am using only an idea from African notions of movement, not the music itself. In Africa, cycles or periods of constantly equal length are supported by a regular beat (which is usually danced, not played). The individual beat can be divided into two, three sometimes even four or five 'elementary units' or fast pulses. I employ neither the cyclic form not the beats, but use rather the elementary pulse as an underlying gridwork. I use the same principle in DĂŠsordre for accent shifting, which allows illusory pattern deformations to emerge: the pianist plays a steady rhythm, but the irregular distribution of accents leads to seemingly chaotic configurations. Another fundamental characteristic of African music was significant to me: the simultaneity of symmetry and asymmetry. The cycles are always structured asymmetrically (e. g. twelve pulses in 7 + 5), although the beat, as conceived by the musician, proceeds in even pulses.73 Moreover, My interest in the music Arom has recorded stems also from the proximity I feel exists between it and my own way of thinking with regards to composition: that is, the creation of structures which are both remarkably simple and highly complex. The formal simplicity of sub-Saharan African music with its unchanging repetition of periods of equal length, like the uniform pearls of a necklace, is in sharp contrast to the inner structure of these periods which, because of simultaneous superpositioning of different rhythmic patterns, possesses an extraordinary degree of complicity. Gradually, through repeated listening, I became aware of this music's paradoxical nature: the patterns performed by the individual musicians are quite different from those which result from their combination. In fact, the ensemble's superpattern is in itself not played and exists only as an illusory outline. I also began to sense a strong inner tension between the relentlessness of the constant, never-changing pulse coupled with the absolute symmetry of the formal architecture on the one hand and the asymmetrical internal divisions of the patterns on the other. What we can witness in this music is a wonderful combination of order and disorder which in turn merges together producing a sense of order on a higher level.74 The opening measures of the first movement of the Piano Concerto, for example, are rhythmically constructed based on a periodicity of two rhythmic patterns. The rhythmic pattern 3-3-3-2-3-3-3-4-2-2-2 is realized by the solo piano and four tom-toms, 73

Ligeti, as translated by Sid McLauchlan in linear notes to "Ligeti: Piano Etudes, BK 1," Pierre-Laurent Aimard, piano (Erato, ECD 75555, 1990), 3. 74 GyĂśrgy Ligeti, forward to African Polyphony and Polyrhythm, xvii.

notated in 12/8 time, with the 8th-note as the basic unit (ex. 1.4.14). The other rhythmic pattern 3-3-3-4-2-2-3-2-2 is realized by the strings in pizzicato, notated in 4/4 time, with also the 8th-note as the basic unit (ex. 1.4.15). However, each 8th-note in the piano solo and percussion rhythmic pattern in 12/8 time is equivalent to a triplet 8th-note in the strings in 4/4 time, and the two rhythmic patterns are superimposed in the opening measures (ex. 1.4.16). These two periodicities are different in duration; the strings pattern is shorter than the piano and the tom-toms pattern. The polyrhythmic ratio between the two is 10:12, which can be reduced to 5:6 (ex. 1.4.17). Each pattern is repeated without any modification or interruption throughout the whole first section, and both patterns unfold simultaneously from the very beginning of the first note, so that taking as a whole, the opening measures creates a periodicity between these two rhythmic patterns.

Ex. 1.4.14. Ligeti Piano Concerto – Piano solo and percussion part 1

Tom-toms

Piano Solo

j j j j 8 œ‰ ‰ œ‰ ‰ œ‰ ‰ œ œ œ / 12 3

j œœ 3

œ œ œ œ œ œ œ & 12 8 œ œ œ œ œ œ œ œ œ œ œ œœ œ œ 3

Tom-toms

Piano Solo

j j j œ ‰‰œ ‰‰ œ ‰ ‰œ 3

j œœ

œ ‰ œ œ œ œ œœ œ œ œ œ J œ œ œ

œœ J œ ‰ œ

j œœ 3

œ J œ ‰ œœœœ œ

œ‰ ‰ ‰ œ J

œ œ œJ ‰

œœœœ œœœ ‰ œ œ œ 4

œœ œ 2

œ œ J ‰ ‰ ‰ œ œ œ œJ ‰ œ ‰ œœ Œ œ œ œœ œœ œ œ œ œ

Ex. 1.4.15. Ligeti Piano Concerto – String pizzicatos

œ & 44 bœœ bœ

Strings

& bœœœ bœ

œj ‰ œœ Œ œ 3

bœœ bœœ

j nœ ‰ bœœ Œ nb œœ bœ b œœ

œj ‰ bœœ Œ bœ

j œ ‰ bœœ Œ bœ

j œ œj œ n œ ‰ b œ ‰ œœ ‰ bœœ Œ bœœ bnœœ n b œœ bœJ œ b œ n b œœ 2 3 2 2 2 j nœ œ ‰ nbœœ ‰ bœœ ‰ bœœœ Œ bnœœ bœœ bœJ n b œœ œ bœJ bœ bœ

Ex. 1.4.16. Two rhythmic patterns superimposed 1

Tom-toms

Piano Solo

Piano and tom-toms cycle

j j j j 8 œ‰ ‰ œ‰ ‰ œ‰ ‰ œ œ œ / 12

œ œ œ œ œ œ œ 8 œ œ œ œ œ œ œ œ œ œ œ œœ œ œ & 12 Strings cycle

Strings

j ‰ œœœ Œ œ

4 œ & 4 bœœ bœ

Piano Solo

Strings

œj ‰ bœœ Œ bœ

bœœœ bœ

j œœœ‰

œ‰ ‰ ‰ œ J

œœ J

œœœ œ œ œ œœ ‰ œ œ

œ ‰ œ

œœ œ

j j œ ‰ bnœœ ‰ bœœœ ‰ bœœ Œ bœJ n b œœ bœ

Piano and tom-toms cycle

j j j j j œ / œ ‰ ‰ œ ‰ ‰ œ ‰ ‰ œ œ œ œ œ œJ œ J ‰ ‰ ‰ œ œ œ œJ ‰ œœœ œœ œ ‰ œ œ œ œ œ œ œ œ ‰ œ œ œ œ œ ‰ œœ Œ œ œ œ & œ œ J œ œ œ œ œ œ œ

Tom-toms

j œœ

Strings cycle

œ & bœœ bœ

nœœ nbb œœ

j ‰ bœœ Œ b œœ

bœœœ bœ

œ nbbœœœ

œj ‰ bœœ Œ bœ

nœ œ ‰ nbbœœœ ‰ bœœ bœJ J

Ex. 1.4.17. The polyrhythmic ratio between the two periodicities ° 4 Pulsation ¢ / 4 œ

° 4œ œ œ œ œ œ œ œ œ œ œ œ ¢/ 4 12

Pulsation

° ¢/ œ ° œ ¢/

œ œ

Since these two rhythmic patterns are different in duration, they create a complex multi-layered periodicity of unsynchronized alignments and cycles of polyrhythm. The first alignment of these two strata of rhythmic patterns begins the movement, and the next alignment occurs at rehearsal letter B, and the next alignment concludes the first section of the first movement (right before rehearsal F). The first section of work is therefore determined by the completion of two cycles of this multi-layered periodicity, and all the rhythmic complexities and syncopations arise through the superimposition between these two streams of rhythms, which also create a byproduct of cross-rhythm between 3 against 2 that is inherent from the two time-signatures. The next cycle of periodicity begins at rehearsal F where the instrumental assignments have shifted. The 3-3-3-3-2-3-3-3-4-2-2-2 rhythmic pattern that was originally realized by the piano and the tom-toms are now transferred to the woodwinds, while the 3-3-3-4-2-2-3-2-2 rhythmic pattern that was realized by the strings are now transferred initially to the left hand of the piano (2 bars before rehearsal H), and then to the whole piano part and the castagnettes at rehearsal I. Meanwhile, brief appearances of rhythmic cycles of different basic units decorate around the two rhythmic strata (for instance, the horn solo is composed based on a 5-5-5-10 pattern with the 16th-note as the basic unity, or the 12 dotted 8th-notes in the strings and woodwinds that appear sporadically throughout the movement). The entire temporal organization of the first movement is structured around the periodicities of these non-synchronizing rhythmic patterns of varying lengths and rhythmic units.

Similar types of rhythmic pattern of periodicities of large-scale cycles are also found subsequently in Études 8 Fém for piano, and in the Violin Concerto. In some other of Ligeti’s works, periodicities frequently go through systematic, chaotic, or fractal manipulations, as seen in the third, fourth, and fifth movement of the Piano Concerto, Étude 1 Désordre for piano (which incidentally, is dedicated to Boulez), Étude 6 Automne à Varsovie for piano, and the Violin Concerto. In some other works, periodicities are assigned to different instrumentations, while all these instruments share the same equal subdivision of beats, and the multiple periodicities arise through different multiples of such equal subdivisions. This technique is exemplified in the third movement of the Piano Concerto, Étude 6, Automne à Varsovie for piano, the third movement of the Violin Concerto, and in the fourth movement of Hamburgisches Konzert. Conversely, periodicities can also arise through simultaneous presentation of unequal subdivisions of beats, as seen in Ramifications, the third movement of Kammerkonzert, and the fourth movement of the Piano Concerto. In the most extreme case of Poème Symphonique, the periodicities arise simply through the multiplicity of a hundred different tempi. The use of ostinato, for example, can be found in Étude 4, and other piano works as well. For Ligeti, the significance and the relevance of periodicity do not rely on its repetitive nature, but Ligeti focused on how these similarities and recurrences are transformed through time in both the micro- and the macro-level, and how they interact with other parameters in the music. For Ligeti, periodicity does not control the overall formal structure of the music, and hardly a single periodicity is carried through without any alteration.

Elliott Carter Carter’s interest in the temporal dimension of music is well known, as it is evident in his own writing: “Any technical or esthetic consideration of music really must start with the matter of time. …All the materials of music have to be considered in relation to their projection in time, and by time, of course, I mean not visually measured ‘clock-time,’ but the medium through which (or way in which) we perceive, understand, and experience events.”75 Among the many temporal innovations Carter has developed over the years, periodicity remained an important temporal device throughout much of his latter periods. Though in comparison to Ligeti’s method of composing with periodicity, Carter’s approach is less obvious and audible. Majority of Carter’s works from Night Fantasies (1979-80) to 90+ (1994) are constructed based on large-scale long-range polyrhythms.76 In these works, the “rhythmic layering is associated with the use of simultaneous speeds generated by the subdivision of the temporal grid into two or more pulse streams: series of equally spaced time-points also known as pulses or pulsations.”77 These time-points that are derived from the long-range polyrhythms will determine both the local rhythmic constructions and configurations, and the structural temporal

Allen Edwards, Flawed Words and Stubborn Sounds: A Conversation with Elliott Carter (New York: Norton, 1971): 90. 76 See John Link’s analysis of the long-range polyrhythms and ratios in Elliott Carter’s works. John Link, “Long-Range Polyrhythms in Elliott Carter’s Recent Music.” See also Guy Capuzzo, “Registeral Constrains on All-Interval Rows in Elliott Carter’s Changes,” Integral 21: 79-108; David Schiff, The Music of Elliott Carter, 2nd ed. (Ithaca: Cornell University Press, 1998); and Jonathan Bernard, “The String Quartets of Elliott Carter,” in Intimate Voices: The Twentieth-Century String Quartet, edited by Evan Jones (Rochester, New York: Rochester University Press, 2009): 238-275. The Fifth String Quartet (1995) only uses longrange polyrhythms in some portions of work. See John Aylward, “Metric Synchronization and Long-Range Polyrhythms in Elliott Carter’s Fifth String Quartet,” Perspectives of New Music 47, no. 2(Summer 2009): 88-99. The following polyrhythm ratios are all taken from Link’s dissertation. 77 Ève Poudrier, “Local Polymetric Structure in Elliott Carter’s 90+ for Piano (1994),” in The Modernist Legacy: Essays on New Music, ed. Björn Heile (Burlington, VT: Ashgate Publishing Company, 2009), 205.

hierarchies in these works. Carter himself has discusses the use of large-scale polyrhythms in his own works: …I was aware that one of the big problems of contemporary music was that irregular and other kinds of rhythmic devices used in it tended to have a very small-scale cyclical organization—you heard patterns happening over one or two measures and no more. For this reason, one of the things I became interested in over the last ten years was an attempt to give the feeling of both smaller and larger-scale rhythmic periods. One way was to set out large-scale rhythmic patterns before writing the music which would then become the important stress points of the piece, or section of a piece. These patterns or cycles were then subdivided in several degrees down to the smallest level of the rhythmic structure relating the detail to the whole.78 Carter’s practice of long-range polyrhythm can already be traced as early as in the Double Concerto for Harpsichord and Piano with Two Chamber Orchestras (1956-61), where Carter employed multiple tempi simultaneously in the form of polyrhythm, with each tempo assigned to a particular interval class in the 12-note chord. Similar process can also be found in the Piano Concerto (1965), Concerto for Orchestra (1969), and A Mirror on Which to Dwell (1975).79 However, it is not until Night Fantasies and after that the long-range polyrhythm governs the whole work systematically. It should not come as a surprise for Carter to employ long-range polyrhythms in a methodical and structural way, since as early as the Cello Sonata, Carter was already interested in exploring local level polyrhythms, multiple streams of tempi unfolding simultaneously, temporal modulation, and varying subdivisions within a steady pulse – all of which are directly related to the concept of long-range polyrhythm.

Allen Edwards, Flawed Words and Stubborn Sounds: A Conversation with Elliott Carter, 111. See Link, “Long-Range Polyrhythms,” 3-4; Jonathan Bernard, “The Evolution of Elliott Carter’s Rhythmic Practice,” Perspectives of New Music 26, no. 2 (1998): 164-203; and Brenda Ravenscroft, “Setting the Pace: The Role of Speeds in Elliott Carter’s A Mirror on Which to Dwell,” Music Analysis 22, no. 3 (2003): 253282. 79

In these works between 1980 to 1994,80 Carter typically employs at least one long-range polyrhythm in each work, and the entire temporal conception and the total duration of the work are determined by the completion of this polyrhythm cycle. Generally the first and the last structural note (or notes that coincide) are typically the starting and the ending convergent points for the entire cycle of the polyrhythm: “When the streams of pulsations are combined they form a polyrhythm. In most cases the streams all coincide once near the beginning of a piece and a second time near the end, so that the overall proportions are determined by the polyrhythm’s cyclic pattern.”81 For example, in Night Fantasies, the ratio of polyrhythm is 216 : 175; in Esprit Rude/Esprit Doux, the ratio of polyrhythm is 21 : 25; and in String Quartet No. 4, the ratio of polyrhythm is 120 : 126 : 175 : 98. The only exception is the polyrhythm ratio in Penthode, which is based on the highly complex ratio of 2,079,168 : 1,786,785 : 1,944,800 : 1,832,600 : 1,793,792 - a cycle of which would take 315 days to complete! In these works, all the pulsation points are not always sounded in the music. Furthermore, the realization of these periodicities varies from work to work, and the approach even changes within a single work. Despite the tempo modulations (a device which Carter has utilized since the Sonata for Cello and Piano in 1948) which alters the notated rhythmic values, the sounding pulsations for these long-range polyrhythms remain consistent throughout. In the rare case of 90+, only one stream of periodicity at MM = 18 is constant 80

Works during this time span includes Night Fantasies (1980), Triple Duo (1983), Changes (1983), Esprit Rude/Esprit Doux (1984), Penthode (1985), String Quartet No. 4 (1986), A Celebration of Some 100x150 Notes (1986), Oboe Concerto (1987), Enchanted Preludes (1988), Remembrance (1988), Anniversary (1989), Violin Concerto (1989), Con leggerezza pensosa (1990), Quintet for Piano and Winds (1991), Scrivo in Vento (1991), Trilogy (1992), Partita (1993), Adagio Tenebroso (1994), Esprit Rude/Esprit Doux II (1994), 90+ (1994), Of Challenge and of Love (1994), and String Quartet No. 5 (1995). Not all the works here are structured on large-scale polyrhythms. 81 Link, “Long-Range Polyrhythms,” 5.

throughout the whole piece, while various surface rhythms, pulses, subdivisions within these pulses, and tempi are superimposed on top of this constant pulsation.82 Similar to those examples found in the Ligeti, the duration of the whole work and the formal structure rely profoundly on the whole cycle of this long-range polyrhythm of multiple periodicities. The long-range polyrhythms and periodicities that Carter chose in these works tend to be complex ratios, and they are usually calculated in specific ways on how the ratios are realized, and the choice of tempo and the various tempo modulations are subordinate to these relations for the ease of the performer. Moreover, Carter typically varies the subdivisions between pulses within a single work, and in order to accommodate for the ease of notation, tempo modulations are employed, but the underlying pulsations remain constant throughout. Similar to Ligeti, the surface complexity and the unequal rhythmic subdivisions within these periodicities obscure the principle periodic nature of periodicity, and they often divide the periodicities into layers of hierarchies; but the temporal conception in large relies solely on these periodicities to structure patterns and reference points in the music. However, it is from these polyrhythm of periodicities that structural rhythmic pulses and tempi are derived. For example, the large-scale polyrhythm of in A Celebration of Some 100x150 Notes is 56:45. The whole work, with the metronome marking of quarter-note equals 150, lasts exactly 2 minutes and 48 seconds. The entire work has 450 quarter notes, while 30 quarter notes do not belong to the large-scale polyrhythm, therefore the whole polyrhythmic cycle consists of 420

The constant 90+ pulsations are numbered by Carter in the score, and they are all notated with a tenuto accent. In the opening, this pulsation occurs every 16 triplet 8th-note, and if the notated tempo is quarternote equals 96, the speed for this pulsation would be MM 96 times 3 divided by 16, which comes to MM = 18.

quarter notes. Therefore, the "56" pulse at a quarter-note equals 150 comes to a pulse of every fifteen 8th-note, while the "45" pulse at a quarter-note equals 150 comes to a pulse of twenty-eight triplet 8th-note (ex. 1.4.18). Ex. 1.4.18. 56:45 polyrhythm in A Celebration of Some 100x150 Notes

Pierre Boulez Besides Dérive 2, Boulez has also used periodicity in his other music. Rituel is an interesting case since the periodicities arise through indeterminacy, as each individual group has its own pulse, while the simultaneities of these groups are directed by the conductor independently.83 In the opening measures of sur Incises, for example, periodicities arise through different combinations of rhythmic patterns and modifications. The opening six measures are notated in 6/4, and taking these six quarternotes as a cycle, harp I plays progressively later in the measure by a quarter-note (in measure 1 harp I plays on beat 2, in measure 2 harp I plays on beat 3, and so on). Similarly, harp II plays progressively later in the measure by an 8th-note (in measure 1 harp II plays on beat 4.5, in measure 2 harp II plays on beat 5, and so on). In these same measures, harp III plays progressively earlier in the measure by an 8th-note (in measure 1 harp III is omitted, in measure 2 harp III plays on beat 3.5, in measure 3 harp III plays on beat 3, and so on). The three pianos also operate on similar patterns. In these same measures, Piano II plays progressively later in the measure by a quarter-note (in measure 1 piano II plays on beat 3, in measure 2 piano II plays on beat 4, and so on). Piano III is based on a fixed incremental expansion of duration, where each chord progressively gets 83

“This ceremony was to have each group congregate more and more and become diverted and diffuse until practically there is a kind of confusion. Yet in Rituel you have always the pulse which is more and more imposing as you go on, where everyone is reassembled after straying away. Also, you have a very strange change of perception. At the beginning of the work you don’t hear very much percussion. You hear very clearly the oboe and two clarinets, but the more the groups come in, the less you pay attention to the pitch and the more you pay attention to the rhythm. So there is a kind of reverse in the ceremony when the big chords like clouds happen; you don’t listen anymore to sentences, but you hear the rumor of the clouds - and that I want to reproduce, a kind of general feeling where the individual is no more important, a measurement in time, which is different in each group and becomes continuous. After all, not a single percussionist can follow the other percussionists. So you have to pay attention to this; it is so irregular, and yet the beat is very even, but even in different ways. So you have the general impression of multiple dimensions of time.” Di Pietro, 51.

longer by an 8th-note, as the first chord lasts 4 quarter-notes, the second chord lasts 4.5 quarter-notes, the third chord lasts 5 quarter-notes, and so on. Piano I has two separate voices, the top voice always plays on beat 1 in every single measure, while the bottom voice plays progressively earlier in the measure by a quarter-note (in measure 1 piano I plays on beat 6, in measure 2 piano I plays on beat 5, and so on). These periodicities produce varying entrances of patterns for these instruments, and through careful calculation, these instruments do not align in measure 1, and the only alignment of the three pianos takes place in measure 6 where all the periodicities conclude the opening section. Example 1.4.19 demonstrates the various stratification of periodicities in the opening measures of sur Incises.

Ex. 1.4.19. A reduction of the opening measures of sur Incises 1

? Piano 1

Piano 2

Piano 3

? Ó

? 1

Harpe 1

Harpe 2

Harpe 3

? Œ ? Ó

Œ œ

Ó™

Ó Œ

∑

Ó Ó

Œ ‰ ¸Ó œ

Ó Ó

? Ó

œ œ

Ó Ó

Ó™

˙™ Œ Ó Œ

˙ Œ Œ Ó

Œ Œ Ó Ó Œ

Ó™

œ ˙™

Œ Ó Ó Ó

Œ Ó Œ

Ó™

Œ Œ Ó

‰ ¸Œ Ó œ

Ó™

Œ Ó

Œ Ó™

Œ Œ

Œ Ó™

∑

¸ œ œ™ w

œ Œ

Œ Ó

Ó Œ

Ó™

Œ Œ Ó™

w™ ∑

Ó Œ ‰ ¸ œ

Ó Œ

w™

∑

Œ ‰ ¸Œ Ó™ œ Œ Œ Ó

Œ Œ Ó™

¸ œ™ œ w

˙™

Ó Œ

Ó ‰ ¸Ó™ œ

∑ 1

Marimba

Ó Ó

Œ Ó™

Œ Ó™ Œ Ó™

∑ ‰ ¸Œ Œ Ó™ œ

Œ Œ Ó™

Similar process of periodicities can also be found in Boulez’s orchestration of Ravel’s Frontispice, where multiple cycles of periodicities of unequal sizes are accentuated by instrumental doublings, dynamic accents, and various sorts of orchestrations. In Dérive 2, Boulez's use of periodicities, similar to those works by Carter and Ligeti, also determine the formal shape of the work. The surface rhythmic and metric formations are very much derived from and structured based on these periodicities. All these three composers are interested in multi-layered periodicities of fixed rhythmic

durations and patterns, and they use different methods to employ periodicity in systematical ways, or systematically vary the rigidity of periodicity, or to intensify such. Periodicity for these composers is no longer comprised of infinite time-points with no apparent directions or structures, rhythmic cycles and patterns become goal-orientated, and propose an arch-like form for the large-scale formal organization. Periodicity in DĂŠrive 2 is functioning in the foreground with the immediate rhythmic patterns, in the middle-ground in determining rhythmic cycles, periods, and phases, and in the background in determining structural and formal organizations â&#x20AC;&#x201C; all of which will be discussed in chapter 3.

CHAPTER 2. FORMAL STRUCTURE Time, like pitch, has three dimensions: horizontal, vertical and diagonal; distribution proceeds similarly by points, groups and groups of groups; these organizations do not necessarily run parallel with those of pitch; finally, time acts as a link between the different dimensions relative to pitch, since the vertical is only the zero time of the horizontal, a progression is from the successive to the simultaneous. Because of this morphology, local and global structures—responsible for the form—no longer obey permanent laws. There is also an absolutely new way of conceiving large forms: homogeneity or otherwise of their different components, causality or isolation of their various events, fixity or relativity in the order of succession and in the hierarchy of classification, potentiality or actuality of the formal relationships. … This almost unexplored aspect of form will be examined in detail when we come to deal with this subject [of form].1 Chapter 2.1. Formal Organization in Boulez’s Early Works Boulez’s conception of form has changed drastically throughout the years, from the earlier works which are still visibly divided into contrasting movements following the classical norm (Sonatine for Flute and Piano2, First Piano Sonata (1946), Second Piano Sonata (1946/48)3, and Structures I), to works where the poem dictates the formal outline (the two cantatas - Le Visage Nuptial (1947/52/89) and Le Soleil des Eaux (1948/65), and much later, in Marteau and Pli selon pli), to “open” or “mobile” forms (Third Piano Sonata, Domaines, …explosante-fixe…, and Dialogues l’ombre double); each phase of his output represents Boulez’s attempt to solve the dilemma of formal structure and large-scale temporal organization.

Boulez, Penser la musique aujourd’hui, 28. Although in the Sonatine the series is treated thematically (not in the fashion of Schoenberg or Berg, but as a virtual theme in which the theme is never stated the same twice), its function in the formal structure in relation to the tonal tradition is absent, if not avoided completely. 3 Though Boulez claims he symbolically destroyed the Sonata form (and traditional formal structures) in the Second Piano Sonata. See Boulez, “Towards the dissolution of classical forms,” in Par volonté et par hasard: entretiens avc Célestin Deliége, 40-42. 2

Boulez’s first remarkable conception of musical form appears in Le Marteau sans maître (1952-55).4 Marteau, a song cycle based on poems by René Char (whose poetry Boulez had set in two earlier cantatas, Le Soleil des Eaux and Le Visage Nuptial), is a work of 9 movements interwoven to form 3 cycles of songs, with each cycle based on a different poem.5 Movements 1, 3, and 7 form the first cycle, l’Artisanat furieux; movements 2, 4, 6, and 8 form the second cycle, Bourreaux de solitude; while movements 5 and 9 form the third cycle, Bel édifice et les présentiments.6 Marteau is Boulez’s first attempt to present a non-linear formal trajectory where the adjacent movements are not directly linked, and it is not until the final song that the whole shape forms a coherent totality (see figure 2.1-1). What is important to realize in Marteau (and directly related to Boulez’s later formal concepts) is that the formal structure manifests itself on multiple levels. Each movement of Marteau has its own internal structure and coherence, yet the overall realization is not foreseeable and is incomprehensible until the arrival of the last movement, which then completes the journey. The macrostructure and the large-scale temporal organization in Marteau are very much bound to a particular ordering in time, and such ordering is 4

The formal structure of Notations (1945) for solo piano is an interesting exception among the early works. Each movement uses the same 12-tone row. The opening pitch material of each movement is determined by the series itself, similarly to the first movement of Schoenberg’s Piano Concerto. See O’Hagen, “Pierre Boulez and the Foundation of IRCAM,” 303-330. 5 Curiously, Schoenberg’s Pierrot Lunaire is also divided into 3 cycles, with 7 songs in each cycle, but the three 3 cycles are organized consecutively. There are numerous resemblances between Schoenberg’s Pierrot Lunaire and Boulez’s Le Marteau sans Maître in terms of text setting, instrumentation (both works employ a different combination of instruments in each movement), and especially the flute and voice duet, which is a direct homage to Schoenberg from Boulez. However, Pierrot lacks an overall formal and pitch coherence, while Marteau is generated from a single 12-tone series. The use of the voice is also significantly different. In Pierrot, the voice is present in all the movements, while in Marteau the voice is only present in some movements, and at the end of the last movement the voice integrates itself into the ensemble as another instrument. 6 See Lev Koblyakov, “Pierre Boulez: A World of Harmony (New York: Harwood Academic Publishers, 1990), and Robert Piencikowski, “René Char et Pierre Boulez. Esquisse analytique du Marteau san maître,” in Schweizer Beiträge zur Musikwissenschaft 4, ed. Jürg Stenzl (Bern: Berlag Paul Haupt Bern und Stuttgart, 1980), 193-264, for detailed analysis of the formal structure of Marteau.

essential to our appreciation and understanding of form and its internal unfolding of events. In the Marteau preface, Boulez writes that: In choosing this order I have tried to interlock the three cycles in such a way that the passage through the work becomes increasingly complex making use of memory and virtual relationships; it’s only the last piece that to some extent, offers the solution, the key to this labyrinth. This concept of the form actually led me much further and completely freed the form from all predetermination; here the first steps were effected by breaking away from ‘one-way’ form.7

1 2 3 4 5 6 7 8 9

avant l’Artisanat furieux Commentaire I de Bourreaux de Solitude l’Artisanat furieux Commentaire II de Bourreaux de Solitude Bel édifice et les pressentiments version première Bourreaux de solitude après l’Artisanat furieux Commentaire III de Bourreaux de Solitude Bel édifice et les pressentiments double

Flute, Vib., Guitar, Viola Flute, Xylo., Perc., Viola Voice, Flute Xylo., Vib., Perc., Guitar, Viola Voice, Flute, Guitar, Viola Voice, Flute, Perc., Guitar, Viola Flute, Vib., Guitar Flute, Xylo., Vib., Perc. Voice, Flute, Xylo., Vib., Perc., Guitar, Viola

Fig. 2.1.1. Formal and instrumental outline of Le Marteau san maître

On the other hand, Boulez’s “open” or “mobile” works are the opposite in their formal conception and in terms of their temporal organization.8 In the Third Piano Sonata (and subsequently in Structures II, Domaines, …explosante-fixe…, and Dialogues l’ombre double) the performer is free to navigate (with certain restrictions) through the many different musical segments, and in the end the formal structure offers several

Pierre Boulez, preface to Le Marteau sans maître (London: Philharmonia Partituren in der Universal Edition (UE 12450): 1957). 8 See Boulez’s own definition and the concept of “open” or “mobile” forms in Pierre Boulez, “Sonate, Que me Veux-tu?” trans. David Noakes and Paul Jacobs, Perspectives of New Music 1, no. 2 (Spring 1963): 32-44; and Pierre Boulez, “Alea,” Perspectives of New Music 3, no. 1 (Autumn-Winter 1964): 42-53.

possible orderings.9 The idea of “open” form comes indirectly from Mallarmé, as well as Cage and Joyce.10 What interested Boulez was to follow “the examples of Joyce and Mallarmé” and “to stop regarding the work as a simple trajectory, traced between point of departure and point of arrival.”11 The form is a “street-map of a town: you don’t change the map, you perceive the town as it is, but there are different ways of going through it, different ways of visiting it. …[The] idea is not to change the work at every turn nor to make it look like a complete novelty, but rather to change the viewpoints and perspectives from which it is seen while leaving its basic meaning unaltered.”12 Yet such aleatoric formal structure is only a conceptual one, since in a single performance it is not possible to experience all the different orderings, nor does the listener perceive an “out-of-order” narration due to the change of the sequence of events on a first hearing. Each piece can only offer one ordering at a time, and therefore it does not make the piece open or mobile on a single presentation; it is only possible to perceive an “open” form if the listener hears more than one ordering of the work, and that the collective experience (or the possibility 9

“A composition is no longer a consciously directed construction moving from a “beginning” to an “end” and passing from one to another. Frontiers have been deliberately “anesthetized”, listening time is no longer directional but time-bubbles, as it were.” Boulez, Orientation, 178. 10 “I read it afterwards and found that the way I had conceived the Third Sonata, without of course being identical, was very close to Mallarmé’s conception of the ‘open book’, and in particular his idea of the threedimensional book – that is, where the developments become more and more complex as one moves into further dimensions of the content. One of the formants of this sonata is based on this principle, … this is a conception that is very much part of my thinking. As you get deeper into a book there ought to be a more or less complex texture because you have gradually been accumulating knowledge; in other words, you do not read page 1 in the same way as page 30. Page 1 is simple, whereas page 30 is complex because it contains all the knowledge you have gained from pages 1 to 29. This is what I sometimes do in music; developments accumulate and become tropes grafted on to other tropes, which in turn are superimposed on yet other tropes so that one gets different accumulations of richness. For me this represents a very special procedure: this accumulation that springs from a very simple principle, to end in a chaotic situation because it is engendered by material that turns in on itself and becomes so complex that it loses its individual shape and becomes part of a vast chaos.” Boulez, Par volonté et par hasard: entretiens avc Célestin Deliége, 51. Mallarmé’s influence on Boulez extends beyond Third Piano Sonata, Boulez’s next large scale work Pli selon pli is set to Mallarmé’s poems. 11 Pierre Boulez, “Sonate, que me veuxtu?” 32. 12 Boulez, Par volonté et par hasard: entretiens avc Célestin Deliége, 82.

of having such experience) makes the piece open and mobile. But what exactly does this “openness” in form do to the music by simply allowing multiple events and durations to be ordered differently and freely within limits? Does the experience of such different ordering really alter our perception of the work, or even challenge our conception of the piece as a whole? The formal structure essentially lies on a higher level, and has yet to be determined, while each performance is merely a realization of such form from a particular perspective; it does not represent the totality or even the definition of the form, but simply a rendering of it.13 The formal structure does not unfold on a straight line, but rather, the multiplicity of these unfoldings defines the formal structure of the work.14 This aleatoric, open, or mobile form is, for the most part, ultimately a conceptual idea, if not an interpretive one; it is certainly not a perceptual one. The form of an open/mobile work as a whole is then analogues to a sculpture, where each angle can only offer one perspective but not its totality. The beauty of the work (and the overall shape) relies on this multiplicity of offering the manifold dimensions, and the differences that arise through such multiple interpretations. Yet one needs to absorb and comprehend all of the possibilities in order to understand and gain an appreciation of the totality of its form. Again, similar to Marteau, the formal structure of an open work is manifested on a higher level, a level of manifold possibility; but unlike Marteau, the formal experience or interpretation of a mobile form is not fixed within one particular direction or order, and therefore it is no longer bound by

“Firstly, I felt that the course of a work ought to be multiple rather than simple; secondly, I found that the typographical layout of music could be renewed by the introduction of parentheses, italics, and so on; and thirdly, I wanted the performer confronted by a work to be able to find himself in a completely fresh situation every time he tackled it. It was these three considerations that persuaded me to depart completely from traditional musical structure.” Ibid., 81. 14 See Pierre Boulez, Michel Fano, and Thomas Respensek, “A Conversation,” October 14 (Autumn 1980): 101-120.

the restriction of “clock time.” Both Marteau and the “open” works restrict the listening experience to discrete moments, where one’s experience is confined within these discontinuous compartments. However, what makes these temporal concepts different from Stockhausen’s “moment time”15 is the emphasis on the unity of the work and the underlying trajectory through this maze of moments. The works of Boulez’s next period concentrate solely on the idea of trajectory.

Karlheinz Stockhausen, “…How Time Passes…,” trans. Cornelius Cardew, Die Reihe 3 (1959): 10-40.

CHAPTER 3. DÉRIVE 2 Chapter 3.1. Pitch Construction in Dérive 2 The underlying pitch structure of Dérive 2 is laid out fully in the very opening measures, as exemplified in the opening viola line up to the first double barline (ex. 3.1.1). Here Boulez has planted the necessary seeds for him to proliferate throughout the entire piece.

44 Rapide (q = 152) nœ œ

Ex. 3.1.1. The opening viola line up to the first double barline Alto

poco

#œ

B nœfi¸

> nœ

B bœfi¸

poco

> fi¸n œ B b˝œ ‰ ‰ nœ ˝ nœ ™ dim.

œ B ˝ ‰ B Œ B

poco

nœ

<n>œ

¸ Ó nœ '

poco

¸ nœ >

œ ˝ 3

nœ ‰#œfi¸

‰ #œfi¸nœ

dim.

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> ‰nœfi¸n˝œ œ

nœfi¸n œ

‰ Œ

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œ >

Œ Œ

nœ

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nœ Œ

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nœ œ

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nœ ‰ ˝

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dim.

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‰

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This opening viola line can be partitioned into 24 discrete trichords (excluding the gracenotes), with the beginning of each trichord articulated and divided by a single grace-note (ex. 3.1.2.). Ex. 3.1.2. The opening viola line partitioned into 24 trichords, labeled with set-classes1

(026)

bœ (026)

& œ (015)

#œ œ

(014)

#œ

nœ

#œ

(014)

nœ

bœ

(016)

(026)

& œ (014)

#œ 17

& #œ (016)

#œ

nœ

bœ (016)

& #œ (012)

#œ

nœ

(015)

bœ

(025)

(013)

#œ

(015)

(012)

& #œ

#œ

(015)

nœ

nœ bœ

nœ 19

œ 23

nœ

(026)

#œ

nœ 8

#œ

nœ

(026)

nœ

(016)

#œ (013)

bœ (026)

#œ (015)

œ 20

#œ nœ

nœ

#œ

nœ

These twenty-four trichords do not form any overall organization, and they do not represent all the possible trichordal set-classes, as Boulez avoided all the diatonic sounding

This example is based on sketches found in the Pierre Boulez Collection, film 0590, p. 0965 (Dérive 2) in the Paul Sacher Foundation in Basel, Switzerland.

combinations â&#x20AC;&#x201C; (024), (027), (036), (037), and (048),2 while putting an emphasis on trichords that involve the tritone â&#x20AC;&#x201C; (026) and (016). One can posit a secondary organization that trichords 7+8 are a reordering of trichords 3+4; and trichords 23+24 are a reordering of trichords 11+12; while trichord 20 and 10, 8 and 4, as well as 21 and 6 are also re-orderings of one another. This chart (ex. 3.1.2) demonstrates that Boulez is not thinking in conventional 12-tone methods, since many adjacent trichords share at least one common tone, suggesting that the exhaustion of the total chromatic is irrelevant to Boulez. Also, there is no apparent implication or limitation of ordering of any kind within these trichords as they appeared in the music. These pitch-classes are not to be understood as a fixed-order series as in the conventional serial technique, but rather, a series of melodies that are built upon internal repetitions and redundancies to generate a greater possibility of relatedness, much similar to his approach in the Third Piano Sonata and Pli selon pli: The series is . . . the germ of a developing hierarchy . . . endowed with a greater or lesser selectivity, with a view to organizing a FINITE ensemble of creative possibilities . . ; this ensemble of possibilities is deduced from an initial series by a FUNCTIONAL generative process (not simply the consecutive exposition of a certain number of objects, permutated according to restrictive numerical data).3 Similar to the viola line, the three woodwinds (cor anglais, clarinet, and bassoon) in the opening measures also feature the same sequence of trichords, with the beginning of each trichord correspondingly articulated by a single grace-note (ex. 3.1.3). By comparing

Throughout this document, round brackets ( ) will denote set-classes, square brackets [ ] will denote sets in normal order, while curly brackets { } will denote ordered sets in their indicated order. 3 Boulez on Music Today, 35-36.

the viola trichords (at times doubled by the other two strings) with the woodwinds, one can observe two important characteristics. First, the harmonic rhythm (the rate of unfolding of these trichords) is different for the two groups, and in this sense they create an illusion of canonic imitation. Second, the duration for each trichord within each respective groups varies for each individual trichord, as some trichords are spread out over a longer duration than others (for example, in the viola part of ex. 3.1.3, the first trichord lasts five 8th-notes, the second trichord lasts eight 8th-notes, while the third trichord lasts five 8th-notes. In the woodwinds in the same example, the first trichord lasts ten 8th-notes, the second trichord lasts eight 8th-notes, while the third trichord lasts six 8th-notes). Yet despite the varying of durations, each grace-note distinctively signals the beginning of each trichord, so that the grouping of three is clearly unambiguous and audible - an essential feature that will continue for the rest of the piece.

Ex. 3.1.3. Trichords in the woodwinds and strings in the opening measures up to the first double barline (boxed numbers refer to trichords in ex. 3.1.2)4

Cor anglais

° &

Basson

¸ bœfi

nœ >

Clarinette en la

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¸ nœfi#œ

Œ ‰ nœ¸ Œ ‰nœfi¸bœ œ ˝ nœ > œ >

¸ #œ '

‰ Ó

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Œ ‰ ¸ ‰nœfi¸bœ œ Œ ˝ nœ #œ nœ > >

¸ #œ '

‰ Ó

¸ #œfi

#œÆ ˝

‰ Ó

> nœfi¸b œ œ Œ ‰ n˝œ nœ ‰ ˝ Œ #>œ

Rapide (q = 152)

Alto

°B ¢

° Cor anglais &

Clarinette en la

Basson

bœ œ ‰ ¸#œfi¸ œÆ ‰ nœ ˝ nœ > '

b œÆ ¸ Œ#œfi¸nœ nœ œ nœ ‰ Œnœfi¸ ˝ ‰ Œ

œ #œ ‰

nœ

œ b >œ œ

‰ ¸#œfi¸nœÆ ‰ œ ˝ '

Æ >œ œ nœÆ # œfi¸n˝œ n ‰ ‰ nœ ˝

#œfi¸

¸ nœfinœ¸

#œ nœ

#œfi¸n˝œ

‰

‰ ¸ ‰ nœ¸ Œ nœ œ Œ # œ n œfi¸ ' #œ '

‰ ¸ Ó nœ '

¸ ‰ ¸ ‰ nœ Œ nœ œ Œ # œ œfi¸ ' # œ '

‰ ¸ Ó nœ '

Ó 6

¸ # œfi

n œÆ ‰ #˝œÆ ‰nœfi¸˝ Œ #œ nœ œ Œ

‰ nœ¸ Ó '

nœfi¸

œ Œ >

nœ

> #œ

∑

nœfi¸#œ

n œfi¸# œ

#œ ‰ fi¸ ¸ œ nœ #œ >

Ó Œ nœfi¸#œ

nœ ‰ Œ ˝

Violon

° fi¸nœ œ > Alto B n>œ #œ ˝œ ‰#œ #œ ¢

∑

¸ nœfin œ¸

¸ nœfinœ¸

> nœ <n>œ nœ Œ

nœfi¸nœÆ

n œ <n> œ b>œ 7

nœ <n>œ

n>œ

' '

‰

˝ ‰

Throughout this document, all the transposing instruments will be notated in and referred to in concert pitch (in C).

Ex. 3.1.3. (cont’d) > ° Œ ‰ œœ Clarinette en la & <#>œ ˝ ¢ nœ ° #œ nœ¸‰ Œ Violon & Œ

Alto

B Œ

Violoncelle

B Œ ¢

= Cor anglais

° & Œ

# œ nœ ˝ ‰ Œ

? Œ

¸ fi¸ Æ œ ‰ ‰nœ #˝œ Œ '

¸ # œfi œ

#œ nœ Ó ¸ n œfi nœ #œ nœ nœ Œ

fi¸

# œfi¸n œ

¸ nœ Œ nœ œ > 13

¸ nœ ‰

#œ

° œ fi¸nœ Alto B ¢ ˝ ‰#œ #œ œ nœ Œ >

nœ bœ œ¸‰ nœ bœ œ ‰ ˝

‰ nœ nœ¸ Ó '

¸ ¸ bœfibœ œ

‰

nœb œ œ¸‰

‰nœfi ¸ Ó nœ '

¸ nœfi

nœ œ nœ > Œ

‰ ¸ nœ '

Basson

¸ #œ nœfin œ

#œ nœ ˝ ‰ Œ

. & ‰ œfi¸n˝œ Œ

Clarinette en la

Œ bœfi¸ œ

> ‰nœfi¸b˝œ œ Œ > fi¸b œ œ Œ ‰nœ ˝

Œ nœ ¸ ' b œfi nœ '

‰ Œ nœb œ œ¸

∑ >

nœfi¸n œ

¸> bœfi nœ

∑

#œ #œ

nœ œ Œ nœ œ

> nœ

¸ bœfi

nœ

¸ ¸ ‰ ‰nœfi¸n˝œ nœ ™ bœ nœ > >

œ Œ

‰ ¸ #œ œ

¸ n>œ ‰ Œ

‰ ¸bœ b>œ n>œ ‰ ˝ #œ

¸ nœfi œ

‰nœfi¸#œ¸ Œ '

‰ nœ¸ 14

¸ ‰ ¸ ‰nœfi¸b˝œÆ ‰ nœ #œ œ '

nœ

∑

nœ

3 ¸n œ > bœ ‰ ‰nœfi ˝ nœ ™ nœ¸ Œ ˝ >

fi¸#œÆ ‰nœ ˝ Œ

‰ n˝œ

Ex. 3.1.3. (cont’d) 17

Cor anglais

nœ ° & œ ˝ ‰#œfi¸#œ #œ nœ ˝œ ‰ & Œ ‰ ¸ Ó nœ '

Clarinette en la

Basson

Violon

Æ # œfi¸# œ #œn œ

B Ó

¸ Œ bœfi œ¸

Œ ‰

¸ ‰ ¸œfin˝œÆ ‰ Œ bœ '

œ ˝‰

nœn œ >

bœfi¸ Æ

∑

#œ ‰ ˝ nœ

¢&

¸ #œ ‰

° Violon & Œ ‰

∑ 17

¸ ¸ n œfin œ œ

B nœ n˝œ ‰ Œ >

¸ #œ ‰

> ‰ Œ Œ nœ bœ ‰ ‰ nœfi¸#œ #œ œ¸‰ ‰ nœ œ ˝ ˝ ˝ 20

∑

Œ bœfi¸bœ

bœ ¸‰ #œfi¸nœ¸‰ Œ ' nœ '

> ‰nœfi¸˝œÆ Œ

nœ

œ ˝ ‰ Œ

‰ ¸nœfi¸bœÆ ‰ œ ˝ '

¸ œ ‰ Œ

bœ

œ ˝ ‰ Œ

‰ ¸ n˝œ ‰ Œ nœ œ >

> w

¸# œ nœfi ˝ 20

Æ n œfi¸b œ ‰ nœÆ ˝ ‰ ˝

∑ ‰ œ¸˙™ >

∑

‰ ¸nœfi¸bœÆ ‰ nœ ˝ '

> fi¸n œ nœ œ ‰nœ ˝ nœ ˝ ‰ 19

‰ ¸ n˝œ ‰ Œ nœ œ >

œ ˝ ‰ Œ

bœ œ > ‰ ˝ nœ Œ

#œ nœ œ nœfi¸ #œ œ ‰ ˝ ˝‰‰ ˝ ˝ ‰

Œ 20

#œ # fi¸ Æ B Œ ‰nœfi¸nœ œ ˝ ‰ nœÆ ‰ œ n˝œ ‰ Œ ¢ ˝ ˝

nœ > bœ # œfi¸ ¸b œ œ > ‰nœfi¸n˝œ œ nœ Œ ‰nœfi ˝ ˝ ‰ Œ ‰ n˝œ ˝ ‰ ‰ #˝œ

#œ ‰ ˝ nœ

bœ ¸ ¸ ¸‰ #œfinœ ‰ Œ ' nœ '

Violoncelle

∑

‰ fi¸˝œ#œ¸ ‰ nœ

∑ 4

∑

¸ ¸ n œfin œ œ

Alto

Ó ¸ ¸ nœfi#œ

¸ ¸ Œ ‰bœfinœ Œ '

∑

° Cor anglais & Œ ‰

Clarinette en la

∑

nœ #œ nœ œ # fi¸ B œ ˝ ‰ œ #œ ˝ ‰

¸ ¸‰ Œ œfi¸#œ ™ ≈Œ <n> œ

Œ ‰ nœ Œ ˝

Violoncelle

> nœ ‰#œ¸ nœfi¸#œ¸ ‰ Œ Œ nœ ˝ ‰ ‰ nœfi¸#œÆ #œ œ¸‰ ‰ nœ œ ‰ ˝ ˝ ˝ nœ '

° &

Alto

Œ nœ >

‰ Œ ‰ ¸nœ #œ

‰ bœ¸ œ nnOœ ‰ Œ > ˝ 0

∑

The pitch content for the horn part in the opening measures is also directly related to this trichordal chart. Its pitches are simply the doublings of the other instruments, either taken from the strings or the woodwinds, depending on a fixed duration and where it aligns with the other instruments.5 Consequently, the cello pizzicatos from 5-R1 to R11, cor anglais notes from R1-3 to R2-2, cello pizzicatos again from R2-3 to R3-4, and bassoon notes from R3-5 to R4-3, are themselves the doublings of the horn part but by an delayed eighth-note, creating a canonic imitation of another kind (ex. 3.1.4). Ex. 3.1.4. Opening horn part, along with cello, cor anglais, and bassoon

Violoncelle

Cor en fa

B &

long

U ∑

bouché

U +

pizz. sec

‰ nœ Œ Ó ˝ + Œ Ó n>œ

Cor anglais

Cor en fa

Violoncelle

Cor en fa

Basson

Cor en fa

& & Ó

+ Œ ‰ ¸ œ

¸ #œfi

B Ó

» Œ ‰#œ¸

& Ó

B Ó & Ó

¸‰ Œ Ó œ. Œ Ó œ

pizz.

#œ . nœfi¸b œ Œ ˝ ‰ + ‰ bœ œ ˝

∑ ¸‰ Œ Ó œ

œ‰ Œ ˝

¸ ‰nœfiœ. ¸Œ

¸‰ Ó œ

pizz. ord. sec

‰ nœ Œ ˝ + Œ nœ Ó

Ó b œfi¸b œ.

Œ #˝œ ‰ Ó + ‰ ¸ ¸‰ Ó #œ œ

∑ ∑

Œ ¸#œ. ‰ œfi ˝ + ‰#˝œ ˝œ ‰

œ ˝ ‰Œ Ó

» Œ #˝œ ‰ Ó + ‰ ¸ #œ ˙ ™

∑ ˙

Œ ‰nœfi¸#œ ‰ Œ Ó + Œ #œ

Œ ‰ n˝œ + Œ nœ

+ Œ ‰ bœ œ ‰ Œ ˝˝

poco poco pp (à peine audible)

∑

bœ ˝ ‰Œ

∑

nœ ˝ ‰Œ Ó

∑

+ Œ ‰œ ˙ ˝

˝ ‰ Œ

+ Œ ‰ bœ œ ˝

‰#œfi¸œ. Œ Ó ˝ + ˙™ Œ

œ Œ ˝‰

. n œfi¸b œ

Œ ‰

+ Œ bœ

‰ nœ¸Œ

nœ

¸ ‰ nœfib œ¸

. + Œ ‰ ¸ ¸‰ Œ bœ œ Œ ‰ nœ Ó ˝ + Œ ¸‰ Œ nœ œ

∑ ˙

œ‰ Œ ˝

The harmonies that arise from the simultaneities and the grace-notes themselves are not derived from these trichords, but they belong to a different pitch structure that

The fixed duration of the horn part will be explained in the context of periodicity.

also includes the harmonies of the four resonant instruments (vibraphone, marimba, harp, and piano). These pitch constructions will be discussed shortly in the next section. The trichordal chart (ex. 3.1.2) that was derived from the viola line (ex. 3.1.1) will then be reordered and combined to form larger collections. Example 3.1.5 demonstrates the combination of two trichords to form 6-element collections (trichord 1+trichord 2, trichord 3+trichord 4, etc.); example 3.1.6 demonstrates the combination of three trichords to form 9-element collections (trichord 1 + 2 + 3, trichord 4+5+6, etc.); and example 3.1.7 demonstrates the combination of four trichords to form 12-element collections (trichord 1+2+3+4, trichord 5+6+7+8, etc.). Similarly to the trichords, these combined collections do not form any systematic set-classes, and there is no specific rule or process on how the trichords are combined. Because these combined collections inevitably contain some duplication of pitches, most of them are in fact weighted hexachords, septachords, and aggregates, respectively. Ex. 3.1.5. Trichords are combined to form 6-element collections6 1+2

bœ

n œ œ n œ #œ 9+10

3+4

#œ

nœ

#œ

11+12

bœ

nœ œ œ #œ #œ œ

17+18

œ œ #œ œ

#œ

nœ #œ #œ #œ œ œ

bœ bœ nœ nœ n œ

bœ nœ

bœ œ

nœ

#œ

7+8

nœ

13+14

19+20

5+6

bœ œ #œ 21+22

#œ œ œ œ

œ œ

#œ nœ œ

#œ

15+16

#œ œ nœ œ nœ 23+24

nœ

#œ nœ œ

#œ

The following three examples (ex. 3.1.5 to ex. 3.1.7) are based on sketches found in the Pierre Boulez Collection, film 0590, p. 0964 (Dérive 2) in the Paul Sacher Foundation in Basel, Switzerland. The ordering within each collection is determined by Boulez.

Ex. 3.1.6. Trichords are combined to form 9-element collections 4+5+6

1+2+3

#œ

bœ #œ nœ nœ #œ œ nœ œ nœ #œ nœ nœ œ #œ nœ nœ

œ bœ nœ nœ nœ œ bœ nœ # œ # œ œ b œ nœ n œ œ œ #œ œ

#œ nœ nœ nœ #œ œ œ œ

#œ nœ

nœ

#œ

nœ

#œ nœ

nœ

#œ

22+23+24

19+20+21

& nœ œ #œ

#œ

16+17+18

13+14+15

10+11+12

7+8+9

œ bœ bœ

nœ

œ nœ nœ

#œ œ nœ œ bœ

nœ #œ œ

Ex. 3.1.7. Trichords are combined to form 12-element collections 1+2+3+4

5+6+7+8

œ #œ bœ #œ n œ # œ nœ n œ n œ nœ nœ œ 9+10+11+12

#œ

#œ nœ nœ #œ œ nœ

#œ œ nœ

13+14+15+16

œ nœ nœ nœ nœ b œ & #œ nœ œ œ # œ n œ œ œ #œ œ #œ bœ œ bœ n œ œ œ #œ 17+18+19+20

& #œ

nœ

bœ

21+22+23+24

#œ œ bœ #œ n œ nœ œ œ

#œ #œ œ bœ

nœ nœ œ n œ nœ œ œ #œ

Notice in these examples Boulez keeps all the redundancies of pitches, and does not avoid or simplify pitch duplications—strongly suggesting that Boulez is conceptualizing and working with pitches instead of pitch-classes.7 These combined 6element, 9-element, and 12-element collections will serve as harmonic and melodic domains for which Boulez constructs his pitch materials, similar to his practice with the pitch-multiplication technique where the pitch domains can be used both melodically and harmonically. The ordering within each combined collection is less important to him (also similar to his earlier serial method where the ordering is less important), and he is not obligated to use these pitches in their fixed registers. Boulez writes: A pitch series can be imagined in a number of different ways. And it is crucial to recognize that it is not the succession of the elements it combines that constitutes the serial phenomenon. The series is not an order of events, but a hierarchy – which can be independent of that order. It is in this sense that harmonic regions – using the same interval relationships – can, for example, within a certain set of transpositions, group series into families. Equally it is in this sense that the horizontal and vertical dimensions become combined under a single principle of distribution. And again it is by virtue of this hierarchy that a note of a series can be taken as a starting-point, without diminishing its organizing power.8 Example 3.1.8 illustrates these combined collections in their ascending order, and all the duplications have been reduced. One notices immediately that not every domain (the combined collection) has the same number of pitch-classes; for instance, the 7+8 domain only has 5 pitch-classes. 7

The practice of permitting redundancies in a pitch collection is different from the pitch-multiplication technique Boulez often uses in the earlier works. In those multiplication charts, Boulez typically reduces the redundancies that arise through the multiplication to the same pitch-class, instead of having two pitches at different registers. 8 Boulez, Stocktakings, p. 149-150.

Ex. 3.1.8. Ascending order for the combined 6, 9, and 12-element collections &

1+2

3+4

œ #œ œ bœ nœ œ

5+6

œ œ #œ nœ #œ œ

œ #œ œ œ #œ

9+10

11+12

17+18

19+20

œ b œ n œ œ bœ nœ n

1+2+3

& nœ #œ œ #œ œ bœ nœ nœ #œ 10+11+12

13+14

15+16

21+22

23+24

œ bœ œ œ œ œ #œ œ nœ œ #œ œ #

4+5+6

7+8+9

13+14+15

16+17+18

#œ nœ #œ œ nœ #œ œ œ nœ #œ nœ #œ œ œ

& œ œ #œ nœ bœ nœ nœ #œ nœ œ bœ nœ nœ #œ œ bœ nœ 19+20+21

& œ bœ nœ nœ #œ nœ bœ nœ

œ nœ #œ œ œ #œ œ œ #œ

& nœ #œ œ #œ nœ #œ œ nœ #œ 9+10+11+12

nœ b œ n œ n œ # œ œ œ nœ #œ 17+18+19+20

œ nœ #œ œ #œ œ œ #œ

22+23+24

1+2+3+4

œ #œ œ œ #œ

bœ nœ #œ œ bœ n œ œ nœ nœ œ #œ

œ b œ n œ # œ œ œ n œ n œ # œ œ #œ

& #œ #œ œ œ œ #œ

7+8

œ #œ nœ nœ #œ & #œ nœ #œ œ œ #œ

5+6+7+8

œ nœ #œ œ nœ #œ nœ #œ nœ 13+14+15+16

nœ œ bœ nœ nœ #œ œ #œ œ #œ 21+22+23+24

n #œ nœ nœ #œ nœ nœ #œ œ œ

Example 3.1.9 demonstrates the horn part after the conclusion of the opening viola line (ex. 3.1.1), starting from R5 to R13, where the principle pitch materials are directly derived from the 6-element collections of example 3.1.3. In the horn part Boulez does not maintain the trichordal ordering that projected by the viola line, nor presents

initial registral constrains. Similar to the viola line, the horn part is also articulated by a grace-note every three notes, undoubtedly formulating a division of trichords. Due to the reordering of the original trichords first established by the viola in the opening measures, however, the trichords that are articulated by the grace-notes in the horn part will be different from those in the viola. What one perceives is a new set of trichords that more or less resembles the previous ones, but with a different ordering through internal shuffling. Despite these re-orderings, the initial trichords are still maintained within their hexachordal boundaries. The original trichords are now embedded intermittently within a new set of orderings and combinations, suggesting a new array of relations and properties. The effect of this multi-layered pitch construction consequently establishes and implies a polyphonic pitch structure.9 What is striking in the horn part is that the harmonic sequence of trichords remains in the same order as it appears in the viola part, i.e., trichord 3 and 4 are still preceded by trichord 1 and 2. This overall harmonic ordering will remain constant for the whole work, regardless of its internal reordering. In most cases, after trichord 24, Boulez typically returns to trichord 1 and starts the process again, though he might go through the list differently by choosing every other combination, or using various other methods.

Boulezâ&#x20AC;&#x2122;s method of trichordal combination in creating polyphonic structure resembles that of Milton Babbitt. However, Babbittâ&#x20AC;&#x2122;s trichords are combined under a specific method to create desired combinatorial hexachords, and specifically to exhaust the aggregate, while Boulez is less interested in the resultant collections, or the exhaustion of the aggregate.

Ex. 3.1.9. Horn part starting from rehearsal no. 5 to rehearsal no. 13 1+2

¸ ‰ #œfi¸nœ œ

Horn in F

& Œ

¸ œfi# œ

> nœ ˝

‰

¸ ‰ œ

‰ #œfi¸nœ¸ œ¸ ‰

¸ ‰ œ

œ bœ

3+4

¸ nœ ‰ >

nœ

#œ

<n>œ #œ

nœfi¸nœ

œ ˝ ˝

‰

¸ œ ‰

nœ >

¸ nœfi#œ

‰

#œ

‰

5+6

7+8

& œ

nœ

‰

¸> bœfinœ

‰

¸ Œ ‰

¸ nœ

‰

¸nœ n œfi >

9+10

¸ bœfin˝œ

11+12

nœ ˝ #œ

‰

¸ ¸ # œfi œ

¸ bœ n˝œ ‰

‰

nœ œ >

‰

bœ

¸ nœ ‰

nœ

¸ ¸ # œfi œ

¸ nœ ‰

nœ

‰

¸ ¸ nœfi# œ

13+14

& Œ

‰

¸ & ‰ nœfi¸nœ œ & Œ

¸ #œfi#œ

¸ ‰ œ

¸ œ ‰ ‰ ˝œ

& Œ

¸ nœfinœ bœ

¸ ‰

¸ n œfin œ

¸ ‰ <#> œ

¸ œ ‰

#œ

¸ ¸ n œfi œ

¸ ¸ nœfibœ

nœ

Ó™

¸ ‰ œ Ó 21+22

¸ ‰ nœ

‰

19+20

¸ bœ nœfin œ¸ >

Œ 12

¸ ¸ b œfin œ

‰

nœ

#œ Ó

‰

15+16

nœ >

#œ

‰

¸ #œ nœ œ >

‰

¸ bœ >

nœ ™ >

¸ #œ Œ

¸ nœfi#œ

‰

¸ # œfin œ

¸ ¸ n œfi# œ

¸ ‰ #œ

#œ

˙™

23+24

nœ

#œ >

9+10

¸ ¸ # œfinœ

‰

nœ

œ# œ œ nœ œ >

1+2

¸ #œ œ b>œ

¸ nœ¸ ‰ nœ >

‰

17+18

œ ‰

bœ

Example 3.1.10 demonstrates a realization of the 6-element collections using the full ensemble from R5 to R6-3. The harmonic domain presents two distinct strata, similar to the opening measures where harmonic domains also unfold independently through two opposing instrumental groups. This example has reordered groups the instruments in the

example to reflect these two strata. The horn, vibraphone, marimba, and harp represent one stratum of harmonic domain, while clarinet, violin, viola, cello, and piano represent the other harmonic dimension. The vibraphone and marimba parts double the horn, but rhythmically they are usually one 8th-note apart from the horn, which again, similarly to the opening measures, creates a canonic imitation. This mixture of instrumentation is especially effective texturally, since the horn plays longer values, while the other percussion instruments only punctuate these held values with the same notes. The cor anglais and bassoon parts create a completely different canon from the horn. The pitch content for these two instruments is not derived from the trichordal collections (ex. 3.1.2), but are transpositions of the horn part. As mentioned earlier, since the horn part itself is a reordering of the trichordal collections set forth by the viola in the opening, (ex. 3.1.2), the trichords that are articulated by the grace-notes in this horn part will also be different from those collections. For example, in example 3.3.10, the first trichord in the horn part is {A, B, Bb}. The first trichords in both cor anglais and bassoon, likewise articulated by the grace-notes, are {C, C#, D} and {B, C#, C}. These three trichords all belong to the same set-class of (012), and they all represent a different ordering of the (012) trichord. The next trichord in the horn part is {F, C, F#}, while the next trichords in cor anglais and bassoon are {E, Eb, A} and {G#, C#, G}. Again, all three trichords belong to the same setclass of (016), and they all represent a different ordering of the (016) trichord. This process will continue for the following trichords for these instruments (ex. 3.1.10).

Ex. 3.1.10. A realization of the 6-element collections from R5 to R6-3

Clarinette en la

° &

Violon

Alto

Violoncelle

1+2

¸ œfiœ

> b >œ ¸

œ ¸nnœ b œfi >'

Cor anglais

Basson

Cor en fa

‰

¸ #œfinœ

‰

¸ nœfin˝œ # œfi¸

nœ ˝

¸ ¸ #œfinœ

‰

¸ fi¸ ##œœ œ.

Marimba

¢&

‰

¸ fi¸ ##œœ œ.

‰

¸ nœfi ¸ # œ nœ.

¸ ‰ nœ¸ #œ¸ ‰ ‰ fi¸ nœ¸#nœœ # œ n œ >' n œ # œ' ' >'

#œ

‰ #>œÆ n# œœ

‰

¸ nœ ‰ Œ #œ '

¸ ‰ fi¸nnœœ #œ '

n>œ #>œ Œ

nœ ‰ #œfi¸nœ

b#œœ nœ >'

n nnœœœ Ó >' ‰ bœfi¸n˝œ ˝œ

‰ Œ

¸ ‰ nœ

fi¸# œ ‰ bœ ˝ #˝œ

‰ Œ

nœ #œ

>Æ n nœœ #œ &

∑

n#œœ Œ >' ¸ bœfinœ # œfi¸

œ bœ

¸ ‰ œ

¸‰ ¸ ‰ nœ bœ

#œ

∑ ¸ nœfi ¸ bœ œ nœ ‰ #œ œ bœ n œ¸ ‰ >

nœ

¸ ‰ bœfinœ¸nœ bœ œ¸ ‰ > 3+4

‰ #œfi¸nœ¸œ¸ ‰ Œ ‰ #œfi¸nœ¸Œ #œ .

‰ ¸nœ¸Œ n fi .

Mi§ Sib

¸nœnœ # œfi

1+2

‰ Ó nœ. ˝

nœ # œ ‰ Œ

> nnœœÆ

nœ œ ‰ ˝

¸ ‰ ‰ ¸Œ nœ bœ Œ

‰

n>œ #>œ

5+6

#>œ n>œ Œ

∑

Assez rapide, stable (q = 138)

‰

#œ nœ ‰ ‰ # œfi¸n œ œ# œ n>œ >

#œœœ ‰ Œ #œœ

Vibraphone

Harpe

nœn œ ‰ >>

bœ bbœœ >'

° &

pizz. nn œœ

‰

∑

nœfi¸ > & nnœœ nœÆ Piano

3+4

nœ #œ œ

‰

¸ ¸nœ # œfi œ

¸ nœ ‰

¸ ‰ n˝œ. ‰ #œ¸‰ nœfi#œ¸‰ n˝œ nœ¸ ‰ . n œ. >

nœ. ‰ ¸‰ ‰ nœ ¸ ¸ nœ Œ ˝ #œ b œfi nœ >

∑

¸ ‰ nnœœfi#œ¸œ > > bœfi¸n˝œ œ ‰n œ

¸ œ ‰

œ ˝

‰

Ex. 3.1.10. (cont’d) Clarinette en la

° #œ & #œnœ nœ Œ > > & ‰ n œ¸ nœ >' #œ B #œ ‰

Violon

Alto

Violoncelle

Piano

Cor anglais

‰ #œ¸ ¸ nœ¸ ‰ n œ n œfi # œ ' '

n œ #œnœ

∑

# œ n#œœ nœ Ó

>Æ bnœbœœ nœ

& ‰ n#n œœœ >' ° & Œ

B Basson ¢ Œ

n œfi¸nœ

¸ #œ ‰ nœ '

¸ bœfibœ

nœ

¸ nœ œ ‰ > b >œ ˝ ‰

œ#œ#œfi¸ #œ #œ ‰ #œnœ #œ Œ nœ nœnœ ‰ nœnœ

œ#œnœ Ó

‰ #œ ¸ nœ ‰ nœ n œfi # œ

9+10

7+8

nœ nœ '

nœ Ó

b œnœn œ # œ

Vibraphone

Marimba

° & Œ & Œ ¢&

& Ó Harpe

Sol#

? Ó

¸ œfi # œ

¸‰ œ

¸ nœ nœfi# œ >

fi¸ nnœœ

> ¸ ‰ nœ ‰ # œ. ˝ Œ

∑

‰ #œ¸

¸#œ #œfi nœ

n œ nœ >

> bœ

‰

¸#œ nœfi ˝

‰ fi¸#œ œ nœ ˝ nœ n œ >

> nœfi¸nœ

˝ Œ

‰

¸ œ

¸ bœ bœ >

5+6

> ‰ n˝œ <n>œ #œ Œ

¸ nœfi#œ

‰ n œ¸ #œ¸ ‰ Œ . # œ. >

¸ nœfi#œ nœ

nœ œ >

bœ Œ

. b œ. n œ #œ. > >.

¸ ‰ #nnœœ <n>nœœ œ> œ > œ ‰ #˝œ

‰

∑

¸nœ nœfi

‰ #œ¸ n >œ

∑

Cor en fa

∑

¸ #œ ‰ ¸ ‰ nœ nnœœ ' '

∑

fi ‰ bœ ¸ nœ nœ ¸

¸bœ nœfi nœ

‰

‰ nœ¸ œ

¸Œ

¸nœ n œfi >

‰ n fi¸ ¸ Ó n œœ n œ >.

¸ nœ >

‰ nœ. nnœœ >.

#œ ¸#œ n œfi . nn œœ

‰ #œfi ¸Ó nn>œœ

¸ nœ. ‰ nœ¸ nb œœ >

‰#nœœ¸ > b>œ ‰ n ˝œ

œœ

‰

œœ

Œ n nœœ¸ n >œ

‰

¸ #œ Ó n>œ

‰

¸‰

n œfi¸ nœ

Fa§

∑

The other harmonic strata in example 3.1.10 are realized by the clarinet, violin, viola, and cello. What is strikingly different from the opening measures is that the harmonic rhythm of this stratum (lead by the clarinet) has doubled in speed. Compared to the other stratum (horn, percussion, and harp) in the same example, it takes two trichords for the clarinet stratum to match the duration of just one trichord in the horn stratum. Consequently, because the harmonic rhythm is now twice as fast in the clarinet, it has rhythmically gone from 8th-notes to 16th-notes. This is a special moment in the piece, as it is the first time that 16th-notes are used in the piece (up to this point, all the rhythmic values have all been 8th-note and its multiples). One not only perceives a quickening of time rhythmically, harmonically the rhythm is also doubled (a 2:1 harmonic rhythm ratio between the two strata). Internally, the clarinet stratum also has its own canonic imitation. Comparing the clarinet and the bassoon lines, the bassoon notes are completely the same as those in the clarinet part, but they are re-ordered. Similarly the upper notes of the piano part are also reordered notes taken from the clarinet part. One not only perceives a canonic imitation among these instruments (not to mention that the violin and the cello also create another level of canonic imitation with the clarinet), but also an illusion of sustained sound, as if creating an artificial resonance for these notes, since they are re-orderings of each other. After three 6-element collections in the horn stratum, and conversely, six 6element collections in the clarinet stratum, Boulez then alters the instrumentation. In example 3.1.11, the horn stratum retains the same instrumentation, while the re-ordering of the horn part that was played by cor anglais and bassoon is now replaced with the violin

and cello. The clarinet stratum has also been switched to viola, cor anglais, and bassoon, while the clarinet plays trills with notes derived from the viola (ex. 3.1.11).

Ex. 3.1.11. A realization of the 6-element collections from R6-3 to R8

Alto

° B

Cor anglais

nœ ‰ ‰ Œ

‰

> nœ #œ nœ nœ fi¸ n>œ b œ #œ nœ bœ nœ nœ #œ #œ ‰#œ nœ ‰ Œ

¸ & fi¸nœ ‰ ‰ Œ

‰

¸ ¸ ‰ nœ œ

‰

#œ nœ ˝ ˝

¸ nœfiœ

Basson

Clarinette en la

11+12

nœfi¸œ

‰ ‰ Œ

Ÿ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ¸ ‰ ¸ ¢& œfin œ¸ œ œ œ œ ‰

Vibraphone

‰

¸ #œ ‰ #nœ. œ n œ. . >

Marimba

¢&

‰

¸ n>œ

‰

° &

¸ ‰ #œ n œ nœ n >œ

‰

nœ nœ œ #œ >

¸ œ

‰

nœ ˝

‰

Violon

Violoncelle

Harpe

bœ

arco

nœ

Fa#

∑

¸ ‰ bœ nœ. > &

nbœœ

& &

¸ œ

#œ nœ >

Piano

>> bœfi¸nœ n œ #œ

‰

‰ Œ

‰

‰ #nœœ

¸ #>œ

‰ Œ

b>œ ˝

‰ Œ

Ÿ~~~~~~~~~~~~~~~~~~~~~~~~~~ nœ ™

¸¸ bœfiœ #œ

9+10

‰

nœ ˝ #œ

‰

¸ bœfin˝œ

‰ #nœœ ¸ Œ n œ.

nœ.

‰

fi¸nœ ##œœ ˝

‰ nœfi¸ ¸ Œ nœ

‰

œ ˝ ‰

¸ #œ

‰

¸ # œfi # œ

‰

¸bœ nœ

nœ

‰

¸nœ #œfi ˝

nœ

‰

#œ ‰ #˝œ Œ

‰

#œfi¸#œ

nœ

‰

Fa§

‰

¸ ¸ # œfi œ

¸ Œ nœ

‰ ‰

¸ nœfi

bœ. ∑ nœ Ó

Œ > nœÆ bnœœ

# œ.

bœ ˝

nœ

∑

∑ ‰

nœfi¸ fi¸

##nœœœ nnn œœœ

fi¸

¸ #nœœ n œ. >

#>œ n>œ ‰ Œ

7+8

° &

Cor en fa

‰

¸ #œfinœ nœ bœ

13+14

¸ #œfi bœ

nœ¸ nœ #œ

Si§

>Æ œ œœ ‰ Œ

nbœœ bœœ #œ

¸‰ # œ.

# œ.

Œ nœ ‰ nœ >'

## œ œ # œœ' >

nœ

Ex. 3.1.11. (cont’d)

Alto

°B Œ

Cor en fa

Violon

Violoncelle

nœfi¸# œ

nœ ˝

‰

nœ ‰ ˝

B œ

bn œfiœ¸

Piano

‰nœnœnœ#œ#œ#œ

¸ ¸ #œ ‰ nœ

‰ ¸ #œ nœ #œ

nœ ˝

#œ #œ ‰ ˝ nœ

#œ œ ‰ ‰ ˝ ˝ ‰

‰

Ÿ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ¸‰ Œ œ œ œ

¸ ¸ n œfi œ bœ

11+12

œ ˝ ‰

¸ ‰ bœ n˝œ ‰

. ‰ nnœ˝œ

bœ. ‰ ˝

. ‰ n˝œ Œ

‰ #bœœfi¸nœ. ˝ n œ.

nœ.

‰nœfi¸nnœ˝œ

bœ ‰ ˝

¸ ‰ fi¸#nœœ ‰ . nn œœ

¸ ¸‰ Œ n œ nœ

‰

¸ bœfi ˝

∑

nœ. n˝œ

n œfi¸

& Ó nœ & Œ

& Harpe

nœfi¸nœ

nœ

#œ

¸ œ ‰ ‰

‰

° œ &

> n>œ n>œ #œ n>œ ‰ ‰ #œ nœ nœ #œ ‰

> ‰ bœfi¸nœ ˝ nœ

. & ‰ n˝œ Œ ¢&

17+18

> > #œfi¸nœ

nœ Œ

#œ

¸ #œfi#œ

‰ nœ¸

¢& œ ° &

#œfi¸#œ #> œ

‰

‰ nœnœ

nœ ˝

<Ÿ>~~~~~~~~~~~~

Vibraphone

Marimba

B Œ

Basson

> nœ n>œ #>œ nœn œ ‰

¸nœ nœfi

& Œ

Cor anglais

Clarinette en la

> n œfi¸n œ

15+16

‰ ##>œœÆ n#nœœœ œ >' n# œ' >

>Æ nnœœœ ‰

nœ

nœfi¸

n œfi¸ nœ nœ

nœ ‰ ˝

‰

∑

nœ ‰ ˝

Sol# Do§

¸ ‰ #nnœœ. œ ¸ #œfi bœ

bœœ ‰ ?

nnœœnœ 'nœ > >'

∑

> nn œbœœÆ

>Æ #bœœ ‰ Ó #nnœœœ nœ >' nœ' >

∑

nœ ‰bœfi¸nœ ˝ ‰ ˝

Œ Ó

nœ # œ nœ

¸ nœ ‰ Œ

‰

¸ nœfin˝œ

<n>œ

#œ

¸ ‰ #nœœ <n>œœ ¸ fi nœ > œ

nœ.

n œ. #nœœ

nnœœ #n œœ

nœ ‰ Œ ˝ ¸ œœ ‰ Œ œ

¸ nœ. ‰ Œ nœœ Œ

Sol§ Sib

Example 3.1.11 also retains the characteristics found in the previous example (ex. 3.1.10). The harmonic rhythm ratio between the two strata is also 2 to 1, while the

canonic imitation among the parts continues to articulate trichords in their various forms. Similarly, after roughly three 6-element collections, Boulez alters the instrumentation. In example 3.1.12, the instrumentation assignment goes back to the same instrumentation that began in example 3.1.10.

Ex. 3.1.12. A realization of the 6-element collections from R8 to R9-3

Clarinette en la

° &

Violon

Alto

Violoncelle

Basson

b>œ ‰ ∑

21+22

bœ bœ nœ nœ

‰ nœ œ

¸ nœfibœ

Æ bbœœ ˝

¸ nœ nœ '

¸ ¸ nœfibœ

‰

∑

‰

‰ nnœœ

bœfi¸

¸ nœ

¸nœ nœfi ˝

‰

>œ ˝

#œ

‰

nœ ˝

> nœ

‰ #œ¸ nœ¸ n œ nœ ' '

Œ ∑

‰ Œ

∑

n#œœ

∑

bœ

‰

¸ #œfi#œ¸

‰

¸nœ #œfi ˝

¸ nœ ‰

œ > nœ nœ ˝ ‰

œ >

‰

Vibraphone

‰

fi¸ ¸ nnœœ # œœ

¸ nœ >

‰

Marimba

¢&

‰

fi¸ bbœœ nœ¸

¸ n œ.

‰

∑

¸ nn>œœ

> ‰ bnœœ ˝ n œ. ˝ ‰

œ ¸ #œ. n>œ œœ nœ.

Fa#

#œ Œ

‰

‰ Œ

‰

#œ Œ

‰

# œ.

¸ bœ nœfin œ¸ >

¸ #œ œ ‰

¸ fi¸ ¸ bœ n œ. n#œœ nn œœ >

‰ #œ¸ ‰ .

b>œ #œ¸ ‰

¸ œœ ‰

fi¸bnœœ #n œœ

n>œ

#n>œœ n >œ nœ

‰

nœ b œ nœ

13+14

nœ

‰ #œ nœ nœ bœ nœ n œ

11+12

¸ ¸ nœfi# œ

∑

‰

Harpe

‰

œ bpizz. nœ

b nœœ ## œœ # œ

° Cor en fa &

nœ '

n fiœ¸

° &

œ ‰ Œ nœ nœ #œ nœ

∑

? #nœœ

‰

bbœœ & nnœœ >' ‰ '>

Piano

Cor anglais

¸> œfiœ

19+20

œ œ ˝ ‰

Ré#

nœ

Ex. 3.1.12. (cont’d)

Clarinette en la

23+24

° &

‰ œfi¸#œ œ œ ‰ Œ n œ n œ # œ n œ nœ ¸ ‰nœfi¸n#œœ Œ '

nœ

Alto

∑

nœ nœ

∑

Piano

Cor anglais

Basson

° & ¢

° Cor en fa & Vibraphone

Marimba

Harpe

¸ nœfi

¸ ‰ Œ bœ >

nœfi¸

¸ bœ ‰ Œ >

¸ ¸ nœfibœ

& nnœfiœ¸

‰ Œ

‰ bœ¸ œ ‰bœfi¸bœ¸

‰ Œ

‰

fi¸ nbœœ ‰

‰ Œ

Ó Œ

‰

bœ >

nœ ‰ bœ >

‰

nœ nnœœ n œ

‰ Ó

¸ bœ nœ

nœfi¸

bœ ˝

¸ ¸ nœ ‰ ¸ nnœœ n œ bbœœ > bn œfiœ >

n>œ bœ

‰ Lab

¸ œ nœfinbœ .

∑

n œfi¸

#œ œ ‰ ˝ ˝

>Æ ‰ #nœœ

¸ #œfinœ

¸ ¸ nœfinœ

‰ Œ

> ‰ ‰ b˝œ

¸ #œfinœ

‰ Œ

‰

¸ ¸ ¸‰ n œfi œ œ

‰

fi¸ ¸ ¸‰ nn œœ n œ# œ

nœ ˝

bœ

‰ ‰ #œ¸ >

œ n>œ œn œfi¸n œ n œ¸ n>œ

¸ #œfi#œ nœ

nb∑œœ >'

bb∑œœ

bœ

> fi¸ ‰ nœfi¸nœbœ nœ nœ ‰ ‰ #œ b>œnœ nœnœ#œnœ Œ

¸ bœfi

¸ œnœnœ # œfi n œ

∑

15+16

nœ #œ >

¸ ‰ œ

¸ ‰ nbœœfi¸ nœ¸ ¸ ‰ nœ Œ bœ. n œ n>œ. b œ. # œ b œ. # œ. nœ. . > >

#nnœœœ >

‰ ¸ nœ¸ ‰ ‰ nœfi¸nœ¸ nœ >

fi¸ ¸ nbœœ bœ

# >œ #œ ˝

‰

¢&

bn œœ

‰

Fa§

‰ nbœœfi¸

¸ ‰ Œ bb œ. œ >

œ ‰

¸ nœ nœ '

¸ ‰ Œ nnœœ '

Violon

Violoncelle

5+6

1+2

¸‰ nœ¸ ‰ œ # b >œ

¸ nn>œœ œœ ‰ > ‰#bœœ œœ

bœ. ˝

‰

. .

¸Œ

fi¸ #œ nn œœ # œ

#œ

nœ œ ¸ nœ. ‰

‰ ¸ ‰ nœ¸ #œ

∑

Likewise, after three 6-element collections, Boulez again alters the instrumentation. This time, the instrumentation is the same as it appears in example 3.1.11 (ex. 3.1.13).

Ex. 3.1.13. A realization of the 6-element collections from R9-3 to R9-4

Alto

° Œ B

n>œ > ‰ #œ

& Œ

Cor anglais

nœ ‰ #>œ > n>œ > #œ ‰

B Œ

Basson

Clarinette en la

5+6

Violoncelle

> nœÆ nœ > n œÆ

Cor en fa

Vibraphone

Marimba

Harpe

? Œ

> ‰ # œÆ > #œÆ nœ ‰

° & ¢

‰

¸ #œ >

¸ nœ

‰

¸ œfi#œ

‰

> nœ ˝

œ ˝

‰

¸ nœfi#œ

#œ ˝

‰

nœ nœnœfi¸nœ bœ Œ

‰

bœ

Œ Œ

> ‰ #nœœÆ

‰

n#nœœœ >'

bœ

nœfi¸

nœ ˝

bœ

nœ ˝

nœ

¸ ‰ œ

‰

b nœœ ‰ >'

‰

¸n œ n œfi n œ'

¸ #œfi

‰

¸ nœ ¸ nœ

‰ nœ œ >

#>œ

‰

>Æ ‰ #n#nœœœœ

¸ bœ nœ '

bœ nn œœ >'

‰

¸ nn œœ '>

‰

∑

‰ 17+18

‰

¸ & nœ. ‰ ‰

‰

>Æ nnœbœœ

‰

∑

° & œ

¢&

n>œ > ‰ nœ

‰

Ÿ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ #œ nœ #œ œ œ œ

& Œ

Violon

#>œ n>œ

¸ œfi

¢& Œ

Piano

9+10

‰

¸ œ ‰

¸ ¸ n œfi # œ

¸ #œfi#œ

fi¸ ¸ #nœœ ##œ œ

¸ #œ # œ¸ ‰

¸ nœfi #œ nœ

¸ ‰ #œ

#œ

fi¸ nbœœ

¸ #œ. # œ. # œ.

¸ nœfi ¸ n œ #n œ œ

‰

¸ œ ‰

‰

œ ˝

‰

nœ. ˝

‰

¸ n œ.

nœ

∑

‰

Example 3.1.14 demonstrates the pitch structure for the next section from 2-R11 to R13, labeled with an analysis of 6-element collections.

Ex. 3.1.14. A realization of the 6-element collections from 2-R11 to R13 ° Cor anglais &

13+14

¸ bœfi

¸ n œ.

‰

¸ bœfi

Clarinette en la

Basson

¸bœ. nœfi ˝

¸ bœfi f

Cor en fa

nœ

n œfi¸ bœ .

nœ ˝

‰

¸ nœfi œ œ n œ bn >

‰

n>œ

bœ ' f

bœ bœÆ ˝ f

‰

19+20

œ ' > n>œ f

¸ nœfinœ

bœ

¸ bœfi

nœ

nœ bœ #œ Œ

¸ nœfi n œ bn œ œ

‰

Æ nnœœ ˝

¸ nœ

‰

¸ bœfinœ¸

Piano

b #œ bœ Œ nœ œ Si§ nœ b œ

nœ. ˝

nœ fi¸n œ

‰

21+22

‰

¸ nœ

¸ ¸ b œfin œ

‰

15+16

> > > bœ bœ nœ b>œ nœ nœ ‰ °

nœ

‰

nœ # œ #œ n œ *°

° ‰

¸ nœ

#œ

21+22

¸ #œfi

17+18

¸nœ #œfi

Œ 17+18

¸ bœfi

17+18

n œ #œ #œ #œ nœ nœ

Œ &

bœ

nnœœ #œ

bnœœ nœ

nœ #œ ‰ ¸nœ #œfi

‰

19+20

#œfi¸

13+14

19+20

‰

¸ nœ '

#œÆ ˝

‰

nœ

21+22

bœ

#œfi¸# œÆ

b œ bœ n>œ > n>œ b>œ ‰ nœ

19+20

17+18

21+22

¸ #œfi f

¸ nœ '

nœ #œ # œ #œ nœ

15+16

17+18

¸œ #œfi

n>œ b>œ #>œ n>œ ‰

‰

bœ n>œ n œ b>œ #>œ ‰ > ff

13+14

fi¸bœ nnœœ #œ

‰

¸ œ nnœ '

13+14

¢&

nœ b œ #œ Œ

13+14

Harpe

‰

Œ 13+14

° Vibraphone &

Marimba

¸ n œ.

bœfi¸

Violoncelle

13+14

œ bœ #œ Œ

n>œ #œ ‰ n>œ b>œ > ff

19+20

° Violon &

Alto

nœ

bœÆ ˝

17+18

¸nœÆ #œfi ˝

21+22

nœ bœ œ nœ bœ nœ

‰ n œ nœ

#œ ‰ ¸ nœfi bœ nœ

#œ œ bœ ‰ Œ ?

Ex. 3.1.14. (cont’d) 17+18

° Ó Cor anglais &

#œ œ ‰ ' mf '

Clarinette en la

#œ mf

nœ

#œ nœ Œ

Violon

° & n œ bœ nœ b >œ > > b >œ b>œ n>œ #œœ'

Alto

21+22

17+18

#œ

B Ó

nœ

¸ œ n#œ '

‰

B n>œ #>œ n>œ #>œ n>œ n>œ nbœœÆ Violoncelle ¢ #œ ff

Cor en fa

21+22

& nœ ™ >

¸ #œ

#œ f

bn>œœ

bb >œœ

#>œ

#œ >

Œ ¸ nœfi#œ

bbnœœœ nn#œœœ > >

bnbœœœ >

bœœœ >

nœ

# œ bn œœ > n>œ

‰

nœ

Œ Œ Œ

¸#œ n œfi f

nœ nœ nœ

#œ nbœœ

#b#œœœ

&nœ

nœ

#œ

21+22

bœ œ

nœ

° *° Œ

23+24

nœ nœ nœ bœ Fa#

œ n#œ

‰

nœ n œ ‰

nœ

23+24

#>œ f nbn œœœ nœ > >

¸ #œfinœ

21+22

n œ ? b#>œœ

23+24

∑

nœ

Fa§ Sol§

23+24

∑

nœ

nœ ‰

° *

21+22

¢& bn œœ >

Harpe

nœ

17+18

Piano

> > > n œÆ #œ n œ #œ ##œœ n>œ #œ nœ #œ nœ

° Ó Vibraphone &

#œ

¸ nœfi#œ

Marimba

#œ

#œÆ

#œfi¸nœ b œ

23+24

nœ n œ nœ nœ

21+22

nœfi¸ #œÆ

nœ nœ

nœ

nœ #œ #>œ #>œ '

‰

#œfi¸n œÆ

23+24

¸ nœfi

21+22

Æ nœÆ #œ ‰

B Ó

¸ bœfinœ

‰

17+18

Basson

21+22

17+18

& Ó

¸ bœfinœ¸

nœ

nnœœ

100

Ex. 3.1.14. (cont’d) ° ¸ Cor anglais & #œ '

3+4

¸ #œfi

‰

¸ nœ '

& #œ bœ ‰

Clarinette en la

¸ nœ '

‰

fi¸ ‰ nœ œ nœ n œ #œ œ#œ œ # œ # œ nœ

#œ n œ ‰ ' ' f

#œ nœ nœ #œ

3+4

Basson

B Ó ¸ #œfi nœ

nœ ‰ n >œ >

B nœ nœ nœ nœ Œ 23+24

Violoncelle

Æ > n>œ #œÆ n œ

¸ nœfi#nœ œ

B ¢ Œ

3+4

23+24

° Violon & Œ

Alto

#œfi¸nœÆ

nœÆ

∑ 1+2

‰

3+4

Cor en fa

° Vibraphone &

Marimba

¢&

nœ

¸ n œfin œ

‰

> #œÆ nœ # œ n œ bnœœ #>œ n>œ n>œ > nœ b>œ Œ

‰

¸ #œ

#œ

fi¸

n œ# œ n œ

fi¸ œ bb œœnn œ f

nœ

#œ nœ n œ nœ

3+4

Œ nœ n œ nœ

3+4

1+2

‰

bœ Ó n>œ n>œ n œ #>œ > ff

nœ nœ #œfi¸nœ nœ #œ ‰ nœ

& Œ

¸ #œ '

bœ >

Œ 3+4

nœ # œ nœ ‰

*°

#œ n œ ‰

1+2

‰

¸ #œ mf

nœ n>œ

nb>œœ

nœ

bn>œœ

1+2

& Harpe

∑

‰

∑

b>œ #œ Ó > n>œ n œ >

Sib

1+2

fi¸ ###œœœ

& Œ Piano

? Œ

nnbœœœ ‰ >

#nnœœœ >

¸ #n>œœ n#œœ > f

‰

#œ nœnnbœœœ n œ > f

101

nn#œœœ

nœ bn>œœ #b >œœ

Ó Ó

nn>œœ

Ex. 3.1.14. (cont’d) ° Cor anglais &

∑

‰

œÆ ˝

7+8

¸ ¸ #œfinœ

‰

7+8

‰

& #œ nœ

Clarinette en la

¸ #œfinœ

B ¢ ‰

° Violon & ‰

¸ nœ '

‰

nœ '

‰

œ #œ Œ nn>œ > ff

‰

nœÆ ˝

‰

¸n œÆ #œfi ˝

Ó ¸ nœfinœ¸

‰ ¸nœ¸ b œfi '

& ‰

9+10

nœ ¸b œ nœ ‰ n fi

¸ # œfin œ

° Vibraphone &

nœ

¸ œ

#œ >

∑

‰

¸ #œfi mf

& Harpe

Piano

nœ

Mi§ Ré§

5+6

Œ 7+8

nœ #>œ f

##>œœ

9+10

‰ 9+10

fi¸ nœ Œ nœ #œ n œ # œ#œ nœ

¸ b œfinœ

*°

nœ

5+6

?nœfi¸

n œ b œ nœ 9+10

bœ

9+10

∑

& ‰ ?

¸ nœ

¸ ¸ # œfinœ

Marimba & ‰ ¢

‰

7+8

1+2 Cor en fa

9+10

n œ #œ n œfi¸ nœ #œ nœ nœ #œ nœ #œ nœ Œ

#œ nœ n>œ b>œ ‰ n#>œœ n>œ > >

7+8

nœÆ ˝

n œÆ ˝

7+8

5+6

¸ #œfinœÆ

B ‰

‰

> n>œ #>œ nœÆ nœÆ #œ nœ Œ nœÆ nœ #n>œœ nœ

Violoncelle

nœ nœ nœ nœ #œ nœ nœ nœ nœ ¸ #œ >'

#œfi¸

B ‰

Alto

5+6+7+8

5+6

¸ #œfi f

nœ #œ fi¸nœ nœ ‰ #œ # œ n œ nœ #œ

5+6

Basson

‰

∑

nœ n œ bn œœ

bœfi¸ nœ

n#œœ

9+10

¸ #œfi

Œ nœ nœ #œ nœ nœ n œ nœ n œ # œ #nœœ ff #œ #n>œœ > Œ

Ó Ó

Œ Œ

nœ n œ b œ # œ

bœfi¸ nœ

nœ #œ

102

Ex. 3.1.14. (cont’d) ° Cor anglais &

7+8

9+10

œ ¸ #>œ n>œ #>œ > n>œ #>œ # œ '

‰

¸ bœ

7+8

Clarinette en la

‰

Basson

‰

≈

n>œ #>œ nœÆ n>œ n>œ

≈ nœ¸™

° Violon &

n œ.

‰

nœ. ‰

¸ nœ 'mp

nœ nœ 11+12

™ nœ nœ

9+10

#œÆ n>œ n>œ ˝

‰

nœ

Violoncelle

œ 11+12

> n>œ n>œ nœ #>œ #œ nœ Œ

‰

9+10

7+8

Alto

#œ #œ œ

nœ p

#>œ #>œ n>œ #>œ n œ '

‰

9+10

nœ #œ nœ nœ nœ nœ #œ Œ #>œ > > > > #>œ n œ f

bœ. #œ. ‰

‰

bœÆ ˝

nœ

bœ

11+12

9+10

Cor en fa

nœ

7+8+9+10

° Vibraphone &

7+8

‰

nœ nœ ‰ #>œ #>œ n œ # œ n œ #œ ff

Marimba

nœ ? ˝

‰

° Œ

# œ. nœ. n œ. b œ. sans

nœ

9+10

Œ nœ. n œ . b œ. n œ. ‰™

#œ

bœ

9+10

Harpe

& n œ nœ ‰ bœ ?

#œ ‰

Œ Œ

Œ f

nœ

& Piano

‰ ff

‰

nœ ˝

11+12

b œ nœ

7+8

# œfi¸

n#œœ ˝

‰

bnœœ

nnœœ

9+10+11+12

‹#œœ #nœœ nœ ‰

b#œœ nœ #>œ bnœœ > ‰ n>œ >

bœÆ nœœ nœÆ #œœ nœ ' '

# œÆ nœ

#n œœ '

bnœœÆ

Æ Æ bœ nœ n œ n œ b# œœ b œ # œ' b œ n œ' ' f

In example 3.1.14, similarly to the previous examples, the harmonic strata are also realized through two separate instrumental groups. The instrumentation, however, is

103

different from the earlier sections. One group consists of the cor anglais, clarinet, bassoon, viola, vibraphone, and harp, while the other consists of the violin, cello, horn, marimba, and piano (ex. 3.1.14). Unlike the previous examples, instruments are not always fixed within their own instrumental groups. For example, the harp in R11-2 switches from one group to the other, while the bassoon in R12-2 plays in both groups. In this example, Boulez has maintained apparently the same instrumentation for the two strata throughout this section, different from the previous sections where the instrumentation changes every three collections. The most obvious difference between this example and the previous examples (ex .3.1.10 to ex. 3.1.13) is that the rhythmic values for both strata are all 16thnotes, whereas the previous section is divided into 8th-notes and 16th-notes. The harmonic rhythm for both strata is now the same, however, one group (cor anglais, clarinet, bassoon, viola, vibraphone and harp) goes through every other 6-element harmonic domain, while the other simply goes through the list consecutively. Example 3.1.15 contains a pitch analysis for the next section. Here one encounters the first instance of 9-element collections and their realizations. Rhythmically, it is also the first time that sextuplets are introduced in the music, also reflecting the change of harmonic scheme. It might be tempting to regard this section (ex. 3.1.15) as being in a faster speed than the previous one (ex. 3.1.14), since this is the first time that sextuplets are heard. However, the metronome marking for the previous section is quarter-note equals 138, which makes a single 16th-note the speed of mm = 552. The metronome marking in this section is quarter-note equals 90, which makes a single 16th-sextuplet at the speed of mm = 540. The fastest speed for both sections is effectively the same, what is

104

different is the larger pulsation and rhythmic values. Due to the change of the larger pulsation, harmonic rhythm nevertheless will be different. In this section, harmonic domains take longer to unfold, and there are also more notes per collection, 9-elements instead of 6-elements (ex. 3.1.15).

105

Ex. 3.1.15. A realization of the 9-element collections from R14 to R16 Cor anglais

° &

>. œ ≈ ‰ ˝

>. & #˝œ ≈ ‰

Clarinette en la

∑

‰™

Basson

>œ. ˝ ≈ ‰

¢&

¸ nœ

‰

¸ nnœœ

nnœœ

¸ nnœœ

nœ bœ

nœœ

‰

#œ.

n#œœ

nœ nœ f

‰

#nœœ

‰

4+5+6

n#œœ

B#œ

4+5+6

nœ

1+2+3

bœ.

n œ.

œ.

nn œÆ˝œ

‰

nœ

#œ f

1+2+3

>¸ #œœ œ˝

¸ œœ œ

œœ œ

¸ n œfin œ

#œ

≈ nœ bœ

‰

‰™

‰ n œ #œ nœ n œ n œ n œ

¸ œ

‰

œœ

œœ ˝

Do§

¸ nœfi

& nœ #œ nœ ff ‰

nœ nœ 3

nœ

œ nœ nœ

bœ nœ #œ¸ ≈ ‰ . n œ. .

nœ

bœ bœ

œ nœ nœ nœ nœ ‰

106

–— —

n#œœ n œ. n œ #œ. n>œ ff

1+2+3

¸ ≈ bœ ‰ n œ. >

Œ 3

#œ. nœ

bœ ≈ ≈ ¸ ‰ #œ bœ ¸ ≈ nœ n œ n œ #œ n œ n œ n œ #œ n œ n œ #œ nn œœ nœ #œ

‰

œ™

1+2+3

¸ #œ > fi¸> œ nnœœ œ ˝

#œ # œ ™ >

n œ.

1+2+3

Piano

Harpe

n>œ. œ. œ. œ. œ. œ. ≈ Ó

≈

Assez rapide (q = 92) Tempo variable en'sassordant á la dynamique. Les indications métronomiques sont à relier avec souplesse

Marimba

‰

1+2+3

° Vibraphone &

n#œœ

>. ˝œ ≈ ¸ nœ >

n œ.

1+2+3

#œ. #œ

4+5+6

nnœœ

nœ

pizz. n>œ. > B ˝ ≈ n#œœ ˝

pizz. n>œ. ˝ ≈ nœ¸ B Violoncelle n >œ ¢

Cor en fa

œ œ™ >

1+2+3

¸ #œ ™ n œ. .

¸ œfi mf

pizz. >. ° ¸ Violon & nœ ≈ #œ ˝ n >œ ff

Alto

1+2+3

Œ Réb

bbœœ œ. ≈ ‰ # œ œ. . ˝ n œ. >

b>œ nœ > n# œœ

n>œ. œ. œ. œ. œ. œ. ≈ Œ ≈ bnbœœœ œœœ œ œ œ ≈

– — – —

10+11+12 3

& ff

#œ

nœ nœ #œ n œ nœ

Ex. 3.1.15. (cont’d) Cor anglais

° &

4+5+6

‰™

& Ó

Clarinette en la

Ó™

∑

‰™

Ó™

¸ nœ nœ # œ ≈ #œ #œ

¸ ¸≈‰ n œ nœ mf

‰™

nœ. nœ ˝ . # œ. œ. n œ . mf

Basson

Violon

° & Ó B

Alto

Ó™

∑ ‰

≈ nœ #œ

#œ

nnœœ ˝

‰

¸ ‰ nnœœ #œ

nœ ¸ #œ ≈ nœ ‰ b œ 4+5+6

#œ nœ ˝

≈

4+5+6

nœ ≈ nœ

Violoncelle

‰

≈ #œ

nnœœ

nœ

nœ ≈ nœ¸ ‰ #nœœ

#œ nœ

bœ nœ. #œ Œ n œ. .

#œ Œ n œ. .

##nœœ. œœ. œ‰ ™ >

¢& Ó &

Harpe

7+8+9 3

& Piano

? bbnœœœ >

nœ ™

¸ œ ≈ ‰ bnbœœ

≈ ‰ nœ

¸ nœ.

bœ

– — —

≈ œ nœ # œ. #n œœ. ## œœ ‹# œœ # œ n#nœœ . . .

‰

#œ

nœ œ

#>œ

nœ.

4+5+6

≈ ≈n fi¸nœ œ #œ #œ 3

≈ fi¸nœ¸ #œ

∑

10+11+12

nœ #œ n œ #œ nnœœ bœ. nœ. n œ #œ ≈ ‰ nœ >

‰

∑

nœ #œ nœ .

4+5+6

Lab Si§

¸ nœ #œ

1+2+3 Marimba

≈ nœ.

° Vibraphone & #œ ‰ ‰ nn œœ œœ >. >.

4+5+6 Cor en fa

n˝œ #œ nœ nœ ≈

4+5+6

¸ bbœœ

‰™

nœ nœ b œ nœ ff

#œ ‰

– —

nn>œœ b>œ

#œ ≈nnœœ n˝œ ≈ ‰ 3 f

‰™

–

¸ Œ nnœœ. La§ f b œ. Sib n˝œ Œ

‰™

Fa§

4+5+6

≈ nœ#œ

#œ œ .

nœ. ‹n#œœœ >.

œœœ

nœ n# œœ >

nœ nœnœ

‰

nœ#œbœ nœ

œ nœ. ‰ 3

?≈

>. . nnœœ œœ ≈ f

≈n b œ¸‰ b œœ. >

107

Ex. 3.1.15. (cont’d) Cor anglais

° &

Basson

¸nœ n œfi

arco 3

#œ

Cor en fa

‰

∑

≈

1+2+3

œ nœ >

#œ.

. nœ. b œ

#œ.

nœ.

‰

œ n œ bœ

‰

≈

nœ nœ nœ ‰

nœ

nœ. nœ.

n œ b œ nœ nœ nœ nœ nœ bœ nœ

‰

¸ nœ nœ œ nbœœ nœœ œ. n œ. # œ. œ. . . ˝

1+2+3

arco

nœ bœ #œ ‰ ™

nœ #œ

¸ œ

nœ

. bœ

nœfi¸nœ

nœ

‰

Marimba

Harpe

Piano

¢&

# œ.

nœ.

¸ ## œœ

# œ. ≈

‰

n œ #œ.

nœ. n>œ œ

‰

#œ nœ

nœ.

‰™

¸ nœfinœ.

fi¸

#œ. bœ n œ

¸ ¸ n œfin œ

∑

108

≈

¸ ¸≈ ‰ n œ. #œ.

≈

# œ. œ nœ #œnœfi¸nœ. nœ.

‰™

‰

b œ #œ nœ #œ

nœ

15 (q = 84)

° Vibraphone &

≈

¸nœ. nœfi

‰

nœ.

œ.

nœ

bœ œ

n œ. #œ. nœ #œ nœ nœ

‰

≈

Violoncelle

‰

≈

nœ œ

#œ nœ

° Violon &

Alto

¸ œ

Clarinette en la

‰

œ nœ #œn œfi¸n œ.

#œ n œ nœ nœ b˝œ

≈

‰

Ex. 3.1.15. (cont’d) Cor anglais

° #œ &

¸≈ ‰ Ó #œ.

3 #œ ¸ & . #œ. œ nœ nœ. ≈ ‰ ™ nœ

Clarinette en la

Basson

Violon

# œ. f

¸#œ n œfi

¸ ≈ Œ nœ

∑

nœ >

#œ

nœ

¸ #œfi

nœ

bœ

#œ

¸ ≈ ‰ nœ

n>œœ. nœ #œ nœ b œ nœ. nœ. # ‰ . nœ ˝ ≈ ‰ ≈ nœ

nœ nœ f

4+5+6

1+2+3

Violoncelle

° #>œ &

Alto

∑

nœfi¸nœ

#œ ≈

‰

nœ nœ nœ > nœfi¸n˝œ.

˝ ‰

>. . nœ n >œ. nœ nœ #œ nœ #œ œ. ‰ ≈ n#œœ œ ≈ ≈ nœ nœ

nœ >

#œ

≈ ≈

# œ. nœ n œ. n>œ #œ n œ bœ f

≈

nn>œœ. œ. nœ nœ nœ nœ. nœœ œ ≈ nœ nœ bœ #œ ≈ . . f

#n œ. œ >

#>œ

19+20+21 Cor en fa

Vibraphone

Marimba

Harpe

Piano

& ‰

#œ nœ nœ. #œ ‰ Ó .

nœ.

nœ œ b œ nœ #œ

≈

° &

∑

¢&

∑

≈ nœ nœ mp

? Ó

‰™

& Ó

‰

? Ó

‰

oo nnœœ œœ ˝ ¸

œ nÆ˝œ nœ ˝ #nœœ ˝

nnœœ

#œ œ nnnœœœ ‰ . mf œœ ‰

# œ nœ

Mib Sol#

109

The melodic structure of the entire piece can be derived directly from the trichords that first appeared in the viola and are methodically manipulated and transformed into larger collections. The vertical/harmonic content, however, has an independent structure of its own. The first two harmonic trichords that appear in the music, the [D, B, Eb} grace-notes and the {G, Ab, A} trichord, actually combine to form an all-trichordal hexachord, a fitting tribute to the favorite hexachord of the dedicatee (ex. 3.1.16). Having an all-trichordal hexachord at the very opening also represents the importance of trichordal pitch orientation and construction in the whole work. Example 3.1.17 analyzes the vertical structure (the simultaneities) of the opening measures up to the first double barline, provided with harmonic reductions in the last system. These simultaneities, similarly to the linear melodies, are also constructed based on trichords. Unlike the process that takes place in the horizontal construction where trichords are combined to form 6, 9, and 12-element collections, the vertical trichords are combined to form tetrachords and pentachords. These chords do not seem to suggest an underlying structure. These simultaneities cannot be directly derived from the trichords that first appeared in the viola, although they are loosely based on those set-classes. The vertical structure does not pose the same kind of organizational formation as that of the horizontal structure.

110

Ex. 3.1.16. The opening all-trichordal hexachord 037 036 027 [012478] - All-trichord Hexachord

B bœœ œ

‹#œœœ

nœ #œ

œ bœ

nœ

025 013

012

#œ

015

bœ

014

016 024 026 048

Ex. 3.1.17. A reduction of the simultaneities in the opening measures Cor anglais

° &

Violon

Vibraphone

Harmonic reduction

‰ nœ¸ > œ

‰ Ó

¸ nœ. ‰ Œ

¸ n œfi#œ.

∑

‰

∑

bœ. Œ

° ¢&

? & ?

Mib Fab Sol§ La§ Ré§ Do§ Si§

fi¸ ¸ ‰ nbœœ n œ –

bœœ – — ' ‰ Œ ®°

nœ n

¸ #œfi

#œÆ ˝

¸ n œfi n œ

¸ #œfi

° ¢&

‰ Ó ‰ Ó

¸ #œ ' ¸ #œ '

Œ Œ

¸ nœfi#œ

fi¸ nœ œ

Œ ‰ nœfi¸bœ œ ˝ nœ > ‰ nœfi¸bœ œ Œ ˝ nœ > b œ œ ¸ fi nœ ‰ ˝ Œ #>œ

& Piano

‰ ¸ #œ nœ > > ‰ n˝œ nœ

& Harpe

nœ >

Clarinette en la

Basson

¸ bœfi

‰ n œ¸ Ó nn œœ

‰ #œ' Ó ®°

bœœ œ œn#œœ

œ œ œœ #œ

fi¸ nn œœ

Œ ‰

‰

¸Œ nœ

nœ ‰n# œœ Œ ' ®° bœ œ

œœ #œ

111

> nœ nœ œ ∑

¸‰ Œ œ

Ó ∑

bbnœ>œœ ˙˙˙ ˝

œœœ ‰ Œ ˝

∑

‰bbœœ¸– —Œ

∑ #œ

#œfi¸

œ b >œ œ

‰ Ó fi¸ ¸ nn œœ n œ

‰ ¸#œfi¸ œÆ ‰ nœ ˝ ' ‰ ¸#œfi¸nœÆ ‰ œ ˝ ' Æ # œfi¸n œ ‰ n˝œÆ ˝ ‰

bœ œ nœ >

œ n œœ #œœ

bnœœ– ‰ '——Œ ®°

bœ œ bœ bœœ

Lab

Ex. 3.1.17. (cont’d) Cor anglais

° &

Clarinette en la

Basson

Violon

Violoncello

° & ¢

Marimba

Harpe

Piano

Harmonic reduction

¸Ó nœ ' ‰ ¸Ó nœ '

‰ nœ¸ Ó '

‰

? nnnœœœ

‰

‰n#bœ˝œœ

#œ

∑ ¸ ¸‰ Œ ## œœ œœ

fi¸ n œ¿ n œ. n ˙

bœ œ ‰ n œ˝ œ˝ ‰ Œ

¸ bœfi n œ.

nœ œ nœ œ nœ œ

Œ Œ

¸ # œfi

#œ Œ

¸ nœfi#œ

n˙ b˙

∑

? nœœœ

#bœœœ

#œ

> nœ <n>œ nœ Œ

nœ nœœœ bœ

#œ

‰

¸ ¸ nœfi#œ

¸ nœfin œ¸

‰

nœfi¸nœÆ

¸ ‰ bnœœ ˙˙ n>œ ˙ &

¸ œ œ ‰ Ó œ œ ∑

n œ <n> œ b>œ

∑

n œfi¸# œ

∑

¸‰ Ó œ œ

∑

nœ

∑

#œ

‰ ¸Œ nœ.

∑ Ó

#œ

n œÆ ‰ #˝œÆ ‰nœfi¸˝ Œ ‰ Œ

¸ # œfin œ

° Vibraphone &

‰ ¸ ‰ nœ¸ Œ # œ n œfi¸ ' ' ‰ ¸ ‰ nœ¸ Œ # œ œfi¸ ' '

‰

¸ ‰ nœ Ó bnœœ > ¸ ‰ bnœœ ˙˙ n>œ ˙

Œ Œ ∑

Mi§ Fa§ Ré#

fi¸ bnnœœœ

Ó #œ

nœ

Œ œ nb>œœ

– # œÆ – nnœœ – — ‰ ˝ ®°

bnœœœ #œœ œ

All-trichordal hexachord [012478]

112

Ex. 3.1.17. (cont’d) Clarinette en la

Basson

Violon

Violoncello

° & <#>œ ¢

B Ó

° Œ & ¢

B Œ & Ó

Piano

Harmonic reduction

nœ

#œ #œ

>œ œ ˝ n>œ bœ ‰ ˝ bœ ¸ nœ ‰

nœ ˝

‰

¸ ‰ b nœœœ bnnœœœ b> nnb œœœ °

¸ nœfin œ

‰

#œ

¸ n œfi

nœ

– – – — — – — —

‰ nœ¸ '

nœ b œ

¸ ‰ œ

œ n#œœ nœ

113

‰ n˝œ

fi¸

n ‰ œ nœ¸ '

–– Œ — – œœœ ‰ — — Œ ˝ ø

¸ nœfi#œÆ

#nnœ>œœ

? œ bbnœœ nnbœœœ bnbœœœ

‰

fi¸ n#n œœœ

¸ œ ‰ '

∑

& &

#œ

œ &

nœ bœ

> ¸ ¸ – ‰nnœœfi #œœ Ó– — n — ‰

nœ

œ ˝

‰

–

Ó—

° ø

#œ œœ œ n œœ

Ex. 3.1.17. (cont’d) Cor anglais

° &

Violon

b œ. B <n>œ ≈nœfi¸ Œ

° & B

¸bœ n œfi > ¸> bœfi nœ

nœ ' b œfi¸ nœ '

Clarinette en la

Basson

∑

Vibraphone

° fi¸ n – œ– Ó & nœ b >œœ— n œ– —œœ — — * ° ¢& b œfi¸n œ > &

Harpe

¸bœ— n œfi >

bnœœ Œ

nœ œœ n œ—œ

Ó ∑

Piano

œ bn œœ

Harmonic reduction

¸ ‰ nœ nn œœ > °

Œ bnœœœ œ

œ œœ ?

#bnœœœ

– – —

¸ # œfi œ

nœ œ nœ > ‰ œfi¸nœ. Œ Œ ˝ ?

# œfi¸n œ

¸ nœ ‰ Œ

¸ bœfi œ

#œ

œ Œ

nœ

> bœ œ ‰ nœfi¸˝

∑

¸ œœ ‰ ‰ ¸ œ #œ œ > ?

> nœ ‰ n˝œ

‰ nnœœ¸ œœ

¸ œœ ‰ Œ

‰ #œ œ ˝

œ ‰ ‰ nœ¸n#œœœ ˝ >

Fa#

Fa§ Do#

‰

¸bœ

¸bœ n œfi >

bœ

Ó *

∑

‰ nœfi¸b>œ Ó n œ ˝ # œ #œ

∑

n œœ ##œœ

nœ

∑

¸ ¸ bœfibœ œ

‰

¸ nœ Œ nœ œ >

‰nœfi¸nœ. Œ ˝

‰ nnœœ¸ œœ #œ œ ∑

–— —

n>œ ‰ ˝ #œ

fi¸ ? #bœœ nœ

‰ nœ¸ #œ >

La§ Sib

‰ nœ b œ œ¸ Œ

∑

Marimba

¸ b œfi

nœ œ

#œ

¸ ‰ nœfib œ¸

n>œ

Violoncello

114

Ex. 3.1.17. (cont’d) Cor anglais

° &

nœ ¸ ¸ ‰ ‰nœfi¸˝ nœ ™ bœ nœ > > dim. Ó & ‰ ¸ #œ œ

Clarinette en la

Basson

Violon

¸ n>œ ‰ Œ

° Œ ¢&

° Vibraphone & Ó Marimba

nœ ™

¸ ¸ b œfi# œ.

Œ Œ

Harpe

Piano

∑

Do§

? Ó

bœ ? b nœœ

poco

‰

poco

nœfi¸# œÆ

#œ

‰ bœ ' #œ

nœ œ ˝ ‰

˝ ‰ Œ Œ

‰nœfi¸ ¸ Ó œ >

¸ Œ bœfi œ¸

‰

¸œfi¸nœÆ

¸ Ó nœ ' f Æ fi¸# œ

nœ n œ >

b fi¸ ‰ œ nœÆ Œ ˝

∑

œœ

revenir progressivement - - - - - - - - - - - - - - - - - - - - - - - - - - - -

¸ ‰ Ó œœ ∑

∑

∑ Œ

‰ nnœœ ˝

œ œ Ó

∑

& Harmonic reduction

‰ ˝ Ó

∑

poco

¸ n>œ

‰ ¸ nn>œœ f nnœ>œ ‰ nœ ˝

¸ ¸ ‰ ¸ ‰nœfi¸b˝œÆ ‰ nœfi œ nœ #œ œ '

‰ Ó

bœ ? b nœœ ‰ Œ ¢ ˝ &

poco

dim.

nœ ‰ nœ¸ œ ˝ ‰#œfi¸#œ #œ nœ œ ‰ ˝

‰nœfi¸#œ¸ Œ '

nœœ œ

Mib Ré§ Si§

œ ‰ Ó œ ˝

– nn œ #nœœ œ Œ — f

b#œœ Œ n#œœ > ° ø nœ #n œœ n œ b#œœ n#œœ

115

nœ¸–

– – Œ ‰ nbnœœœ– —Ó — — '— ®°

nbœnœœœ

bnœÆ –– œ ‰ nnœœ – ˝—Œ ®°

bœ nœœœ

> nœœ b ‰ nœ ˝ °

bœœœ

Ex. 3.1.17. (cont’d) Cor anglais

° ¸ ¸ & ‰ #œ nœfi¸#œ ‰ Œ nœ ' ¸ & ¸ ‰ Œ œfi¸#œ ™ <n> œ

Clarinette en la

Basson

Violon

° & Ó

dim.

≈Œ

¸ ¸ nœfi#œ

¸ nœ œ œ ‰ ‰ ˝ ˝ ‰ poco

‰

¸ ¸ n œfin œ œ

‰ fi¸ œ #œ¸ ‰ nœ ˝

¸ #œ

‰

¸ #œ

‰

#œ ˝

‰

∑

> Œ nœ bœ ‰ ‰ nœfi¸#˝œ #œ ˝

‰ Œ

poco

dim.

∑

poco

∑

Viola

Violoncello

#œ B ‰ ˝ nœ

n>œ Œ nœ ˝ ‰ ‰ nœfi¸#œÆ #œ ˝

∑

¸ œ ‰ ‰ n˝œ œ poco

∑

¸ ¸ n œfin œ œ

∑

> # œ #œ œ nœ œ nœ n˝œ nœfi¸ Œ ˝ ‰ ‰ ˝ ˝ ‰ ‰ ‰ ˝ arco

∑

‰

poco

dim.

poco

‰nœfi¸n˝œ œ

(revenir progressivement) jusqu'au - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Rapide (q = 152)

Vibraphone

° & Ó

fi¸ ¸ ¿ nœ œ nn œ.œn >

‰

Marimba

Harpe

Piano

? Ó

& Ó

Œ ¸ nbnœœfiœ

—Œ ? nnœ˝œ ‰–

∑

nœ

‰ ¸ ¸ ‰ Œ bnn œœœ œœœ >

‰

¸ bœfi#œ

Lab Do#

#nœœ

∑

n>œ bœ

Ó ∑

– nœÆ – – b œ & nœ — ‰ Œ ˝ mp

∑

œ bœœ

‰

œ ‰ Œ ˝œ

‰

La§

∑

#œœ bœ

fi¸ ¸ ¿ nœ œ nn œ.œ n >

‰

∑

b œœœ

œœœ bœœ

¸ ¸ ‰ Œ #n œœ œœ p

> b n œœ ‰ nœ˝ Ó

®°

& Harmonic reduction

œœ

‰ ˝

œœ

¸ nn>œœ

nœfi¸ #œ

nœœœœ #œ

116

Ex. 3.1.17. (cont’d) Cor anglais

° & &

Clarinette en la

Basson

Violon

Marimba

& Harpe

Piano

bœ

œ ˝ ‰Œ

bœ

nœÆ ‰ ˝ ‰Œ ˝

dim.

nœ‰Œ Ó ˝

Æ ˝ ‰ ‰ ¸ bœ œ > mp

˙™

n>œ œ ‰˝

sfp

∑

˙™

n>œ œ ‰˝

n >œ ‰#b œ˝œ Œ

n˙ p

¸‰Ó œœ Fa#

Œ ‰#nœœ¸ Ó > p

¸ œœ ‰ Œ œ Fa§

∑

#œœ œ b#œœœ

∑ nb œœ #œ

nnœ>œ ‰ b˝œ

– –

nnn œœœ – Ó— — * ø

˙™

∑

‰ #œnœnœ œ ‰ > f

˙™

¸ œ ‰

Œ n˙ ™ >

117

œ™ œ. > ff

Fa#

∑ ∑ ∑

∑ &

n>˙ ™

f nnœœ. ‰ bœ. Œ

#œœ œœ œœ b œœœ œœœ bœ #

¸ ‰n#œœ n˙ .

Ó—

ralentir très légèrment à - - Moins rapide (q = 96) accél. - - -

Do§

∑

¸ œ ‰ ≈ nœ. œ. œ. >

Œ ‰ ¸ n˙ nœ.

∑

Œ ‰#nœœ¸ ˙˙ n>œ ˙ p

sfp

avec sourd.

¸‰ Œ œ

œ œ

mf sfp

¸‰Ó œœ

sfp

Œ nœ >

∑

˙™

sfp

nnOœ‰Œ ˝

‰™ ¸ nœ œ nœ # œ. œ. . >

sfp

> w

∑

¸b œ Æ n œfi

‰ nœ ˝

sfp

‰ ¸nœfi¸bœÆ ‰ ‰ ¸ nœ œ nœ ˝ > ' mp

∑

& Harmonic reduction

œ ˝ ‰Œ

∑

<n><n> œœ

‰ œ¸˙™ > Ó

∑

dim.

# œfi¸nœÆ

nœ‰Œ ˝

dim.

∑

° & <n> œ <n> œ ?

¸‰Œ nœ œ

¸ ¸‰#œfi¸nœ ‰Œ ' nœ '

Horn in F

Vibraphone

dim.

‰ ¸nœfi¸bœÆ ‰ ‰ ¸ nœ œ œ ˝ > ' mp ¸ ¸ fi Œ ‰bœ œ ∑ .

œ ˝ ‰Œ

¸ bœfibœ

∑

B ¢

bœ

° &

Viola

Violoncello

¸‰#œfi¸nœ¸‰Œ ' nœ '

œœœ bœ

Through these examples, one clearly sees the process of proliferation from such simple material, as a majority of the pitch structure of the whole work is derived from the initial trichords that first appeared in the viola. The idea and the importance of proliferation is essential to Boulez's method of composition: To my mind, a chord or an initial interval can generate many things, either when it proliferates or when it becomes distorted. In my experience, the myriad possible combinations must always already exist within the cell, so to speak. I personally work from proliferating or symmetrical chords organized around an axis; in my view, what is important is the parallel reproduction, the symmetrical reproduction or the defective reproduction in the actual structure of the harmonic writing. To these, you could also add unrecognizable reproduction, where the same elements are placed in a completely different order, in inverted registers: you can just about hear it’s the same combination, but you cannot say why. So there are several stages: total identification, partial identification which changes progressively and variation which alters the position of the elements and where it is as much as you can do to recognize the density. Detection, in these circumstances, comes too early or too late: if one knows there will be fifths and seconds, it is too early and if one says to oneself: ‘Yes, that’s right, there were fifths and seconds all over the place’, realization has occurred after the event, too late.10 In the first section (5-R1 to 2-55), these trichords are realized through two strata of instrumental groups, and the harmonic rhythm between the two strata is a one to one ratio, though the strata are independent. In the second section (R5 to R10), Boulez introduces 16th-note rhythmic values, and doubles the harmonic rhythm of one stratum, so that the harmonic rhythm ratio between two strata is a two to one. The original trichords similarly double their length and change from simple trichords to 6-element collections through the process of reordering and combination. The third section (2-R11 to R13) uses the 16th-note rhythmic value with its multiples in both strata, so that the

Philippe Albéra, and Pierre Boulez, “Pierre Boulez in Interview (2): On Elliott Carter, ‘A Composer Who Spurs Me On’,” Tempo 217 (July 2001): 3.

118

harmonic rhythm between the two is back to a one to one ratio. The harmonic domains are still divided between two strata of instrumental forces. The fourth section (R14 to R24) introduces the sextuplet rhythmic value and its multiples in one stratum, while the other continues the 16th-note rhythmic value and its multiples from the previous sections. The harmonic rhythm between the two is now a complex ratio of three to two. Similarly, just as the rhythmic value increases its subdivision, each harmonic domain also increases its size, changing from 6-element to 9-element collections through the process of more reordering and combination. The fifth section (R26 to R66) complicates these relationships further, as Boulez introduces 32nd-note rhythmic values for the first time. Harmonically it also increases from 9-element to 12-element collections. This process of proliferation and the constant manipulation of the pitch material continue for the rest of the piece, to a point that the initial trichords are no longer recognizable. These trichords not only have been disguised and integrated into the background of the pitch structure, but their identity has been completely transformed and distorted. One no longer sees traces of these trichords, but only a glimpse of their shadows and imprints on all levels of the pitch structure and construction. Yet it is these initial trichords and their evolution and development that tie the pitch structure together as a coherent whole. Another level of manipulation is the canonic illusion that arises through the realization of these pitch materials. This is already evident in the two harmonic strata of unfoldings since both strata of instrumental forces share the same harmonic domain, but the two are juxtaposed while each has its own harmonic rhythm and way of realizing these materials. As the work evolves, despite the fact that these strata go through different

119

process of development and manipulation, what one perceives is a high degree of resemblance of pitch relation between the two forces, and the continuous reordering of the same pitch material in all of the voices. But what is more remarkable, is the subtle pitch doubling and reordering that happen at the local level, the surface level of these realizations. In the first section, for instance, the horn subtly doubles both strata at a fixed duration, while the resonant instruments also double both strata at various points. The result is that certain pitches are highlighted from these unfoldings, which in turn creates a sub-level of canonic imitation and sustained notes from these polyphonic textures. This method of doubling and canonic imitation is reinforced in the subsequent sections through additional surface re-orderings and doublings. For example, both the viola and the piano music in 2-R5 (ex. 3.1.10) is a reordering of the pitch material derived from the clarinet, which itself is a reordering and a realization of the underlying 6-element collection. In the same example cor anglais and bassoon transpose and reorder materials derived from the horn part, further disguising the identity and the origin of these trichords. In addition there are the grace-notes, which continue to articulate trichordal groupings in all the voices. Even if the rhythmic values have gone from 8th-note to 16th and 32nd-note, gracenotes still provide an important divider to articulate the boundaries of the various harmonic domains. These microscopic details of doubling and surface reordering eventually take over the underlying pitch material, and as the work develops, more levels of manipulation in both the underlying structure and the surface detail are added to the music. Moreover, by varying the combination of instruments that participate in the realization of these

120

harmonic domains, Boulez is able to offer unique instrumental combinations in each section for realizing the stratification of the pitch structure. The work inevitably becomes more complex, even incomprehensible; yet at the same time, such methodical proliferation nevertheless creates more layers of inter-relationships and a network of connections â&#x20AC;&#x201C; a labyrinth out of a labyrinth.

121

Chapter 3.2. Periodicity in Dérive 2 Looking back at the opening viola line (ex. 3.1.1), one notices that it is rhythmically composed under a particular pattern—2-1-3-1-1-2-1 where the basic unit is the 8th-note (ex. 3.2.1). Ex. 3.2.1. Rhythmic pattern of 2-1-3-1-1-2-1

118 2

¸ œ 1

¸ œ

œ™

¸ œ

If the opening viola line is re-barred and re-grouped under this pattern, one sees the rhythmic construction of the viola corresponding to it (ex. 3.2.2). Example 3.2.2 does not fully explain the relationship between this rhythmic pattern and its actual realization, because this rhythmic pattern is partitioned into two strata: one with the three woodwinds, and the other with three strings (ex. 3.2.3). In other words, the rests that are present in the viola (ex. 3.2.2) are now filled with rhythms from the woodwinds, with both parts combined and interlocked to form the rhythmic pattern 2-1-3-1-1-2-1 (ex. 3.2.3). Ex. 3.2.2. The opening viola line re-barred under the pattern of 2-1-3-1-1-2-1 Rhythmic pattern

Alto

/ œ

B #œ

118

¸ œ œ™ 1

¸ ¸ œ œ œ

Œ #œfi¸nœ¸ nœ ™

Rapide (q = 152)

B ¸ œ œ™ 1

‰

nœ ™

¸ #œfi

¸ ¸ œ œ œ 1

¸ nœ #>œ ˝ Œ

¸ œ œ 1

¸ ‰ Œ ¸b˝œÆ nœ nœfi

¸ œ 1

‰

œ ¸ nœfi

œ >

¸ œ œ™ 1

‰ Œ™

122

¸ œ œ

¸ œ œ™

#œ ‰nœfi¸ #˝œ nœ ˝ nœ >

‰ Œ™

‰ nœ ™

¸ œ œ™ 1

¸ ¸ œ œ œ

¸ ¸ œ œ œ 1

¸ œ œ

#>œ nœ ˝ Œ ˝

¸ œ œ™ 1

n>œ #œ ˝ nœ ˝ Œ ™

nœfi¸

¸ ¸ œ œ œ

¸ œ 1

n œÆ ˝ ‰ Œ nœ¸ >

#œfi¸

¸ ¸ œ œ œ

¸ œ

‰ nœfi¸nœ¸ Œ '

‰

Ex. 3.2.2 (cont’d). ¸ œ œ™

/ œ

#œ

nœ Œ ™ ˝

¸ œ œ™

B Œ 2

bœ

¸ nœfi

œ nœ ˝ nœ ˝

‰

¸ bœ ™ b>œ

‰ n˝œ #œ >

¸ nœ Œ

¸ œ œ™

¸ ¸ œ œ œ

¸ œ œ

¸ #œfi

¸ œ œ™

¸ œ œ

1 4 2

¸ ¸ œ œ œ

1 3

> bœ™

> bœ ˝

¸ œ œ

> n œ fi¸ #œ ‰ Œ ™ n˝œ ˝ Œ #˝œ nœ n˝œ Œ ™ >

¸ nœfi

nœ ˝ nœ ™

#œ ˝ nœ

¸ bœfi

nœfi¸

‰ ¸ œ

#œÆ ˝

nœfi¸

¸ ¸ œ œ œ

bœ ˝ nœ¸ Œ

> ‰ nœ ™

1 1 2

¸ nœ ‰ Œ >

¸ ¸ ¸ ¸ œ œ œ œ œ œ œ™

1 3

> nœ

¸ ¸ œ œ œ

¸ œ

¸ œ œ™

¸ nœ ‰ Œ '

¸ bœfi

‰ ‰

¸ œ œ™

¸ nœ ™ #œ

¸ œ œ

¸ ¸ œ œ œ

1 2

> nœ

¸ ¸ ¸ ¸ œ œ œ œ œ œ œ™

1 3

¸ ¸ œ œ œ

‰ Œ™

¸ œ œ™

nœfi¸

‰

‰ ‰ nœ

¸ œ œ™

¸ œ

¸ ¸ œ œ œ

nœfi¸

‰

‰ n˝œ bœ

¸ œ œ™ 1

¸ ¸ œ œ œ œ™

1 1 2

¸ Œ™ nœ '

1 2

¸ ¸ œ œ œ

¸ nœfi

nœ ¸# œ #œ n œ ˝ ‰#œfi ˝

¸ œ œ œ™ 2

¸ œ œ

¸ nœfi

‰ nœ ™

B Œ

/ œ

¸ ¸ œ œ œ

/ œ

¸ ¸ œ œ œ

1 3

Æ > b œ ¸ nœfi¸n œ ¸œ n> ‰ nœfi¸#˝œ ™ ˝ ™ Œ Œ ‰nœfi¸n˝œÆ Œ ‰ Œ nœ n>œ

¸ œ 1

‰

¸ ¸ ¸ œ œ œ œ 1 1 2

> ‰ #œ¸nœ ˝œ

Ex. 3.2.3. The opening instrumental parts re-barred under the rhythmic pattern of 2-1-31-1-2-1. The E and N letters will be explained in the next section

Rhythmic pattern

Cor Anglais

° & &

Clarinet in A

Bassoon

° Viola B ¢ Horn in F

118

E 2

œ ¸ bœfi

nœ >

fi¸ œ nœ > ¸ nœfi#œ

E E ¸ 1¸ 2 œ œ œ N

¸ 3 œ œ™ N N

‰ Œ™ ‰ Œ™

¸ œ N

‰ nœ¸ > œ

‰

¸ nœfibœ

‰

¸ nœfibœ

¸ #œ nœ > > n ‰ ˝œ nœ

‰ Œ™

‰

nœfi¸b œ

Rapide (q = 152)

Œ +

n>œ

¸ ¸ nœ ™ #œfinœ

b œÆ ¸ nœ ‰ Œ nœfi¸ ˝

‰

E 1 3 ¸ œ œ™ N

¸ Œ™ n>œ ¸ Œ™ n>œ

¸ Œ™ #>œ

‰ nœ ™

+ bœ

123

E ¸ 1¸ 2 œ œ œ N N

¸ 2 œ œ N N

‰ #œ¸ Œ '

‰

‰ #œ¸ Œ ' #œÆ ‰ ˝ Œ

‰

Œ #œfi¸ ¸ œ b >œ ™

‰

‰ Œ™

#œfi¸n˝œ

‰

#œ ‰nœfi¸ #˝œ nœ ˝ nœ > + Ó Œ ‰ nœ

E E 1 3 ¸ œ œ™

E ¸ 1¸ œ œ N 1

¸ bœ ™ nœ >

¸ #œfi

‰

1 ¸ œ N

¸ fi¸nœÆ ‰ ‰ n œ #œ ˝ '

¸ fi¸nœÆ ‰ ‰ œ #œ ˝ ' Æ # œfi¸n œ ‰ n˝œÆ ˝ ‰ ‰

#œfi¸

¸> nœ nœ ™

‰

¸ n>œ

‰

Ex. 3.2.3. (cont’d)

Rhythmic pattern

Cor Anglais

Violin

& Œ ¢

Horn in F

B Œ

° & B #œ

Viola

Violoncello

œ N

° & Œ

Clarinet in A

Bassoon

E 1 3 ¸ œ œ™ N

¸ œ N

¸ Œ™ nœ '

E ¸ 2 œ œ N

E 1 ¸ œ

œ N

¸ ¸ ‰ nœ # œ n œfi¸ ' ' ¸ ‰ ‰ ¸ ‰ nœ ¸ # œ œfi ' ' n œÆ ‰ ‰ #˝œÆ ‰nœfi¸˝

nœ ™ ‰#œfi¸

‰

+ & #œ

‰

Rhythmic pattern

Clarinet in A

Violin

Viola

œ N

° Œ ¢&

E ¸ 3 œ œ™ N 1

> ‰ œ œ ˝

° #œ ¸ & nœ Œ ™ # œ nœ B ˝ Œ™

¸ œ N

¸ 2 œ œ N N

‰ ‰ Œ ¸ n œfi˝

¸ nœ nœ

¸ nœfi˝

nœ ˝

œ œ

nœ

E E 1 ¸ 2 œ œ ¸ nœfin œ¸

#œ

‰

Œ Œ

#œ nœ ˝ Œ™ Violoncello ¢

‰ ‰ Œ

‰

œ ™ ˝ Œ

‰ ‰ Œ

+ ‰ #œ

Horn in F

+ & #œ

nœ

¸ # œfi

‰ ‰

‰ Œ™

œ >

∑

‰ ‰

nœfi¸

E ¸ 3 œ œ™ N 1

E ¸ 2 œ œ N 1

‰ ‰

¸ œ™ #œ

¸ nœ #>œ ˝ Œ

¸ œ N

¸ œ™ #œ

∑

¸ œ™ #œ

‰ ‰

¸ Œ™ nœ ' ¸ nœ Œ ™ '

E E 1 3 ¸ œ œ™

¸ œ N

‰

¸ ¸ nœfi#œ

E ¸ 3 œ œ™ N

œ N

E 1 ¸ œ

‰

¸ nœ '

n>œ Œ ™ ˝

œ N

‰

‰ n˝œ bœ

‰

¸ ™ nœ Œ '

‰ n˝œ bœ

‰

¸> bœfinœ

‰ Œ™

‰ ‰ Œ

‰

+ ˙™

¸ Œ™ nœ '

nœfi¸

124

¸ nœfin œ¸

‰

Œ ' ¸ ‰nœfinœ¸ Œ '

¸bœ n œfi >

nœfi¸n œ

¸ 3 œ œ™ N N

‰ Œ™ ‰ Œ™

¸ nœ ™ #œ ¸ nœ ™ #œ

‰ ‰

fi¸ ‰nœ n˝œÆ Œ

‰

¸ ‰ Ó œ

‰ nœ¸ bœ

nœfi¸

#œ ˝ ‰nœfi¸#œ ¸ > nœ ∑

‰

E 1 1 2 ¸ ¸ œ œ œ N N

‰ nœ ™

n œfi¸# œ

E 1 ¸ œ

∑

¸ Œ™ œ b>œ n>œ œ ˝ Œ™

nœfi¸#œ

E E 1 1 2 ¸ ¸ œ œ œ N

‰

#>œ nœ ˝ Œ ˝

œ ¸ ‰ ‰nœfi¸#œÆ Œ ˝ œ' ˝ ‰ Œ™

E E 1 ¸ 1¸ 2 œ œ œ N

E 1 ¸ œ

nœÆ ¸ Œ ¸ ˝ b œfi¸ nœ nœ ' > ‰ ‰ ¸nœ ‰ b œfi

> nœ

‰

‰bœfi¸

‰

‰ Œ

‰

‰ Œ

‰

Ex. 3.2.3. (cont’d) E Rhythmic pattern

Cor Anglais

/ œ

° Œ & &

Clarinet in A

Bassoon

Violin

B Œ

° & Œ

Horn in F

bœ E

Rhythmic pattern

Cor Anglais

/ œ

° & Œ

Clarinet in A

Bassoon

Violin

Violoncello

Horn in F

‰

‰ Œ

‰ œfi¸nœ. ˝

‰ Œ™ ¸ nœ # œ >

¸ 2 œ œ N N

‰ nœ ™

B &

œ™ N

‰ Œ ¸ #œfin˝œ

‰

¸ #œ nœ Œ >

‰ Œ

‰

‰ Œ

‰

E ¸ 1¸ 2 œ œ œ N N 1

¸ 2 œ œ N N

¸ Œ nœ ' ¸# œÆ # œfi

nœ

‰ Œ™

¸ œ N

¸ 3 œ œ™ N N

™ ‰ nœ

‰ ‰ nœ nœ¸ Œ > ∑

> ‰ ‰ nœ b˝œ

‰ Œ

‰

Œ™

‰

E E ¸ 1¸ 2 œ œ œ N 1

¸ bœfi œ¸

nœ

nœfi¸˝

‰ Œ

#œ

¸ 2 œ œ N N

¸ #œ

∑ ‰

Œ™

‰

+ œ

œ ˝

œ™

125

1 ¸ œ N

‰nœfi¸#œ¸ ¸Œ ' nœ > ‰ fi¸ œ¸ ‰ nœ #œ nœfi¸# œÆ ‰ ˝ ¸Œ n>œ

nœ ™

nœfi¸#œÆ

¸ nœ ‰ Œ >

∑ Ó

E E 1 ¸ 3 œ œ™

E E ¸ 1¸ 2 œ œ œ N 1

∑

nœ

bœfi¸nœ¸

‰ Œ

¸ œ œ™

‰ ‰ Œ

¸‰ Œ œ

‰

E 1 3 ¸ œ œ™ N

¸ œ N

‰ ‰ nœ

n>œ ˝

‰ ‰ Œ

‰

‰ ‰ Œ

‰

¸ ¸ nœfi#œ

Œ™

¸ ¸ nœfi#œ

Œ™

E ¸ 2 œ œ N 1

> ‰nœfi¸nœ ™

‰ Œ™

‰ ‰

¸ Œ™ œ

‰ ‰ Œ

‰

nœ

E 1 ¸ œ

> ‰ ‰ nœ bœ ˝ bœ ˝ nœ¸ Œ ‰

#œ ˝ nœ

∑ Œ

¸ œfi¸nœÆ ¸ ¸ ¸ ™ bœ ˝ ‰ ‰ ‰ nœ n œ ‰ Œ œfi#œ Œ ' > n œ #œ fi¸ Æ #œ b œ nœ ‰ Œ ˝ nœ ‰ Œ ™ ˝ ‰ Œ™ ˝ ‰

nœ ¸# œ #œ n œ ˝ ‰#œfi ˝

‰ ‰

™ nœfi¸b œ

¸ œ

#œ

∑

nœ ˝ ‰#œfi¸#œ #œ¸ nœ ‰

‰

E 1 2 ¸ œ œ N

¸ Œ nœfi¸n˝œ nœ ™ nœ bœ > ¸œ ¸ fi Œ™ bœ ˝ œ Œ ‰ # œ œ¸

¸ Œ ‰ bœfi¸bœ ™ n œ > nœ > ¸ Œ™ ‰ Œ ‰ nœ #œ > # œfi¸n œ nœ Œ ‰ B n fi¸bœ ™ ? ˝ œ ˝ n>œ

∑ +

E 1 ¸ œ

‰nœfi¸nœ. ˝

E 1 3 ¸ œ œ™ N

E 1 ¸ œ

¸ œ N

¸ # œfiœ

‰

‰ Œ™

∑

B Œ ¢

‰ Œ™

& ¸ ‰nœfi¸b˝œÆ Œ ™ nœ ' B Œ ‰ Œ™

° &

Viola

¸ 3 œ œ™ N N

¸ bœ ™ ‰ b>œ > B Œ n˝œ #œ ‰ ‰ B Œ

Viola

Violoncello

bœ

E 1 ¸ œ

> nœ n˝œ ‰

Ex. 3.2.3 (cont’d)

Rhythmic pattern

Cor Anglais

Violin

° ¢& ° &

Viola

Violoncello

Horn in F

œ N

EE 1 3 ¸ œ œ™

E E ¸ ¸ 2 1¸ 2 1¸ 3 œ œ œ œ œ œ œ™ NN N N N 1 1

E E 1 1 2 ¸ ¸ œ œ œ N

E E 1 2 1 3 ¸ ¸ œ œ œ œ™ N N

¸ ‰ ‰ fi¸ ¸ Œ ‰ bœ ™ Œ nœfi¸#œÆ #œ ™ ‰ ‰ nœ ‰ Œ ‰ ˝ ¸n œ ™ #œ n œ¸ #œ nœ' fi nœ '

Œ nœfi¸#˝œ #œ ™ ‰ ‰ nœ ˝œ Œ ‰ ¸ n œfin œ ™ >nœ # œfi¸ ¸b œ B nœfi ‰ Œ ™ n˝œ ˝ Œ #˝œ nœ n˝œ Œ ™ > nœ fi¸# œ B Œnœ ˝ #œ ™ ‰ ‰ ‰ Œ ‰ nœfi¸nœ ™ + & bœ ˝œ œ ™ ˝œ ˝œ Œ

‰

¸ Œ™ œ œ +

E 1 ¸ ¸ ¸ œ œ œ œ NN N 1 1 2

‰ ‰ Œ

bœ ™ ‰ ‰ Œ ¸ ¸ ¸ #œ ‰ ¸‰ #œfinœ Œ ‰ ' nœ ' > >Æ b œ ¸ nœfi¸n˝œ ¸œ n ‰nœfi¸œ Œ ‰ nœ n>œ Œ ™ ˝ bœ™ #œ # œfi¸nœÆ ˝ ‰ nœÆ ‰ ˝ Œ ‰ ‰ ‰ Œ ˝ ‰ ‰ Œ

+ ‰ bœ œ ‰ Ó ˝

E 2

E ¸ œ œ™ N 1 3

¸ ¸ œfibœÆ ‰ ‰ nœ ™ œ ˝ > '

E E 1 1 2 1 ¸ ¸ ¸ œ œ œ œ NN

nœ ‰ Œ ‰ ˝

nœ ¸ ¸ œfibœÆ ‰ ‰ ™ ˝ ‰ Œ n>œ nœ ˝ ' Æ ¸# œ ‰ Œ nœfi ˝ ‰ Œ ‰ #œ¸nœ Æ œfi¸b œ nO ˝ ‰ ‰ nœÆ b>œ ™ n ˝œ ‰ Œ ˝

Œ ‰

+ bœ

œ œ™ ˝

‰ >œ ˝ ‰

œ‰ Œ ‰ ˝

Example 3.2.3 demonstrates the re-barring of these two instrumental groups under the rhythmic pattern of 2-1-3-1-1-2-1 from the opening measures up to the first double barline. This partitioning of forces corresponds precisely to the trichordal harmonic domains that also feature these two instrumental groups in the previous example (ex. 3.2.3). In other words, the opening measures consist of a duality of forces - the three woodwinds, and the three strings. Each force has its own harmonic rhythm of trichords, and each force has its own rhythmic construction that is extracted from the rhythmic pattern of 2-1-3-1-1-2-1. Even though both forces share the same harmonic source, and extract their rhythmic construction from the same rhythmic pattern, the two forces never align, and they are meticulously interlocked. Similarly to their pitch content, the rhythmic construction of the two strata nevertheless also conveys the illusion of a canon. The letters E and N that are placed above and below the rhythmic pattern in example 3.2.3 will be explained below in example 3.2.5.

126

In example 3.2.3, one sees the role of the horn in relation to other instruments and the rhythmic pattern of 2-1-3-1-1-2-1. As mentioned earlier, the horn simply doubles another instrument (whether it is the woodwinds or the strings) in the opening measures. Its true function, however, is actually to articulate each cycle of the rhythmic pattern of 21-3-1-1-2-1. That is, at the beginning of each rhythmic cycle, the horn amplifies this (ex. 3.2.4). Consequently, the cello pizzicatos (5-R1 to R1-2), cor anglais (R1-3 to R2-2), cello pizzicatos again (R2-3 to R3-4), and bassoon (R3-5 to R4-3) that share the same pitches as the horn but are an 8th-note delayed also help to articulate the underlying rhythmic pattern (ex. 3.2.4). Essentially, the horn articulates the starting point of each cycle of the rhythmic pattern, accompanied by an instrument that switches every 5 cycles.

Ex. 3.2.4. Rhythmic cycle of 2-1-3-1-1-2-1 articulated by horn and other instruments Rhythmic pattern

U ∑

∑

long

Violoncelle

B U ∑ &

U +

pizz. sec

‰ nœ Œ Ó ˝ +

n>œ

Œ Ó

bœ ˝‰Œ Ó

+ Œ ‰bœ œ ‰ Œ Ó ˝˝

poco poco pp (à peine audible)

112 1 2

Rhythmic pattern

/ œ œœœ œ œ

Cor anglais

Cor en fa

2 1 3 11 2 1 2 1 3 11 2 1 2 ¸ ¸ ¸ ¸ ¸ œ œ œ™ œ œ œ œ œ œ œ œ œ œ œ œ œ

bouché

Cor en fa

& Ó

∑ + Œ‰ ¸ œ

112

1 2

œ œ œ™ œ œ œ œ œ œ ¸ #œfi

¸‰ Œ Ó œ. œ

Œ Ó

1 3 ¸ œ œ™

œ ˝ ‰Œ Ó

127

Œ ‰ n˝œ

∑

nœ

+ ‰#˝œ ˝œ ‰

∑ ∑

nœ

1 3 11 2 1 2 ¸ ¸ œ œ™ œ œ œ œ œ

‰#œfi¸œ. Œ Ó ˝ + ˙™

œœ

‰ nœ¸Œ

+ ‰ ¸ ¸‰ Ó #œ œ

Œ ¸#œ. ‰ œfi ˝

1 3 1 1 2 1 2 ¸ ¸ ¸ ¸ œ œ™ œ œ œ œ œ

Œ #˝œ ‰ Ó

112 1 2 1 3 1 1 2 1 ¸ ¸ ¸¸ ¸ 2 œ œ œ œ œ œ œ™ œ œ œ œ œ

Œ ‰nœfi¸#œ ‰ Œ Ó + Œ #œ

1 3 112 1 2 ¸ ¸ œ œ™ œ œ œ œ œ

Ó Œ

1 3 ¸ œœ

¸ ‰ nœfib œ¸

+ Œ ‰ ¸ ¸‰ Œ bœ œ

Ex. 3.2.4. (cont’d) Rhythmic pattern

Violoncelle

Cor en fa

= Rhythmic pattern

Basson

Cor en fa

11 2 1 2 ¸ ¸ / œœœ œ œ œ

B Ó

» Œ ‰#œ¸

& Ó

pizz.

1 1 2 1 2 ¸ / œœœ œœ

1 3 1 1 2 1 2 13 112 12 13 112 1 2 1 3 ¸ ¸ ¸ ¸ ¸ œ œ™ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ™ œ œ œ œ œ œ œ œ™

» Œ #˝œ ‰ Ó

∑

nœfi¸b˝œ

B Ó

& Ó

+ ‰ bœ œ ˝

1 3 1 1 2 1 ¸ ¸¸ ¸ 2 œ œ™ œ œ œ œ œ

∑

‰ ˙

+ ‰ ¸ #œ ˙ ™

¸‰ Œ Ó œ

#œ

1 3 112 1 2 ¸ ¸ œ œ™ œ œ œ œ œ

œ‰Œ ˝

1 3 ¸ œ œ™

¸ ‰nœfiœ. ¸Œ

pizz. ord. sec

‰ nœ Œ ˝ +

nœ

1 1 2 1 2 ¸ œœ œ œœ

¸‰ Ó œ

+ Œ ‰ ˝œ ˙

b œfi¸b œ.

nœ ˝ ‰Œ Ó

∑

1 3 ¸ œœ

œ‰ ˝ Œ

1 1 2 1 2 ¸ ¸ œœœœ œœ

. n œfi¸b œ

˝ ‰Œ

Œ ‰

+ Œ ‰ bœ œ Œ ˝

+ Œ bœ

Œ ‰ nœ Ó ˝ +

¸‰ Œ nœ œ

1 3 1 1 2 ¸ œ œ™ œ œ œ

∑ ˙

œ‰Œ ˝

Referring back to Example 3.2.3, the rhythmic pattern of 2-1-3-1-1-2-1 is essentially partitioned into two strata of rhythmic layers under a specific scheme of 11element pattern E-n-n-n-E-E-n-E-E-n-n,11 where E denotes “events” and N denotes “nonevents” following the convention found in Boulez’s sketches.12 Rhythmic values that correspond to E’s will be assigned to one line, while rhythmic values that correspond to N’s will be assigned to another line (ex. 3.2.5). For the opening measures, E’s are assigned to a trio of instruments, while N’s are assigned to a solo instrument. Furthermore, Boulez alters the instrumental assignment every 3 cycles: for the first 3 cycles of EN, N’s are assigned to viola, while E’s are assigned to the three woodwinds; in the next 3 cycles, N’s are assigned to clarinet while E’s are assigned to the three strings; in the next 3 cycles after that, N’s are assigned to clarinet while E’s are assigned to cor anglais, bassoon, and viola; 11

Curiously, in the first movement of Ligeti’s Piano Concerto, the solo piano part also uses an 11-element rhythmic pattern 3-3-3-2-3-3-3-4-2-2-2. 12 Pierre Boulez Collection, Paul Sacher Foundation in Basel, Switzerland. These letters were found in Boulez's sketches throughout, though the term "non-event" might be misleading since Boulez does assign Ns to actual events in the music. The E and N letters might be best understood as principal roles for the Es and supporting roles for the Ns in the music.

128

and the 3 cycles after that, N’s are assigned to viola while Es are assigned to cor anglais, violin, and violoncello. Figure 3.2.6 illustrates such instrumental assignments of E-n-n-nE-E-n-E-E-n-n in the opening that appears in example 3.2.3. Ex. 3.2.5. Partitioning the rhythmic pattern of 2-1-3-1-1-2-1 with E-n-n-n-E-E-n-E-E-nn. Rhythmic pattern

Lyne 1

Lyne 2

118

E 2

œ œ

¸ 3 œ œ™ N N

¸ œ œ™

E N

¸ œ

E 1 3 ¸ œ œ™ N ¸ œ œ™

3 cycles E

¸ œ

¸ œ œ

¸ 2 œ œ N N

¸ œ œ

E E 1 3 ¸ œ œ™

E E ¸ 1¸ 2 œ œ œ N

¸ œ œ™

¸ œ œ

CA., Cl., Bsn. Vla.

Vln., Vla., Vc Cl.

CA., Bsn., Vla Cl.

CA., Vln., Vc. Vla.

¸ œ

1 ¸ œ N

¸ œ

Vln. E

Bsn. E

E ¸ 1¸ 2 œ œ œ N N 1

E N

¸ œ œ

E N

E ¸ 2 œ œ N

¸ œ

E N

E E ¸ 1¸ 2 œ œ œ N 1

Vln. E

3 cycles

Cl.

Fig. 3.2.6. Instrumental assignments according to the E-n-n-n-E-E-n-E-E-n-n partitioning

Similar to the rhythmic cycle where the horn and one other instrument articulates its periodicity every cycle and every fifth cycle, the EN cycle also assigns a specific instrument to articulate its periodicity: the first 3 cycles are articulated by violin, then bassoon, then back to violin, and finally clarinet. Example 3.2.7 demonstrates the opening measures up to the first double barline, re-barred and re-grouped under the instrumental assignment of E-n-n-n-E-E-n-E-E-n-n. In this example, one undoubtedly sees the pattern

129

of alternation between the various instrumental forces, along with each EN cycle being marked by the boxed notes. Example 3.2.7 also demonstrates the structural function of dynamics: the first cycle of EN is marked piano in all the instruments; the second cycle is marked with a crescendo; the third cycle is marked mezzo-forte; the fourth cycle is marked with a crescendo; the fifth cycle is marked forte; the sixth cycle is marked with another crescendo; the seventh cycle is marked fortissimo; the eighth cycle is marked with a diminuendo; the ninth cycle is marked forte; the tenth cycle is marked with a diminuendo; the eleventh cycle is marked mezzo-forte; and the last cycle is marked with a diminuendo. The dynamics actually do not articulate the cycles of EN, rather, they divide the opening measures into a symmetrical structure, with a smooth continuation from soft to loud and back to soft (ex. 3.2.7). In this example, the opening measures have been re-barred according to the instrumental partitioning of E-n-n-n-E-E-n-E-E-n-n, with the beginning of each cycle articulated by the boxed notes in the example. These two schemesâ&#x20AC;&#x201D;the rhythmic pattern of 2-1-3-1-1-2-1 and the instrumental assignment of E-n-n-n-E-E-n-E-E-n-nâ&#x20AC;&#x201D;constitute a different number of elements, 7 elements for the rhythmic pattern and 11 elements for the instrumental assignment. The two schemes will then take a total of 77 elements (11x7) to complete an entire cycle, since it takes 11 cycles for the rhythmic pattern and 7 cycles for the instrumental assignment cycle to align. This important aspect of periodicity will be demonstrated below.

130

Ex. 3.2.7. Opening measures re-barred according to the instrumental partitioning of E-nn-n-E-E-n-E-E-n-n E Rhythmic pattern

Cor Anglais

° & & ¢

° Violin & Viola

B &

Horn in F

= Rhythmic pattern

Cor Anglais

° & &

Clarinet in A

Bassoon

Violin

° & B

Viola

Violoncello

Horn in F

B &

E E 1 ¸ ¸ ¸ œ œ œ œ N N

1 3

‰ Œ™

¸ bœfi

nœ >

‰ nœ¸ ‰ > œ

‰ Œ™

fi¸ œ nœ > ¸ nœfi#œ

‰ ¸ ‰ #œ nœ > > ‰ n˝œ nœ ‰

‰ Œ™

E E 2 1 3 ¸ œ œ œ™ N

1 1 2

¸ œ œ™ NN

Clarinet in A

Bassoon

¸ nœfibœ ¸ nœfibœ

nœfi¸b œ

Rapide (q = 152)

¸ n œfi n œ bœ.

¸ ¸nœ ™ #œfinœ

n>œ p

E 1 3 ¸ œ œ™ N

nœ ™ ‰#œfi¸

Œ™

œ œ œ NN

Œ™

∑

‰nœfiœ ‰ Œ ™ > ¸

nœ Œ

Œ™

∑

¸ œ N

‰ ‰ ‰ ‰

Œ ‰ nœ ™ #˝œ + bœ ‰ Œ ™ ‰

¸ ¸ ‰ nœ Œ ¸ œ ™ # œ n œfi¸ ' #œ ' ¸ ¸ ‰ nœ Œ ¸ œ ™ # œ œfi¸ ' #œ ' Æ nœ #œÆ ‰nœfi¸ ˝ Œ #œ¸ œ ™ ˝

#œ > Œ™

¸ Œ™ #>œ

E E E 1 2 1 3 ¸ ¸ œ œ œ œ™ N

1 1 2

¸ ‰ ‰ Œ™ œ ' ¸ ‰ ‰ Œ™ œ ' ¸ œ ‰ ‰ Œ™ '

¸ # œfi n œ

Œ ‰

¸ Œ™ n>œ

Œ ‰ Œ™

Œ ‰ b œÆ ¸ nœ ‰ Œnœfi¸ ˝

¸ Œ™ n>œ

+ nœ

‰ Œ™

131

1 1

E E E 1 2 1 2 1 ¸ ¸ ¸ 3 œ œ œ œ œ œ™ N NN

¸ #œ Œ ‰ Œ œfi¸ ¸ b>œ ™ ' nœ

œ œ N

¸ Œ ‰ Œ ‰

#œfi¸n˝œ

Œ™

‰nœfi¸nœ #˝œ nœ ‰ > + ‰ Œ ‰ ‰ nœ cresc. 2

‰ Œ ‰ Œ

¸ n>œ #œ + ‰ #œ

‰

1 3

¸ œ œ NN

‰ Œ

‰ Ó

E ¸ œ œ™ N

1 2

1 2 ¸ œ œ N N

∑

¸ n œfi nœ.

‰ ¸#œfi¸nœÆ ‰ œ ˝ ' Æ # œfi¸n œ ‰ n˝œÆ ˝ ‰

E 1 2 ¸ œ œ N

œ œ N

1 ¸ œ N

∑

¸ # œfi

nœ

> #œ

#œ Œ

¸ ¸ nœfi#œ

nœ ™

¸ Œ™ œ b>œ

n>œ ˝ œ ˝ Œ™

nœfi¸#œ n œfi¸# œ

˝ œ n˝>œ Œ ™ + Œ™ Œ™ œ™ cresc. Œ

‰ ¸ #œfi¸œÆ ‰ nœ ˝ '

¸ #œ Œ ‰ Œ #œfi¸ ¸ b œ ™ ' œ > ¸ Æ fi #œ œ ¸ > ˝ Œ ‰ Œ nœ nœ ™

œ œ NN

nœ

1 1

#œ ‰ fi¸ ˝ nœ #œ > Ó

¸‰ Œ nœ Ó

¸ nœfin œ¸

‰

¸ ‰nœfinœ¸ Œ '

fi¸ ‰nœ n˝œÆ Œ

Œ™

‰ ‰ #œ

¸ nœ

#œ

nœ ˝

#œ

nœ ˝

‰ ‰

+ ‰ ‰ #œ ™

Ex. 3.2.7. (cont’d) E Rhythmic pattern

Clarinet in A

Bassoon

Violin

Viola

Violoncello

Horn in F

œ™

œ œœ NNN

° n>œ ™ & ¢

n>œ bœ bœ ˝

° & Œ

‰ Œ Ó

#œ nœ ‰nœfi¸ nœ

B Œ™ ¢

Œ™

& Œ™

Œ™

E ¸ œ œ™ N

¸ fi¸ Æ nœ ‰ ‰nœ #˝œ Œ '

‰ ∑

Œ™ Œ

‰ nœ¸ bœ

¸‰ ‰ nœ '

‰ n˝œ bœ

nœfi¸

fi¸

+ #œ

n nœ ‰ œ nœ¸‰ ‰ ‰ ‰ ˝ bœ '

1 3

¸ nœ ‰ '

‰ Ó

‰ fi¸bœ nœ > > nœfi¸n œ ‰

¸nœ ™ #œ ¸nœ ™ #œ

¸> ‰ bœfinœ

+ ˙

EE ¸ ¸ œ œ œ œ œ™ N N N 2 1 2 1 3

Œ ¸ Œ ‰ nœ ¸ nœ œ ' b œfi nœ ' ∑

. nœ <n>œ ≈nœfi¸b˝œ ‰ Ó

‰

E 1

¸ œ œ™ NN

‰ ‰

+ œ ‰

° &

‰

E E 1 ¸ ¸ œ œ œ œ œ NN N 1 2

1 2

EE 1 1 2 ¸ ¸ œ™ œ œ œ N N 3

¸Œ ‰ bœfi¸bœ ™ ‰ > nœ nœ > ¸. ‰ Œ ‰ nœ¸ Œ ™ bœfi¸œ Clarinet in A & œfinœ ˝ ˝ #œ > # œfi¸n œ nœ Œ ‰ B n fi¸bœ ™ ‰ B ? ˝ ‰ œ ˝ Bassoon ¢ n>œ ° ¸. ‰ Œ Ó ∑ Violin & nœfinœ ˝ > fi¸b œ ™ ¸nœ ¸ n œ fi ‰ ‰ #œ ˝ #œ nœ Œ ‰ Viola B ¢ > + Ó ‰ ∑ Horn in F & #œ ™ ff ¸ # œfi œ

‰

nœ

¸ œ N

¸ bœ >

¸ œ‰ ‰ ‰ ‰ nœ

‰ nœ

¸ n>œ

> bœ ˝

E 2

bœ

E 1 3 ¸ œ œ™ N

‰nœfi¸#œ¸Œ '

‰ nœ ™

‰ Ó

Æ nœ ™ nœ¸ ‰ ‰ ‰nœfi¸#˝œ Œ > + ‰ ¸‰ Œ œ nœ

‰ nœ ™

¸ œ œ™ NN

‰ ‰nœfi¸n˝œ ™ nœ Œ™

#œ ™

nœ ™

Œ™

¸ ¸ b œfi # œ.

‰ ‰ ∑

‰ ‰ +

nœ

nœfi¸ ˝

#˙ ™ dim.

E ¸ œ œ N

1 3

1 1 2

œ œ œ N

¸Œ nœ >

‰n fi¸œ¸ œ #œ

¸Œ n>œ

Œ Œ

> ‰ ‰ ¸nœ ‰ Œ nœ¸ #œ b œfi > > n¸ œ ‰bœfi ‰ Œ ¸bœ ™ b>œ n>œ Ó Œ ˝ #œ ‰

cresc.

Cor Anglais

‰ Œ

‰ Ó ‰ Œ

E 1 ¸ 2 œ œ N

E 1 1 2 ¸¸ œ œ œ N N

1 3

‰n fi¸ ¸#œ ™ œ nœ >

‰ ¸#œ nœnœ n œfi

B Œ

= Rhythmic pattern

EE 1 2 ¸ œ œ

1 1 2

1 2

1 ¸ œ N

nœ ˝

‰ ¸‰nœfi¸bœÆ ‰ Œ ™ ˝ nœ ' nœfi¸# œÆ ‰ ˝ ‰ Œ™ Œ™

nœ ˝

‰ ‰ Œ™

132

Ex. 3.2.7. (cont’d) Rhythmic pattern

Cor Anglais

° & &

Clarinet in A

Bassoon

Violin

° & B

Viola

Violoncello

¸ nœfi ¸

‰ Œ

Clarinet in A

Violin

œ > ‰

#œfi¸# œ

Violoncello

Œ™

Horn in F

1 2

¸ ‰bœfinœ¸‰

E ¸ 3 œ œ™ N

¸bœ nœfi nœ

B &

nœ ™

fi¸

¸ n œfi n œ ™

‰ ‰ Ó Œ™

¢&

¸ ¸ nœfi#œ

1 1 2

œ œ œ N

‰ Ó

¸ œ N

¸ ¸ #œ ‰ ¸‰ #œfi¸nœ ‰ Œ ' nœ ' >Æ bœ ¸ ¸ Œ ™ ‰nœfi˝œ ‰ ‰ ‰ n>œ

¸ n œfi n œ ™

#œ Œ #˝œ nœ n˝œ > nœ #œ # œfi¸nœÆ ‰ ‰ ‰ ‰ nœfi¸nœ ™ ˝ ‰ nœÆ ‰ ˝ ‰ Œ ˝ + + Œ™ Œ™ Œ™ ‰ ‰ bœ ™ œ™ mf

133

Œ™

bœ

‰ Œ™

œ œ œ NNN

bœ ™ bœfi¸bœ

nœ

bœ ™

Œ™

‰

bœ ‰ Œ Œ™ fi¸n œ

‰ ‰ ‰nœ

bœ™ Œ™

dim.

Œ™

Œ ‰

nœ

‰

> nœ b˝œ Œ nœfi¸#˝œ #œ ™

œ '

nœ '

> nœ ‰

¸b œ ‰nœfi

nœ Œ

> nœ nœ

‰ Œ™

> #œ nœ n˝œ nœfi¸ ˝ #œ ™ Œ Œ™

E E 1 ¸ 2 œ œ

1 1 2

œ™

1 1

œ œ NN

∑

¸ ‰ ‰ Œ™ œ dim.

nœ

E 1 2 ¸ œ œ N

‰nœfi¸nœ ™

¸ ¸ #œ ‰ ¸‰ #œfi¸nœ ‰ Œ ' nœ '

¸ ¸ nœfi#œ

EE 1 3 ¸ œ œ™

> nœ ˝ Œ nœfi¸#˝œÆ #œ ™ nœ

Œ ≈Œ

‰ ‰ ˝œ nœ

‰ ‰ Œ™

E E 1 2 ¸ œ œ œ œ œ NN N 1 1 2

Œ™

¸ bœfinœ ™

‰

+ ‰ nœ ˝œ œ™

‰ ‰ ‰ ‰

∑

nœ

Viola

‰ b fi¸ ¸‰ ‰ ‰ œ¸ œœ nœ '

#œ n œ Œ ™ ˝

° &

¸ œ œ N N

œ œ œ N

E 1 3 ¸ œ œ™ N

1 2

fi¸ ‰ Œ ¸ ‰ ‰ Œ ™ bœnœ nœ' ‰ ‰ ‰ nœ¸ nœ nœ ' ' Æ > # œfi¸# œ #œ n œ œ fi¸ Æ b œ ˝ Œ™ ‰ n˝œ ‰ ‰ ‰ ˝ nœ ‰

¸ œ œ NN

Rhythmic pattern

E E

1 1 2

¸ #œ nœ Œ ™

‰ #œfi¸#œ

Cor Anglais

E E 1 3 ¸ ¸ œ œ œ œ™ NN 1 2

B &

Horn in F

E 1 ¸ 2 œ œ N

¸ nœfibœ'

+ b˙

E ¸ œ œ™ N

¸ nœfibœ'

‰ ‰

nœ Œ ˝

n>œ ™

‰ ‰ Œ

¸b œÆ

+ ‰ ‰ b˙

‰ ‰ bœ ™ >

‰ ‰ #œ¸nœ

‰ ‰ ‰ >œ ˝

nnOœ Œ ˝

‰ ‰

œ™

E 1 ¸ œ

‰ ‰

nœ Œ ˝

n>œ ™

Æ ¸# œ Œ nœfi ˝ Œ

n œfi ‰ nœÆ

E 1 1 2 ¸ ¸ œ œ œ NN

1 3

‰ ‰

Taking the EN scheme as the basic unit, Boulez then adds yet another level of periodicity, where every fourth composite element (counting both Eâ&#x20AC;&#x2122;s and Nâ&#x20AC;&#x2122;s) is articulated by the resonant instruments: if the cycle of every fourth element aligns with E, then the piano plays; if the cycle of every fourth element aligns with N, then the other three resonance instruments play (ex. 3.2.8). The result is that every 4 composite events (counting both the strings and the woodwinds) the resonant will align, and double the pitches taken from those instruments. The effect is a kind of composed-in sustain for these monophonic instruments, as if they possess the ability to resonant, equivalent to the pedal effect on the piano. Moreover, if the resonant instruments align with an 8th-note from the rhythmic pattern of 2-1-3-1-1-2-1, then those instruments would only play one 8th-note on that beat; if they align with a quarter-note from the rhythmic pattern, then they either play two 8th-notes, or play staggered entrances to articulate two 8th-notes in that beat; and if they align with a dotted quarter-note from the rhythmic pattern, then they either play three 8th-notes, or play staggered entrances to articulate three 8th-notes. Because these instruments are all resonant instruments, each instrument is sustained until the next entrance of the resonant instruments (ex. 3.2.8).

134

Ex. 3.2.8. Periodicities of the resonant instruments in alignment with the rhythmic pattern of 2-1-3-1-1-2-1 and the pattern of E-n-n-n-E-E-n-E-E-n-n 1

Viola

Horn in F

° Vibraphone & Marimba

n>œ

44 Rapide (q = 152)

fi¸ ¸ nbœœ n œ

‰

®°

Viola

Horn in F

Vibraphone

Marimba

Harp

° &

E E 1 ¸ 3 œ œ™

‰

1 ¸ œ N

4 2

b œÆ ¸ nœfi ˝

‰

+ bœ ˝

œ ˝

∑

nœ

E 1 ¸ œ

œ N

#œ

¸ nœfi nœ¸

‰

fi¸ ¸ nn œœ n œ

fi¸ nn œœ

‰

#œ '

‰ n#nœœœ '

¸ nœ ®

∑

E 1 ¸ 3 œ œ™ N

1 ¸ 2 œ œ N N

1 Œ

n>œ ‰

‰

#œ + ¸ #œ

œ ˝

‰

¸ nœ #œfi

¸ œ

‰

Ó œ ? nnnœœ

‰

nœ +

bbnœ>œœ ˝

˙˙˙ ∑

∑

1 ¸ œ N

1 œ

> #œ

E 1 ¸ 2 œ œ N nœ ˝

‰

E 1 ¸ 2 œ œ N

‰

bœ ‰n# œ˝œ

¸ ## œœ

¸ ‰ œœ

‰ nbœ˝œ

œ > nœ

fi¸ n¿œ n œ. n ˙

nœ

> #œ

œœ ˝

¸ ‰ œ

∑ ∑ ¸ œ ‰ œ

∑

135

n˙ b˙

œ œ

œ œ N N nœfi¸ #œ

‰

nœ + ¸ œ

¸ Œ ‰ bbœœ – —

‰

∑ ¸ bœfi n œ.

œ œ œ N N

1 +

¸ nœfi

E E

bnœœ – '— — Œ

nœ

‰

œ N

∑

∑ Lab

1 ¸ œ N

[missing?]

E E 1

#œ

¸ nnnœœœ °

œ N

‰

E 1 ¸ œ

∑

‰

¸ nœ

œ œ N

? œœœ ˝

Æ #œfi¸ n˝œ

¸ ‰ œ

œ N

etc.

¢&

& Piano

nœ

Mib Fab Sol§ La§ Ré§ Do§ Si§

– – b œ ? nœ n œ —‰ ' &

Piano

¸ #œfi nœ

œ™ N

¢& &

Harp

1 ¸ œ N

Ex. 3.2.8. (cont’d) 1 1

/ œ œ œ™ N

Viola

Horn in F

° ‰ nœ¸ ˙ & bnœœ ˙˙ > ¸ œ Ó n Marimba & ‰ ¢ bnœœ >

¸ nœfi nœ¸

Harp

œ œ œ N

‰

∑

#œ + #œ

¸ 3 œ œ™ N 1

œ œ œ N N N

nœ ˝

‰

œ ˝

‰

¸ nœfi

#œ

œ N

Viola

Horn in F

nœfi¸n œ

#œ

Marimba

Harp

Piano

∑

¸ ‰ bbnœœœ bnnœœœ > nnb œœœ

®°

– – – — — – — —

E E 1

œ N ¸ bœfi

> nœ

b fi¸ nœ

?# œœ

¸ ‰ nœ nn œœ

–— — – – —

Œ ‰

+ ¸ bœ

‰

¸ bœ b>œ

œ ˝

¸ œ

‰

‰ nnœœ¸ œœ #œ œ ∑

La§ Sib

E 2 ¸ œ œ N

¸ œ œ œ N N 1

‰

¸ #œfi nœ

#œ >

œ Œ

¸ œœ ‰ œ

‰

> ‰ n˝œ nœ

¸ #œ œ >

‰ nnœœ¸ œœ

¸ œœ ‰

‰ #œ ˝

œ ˝

‰ nœ¸ n#œœœ > ∑

∑ ∑

‰

Fa#

∑

136

¸ ¸ ‰nnœœfin#œœ

‰

¸ œ œ N N 1

E E 1 2 ¸ œ œ

#nnœ>œœ

–– Œ — – œœœ ‰ — — Œ ˝

fi¸ n#n œœœ

nœ

Œ + #œ

Œ Œ

Fa§ Do#

E 1 ¸ œ

¸ œ

bœ

‰

∑

‰

& n fi¸bœ— nnœœ — œœœ œ >

œ ˝

∑

° fi¸ n – œœ – Ó & nœ b >œœ— – nœ — œ — — * ° ¢& b œfi¸n œ >

+ #œ ˝

‰

nœ

∑

+ ˙™

bnœœ Œ

∑

Vibraphone

‰ nœfi ¸ nœ '

¸ 3 œ œ™ N 1

nœ

E 1 ¸ 1¸ 2 œ œ œ N N

∑

¸ œ œ™ N N 1

nœ

1 2

∑

= 1

E E 1 ¸ 2 œ œ

∑

– # œÆ – nnœœ – ‰ ˝—

fi¸ bnnœœœ

? Ó

Mi§ Fa§ Ré#

& Piano

Vibraphone

¸ & ‰ bnnœœœ ˙˙˙ > ?

œ œ N

n>œ

B <n>œ

nœ

∑

– Ó– —

— – —

° ø

E 1 3 ¸ œ œ™ N

‰

E E 1

nœfi¸ b˝œ

‰

¸ œ

Ó *

∑

‰ nœfi¸b>œ n œ ˝ # œ #œ

Ó ∑ ∑ ∑

œ N nœ

‰ bœ¸ bœ bœ ¸ n œfi > °

Ex. 3.2.8. (cont’d) E

1 ¸ 2 œ œ N

1 ¸ 3 œ œ™ N N

> bœ ˝

‰

Horn in F

‰

+ ¸ #œ

Vibraphone

Marimba

° & Ó ¢

& Harp

¸ nœ >

˙™

fi¸ #œÆ ‰ nœ ˝

¸ ‰ œœ

‰ nnœœ ˝

œ œ

∑

œ œ œ N N

E 1 ¸ 3 œ œ™ N

Viola

‰

Horn in F

‰

nœ ˝

1 E E

œ ‰ œ ˝

Mib Ré§ Si§

nœ ˝

# fi¸ # œ ‰ œ

#œ

‰

nœ

œ ˝

+ œ ˝

‰

E E 1

œ œ œ N

¸ ‰ bœfi nœ¸ '

œ ˝

‰

∑

∑ ∑

bnœœÆ ––

– —Ó ‰ nbnœœœ— '— ®°

∑

nœ¸–

– – —

b#œœ Œ n#œœ > ° ø

œ œ œ™

∑

– —

nn œœ #nœœ Œ

œ œ N N

œ œ œ N N

∑ ∑

1 E

∑ Do§

? Ó

Piano

Œ +

¸ nn>œœ œœ nnœ>œ ‰ nœ ˝

œ œ N

nœ

‰

1 ¸ 2 œ œ N

bœ ? b nœœ ‰ ˝

nœ ™

nœ n fi¸ ‰ œ ˝

Viola

E E 1 ¸ 1¸ 2 œ œ œ N

‰ nnœœ – ˝—Œ ®°

> nœ ‰ bnœœ ˝ °

Vibraphone

Marimba

Harp

Piano

° & Ó ¢

#œ ˝

nœ +

nœ

‰

‰ ¿fi¸ nœ¸ œ œ œ nn œ.œn >

œœ

& Ó

Ó ∑

nœ

E 1 ¸ 2 œ œ N

Lab Do#

E E 1 ¸ 3 œ œ™

¸b œ ‰ nœfi ˝

œ ˝

+ bœ ˝

‰

‰ bœfi#˝œ

¸ nbnœœfiœ

bœ

¸ œ

œ œ œ N N

> ‰ nœfi¸ n˝œ

? Ó

—Œ ? nnœ˝œ ‰–

E 1 ¸ 1¸ 2 œ œ œ N N

‰

> nœ ˝

∑

nœ ˝

‰

¸ ‰ œœœ

œ ˝

‰

¸ bnn œœœ > > b n œœ ‰ nœ˝

1 ¸ œ N # œfi¸

#œ ˝

œ N

n>œ +

‰

#nœœ

∑

n>œ bœ

–

nœÆ – – —‰ ˝

&bnœœ

∑

137

E 1 ¸ 3 œ œ™ N

E 1

œ N

nœ ˝

‰

¸ œ

‰

¸ ¸ ‰ #n œœ œœ

∑

¸Æ nœfi˝œ

fi¸ ¸ ¿ nœ œ nn œ.œ n >

‰

La§

œ ‰ ˝œ

‰

¸ nn>œœ

nœfi¸ #œ

Ex. 3.2.8. (cont’d) 1 E /

E 1 ¸ œ

¸ œ N

œ N

Viola

‰

Horn in F

‰

+ bœ ˝

‰

Marimba

° & <n> œ <n> œ ¢

& Harp

Piano

¸ œœ

> nœ

> ‰ #bnœœœ ˝

<n><n> œœ

bœ ˝

Vibraphone

‰

Fa#

∑

‰

œ N nœ nœfi¸ ˝

¸ 2 œ œ N N 1

‰

¸# œ nœfi ˝

+ bœ

¸ œ N

nœ

> nœ

œ ˝ Œ

Ó Œ

E 1 ¸ œ

n˙

¸ #n>œœ

‰

‰ #nœœ¸ n>œ

˙˙ ˙

œ™

œ N

‰

¸ #œ

nœ

œ ˝

‰

¸ œ

‰

¸ œœ œ

‰

∑

Fa§ Do§

Ó ∑

nb œœ #œ

‰

∑

nnœ>œ bœ ˝

Ó nnn œœœ

– – —

– — —

To summarize the very opening measures up to the first double line, there are several periodicities at work. Rhythmically the opening measures are constructed based on the pattern of 2-1-3-1-1-2-1 (ex. 3.2.1) where the 8th-note is the basic unit (ex. 3.2.2 and ex. 3.2.3). Each cycle of this rhythmic pattern is articulated by the horn, followed by an 8th-note delay with the violoncello pizzicato (the only instances in the opening where the violoncello plays pizzicato), cor anglais, violoncello pizzicato again, and then bassoon (ex.3.2.4). The space between each horn entrance lasts exactly eleven 8th-notes, since it takes eleven 8th-notes to complete the rhythmic pattern of 2-1-3-1-1-2-1. Instrumentally, the forces are determined by the partitions of the rhythmic pattern under the scheme of En-n-n-E-E-n-E-E-n-n (ex. 3.2.5), where Es are assigned to three instruments, and Ns are assigned to a solo instrument (fig. 3.2.6). Each cycle of EN is articulated first by the violin, then bassoon, violin, and clarinet; and every 3 cycles the instrumentation changes (ex

138

3.2.7). Furthermore, within each instrumental force derived from the pattern of E-n-n-nE-E-n-E-E-n-n, a single grace-note articulates every 3 notes to indicate the trichordal division (ex. 3.2.3). Taking the EN pattern as a basis and imposing a scheme on top, counting every 4 composite elements, the beginning of each 4-element cycle is articulated by the resonant instruments: if the cycle of 4 falls on E, the piano plays; if the cycle of 4 falls on N, then the other three instruments (harp, marimba, vibraphone) play (ex. 3.2.8). The opening measures are essentially composed under these multiple periodicities â&#x20AC;&#x201C; 7 elements of a rhythmic pattern which adds up to eleven 8th-notes, 11 elements of alternation between different instrumental forces (EN), 3 elements of pitch grouping (trichords) within each instrumental force, and finally, the composite cycle of 4 elements which determine the music played by the resonant instruments. This entire array of periodicities is represented and reduced into simplified notation in figure 3.2.9. In figure 3.2.9, each line represents one cycle of E-n-n-n-E-E-n-E-E-n-n, where Es are notated as notes and Ns are notated as rests. Each rhythmic value, both notes and rests, correspond to the rhythmic pattern of 2-1-3-1-1-2-1, and the basic unit here is the 16th-note instead of an 8th-note. This rhythmic pattern (or rhythmic duration) repeats itself every 7 lines: it takes 7 lines to complete a cycle to line up with the cycle of EN since this rhythmic pattern only has 7 elements, therefore 7 (elements of the rhythmic pattern) x 11 (cycles) = 77, which is equivalent to 11 (elements of the EN pattern) x 7 lines. This entire cycle of the rhythmic pattern is bracketed as D (duration) on the left side: the first 7 lines are labeled as D1 (duration cycle 1), the next seven lines are labeled as D2 (duration cycle 2), etc. The accents that appear on both notes and rests represent the composite cycle

139

of every fourth element. If the accent aligns with a note, then the accent is placed above the note; if the accent aligns with a rest, then the accent is placed below the rest. This cycle takes 4 lines for the accents and 11 cycles to align with the pattern of EN, since each line has 11 elements, and 11 (elements) x 4 (lines) = 44 , which is equivalent to 4 (accents) x 11 (cycles). This entire cycle of the accent is bracketed as A (accent) on the left side: the first 4 lines are labeled A1 (accent cycle 1), the next four lines are labeled A2 (accent cycle 2), etc. The Roman letters (ABC) above the notes and the Greek letters (αβγ- alpha, beta, gamma) below the rests represent the trichords and the cycle of 3, which Boulez o™ >j / œJ

≈

r / œ

‰ >

E1 A1

Į c

r / œ

‰ >

≈

r / œR

≈

‰ >

j / œ

≈

>br / œ

‰

r / œR

‰™ >

j / œ

≈

j / œ™

≈

ȕ c

100

D2 A3 E4

≈ >

r / œ

‰

≈

Ȗ >c

111

122

Į c

‰ ≈

j / œ™

≈

‰™

>br œ

≈

≈ >

≈

‰™

≈

j / œJ

‰™

‰ ‰™ ≈

r œ

a j œ J

≈ >

‰

b 2 17 x

r œ

j œ™

≈

≈ >

r œ

‰

≈

d 4 17 x

j œ™

>ar œ R

≈

‰

e 5 18 x

‰

r œ

j œ

‰™

f 6 18 x

‰™ >

≈ >

r œ R

r œ

‰

≈

j œ

r œ R

j œ

r œ R

‰ >

r œ

≈

j œ

≈

Ȗ c

j œ™ b

>ar œ R ȕ c

ȕ c

r œ

>ar œ R

j œ

r œ

Į c

≈

‰

r œ

j œ ™™ J

≈

Į >c

j œ b

j œ ™™ J a Ȗ

‰ >

a 1 18 x

≈

j œ

≈

>bj œ™

r œ R

≈

‰™

r œ R

≈ >

‰ >

>br œ

≈

c j œ

c r œ

j œ J

≈

Fig. 3.2.9. Periodicities of Section A

140

>aj œ J Į c

Ȗ >c

‰ ≈ >

r œ

r œ b

ȕ c

≈

>cr œ

j œ J

Į c

Ȗ c

‰™

ȕ >c

≈ >

c 3 17 x

g 7 16 x

≈

h 8 18 x

j œ

≈

‰

i 9 17 x

r œ R

j œ™

≈ >

≈

j 10 17 x

j œ

r œ R

‰

≈ >

ȕ c

>bj œ™

Į c

r œ

≈

‰

k 11 17 x

™ l 12 o

18 x

labeled “events.” The individual cycle within both the notes and rests takes 3 lines to line up with the cycle of EN, since each line has 5 elements for the notes and 6 elements of rests, therefore 5 (notes) x 3 (lines) + 6 (rests) x 3 (lines) = 33 (composite elements) which is equivalent to 11 (elements per line) x 3 lines. This large cycle of events is bracketed as E (event) on the left side: the first 3 lines are labeled E1 (event cycle 1), E2 (event cycle 2), etc. At the end of each line, a numerical and alphabetical labeling system is provided for the ease of identification, and a total number of rhythmic values are also provided to display the total duration for each line. These periodicities of rhythmic pattern, pulsation, pitch structure, and the alternation of instrumentation come to a full stop at the first double barline (2-R5), where the first interruption occurs and interrupts the periodicity, as all the processes are suspended. The first interruption also concludes Section A of the piece, where the first cycle of A (accents) and E (events) aligns (see fig. 3.2.9). Section B continues after two measures of interruptions (with the beginning and the ending of these marked by double barlines), and resumes all the activities from Section A, where the periodicities continue to unfold based on the process that began the piece. Figure 3.2.10 shows the underlying periodicities of Section B. Figure 3.2.11 illustrates a summary of the instrumentation in accordance with various periodicities in Section B. Example 3.2.12 shows the first three instrumental cycles of E-n-n-n-E-E-n-E-E-n-n, where E’s are assigned to clarinet and violin, and N’s are assigned to cor anglais, bassoon, horn, vibraphone, and marimba. Each cycle of EN is articulated by viola, each cycle of the rhythmic pattern is articulated by violoncello, and each cycle of accents is articulated by

141

the piano and harp. The next two and a half cycles of EN, similar to those in Section A, alters the instrumental assignment. During these two and a half cycles, Eâ&#x20AC;&#x2122;s are assigned to clarinet, viola, and bassoon, while Nâ&#x20AC;&#x2122;s are assigned to violin, violoncello, horn, vibraphone, and marimba. Each cycle of EN is articulated by harp, while the clarinet trills articulate the rhythmic cycle (ex. 3.2.13). The next 2 cycles of EN resume the original instrumentation that started Section B (ex. 3.2.14), while the following one and a half cycles starting at R9 (ex. 3.2.15) return back to the instrumentation of example 3.2.13. Unlike Section A where the instrumentation changes every 3 cycles, in Section B the instrumental forces change progressively faster, starting with 3 cycles of EN, then two and half, 2, and finally down to just one and a half cycles of EN.

142

>ar / œR

‰™

≈

‰

133

>br œ

D2 E5

c r / œ

≈ > Į

j / œJ

≈

144

155

r / œR

166

D3 A5

‰™

≈

c r / œ

>br / œ

≈

>aj œ J

>br œ

‰™

j œ J

≈ > Į

j œ

≈

>ar œ R

c j œ™

r œ b

c j œ™

r œ b

>ar œ R

c r œ

j œ ™™ J

r œ

‰

>cr œ

‰™

r œ

≈

>cj œ

r œ R

≈

≈ >

≈

‰ >

≈

q 17 17 x

≈

r 18 17 x

‰ >

s 19 18 x

‰™

≈

‰

≈

Fig. 3.2.10. Periodicities of Section B

Rehearsal Numbers EN cycle length E N

En: cycle of 7 2131121: cycle of 11 Accent: cycle of 4

R5 3 Cl., Vln, CA., Bsn., Cor., Vib., Mar. Vla. Vc Pno. Harp

2-R7 2.5 Cl., Vla, Bsn. Vln., Vc., Cor., Vib., Mar. Harp Cl. trill Pno. Harp

R8 2 Cl., Vln, CA., Bsn., Cor., Vib., Mar. Vla./Pno. Vc Harp

R9 1.5 Cl., Vla, Bsn. Vln., Vc., Cor., Vib., Mar. Cl. 16th-notes Cl. trill Pno. Harp

Fig. 3.2.11. Instrumental and Periodicity assignment in Section B

143

t 20 18 x

>cr œ

p 16 17 x

j œ J

‰

≈

o 15 17 x

≈

n 14 16 x

>aj œ ™™ J

m 13 18 x

‰™

≈ > Į

j œ

c j œ

‰ >

c r œ

‰

≈

‰™

≈

r œ R

r œ

>bj œ

r œ R

c r œ

r œ R a

‰

j œ

‰™ > Ȗ

r œ

‰

≈

‰

c r œ

≈ >

≈

‰

210

221

≈

j / œJ ™™ a

‰ > Ȗ

≈ > Į

199

≈

j / œ b

‰™ >

‰

188

j œ J

≈

>cr / œ

177

‰

c r œ

o™

™ o

u 21 16 x

Ex. 3.2.12. The first 3 cycles of EN with annotated rhythmic patterns in Section B 1

E /

Periodicity

Clarinette en la

Violon

Cor anglais

° &

Violoncelle

‰

¢&

‰

nœfi¸n˝œ

#œ

nœ ˝

#œ

¸ #œfin˝œ

# œfi¸

#œ nœ

nœ

¸ fi¸ ##œœœ.

‰

¸ œ 1

E E ¸ 2 œ œ

‰ bœ nœ nœ ‰ ‰ nœfi¸ bœ nœ b>œ nœ > > n>œ ¸ ‰ nœ¸ #œ¸ ‰ ‰ fi¸ nœ¸ #nœœ œ # # œ ' nœ > nœ ' ' >'

nœ œ ˝ ‰ ¸‰ nœ ¸ œ ‰

‰ ¸Œ bœ ¸‰ ¸‰ nœ bœ

°B ¢

5 Assez rapide, stable (q = 138) ¸ ¸ ‰ & ‰ ##œfiœ¸ œ.

∑

nœ B nœ

œ œ

E œ

nœfi¸n˝œ

œ ˝

‰

fi¸# œ ‰ bœ ˝ #˝œ

‰

¸ #œfinœ

‰

œ ˝ ˝

‰ ‰

fi¸ ¸ Œ ##œœ nœ.

‰

¸nœ n œfi .

¸Œ

Œ n>œ #>œ

‰

Œ ¸ nœfi ¸ # œ nœ.

œ œ

bœ nœ ‰ Œ

‰ n fi¸nœbœ

¸ n ‰ Œ # œœ '

‰ fi¸nnœœ¸ #œ '

‰

nœfi¸n œ # œfi¸

#œ nœ

nœ #œ œ

& nnœœnœ >'

¸ #œfinœ

n>œ #>œ Œ

‰ Ó

∑

bœ bbœœ >'

#œœ

#œ

‰ #œ n# œœ >'

n nœœ nnœœ #œ '> >' # œ bœ nœ & n#œœ Œ >' >'

n nnœœœ Ó >'

144

nœ ‰ #œfi¸#œ¸ >

nœ ˝

‰

nœ. ‰ #œ¸ ‰ ‰ ¸nœ nœ Œ ¸ ˝ b œfi nœ >

#œœœ ‰ Œ

¸ ‰ bœfinœ¸nœ bœ œ¸ ‰ >

¸ ‰ nœ. ‰ #œ¸ ‰nœfi #œ¸ ‰ nœ nœ¸ ‰ ˝ . n œ. ˝ >

∑

Mi§ Sib

> nœfi¸ nœ bœ œ nœ ‰ #˝œ œ #œ¸ ‰

nœ. ˝

n>œ b>œ Œ

pizz.

nœfi¸

Piano

> b>œ

& Harpe

¸ bœfinœ

‰

Marimba

¸ œ

° &

Vibraphone

Alto

¸3 œ œ™

¸ ¢& b œfi¸nnœœ' >

Basson

Cor en fa

¸ œ 1

n#œœ

‰ #œfi¸nœ

¸ ‰nnœœfi#œ¸ œ > > bœfi¸n œ œ ‰n œ ˝

∑ ∑

¸ œ ‰ œ ˝

‰

nœ

Ex. 3.2.12. (cont’d) 1

E / œ

Periodicity

œ™

œ œ

[mistake?}

¸3 œ œ

Clarinette en la

Violon

Cor anglais

‰ nœnœ fi¸nœbœ ‰

¸ ¢& ‰ nn œœ '>

‰ #œ¸ ¸ nœ¸ ‰ n œ n œfi# œ ' '

° Œ & B Œ

Basson

¸ nœ nœfi# œ >

& Œ

fi¸ nnœœ

Marimba

Harpe

b >œ ˝ ‰

& Œ

Vibraphone

#œ °B #œ ‰ ¢

n œfi¸nœ

> nœ œ ‰ ˝

¸ œ ‰

Œ ¢&

Violoncelle

nœ

¸ #œfi#œ

Cor en fa

Alto

nœfi¸b œ

> ¸ ‰ nœ ‰ # œ. ˝

b œfi¸

nœ

‰ nœ ˝

nœ

¸ #œ ‰ nœ '

¸nœ nœfi ¸#œ #œfi nœ

> nœ ‰ ˝

<n>œ #œ

nœfi¸#œ

¸ nœfi#œ nœ

¸ #œ. ‰ Ó

∑

# œ n#œ nœ œ Ó

& Ó

¸ ‰ #nnœœœ <n>nœœœ >

? Ó

bnbœœœ nœ >' & ‰ n#n œœœ Œ >' &

Piano

1 ¸ œ

¸ ‰#nœœ œœ > b>œ œ ‰n ˝œ œ

¸ #œ ‰ ¸ ‰ nœ nnœœ ' '

∑

¸bœ nœfi nœ

Œ ‰

¸#œ nœfi ˝

#œ œ > ‰ nœfi¸˝ nœ nœ

‰

bœ Œ

> ‰ fi¸nœ Œ nœ ˝

‰ n˝œ œ

‰ n fi¸ ¸ Ó n œœn œ . >

¸‰

#œ ¸#œ n œfi . nn œœ

> ‰ #˝œ

nœ œ >

fi¸ > ‰ nœ n˝œ Œ

. b œ. n œ #œ. > >.

Sol#

> bœ

∑

#œ nœ Œ >

n œ #œnœ nœ Ó

bœnœnœfi¸ nœ #œ ‰ nœbœ #œ nœ nœnœ ‰ nœnœ nœ nœ '

¸2 œ œ

bœnœbœ Ó

‰ #œ¸

‰ n œ¸ # œ. > ‰ n#œœ¸ >

° nœ > > & nœnœ bœ Œ Œ

¸ 1¸ œ œ

¸ nœ. ‰ nœ¸ nb œœ >

b œnœn œ # œ

∑ Œ

‰

∑ ∑

¸ ‰ #œ Ó n>œ n œfi¸ nœ

> nœ ˝

‰ nœ. nnœœ >.

‰#œfi ¸Ó nn>œœ

¸ œ

¸ bœ bœ >

Fa§

¸ Œ nnnœœœ > ∑

∑

145

Ex. 3.2.13. The next 2.5 cycles of EN with annotated rhythmic patterns in Section B E /

Periodicity

1 ¸ œ

N N

œ™

1 ¸ œ

œ œ

N N

œ™

Alto

° B

Cor anglais

Basson

Cor en fa

¸ nœfiœ

nœ ‰ ‰ Œ

‰

> nœ #œ nœ nœ fi¸ n>œb œ #œ nœ bœ ‰ nœ nœ #œ #œ ‰#œ nœ Œ

& #œfi¸n˝œ

‰ ‰ Œ

‰

¸ ¸ nœ #œ ‰

‰ ‰ Œ

‰

#œ nœ ˝ ˝

nœfi¸œ

° &

‰

#œ

‰

nœ nœ œ #œ >

Violon

#œ n>œ

bœ

arco

nœ

‰

Vibraphone

‰

¸ #œ. ‰ #nœ. œ n œ. >

Marimba

‰

¸ ‰ #œ n œ nœ n >œ

Violoncelle

Clarinette en la

Harpe

‰

‰ #œfi¸#œ¸ Œ

¸ œ

‰

nœ ˝

‰

#œ ‰ #˝œ Œ

¸ n>œ

Ÿ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ¸ ‰ ¸ & bœfinœ¸ bœ œ œ œ

Fa#

∑

¸ ‰ bœ nœ. > &

‰

#n œœ

Fa§

¸ #nœœ n œ. >

nbœœ

∑

‰

‰ Œ

b>œ ˝

‰ Œ

‰ ‰ 7

¸ nœfi

¸ Œ nœ

nœ ˝ #œ ¸ nœfi

¸bœ nœ

146

nœ

bœ ˝

nœ

nœ.

‰

œ ˝ ‰

# œ. ¸ #œ

# œfi¸ bœ

nœfi¸n˝œ

‰

¸nœ #œfi ˝

nœ

‰

#œfi¸#œ

nœ

‰

fi¸nœ ##œœ ˝

‰

¸ # œfi# œ

nœ

¸‰ # œ.

nœ¸ nœ #œ

Ÿ~~~~~~~~~~~~~~~~~~~~~~~~~

bœ n œ. # œ. nbœœbœœ

Œ nœ nœ >' ‰

‰

¸ ¸ bœfibœ nœ bœ ™

nœ bnœœ >'

‰

#>œ ˝

∑

##nœœœ nnnœœœ

¸ Œ nœ

fi¸

‰

nœfi¸

‰ #nœœ ¸ Œ n œ.

‰

fi¸

bœfi¸nœ n œ #œ

œ ˝

‰

& Piano

‰

¸> nœ #œfinœ n> œ

#>œ n>œ ‰ Œ

#œ

Si§

## œ œ # œœ' >

œœ ‰ Œ œ >'

Ex. 3.2.13. (cont’d) 1

Periodicity

Basson

œ œ œ

> n œfi¸n œ >

Violoncelle

Vibraphone

Marimba

Clarinette en la

nœ ‰ nœ

œ™

#œfi¸#œ

¸ 1¸ 2 œ œ œ

¸ œ 1

¸ œ œ™

1 3

> #œ #>œ n>œ n>œ #œn>œ ‰ ‰ #œ nœnœ ‰

#œ ‰ nœnœnœ#œ#œ

‰

nœ ˝

#œfi¸nœ

n œfi¸n œ

œ ˝

‰

nœ ‰ b˝œ ˝ ‰

> fi¸ ‰ nœ #˝œ nœ

nœ ˝

¸ ‰ #œ

‰ nœ¸œ¸ ‰

‰ #œfi¸#œ¸ > #œ

¸ bœfi ˝

nœ

nœ ˝

‰

nœ ‰bœfi¸nœ ˝ ‰ ˝

‰

. & ‰ n˝œ Œ

. ‰ nnœ˝œ

bœ. ˝

‰

. ‰ n˝œ Œ

& Ó

‰ nœfi¸nnœ˝œ

bœ ˝

‰

nœ ‰ ˝

‰ #bœœfi¸nœ. ˝ n œ.

Ÿ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ¸ ‰ fi¸bœ¸bœ œ œ œ‰ Œ nœ

#œ

B œ

‰

<Ÿ>~~~~~~~~~~~ Œ & œ

n œfi¸

nœ

bn œfiœ¸ nœ

¸ #œfi#œ

‰

& Ó

nœ. n˝œ

& Œ Piano

nœ ˝

B Œ

nœfi¸# œ

& Harpe

¸ œ

nœ n>œ #>œ n>œn œ ‰

‰ n˝œ

& œ ¢

¸ œ 1

‰

nœ nœfi¸

nœ ° Œ Cor en fa & #œ

Violon

nœ ˝

& Œ ¢

N N

° Alto B Œ

Cor anglais

##œœ ' nœœ ‰ > n >œ' n#n œœœ # œ >' n>œ'

nœfi¸

n œfi¸ nœ nœ

Sol# Do§

¸ ‰ #nnœœ. œ ¸ #œfi bœ

bœœ ‰

nœ # œ nœ

nnœœnœ 'nœ > >'

œ #œ ˝ ‰ ‰ nœ¸ ˝

‰

‰ nœ #œ #œ

#œ œ ‰ ‰ ˝ nœ ˝ ˝

‰

#œ #œ ‰ nœ

∑ nnbœœœ >'

#bœœ ‰ Ó >'

#nnœœœ nœœ >' n ' >

¸ nœfin˝œ

nœ ‰ Œ ˝ nœ

<n>œ

#œ

nœ.

¸ ‰ fi¸#nœœ ‰ nn œœ .

œ‰ Œ nœ ‰ ‰ Œ

‰

‰ Œ n œ nœ

¸ ‰ #nœœ <n>œœ ¸ n œfi > œ

nœ.

n œ. #nœœ

¸ œœ ‰ ‰ œ

nœ. ‰ ‰ nœœ Œ

‰

Sol§ Sib

‰

nnœœ #n œœ

147

Ex. 3.2.14. The next 2 cycles of EN with annotated rhythmic patterns in Section B 1

E Periodicity

Clarinette en la

Violon

° &

¸> œfibœ

¢&

¸ ¸ œfi# œ œ

° Cor anglais &

‰

b>œ

¸ œ

bœ bœ nœ nœ

‰

Æ bbœœ ˝

¸ nœ

b œfi¸

‰

¸ ¸ nœfibœ

‰

>œ ˝

#œ

‰

#œfi¸#œ¸

nœ ˝

> nœ

‰

¸nœ #œfi ˝

#œ Œ

‰

¸ nœ nœfinœ¸

#œ œ ‰ ˝

‰ Œ

‰

¸ fi¸ ¸ bœ n œ. n#œœ nn œœ >

‰ #œ¸ ‰ .

#œ Œ

‰

b>œ #œ¸ ‰

¢&

‰

¸ ¸ nœfi#œ

‰

¸ nœ >

Vibraphone

‰

fi¸ ¸ nnœœ # œœ

‰

¸ n œ. ¸ œ nn>œ

Alto

∑

Violoncelle

Piano

Harpe

bbœœ & nnœœ >' ‰ >'

Cor en fa

nœ

bœ ‰ Œ bœ nœ nœ nœ

nœ '

bœ

¸ #œ. n>œ

‰

n fiœ¸

∑

b nœœ ## œœ # œ

∑

> ‰ bnœœ ˝ n œ. ˝ ‰

œ > nœ nœ ˝ ‰

fi¸bnœœ #n œœ

n>œ

bpizz. œ nœ ‰ Œ

œœ

∑

nœ.

Fa#

148

∑ Œ

# œ.

Œ ?

#n>œœ

n >œ nœ

¸ œœ ‰ œ œ ˝ ‰

‰

nœ b œ nœ

∑

n#œœ

nnœœ

? #nœœ

Ó Œ

‰

nœ ‰ ˝

#>œ

‰ nœ nœ bœ bœ bœ nœ ‰ #œ¸ nœ¸ n œ nœ ' '

∑

‰

N N N

¸ nœ nœ '

¸ nœfibœ

¸nœ nœfi ˝

Marimba

¸ œ

‰

fi¸ bbœœ nœ¸

¸ 2 œ œ

nœ bœ ‰

Basson

‰

¸ 3 œ œ

Œ Ré#

nœ

Ex. 3.2.14. (cont’d)

Periodicity

¸ œ 1

œ™

Violon

Cor anglais

Basson

Cor en fa

Vibraphone

Marimba

Alto

Violoncelle

Piano

° &

‰

¢&

¸ ‰ nœfi¸n#œœ '

1 ¸ œ

‰ bœ œ bœ nœ bœ bœ nœ

nœ

¸ nnœœ '

‰

¸ nœ nœ '

‰

bœ ˝

‰

¸ bœ ‰ >

‰

¸ ¸ bœfibœ

‰

& nnœfiœ¸

¸ ‰ bb œ. œ

‰

fi¸ & nbœœ bœ¸ ‰ >

‰

° & B ¢&

nœfi¸

¸ bœ ‰ >

nœfi¸

¸> nœfibœ

b>œ

bœ

nœ >

‰

¸ n>œ nœ ˝

‰

¸ bœ. n œ n>œ. b œ. > ¸ bœ nœ b œ >

n>œ

‰ ‰

∑

nœ nœ

∑

‰

∑

fi¸ nbœœ ‰

# >œ #œ ˝

‰ ‰

Fa§

Œ Œ

#nnœœœ > n>œ bœ

‰ Lab

œ™

N 1

bœ ‰

‰

¸ bœ

nœ

¸ > œ nœ

bœ ˝

bœ

n>œ ˝

¸ nœfinœ

¸ #œ n>œ

nœfi¸ nœfi¸

fi¸ nbœœ bnœ œ

fi¸ nnœœ bn œœ

‰

# œ.

¸ œ

¸ nœ nœ¸ ‰ . # œ. >

¸ ‰ # œ b >œ

Œ fi¸ nbœœ

bn œœ

1 ¸ œ

& Harpe

¸ bœfinœ

Clarinette en la

¸ œ nœfinbœ .

‰

nœ nnœœ n œ

‰

¸ nœ ‰

bbœœ∑ Œ bœ. ˝

nn>œœ

> ‰ #bœœ

¸ œœ œœ ˝

149

Ex. 3.2.15. The next 1.5 cycles of EN with annotated rhythmic patterns in Section B E /

Periodicity

Alto

1 ¸ œ

> ‰ nœ ˝

œ ˝

¸‰ nœ œ

‰

¸¸ #œ nœ ‰

¸ #œfinœ

bœ

#œ œ ‰ ˝ ˝

¸ #œfi#œ nœ

∑

‰

Cor en fa

¢&

‰

¸ ¸ ¸ nœfi#œ œ ‰

Vibraphone

∑

‰

fi¸ ¸ ¸‰ nn œœ œ # œ

Marimba

‰

Piano

nœ ˝

‰

Harpe

> ‰ #˝œ

‰ Œ

∑

nœfi¸

n œfi¸

. .

¸Œ

fi¸ #œ nn œœ# œ

#œ

nœ ‰ Œ #>œ > n>œ > #œ ‰ Œ

¸ #œfinœ

‰

bœ œ ‰

nœ œ

‰

¸ nœ. ‰ ‰

œ œ

> n>œ n>œ ‰ #œ

> ‰ ‰ bœ ˝

n>œ > ‰ Œ #œ

‰ Œ

° &

¸ bœfi#œ¸nœ nœfi¸

œ œ

nœ œ n˝œ ‰ ˝ bœ

¸ ¸#œ nœfi#œ

nœ ˝

Clarinette en la

¸ nœfinœ

¸ #œfiœ

Violoncelle

> ¸ & nœfi¸nœ #œ¸ ‰ ‰ #˝œ B

Violon

œ œ

> fi¸nœ ° fi¸ B nœ nœbœ nœ nœ ‰ ‰ #œ b>œnœ nœ#œnœ Œ

Cor anglais

Basson

E E

œ™

n>œ ‰ Œ fi¸

# ‰ œ #œ nœ œ¸‰ ‰

#œ ˝ ‰ ‰

‰nœfi#œ ¸

¸ Œ

¸n œ n œfin œ'

#œfi¸#œ

fi¸ nbœœ

¸#œ Œ # œ. . # œ.

‰ ¸ ‰ nœ¸‰ nœfi¸ ¸‰ #œ¸ ¸ ‰ #œ œ n œ #n œ #œ

fi¸ ¸‰ #nœœ ##œ œ

¸ nœfi#œ nœ nœ

Ÿ~~~~~~~~~~~~~~~~~~~~~~~~~~~ Œ bœfi¸ bœ ¸nœ nœ nœ œ œ œ

bœnœfi¸nœbœ Œ

∑

‰

∑

? #bœœ ˝

∑

>Æ ‰ #nœœ

Ó > nbœœÆ

Œ nnœœ >' >Æ nœ Œ

150

‰ Œ #œ '> > #œÆ nœ‰ Œ

‰ #nœœ >' ‰&

>Æ nnœbœœ n#nœœœ >'

>Æ

? bnœ œ

#œfi¸

¸ nœ ¸ nœ

¸ ‰ bœ ¸ nœ nn œœ ' >'

‰

œ ‰ ˝ ‰

#œ ˝

nœ. ‰ ‰ ˝

¸ n œ.

Ó ‰ bœnœbœ >

¸ bœnœbœ Œ œfin œ

∑

¸ #œ ‰

‰ Œ

bœ nn œœ >'

n>œ ‰

‰ #n#nœœœœ >' ‰

In Section B, Boulez also imposes a permutation structure on the dynamics based on the EN cycle. The first three sets of E’s are assigned to the dynamic markings of ff, f, mf (R5 to R6-3 in the clarinet and viola parts); the next three sets of E’s are assigned to the dynamic markings of mf, ff, f (R6-4 to R8 in the viola part); the next three sets of E’s are assigned to the dynamic markings of f, mf, ff (R8 to R9-4 in the clarinet and viola part). The N’s also go through a permutation of dynamic markings. Figure 3.2.15 illustrates the dynamic scheme of Section B. Notice that both the piano and harp do not participate in this dynamic scheme, since their dynamic markings are determined by the instruments with which they align. Rotation E:

mf mf ff

f f

Rotation N:

mf ff

mp p mp p

mp p mp p p

Rehearsal Instruments numbers R5 Cl., Vln., Vla., Vc. Ca., Bsn. Vib., Mar. Horn R5-2 Cl., Vln., Vla., Vc. Ca., Bsn. Vib., Mar. Horn R6 Cl., Vln., Vla., Vc. Ca., Bsn. Vib., Mar.

mp mp mp E ff

nnn

p mp mf p mp mp p p

EE ff

151

p mp mp p p p p mp

EE ff

p p p p mp mf p p?

Horn R6-2

R8-2

R9-3

Vla. Vib., Mar. Ca., Bsn. Horn Vla. Vib., Mar. Ca., Bsn. Horn Vla. Vib., Mar. Ca., Bsn. Horn

Cl., Vln., Vla., Vc. Ca., Bsn. Vib., Mar. Horn Cl., Vln., Vla., Vc. Ca., Bsn. Vib., Mar. Horn

Vla. Vib., Mar. Ca., Bsn. Horn

p mp mp p f p mp f mf p p p p mp mf p ff mp

mf mf

p mp p mp f mf p mf mp p mp mf mf mp mp p f p

mp mf

mp mf mf p f mp p p p p p mp mf p p p p mf

Fig. 3.2.16. Dynamic scheme of Section B Section B comes to a full stop at R10, where the second interruption occurs for a total of four measures from R10 to 3-R11. Section C continues from 2-R11, and all the periodicities resumes. Figure 3.2.17 shows the underlying periodicities of Section C. In this section, there are only three lines of EN cycle, labeled as V22, W23, and X24. The first cycle (V) is presented normally in the music, with no rhythmic transformations. The

152

second cycle (W) is transformed rhythmically through a simple permutation, as shown in example 3.2.18. j / œJ

≈ > Į

c r / œ

‰

232

243

r / œ

254

‰™

≈

≈ >

‰™

≈

≈ > Ȗ

r œ R

j œ™

j œ J

≈ > Į

c r œ

‰

>br œ

c j œ

≈

‰

>cj œ

r œ

r œ R

r œ b

j œ J

c j œ™

Ex. 3.2.18. Rhythmic transformation in Section C r / œ

‰

Doubles the value (x2)

j / œ

≈

‰

r œ

j œ™

≈

r œ

j œ

≈

‰

j œ

œ™

‰

j œ

‰

Rhythmic transformation and permutation

j œ

/ œ

Œ™

r / œ

‰™

≈

j / œ

Œ™

‰

œ™

Rhythmic pattern for R11 to R12 E

Rhythmic pattern for X

Doubles the value (x2)

/ œ™

Œ™

Rhythmic pattern for R12 to R13

j œ œ™

Œ™

j œ

r œ

‰

r œ

j œ™

≈

j œ

œ™

‰

j œ

Œ™

œ™

Rhythmic transformation and permutation

j œ

œ™

œ™ j œ œ™

œ ‰

œ™ 153

Œ™

≈

Fig. 3.2.17. Periodicities of Section C

Rhythmic pattern for W

‰™ >

≈

o™

‰ >

v 22 18 x

w 23 17 x

≈

™ o

x 24 17 x

This rhythmic permutation and transformation is an important departure from the rhythmic pattern of 2-1-3-1-1-2-1. The original rhythmic values are noticeably obscured, but the pattern is more or less recognizable since these numbers operate in a mod-3 permutation. Unlike Messiaen who increases each rhythmic value by an added value, Boulez treats these values in a closed operation. Because of these transformations and permutations, the total rhythmic value for each line will ultimately be different. Example 3.2.19 demonstrates the horn part in Section C with annotated periodicities and the transformed rhythmic pattern. Ex. 3.2.19. Horn part with annotated periodicities in Section C V E Periodicity

1 ¸ œ

¸ nœfinœ

bœ

Cor en fa

Periodicity

Cor en fa

¸ & œ & <#>œ

N #œ >

¸ # œfin œ

nœ

œ™

N œ

nœ œ

N œ >

N ¸

E E 1

‰

nœ ™ >

¸ #œ

‰

œ™

N ¸ nœfi#œ

X E 3

N bœ >

œ 3

N Œ

œ™

N ¸ #œ

W E

‰

¸ œ

¸ ¸ # œfinœ

¸ œ

¸ œ œ Œ

[mistake?] 1 3

œ œ œ™ ¸ ‰ œ

‰

1 ¸ œ

œ™

¸ n œfin œ

1 ¸ œ

¸ ¸ b œfin œ

1 ¸ œ

[mistake?]

E 2

œ ‰

¸ nœ

¸ & œ œ & ‰

œ™

Cor en fa

Periodicity

œ™

nœ

œ œ œ

bœ œ

Example 3.2.20 demonstrates the whole ensemble with annotated periodicities in Section C.

154

Ex. 3.2.20. Annotated periodicities in Section C E

Cor anglais

Basson

¸ bœfi

¸bœ. nœfi ˝

Violon

¸ bœfi

Violoncelle

Cor en fa

Vibraphone

Marimba

¸ n œ. nœ

° &

¸ n œ.

bœfi¸

Alto

nœ

n œfi¸ bœ .

° &

nœ ˝

¸ bœfi

nœ °

Piano

œ bœ #œ Œ

‰

bœ bœ nœ > nœ > > Æ nœ bœ n>œ ˝

‰

¸ nœfi œ œ n œ bn >

¸ œ nnœ '

‰

Æ nnœœ ˝

nœ b œ #œ Œ ‰

bœ '

œ ' > n>œ

¸ nœfinœ

bœ

nœ bœ #œ Œ

¸ nœ ‰

* bœ

nœ b œ

nœ. ˝

n>œ

fi¸bœ nnœœ #œ

bœ #œ bœ Œ

n>œ #œ ‰ n>œ b>œ >

bœÆ ˝

¸ bœfi

‰

nœ

‰

nœ

¸ nœ

nœ

‰ bœ

¸ œ

¸ nœfi n œ bn œ œ

‰

nœ

œ™

¢&

Harpe

° &

Clarinette en la

Periodicity

¸ œ

‰ b>œ #>œ b>œ #>œ

‰

¸ bœfinœ¸

¸ ¸ b œfin œ

> > n>œ ‰ bœ bœ nœ b>œ nœ °

‰

¸ #œfi

¸ nœ

‰

1 ¸ œ

œ ¸nœÆ #œfi ˝ ¸œ #œfi #œfi¸# œÆ

¸ #œfi

¸ nœ '

> > bœ nœ n>œ n>œ bœ ‰

‰

‰ #œ

¸ nœ '

nœ # œ #œ n œ

‰

#œÆ ˝

n œ #œ #œ #œ nœ nœ

#œfi¸

‰

¸nœ #œfi

nœ #œ # œ #œ nœ *° * ° ‰

Œ nœ #œ ‰

Si§

œ fib¸n œ

nnœœ #œ

nœ bœ nœ nœ bœ nœ bnœœ nœ

¸nœ #œfi

#œ ‰ &

‰ n œ nœ

¸ nœfi bœ nœ

#œ œ bœ ‰ Œ ?

155

Ex. 3.2.20. (cont’d) ¸ œ

& œ™

Periodicity

Cor anglais

° & Ó

Violon

° & n œ bœ nœ b >œ > > b >œ b>œ n>œ #œœ'

Violoncelle

Cor en fa

B Ó B n>œ #>œ n>œ #>œ n>œ n>œ nbœœÆ ¢ #œ ¸ #œ

& nœ ™ >

¸ œ

Marimba

° & Ó

bn>œœ

bb >œœ

¸ bœfinœ¸

#œ n œ #œ nœ Œ

¸ bœfinœ

Œ #œ n œ ‰ Œ Œ

¸ nœfi

¸ œ n#œ '

‰

nœfi¸ #œÆ

#œ

¸ nœfi#œ

nœ n œ nœ nœ

#>œ

¸ nœfi#œ

nœ

n œ ? b#>œœ

bnbœœœ >

Œ #>œ nbn œœœ nœ > > Œ

nœ nœ nœ

#œ >

nœ

#œ n>œ

‰

nœ

156

#œ nbœœ

‰ nœ nœ

#œÆ

#œ

nœ > > > n œÆ #œ n œ Œ #œ ##œœ n>œ #œ nœ #œ nœ nœ

¸ #œfinœ

nœ

#œ

nœ nœ nœ bœ

#œ nœ

nnœœ

bn>œœ nœ

Œ ¸#œ n œfi

nœ

#œfi¸nœ b œ

∑ bœœœ >

#œfi¸n œÆ

Fa§ Sol§

bbnœœœ nn#œœœ > >

nœ #œ #>œ #>œ '

∑

Piano

° *

#œ nœ ‰

¢& bn œœ >

Harpe

œ™

Vibraphone

Æ nœÆ #œ ‰

B Ó

Basson

Alto

‰ #œ ' œ'

& Ó

Clarinette en la

W E

Fa#

‰

nœ n œ ‰

nœ #b#œœ œ &nœ

nœ

bœ œ

nœ *°

Ex. 3.2.20. (cont’d) [mistake?]

E 2

Clarinette en la

Violon

Alto

N ‰

& #œ bœ ‰

Vibraphone

Marimba

œ ¸ #œfi

‰

¸ nœ '

¸ #œfinœÆ

X E

œ™

N 1

fi¸ nœ œ nœ ‰ nœ n œ #œ œ#œ œ # œ #œ

nœÆ

° Œ &

¸ œ

¸ nœ '

B Ó

‰ #œ ' n œ'

#œ nœ nœ #œ

∑

¸ #œfi nœ

nœ Œ B nœ nœ nœ

nœ

¸ nœfi#nœ œ

‰

> #œÆ nœ # œ n œ bnœœ #>œ n>œ n>œ > nœ b>œ Œ

‰

¸ #œ

nœ ‰ n >œ >

Æ > n>œ #œÆ n œ

B Violoncelle ¢ Œ Cor en fa

1 Œ

° ¸ Cor anglais & #œ '

Basson

& œ

Periodicity

& Œ ° &

¢& &

Harpe

nœ Œ

¸ n œfin œ

nœ n œ

‰

nœ Œ

nœ

‰ fi¸

nœ #œ nœ

¸ # œfin œ

nœ #œ n œ

nœ # œ nœ ‰

nœ ° *°

* #œ

fi¸ œ bb œœnn œ

‰

¸ #œ '

bœ Ó n>œ n>œ n œ #>œ >

bœ >

& Œ Piano

? Œ

fi¸ ###œœœ

nnbœœœ ‰ >

#nnœœœ >

#œ n œ ‰

¸ #œ

‰

∑

‰

∑

b>œ #œ Ó > n>œ n œ >

‰

nb>œœ

nœ

nœ n>œ

¸ #n>œœ n#œœ >

#œ nœnnbœœœ n œ >

nn#œœœ

bn>œœ #bn>œœœ

nœ

Sib

#œ nœ n œ

bn>œœ nn>œœ

Œ Œ

Ó Ó

157

Ex. 3.2.20. (cont’d) ¸ & œ

Periodicity

Cor anglais

Violon

Cor en fa

Vibraphone

Marimba

B ¢ ‰

¸ #œfinœ

nœ nœ

° & ‰

¸ #œfi

B ‰ B ‰ & ‰

¸ nœ '

#œfi¸

nœÆ

¸ #œfinœÆ

¸ # œfin œ

Ó nœ

nœ #œ nœ nœ nœ ¸ nœ œ >'

nœ '

‰

nœÆ ˝

nœ

° &

Œ œ #œ nn>œ >

¸ #œfi

¸ nœ

nœ

œ™

‰

œÆ ˝

‰

nœ #œ fi¸nœ nœ ‰ #œ # œ n œ nœ #œ

‰ ‰

nœ #>œ

n œÆ ˝ nœÆ ˝

1 ¸ œ

‰ Œ

¸ ¸ #œfinœ

¸n œÆ #œfi ˝

& ‰ ?

Ó ¸ nœfinœ¸

‰

¸ b œfinœ'

¸b œ nœfi

‰

¸ ¸ # œfinœ

¸ œ

∑

¸ #œfi

Œ nœ nœ #œ nœ nœ n œ nœ n œ # œ #nœœ #œ Œ #n>œœ >

158

‰

n œ #œ n œfi¸ nœ #œ nœ nœ #œ nœ #œ nœ Œ Œ

∑

‰

fi¸ nœ Œ nœ #œ n œ # œ #œ nœ *° * °

Mi§ Ré§

Harpe

œ >

##>œœ

> #œ nœ nœ b>œ ‰ n#>œœ n>œ > >

∑

‰ ¢&

‰

> n>œ #>œ nœÆ nœÆ #œ nœ Œ nœ #n>œœ nœ ‰

∑

Piano

œ™

& #œ nœ ‰

Alto

Violoncelle

° &

Clarinette en la

Basson

nœ nœ ‰ ‰

¸ b œfinœ

n œ œ nœ * ° nœ

?nœfi¸

bœ

nœ n œ bn œœ bœfi¸ nœ

n#œœ

nœ n œ b œ # œ

bœfi¸ nœ

nœ #œ

Ex. 3.2.20. (cont’d) 1

¸ œ

° &

‰

œ ¸ #>œ n>œ #>œ > n>œ #>œ # œ '

‰

° &

‰

Periodicity

Cor anglais

Clarinette en la

Basson

Violon

Alto

Violoncelle

Cor en fa

° Vibraphone &

Marimba

Harpe

Piano

œ™

‰

nœ nœ #œ Œ #>œ n>œ #>œ > n>œ #>œ n œ nœ

≈

> nœ nœ #>œ nœÆ n>œ >

≈

> #>œ # œ nœ Ó > #œ nœ > ' > n>œ n>œ nœ #>œ #œ nœ Œ #œÆ n>œ n>œ ˝

‰

n œ.

‰

#œ ‰

bn œœ nœ

‰

‹#œœ #nœœ nœ ‰

‰

b#œœ nœ #>œ bnœœ > ‰ n>œ >

¸ bœ

#œ #œ œ

nœ ¸ nœ ™

# œ. nœ. n œ. b œ. ° nœ

nœ

nœ n œ nœ

bœ. #œ. ‰

‰

nœ

nœ nœ ‰ #>œ #>œ n œ # œ n œ #œ * ° ‰

nœ. ‰

¸ nœ '

‰

nœ ? ˝

‰

bœÆ ˝

nœ

bœ

Œ nœ. n œ . b œ. n œ. #œ

bœ

b œ nœ

bœÆ nœœ nœÆ #œœ nœ ' '

™ nœ nœ

n#œœ ˝

‰

# œÆ nœ

#n œœ '

bnœœÆ

‰™

bnœœ

# œfi¸

nœ ˝

nnœœ

Æ Æ bœ nœ n œ n œ b# œœ b œ # œ' b œ n œ' '

159

The underlying periodicities continue to transform and expand for the remainder of the work. Rhythmic values go through various methodical permutations, while the EN cycle is partitioned into two strata of rhythmic layers, where each layer is then transformed and expanded independently. The accent cycle (cycle of every fourth composite element) is also expanded by enlarging its duration. As the periodicities continue to unfold, more layers of transformations and transfigurations continue to unfold through the manipulation of these periodicities, so that at the end they are no longer recognizable. It is not until R221, that the periodicities reach their last section, Section L, and the opening rhythmic configuration returns. In this section, the realization of these periodicities is exactly the same as that in Section A. Rhythmic values are clearly articulated and they follow precisely the rhythmic values that appear in the underlying periodicity chart (ex. 3.2.21). The EN cycle is also clearly articulated by instrumental groups, while the accents (a cycle of a fourth composite element) are again articulated by the piano and harp (ex. 3.2.22). The moment of return to rhythmic and periodic clarity, and the way these periodicities are realized with only 8th-notes in Section L, immediately echo and recall the opening measures of Section A, where both sections are not only closely related, but in fact, are retrograde of one another. This is the nature of any multi-layered periodicity cycle since the full structure, from the first alignment to the next alignment, is completely symmetrical. For instance, take a simple multi-layered periodicity of the rhythmic pattern 5 against 4, where the third beat of 5 is the center of this pattern, and from this midpoint, the rest of the rhythms are the exact retrograde of the first half. The symmetrical structure of the underlying periodicity will be discussed in chapter 3.3.

160

Ex. 3.2.21. Periodicities of Section L >r / œR

793

E25 A19

j / œ™

≈

r / œR

j / œ

≈ > Į

r / œR

‰

c r / œ

870

‰™

j / œJ ™™

≈ > Į

c r / œ

‰

892

E28

≈

>bj / œ

881

A21

859

D12

‰

>cr / œ

848

E27

837

A20

≈ > Į

826

E26

‰™

c j / œ

804

815

D11

≈

903

r / œ

914

≈

E N Every 4th element EN cycle (a complete statement of EnnnEEnEEnn) Rhythmic pattern cycle (a complete statement 0f 2-1-3-1-1-2-1)

≈

‰

≈ >

‰

‰™

r œ

j œ ™™ J

>br œ

j œ J

≈ > Į

j œ™

‰

>ar œ R

c r œ

‰

r œ

≈

>aj œ J

r œ

≈

c r œ

>aj œ J

j œ™

≈

≈ > Ȗ

j œ J

r œ

‰

≈

‰

≈ >

‰

‰™

≈ > Ȗ

>cj œ™

≈ > Į

r œ R a

r œ R

r œ

c j œ

r œ R

c j œ b

c r œ

j œ b

r œ R a

j œ™

c r œ

j œ b

j œ

>cr œ

j œ ™™ J

r œ

≈

‰ > Į

c r œ

>br œ

‰™

Short notes of horn

Fig. 3.2.22. Instrumental assignment in Section L

‰ >

≈

a j œ ™™ J

c r œ

≈

yyy 77 16 x

‰™ >

≈

zzz 78 18 x

≈

‰ >

aaaa 79 17 x

≈

bbbb 80 17 x

≈

cccc 81 17 x

‰

dddd 82 18 x

‰

≈ >

‰™ >

≈

eeee 83 18 x

xxx 76 18 x

‰

www 75 18 x

‰™

≈

a j œ J

vvv 74 17 x

‰ >

‰

uuu 73 17 x

o™

≈

>ar œ R

≈

j œ

≈

≈ ȕ

a r œ R

>cr œ

Ca., Vla, Vc., Vib., Harp Cl., Bsn., Vln., Mar. Piano Long notes of horn

161

r œ R

‰

>cj œ

‰™

r œ b

≈

‰

≈ >

j œ

r œ

‰™ > Ȗ

≈

‰

≈

™ o

ffff 84 16 x

Ex. 3.2.23. Annotated periodicities in Section L UUU73

Cor Anglais

¸ ¸ œ œ œ

¸ bœfi

nœ

n˙ ™ > ff

Bassoon

Violin

‰ œfi¸#˝œ #œ

Clarinet in A

#œÆ

‰ nœfi¸#œÆ Œ ˝ arco

nœfi¸

Horn in F

n˙ ™

221

Marimba

¢&

Très rapide (q = 164)

¸ nœfi nœfi¸#œ nœ ff

b n˙>˙ ™ ? n ˙ ™™ ff

n >˙ ™ ? nn ˙˙ ™™

> nœfi¸nœ

b >œ ™ ‰

nœ

Ó ¸ bœfi ff

nœ

nœÆ ˝

nœ ™ >

n˙ > ff > n œfi¸ nœ b ˙ ˝

¸ ¸ ‰ œ nœ

#œ

nœ Œ

nbbœœœ ™™™

Œ œœœ ‰ ˝

Solb

n œ¸ bœ

nœ ‰ #œ

fi¸#œÆ ‰ nœ ˝

¸ ¸ n œfinœ'

162

¸ nœ ‰ ' bœ

bœ

b œÆ ˝ ‰ nœÆ ‰ ˝

Æ ‰ bœfi¸b˝œ bœ nœ

nœ

Œ ¸ nœ

nœ œ™

‰ nœ ˝

nœ ff

‰ b˝œ œ

bœ nœ Œ f

¸ œ ‰ Œ œ *

œ œ

nœ ff

¸ ‰ #bœœ nœ ff

° bœ ‰ ¸ bœ ‰ ¸ ‰ nœ nœ ¸œ ˝ n œfin> nœ f

œ bbœœ ˙˙ œ nœ ˙

b˙ b˙

œœ

œ bœ ˙

¸ ‰ nœ œ ff

n˙ ‰

¸ bœfinœ

œfi¸

œ ‰ ‰ ¸nœfi¸b˙ nœ nœ ˝œ b œ n>˙ * ff ° ° ‰ bœfi¸ ¸ Œ Œ nœ

fi¸

œ ‰ ˝ bœ

& nnbœœœ Piano

> nœ

¸nœ bœfi n œ nœ

fi¸ nbœœ

& Harp

fi¸ nbœœ n ˙ ™

n >˙ ™ ff °

¸bœ bœfi

N N

bœ

¸ ¸ œ œ

¸ nœfin˙ ™

° Vibraphone &

> nœfi¸

nœ ‰ ˝

¸ œ œ

N N

fi¸ ‰ nœ nœÆ

bœfi¸

Violoncello

nœÆ

¸ nœfin œ¸

¸ ¸ œ œ œ

¸ fi¸ nœ bœ n œ nœ œ >

bœ ˝ nœ ‰

n˙ ™ >

nœ

VVV74 E

¸ œ œ™

° &

Viola

#œ

¸ œ œ

N N

° &

¸ œ œ™

Periodicity

œ bœ œœ b œ

∑ Œ

¸ ‰ #œœ ff

‰

nœ °

Ex. 3.2.23. (cont’d) 1

Periodicity

° Cor Anglais & &

Clarinet in A

Bassoon

Violin

° &

Violoncello

Horn in F

Vibraphone

Marimba

Harp

Piano

¸ ¸ ¸ ¸ œ œ œ œ œ™

¸#œ œfi >

nœ nœ nœ

Ó # œfi¸nœÆ

‰ ˝

Œ f

Viola

¸ ¸ œ œ œ

#œ ‰

¸ ¸ n œfi œ

> ‰ nœ ˝ nœfi¸ Æ

> fi¸nœ

nœ

#œ f

œ #œ > #œ # œ nœ nœ

> > fi¸#œ

bœ

¸ #œfi

¸ n œfin ˙

nœ nœ œ

WWW75 E

nœ

¸ #œ ‰

N N N œ

#˙

#œ #œ >' ¸Æ nœfin˝œ ‰

Æ > ‰ fi¸b˝œbœfi¸#œÆ bœÆ nœ f

#˙ ™ f

#œ

nœ ˙

¸ ‰ nœ

¢&

‰ #œfi¸nœ¸ Œ Réb

& <#>œœ

¸ œœ ‰

œ ‰

¸ ‰ #˙ ™ œ

nœ

¸ œfi

¸ ‰ nœ f > > bœfi¸ Æ nœ ‰ ˝ f

nœ #œ Œ Œ

¸ nœfi

¸ #œ ‰ Œ '> f nœ ™ n œ #œ n œ œ #œ ˝ f nœ #œ nœ œ nœ nœ œ Ó

œ œœ

bnœœ #œ Œ

bœ bœ

¸ ‰ #œ¸ ¸ ‰ n>œ nn œœ f œ œ

‰ #œfi¸n˝œ f

b˙ ™

Ré§

n˙ ™

#œ n#œœ

œ œœ

¸ œ ‰ œ œ

fi sim.

sim.

163

Ó f

¸ nœfi

¸ ‰ n>œ

bœ ™

Sib

<n>œ nbœœ bnœœ

∑

b>œ

œ #œ

¸œ n œfin>

* °

¸ nœ ™ œ bbœœ ™™ œœ

¸ <#>œ ‰ #œ <n><n> œœ nn œœ

¸ ‰ #œfi

¸ ‰ œ >'

#˙ ™ nn ˙˙ ™™ °

¸ nœfi

#œ

¸ ¸ n œfin œ

¸ œ ‰ n##œœœ œœ

¸#œ n œfin œœ

¸ Si# & œœ ‰b fi¸ œ nœ f > ?

‰

° œ & <#> œ œ

¸ œ œ™

œ nœ #œ

¸ nœfi

E E

¸ ¸ œ œ œ

#œ

#œÆ ‰ ˝ Œ ¸ #œ ‰

¸ ¸ ¸ œ œ œ

œœ¸

#˙

¸ œ ‰

nn˙˙

œœ ‰

Ex. 3.2.23. (cont’d) &

Periodicity

Cor Anglais

° &

XXX76

¸ ¸ œ œ œ

¸ œ œ

N N

¸ nœ #œ ™

Bassoon

Violin

° &

Violoncello

Horn in F

nœ

¸ ¸ ‰ #œ ' nœ '

¸ #œfi

¸n˙ ‰ nœfi

¸ ‰

¸ # œfi œ'

‰ fi¸#œÆ Œ nœ ˝

‰

œ™

> ˝

b œfi¸b œ

¸ n œfi

nœ

#œ ‰

#œfi¸

n˙

œ œfi¸n œ '

¸ #œfi

n˙

nœ

¸ œfinœÆ

#œ ‰ ˝ ‰ #œ¸ '

¸ œfi ¸

œ >'

‰

nœ nœ n œÆ

nœfi¸ ˝

> n œfi¸b œ

¸> ‰ #œfi nœ

¸ œ œ™

N N

¸ nœfibœ

n˙ ™

> nœ

#œ '

nœ

#œ Œ '

bœ n˙

¸ fi¸ bœfinœ nœ #œ mf

> nœfi¸ ‰ #œ #œ nœ bœ ˝ mf

nœ œ

∑

nœfi¸bœ

bœ ˙ Æ #œÆ # œ

> b œfi¸n œ

Marimba

¢& &

Harp

? &

Piano

¸ nœfibœ

n˙ ™

¸ ¸ œœ œœ ‰ – – # œ nœ— n œ —œœ œœ

nœ f

* ° ¸ ‰ b˝œnœfi¸ ¸ ‰ #œ ‰ n>œ n œ n œfi¸n œ mf

b˙ nn˙˙

Ó Ó

Lab

nbb˙˙˙ ∑

#˙˙ f

nn ˙˙

¸œ # œfin >

œœ ™™ œ ™

#œ

¸ ‰ nœfinœ

¸ ‰ nœ

¸ ‰ fi¸#bœœ ˙˙ ™™ #œ > n˙ ™ mf

> nœfi¸bœ ˝

œœ ™™

nnœœ œœ

¸ œœ ‰ œ

‰ #œ #œ ˝ mf

b˙ ™

fiÆ¸ nœ

¸ nw ‰ #œ b w w

fi¸ ¸ nœnœ nœ nœ b œ

‰

bnœœ

∑ ¸ n#œœ œœ

œœ ™™

‰

#œ

¸ ™ bœ nbnœœœ ™™

Do§

nœ

222 ralentir brusquement - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

Vibraphone

> bœfi¸bœ

#œ #œ #œ –

YYY77

1 E ¸ ¸ œ œ

¸ ¸ ‰ nœfinœ #œ b fi¸bœ œ >

¸ ‰ nœ

nœ #œ

œ ° & œ˝œ ‰

¸> nœfibœ

¸ nœ nœ ˝ nœ

1 œ™

#œ ‰

¸ œ

n˙

Æ ‰ b˝œbœfi¸nœ¸ ‰ >' mf

Viola

¸ nœfi#œ¸

¸ ¸ œ œ œ

¸ #œfi#œ

E E

œ™

¸ nœfi

Clarinet in A

¸ œ

&nœfi¸bœ

‰

#b œœ nœ

164

Ex. 3.2.23. (cont’d) 1

Periodicity

Cor Anglais

° & &

Clarinet in A

Bassoon

Violoncello

Horn in F

Vibraphone

Marimba

¢&

<n> œ ™

Piano

nœÆ ˝ ‰ nœ¸ ‰ '

¸ nœfi#œ¸

nœ #œ #œ

¸ nœfi

¸ nœ œ >

nœ

bœ™

n œ nœ

¸nœ nœfi ˝

n œÆ ˝ ‰

#œ

fi¸ ‰ bœ #œ¸ œ mf > ¸ ¸ bœfi nœ

¸ œ

bœ nœ

nœ

‰

nœ ‰ nn œœ ˝

#œfi¸n˙

nœ '

n œ nœ

nœ

¸Æ #œfinœ

nœ

> Œ

‰

¸ nœfi#œ

bœ bœ Œ

nœÆ > nnœœ ‰ ˝ '

bœÆ ˝

> œ #œ nœfi¸#œ #œ ˝

n˙

nœÆ Œ

¸ ‰ œ

#œ ™

¸ n œfi#œ'

¸nœÆ nœfi ˝

#œfi¸#œ

œ ˝ # œn fi¸ # œ œ #œ

¸Æ nœfinœ

1 ˙

¸nœ n œfi >

¸ ¸ œ œ œ

N N

‰ n˝œÆ ‰ nœ¸ Œ ' > mf nœ nœ ¸ œ nœ # œfin œ nœ œ

nœ ‰ ˝

E E

¸ œ œ™

nœ

> nœ #œ

‰ nœfi¸

¸ Œ ¸ # œfin œ'

nœ

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165

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¸ nœ n>œ Æ fi¸ ¸ ‰bbœ nœ œ

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Chapter 3.3. Formal Structure in Dérive 2 The formal conception of Dérive 2 is summarized by Boulez: I know I indicated the form [coda, reprise, and transition] which I had in mind. But, you know, I’m not fervent about the form I discovered or used there, because I think the form is stronger than you are – certainly in terms of what you can do until about halfway through the work. You have strict forms, strict rhythmical forms and canonic forms and so on – whatever you want to call them. They are strict forms, or obbligato forms, which are then interrupted more and more by free forms. And you feel that: “A-B-A-B-AB-A” … it’s always the same alternation, with the interruptions at the beginning being extremely short. They’re barely to be noticed, and then the interruptions grow progressively longer and longer until they become more important than the text itself. After that, things get more complex. I can’t really explain it … again you have the rhythmical structures of the first half, which are very strict. But despite this strictness, they are freer than in the first half. So you have a kind of balance between both halves, and then there’s a long coda. That was how I operated there.13 The entire architectural periodicity of the whole work is reproduced in appendix I. As previously mentioned in the last chapter, each section of Dérive 2 is determined by an alignment of any two of the three cycles in this blueprint. For example, Section A concludes after an ending synchronization between cycle E (event) and cycle A (accent), as shown by the E and A brackets on the left hand side of the graph, where 4 cycles of E align with 3 cycles of A. Section B concludes after an ending synchronization between cycle E (event) and cycle D (duration), as shown by the E and D brackets on the left hand side. This process continues until all three cycles of E, A, and D finally align at their conclusion, where the whole periodicity cycle also concludes. Through these alignments, the periodicity table generates 12 sections (12 instances where two of these three cycles align), and they are lettered from A to L. As a cycle any multi-layered periodicity guarantees a

Pierre Boulez, “Interview Pierre Boulez: Sometimes You Discover Yourself!” in Musikblätter 1, ed. Wolfgang, Schaufler (Vienna: Universal Edition, 2011): 8-9. 13

166

symmetrical structure, therefore Section F is also labeled as Section G after the midpoint of the periodicity. Section E is then equivalent (the retrograde, to be precise) to Section H, and Section D is also equivalent to Section I, and so on. These symmetrical pairs share the same number of lines, but the total duration for each line is different within each pair. The whole periodicity chart has a total of 84 lines (or 84 EN cycles), since 3-cycles of E times 4-cycles of A times 7-cycles of D equals 84. The axis of symmetry (midpoint) occurs between line 42 and 43. Using the periodicity table provided in appendix I, the formal structure of the whole work can be divided into 11 sections, as demonstrated in figure 3.3.1. In this chart, the first column refers to the lettered sections from A to L, along with interruptions and Codas. The second column refers to line letters/numbers according to the periodicity chart in appendix I. The third column refers to the corresponding rehearsal numbers in the score. The fourth column provides tempo markings in the corresponding sections. The last column provides the speed of the fastest subdivision within sections of the periodicity.

167

Section

Line

anacrusis A

Rehearsal

Tempo Marking

long (fermata), levee rapide

a b c d e f g h i j k l Interruption I B m n o p q r s t u Interruption II C v w x Interruption III D y z a2 b2 Interruption IV E c2 d2 e2 Interruption Va f2 g2 Interruption Vb h2 i2 Interruption Vc

1 2 3 4 5 6 7 8 9 10 11 12

j2 Interruption Vd

13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35

5-R1 3-R1 1-R1 R1-2 R1-4 1-R2 2-R3 1-R3 R3-2 3-R4 1-R4 R4-2 2-R5 R5 R5-3 R6 R6-3 R7 R8 R8-2 R9 R9-3 R10 2-R11 R11 R12 R13 R14 R17-2 R19 R22-1 R24 R26 R29 R33 R35 R37 R41 R45 R49 R52 R56

Rapid (q = 152)

R63 R66

To Vif (e = 142/152)

304

accélérer très progressivement

Très rapide (q = 164) revenir progressivement jusqu àu Rapid (q = 152) Moins rapide (q = 96) Assez rapide, stable (q = 138)

552

(q = 92) Assez rapide, stable (q = 138)

552

Assez rapide (q = 90 – 84 - 80) (q = 90 – 84 - 80) (q = 90 – 80 – 84 – 80 – 84 – 90 – 84 – 90 – 80) (q = 84 – 80 – 90 – 84 – 90 – 80)

540

Modéré (e = 102 – 92)

568/608

Vif (e = 142/152)

(e = 108) Vif, comme précédemment (e = 142/152)

568/608

(e = 96 – 108 – 102 – 96 – 92 - 108)

Vif, comme précédemment (e e = 142/152)

568/608

(e = 108 – 138 – 118 – 114 – 110 – 106 – 102)

568/608 Assez vif (e = 120 – 132 – 138 – 132 – 120 – 132 – 112 – 104 – 96 – 90 – 84 – 88 – 92 – 88 – 84 – 78 – 98 – 132 – 120 –

168

98 – 90 – 82 – 120) 2

F k Interruption VIa l2 Interruption VIb m2 Interruption VIc n2 Interruption VId o2 Interruption VIe p2 Interruption VIf Midpoint

38 39 40 41 42

49 50

R83 R85 R87 R89 R91 R94 R102 R105 R111 R114 R126 R129 R133

R167 R175 R176 R180 R181 R185

588

Plus modéré (e = 98) Plus calme (e = 76)

588

Plus modéré (e = 98) Plus calme (e = 74)

588

Plus modéré (e = 98) (e = 84 – 92/88)

588

Plus modéré (e = 98)

Tempo détendu (e = 74), très régulier

588

Plus modéré (e = 98) Large (e = 62), très régulier

588

Plus modéré (e = 98)

Assez modéré (e = 76–74-72–70–68-66)

Lent, avec souplesse (q = 66) (les groups de petites notes à l’aise) Tendu, sans traîner (q = 92) Lent, avec souplesse, sans trainer (q = 80) Tendu, sans traîner (q = 76) Plus tendu (q = 92) Tempo, stable et régulier (q = 76) Plus tendu (q = 92) Tempo (q = 76) Lent, avec souplesse (q = 76) Stable, sans traîner (q = 76) Lent, avec souplesse (q = 68) Stable, sans traîner (q = 76) Lent, avec souplesse (e = 104) Tendu, sans trainer (e = 140) Un peu moins serré (e = 134) Un peu plus détendu (e = 128) Un peu plus serré (e = 134) Un peu plus tendu (e = 140) Assez serré (e = 134) Tendu, sans trainer (e = 140) Plus détendu (e = 128) Plus serré (e = 134) Plus détendu (e = 128) Tendu sans trainer (e = 140) (q = 88/92) Lent, avec soupesse (e = 98) Allant, sans trainer (q = 68) Rythmique, énergique (q = 74, e = 148)

Très calme, suspend, régulier (q = 60)

Rythmique, énergique (e = 150) Très calme, suspend, régulier (q = 60) Rythmique, énergique (e = 156) Sub. Très calme, suspend, régulier (q = 60)

169

51 52 53 54 55 56 I

57 58 59 60

61 62 63

64 65 66 67 68 69 70 71

u3 v3 w3 x3 y3 z3

73 74 75 76 77 78

R186 R189 R190 R193 R194 R196 R197 R199 R200 R202 R203 R205 R206 R208 R209 R210 R211 1-R213 R213 R213-4 R214 R214-5 R215 R215-4 R216 R216-3 R217 R217-3 R217-4 R217-5 R218 R218-3 R218-5 R218-6 R219 R219-3 R219-4 R219-5 R220 R220-3 R220-5 R220-6 R221 R221-3 R221-5 R222 R222-3 R222-5

Rythmique, énergique (e = 160) Très calme, suspend, régulier (q = 60) Rythmique, énergique (e = 166) Très calme, suspend, régulier (q = 58) Rythmique, énergique (e = 180) Très calme, suspend, régulier (q = 58) Rythmique, énergique (e = 180) Très calme, suspend, régulier (q = 56) Rythmique, énergique (e = 184) Très calme, suspend, régulier (q = 56) Rythmique, énergique (e = 184) Très calme, suspend, régulier (q = 56) Rythmique, énergique (tempo rigide) (e = 188) Très calme, suspend, régulier (q = 54) Rythmique, énergique (e = 188) Très calme, suspend, régulier (q = 54) Rythmique, énergique (e = 192) Très calme, suspend, régulier mais tendu (q = 52) Rythmique, énergique (e = 192) Très calme, suspend, régulier mais tendu (q = 52) Rythmique, énergique (e = 192) Très calme, suspend, régulier mais tendu (q = 52) Rythmique, énergique (e = 196) Très calme, suspend, régulier (étirer le tempo) (q = 52)

Rythmique, énergique (q = 108, e = 216)

Très calme, suspend, régulier mais tendu (q = 52) Rythmique, énergique (q = 114) Sub. Très calme, plus étiré (q = 52) Rythmique, énergique (q = 120) Sub. Très calme, toujours plus étiré (q = 50) Rythmique, vif (q = 126) Sub. Très calme, tendu (q = 50) Rythmique, vif (q = 132)

Très calme, tendu (q = 50, e = 100)

Rythmique, vif (q = 138)

Très calme, tendu (e = 100) Rythmique, vif (q = 144) Très calme, tendu (e = 100) Rythmique, vif (q = 150) Très calme, tendu (e = 96) Rythmique, vif (q = 156) Très calme, tendu (e = 96) Très rapide (q = 164)

ralentir

170

328

a4 b4 c4 d4 e4 f4 Coda I Coda II Coda III Coda IV

79 80 81 82 83 84

R223 R223-2 R223-4 1-R224 R224-4 3-R225 R225 R234 R244 R251

jusqu’à

Sub. Moins rapide (q = 152, 164, 152, 164, 152, 164) Méme tempo Toujours plus rapide (q = 172) (q = 200, 224/240, )

Fig. 3.3.1. Formal Structure of Dérive 2 Example 3.3.2 demonstrates the entire periodicity cycle of 3 x 4 x 7 of the whole piece in a simplified notation, as one clearly sees the large-scale polyrhythm within this periodicity cycle. The rehearsal marks correspond to the lettered sections in figure 3.3.1, and each section is determined by the synchronization of any two of the three cycles when they align. These structural sections are articulated and inserted by interruptions–music that is not composed based on this periodicity, and harmonically tends to be static. As the work unfolds up to the midpoint before Section G, these interruptions become longer and longer, and eventually outgrow the periodicity sections. However, the entire process is reversed the midpoint. Each interruption becomes shorter and shorter, and eventually it is reduced to only a few beats. Each Arabic number in example 3.3.2 corresponds to the line numbers in figure 3.3.1, as well as the line numbers in appendix I. These numbers represent the smaller "sections" or "phrases" in the work, and typically articulated by a change of instrumentation and their assignments. Notice the entire cycle ends at number 84, since 3 x 4 x 7 = 84. The symmetry of the periodicity cycle is clearly shown in this realization, as Section G to Section L is the retrograde of the first half. Figure 3.3.3 is another representation of the periodicity cycle of 3 x 4 x 7.

171

Fig. 3.3.2. The periodicity cycle of 3 x 4 x 7

172

Sections 3 4 7

A   

Sections 3 4 7

D  

Sections 3 4 7

H  

Sections 3 4 7

L  



B  



 

  

F  

 



I  

G 

 



 

 J  

  



 



C 



E 



K 

 



  

All 3 finally align   

 



Fig. 3.3.3. The periodicity cycle of 3 x 4 x 7

Example 3.3.4 illustrates the form of Dérive 2, with both the 3 x 4 x 7 periodicity and the interruptions.

173

Example 3.3.4. The form of DĂŠrive 2, with the 3 x 4 x 7 periodicity and the interruptions

174

Example 3.3.4. (cont'd)

175

The formal structure of DĂŠrive 2 as shown in figure 3.1.1 demonstrates the symmetrical formation of the whole piece. Sections A to F constitute the first half, while the second half consists of Sections G to L. The two halves are separated at the exact midpoint of the periodicity cycle by the longest duration of music that is not derived from this underlying periodicity. This particular interruption at the midpoint is drastically different from other interruptions, as harmonically this is not static. Furthermore, this section extends the registral span of the entire ensemble to its maximum, featuring the lowest and the highest notes from all the instruments. From Section A to D, each section is separated by a progressively longer interruption. These interruptions continue to disrupt the periodicity, and divide Section E into four sub-sections and Section F into six sub-sections. After the midpoint, Section G is not interrupted until R175, where it is interrupted for a brief moment. From Section H to the end of Section K, interruptions occur at the end of each line. As was already mentioned in chapter 1.3, these sections (from line 49 to 52) are progressively shorter in their total notated rhythmic values, and their tempi gradually increase for each periodicity line and gradually decrease for the interrupting sections. Section L returns to the same realization of rhythmic value (8thnote) and tempo as in Section A. A brief pause (four 8th-note rests) occurs right before R225, and the Coda begins immediately after the pause. These interruptions, in a way, destroy the symmetrical structure of the periodicity. On the one hand, these interruptions serve as important signals that articulate the various alignments of periods/cycles in the periodicity, while on the other, they deliberately and subsequently abolish the flow and the unfolding of these periodicities. As the interruptions

176

progressively grow and expand successively toward the midpoint of the work, ultimately overshadowing and outweighing the actual text itself, the role of the interruptions has been reversed and transformed: it is now the periodicity sections that articulate the various expansive segments of the "interruptions," and these interruptions occur not only at the juncture alignment points in the periodicity structure, but they actually divide and separate each individual periods in the periodicity (such as in Section E and F), as the periodicity sections are no longer the main subject of the work. The most crucial moment of the piece, the exact midpoint of the work and the exact midpoint of the periodicity structure, Boulez inserts the longest duration and largest segment of music that is neither part of the periodicity section or the interruptions (see ex. 3.3.4 where the midpoint is in fact preceded by an insertion of an interruption). Moreover, the considerable and increasingly manipulation that Boulez imposed on the periodicity sections has completely alter the recognition and the pure properties of these periodicities. The periodicity sections, similar to their counterpart insertion of interruptions, likewise progressively grow longer and more complex, at a point where one loses the stratification of these layered periodicities, and most importantly, the polyrhythmic nature of these periodicities. After the midpoint, the process of development and expansion is reversed, reflecting the symmetrical structure of the periodicity, where both the periodicity sections and the interruptions grow progressively shorter until Section L. At Section L, the music returns to the same realization approach as in Section A, and immediately one recognizes the return of the periodicity. Thus the overall form of DĂŠrive 2 is an alternation of two

177

equally composed structures: one is bound by the periodicities, and the other is bound by aperiodicity fixed static harmonic progressions. This formal plan closely mimics Carter's Boston Concerto, which is in fact composed of a rondo of ritornello and developmental sections. The Coda is divided into four sub-sections, all notated in 8th-notes and their multiples. In the first sub-section, Coda I from R225 to R234, the tempo gradually increases, followed by a measured pause. Rhythmically it is composed of two strata of rhythmic patterns, and within each stratum there is a further partitioning of two rhythmic layers. Each layer is assigned to different instruments (ex. 3.3.4). In the earlier versions, these two strata are realized simultaneously. In the later versions, however, one stratum is realized progressively an 8th-note later. In the first half of Coda I from R225 to R231, Boulez employs a dynamic scheme based on varying durations to create an asymmetrical structure from soft to loud and back to soft (fig. 3.3.5). Dynamically in the second half of Coda I from R231 to R234, however, this whole sub-section is divided into twenty phrases, each lasting exactly seven 8th-note long, with a different dynamic marking in each, while creating a gradual crescendo in the overall scheme (fig . 3.3.6). Starting from R233, a progressively longer rest interrupts the music, as it starts as an 8th-note in R2331, then two 8ths, three 8ths, four 8ths, five 8ths, and finally six 8th-notes which function as a measured pause before going into Coda II.

178

Ex. 3.3.5. Instrumental assignment in Coda I

j œ œ œ œ™ j œœ™ œ œ œ œ œ œœ™ œ œ œJ œ œ œj œœ™ œ œ œ J œœ™ œ œ œ œ œ œ œ J œœ™ œ J

Duration 33 8th-note 36 8th-note 19 8th-note 19 8th-note 22 8th-note 19 8th-note 24 8th-note

œj J

Cl., Bsn., Vla., Vib.

Vln., Pno.

Hr., Harp Ca., Vc., Mar.

Dynamic Marking pp - p mp - mf f ff f mf mp - p

Fig. 3.3.6. Dynamic Scheme in the first half of Coda I from R225 to R231 pp-mp mp-pp p-mf mf-p mp-f f-mf mf-ff ff-mf mf-f f-mp mp-mf mf-p p-mp mp-pp pp p mp mf f ff Fig. 3.3.7. Dynamic scheme in the second half of Coda I from R231 to R234

179

The second sub-section, Coda II from R234 to R244, is composed of ten phrases of rhythmic durations. At the end of each phrase, an interruption occurs. The duration of each interruption progressively gets shorter for the first half, and progressively gets longer for the second half - a process which is exactly the opposite from the non-interruptive phrases (fig. 3.3.7). These interruptions are similar to those interruptions found in the beginning of the work, as the pitch G# is featured as the focal point in these interruptions, decorated by grace-notes. This G# is very important, as it continues to be the pitch center for the remaining of the Coda, and it is also the last note of the entire work. Rehearsal numbers R234 R235 R236 R237 R238 R239 R240 R241 R242 R243

Durations for the noninterruptions in 8th-notes 11 15 17 19 19 25 16 13 10 6

Durations for the interruptions in 8th-notes 6 5 4 3 2 1 1 4 7 11

Fig. 3.3.8. Duration of phrases in Coda II The third sub-section, Coda III from R244 to R251, features constant running 8th-notes throughout in the entire ensemble. This is perhaps the most intense and rhythmically driven section of the whole piece. The piano in this sub-section also breaks its registral constraints, and explores the entire range of the piano, with sudden and wide register jumps. As the tempo increases in this sub-section, the intervals are compressed and become progressively smaller, and by R249-4 each non-resonant instrument is

180

reduced to only repeated notes, as the whole ensemble is unified rhythmically and reduced to playing block chords. These chords are juxtaposed rhythmically with block chords from the resonant instruments. The two contrasting ensembles are paired against each other and they never align rhythmically. The metric tension and alternation between these two instrumental groups come to a full stop at R251 (Coda IV), where a series of large chords are fractured and displaced with short instrumental bursts, amidst a sustained G# that is progressively joined in unison by the full group (ex. 3.3.9).

181

Ex. 3.3.9. The concluding measures of Dérive 2. Cor anglais

° ‰ &

^¸Œ œ '

∑

sfffz

& ‰

Clarinette en la

sfzp

^¸ ¸ #œ ‰ œ ˙ ' >

¸ œ ‰ Œ Ó >.

sfffz sfzp

Basson

Violon

B ‰

° Œ &

^ œÆ ˝ Œ

B Œ

Violoncelle

Cor en fa

^¸ œ ‰ Ó #œ '

∑

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sfzp mf

? Œ

Piano

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sfzp

∑

‰

sfzp

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>œ. ˝ ‰

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∑

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^ #œœÆ ˝

> ‰ #œ

^ n#œœÆ ˝

‰ Œ

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∑

^¸ #œ ‰ '

∑

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^ ‰ ##œœ¸ Œ œœ '

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^ Œ ##œœ¸ ‰ œœ '

∑

^¸ ‰ Œ n œ ## œœœ '

∑

sfffz

^¸ ‰ Ó n œ ## œœœ '

sfffz sempre

Harpe

^¸ #œ Œ '

sfffz

^ # œœÆ Œ ˝ ‰ Œ

^¸ & ‰ ##œœœœ Œ '

& Œ

∑

^ œÆ #œ ‰ Ó ˝

& Œ

¸ œ œ. ‰ Ó >

sfffz

sfffz sempre

Vibraphone

^ œÆ #˝œ ‰

∑

sfffz

h = 112

Marimba

‰

^¸Œ œ '

sfffz

>œ. ˝ ‰ Œ Ó

∑

sfffz

B Œ

^¸ ‰ #œ '

^ #œœÆ ‰ Œ #>œ ˝

‰

^ #œÆ ‰ ˝

sfffz

¸ œ. ‰ Ó

sfffz

Alto

‰ #œ¸ ˙ ™ >

^¸ #bœœœœ ‰ Ó ' sfffz sempre ^Æ bœœœ ˝ ‰ Ó

^¸ & ‰ ##‹bœœœœœ Œ Ó ' sfffz sempre ^Æ ∫b œœ ? ‰ #œ˝œœ Œ Ó

sffz

^¸ Œ #bœœœœ ‰ ' ^Æ bœœœ Œ ˝ ‰ ^ Œ # bœœœœ¸ ‰ #‹œ ' ^Æ ∫b#œœœœ Œ ˝œ ‰

182

#>œ œ >œ. ˝ ‰ Ó ‰ ˝ sfzp

sfffz

sffz

^¸ #bœœœœ ‰ Œ ' ^Æ bœœœ ˝ ‰ Œ ^ ‰ # bœœœœ¸ Œ #‹œ ' ^Æ ∫b#œœœœ ‰ ˝œ Œ

>œ. ˝ ‰

sfzp

∑ ∑

Ex. 3.3.9. (cont’d) ° Œ Cor anglais & & Œ

Clarinette en la

Basson

B Œ

° Œ Violon &

^¸ #œ ‰ '

B Œ

∑

^¸ nœ ‰ ' sfffz

^ œÆ ˝ ‰

^¸ #œœ ‰ '

sfffz

#œ >

sfzp

Cor en fa

Marimba

Vibraphone

Harpe

Piano

B Œ ¢

non vib.

#œ

#œ >

sfzp

U w

fff

long

arraché

long

arraché

U w

w ∑

∑

fff

U w

> #w

sfzp

fff

U w

long

sfzp

fff

^Æ #nœœ ‰ ˝

∑

^Æ #œœ ‰ ˝

∑

sfffz

Violoncelle

sfffz

Alto

∑

arraché

long

#>˙

non vib.

U w

sfzp

∑

fff

sfffz

#>˙

non vib.

U w

sfzp

fff

& Œ

^ ‰ #œ¸ ˙

^¸ & # nœœœ ‰ #œ '

∑

^¸ & ##œœœœ ‰ '

sfffz p

^¸ & ‰ #bœœœœ Œ ' ^ bœœœÆ ? ‰ ˝ Œ

^¸ & ‰ ##‹bœœœœœ Œ ' ^ ∫b#œœœœœÆ ? ‰ ˝ Œ

U w

long

fff

183

The emphasis on a single pitch in the last few measures is undoubtedly a direct reference to the opening held A, which was played by the solo horn (ex. 3.3.10). The G# in the last few measures is played by the whole group. Both held notes have a rapid crescendo. This echoing effect of the last note evoking the first note of the piece not only reinforces the symmetrical structure of the whole work, as the ending balances the opening, but also recalls the interrupting sections – sections where one pitch or a group of pitches are usually emphasized as the focal pitches of that section. For example, the first interruption focuses on F (ex. 3.3.11). The second interruption focuses on the chord {E, A, C, D, G#, B} (ex. 3.3.12). In retrospect, the opening A is more than just an anacrusis to the following measures. It can actually be conceptualized as a shortened version of the subsequent interruptions, a preparation for the characteristics of centering on a single pitch. Boulez's fascination on centering a harmonic pole on a single pitch can be traced back to Structures: It is the kind of phenomenon that guided me. In order to produce a standard object you must have not only the same components but also what mathematicians call a ‘class of objects’ answering directly to one’s perception of them. This is what I have tried to do; and, in particular, I have tried to polarize certain classes of object around certain very perceptible and conspicuous centres. At the end of the first chapter of the second book of Structures there is such a polarization around a B flat that is repeated three times. The entire peroration is based solely on chords and sound objects pivoting round this B flat – in other words, the ear is progressively polarized by certain constants.14

Boulez, Par volonté et par hasard: entretiens avc Célestin Deliége, 91-92.

184

Ex. 3.3.10. Opening horn solo Cor anglais

° &

U ∑

Clarinette en la

Basson

Violon

° &

Alto

Violoncelle

Cor en fa

nœ >

fij nœ p

bœ >

j nœfi p j n œfi

#œ >

U ∑

‰

+ n>œ

poco

U ∑

j #œfi

pizz. sec

bouché

U + & nw

bœ.

n œ.

U ∑

pp (à peine audible) long

Piano

j bœfi

nœ J p

n>œ J

‰

j nœ > > nœ J

Ó j nœ

‰

nœ

‰

j nnnœœœ

‰

#œ '

poco sfp

– —‰ nbœœ – nœ

j bœfi nœ p

‰

j nœ

' ®°

185

Ex. 3.3.11. The first interruption ° Cor anglais &

Violon

‰

∑ œ ˝‰ Œ

¸ nœ œ > mp

Violoncelle

Cor en fa

Marimba

Harpe

˙™

œ‰ Œ ˝

nœ ˝

#œ

‰

¸ b œfi ff

¸ œ ‰ ≈ nœ. œ. œ. >

mf sfp

¸‰ nœ ‰ nœ¸ nœ >' ff

pizz. nn œœ

sfp

Œ ‰ ¸ n˙ nœ.

‰ nœ > n>œ

∑

n>œ œ ‰˝

# œfi¸

avec sourd.

¸ nnœœ >'

nœ œ ‰ œ ˝

‰ nœ > b >œ ff

¸ œfi

∑

#œ

˙™

nœ ˝

¸ nœfi

‰

sfp

¸ nœ

‰

¸ #œfi mf

¸‰ œb œ œ

ralentir très légèrment à - - Moins rapide (q = 96) accél. - - - Assez rapide, stable (q = 138)

¸‰ Œ œ

¸ ‰n#œœ n˙ .

˙™

∑

‰

œ™ œ. > ff

Œ n˙ ™ >

‰ #œnœnœ œ ‰ >

‰

¸ œ ‰

‰

∑

n œœ p b# œ

nnœ>œ ‰ b˝œ

˙™

– –

Ó— nnn œœœ – Ó— — ø

∑

¸ ‰ ‰ ¸Œ nœ bœ

¸ œ.

¸ nœ.

‰ Ó nœ. ˝

nnœœ. ‰ bœ. Œ

186

nœfi¸ nnœœ

n>˙ ™

nœ >' f

¸‰ ¸ ‰ nœ bœ

¸ nœfi #œ

∑

fi¸ ##œœ

Fa#

∑

¸ œ.

fi¸ ##œœ mp

Do§

Œ nœ >

¸ & #œœ ‰ Œ œ Fa§

Piano

sfp

nnOœ ‰ Œ ˝

‰ bœ¸ œ > mp

¢&

˙™

> ‰ ¸nœ w #œ

& b˙

° Vibraphone &

n>œ œ ‰˝

sfp

˙™

sfp

‰

‰™ ¸ nœ œ # œ. œ. n œ. >

‰ œ¸ ˙™ >

# œÆ B nœfi¸ ˝ ‰ Œ

Alto

∑

° &

œ ˝‰ Œ

¸ nœ œ > mp

Clarinette en la

Basson

ralentir très légèrment à - - Moins rapide (q = 96) accél. - - - Assez rapide, stable (q = 138)

#œœœ ‰ Œ

nœ

¸ #œfi

bœ bbœœ >'

#œœ

#œ

‰ #œ n# œœ >' f

nnnœœœ >'

Ex. 3.3.12. The second interruption Cor anglais

° &

¸ ‰ Ó™ œ

nœ

¸ #œfi

¸ n œfin ˙

f sfpp

Basson

Violon

¸ œ ‰ Ó™

fi¸ ¸n œ > n œ fi¸ nœfi nœfi¸nœ œ nœnœ nœ œ pp

Cor en fa

f sfpp

nœ

Marimba

¢& &

Harpe

Piano

f sfpp

> # œfi¸n œ œ ‰ ˝

° Vibraphone &

˙™

nœ

n˙

nœfi¸

‰ nœ¸œ > f

œ™

sfpp

¸ nœfinœ ™ fi¸nœ¸ nœ

¸ nœfin˙

œn œfi¸œ ˙

œ™

nœ

¸ nœfin˝œ ˙

Ó™

¸ ‰ Œ nœ

Mib Fa# Sol# La§ Réb Si§

˙™

‰ ¸ bœ >

ralentir très brusquement - - - - - - - - (q = 92)

∑

poco

∑

Ó™

∑

¸ #œfi poco

n œ œ. œ. œ. œ. œ. >' mf

œ nœfi¸œ œ n œ œ œ œ œ œ œ œ œ œ œ œ . . . . . . . . . . . . . >' cresc. mp cresc. ¸ nœfiœ œ œ œ

cresc.

˙™ poco

cresc.

poco

˙™ a

. . >' .

poco

revenir peu à peu au - - - - - - - - - - - - - - - - - - - - - - - - - -

‰ fi¸#œbœnœ nœ nœn œ p > mp

œ™

∑

¸nœ n œfi n œ n œ

∑ ‰ nœfi¸ #œ bœ nœbœ #œ nœ nœ nœ

∑

¸ #œfinœ pp

poco

cresc.

sffp

œ™

˙™

¸ nœfi

˙™

¸nœ nœfi

˙™

poco

> ¸. . . . . . . . . . . . . . . . . . nœfinœ œ œ œ œ œ nœ œ œ œ œ œ œ œ œ œ œ œ

œ nw

nœfi¸

cresc.

nœfi¸

Ó™

f sfpp

Violoncelle

f sfpp

° &

Alto

¸ nœfi#˙ ™

∑

Clarinette en la

˙™

œ nœ ˙

n œ #œnœbœ

nœ nœ œ >

nœ >

Sib

¸nœ œ bœfi œ œ f

The fourth interruption focuses on D, with more elaborated ornaments around this centered note from the entire ensemble (ex. 3.3.13). 187

Ex. 3.3.13. The fourth interruption Cor Anglais

° &

≈#œnœ nœ 3

œ 3

≈#œ n œnœ

∑

Clarinet in A

Bassoon

Violin

≈nœ#œ nœ

‰

° &

‰

∑

≈ nœnœ #œ p

Violoncello

Horn in F

nœ ≈ nœ nœ

‰

° ¸ ≈‰ Vibraphone & nœ.

‰

¸ ≈‰ ¢& nœ

& Harp

Piano

¸ œ

nœ ™

¸ n œfi pp

nœ—

¸ nœfi #œ

*°

∑ ∑

.¸ bœfi #œ

n˙

fi¸ nnœœ Fa§

¸ nœ

fi¸ #n œœ pp

. ¸ nœfi

¸ nœ

* Œ

œ™

‰

#œ.

fi¸ n# œœ

¸ nœfi #œ

. fi¸ nbœœ

¸ nœ

nn œœ

nœ. nœ bœ. .

sans

° ≈

nœ—

*°

¸ nœ.

#œ

nœ—

&nœ

‰

≈

accélérer très peu - - - -

nœ ™

¸ nœfi #œ

fi¸ nbœœ p

¸ œ

œ#œnœ# œnœ

* ° ‰

* °

∑

& nœ

≈ Œ

¸ œ

≈

¸ œ n# œfi¸œ nœ ™

¸ nœ °

Marimba

bœ ¸ œ # œn œnœ ™

calmer - - - - - - Très modéré (e= 92)

¸ n œfi p

œ ‰ ˝ ≈

bœ œnœnœ nœ ™ ˝

œ ˝ ≈ ‰

œnœ#œ œ #œ

∑

œnœnœn œnœ

≈nœ # œnœ

48 Modéré (e = 102)

≈

Viola

¸ œ #œn œnœ ™ #œ

≈nœ n œnœ

≈ Œ

∑

¸ œ# œnœn œ nœ ™

#œ nœ nœ

Fa#

?œ

. ¸ bœfi mp

nœ

n# œœ

* °

¸ œ

nœ.

. ≈ ?nœ

#œ. nœ. bœ. . bœ. n œ# œ.

188

The interrupting sections are not composed with any systematic or periodic rhythms. Metrically these sections are much more clearly articulated, and most often all the instruments are participating in the same metrical activity. The rhythmic clarity in these sections dramatically contrasts the highly complex rhythmic patterns of the periodicity. The interrupting sections are also harmonically orientated, rather than being polyphonic or heterophonic. These static and sustained harmonies slow down the harmonic rhythm of the periodicity, as if these harmonies are frozen for a period of time. These interruptions also serve as important preparations for introducing new rhythmic values. For example, Section A is composed with only 8th-notes. The first interruption introduces the 16th-note rhythmic value that will be used in Section B. The fourth interruption likewise introduces the 32nd-note rhythmic value that will be used in Section E. The pitch constructions in these sections are not derived from the trichords that serve as the basis for the pitch constructions in the periodicity sections. Each interrupting section seems to have its own harmonic structure and intervallic relations. The beginning and the ending notes in these sections tend to come from the preceding or the succeeding pitches in other sections by way of common tones. These sections are also the only sections where the full ensemble projects harmonically in tutti. Certain harmonies can be analyzed an enlarged collection of the SACHER chord (ex. 3.3.14), or the all-trichordal hexachord. The harmonic structures in these sections are independently constructed, but all the harmonies are almost always symmetrical over an axis pitch (ex. 3.3.15).

189

Ex. 3.3.14. SACHER pitch sequence S

e(s) a

& bœ

r(e)

œ nœ

By constructing the whole piece based on a periodicity structure Boulez achieves long-range formal coherence, and the architectural design of periodicity in return gives Boulez the opportunity to disrupt this continuous trajectory by ways of interruptions. This seemingly discontinuous formal outline in Dérive 2 is united by the internal cycles of the periodicity, so that these discrete sections actually contribute to a collective understanding of the whole piece. With regard to large-scale form, Boulez states, My last piece, sur Incises - for three pianos, three harps, and three percussion - is a continuous work because I want to have more and more of these vistas or panoramas, as you called them-actually a long trajectory (like Répons, which is also about forty-five minutes). I want to get rid of the idea of compartments in a work-once again, similar to Proust, where you find that the narration is continuous. You have, of course, chapters in Proust, but the work has to be read in one go. That for me is one of my main goals in music (for large works). I don’t want any breaks in the music, but you can introduce new ideas and abandon some other ideas, like that characters in a novel.15

Di Pietro, Dialogues with Boulez, 70.

190

Ex. 3.3.15. Symmetrical pitch structure in the interrupting sections

Cor anglais

Clarinette en la

Basson

Violon

U ∑

Alto

Violoncelle

∑

¸ œ ˙™ >

‰

> w

n>œ ˝

Vibraphone

Marimba

U +

long pp (à peine audible)

˙™

nœ >

˙™

¸ ‰ n#œœ n˙ .

poco

U ∑

¢&

U ∑

∑

U ∑

∑

U ∑

∑

¢&

Piano

‰

avec sourd.

191

‰

n>˙ ™

Œ ‰

nnœœ. . bœ Œ

Eb/E axis SACHER chord

nœ œ œ. >. .

œ™

œ >.

¸ œ ∑

≈

#œ nœ nœ œ ‰ >

‰

˙™

nœ

¸ œ

#œœœœ œ

? bbnœœœœ

n>œ ˝

˙™

˙™ >

œ & #œœœœ

‰

¸ n˙ nœ.

∑

œ œ

∑

U ∑

Harpe

Harmonic reduction

¸ n œ nœ # œ. œ. . >

˙™

bouché Cor en fa

‰™

œ œ

bœ œ

œ bœ #œ

œ œ bœ

bœ

‰ Fa#

Ex. 3.3.15. (cont’d)

Cor anglais

¸ ‰ Ó™ œ

ralentir très brusquement - - - - - - - - (q = 92)

& nœ

n˙ >

Basson

Violon

& B

Alto

Violoncelle

Cor en fa

Vibraphone

Marimba

¢& &

Harpe

˙™

f sfpp

¸ ™ œ ‰ Ó

f sfpp

> œ n˙ ˝ nœfi¸

# œfi¸n œ

‰

f sfpp

nœ

¸ nœfi nœ ™ fi¸nœ¸ nœ

œ n œfi¸œ ˙

¸ nœnœfi nœ

œ™

sfpp

nœ

nœfi¸

¸ nœfi n˙

¸ nœfi nœ

cresc.

poco

‰ bœ¸ > sffp

∑

Ó™

˙™

∑

œ nœfi œ œ n œ œ œ œ œ œ œ œ œ œ œ œ . . . . . . . . . . . . . >'

cresc.

˙™

poco

cresc.

¸ nœfi œ œ œ

cresc.

poco

. . œ. >'

poco

˙™

‰ fi¸#œ nœ nœ bœ nœn œ p > mp

œ™

∑

poco

‰nœfi

nœ n œ #œ bœ n œ bœn œ #œ

n œ #œ nœbœ

Sib

nœ >

∑

¸ #œfi nœ

¸nœ n œfi n œ n œ

∑

n œ œ. œ. œ. œ. œ. >'

poco

revenir peu à peu au - - - - - - - - - - - - - - - - - - - - - - - - - -

∑

¸ #œfi

cresc.

∑

œ™

poco

> . . . . . . . . . . . . . . . . . œ œ œ œ œ nœ œ œ œ œ œ œ œ œ œ œ œ

cresc.

˙™

¸. nœfi nœ

˙™

∑

Mib Fa# Sol# La§ Réb Si§

˙™

œ ¸n w œfi

poco

Ó™

cresc.

˙ ¸nœ nœfi

10 ralentir très brusquement - - - - - - - - (q = 92)

¸ ‰ Œ nœ

œ™

sfpp

œ ˝

fi¸ ‰nœ nœ¸œ > f

Ó™

n œfi¸

¸n œ > nœfi nœfi¸nœ pp

¸ nœfi#˙ ™

∑

˙™

œ nœ ˙

f sfpp Clarinette en la

revenir peu à peu au - - - - - - - - - - - - - - - - - - - - - - - - - -

¸ #œfi

¸ n œfi

¸nœ œ bœfi œ

Piano

Harmonic reduction

& bbœœœœ œ ? œnœœ

nœ nœ >

∑

C/B axis

œ bœ

œ #œ

bœ œ œ œ bœ All-trichordal hexachord

œ bœ œ bœ œ nœ All-trichordal hexachord

192

Ex. 3.3.15. (cont’d) Cor anglais

& &

Clarinette en la

Basson

Violon

ralentir brusquement - - -puis très progressivement - - - - - - - - - - - - - - - - - - - - - - - - - - (ne pas presser) - - - - - - - - - - - - - - - - - - - - - - -

#˙ ™

˙™

¸ œ ™ #œfi nœ- œ p

˙™ fi¸

œ nœ # -œ ™

p pp

˙™

Cor en fa

œ™

™ B <n>˙

˙™

B <n>˙ ™

˙™

œ#œfi¸n-œ ™

¸#œfi nœ p

nœfi¸nœ

‰ fi¸ ¸ nœ # œ-

¢&

œ. œ. œ. œ. œ. œ.

#-œ ™

œ. œ. œ. œ. œ. œ. #-œ œ. œ. œ. œ. œ. # œ.

œ nœ ™

p pp

nœ Œ nœ ¸#œnœ # œ# œb œn œ # œfi

∑

≈

‰™

œ–

œ– #œ f

n˝œ œ

¸ #œ nœ nœ n œfi nœ. b œ# œ. . . n œ. .

#œ. Œ ˝

& nbbœœœœœ ? #bœœœœ

≈

bœ

œ– n œ ™ mf

#œ ¸nœ nœ ‰ n œ. nœ. # œ. . n œfi . . mf

nœ #œ #œ œ œ. œ. n œ. . . .

G/C# axis

‰™

nœ #œ #œnœ ‰ ˝ n œn œ nœ p

œ # œ–™

Do#

nœ

— — nœb—œ– – — mp p

– – — —

œ #œ

bœ # œ œ bœ #œ œ

193

fi nœ.nœ #œ. nœ. ≈ Œ

∑

‰

La§

‰

‰ nœ #œ #œ nœ œœ ... nœ p

nœ nœ

∑

œ œ bœ œ

∑

Piano

nœ ‰ #œ ¸nœ # œ # œfi nœ mf

∑

Harpe

Harmonic reduction

œ™

13 ralentir brusquement - - -puis très progressivement - - - - - - - - - - - - - - - - - - - - - - - - - - (ne pas presser) - - - - - - - - - - - - - - - - - - - - - - ¸ . ¸ fi¸nœ. ≈nœfi #œnœ ∑ Œ ≈ ‰ nœ#œfi œ ¸nœ ≈ ‰ nœ. nœ.#œ.nœ ˝ . . b œ. . # œ. nœ.n œ. .b œfi ˝

˙™

Marimba

p pp

ouvert

∑

Vibraphone

œ œ œ. œ. œ. œ. #œ. #œ. pp . . n œ. # œ. # œ nœ. n œ œ nœ- œ.

Violoncelle

p pp

˙™

#œ n œ- œ œ. œ. œ. nœ. #œ. .

œ™

˙™

& <n>˙ ™

Alto

¸ n œfi b œ

b˙™

SACHER + All-trichordal hexachord

bbœœœœ œœ œ

A/G# axis

œ œ

œ bœ

#œ #œ nœ

Ex. 3.3.15. (cont’d)

Cor anglais

Clarinette en la

Basson

Violon

n˙ ™ >

> b˙ ™

> œ

ffp

#>˙ ™

>œ

ffp

arco

œ œ

œ œœœœœœœ œ œ. œ. œ. œ. œ. œ. œ.

Violoncelle

Cor en fa

44 ° & ¢&

˝œ ≈ ‰

‰ b˝œ. nœ nœ . . nœ. # œ. n œ n œ . . 3

‰

∑

∑ œœ™™

nœnœ bœ n œ

bœ bœbœ bœ bœ

& bn#nœœœœœ

bnœœ

nœ

œ œ

D axis

n œ bœ #œnœnœ

nœ nœ

œ bœ bœ * °

bœ

#œ

bœ nœbœœ

bœ nœ f

nœ bœ bœ n œ n œ

¸ ≈ ‰ # œ.

#œ

nœ. nœbœbœnœ . . . . nœ.

nœ. #œ. nœ. bœnœ . . n œ.

¸≈ ‰ n œ.

nœ. bœ. nœ ¸ ≈ ‰ bœ. ¸ . n œ.

nnœœ

œ œ #œ #bnœœœ

3 3

nœ

œnbœœ nœbœœ nœbœœ

∑

nœ n œ nœ bœn œ

œœ nœ

bœ n œ n œ n œ

*°

D axis

*°

bœn œ # œ nœ #œ nœ

nœ

b#œœœ œ bœ œ œ

n œœ

bœ. bœ nœ nœ¸ ≈ ‰ bœ¸ . . . .

nœ #œ

n œ n œ nœbœ

¸≈ ‰ nœ.

Si§

œœ

œœ nnœœ.

œnœnœ nœnœnœ

> œœ

œbœnœ

œ >

bœ nœn œ œ œ

#nœœ

Fa# Sib

¸ ≈ ‰ nœ.

nœnœnœ nœ

œ œ

nœ

bœ #œ n œ n œ n œ nœ

#œ œnœnœ nœn œnœ

Sol§

? b#œœœœ

œ #œ#œ œ œ œ

3 . ‰ #˝œ #œ. #œ. #œ. nœ. nœ nœ . . 3

Piano

Harmonic reduction

∑

‰

nœ. nœ. nœ ¸≈ ‰ . n œ. n œ. # œ n œ . . 3

¸≈ ‰ œœ

#œ. nœ. #œ. . . #œ #œ #œ. nœ¸ ≈ ‰ . 3

˙˙

Harpe

Marimba

n˙ ™ >

Vibraphone

œ bœ bœ œ œ œ

ffp

arco > nn˙˙ ™™

œ œœœœœœœ œ œ. œ. œ. œ. œ. œ. œ.

nœ b >œ

œ œ

Alto

#œ n>œ

œ #œ nœ Œ

œ >

*°

°*

194

Ex. 3.3.15. (cont’d)

Cor anglais

Violon

≈#œnœ nœ 3

≈#œ #œ nœ ≈nœ

œ pp

≈nœ n œ nœ 3

≈#œnœnœ

‰

∑

Violoncelle

Cor en fa

48 ° Vibraphone & Marimba

‰

¢&

¸≈ ‰ nœ

¸ œ n œfi¸nœ ™

nœ

nœ—

* °

‰

¸ ¸ n œfi nœ

#œ

Fa§

fi¸ nbœœ nœ¸

fi¸ nnœœ

œ™ ∑

∑

.¸ nœfi nœ¸

bœ #n œœœœœn#nn nœœœœœœ # #

195

‰

¸ ≈ œ

#œ. nœ. nœbœ. . pp

sans

° ≈

# œ #œnœ nœ pp

nœ—

fi¸ n# œœ

fi¸ #nœœ

?œ

. nbœœ nœ¸ mf fi¸

nbœœœ # œœ

∑

* ° ‰

& nœ

≈

‰

bœ bœ œœ #nœœ œ # œœœ # œœ #n œœœ nœœœœ

bœ ¸ œ # œ n œ nœ ™

* °

#œ

. fi¸ #bœœ n˙

D axis

¸ fi¸ œ nœ nœ ™

fi¸ #nœœ

¸ œ n# œfiœ¸ nœ ™ Œ

‰

œ#œnœ# œ nœ

≈

˝œ ≈

Piano

Harmonic reduction

¸ n œfi nœ

∑

≈

≈nœ # œ nœ

‰

nœ ™ ˝

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Chapter 3.4. Revisions It was a period where a long-held doubt had passed. Without doubt, you finish – and with doubt, you have a tendency never to finish. And that’s what moved me to refrain from touching certain works again. But there are some works which are unfinished not because I gave up, but because the reflection on the content of the work, on the structure of the work, was not very clear to me. Therefore I do return to works, but there are also some works which I don’t touch again – Derive II, for instance, I will not touch again. It’s finished because I worked on it for quite a few years, and then I discovered of way of structuring, of composing the work which was totally different than what I had done up to that point, a sort of narrative aspect of the work. It was then that I saw that this narration was finished, and that I could not add anything – the addition would have been totally artificial. There are other works which I want to finish, and some other works which I don’t want to finish. You know, I have a relationship with my work which is very sentimental.16 Throughout his life, Boulez has revised almost every work of his output, and Dérive 2 is no exception.17 The compositional history of Dérive 2 can be divided into four periods with three principal publications of different versions. The first period starts from the initial conception of the piece in 1988 (or perhaps even earlier) to its premiere in 1990. The 1990 version of the work lasts only about 3 to 4 minutes. Boulez then expanded the piece to 7 minutes in 1993, and continued to do so until 1996 when it was published for the first time. This first version was published as UE19128, and it was performed on September 29th, 1996. The second version was published in 2002, and this version was premiered on September 2nd, 2002 at the Lucerne Festival in Switzerland. This version lasts approximately 24 to 25 minutes. This is also the same version as what is available on 16

Pierre Boulez, “Interview Pierre Boulez: Sometimes You Discover Yourself!” in Musikblätter 1, ed. Wolfgang, Schaufler (Vienna: Universal Edition, 2011): 6-7. 17 “All the works I write are basically different facets of one central work, of one central concept. In any case I cannot easily detach myself from a particular musical universe; once I have set it in motion it has a tendency to become independent and to grow. Unfortunately, circumstances sometimes prevent me from expanding it sufficiently, and this is why there are still ‘expanding universes’ that have not reached a completed state. I cannot bring myself to let go of material that is still alive for me.” Boulez, Par volonté et par hasard: entretiens avc Célestin Deliége, 50.

196

the first commercial recording that was recorded a few days earlier than this premiere in Lucerne. Boulez continued to revise and expand the piece. The next major publication of Dérive 2 took place in 2006; this version was premiered on July 13th, 2006, at the Aix-enProvence Opera Festival. This version lasts approximately 45 minutes, and so is significantly longer than its predecessors. This third edition was published as UE 32528, and all the subsequent revisions and additions have been made based on this version. This latest version contained Boulez’s last major addition to the work, as all the subsequent revisions to follow were minor revisions to the Coda and the sections at R135. According to Universal Edition, the latest version was finalized in 2009, and this version lasts approximately 45 to 46 minutes, though Boulez continues to make adjustments to the piece after 2009. During these intervening years of revisions, the central concept of Dérive 2 remained unaltered. As mentioned previously, the overall formal structure that is derived from the underlying periodicity was already predetermined by its premiere in 1990, and what is left to be determined is how these materials will be realized and transformed. What is remarkable about the compositional history of Dérive 2 is that all the earlier versions did not include any interrupting sections. These interruptions were not added until 2001 (when it was performed on December 1st, 2001). From 2001 to 2004, the interruption section at R10 only contained three measures, and this interruption section was revised into its current version on October 13th, 2004, where the last horn solo was also heard for the first time. This entire version lasts approximately 25 and half minutes. The opening sustained horn note was absent in the earlier versions, and it was not added

197

until 1993. Sections A and B (referring the sections in the periodicity) in the earlier versions are the same as those in the most current version, but Sections C and D were realized completely differently in these earlier performances. These two sections were revised between 2001 and 2002, and the commercial recording reflects their changes as they match the most current version. The midpoint section, majority of sections after the midpoint, and their respective interruptions were composed between 2005 and 2006, at the time when the work was expanded from 25 minutes to 45 minutes.18 Besides the drastic expansions that Boulez has made during these intervening years, most of the revisions and the editing processes were focused on areas of tempo, dynamics, articulations, phrasing, and even notational issues. These changes most likely reflect Boulez’s experience as conductor as he rehearsed and performed the work with the same ensemble over the years, and many of the changes were probably made during those rehearsals. Through these rehearsals, Boulez had first hand experience regarding how the work sounded, and how the work was performed by the musicians – all these experiences informed him on how to re-notate certain passages, or revise certain sections. The most obvious case is in Coda I: here the two rhythmic strata were initially realized simultaneously, but were recomposed in the later versions so that one of the two strata progressively falls an 8th-note later to allow more room between the two strata. Another example is in Section C, where the earlier versions were composed with extremely complicated rhythmic juxtapositions, and the density of texture was overwhelming. In the 18

This process of continuing revision to a work can be best compared to Répons, as Boulez comments, “[a]s for Répons, where I progressively added elements, you are in the midst of a spiral form. Even if you add new sections, it does not change the fact that it is still a spiral; you only have to mark the beginning and end of this process, so in a sense it does not matter where you stop. I am very found of this type of form that is infinite, with its double meaning, which can be extended indefinitely.” Di Pietro, Dialogues with Boulez, 69.

198

later versions, Section C is filled with rests and spaces, which allowed the underlying rhythmic patterns to be more clearly projected. Some of the notational changes made during these years were intended to simplify the score, but these changes are minute compared to the changes Boulez has made in other sections of the work. All the changes after 2009 generally reflect Boulez's desire to expand and to proliferate musical materials to their maximum potential. A clear example can be found in R135, where the solo violin part in version 2009 is stated as a single flowing melody, while the most current version is decorated with full of grace-notes and ornamentations to this simple melody. Even though the work is done, motifs can still be developed: This does not mean that works are never finished but they that it may make twenty years to find solutions to the problems the work itself has proposed is very real for me. I think the most interesting example I can give is the case of Proust, in which I find a model for the kind of thing we are talking about â&#x20AC;&#x201C; I mean the work being one long process throughout life. In the Recherche du temps, you find that Proust gives one character a particular theme or motif, only to find that the motif has been taken up much later by another character and entirely transformed, even if it is several volumes later. I was inspired by this in my own work Notations, where a seed from a piece is transformed and developed thirty years later into a gardenlike expansion of the orchestral Notations. This process of growth is very much like Proust, which I found to be very interesting as a model for my own evolution as perhaps the closest of anyone to my own way of working.19 Since the entire formal structure of DĂŠrive 2 is based on periodicity, the form was predetermined from the very beginning. Section A was composed first, and the sketches show that Boulez had composed the opening measures in many versions, trying out different realizations of the periodicity. It was clear from the very beginning that the rhythmic pattern 2-1-3-1-1-2-1 was conceived before other parameters and the idea of EN 19

Rocco Di Pietro, Dialogues with Boulez (Lanham: Scarecrow Press, 2001), 3.

199

cycle came relatively later. The function of the resonant instruments doubling every fourth composite element was also conceived later, as many of the earlier drafts contained different functions for these instruments. After Boulez had composed the first two sections, Section A and Section B, he then composed Section L instead of continuing consecutively to Section C. This is only possible if the periodicity chart had already been conceived and constructed in the beginning stages of the compositional process. In an interview, Boulez states that “[t]he concept of the end of Dérive II was also already there fifty years ago, but it was too soon: so I composed it, and knowing that I would compose a long development in between, well, I jumped to the end – because the end was already there. Sometimes you think of the end long before the rest of the piece.”20 Even though the interruptions, the midpoint section, and the Codas were composed and revised most recently, these sections do not alter the underlying formal structure of the piece, as their rhythmic and pitch constructions are not derived from the periodicity chart. These additions were added simply to expand and further allow materials to proliferate. The idea of these interruption passages, and their contrast to the periodicity architecture came during the compositional process sometimes after 1993. In an interview, Boulez states, “in Derive II … there are some moments where I decided I must have a sort of static development. I thought of that long ahead of time, when it would come in the structure of the piece, but I discovered at the last moment a satisfying way of dealing with it. I didn’t foresee how to fill this section; it was conceived in the process of composing.”21 These interruptions, similar to the homophonic sections in Rituel, or the

20 21

Pierre Boulez, “Interview Pierre Boulez: Sometimes You Discover Yourself!” 7. Paul Steenhuisen, “Interview with Pierre Boulez,” 67.

200

periodic signals in Mémoriale22 and Anthèmes, were composed and inserted later in the compositional process.23 It is unclear whether Boulez had planned these interruptions in the early stages of composing the work (even though the earlier versions did not include interruptions, Boulez could have already had the conception in his mind, though they had yet to be written),24 the interruptions nevertheless serve as pillars that articulate the architecture of the periodicity, or simply, to deliberately destroy and alter the formal symmetry inherent in the periodicity structure. However, unlike Rituel and the earlier works where the interruptions were initially meant to serve as sectional dividers, in Dérive 2 they actually overshadow and out-grow the periodicity sections and become an integral part of the work. These interrupting sections, then, create an entirely new path of their own, deviating from the periodicity and the overall structure.25 The ending of Dérive 2 is perhaps the most complicated one to decipher, since Boulez has made most of his revisions to this last section. The earlier versions before 1993

In Mémoriale, the opposition between groups of appoggiaturas and long polar notes is decisive for the articulation of form. This in turn is related to one of the possible meanings of the title ‘… explosante-fixe…’.” Jonathan Goldman, “Charting Mémoriale: Paradigmatic Analysis and Harmonic Schemata in Boulez’s …Explosante-Fixe…,” Music Analysis 27, no. 2-3 (July/October 2008): 217. 23 This according to Paolo Dal Molin who proposes that these homophonic sections in Rituel were composed and inserted later in order to better articulate the heterophonic sections. See Paolo Dal Molin, “À propos des versets de Rituel in memoriam Bruno Maderna,” Mitteilungen der Paul Sacher Stiftung 17 (March 2004): 2934. Goldman also makes the similar proposition in his book. See Jonathan Goldman, The Musical Language of Pierre Boulez, 105. 24 "Form can be completely viewed in a precise and complete vision from the beginning, and little by little realized over the course of its development. On the other hand, certain accidents or gestures can modify the transcription of this form, deforming it to a point in which it has very little in common with the form conceived in a vision," Pierre Boulez, 1965 Darmstadt lectures, transcribed by Jonathan Goldman, in Goldman, "Exploding/Fixed: Form as Opposition in the Writings and Later Works of Pierre Boulez" (diss., University of Montreal, 2006): Appendix I. 25 Boulez makes a comparison of his method of non-linear approach to structure to Proust’s: “Just as in the course of Remembrance of Things Past the structure does not keep to the straight and narrow, my work starts from a fairly straightforward path and goes through some complications before arriving. In devising it I came to thinking about the sheets Proust adds to the first drafts of his work … In a word, I know where I’m going but I make many detours in getting there. …Proust represents an ideal of initial simplicity realized through complexity.” Boulez, “Interview with Philippe Olivier,” Libération, 25 Septemper, 1984.

201

all finished on a single block chord played by the full ensemble, and this roughly corresponds to the end of R249-3 in the current version, with a single block chord on this measure (ex. 3.4.1). The 2001 version concludes the piece with an added long held G# played by the solo horn after the 1993 version (ex. 3.4.2). By the end of 2001, this long held G# note was followed by a tutti chord (ex. 3.4.3). In the commercial recording, the ending corresponds roughly to the end of R251 in the current version (ex. 3.4.4). By 2004 and 2005, Boulez added a few more block chords for the full ensemble, and it is followed by a single sustained G# played by the horn. Compare to the most current version, this version did not have the doubling G#s between the block chords, and the ending was only played by the horn (ex. 3.4.5). In the 2006 and 2007 version, the sustained G# after R251 was also absent between the block chords, however, the last sustained G# is now supported by the full ensemble (ex. 3.4.6). In 2007, between these block chords after R251, the piano was added playing chords of its own. It is not until 2009 that these changes to the Coda were finalized to its most current version, as appears in example. 3.4.7. From careful observation of the sketches and the earlier versions of DĂŠrive 2, it is safe to assemble a chronology for this unusual compositional expansion: Sections A, B, the earlier versions of C and D, Section L, and the earlier versions of Coda were composed first. Boulez then progressively filled in the middle sections, while expanding the Coda sections. It was not until much later that Boulez inserted interruptions as a way to intensify the formal structure of the periodicity. All these revisions and additions were completed by 2006, and the subsequent revisions were incidental improvements to the

202

work, mostly on the Coda and the added grace-notes in the violin part in R135. Nothing relating to formal structural was added or altered in these recent revisions.

203

Ex. 3.4.1. Ending for the 1993 version Cor anglais

° &

Violon

Cor en fa

¸> nœfi#œÆ

¸ bœfinœ¸

œ nœ nœ' ' nœ' '

nœ ' nœ #œ # œ nœ ' ' ' '

¸ ¸ b œfi # œ

Æ Æ nœÆ n œ # œ nœÆ

#œÆ nœÆ

Æ #œÆ nœÆ nœ

bœÆ #œ nœfi¸##œœ nœÆ bœÆ '> #œ ' ' ff

Æ Æ nœÆ nœ #œ nœÆ

Æ nœÆ nœ #œÆ

Æ n œÆ #œÆ #œÆ #œ

B #œ nœ ' ' ¢

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Alto

Violoncelle

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¸ ¸ nœ n œfi œ '

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nœ nœ bœ' nœ ' ' ' ff

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n œ # œ' # œ ' '

nœ #œ #œ nœ ' ' ' ' ff

nœ nœ #œ ' ' '

#œ #œ' # œ nœ ' ' '

nœ #œ nœ ' ' '

nœ nœ' #œ #œ' ' ' ff

# œ nœ' nœ' '

Æ #œÆ nœÆ #œÆ bœ

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° Vibraphone & #œ #œ ' '

Marimba

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n#œœ >'

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& nœ bœ

Piano

Æ nœÆ bœ bœÆ œÆ

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nœ nœ ' #œ #œ nœ ' ' ' '

nœ nœ nœ nœ

#œ

nœ #œ nœ nœ

nbœœ nœ #œ nœ

bnœœ bnœœ

bœ n œ b œ

bn œœ

“” >Æ b œÆ b œ nb œœ n œ ˝

b œÆ nœ

nbœœ #œ nœ bœ nœ œ nœ b œÆ Æ nbœœ

n œÆ nœ

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204

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> n nœœÆ nœ ‰ ‰ ? ˝

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° &

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nœ ' b œ'

nœ '

¸ #œ > '

nœ # œ ' '

nœ '

fff sempre

#œ '

#œÆ

nœÆ

fff

Clarinette en la

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fff

Basson

Violon

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# œÆ

#œÆ

¸ ¸ nœfi #œ

n œ #œ ' >' fff

¸ œ¸ nœfi##œ

Violoncelle

Cor en fa

b œfi¸n> œÆ

nœ ˝

nœÆ

¸ ¸ n œfi œ ff

° Vibraphone &

nœÆ

¢&

¸ ¸ #œfi #œ

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n œÆ

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249

fff sempre

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fff

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205

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Ex. 3.4.2. Ending for the 2001 version Cor anglais

° &

Violon

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> Æ Æ n œ nœ nœ ˝

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nœ ' # œ' n œ' nœ' œ ' f

nœ nœ bœ' nœ ' ' ' ff

bœ '

n œ # œ n œ nœ' ' ' ' ff

n œ # œ' # œ ' '

nœ #œ #œ nœ ' ' ' ' ff

nœ nœ #œ ' ' '

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nœ #œ nœ ' ' '

nœ nœ' #œ #œ' ' ' ff

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∑

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n œÆ b œÆ n œÆ #œÆ

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n œÆ # œÆ n œÆ

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248 (accél) - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

° Vibraphone & #œ #œ ' '

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¢& nœ

#œ

¸ #œfi ff

n#œœ >'

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& nœ bœ

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#œ nœ bœ

nœ nœ ' #œ #œ nœ ' ' ' '

nœ nœ nœ nœ

#œ

nœ #œ nœ nœ

nbœœ nœ #œ nœ

bnœœ bnœœ

bœ n œ b œ

bn œœ

“” >Æ b œÆ b œ nb œœ n œ ˝

b œÆ nœ

nbœœ #œ nœ bœ nœ œ nœ b œÆ Æ nbœœ

n œÆ nœ

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¸ n#œœ #œ' #œ' '> ¸

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bbœœ

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¸ bœfi bn œœ

& Harpe

nœÆ nœÆ #œÆ

‰

¸ ¸ bœfibn œœ

#œ

nœ

bœ b œ

œ bnnœœ '

206

nœÆ bœ #œœ

nœ nœ #œ ' ' ' #œ ' ff nœ

> n nœœÆ nœ ‰ ‰ ? ˝

#œ nœ nœ

nn œœ

‰ ‰?

> n œÆ ## œœ ˝ ‰ ‰?

bœ ' n œ'

nnœœ bb œœ b#œœ

#œ n œ # œ ' ' ' nœ n œ #œ

bnœœ

bn œœ

nœ '

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nbœœ '

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#nœœ

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bœ œ ' '

Ex. 3.4.2. (cont’d) Cor anglais

° &

¸ ¸ nœfinœ

nœ '

nœ b œ ' '

nœ '

¸ #œ > '

nœ # œ ' '

nœ '

#œ '

#œÆ #œÆ #œÆ

nœÆ

fff

Clarinette en la

¸ # œfinœ'

fff

Basson

Violon

¸Æ nœfi#> œ

# œÆ

fff sempre

¸ ¸ nœfi #œ

Violoncelle

Cor en fa

b œfi¸n> œÆ

nœ ˝

° Vibraphone &

nœÆ

nœÆ #œÆ

n œÆ

# œÆ

¢&

œ œ '

n#œœ '

nœÆ #œÆ nnœœÆ

œœÆ

Æ nnœœ

œœÆ

Æ nœÆ nnœœ n œ

##œœÆ

œœÆ

#>˙ ™ #˙ ™

#œ '

nœ '

#œ '

fff sempre

nœÆ

nœÆ ##œœÆ

Piano

œœÆ

fff sempre

Œ™

249

¸ ¸ #œfi #œ

# œ #œ' #œ' #n#œœ'œ œœ'œ #n#œœ'œ œœ'œ œœ'œ œœ'œ œœ'œ fff sempre >' ¸

œ ¸œ # œfin>

nœ

fff

n nœœ œœ #œ #œ œ

fff sempre

¸ ¸ bœfibn œœ

n nœœ œœ œœ œœ œœ #œ œ œ œ œ

^ Œ œœ œœ '

^ œ #œ ' ff

^ œÆ

œœ '

œœ nn œœ nnœœ ' ' >'

œœÆ #nœ>œ œœ

^ œœ #œœÆ

^ œ # œÆ œ #œ

nœ '

nœ # œ ' '

œœ '

œ '

n nœœ nnœœ nœ #œ

nœ '

¸ ‰ œ '>

U + #˙ ™ ff

#n#œœœ n#œœ ' '

^¸ Œ n œœ

¸ œ ‰

‰

∑

> n œÆ ‰ #œnœ ˝

>Æ ‰ nn#œœœ ˝

> bœÆ ‰ bnœœ ˝

^¸ Œ bœ bb œœ '

bœ

Œ™ ^¸ ‰ nb#b œœœœ <“> '

fff

^ œ '

fff sempre

n œ #œ' ' >'

nœ >

(accél) - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - (q = 200)

fff

Harpe

œ '

fff

Marimba

n>œ

œ œ '

¸ ¸ n œfi œ ff

nœ #œ #>˙ ' '

œ œ '

fff

nœ '

#œœ '

fff

nœ '

œ #œ ' '

œ œ '

fff

¸ œ¸ nœfi##œ

#œ '

#œ #œœ ' n'

n œ #œ ' >'

nœ '

fff sempre

fff

° &

Alto

fff sempre

nœ #œ ' #œ ' '

nœ '

nœ #œ ' '

‰

“” b œÆ n œÆ nœ nnœœ fff

Mi§ Si§

∑

nn œœÆ n œÆ n œ #œ n#œœÆ #œÆ nœ #œ #nœœÆ #œ nœ nœ #œ # œ #œ b œ bn œœ # œ #œ' ' ' nœ ' '

¸ ‹#œœ ‰ '

^ # nœœœÆ n œ ‰ ˝

‰

#nœœÆ ˝

207

Ex. 3.4.3. Ending for the second 2001 version Cor anglais

° &

Violon

Cor en fa

¸> nœfi#œÆ

¸ bœfinœ¸

œ nœ nœ' ' nœ' '

nœ ' nœ #œ # œ nœ ' ' ' '

¸ ¸ b œfi # œ

Æ Æ nœÆ n œ # œ nœÆ

#œÆ nœÆ

Æ #œÆ nœÆ nœ

bœÆ #œ nœfi¸##œœ nœÆ bœÆ '> #œ ' ' ff

Æ Æ nœÆ nœ #œ nœÆ

Æ nœÆ nœ #œÆ

Æ n œÆ #œÆ #œÆ #œ

B #œ nœ ' ' ¢

° &

Alto

Violoncelle

nœ ' #œ nœ nœ bœ ' ' ' '

nœÆ œÆ bœÆ nœÆ

¸ # œfinœ'

Basson

bœÆ nœÆ Æ bœ

¸ #œ #œ ' ' nœ' >

Clarinette en la

¸ ¸ nœfinœ

Æ B nœÆ nœ

¸ œfi##œ œ

> b œfi¸ nœÆ nœ

# œÆ Æ Æ nœ nœ

∑

¸ ¸ nœfi #œ #œ ff

n œ ' #œ ' ' > >Æ

¸ nœ #œfin ˝œ

nœÆ nœÆ

> Æ Æ n œ nœ nœ ˝

n œfi¸nœÆ ff

¸ ¸ nœ n œfi œ '

nœ ' # œ' n œ' nœ' œ ' f

nœ nœ bœ' nœ ' ' ' ff

bœ '

n œ # œ n œ nœ' ' ' ' ff

n œ # œ' # œ ' '

nœ #œ #œ nœ ' ' ' ' ff

nœ nœ #œ ' ' '

#œ #œ' # œ nœ ' ' '

nœ #œ nœ ' ' '

nœ nœ' #œ #œ' ' ' ff

# œ nœ' nœ' '

Æ #œÆ nœÆ #œÆ bœ

Æ nœÆ nœ bœÆ

œ bœ ' ' nœ '

nœ # œ # œ' ' '

b œÆ Æ Æ Æ Æ n œ b œ #œ n œ

∑

‰ ‰

#œÆ nœÆ #œÆ nœ bœ' '

∑

‰ ‰

n œÆ Æ Æ Æ Æ bœ nœ nœ nœ

n œÆ b œÆ n œÆ #œÆ

‰ ‰

n œÆ # œÆ n œÆ

¸> nœfi#œÆ

nœ '

nœ ' ff

248 (accél) - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

° Vibraphone & #œ #œ ' '

Marimba

¢& nœ

#œ

¸ #œfi ff

n#œœ >'

¸œ œ # œfi n >

& nœ bœ

Piano

Æ nœÆ bœ bœÆ œÆ

#œ nœ bœ

nœ nœ ' #œ #œ nœ ' ' ' '

nœ nœ nœ nœ

#œ

nœ #œ nœ nœ

nbœœ nœ #œ nœ

bnœœ bnœœ

bœ n œ b œ

bn œœ

“” >Æ b œÆ b œ nb œœ n œ ˝

b œÆ nœ

nbœœ #œ nœ bœ nœ œ nœ b œÆ Æ nbœœ

n œÆ nœ

nœ bœ '

#œ œ ' '

¸ bœfi ff

¸ n#œœ #œ' #œ' '> ¸

¸#œ œ # œfi# >

bbœœ

Æ nn œœ

Æ n#œœ

“” > nnœœÆ n˝œ

¸ bœfi bn œœ

& Harpe

nœÆ nœÆ #œÆ

‰

¸ ¸ bœfibn œœ

#œ

nœ

bœ b œ

œ bnnœœ '

208

nœÆ bœ #œœ

nœ nœ #œ ' ' ' #œ ' ff nœ

> n nœœÆ nœ ‰ ‰ ? ˝

#œ nœ nœ

nn œœ

‰ ‰?

> n œÆ ## œœ ˝ ‰ ‰?

bœ ' n œ'

nnœœ bb œœ b#œœ

#œ n œ # œ ' ' ' nœ n œ #œ

bnœœ

bn œœ

nœ '

#œ ' “‘

nbœœ '

œ ' “‘

#nœœ

#œ '

bœ œ ' '

Ex. 3.4.3. (cont’d) Cor anglais

° &

¸ ¸ nœfinœ

nœ nœ >' ' ' b œ' n œ'

n œ # œ nœ' nœ' nœ' ' '

¸ # œ nœ # œ n œ > ' ' ' '

œ # œ nœ' ' '

fff sempre

fff

Clarinette en la

¸ # œfinœ'

Basson

Violon

¸Æ nœfi#> œ

# œÆ

¸ ¸ nœfi #œ

¸ œ¸ nœfi##œ

Violoncelle

Cor en fa

° Vibraphone &

¢&

œœÆ

Æ nœÆ nnœœ n œ

##œœÆ

œœÆ

#>˙ ™ #˙ ™

^ œ # œÆ œ #œ

#œ nœ nœ nœ # œ n œ ' ' ' ' ' '

¸ ‰ œ '>

Piano

^ œœ #œœÆ

#œ nœ ' ' ff

^ œÆ ˝

^¸ œœ '

‰

fff sempre

^ œœÆ ˝ ff

‰

^ œÆ œ ˝

U + #˙ ™ ff

(accél) - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - (q = 200)

249

¸ ¸ #œfi #œ

# œ #œ' #œ' #n#œœ'œ œœ'œ #n#œœ'œ œœ'œ œœ'œ œœ'œ œœ'œ fff sempre >' ¸

œ ¸œ # œfin>

nœ

fff

n nœœ œœ #œ #œ œ

fff sempre

¸ ¸ bœfibn œœ

n nœœ œœ œœ œœ œœ #œ œ œ œ œ

#n#œœœ n#œœ ' '

œœ '

œ '

n nœœ nnœœ nœ #œ

b œ bœ

Œ™ ^¸ ‰ nb#b œœœœ <“> '

‰

“” b œÆ n œÆ nœ nnœœ fff

Mi§ Si§

∑

¸ œ ‰

‰

^¸ Œ n œœ

>Æ ‰ nn#œœœ ˝

> bœÆ ‰ bnœœ ˝

¸ ‹#œœ ‰ '

^ # nœœœÆ n œ ‰ ˝

‰

#nœœÆ ˝

#œ nœ nœ #œ # œ #œ b œ bn œœ # œ #œ' ' nœ ' ' '

> œœÆ nœ ˝ ff

‰

^¸ n œœ ff

∑

nn œœÆ n œÆ n œ #œ n#œœÆ #œÆ Æ nœ #œ #nœœ

‰

∑

> n œÆ ‰ #œnœ ˝

>Æ ‰ nnœœœ ˝ ff

fff

œœÆ #nœ>œ œœ

^¸ œ '

‰

œœÆ

Œ™

‰

œœÆ

^ Œ œœ œœ '

Æ nnœœ

fff sempre

‰

œœÆ

œœ nn œœ nnœœ ' ' >'

^¸ œ '

nœÆ #œÆ nnœœÆ

œœÆ

‰

n#œœ '

n œ #œ' >' '

^ œÆ

œ œ '

¸ ¸ n œfi œ

fff

Harpe

œ œ '

fff

Marimba

# œÆ n>œ

fff sempre

œ œ '

fff

^ œ #œ '

œ œ '

Æ Æ n œ nœ nœ ##œœÆ ˝

nœ #œ #>˙ ' '

œœ '

#œœ '

b œfi¸n> œÆ

^ œ '

œ œ '

fff

#œ #œœ ' n'

n œ #œ >' '

œ n>œ '

fff sempre

fff

Alto

Æ nœÆ nœÆ nœÆ #œÆ n œ

#œÆ #œÆ #œÆ

fff

° &

# œ n œ' '

fff sempre

fff

#œ nœ' #œ' ' #œ '

>Æ œ ‰ nœœ ˝

^ œœœÆ n n œ ‰ ˝

^¸ Œ bœ bb œœ '

‰

^¸ œ œœ '

209

Ex. 3.4.4. Ending for the 2002 version Cor anglais

° &

Violon

Cor en fa

¸> nœfi#œÆ

¸ bœfinœ¸

œ nœ nœ' ' nœ' '

nœ ' nœ #œ # œ nœ ' ' ' '

¸ ¸ b œfi # œ

Æ Æ nœÆ n œ # œ nœÆ

#œÆ nœÆ

Æ #œÆ nœÆ nœ

bœÆ #œ nœfi¸##œœ nœÆ bœÆ '> #œ ' ' ff

Æ Æ nœÆ nœ #œ nœÆ

Æ nœÆ nœ #œÆ

Æ n œÆ #œÆ #œÆ #œ

B #œ nœ ' ' ¢

° &

Alto

Violoncelle

nœ ' #œ nœ nœ bœ ' ' ' '

nœÆ œÆ bœÆ nœÆ

¸ # œfinœ'

Basson

bœÆ nœÆ Æ bœ

¸ #œ #œ ' ' nœ' >

Clarinette en la

¸ ¸ nœfinœ

Æ B nœÆ nœ

¸ œfi##œ œ

> b œfi¸ nœÆ nœ

# œÆ Æ Æ nœ nœ

∑

¸ ¸ nœfi #œ #œ ff

n œ ' #œ ' ' > >Æ

¸ nœ #œfin ˝œ

nœÆ nœÆ

> Æ Æ n œ nœ nœ ˝

n œfi¸nœÆ ff

¸ ¸ nœ n œfi œ '

nœ ' # œ' n œ' nœ' œ ' f

nœ nœ bœ' nœ ' ' ' ff

bœ '

n œ # œ n œ nœ' ' ' ' ff

n œ # œ' # œ ' '

nœ #œ #œ nœ ' ' ' ' ff

nœ nœ #œ ' ' '

#œ #œ' # œ nœ ' ' '

nœ #œ nœ ' ' '

nœ nœ' #œ #œ' ' ' ff

# œ nœ' nœ' '

Æ #œÆ nœÆ #œÆ bœ

Æ nœÆ nœ bœÆ

œ bœ ' ' nœ '

nœ # œ # œ' ' '

b œÆ Æ Æ Æ Æ n œ b œ #œ n œ

∑

‰ ‰

#œÆ nœÆ #œÆ nœ bœ' '

∑

‰ ‰

n œÆ Æ Æ Æ Æ bœ nœ nœ nœ

n œÆ b œÆ n œÆ #œÆ

‰ ‰

n œÆ # œÆ n œÆ

¸> nœfi#œÆ

nœ '

nœ ' ff

248 (accél) - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

° Vibraphone & #œ #œ ' '

Marimba

¢& nœ

#œ

¸ #œfi ff

n#œœ >'

¸œ œ # œfi n >

& nœ bœ

Piano

Æ nœÆ bœ bœÆ œÆ

#œ nœ bœ

nœ nœ ' #œ #œ nœ ' ' ' '

nœ nœ nœ nœ

#œ

nœ #œ nœ nœ

nbœœ nœ #œ nœ

bnœœ bnœœ

bœ n œ b œ

bn œœ

“” >Æ b œÆ b œ nb œœ n œ ˝

b œÆ nœ

nbœœ #œ nœ bœ nœ œ nœ b œÆ Æ nbœœ

n œÆ nœ

nœ bœ '

#œ œ ' '

¸ bœfi ff

¸ n#œœ #œ' #œ' '> ¸

¸#œ œ # œfi# >

bbœœ

Æ nn œœ

Æ n#œœ

“” > nnœœÆ n˝œ

¸ bœfi bn œœ

& Harpe

nœÆ nœÆ #œÆ

‰

¸ ¸ bœfibn œœ

#œ

nœ

bœ b œ

œ bnnœœ '

210

nœÆ bœ #œœ

nœ nœ #œ ' ' ' #œ ' ff nœ

> n nœœÆ nœ ‰ ‰ ? ˝

#œ nœ nœ

nn œœ

‰ ‰?

> n œÆ ## œœ ˝ ‰ ‰?

bœ ' n œ'

nnœœ bb œœ b#œœ

#œ n œ # œ ' ' ' nœ n œ #œ

bnœœ

bn œœ

nœ '

#œ ' “‘

nbœœ '

œ ' “‘

#nœœ

#œ '

bœ œ ' '

Ex. 3.4.4. (cont’d) Cor anglais

° &

¸ ¸ nœfinœ nœ

nœ >' ' ' b œ' n œ'

fff sempre

fff

¸ nœ # œ n œ ' ' ' >

Clarinette en la

¸ # œfinœ' # œ'

fff sempre

fff

Basson

Violon

¸ Æ #œ Æ nœfi#> œ #œ

¸ ¸ nœfi #œ

n œ # œ #œ' n#œœ œœ ' ' >' '

fff sempre

fff

¸ œ¸nœÆ #œÆ nœfi##œ

Alto

Cor en fa

nœ #œ #>˙ ' '

Æ nœÆ nœÆ nœÆ #œÆ n œ

# œÆ n>œ ˙

Æ Æ nnœœ œœ

>' '

¢&

Piano

‰

^Æ ^ nœÆ œÆ œÆ œÆ œÆ ‰ #˝œ ‰ ˝ p

Æ Æ Æ Æ Æ nnœœ œœ œœ œœ œœ

Æ nœÆ œÆ > nnœœ n œ œ#nœœ œœ

^Æ ^ Æ Æ Æ Æ Æ œœ#nœœÆ œœÆ œœÆ œœÆ œœÆ œœÆ ‰ ##œœ œœ œœ œœ ˝œœ ‰ n#˝œœ ‰

Œ™

¸ ¸ #œfi #œ #œ #œ#n#œ œœ

#œ nœ ' ' ff

249

# # œ ' ' ' œœ'œ #n œœ'œ œœ'œ œœ'œ œœ'œ œœ'œ fff sempre >'

¸ œ nœ n nœœ œœ n nœœ œœ œœ œœ œœ ¸œ # œfin> #œ #œ œ #œ œ œ œ œ fff

fff sempre

¸ ¸ bœfibn œœ

p sub.

#>˙ ™ #˙ ™

^ ^ n œÆ œ # œÆ œÆ œÆ œÆ œÆ œÆ Æ Æ Æ Æ ^Æ œ #œ œ œ œ œ œ ‰ #nœœ œœ œœ œœ œœ ‰ #˝œ ‰ ˝

#œ nœ nœ nœ # œ n œ ' ' ' ' ' '

^¸ ‰ œ œ œ œ œ œ œ '> ' ' ' ' ' ' p sub. ff

p sub.

^ ¸ ‰ ^¸ ‰ nœ œ œ œ œ bœ ' ' ' ' ' ' p

(accél) - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - (q = 200)

fff

Harpe

^ œ n œÆ œÆ œÆ œÆ œÆ œÆ

^¸ ^¸ ^ œœ nnœœ œœ œœ œœ œœ œœ ‰ n#œœ œœ œœ œœ œœ ‰ nnœœ ‰ ' ' ' ' ' ' ' ' ' ' ' ' p sub. ff

fff

Marimba

n#œœ œœ œœ nn œœnnœ'œ ' ' ' ' >

fff sempre

¸ ¸ #œ n œfi œ n œ '

p sub.

#œ œ œ œ œ œ œ œ œ œ ' ' ' ' '

Æ Æ n œ nœ nœ ##œœÆ œœÆ ##œœÆ œœÆ œœÆ œœÆ œœÆ ˝ fff

p sub.

b œfi¸n> œÆ

° Vibraphone &

# œ n œ' œ' # œ' nœ' '

^ ^¸ ‰ ^¸ ‰ œ #œ œ œ œ œ œ ‰ nœ nœ œ œ œ œ ' ' ' ' ' ' ' ' ' ' ' '

p sub.

fff sempre

fff

Violoncelle

¸ ^ ‰ #œ œ œ œ œ ‰ ¸ ‰ œ nœ œ œ œ œ œ œ ' ' ' ' ' ' ' ' ' ' ' '

fff sempre

fff

° &

#œÆ #œÆ

#œ nœ' #œ' ' # œ œ' n>œ '

n œ # œ nœ' nœ' nœ' ' '

b œ bœ

Œ™ fff

#n#œœœn#œœ œœ œ Œ ' ' ' ' ¸ n nœœ nnœœnœ œ œ ‰ #œ ∑

∑ Mi§ Si§

∑

“” b œÆ n œÆ nn œœÆ n œÆ Æ Æ nœ nnœœ n œ #œ n#œœ #œ Æ nœ #œ#nœœ & fff

^¸ ? ‰ ‰ Œ nb#b œœœœ <“> '

#œ nœ nœ #œ# œ #œ b œbn œœ # œ #œ' ' nœ ' ' '

#nœœÆ ˝ ‰ Œ

‰ n nœœ¸ Œ # # œœ >

## œ>œ f

> ^Æ ¸2 n œÆ Œ # #œœ¸ ‰n nœœ¸ ‰ ‰ #œnœ #œ œ œ œ œÆ ###œœœ ‰ nbœœ n n œœ # œ ˝ ' ' ' ' nœ ' >' >'

∑

¸ ‹#œœ ‰ Œ '

nnœœ^

>Æ ‰ # nœœ¸ ‰#n bœœœ¸ ‰ nn#œœœ nœ nœ ˝ >' >' ff f > Sib > nnbœœœÆ bœÆ ‰ ˝ Œ ‰ bnœœ ˝

^¸ #n#‹bœœœœœ ‰ Œ fff ' ^ ∫ bn n#œœœœœÆ ˝ ‰ Œ

^ #n œœÆ n œ n œ ‰ ˝ ff

‰

^ ^ nœ ‰ nnœnœœ ‰bnœ˝œ ˝ f

Ó Si§

‰ nbœœ¸ ‰ ¸ ' bn œœ ' f

‰ nnœœ¸ ‰ '

nœÆ n ˝œ

^ n# œœÆ ^ n œÆ # œ ‰ ˝ ‰ #œnœ ˝ f

^¸ ^ ‰ bœ¸ ‰ n##œœœ bn œœ ' '

^¸ Ó bœ bb œœ ' Pierre Boulez „Dérive 2|für 11 Instrumente“

211

Ex. 3.4.4. (cont’d) Cor anglais

° &

Clarinette en la

Ÿ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ¸ ‰ # w nœ œ #œ

Violon

^¸ ^¸ #œ ‰ ' nœ œ œ ' ' '

Flatt. ^ ‰ #œ œ #œ œ œ œ œ œ œ œ #œ¸ ‰ nœ œ œ œ œ œ nœ^¸ ‰ ææ ˙™ ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' '

nœÆ œÆ œÆ œÆ œÆ œÆ œÆ œÆ

^ ^ nœÆ ‰ nœÆ œÆ œÆ ˝ ˝

Ÿ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ œ ‰ #˙ ™ ‰ nœ #œ w ˝

^¸ æ #œ œ œ œ œ œ œ œ #œ ‰ nœæ # œ œ œ œ œ œ œ œ # œ nœ ' ' ' ' ' ' ' ' ' ff

æ¸ œœ f

^ ^Æ Æ Æ Æ Æ Æ Æ Æ Æ bœÆ œÆ œÆ B nnœœ œœ œœ œœ œœ œœ œœ œœ nnœ˝œ ‰n œ œ ˝œ p

^ ^ B nnœœÆ œœÆ œœÆ œœÆ œœÆ œœÆ œœÆ œœÆ nnœœÆ ‰#nœœÆ œœÆ œœÆ Violoncelle ˝ ¢ ˝

Cor en fa

nœ œ œ œ œ œ œ œ ' ' ' ' ' ' ' '

^¸ ^ œ ‰ # œ œ œ¸ ' ' ' '

° Vibraphone &

Harpe

‰ #œææ nœ p

ææ w w

n˙ ™

Ÿ~~~~~~~~~~~~~~~~~~~~~~ bœ #œ nœ ‰ n˙ ™ ff

æ¸ œœæ ‰ n˙ ™

n˙ ™

nœ ‰##˙˙ ™™

pos. nat.

nœ

Æ Æ Æ Æ Æ Æ Æ Æ Æ Æ ^Æ ‰ nnœœ œœ nnœœ œœ œœ œœ œœ œœ œœ œœ nn˝œœ

‰nn˙˙ ™™ ææ

Æ Æ Æ Æ Æ Æ Æ Æ Æ Æ ^Æ ‰ bnœœ œœ bnœœ œœ œœ œœ œœ œœ œœ œœ bnœœ ˝

^ Æ Æ Æ Æ Æ Æ #œÆ œÆ œÆ œÆ œÆ œÆ #œÆ ‰ # œ œ œ œ œ œ # œ ‰bnœœ œœ œœ œœ œœ œœ ˝

sul pont.

non legato

∑

‰ nbœœ Œ n n œœ '

∑

^ ^ ‰ nœ œ nœ œ œ œ œ œ œ œ nœ¸ ‰ nœ œ œ œ œ œ nœ¸ ‰ #œ œ œ œ œ œ ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' p

‰

nnœœ^ ‰ n #œœ Œ ˝

^Æ ‰ bn#œœœ Œ ˝

∑

^ ‰ bœ¸Œ bn œœ ' ff ^Æ n œ ‰n# œ˝œ Œ

∑

^ n œÆ bnœœ ‰ ˝ Œ

∑

^ ‰ nœ¸Œ bb œœ '

^Æ ‰ ###œœœ Œ ˝

212

^¸ ‰ nbœœ Œ nœ '

∑

^ #n œœÆ # œ ‰ ˝ Œ

∑

‰

nnœœ^ ‰ n #œœ Œ ˝

^¸ Œ bbb œœœ '

‰ nbœœ Œ b b œœ ' ff

∑

^Æ ‰ #‹#œœœ Œ ˝

^ nœÆ ‰ bnœœ Œ ˝

˝ Œ

∑

^¸ ‰ & # nœœ Œ b b œœ

Piano

‰ bnœœ Œ ˝

^ ‰#bnnœœœœ Œ ˝ ff

n˙ ™

œœ ‰ n˙ ™ æ˝

22 250h = h = 100 accélérer - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -nb- œ-œ^Æ- - - - - - - - - - - - - - - - - - - - ^ ^¸ ^¸ b nœœ n nœœÆ ff

Marimba

¸ œ ‰ n˙ ™

#œ œ œ œ œ œ œ œ ' ' ' ' ' ' ' '

Alto

Ÿ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ¸ ‰ #˙ ™ #œ œ w œ

° &

‰

Basson

¸ œ

^ ‰ nbœœ¸Œ nœ ' ff ^¸ ‰ nbœœ Œ ' ^ ‰ #nœœ¸Œ nœ ' ff ^Æ n œ ‰ ##œœ Œ ˝

Ó Ó

Ex. 3.4.4. (cont’d) Cor anglais

° &

¸ œ

fff

Clarinette en la

Basson

Violon

æ wæ

¸ œæ

^¸ ¸ #œ Œ œ '

Ÿ~~~~~~~~~~~~~~~~~~~ ^ ¸ œ bœ œ ‰ nœ

¸ œ

^¸ œ Œ '

œ ˝

^ nœÆ Œ ˝

‰

fff

^ œ ‰ #œ ˝ ^ ¸ œœ ‰ nnœ œ

w w

¸ ^¸ œœ n œ Œ #œ '

œœ ‰ nœ^ #œ ˝

w w

œ ˝œ

^Æ #nœœ Œ ˝

^Æ bnœœ ˝

n#œœ^ nsul œ pont. œœ ‰ nœ ˝ ‰

w w

œœ ˝

^ #nœœÆ ˝ Œ

^¸ #œ '

‰

^¸ œ Œ #n#nœœœ '

^ ‰ # #œœ¸ Œ n n œœ '

¸ œœ

B bnœœÆ œœÆ œœÆ œœÆ œœÆ œœÆ œœÆ œœÆ

nœ nœ ‰ ##œœ

° Vibraphone &

#œ œ œ œ œ œ œ œ ' ' ' ' ' ' ' '

Harpe

sffp

fff

^ n œÆ ‰ nn#œœœ ˝

æ wæ

æ¸ œ

sff

∑

^ n nœœÆ ‰ bnœœ ˝

^¸ ‰ nbœœ Œ n n œœ '

∑

^ n bœœÆ ‰ nbœœ Œ ˝

∑

^¸ n#bœœœ Œ ' fff ^Æ nnbœœœ ˝ Œ ‰

∑

‰

fff

Piano

∑

‰

^¸ #nœœ Œ nœ '

^ ‰ n#bœœœ¸ Œ ' ff ^Æ nnbœœœ ‰ ˝ Œ

sfffz

‰

251

‰

sfffz

∑

^ b œÆ bnœœ ˝

∑

sfffz

fff

(accél.) jusqu'à - - - - - - - - - - - - - - - - - - - - (h = 112/120)

∑

sfffz

sffp

Flatt. ^ ^¸ ‰ #œææ nœ œ œ ' ' '

fff

sffp

pos. nat.

fff

Marimba

sffp

pos. nat.

fff

Cor en fa

‰ #œ nœ

fff

sffp

sfffz

fff

sffp

Violoncelle

<Ÿ>~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ œ B w ˝ ‰ bœ

° & <#><#>w w

Alto

^ œ

¸ œ ‰

‰ nœ

sfffz

∑

^ nœÆ ‰ bnœœ Œ ˝

∑

^ n œÆ ‰ bnœœ Œ ˝

∑

^ ‰ #n bœœœ¸ Œ nœ ' sfffz ^Æ nnbœœœ ‰ ˝ Œ ^ ‰ #n bœœœœ¸ Œ #‹œ ' sfffz ^Æ bn∫n œœœœœ ‰ #˝ Œ

Ó Ó

213

Ex. 3.4.5. Ending for the 2004 and 2005 version Cor anglais

° &

Violon

Cor en fa

¸> nœfi#œÆ

¸ bœfinœ¸

œ nœ nœ' ' nœ' '

nœ ' nœ #œ # œ nœ ' ' ' '

¸ ¸ b œfi # œ

Æ Æ nœÆ n œ # œ nœÆ

#œÆ nœÆ

Æ #œÆ nœÆ nœ

bœÆ #œ nœfi¸##œœ nœÆ bœÆ '> #œ ' ' ff

Æ Æ nœÆ nœ #œ nœÆ

Æ nœÆ nœ #œÆ

Æ n œÆ #œÆ #œÆ #œ

B #œ nœ ' ' ¢

° &

Alto

Violoncelle

nœ ' #œ nœ nœ bœ ' ' ' '

nœÆ œÆ bœÆ nœÆ

¸ # œfinœ'

Basson

bœÆ nœÆ Æ bœ

¸ #œ #œ ' ' nœ' >

Clarinette en la

¸ ¸ nœfinœ

Æ B nœÆ nœ

¸ œfi##œ œ

> b œfi¸ nœÆ nœ

# œÆ Æ Æ nœ nœ

∑

¸ ¸ nœfi #œ #œ ff

n œ ' #œ ' ' > >Æ

¸ nœ #œfin ˝œ

nœÆ nœÆ

> Æ Æ n œ nœ nœ ˝

n œfi¸nœÆ ff

¸ ¸ nœ n œfi œ '

nœ ' # œ' n œ' nœ' œ ' f

nœ nœ bœ' nœ ' ' ' ff

bœ '

n œ # œ n œ nœ' ' ' ' ff

n œ # œ' # œ ' '

nœ #œ #œ nœ ' ' ' ' ff

nœ nœ #œ ' ' '

#œ #œ' # œ nœ ' ' '

nœ #œ nœ ' ' '

nœ nœ' #œ #œ' ' ' ff

# œ nœ' nœ' '

Æ #œÆ nœÆ #œÆ bœ

Æ nœÆ nœ bœÆ

œ bœ ' ' nœ '

nœ # œ # œ' ' '

b œÆ Æ Æ Æ Æ n œ b œ #œ n œ

∑

‰ ‰

#œÆ nœÆ #œÆ nœ bœ' '

∑

‰ ‰

n œÆ Æ Æ Æ Æ bœ nœ nœ nœ

n œÆ b œÆ n œÆ #œÆ

‰ ‰

n œÆ # œÆ n œÆ

¸> nœfi#œÆ

nœ '

nœ ' ff

248 (accél) - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

° Vibraphone & #œ #œ ' '

Marimba

¢& nœ

#œ

¸ #œfi ff

n#œœ >'

¸œ œ # œfi n >

& nœ bœ

Piano

Æ nœÆ bœ bœÆ œÆ

#œ nœ bœ

nœ nœ ' #œ #œ nœ ' ' ' '

nœ nœ nœ nœ

#œ

nœ #œ nœ nœ

nbœœ nœ #œ nœ

bnœœ bnœœ

bœ n œ b œ

bn œœ

“” >Æ b œÆ b œ nb œœ n œ ˝

b œÆ nœ

nbœœ #œ nœ bœ nœ œ nœ b œÆ Æ nbœœ

n œÆ nœ

nœ bœ '

#œ œ ' '

¸ bœfi ff

¸ n#œœ #œ' #œ' '> ¸

¸#œ œ # œfi# >

bbœœ

Æ nn œœ

Æ n#œœ

“” > nnœœÆ n˝œ

¸ bœfi bn œœ

& Harpe

nœÆ nœÆ #œÆ

‰

¸ ¸ bœfibn œœ

#œ

nœ

bœ b œ

œ bnnœœ '

214

nœÆ bœ #œœ

nœ nœ #œ ' ' ' #œ ' ff nœ

> n nœœÆ nœ ‰ ‰ ? ˝

#œ nœ nœ

nn œœ

‰ ‰?

> n œÆ ## œœ ˝ ‰ ‰?

bœ ' n œ'

nnœœ bb œœ b#œœ

#œ n œ # œ ' ' ' nœ n œ #œ

bnœœ

bn œœ

nœ '

#œ ' “‘

nbœœ '

œ ' “‘

#nœœ

#œ '

bœ œ ' '

Ex. 3.4.5. (cont’d) Cor anglais

° &

¸ ¸ nœfinœ nœ

nœ >' ' ' b œ' n œ'

fff sempre

fff

¸ nœ # œ n œ ' ' ' >

Clarinette en la

¸ # œfinœ' # œ'

fff sempre

fff

Basson

Violon

¸ Æ #œ Æ nœfi#> œ #œ

¸ ¸ nœfi #œ

n œ # œ #œ' n#œœ œœ ' ' >' '

fff sempre

fff

¸ œ¸nœÆ #œÆ nœfi##œ

Alto

Cor en fa

nœ #œ #>˙ ' '

Æ nœÆ nœÆ nœÆ #œÆ n œ

# œÆ n>œ ˙

Æ Æ nnœœ œœ

>' '

¢&

Piano

‰

^Æ ^ nœÆ œÆ œÆ œÆ œÆ ‰ #˝œ ‰ ˝ p

Æ Æ Æ Æ Æ nnœœ œœ œœ œœ œœ

Æ nœÆ œÆ > nnœœ n œ œ#nœœ œœ

^Æ ^ Æ Æ Æ Æ Æ œœ#nœœÆ œœÆ œœÆ œœÆ œœÆ œœÆ ‰ ##œœ œœ œœ œœ ˝œœ ‰ n#˝œœ ‰

Œ™

¸ ¸ #œfi #œ #œ #œ#n#œ œœ

#œ nœ ' ' ff

249

# # œ ' ' ' œœ'œ #n œœ'œ œœ'œ œœ'œ œœ'œ œœ'œ fff sempre >'

¸ œ nœ n nœœ œœ n nœœ œœ œœ œœ œœ ¸œ # œfin> #œ #œ œ #œ œ œ œ œ fff

fff sempre

¸ ¸ bœfibn œœ

p sub.

#>˙ ™ #˙ ™

^ ^ n œÆ œ # œÆ œÆ œÆ œÆ œÆ œÆ Æ Æ Æ Æ ^Æ œ #œ œ œ œ œ œ ‰ #nœœ œœ œœ œœ œœ ‰ #˝œ ‰ ˝

#œ nœ nœ nœ # œ n œ ' ' ' ' ' '

^¸ ‰ œ œ œ œ œ œ œ '> ' ' ' ' ' ' p sub. ff

p sub.

^ ¸ ‰ ^¸ ‰ nœ œ œ œ œ bœ ' ' ' ' ' ' p

(accél) - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - (q = 200)

fff

Harpe

^ œ n œÆ œÆ œÆ œÆ œÆ œÆ

^¸ ^¸ ^ œœ nnœœ œœ œœ œœ œœ œœ ‰ n#œœ œœ œœ œœ œœ ‰ nnœœ ‰ ' ' ' ' ' ' ' ' ' ' ' ' p sub. ff

fff

Marimba

n#œœ œœ œœ nn œœnnœ'œ ' ' ' ' >

fff sempre

¸ ¸ #œ n œfi œ n œ '

p sub.

#œ œ œ œ œ œ œ œ œ œ ' ' ' ' '

Æ Æ n œ nœ nœ ##œœÆ œœÆ ##œœÆ œœÆ œœÆ œœÆ œœÆ ˝ fff

p sub.

b œfi¸n> œÆ

° Vibraphone &

# œ n œ' œ' # œ' nœ' '

^ ^¸ ‰ ^¸ ‰ œ #œ œ œ œ œ œ ‰ nœ nœ œ œ œ œ ' ' ' ' ' ' ' ' ' ' ' '

p sub.

fff sempre

fff

Violoncelle

¸ ^ ‰ #œ œ œ œ œ ‰ ¸ ‰ œ nœ œ œ œ œ œ œ ' ' ' ' ' ' ' ' ' ' ' '

fff sempre

fff

° &

#œÆ #œÆ

#œ nœ' #œ' ' # œ œ' n>œ '

n œ # œ nœ' nœ' nœ' ' '

b œ bœ

Œ™ fff

#n#œœœn#œœ œœ œ Œ ' ' ' ' ¸ n nœœ nnœœnœ œ œ ‰ #œ ∑

∑ Mi§ Si§

∑

“” b œÆ n œÆ nn œœÆ n œÆ Æ Æ nœ nnœœ n œ #œ n#œœ #œ Æ nœ #œ#nœœ & fff

^¸ ? ‰ ‰ Œ nb#b œœœœ <“> '

#œ nœ nœ #œ# œ #œ b œbn œœ # œ #œ' ' nœ ' ' '

#nœœÆ ˝ ‰ Œ

‰ n nœœ¸ Œ # # œœ >

## œ>œ f

> ^Æ ¸2 n œÆ Œ # #œœ¸ ‰n nœœ¸ ‰ ‰ #œnœ #œ œ œ œ œÆ ###œœœ ‰ nbœœ n n œœ # œ ˝ ' ' ' ' nœ ' >' >'

∑

¸ ‹#œœ ‰ Œ '

nnœœ^

>Æ ‰ # nœœ¸ ‰#n bœœœ¸ ‰ nn#œœœ nœ nœ ˝ >' >' ff f > Sib > nnbœœœÆ bœÆ ‰ ˝ Œ ‰ bnœœ ˝

^¸ #n#‹bœœœœœ ‰ Œ fff ' ^ ∫ bn n#œœœœœÆ ˝ ‰ Œ

^ #n œœÆ n œ n œ ‰ ˝ ff

‰

^ ^ nœ ‰ nnœnœœ ‰bnœ˝œ ˝ f

Ó Si§

‰ nbœœ¸ ‰ ¸ ' bn œœ ' f

‰ nnœœ¸ ‰ '

nœÆ n ˝œ

^ n# œœÆ ^ n œÆ # œ ‰ ˝ ‰ #œnœ ˝ f

^¸ ^ ‰ bœ¸ ‰ n##œœœ bn œœ ' '

^¸ Ó bœ bb œœ ' Pierre Boulez „Dérive 2|für 11 Instrumente“

215

Ex. 3.4.5. (cont’d) Cor anglais

° &

Clarinette en la

Ÿ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ¸ ‰ # w nœ œ #œ

Violon

^¸ ^¸ #œ ‰ ' nœ œ œ ' ' '

Flatt. ^ ‰ #œ œ #œ œ œ œ œ œ œ œ #œ¸ ‰ nœ œ œ œ œ œ nœ^¸ ‰ ææ ˙™ ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' '

nœÆ œÆ œÆ œÆ œÆ œÆ œÆ œÆ

^ ^ nœÆ ‰ nœÆ œÆ œÆ ˝ ˝

Ÿ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ œ ‰ #˙ ™ ‰ nœ #œ w ˝

^¸ æ #œ œ œ œ œ œ œ œ #œ ‰ nœæ # œ œ œ œ œ œ œ œ # œ nœ ' ' ' ' ' ' ' ' ' ff

æ¸ œœ f

^ ^Æ Æ Æ Æ Æ Æ Æ Æ Æ bœÆ œÆ œÆ B nnœœ œœ œœ œœ œœ œœ œœ œœ nnœ˝œ ‰n œ œ ˝œ p

^ ^ B nnœœÆ œœÆ œœÆ œœÆ œœÆ œœÆ œœÆ œœÆ nnœœÆ ‰#nœœÆ œœÆ œœÆ Violoncelle ˝ ¢ ˝

Cor en fa

nœ œ œ œ œ œ œ œ ' ' ' ' ' ' ' '

^¸ ^ œ ‰ # œ œ œ¸ ' ' ' '

° Vibraphone &

Harpe

‰ #œææ nœ p

ææ w w

n˙ ™

Ÿ~~~~~~~~~~~~~~~~~~~~~~ bœ #œ nœ ‰ n˙ ™ ff

æ¸ œœæ ‰ n˙ ™

n˙ ™

nœ ‰##˙˙ ™™

pos. nat.

nœ

Æ Æ Æ Æ Æ Æ Æ Æ Æ Æ ^Æ ‰ nnœœ œœ nnœœ œœ œœ œœ œœ œœ œœ œœ nn˝œœ

‰nn˙˙ ™™ ææ

Æ Æ Æ Æ Æ Æ Æ Æ Æ Æ ^Æ ‰ bnœœ œœ bnœœ œœ œœ œœ œœ œœ œœ œœ bnœœ ˝

^ Æ Æ Æ Æ Æ Æ #œÆ œÆ œÆ œÆ œÆ œÆ #œÆ ‰ # œ œ œ œ œ œ # œ ‰bnœœ œœ œœ œœ œœ œœ ˝

sul pont.

non legato

∑

‰ nbœœ Œ n n œœ '

∑

^ ^ ‰ nœ œ nœ œ œ œ œ œ œ œ nœ¸ ‰ nœ œ œ œ œ œ nœ¸ ‰ #œ œ œ œ œ œ ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' p

‰

nnœœ^ ‰ n #œœ Œ ˝

^Æ ‰ bn#œœœ Œ ˝

∑

^ ‰ bœ¸Œ bn œœ ' ff ^Æ n œ ‰n# œ˝œ Œ

∑

^ n œÆ bnœœ ‰ ˝ Œ

∑

^ ‰ nœ¸Œ bb œœ '

^Æ ‰ ###œœœ Œ ˝

216

^¸ ‰ nbœœ Œ nœ '

∑

^ #n œœÆ # œ ‰ ˝ Œ

∑

‰

nnœœ^ ‰ n #œœ Œ ˝

^¸ Œ bbb œœœ '

‰ nbœœ Œ b b œœ ' ff

∑

^Æ ‰ #‹#œœœ Œ ˝

^ nœÆ ‰ bnœœ Œ ˝

˝ Œ

∑

^¸ ‰ & # nœœ Œ b b œœ

Piano

‰ bnœœ Œ ˝

^ ‰#bnnœœœœ Œ ˝ ff

n˙ ™

œœ ‰ n˙ ™ æ˝

Marimba

¸ œ ‰ n˙ ™

#œ œ œ œ œ œ œ œ ' ' ' ' ' ' ' '

Alto

Ÿ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ¸ ‰ #˙ ™ #œ œ w œ

° &

‰

Basson

¸ œ

^ ‰ nbœœ¸Œ nœ ' ff ^¸ ‰ nbœœ Œ ' ^ ‰ #nœœ¸Œ nœ ' ff ^Æ n œ ‰ ##œœ Œ ˝

Ó Ó

Ex. 3.4.5. (cont’d) Cor anglais

° &

¸ œ

fff

Clarinette en la

Basson

Violon

æ wæ

¸ œæ

^¸ ¸ #œ Œ œ '

∑

Ÿ~~~~~~~~~~~~~~~~~~ ^ ¸ œ bœ œ ‰ nœ

^ ¸ ¸Œ œ œ'

∑

œ ^ ˝ n˝œÆ Œ

∑

‰

fff

^ ¸ œœ ‰ nnœ œ

w w

¸ ^¸ œœ n œ Œ #œ '

œœ ‰ nœ^ #œ ˝

w w

^Æ œœ #nœœ Œ ˝ ˝

n#œœ^ nsul œ pont. œœ ‰ nœ ˝ ‰

w w

^ œœ #nœœÆ ˝ ˝ Œ

∑

æ¸ œ ‰ Œ

∑

^ ‰ n nœœ¸ Ó # # œœ '

∑

^ ‰ # #œœ¸ Ó n n œœ '

∑

‰ #œ nœ

nœ nœ ‰ ##œœ

pos. nat.

Violoncelle

^ B bnœœÆ œœÆ œœÆ œœÆ œœÆ œœÆ œœÆ œœÆ bnœœÆ ˝ ¢

fff

Cor en fa

° Vibraphone &

#œ œ œ œ œ œ œ œ ' ' ' ' ' ' ' '

^¸ #œ '

Harpe

‰

fff

sfffz

sffp

sfffz

sffp

sfffz

Flatt. ^ ^¸ ‰ #œææ wææ nœ œ œ ' ' ' ff

æ wæ

sff

fff

^ n œÆ ‰ nn#œœœ Œ ˝

^¸ ‰ nbœœ Œ n n œœ '

∑

^ n nœœÆ ‰ bnœœ Œ ˝

^ n bœœÆ ‰ nbœœ Œ ˝

∑

^ ‰ n#bœœœ¸ Œ ' ff ^Æ nnbœœœ ‰ ˝ Œ

(accél.) jusqu'à - - - - - - - - - - - - - - - - - - - - (h = 112/120)

∑

fff

∑

^¸ ‰ n#bœœœ Œ ' fff ^Æ nnbœœœ ‰ ˝ Œ

∑

‰

^ b œÆ bnœœ ˝ Œ fff

Piano

pos. nat.

fff

Marimba

sffp

sfffz

^ œ ‰ #œ ˝

fff

sffp

fff

¸ œœ

<Ÿ>~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ œ B w ˝ ‰ bœ

° & <#><#>w w

Alto

^ ¸ œ ‰ œ

‰ nœ

∑

^¸ ‰ #nœœ Œ nœ '

∑

251

sfffz

∑

sfffz

∑

^ nœÆ ‰ bnœœ Œ ˝

∑

^ n œÆ ‰ bnœœ Œ ˝

∑

^ ‰#n bœœœ¸ Ó nœ ' sfffz ^Æ nnbœœœ ‰ ˝ Ó

^¸ #n#‹bœœœœœ ‰ Ó ' sfffz ^Æ bn∫n#œœœœœ Œ ˝ ‰ Ó

∑ ∑

217

Ex. 3.4.5. (cont’d) Cor anglais

° ‰ &

^¸ Œ œ '

sfffz

Clarinette en la

Basson

Violon

& ‰ B ¢ ‰ ° Œ &

∑

sempre

^¸ #œ ‰ ‰ Ó ' sempre sfffz

^ œÆ ˝ ‰ ‰ Ó

sfffz sempre

^¸ ‰ Ó # œœ '

∑

B Œ

B Violoncelle ¢ Œ

Cor en fa

Vibraphone

& Œ ? Œ

Piano

‰

^ #œÆ ‰ ˝

∑

‰

^¸ ‰ # œœ '

∑

^Æ #˝œœ ‰

∑

^¸ #œ Œ '

∑

^ œÆ ˝ Œ

^¸ ‰ Œ # œœ '

sfffz

^ œÆ ˝ ‰ Ó

sfffz

^¸ #œœ ‰ Œ '

^Æ #œœ ‰ Ó ˝

sfffz

^¸ #œ ‰ '

∑

^¸ #œ ‰ Œ '

∑

^¸ ‰ ##nœœœœ '

∑

^ ‰ ##œœ¸ Œ œœ '

∑

^¸ ‰ Œ ##nœœœœ '

^¸ ‰ Ó n œ ## œœœ '

∑

^ Œ ##œœ¸ ‰ œœ '

∑

^¸ ‰ Œ n œ ## œœœ '

∑

^¸ ##œœœœ ‰ Œ '

^¸ #bœœœœ ‰ Ó '

∑

^ Œ #bœœœ¸ ‰ œ '

∑

^¸ #bœœœœ ‰ Œ '

∑

^ ‰ #bœœœ¸ Œ œ '

∑

‰ Œ

∑

‰

∑

sfffz sempre

^ bœœœÆ ˝ ‰ Ó

^¸ Ó & ‰ ##‹bœœœœœ Œ ' sfffz sempre ^ ∫b#œœœœœÆ ? ‰ ˝ Œ Ó

∑

sffz

^ bœœœÆ ˝ ‰

^ Œ # bœœœœ¸ ‰ #‹œ ' ^Æ ∫b œœœœœ Œ #˝ ‰

218

‰ Œ

^¸ œ ‰ Ó '

^ # œœÆ ˝ ‰

^Æ #œœ ˝

sfffz

∑

‰ Œ

^¸ #œ ‰ Ó '

∑

sfffz sempre

^ #œœÆ ˝

∑

sfffz sempre

Harpe

∑

^¸ Œ œ '

^Æ #œœ ‰ Ó ˝

sfffz sempre

^¸ ° & ‰ ##œœœœ Œ ' Œ

‰

∑

¢&

^¸ ‰ #œ '

^ #œœÆ ‰ Ó ˝

sfffz sempre

Marimba

∑

sfffz

sfffz sempre

Alto

^¸ #œ ‰ '

sfffz

∑

sfffz

sffz

^ bœœœÆ ˝

∑

^ ‰ # bœœœœ¸ Œ #‹œ ' ^Æ ∫b œœœœœ ‰ #˝ Œ

∑ ∑

^ U ‰ #œ¸ ˙

long

sfffz

^ bœœœÆ ˝ Œ

^ ‰ # bœœœœ¸ Œ #‹œ ' ^Æ ∫b œœœœœ ‰ #˝ Œ

Ó Ó

Ex. 3.4.6. Concluding measures in an earlier version of 2006 and 2007 (UE31940) Cor anglais

° &

Violon

Cor en fa

¸> nœfi#œÆ

¸ bœfinœ¸

œ nœ nœ' ' nœ' '

nœ ' nœ #œ # œ nœ ' ' ' '

¸ ¸ b œfi # œ

Æ Æ nœÆ n œ # œ nœÆ

#œÆ nœÆ

Æ #œÆ nœÆ nœ

bœÆ #œ nœfi¸##œœ nœÆ bœÆ '> #œ ' ' ff

Æ Æ nœÆ nœ #œ nœÆ

Æ nœÆ nœ #œÆ

Æ n œÆ #œÆ #œÆ #œ

B #œ nœ ' ' ¢

° &

Alto

Violoncelle

nœ ' #œ nœ nœ bœ ' ' ' '

nœÆ œÆ bœÆ nœÆ

¸ # œfinœ'

Basson

bœÆ nœÆ Æ bœ

¸ #œ #œ ' ' nœ' >

Clarinette en la

¸ ¸ nœfinœ

Æ B nœÆ nœ

¸ œfi##œ œ

> b œfi¸ nœÆ nœ

# œÆ Æ Æ nœ nœ

∑

¸ ¸ nœfi #œ #œ ff

n œ ' #œ ' ' > >Æ

¸ nœ #œfin ˝œ

nœÆ nœÆ

> Æ Æ n œ nœ nœ ˝

n œfi¸nœÆ ff

¸ ¸ nœ n œfi œ '

nœ ' # œ' n œ' nœ' œ ' f

nœ nœ bœ' nœ ' ' ' ff

bœ '

n œ # œ n œ nœ' ' ' ' ff

n œ # œ' # œ ' '

nœ #œ #œ nœ ' ' ' ' ff

nœ nœ #œ ' ' '

#œ #œ' # œ nœ ' ' '

nœ #œ nœ ' ' '

nœ nœ' #œ #œ' ' ' ff

# œ nœ' nœ' '

Æ #œÆ nœÆ #œÆ bœ

Æ nœÆ nœ bœÆ

œ bœ ' ' nœ '

nœ # œ # œ' ' '

b œÆ Æ Æ Æ Æ n œ b œ #œ n œ

∑

‰ ‰

#œÆ nœÆ #œÆ nœ bœ' '

∑

‰ ‰

n œÆ Æ Æ Æ Æ bœ nœ nœ nœ

n œÆ b œÆ n œÆ #œÆ

‰ ‰

n œÆ # œÆ n œÆ

¸> nœfi#œÆ

nœ '

nœ ' ff

248 (accél) - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

° Vibraphone & #œ #œ ' '

Marimba

¢& nœ

#œ

¸ #œfi ff

n#œœ >'

¸œ œ # œfi n >

& nœ bœ

Piano

Æ nœÆ bœ bœÆ œÆ

#œ nœ bœ

nœ nœ ' #œ #œ nœ ' ' ' '

nœ nœ nœ nœ

#œ

nœ #œ nœ nœ

nbœœ nœ #œ nœ

bnœœ bnœœ

bœ n œ b œ

bn œœ

“” >Æ b œÆ b œ nb œœ n œ ˝

b œÆ nœ

nbœœ #œ nœ bœ nœ œ nœ b œÆ Æ nbœœ

n œÆ nœ

nœ bœ '

#œ œ ' '

¸ bœfi ff

¸ n#œœ #œ' #œ' '> ¸

¸#œ œ # œfi# >

bbœœ

Æ nn œœ

Æ n#œœ

“” > nnœœÆ n˝œ

¸ bœfi bn œœ

& Harpe

nœÆ nœÆ #œÆ

‰

¸ ¸ bœfibn œœ

#œ

nœ

bœ b œ

œ bnnœœ '

219

nœÆ bœ #œœ

nœ nœ #œ ' ' ' #œ ' ff nœ

> n nœœÆ nœ ‰ ‰ ? ˝

#œ nœ nœ

nn œœ

‰ ‰?

> n œÆ ## œœ ˝ ‰ ‰?

bœ ' n œ'

nnœœ bb œœ b#œœ

#œ n œ # œ ' ' ' nœ n œ #œ

bnœœ

bn œœ

nœ '

#œ ' “‘

nbœœ '

œ ' “‘

#nœœ

#œ '

bœ œ ' '

Ex. 3.4.6. (cont’d) Cor anglais

° &

¸ ¸ nœfinœ nœ

nœ >' ' ' b œ' n œ'

fff sempre

fff

¸ nœ # œ n œ ' ' ' >

Clarinette en la

¸ # œfinœ' # œ'

fff sempre

fff

Basson

Violon

¸ Æ #œ Æ nœfi#> œ #œ

¸ ¸ nœfi #œ

n œ # œ #œ' n#œœ œœ ' ' >' '

fff sempre

fff

¸ œ¸nœÆ #œÆ nœfi##œ

Alto

Cor en fa

nœ #œ #>˙ ' '

Æ nœÆ nœÆ nœÆ #œÆ n œ

# œÆ n>œ ˙

Æ Æ nnœœ œœ

>' '

¢&

Piano

‰

^Æ ^ nœÆ œÆ œÆ œÆ œÆ ‰ #˝œ ‰ ˝ p

Æ Æ Æ Æ Æ nnœœ œœ œœ œœ œœ

Æ nœÆ œÆ > nnœœ n œ œ#nœœ œœ

^Æ ^ Æ Æ Æ Æ Æ œœ#nœœÆ œœÆ œœÆ œœÆ œœÆ œœÆ ‰ ##œœ œœ œœ œœ ˝œœ ‰ n#˝œœ ‰

Œ™

¸ ¸ #œfi #œ #œ #œ#n#œ œœ

#œ nœ ' ' ff

249

# # œ ' ' ' œœ'œ #n œœ'œ œœ'œ œœ'œ œœ'œ œœ'œ fff sempre >'

¸ œ nœ n nœœ œœ n nœœ œœ œœ œœ œœ ¸œ # œfin> #œ #œ œ #œ œ œ œ œ fff

fff sempre

¸ ¸ bœfibn œœ

p sub.

#>˙ ™ #˙ ™

^ ^ n œÆ œ # œÆ œÆ œÆ œÆ œÆ œÆ Æ Æ Æ Æ ^Æ œ #œ œ œ œ œ œ ‰ #nœœ œœ œœ œœ œœ ‰ #˝œ ‰ ˝

#œ nœ nœ nœ # œ n œ ' ' ' ' ' '

^¸ ‰ œ œ œ œ œ œ œ '> ' ' ' ' ' ' p sub. ff

p sub.

^ ¸ ‰ ^¸ ‰ nœ œ œ œ œ bœ ' ' ' ' ' ' p

(accél) - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - (q = 200)

fff

Harpe

^ œ n œÆ œÆ œÆ œÆ œÆ œÆ

^¸ ^¸ ^ œœ nnœœ œœ œœ œœ œœ œœ ‰ n#œœ œœ œœ œœ œœ ‰ nnœœ ‰ ' ' ' ' ' ' ' ' ' ' ' ' p sub. ff

fff

Marimba

n#œœ œœ œœ nn œœnnœ'œ ' ' ' ' >

fff sempre

¸ ¸ #œ n œfi œ n œ '

p sub.

#œ œ œ œ œ œ œ œ œ œ ' ' ' ' '

Æ Æ n œ nœ nœ ##œœÆ œœÆ ##œœÆ œœÆ œœÆ œœÆ œœÆ ˝ fff

p sub.

b œfi¸n> œÆ

° Vibraphone &

# œ n œ' œ' # œ' nœ' '

^ ^¸ ‰ ^¸ ‰ œ #œ œ œ œ œ œ ‰ nœ nœ œ œ œ œ ' ' ' ' ' ' ' ' ' ' ' '

p sub.

fff sempre

fff

Violoncelle

¸ ^ ‰ #œ œ œ œ œ ‰ ¸ ‰ œ nœ œ œ œ œ œ œ ' ' ' ' ' ' ' ' ' ' ' '

fff sempre

fff

° &

#œÆ #œÆ

#œ nœ' #œ' ' # œ œ' n>œ '

n œ # œ nœ' nœ' nœ' ' '

b œ bœ

Œ™ fff

#n#œœœn#œœ œœ œ Œ ' ' ' ' ¸ n nœœ nnœœnœ œ œ ‰ #œ ∑

∑ Mi§ Si§

∑

“” b œÆ n œÆ nn œœÆ n œÆ Æ Æ nœ nnœœ n œ #œ n#œœ #œ Æ nœ #œ#nœœ & fff

^¸ ? ‰ ‰ Œ nb#b œœœœ <“> '

#œ nœ nœ #œ# œ #œ b œbn œœ # œ #œ' ' nœ ' ' '

#nœœÆ ˝ ‰ Œ

‰ n nœœ¸ Œ # # œœ >

## œ>œ f

> ^Æ ¸2 n œÆ Œ # #œœ¸ ‰n nœœ¸ ‰ ‰ #œnœ #œ œ œ œ œÆ ###œœœ ‰ nbœœ n n œœ # œ ˝ ' ' ' ' nœ ' >' >'

∑

¸ ‹#œœ ‰ Œ '

nnœœ^

>Æ ‰ # nœœ¸ ‰#n bœœœ¸ ‰ nn#œœœ nœ nœ ˝ >' >' ff f > Sib > nnbœœœÆ bœÆ ‰ ˝ Œ ‰ bnœœ ˝

^¸ #n#‹bœœœœœ ‰ Œ fff ' ^ ∫ bn n#œœœœœÆ ˝ ‰ Œ

^ #n œœÆ n œ n œ ‰ ˝ ff

‰

^ ^ nœ ‰ nnœnœœ ‰bnœ˝œ ˝ f

Ó Si§

‰ nbœœ¸ ‰ ¸ ' bn œœ ' f

‰ nnœœ¸ ‰ '

nœÆ n ˝œ

^ n# œœÆ ^ n œÆ # œ ‰ ˝ ‰ #œnœ ˝ f

^¸ ^ ‰ bœ¸ ‰ n##œœœ bn œœ ' '

^¸ Ó bœ bb œœ ' Pierre Boulez „Dérive 2|für 11 Instrumente“

220

Ex. 3.4.6. (cont’d) Cor anglais

° &

Clarinette en la

Ÿ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ¸ ‰ # w nœ œ #œ

Violon

^¸ ^¸ #œ ‰ ' nœ œ œ ' ' '

Flatt. ^ ‰ #œ œ #œ œ œ œ œ œ œ œ #œ¸ ‰ nœ œ œ œ œ œ nœ^¸ ‰ ææ ˙™ ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' '

nœÆ œÆ œÆ œÆ œÆ œÆ œÆ œÆ

^ ^ nœÆ ‰ nœÆ œÆ œÆ ˝ ˝

Ÿ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ œ ‰ #˙ ™ ‰ nœ #œ w ˝

^¸ æ #œ œ œ œ œ œ œ œ #œ ‰ nœæ # œ œ œ œ œ œ œ œ # œ nœ ' ' ' ' ' ' ' ' ' ff

æ¸ œœ f

^ ^Æ Æ Æ Æ Æ Æ Æ Æ Æ bœÆ œÆ œÆ B nnœœ œœ œœ œœ œœ œœ œœ œœ nnœ˝œ ‰n œ œ ˝œ p

^ ^ B nnœœÆ œœÆ œœÆ œœÆ œœÆ œœÆ œœÆ œœÆ nnœœÆ ‰#nœœÆ œœÆ œœÆ Violoncelle ˝ ¢ ˝

Cor en fa

nœ œ œ œ œ œ œ œ ' ' ' ' ' ' ' '

^¸ ^ œ ‰ # œ œ œ¸ ' ' ' '

° Vibraphone &

Harpe

‰ #œææ nœ p

ææ w w

n˙ ™

Ÿ~~~~~~~~~~~~~~~~~~~~~~ bœ #œ nœ ‰ n˙ ™ ff

æ¸ œœæ ‰ n˙ ™

n˙ ™

nœ ‰##˙˙ ™™

pos. nat.

nœ

Æ Æ Æ Æ Æ Æ Æ Æ Æ Æ ^Æ ‰ nnœœ œœ nnœœ œœ œœ œœ œœ œœ œœ œœ nn˝œœ

‰nn˙˙ ™™ ææ

Æ Æ Æ Æ Æ Æ Æ Æ Æ Æ ^Æ ‰ bnœœ œœ bnœœ œœ œœ œœ œœ œœ œœ œœ bnœœ ˝

^ Æ Æ Æ Æ Æ Æ #œÆ œÆ œÆ œÆ œÆ œÆ #œÆ ‰ # œ œ œ œ œ œ # œ ‰bnœœ œœ œœ œœ œœ œœ ˝

sul pont.

non legato

∑

‰ nbœœ Œ n n œœ '

∑

^ ^ ‰ nœ œ nœ œ œ œ œ œ œ œ nœ¸ ‰ nœ œ œ œ œ œ nœ¸ ‰ #œ œ œ œ œ œ ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' p

‰

nnœœ^ ‰ n #œœ Œ ˝

^Æ ‰ bn#œœœ Œ ˝

∑

^ ‰ bœ¸Œ bn œœ ' ff ^Æ n œ ‰n# œ˝œ Œ

∑

^ n œÆ bnœœ ‰ ˝ Œ

∑

^ ‰ nœ¸Œ bb œœ '

^Æ ‰ ###œœœ Œ ˝

221

^¸ ‰ nbœœ Œ nœ '

∑

^ #n œœÆ # œ ‰ ˝ Œ

∑

‰

nnœœ^ ‰ n #œœ Œ ˝

^¸ Œ bbb œœœ '

‰ nbœœ Œ b b œœ ' ff

∑

^Æ ‰ #‹#œœœ Œ ˝

^ nœÆ ‰ bnœœ Œ ˝

˝ Œ

∑

^¸ ‰ & # nœœ Œ b b œœ

Piano

‰ bnœœ Œ ˝

^ ‰#bnnœœœœ Œ ˝ ff

n˙ ™

œœ ‰ n˙ ™ æ˝

Marimba

¸ œ ‰ n˙ ™

#œ œ œ œ œ œ œ œ ' ' ' ' ' ' ' '

Alto

Ÿ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ¸ ‰ #˙ ™ #œ œ w œ

° &

‰

Basson

¸ œ

^ ‰ nbœœ¸Œ nœ ' ff ^¸ ‰ nbœœ Œ ' ^ ‰ #nœœ¸Œ nœ ' ff ^Æ n œ ‰ ##œœ Œ ˝

Ó Ó

Ex. 3.4.6. (cont’d) Cor anglais

° &

¸ œ

fff

Clarinette en la

Basson

Violon

æ wæ

¸ œæ

^¸ ¸ #œ Œ œ '

∑

Ÿ~~~~~~~~~~~~~~~~~~ ^ ¸ œ bœ œ ‰ nœ

^ ¸ ¸Œ œ œ'

∑

œ ^ ˝ n˝œÆ Œ

∑

‰

fff

^ ¸ œœ ‰ nnœ œ

w w

¸ ^¸ œœ n œ Œ #œ '

œœ ‰ nœ^ #œ ˝

w w

^Æ œœ #nœœ Œ ˝ ˝

n#œœ^ nsul œ pont. œœ ‰ nœ ˝ ‰

w w

^ œœ #nœœÆ ˝ ˝ Œ

∑

æ¸ œ ‰ Œ

∑

^ ‰ n nœœ¸ Ó # # œœ '

∑

^ ‰ # #œœ¸ Ó n n œœ '

∑

‰ #œ nœ

nœ nœ ‰ ##œœ

pos. nat.

Violoncelle

^ B bnœœÆ œœÆ œœÆ œœÆ œœÆ œœÆ œœÆ œœÆ bnœœÆ ˝ ¢

fff

Cor en fa

° Vibraphone &

#œ œ œ œ œ œ œ œ ' ' ' ' ' ' ' '

^¸ #œ '

Harpe

‰

fff

sfffz

sffp

sfffz

sffp

sfffz

Flatt. ^ ^¸ ‰ #œææ wææ nœ œ œ ' ' ' ff

æ wæ

sff

fff

^ n œÆ ‰ nn#œœœ Œ ˝

^¸ ‰ nbœœ Œ n n œœ '

∑

^ n nœœÆ ‰ bnœœ Œ ˝

^ n bœœÆ ‰ nbœœ Œ ˝

∑

^ ‰ n#bœœœ¸ Œ ' ff ^Æ nnbœœœ ‰ ˝ Œ

(accél.) jusqu'à - - - - - - - - - - - - - - - - - - - - (h = 112/120)

∑

fff

∑

^¸ ‰ n#bœœœ Œ ' fff ^Æ nnbœœœ ‰ ˝ Œ

∑

‰

^ b œÆ bnœœ ˝ Œ fff

Piano

pos. nat.

fff

Marimba

sffp

sfffz

^ œ ‰ #œ ˝

fff

sffp

fff

¸ œœ

<Ÿ>~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ œ B w ˝ ‰ bœ

° & <#><#>w w

Alto

^ ¸ œ ‰ œ

‰ nœ

∑

^¸ ‰ #nœœ Œ nœ '

∑

251

sfffz

∑

sfffz

∑

^ nœÆ ‰ bnœœ Œ ˝

∑

^ n œÆ ‰ bnœœ Œ ˝

∑

^ ‰#n bœœœ¸ Ó nœ ' sfffz ^Æ nnbœœœ ‰ ˝ Ó

^¸ #n#‹bœœœœœ ‰ Ó ' sfffz ^Æ bn∫n#œœœœœ Œ ˝ ‰ Ó

∑ ∑

222

Ex. 3.4.6. (cont’d) Cor anglais

° & ‰

^ ¸Œ œ '

sfffz

& ‰

Clarinette en la

Basson

Violon

B ‰

sfffz

^ œÆ ˝ Œ ‰ Ó

° Œ &

∑

sempre

^¸ #œ ‰ ‰ Ó ' sempre

sfffz

∑

sempre

^¸ ‰ Ó œ #œ '

Violoncelle

Cor en fa

^ ¸Œ œ '

∑

‰

^¸ #œ Œ '

∑

^ #œÆ ‰ ˝

∑

‰

^¸ ‰ œ #œ '

∑

^Æ #˝œœ ‰

∑

^ #œœÆ ˝

‰ Œ

∑

^ œÆ #œ ˝

‰ Œ

∑

^¸ ‰ #œ '

^ œÆ ˝ Œ

^¸ ‰ Œ œ #œ '

∑

B Œ

^ œÆ #œ ‰ Ó ˝

∑

^¸ #œ ‰ '

∑

^¸ #œ ‰ Œ '

∑

^¸ ‰ n œ ## œœœ '

∑

^ ‰ ##œœ¸ Œ œœ '

∑

^¸ ‰ Ó n œ ## œœœ '

∑

^ Œ ##œœ¸ ‰ œœ '

∑

^¸ ‰ Œ n œ ## œœœ '

∑

sfffz sempre

& ^¸ & ‰ ##œœœ Œ œ ' & Œ

sfffz sempre

& Œ Harpe

? Œ

Piano

‰

^ #œœÆ ‰ Ó ˝

sfffz sempre

Vibraphone

B Œ

h = 112

Marimba

∑

sfffz

sfffz sempre

Alto

^¸ #œ ‰ '

^¸ #bœœœœ ‰ Ó ' sfffz sempre ^Æ bœœœ ˝ ‰ Ó

^¸ Ó & ‰ ##‹bœœœœœ Œ ' sfffz sempre ^ ∫b#œœœœÆ ? ‰ ˝œ Œ Ó

^ # œœÆ ˝ ‰

sffz

^¸ Œ #bœœœ ‰ œ ' ^Æ bœœœ Œ ˝ ‰ ^ Œ # bœœœœ¸ ‰ #‹œ ' ^Æ ∫b#œœœœ Œ ˝œ ‰

223

sffz

^¸ #bœœœœ ‰ Œ ' ^Æ bœœœ ˝ ‰ Œ ^ ‰ # bœœœœ¸ Œ #‹œ ' ^Æ ∫b#œœœœ ‰ ˝œ Œ

∑ ∑

Ex. 3.4.6. (cont’d) ° Œ Cor anglais & & Œ

Clarinette en la

Basson

B Œ

° Œ Violon &

^¸ #œ ‰ '

B Œ

∑

^¸ nœ ‰ ' sfffz

^ œÆ ˝ ‰

^¸ #œœ ‰ '

sfffz

#œ >

sfzp

Cor en fa

Marimba

Vibraphone

Harpe

Piano

B Œ ¢

non vib.

#œ

#œ >

sfzp

U w

fff

long

arraché

long

arraché

U w

w ∑

∑

fff

U w

> #w

sfzp

fff

U w

long

sfzp

fff

^Æ #nœœ ‰ ˝

∑

^Æ #œœ ‰ ˝

∑

sfffz

Violoncelle

sfffz

Alto

∑

arraché

long

#>˙

non vib.

U w

sfzp

∑

fff

sfffz

#>˙

non vib.

U w

sfzp

fff

& Œ

^ ‰ #œ¸ ˙

^¸ & # nœœœ ‰ #œ '

∑

^¸ & ##œœœœ ‰ '

sfffz p

^¸ & ‰ #bœœœœ Œ ' ^ bœœœÆ ? ‰ ˝ Œ

^¸ & ‰ ##‹bœœœœœ Œ ' ^ ∫b#œœœœœÆ ? ‰ ˝ Œ

U w

long

fff

224

Ex. 3.4.7. Ending for the most current version Cor anglais

° &

Violon

Cor en fa

¸> nœfi#œÆ

¸ bœfinœ¸

œ nœ nœ' ' nœ' '

nœ ' nœ #œ # œ nœ ' ' ' '

¸ ¸ b œfi # œ

Æ Æ nœÆ n œ # œ nœÆ

#œÆ nœÆ

Æ #œÆ nœÆ nœ

bœÆ #œ nœfi¸##œœ nœÆ bœÆ '> #œ ' ' ff

Æ Æ nœÆ nœ #œ nœÆ

Æ nœÆ nœ #œÆ

Æ n œÆ #œÆ #œÆ #œ

B #œ nœ ' ' ¢

° &

Alto

Violoncelle

nœ ' #œ nœ nœ bœ ' ' ' '

nœÆ œÆ bœÆ nœÆ

¸ # œfinœ'

Basson

bœÆ nœÆ Æ bœ

¸ #œ #œ ' ' nœ' >

Clarinette en la

¸ ¸ nœfinœ

Æ B nœÆ nœ

¸ œfi##œ œ

> b œfi¸ nœÆ nœ

# œÆ Æ Æ nœ nœ

∑

¸ ¸ nœfi #œ #œ ff

n œ ' #œ ' ' > >Æ

¸ nœ #œfin ˝œ

nœÆ nœÆ

> Æ Æ n œ nœ nœ ˝

n œfi¸nœÆ ff

¸ ¸ nœ n œfi œ '

nœ ' # œ' n œ' nœ' œ ' f

nœ nœ bœ' nœ ' ' ' ff

bœ '

n œ # œ n œ nœ' ' ' ' ff

n œ # œ' # œ ' '

nœ #œ #œ nœ ' ' ' ' ff

nœ nœ #œ ' ' '

#œ #œ' # œ nœ ' ' '

nœ #œ nœ ' ' '

nœ nœ' #œ #œ' ' ' ff

# œ nœ' nœ' '

Æ #œÆ nœÆ #œÆ bœ

Æ nœÆ nœ bœÆ

œ bœ ' ' nœ '

nœ # œ # œ' ' '

b œÆ Æ Æ Æ Æ n œ b œ #œ n œ

∑

‰ ‰

#œÆ nœÆ #œÆ nœ bœ' '

∑

‰ ‰

n œÆ Æ Æ Æ Æ bœ nœ nœ nœ

n œÆ b œÆ n œÆ #œÆ

‰ ‰

n œÆ # œÆ n œÆ

¸> nœfi#œÆ

nœ '

nœ ' ff

248 (accél) - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

° Vibraphone & #œ #œ ' '

Marimba

¢& nœ

#œ

¸ #œfi ff

n#œœ >'

¸œ œ # œfi n >

& nœ bœ

Piano

Æ nœÆ bœ bœÆ œÆ

#œ nœ bœ

nœ nœ ' #œ #œ nœ ' ' ' '

nœ nœ nœ nœ

#œ

nœ #œ nœ nœ

nbœœ nœ #œ nœ

bnœœ bnœœ

bœ n œ b œ

bn œœ

“” >Æ b œÆ b œ nb œœ n œ ˝

b œÆ nœ

nbœœ #œ nœ bœ nœ œ nœ b œÆ Æ nbœœ

n œÆ nœ

nœ bœ '

#œ œ ' '

¸ bœfi ff

¸ n#œœ #œ' #œ' '> ¸

¸#œ œ # œfi# >

bbœœ

Æ nn œœ

Æ n#œœ

“” > nnœœÆ n˝œ

¸ bœfi bn œœ

& Harpe

nœÆ nœÆ #œÆ

‰

¸ ¸ bœfibn œœ

#œ

nœ

bœ b œ

œ bnnœœ '

225

nœÆ bœ #œœ

nœ nœ #œ ' ' ' #œ ' ff nœ

> n nœœÆ nœ ‰ ‰ ? ˝

#œ nœ nœ

nn œœ

‰ ‰?

> n œÆ ## œœ ˝ ‰ ‰?

bœ ' n œ'

nnœœ bb œœ b#œœ

#œ n œ # œ ' ' ' nœ n œ #œ

bnœœ

bn œœ

nœ '

#œ ' “‘

nbœœ '

œ ' “‘

#nœœ

#œ '

bœ œ ' '

Ex. 3.4.7. (cont’d) Cor anglais

° &

¸ ¸ nœfinœ nœ

nœ >' ' ' b œ' n œ'

fff sempre

fff

¸ nœ # œ n œ ' ' ' >

Clarinette en la

¸ # œfinœ' # œ'

fff sempre

fff

Basson

Violon

¸ Æ #œ Æ nœfi#> œ #œ

¸ ¸ nœfi #œ

n œ # œ #œ' n#œœ œœ ' ' >' '

fff sempre

fff

¸ œ¸nœÆ #œÆ nœfi##œ

Alto

Cor en fa

nœ #œ #>˙ ' '

Æ nœÆ nœÆ nœÆ #œÆ n œ

# œÆ n>œ ˙

Æ Æ nnœœ œœ

>' '

¢&

Piano

‰

^Æ ^ nœÆ œÆ œÆ œÆ œÆ ‰ #˝œ ‰ ˝ p

Æ Æ Æ Æ Æ nnœœ œœ œœ œœ œœ

Æ nœÆ œÆ > nnœœ n œ œ#nœœ œœ

^Æ ^ Æ Æ Æ Æ Æ œœ#nœœÆ œœÆ œœÆ œœÆ œœÆ œœÆ ‰ ##œœ œœ œœ œœ ˝œœ ‰ n#˝œœ ‰

Œ™

¸ ¸ #œfi #œ #œ #œ#n#œ œœ

#œ nœ ' ' ff

249

# # œ ' ' ' œœ'œ #n œœ'œ œœ'œ œœ'œ œœ'œ œœ'œ fff sempre >'

¸ œ nœ n nœœ œœ n nœœ œœ œœ œœ œœ ¸œ # œfin> #œ #œ œ #œ œ œ œ œ fff

fff sempre

¸ ¸ bœfibn œœ

p sub.

#>˙ ™ #˙ ™

^ ^ n œÆ œ # œÆ œÆ œÆ œÆ œÆ œÆ Æ Æ Æ Æ ^Æ œ #œ œ œ œ œ œ ‰ #nœœ œœ œœ œœ œœ ‰ #˝œ ‰ ˝

#œ nœ nœ nœ # œ n œ ' ' ' ' ' '

^¸ ‰ œ œ œ œ œ œ œ '> ' ' ' ' ' ' p sub. ff

p sub.

^ ¸ ‰ ^¸ ‰ nœ œ œ œ œ bœ ' ' ' ' ' ' p

(accél) - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - (q = 200)

fff

Harpe

^ œ n œÆ œÆ œÆ œÆ œÆ œÆ

^¸ ^¸ ^ œœ nnœœ œœ œœ œœ œœ œœ ‰ n#œœ œœ œœ œœ œœ ‰ nnœœ ‰ ' ' ' ' ' ' ' ' ' ' ' ' p sub. ff

fff

Marimba

n#œœ œœ œœ nn œœnnœ'œ ' ' ' ' >

fff sempre

¸ ¸ #œ n œfi œ n œ '

p sub.

#œ œ œ œ œ œ œ œ œ œ ' ' ' ' '

Æ Æ n œ nœ nœ ##œœÆ œœÆ ##œœÆ œœÆ œœÆ œœÆ œœÆ ˝ fff

p sub.

b œfi¸n> œÆ

° Vibraphone &

# œ n œ' œ' # œ' nœ' '

^ ^¸ ‰ ^¸ ‰ œ #œ œ œ œ œ œ ‰ nœ nœ œ œ œ œ ' ' ' ' ' ' ' ' ' ' ' '

p sub.

fff sempre

fff

Violoncelle

¸ ^ ‰ #œ œ œ œ œ ‰ ¸ ‰ œ nœ œ œ œ œ œ œ ' ' ' ' ' ' ' ' ' ' ' '

fff sempre

fff

° &

#œÆ #œÆ

#œ nœ' #œ' ' # œ œ' n>œ '

n œ # œ nœ' nœ' nœ' ' '

b œ bœ

Œ™ fff

#n#œœœn#œœ œœ œ Œ ' ' ' ' ¸ n nœœ nnœœnœ œ œ ‰ #œ ∑

∑ Mi§ Si§

∑

“” b œÆ n œÆ nn œœÆ n œÆ Æ Æ nœ nnœœ n œ #œ n#œœ #œ Æ nœ #œ#nœœ & fff

^¸ ? ‰ ‰ Œ nb#b œœœœ <“> '

#œ nœ nœ #œ# œ #œ b œbn œœ # œ #œ' ' nœ ' ' '

#nœœÆ ˝ ‰ Œ

‰ n nœœ¸ Œ # # œœ >

## œ>œ f

> ^Æ ¸2 n œÆ Œ # #œœ¸ ‰n nœœ¸ ‰ ‰ #œnœ #œ œ œ œ œÆ ###œœœ ‰ nbœœ n n œœ # œ ˝ ' ' ' ' nœ ' >' >'

∑

¸ ‹#œœ ‰ Œ '

nnœœ^

>Æ ‰ # nœœ¸ ‰#n bœœœ¸ ‰ nn#œœœ nœ nœ ˝ >' >' ff f > Sib > nnbœœœÆ bœÆ ‰ ˝ Œ ‰ bnœœ ˝

^¸ #n#‹bœœœœœ ‰ Œ fff ' ^ ∫ bn n#œœœœœÆ ˝ ‰ Œ

^ #n œœÆ n œ n œ ‰ ˝ ff

‰

^ ^ nœ ‰ nnœnœœ ‰bnœ˝œ ˝ f

Ó Si§

‰ nbœœ¸ ‰ ¸ ' bn œœ ' f

‰ nnœœ¸ ‰ '

nœÆ n ˝œ

^ n# œœÆ ^ n œÆ # œ ‰ ˝ ‰ #œnœ ˝ f

^¸ ^ ‰ bœ¸ ‰ n##œœœ bn œœ ' '

^¸ Ó bœ bb œœ ' Pierre Boulez „Dérive 2|für 11 Instrumente“

226

Ex. 3.4.7. (cont’d) Cor anglais

° &

Clarinette en la

Ÿ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ¸ ‰ # w nœ œ #œ

Violon

^¸ ^¸ #œ ‰ ' nœ œ œ ' ' '

Flatt. ^ ‰ #œ œ #œ œ œ œ œ œ œ œ #œ¸ ‰ nœ œ œ œ œ œ nœ^¸ ‰ ææ ˙™ ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' '

nœÆ œÆ œÆ œÆ œÆ œÆ œÆ œÆ

^ ^ nœÆ ‰ nœÆ œÆ œÆ ˝ ˝

Ÿ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ œ ‰ #˙ ™ ‰ nœ #œ w ˝

^¸ æ #œ œ œ œ œ œ œ œ #œ ‰ nœæ # œ œ œ œ œ œ œ œ # œ nœ ' ' ' ' ' ' ' ' ' ff

æ¸ œœ f

^ ^Æ Æ Æ Æ Æ Æ Æ Æ Æ bœÆ œÆ œÆ B nnœœ œœ œœ œœ œœ œœ œœ œœ nnœ˝œ ‰n œ œ ˝œ p

^ ^ B nnœœÆ œœÆ œœÆ œœÆ œœÆ œœÆ œœÆ œœÆ nnœœÆ ‰#nœœÆ œœÆ œœÆ Violoncelle ˝ ¢ ˝

Cor en fa

nœ œ œ œ œ œ œ œ ' ' ' ' ' ' ' '

^¸ ^ œ ‰ # œ œ œ¸ ' ' ' '

° Vibraphone &

Harpe

‰ #œææ nœ p

ææ w w

n˙ ™

Ÿ~~~~~~~~~~~~~~~~~~~~~~ bœ #œ nœ ‰ n˙ ™ ff

æ¸ œœæ ‰ n˙ ™

n˙ ™

nœ ‰##˙˙ ™™

pos. nat.

nœ

Æ Æ Æ Æ Æ Æ Æ Æ Æ Æ ^Æ ‰ nnœœ œœ nnœœ œœ œœ œœ œœ œœ œœ œœ nn˝œœ

‰nn˙˙ ™™ ææ

Æ Æ Æ Æ Æ Æ Æ Æ Æ Æ ^Æ ‰ bnœœ œœ bnœœ œœ œœ œœ œœ œœ œœ œœ bnœœ ˝

^ Æ Æ Æ Æ Æ Æ #œÆ œÆ œÆ œÆ œÆ œÆ #œÆ ‰ # œ œ œ œ œ œ # œ ‰bnœœ œœ œœ œœ œœ œœ ˝

sul pont.

non legato

∑

‰ nbœœ Œ n n œœ '

∑

^ ^ ‰ nœ œ nœ œ œ œ œ œ œ œ nœ¸ ‰ nœ œ œ œ œ œ nœ¸ ‰ #œ œ œ œ œ œ ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' p

‰

nnœœ^ ‰ n #œœ Œ ˝

^Æ ‰ bn#œœœ Œ ˝

∑

^ ‰ bœ¸Œ bn œœ ' ff ^Æ n œ ‰n# œ˝œ Œ

∑

^ n œÆ bnœœ ‰ ˝ Œ

∑

^ ‰ nœ¸Œ bb œœ '

^Æ ‰ ###œœœ Œ ˝

227

^¸ ‰ nbœœ Œ nœ '

∑

^ #n œœÆ # œ ‰ ˝ Œ

∑

‰

nnœœ^ ‰ n #œœ Œ ˝

^¸ Œ bbb œœœ '

‰ nbœœ Œ b b œœ ' ff

∑

^Æ ‰ #‹#œœœ Œ ˝

^ nœÆ ‰ bnœœ Œ ˝

˝ Œ

∑

^¸ ‰ & # nœœ Œ b b œœ

Piano

‰ bnœœ Œ ˝

^ ‰#bnnœœœœ Œ ˝ ff

n˙ ™

œœ ‰ n˙ ™ æ˝

Marimba

¸ œ ‰ n˙ ™

#œ œ œ œ œ œ œ œ ' ' ' ' ' ' ' '

Alto

Ÿ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ¸ ‰ #˙ ™ #œ œ w œ

° &

‰

Basson

¸ œ

^ ‰ nbœœ¸Œ nœ ' ff ^¸ ‰ nbœœ Œ ' ^ ‰ #nœœ¸Œ nœ ' ff ^Æ n œ ‰ ##œœ Œ ˝

Ó Ó

Ex. 3.4.7. (cont’d) Cor anglais

° &

¸ œ

fff

Clarinette en la

Basson

Violon

æ wæ

¸ œæ

^¸ ¸ #œ Œ œ '

∑

Ÿ~~~~~~~~~~~~~~~~~~ ^ ¸ œ bœ œ ‰ nœ

^ ¸ ¸Œ œ œ'

∑

œ ^ ˝ n˝œÆ Œ

∑

‰

fff

^ ¸ œœ ‰ nnœ œ

w w

¸ ^¸ œœ n œ Œ #œ '

œœ ‰ nœ^ #œ ˝

w w

^Æ œœ #nœœ Œ ˝ ˝

n#œœ^ nsul œ pont. œœ ‰ nœ ˝ ‰

w w

^ œœ #nœœÆ ˝ ˝ Œ

∑

æ¸ œ ‰ Œ

∑

^ ‰ n nœœ¸ Ó # # œœ '

∑

^ ‰ # #œœ¸ Ó n n œœ '

∑

‰ #œ nœ

nœ nœ ‰ ##œœ

pos. nat.

Violoncelle

^ B bnœœÆ œœÆ œœÆ œœÆ œœÆ œœÆ œœÆ œœÆ bnœœÆ ˝ ¢

fff

Cor en fa

° Vibraphone &

#œ œ œ œ œ œ œ œ ' ' ' ' ' ' ' '

^¸ #œ '

Harpe

‰

fff

sfffz

sffp

sfffz

sffp

sfffz

Flatt. ^ ^¸ ‰ #œææ wææ nœ œ œ ' ' ' ff

æ wæ

sff

fff

^ n œÆ ‰ nn#œœœ Œ ˝

^¸ ‰ nbœœ Œ n n œœ '

∑

^ n nœœÆ ‰ bnœœ Œ ˝

^ n bœœÆ ‰ nbœœ Œ ˝

∑

^ ‰ n#bœœœ¸ Œ ' ff ^Æ nnbœœœ ‰ ˝ Œ

(accél.) jusqu'à - - - - - - - - - - - - - - - - - - - - (h = 112/120)

∑

fff

∑

^¸ ‰ n#bœœœ Œ ' fff ^Æ nnbœœœ ‰ ˝ Œ

∑

‰

^ b œÆ bnœœ ˝ Œ fff

Piano

pos. nat.

fff

Marimba

sffp

sfffz

^ œ ‰ #œ ˝

fff

sffp

fff

¸ œœ

<Ÿ>~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ œ B w ˝ ‰ bœ

° & <#><#>w w

Alto

^ ¸ œ ‰ œ

‰ nœ

∑

^¸ ‰ #nœœ Œ nœ '

∑

251

sfffz

∑

sfffz

∑

^ nœÆ ‰ bnœœ Œ ˝

∑

^ n œÆ ‰ bnœœ Œ ˝

∑

^ ‰#n bœœœ¸ Ó nœ ' sfffz ^Æ nnbœœœ ‰ ˝ Ó

^¸ #n#‹bœœœœœ ‰ Ó ' sfffz ^Æ bn∫n#œœœœœ Œ ˝ ‰ Ó

∑ ∑

228

Ex. 3.4.7. (cont’d) Cor anglais

° ‰ &

^¸Œ œ '

∑

sfffz

& ‰

Clarinette en la

sfzp

^¸ ¸ #œ ‰ œ ˙ ' >

¸ œ ‰ Œ Ó >.

sfffz sfzp

Basson

Violon

B ‰

° Œ &

^ œÆ ˝ Œ

B Œ

Violoncelle

Cor en fa

^¸ œ ‰ Ó #œ '

∑

^¸ œ ‰ ‰ #œ ™ #œ > '

sfzp mf

? Œ

Piano

‰ #œ ™ œ œ¸ ‰ > >.

sfzp

∑

‰

sfzp

^ œÆ #>œ ˙ ™ ˝ ‰ ˝

>œ. ˝ ‰

sfffz sfzp

^¸ œ ‰ Œ #œ '

∑

sfffz

^ #œœÆ ˝

> ‰ #œ

^ n#œœÆ ˝

‰ Œ

sfffz

˙™

>œ. ˝ ‰

sfzp

#>˙ ™

∑

^¸ #œ ‰ '

∑

^¸ #œ ‰ Œ '

∑

^¸ ‰ n œ ## œœœ '

∑

^ ‰ ##œœ¸ Œ œœ '

∑

^ Œ ##œœ¸ ‰ œœ '

∑

^¸ ‰ Œ n œ ## œœœ '

∑

sfffz

^¸ ‰ Ó n œ ## œœœ '

sfffz sempre

Harpe

^¸ #œ Œ '

sfffz

^ # œœÆ Œ ˝ ‰ Œ

^¸ & ‰ ##œœœœ Œ '

& Œ

∑

^ œÆ #œ ‰ Ó ˝

& Œ

¸ œ œ. ‰ Ó >

sfffz

sfffz sempre

Vibraphone

^ œÆ #˝œ ‰

∑

sfffz

h = 112

Marimba

‰

^¸Œ œ '

sfffz

>œ. ˝ ‰ Œ Ó

∑

sfffz

B Œ

^¸ ‰ #œ '

^ #œœÆ ‰ Œ #>œ ˝

‰

^ #œÆ ‰ ˝

sfffz

¸ œ. ‰ Ó

sfffz

Alto

‰ #œ¸ ˙ ™ >

^¸ #bœœœœ ‰ Ó ' sfffz sempre ^Æ bœœœ ˝ ‰ Ó

^¸ & ‰ ##‹bœœœœœ Œ Ó ' sfffz sempre ^Æ ∫b œœ ? ‰ #œ˝œœ Œ Ó

sffz

^¸ Œ #bœœœœ ‰ ' ^Æ bœœœ Œ ˝ ‰ ^ Œ # bœœœœ¸ ‰ #‹œ ' ^Æ ∫b#œœœœ Œ ˝œ ‰

229

#>œ œ >œ. ˝ ‰ Ó ‰ ˝ sfzp

sfffz

sffz

^¸ #bœœœœ ‰ Œ ' ^Æ bœœœ ˝ ‰ Œ ^ ‰ # bœœœœ¸ Œ #‹œ ' ^Æ ∫b#œœœœ ‰ ˝œ Œ

>œ. ˝ ‰

sfzp

∑ ∑

Ex. 3.4.7. (cont’d) ° Œ Cor anglais & & Œ

Clarinette en la

Basson

Violon

B Œ

° & Œ

^¸ #œ ‰ '

B Œ

∑

^¸ nœ ‰ '

Cor en fa

Marimba

Vibraphone

Harpe

Piano

B Œ ¢

sfffz

^ œÆ ˝ ‰

^¸ #œœ ‰ '

sfffz

^Æ #nœœ ‰ ˝

#œ >

sfzp

non vib.

^ œÆ #œ ‰ ˝

#œ >

sfzp

U w

fff arraché

U w

w ∑

∑

#œ

arraché

long

fff

long

U w

> #w

arraché

sfzp

fff

U w

long

sfzp

fff

∑

sfffz

Violoncelle

sfffz

Alto

∑

#>˙

non vib.

sfzp

∑

fff

sfffz

#>˙

non vib.

U w

sfzp

fff

& Œ

^ ‰ #œ¸ ˙

^¸ & # nœœœ ‰ #œ '

∑

^¸ & ##œœœ ‰ œ '

sfffz p

^¸ & ‰ #bœœœœ Œ ' ^Æ bœ ? ‰ œ˝œ Œ

^¸ & ‰ ##‹bœœœœœ Œ ' ^ ∫b#œœœœœÆ ? ‰ ˝ Œ

U w

long

fff

230

Figure 3.4.8 displays the earlier version of the musical material at R129. This version was used in the first commercial recording, as well as in the subsequent performances between 2002 to 2004. Notice in this version it is only composed for solo piano. In the current version, however, this is expanded and supplemented with the full ensemble.

Figure 3.4.8 Earlier version at R129 “” 129 ^ W 8 &4

aussi rapide que possible simile

Très large

sfffz

Piano

^ 8W &4 °

43 43

fi œ #œ œ œfi

^ ˙™

sfffz sempre œ

^ ˙™

44 44

fi b œ #œ œ bœ

^ w

fiœ b œ œ bœ

^ w

24 24

fi# œ œ n œfi

nœ

^ ˙ ^ ˙ °

43 43

fi n œ œ bœ

^ ˙™

# œfi #œ

˙^ ™

nœ

fi n œ œ #œ bœ bœ

^ w™

fib œ n œ bœ

^ w™

nœ 46

Figure 3.4.9 illustrates selected chronology of the compositional history of Dérive 2, with performance durations, annotations, corresponding musical examples, and all the relevant details for each version from its premiere in 1990 to 2012.

231

DateMonthYear 21-06-90 07-02-93 08-02-93 15-07-96 28-10-98 17-03-01 04-12-01 xx-09-02 29-09-02 13-10-04 22-10-04 15-02-05 07-11-06 09-11-06 11-11-06 31-07-07 02-10-07 26-04-09 21-09-09 04-03-10 12-03-10

Duration

Interruptions

Sections C and D

R129

Midpoint and sections after

R130 to Section L

Ending

6'36" 6'55" 7'30" 6'55" 6'36" 6'21" 14'00" 24'33" 23'26" 25'25" 26'00" 26'47" 44'50" 45'00" 44'50" 44'52" 44'13 45'39" 44'11" 44'25" 44'37"

Opening horn note Absent Present Present Present Present Present Present Present Present Present Present Present Present Present Present Present Present Present Present Present Present

Absent Absent Absent Absent Absent Absent Earlier version Earlier version Earlier version Current version Current version Current version Current version Current version Current version Current version Current version Current version Current version Current version Current version

Earlier version Earlier version Earlier version Earlier version Earlier version Earlier version Earlier version Current version Current version Current version Current version Current version Current version Current version Current version Current version Current version Current version Current version Current version Current version

Absent Absent Absent Absent Absent Absent Absent Ex. 3.4.8 Ex. 3.4.8 Ex. 3.4.8 Ex. 3.4.8 Current version Current version Current version Current version Current version Current version Current version Current version Current version Current version

Absent Absent Absent Absent Absent Absent Absent Absent Absent Absent Absent Absent Current version Current version Current version Current version Current version Current version Current version Current version Current version

Ex. 3.4.1 Ex. 3.4.1 Ex. 3.4.1 Ex. 3.4.2 Ex. 3.4.2 Ex. 3.4.2 Ex. 3.4.3 Ex. 3.4.4 Ex. 3.4.4 Ex. 3.4.5 Ex. 3.4.5 Ex. 3.4.5 Ex. 3.4.6 Ex. 3.4.6 Ex. 3.4.6 Ex. 3.4.6 Ex. 3.4.6 Ex. 3.4.6 Ex. 3.4.7 Ex. 3.4.7 Ex. 3.4.7

51'10" 44'42" 27'26" 49'42"

Present Present Present Present

Current version Current version Current version Current version

Current version Current version Absent Current version

Absent Absent Absent Absent Absent Absent Absent Absent Absent Absent Absent Absent Earlier version Earlier version Earlier version Earlier version Earlier version Earlier version Earlier version Earlier version Current version (At R135 and R140 the violin and viola solos are much more flourished and ornamented) Earlier version Current version Absent Earlier version

01-04-10 26-06-10 03-09-10 14-11-10

Ex. 3.4.7 Ex. 3.4.7 Ex. 3.4.7 Ex. 3.4.7

06-12-10 21-01-11 18-06-11 14-12-11

51'37" 48'"31 50'44" 7'36"

Present Present Present Present

Current version Current version Current version Absent

Current version Current version Current version Earlier version

Current version Current version Current version Absent

Earlier version Earlier version Current version Absent

14-12-11 18-05-12 20-07-12

47'30" 44'12" 49'00"

Present Present Present

Current version Current version Current version

Earlier version Earlier version Earlier version

Figure 3.4.9. Chronological History of DĂŠrive 2.

233

Ex. 3.4.7 Ex. 3.4.7 Ex. 3.4.6 Ex. 3.4.2 Version 1988 Ex. 3.4.7 Ex. 3.4.7 Ex. 3.4.7

EPILOGUE The formal conception of Dérive 2 can be compared to some of Boulez’s earlier works. Similar to Marteau and the “variable-form” works, the form of Dérive 2 presents itself in non-linear manner, and the perception of this form lies within this temporal richness. On the one hand, there are the distinct cyclic-bound sections (lettered sections from A to L): these are already predetermined in duration and rhythmic structure by the architectural design of the periodicity from the very beginning, along with the stratification of different processes and layers of inner durational and rhythmic cycles, patterns, and repetitions of organizations. On the other hand, there are interruptions inserted in the long-range polyrhythms between the convergence points of two of the three cycles: these are external disturbances to the design, the continuity, and the integrity of the periodicity cycle. The two levels unfold and interlock simultaneously through progressive varied presentations. The form of Dérive 2 is both sectional yet continuous. The length of each periodic cycles of the convergence points in the periodicity, unlike the interruptions, do not follow a dynamic shape where progressively the sections become longer. However, in the actual music, because of Boulez's transformation of these sections that these periodicity sections become longer and more varied, mirroring the processes that occur in the interruption sections. If Boulez were to follow the periodicity design that was conceived in its initial state without any alteration, then the periodicity sections become extremely predicable and perceptible as they establish multiple layers of order in various domains in the music. And because of the inserted interruptions and the transformations, the fundamental order and rigid formula of the periodicity become unpredictable. In the former, musical time in the large-scale becomes static (even though time is directional in 234

all the network of intricate unfolding of periodicities), however, musical time as it stands now in its current version of Dérive 2, is no longer static, but a combination of static time and progressive time, where order and control (both temporal manipulation and construction in the small within the periodicities, or the large in the overall form) is transformed to disorder and chaos. Dérive 2, then, is realized and composed on two distinctive temporal conceptions: the periodicity sections are based on ontological or chronometric time – time that is measurable, like the clock, time that is divisible as in striated time. The interruptions are then based on psychological time, or chronoametric since the sense of real and absolute time is suspended, stretched, and frozen – hence smooth time. These multi-layered temporalities in Dérive 2 are analogous to the way one experiences time in reality, where time is comprised of multiple and complex yet coexisting, structurally conceived and hierarchically nested temporalities. Dérive 2 is constructed with both a static structure and a dynamic structure. Similar to Rituel, each internal structure holds its own coherence, and each is static, with no apparent scheme of duration, with no beginning and ending, therefore the music is static and non-developmental. Yet, viewing the whole structure as a unit, it is developmental, it is non-static. This is the same for Dérive 2, where the internal static time is resolved and satisfied in the overall scheme of the work. The formal structure is so large that the listener will lose the sight of the whole totality, since from the very beginning the music does not suggest the form itself, nor does anywhere of piece suggest an overall unfolding of the work. It is only until the very end that one sees how the form has unfolded. The listener will lose track of the overall shape since the form has gotten too long, too dense, and too complex to perceive. 235

The foreground of DĂŠrive 2 is made of discontinuous passages, as one experiences the disruption of continuity with the change of tempo, instrumentation, harmony, rhythm and so forth. Yet the middleground is a conflict between continuity and discontinuity, as the music seems to suggest a sense of flow and development, but it is quickly destroyed by another interruption. Moreover, one begins to experience that the interruptions themselves are slowly forming another sense of continuity as well. The background, however, forms yet another dichotomy of continuity and discontinuity. The continuity is formed by the use of tempo, in which these sections, both the periodicity and the interruptions sections, are connected with the same speed in the local level. However, because of the increasing transformation in which Boulez has introduced, the music can not by all means feel continuous anymore; in fact, the listener (due to the previous listening experience) attempts to and have a desire to form a sense of continuity, but only to discover that the original pulse (or periodicity) is no longer there, but it is now transformed and integrated into something else entirely completely. This multi-layered periodicities fundamentally build an expectation of a kind for the listener, similar to the temporal and formal structure in Carterâ&#x20AC;&#x2122;s music where the long-range polyrhythms is outlines the whole structure of the work, the listener, whether consciously or unconsciously, expects the completion of the whole cycle where everything will be aligned finally. But such periodicity in principle is static and an uninteresting one to Boulez, but the focus should be on the manipulation of these materials. The pitch content and the harmonic rhythm, also elements of periodicity, undergo various transformations and developments as demonstrated in chapter 3.1 within the first half of the work. The rhythmic cycles of periodicity, along with instrumentation, 236

dynamics, and grouping of various sorts likewise go through a methodical process of alterations as demonstrated in chapter 3.2 in these same sections. Just as the periodicity sections become more elaborated, the interruptions similarly become progressively longer, even to the point of outweighing the periodicity sections. The division between the two alternations gradually disappears. The listener is no longer capable of distinguishing the two, and the role between them is reversed: it is unclear which section interrupts which, or which section belongs to which trajectory in the overall design. The interruptions involuntarily project an expectation and direction for the listener as they slowly attain momentum, just as the periodicity cycle itself also projects expectancy for the cycles to be completed. These alternations reach their maximum complexity of proliferating both in the pitch and temporal domains when the work reaches its midpoint. The midpoint section is significantly if not deliberately different from everything else that has happened in the music. It employs the largest registral span available of the whole ensemble; it is the only section that combines three textures of music that were previously presented separately: heterophony, polyphony, and homophony. It is also the only section that lies between smooth and striated time, where the pulse is ambiguous in some passages while clearly straightforward in others. If the work were to conclude at the end of the midpoint section, the listener would undoubtedly assume that the periodicity sections are the interruptions to this increasingly expanded material. Yet the formal structure of DĂŠrive 2 continues in reverse process after the midpoint. Both the periodicity sections and the interruptions gradually become shorter in duration and faster in tempo, countering the first half of the work. Similarly in the pitch and temporal domains, materials of constructions are comparably reduced and deducted. It is not until the arrival of Section L 237

that it suddenly jolts the listener into remembering how the work began. The reference from Section L to Section A is not only intentional because they are related symmetrically in the architectural design of the periodicity, or that they share the same number of EN cycles; the return to the opening measures intended to trigger the listener’s involuntary memory after this long journey of expansion and contraction. One immediately recalls the opening, and the symmetrical design of the formal structure is achieved and reassured. The harmonic rhythm moves much faster in the periodicity sections, but the number of activities is restricted to simply realizing the various periodic cycles in the structure. However, the harmonic rhythm moves much slower in the interruptions, but the number of activities is much more varied than the periodicity sections, and because of the harmonic simplicity in the interruptions, one immediately focus more on the activities of the movements that occur in the interruptions, whereas in the periodicity sections, one focus nothing but the speed of harmonic rhythm. Tempo, another external temporal manipulation, not only articulates the various sections and strata in the periodicity design, but similar to Dérive 1, tempo adds another level of temporal fluctuation and fluidity to the fabric of the music. Both Jonathan Kramer and Albert Einstein thought that the perception of time is relative, in which time or duration is measured against another, time exists in between points, it is relative to the reference points. Thus in Dérive 2, measuring from the interruptions to the periodicity sections, the time is steady, coherent, measurable, and directional, yet when the position is changed from the periodicity sections to the interruptions, the temporal perspective has altered: the periodicity sections seem coherent metrically, and it’s measurable, directional, much more steady. 238

Dérive 2 incorporates numerous musical conceptions that were found in many of the earlier works. In a way, Dérive 2 is an amalgamation of Boulez’s musical thought throughout the years. These conceptions inevitably also include Boulez’s experience as a composer and as a conductor. The periodicity that lays the temporal foundation of Dérive 2 is an automatism of a kind, since from the very opening measures each periodicity has already been determined and assigned, and the work as a whole can simply be reduced to an unfolding of such automatism. Yet, from the experience of Structures, Boulez is less interested in limiting the musical discourse to just a straightforward realization of this structure; rather, it is the transformations and the proliferations of these materials that interest Boulez. Boulez’s Dérive 2 is a work that requires repeated hearings, and through these multiple hearings, one can truly experience and re-conceptualize the multiple temporal dimensions that the work offers.

239

Appendix I. Periodicity Chart This periodicity chart is based on Boulezâ&#x20AC;&#x2122;s sketches from the Pierre Boulez Collection at the Paul Sacher Foundation in Basel, Switzerland. Boulez made several versions of the same periodicity chart but each uses different annotation and notation of symbols. Most of these charts were written on graph paper, though some portions of this chart were reproduced on loose leaf or staff paper. The periodicity chart in appendix I is constructed based on sketches taken from film number 590, pages 0930 to 0933. The notes and rests, the A, E, and D brackets, the Roman and Greek letters, the numbering and lettering system at the end of each staff, and the total rhythmic value of each staff are taken directly from these sketches. Each line represents a complete statement of an EN cycle. Each cycle of the rhythmic pattern 2-1-3-1-1-2-1 is divided by tiny barline marks provided by the author. The E (event) bracket represents the events cycle (notated by Roman and Greek letters). This cycle aligns with the EN cycle every three lines, the shortest cycle of all three cycles. The A (accent) bracket represents the accent cycle (notated by the accent symbol). This cycle aligns with the EN cycle every four lines. The D (duration) cycle represents the duration cycle (notated by notes and rests). This cycle aligns with the EN cycle every seven lines. Each section of this periodicity chart, lettered A to L, is divided by the alignment of two of these three cycles. For example, Section A ends with the alignment of A and E cycle; Section B ends with the alignment of E and D cycle; Section C ends with the alignment of A and E cycle again; while Section D ends with the alignment of A and D cycle, and so on. This periodicity chart finishes after 84 lines of the EN cycle (3 times 4 times 7 equals 84) and 12 sections of synchronization, where all three cycles of A, D, and E are finally aligned. 240

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251

Appendix II. Performance History Appendix II provides a list of performances of Dérive 2, dating from its premiere in 1990 up to the present day. Each entry lists performance date (day-month-year), conductor, ensemble, and the location of the performance. Performances are provided with their duration when available. The xx symbols refer to dates that cannot be verified by the author. The EIC abbreviation stands for Ensemble Intercontemporain. 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36.

21-06-90 – Boulez – EIC – Conservatorio Giuseppe Verdi, Milan, Italy (PREMIÈRE: 6 minutes and 36 seconds) 09-11-90 – Boulez – EIC – Hôtel Römerbad-Musiktag, Badenweiler, Germany 21-02-91 – Boulez – EIC – Carnegie Hall, New York, US 16-03-91 – Boulez – EIC – Théâtre du Châtelet, Paris, France 18-03-91 – Boulez – EIC – Montbrison, France 19-03-91 – George Benjamin – Ensemble Oblique – Kontraste-Tage Linz, Stiftersaal im Brucknerhaus, Linz, Austria 21-03-91 – Boulez – EIC – Rome, Italy xx-03-91 – George Benjamin – Ensemble Oblique – Brüssel, Belgium 16-05-91 – Boulez – EIC – Roy Thompson Hall, Toronto, Canada 20-05-91 – Boulez – EIC – National Arts Centre, Ottawa, Canada 24-05-91 – Boulez – EIC – Salle Claude-Champagne, Montréal, Canada 25-05-91 – Boulez – EIC – Grand Théâtre, Québec, Canada 29-05-91 – Boulez – EIC – Dalhousie Arts Centre, Canada 14-01-92 – Boulez – EIC – Théâtre du Châtelet, Paris, France 15-01-92 – Boulez – EIC – Théâtre du Châtelet, Paris, France 07-02-93 – Boulez – EIC with members of the Philharmonia Orchestra – London Royal Festival Hall, London, UK (6 minutes and 55 seconds) 08-02-93 – Boulez – EIC with Members of the Philharmonia Orchestra – Théâtre du Châtelet, Paris, France (7 minutes and 30 seconds) 12-02-93 – Peter Eötvös – Koechlin Ensemble Stuttgart – SDR, Sendessal der Villa Berg, Konzert und Rundfunkproduktion, Stuttgart, Germany 14-11-93 – Boulez – EIC – Chicago, US 15-11-93 – Boulez – EIC – St. Louis, US 17-11-93 – Boulez – EIC – Los Angeles, US 19-06-94 – Boulez – EIC – Triennale-Schlußkonzert, WDR, Kölner Philharmonie, Köln, Germany 21-06-94 – Boulez – EIC – Sala Europa, Bologna, Italy 22-06-94 – Boulez – EIC – Milano Musica, Teatro alla Scala, Milan, Italy 27-09-94 – Boulez – EIC – Festival de Berlin, Berlin Philharmonie Kammermusiksaal, Berlin, Germany 14-10-95 – David Robertson – EIC – Oslo, Ultima Contemporary Music Festival, Universitetets Aula, Skandinavien-Tourné, Oslo, Norway 28-05-96 – Boulez – Los Angeles Philharmonic New Music Group – Los Angeles, US 31-05-96 – Boulez – Los Angeles Philharmonic New Music Group – Los Angeles, US 01-06-96 – Boulez – Los Angeles Philharmonic New Music Group – Los Angeles, US 15-07-96 – Boulez – EIC – Centre Acanthes at Avignon, France (6 minutes and 54 seconds) 29-09-96 – Boulez – EIC – Rouen, Lycée Corneille, France (FIRST published version - UE19128) 01-10-96 – Boulez – EIC – Basel, Switzerland 18-10-96 – Boulez – EIC – Buenos Aires, Argentina 22-10-96 – Boulez – EIC – Sao Paulo, Brazil 03-11-96 – Don Ross – St. Crispin’s Chamber Ensemble – Edmonton, Canada 16-11-96 – Boulez – EIC –Hôtel Römerbad-Musiktag, Badenweiler, Germany

252

37. 38. 39. 40. 41. 42. 43. 44. 45. 46. 47. 48. 49. 50. 51. 52. 53. 54. 55. 56. 57. 58. 59. 60. 61. 62. 63. 64. 65. 66. 67. 68. 69. 70. 71. 72. 73. 74. 75. 76. 77. 78. 79. 80. 81. 82. 83. 84.

29-08-98 – Boulez – EIC – Edinburgh, UK 23-10-98 – Boulez – EIC – Alte Oper-Frankfurt am Main, Frankfurt, Germany 28-10-98 – Boulez – EIC – Cité de la musique, Paris, France (6 minutes and 36 seconds) 31-10-98 – Boulez – EIC – Konzerthaus, Großer Sall-Wien, Vienna, Austria 02-11-98 – Boulez – EIC – Teatro alla Scala, Milan, Italy 03-11-98 – Boulez – EIC – Rome, Italy 06-11-98 – Boulez – EIC – Paris, France 12-12-99 – Boulez – Chicago Chamber Musicians – Chicago, US 17-03-01 – Boulez – EIC – Paris, France (6 minutes and 21 seconds) 01-12-01 – Boulez – EIC – Concertgebouw, Amsterdam, Netherlands (14 minutes) – Interruptions are added 04-12-01 – Boulez – EIC – Cité de la musique, Paris, France (15 minutes and 18 seconds) 05-12-01 – Boulez – EIC – Cité de la musique, Paris, France (17 minutes) 14-09-02 – Boulez – EIC – Luzern Festival, Lucerne, Switzerland (SECOND published version, 20 minutes) 20-09-02 – Boulez – EIC – Ircam, Paris, France - (DG recording session) (24 minutes and 33 seconds) 29-09-02 – Jonathan Nott – EIC – Strasbourg Musica, Strasbourg, France (23 minutes and 26 seconds) 27-10-02 – Jonathan Nott – EIC – Dortmund Konzerthaus, Fortmund, Germany 10-11-02 – Boulez – EIC – Genève, Switzerland 17-11-02 – Boulez – EIC – Köln, Germany 23-03-03 – Boulez – EIC – Carnegie Hall, New York, US 28-08-03 – Jonathan Nott – Ensemble Modern – Salzburg, Austria 15-09-04 – Cliff Colnot – Lucerne Festival Orchestra – Luzern, Switzerland 25-09-04 – Boulez – EIC – Torino, Italy 13-10-04 – Boulez – London SO – Barbican, London, UK (25 minutes and 25 seconds) 22-10-04 – Boulez – London SO – Paris, France (26 minutes) 26-10-04 – Boulez – London SO – Bruxelles, Belgium 29-10-04 – Boulez – London SO – Madrid, Spain 02-11-04 – Boulez – EIC – Milano, Italy 29-01-05 – Boulez – London SO – Carnegie Hall, New York, US 15-02-05 – Boulez – EIC – Paris, Cité de la musique, France (25 minutes and 35 seconds) 24-02-05 – Boulez – EIC – Orléans, France 14-03-05 – Boulez – Members of Chicago Symphony Orchestra – Chicago, US 16-08-05 – Tito Ceccherini – Ensemble Risognanze – Kufstein, Austria 20-04-06 – Daniel Kawka – EIC – Nantes, France 21-04-06 – Daniel Kawka – EIC – Angers, France 13-07-06 – Boulez – EIC – Aix-en-Provence Opera Festival, Aix-en-Provence, France (THIRD published version - UE 31940) 27-09-06 – Jean Deroyer – New Music Concerts Ensemble – Warsaw, Poland 07-11-06 – Boulez – EIC – Cité de la musique, Paris, France (44 minutes and 50 seconds) 09-11-06 – Boulez – EIC – La Scala, Milano, Italy (45 minutes) 11-11-06 – Boulez – EIC – Badenweiler, Germany (44 minutes and 50 seconds) 13-11-06 – Boulez – EIC – Bern, Switzerland 12-01-07 – Jean Deroyer – EIC – Nanterre, France 19-02-07 – Francois-Xavier Roth – EIC – Madrid, Spain 24-03-07 – Peter Rundel – MusikFabrik – Bruxelles, Belgium 31-07-07 – Susanna Mälkki – London Sinfonietta – London, UK (44 minutes and 52 seconds) 02-10-07 – Reinbert de Leeuw – Asko Ensemble – Amsterdam, Netherlands (44 minutes and 13 seconds) 25-10-07 – Joan Cervero – Group Instrumental de Valencia – Valencia, Spain 04-11-07 – Bill Linwood – Aventa Ensemble – Victoria B.C. Canada 06-12-07 – Daniel Phillips – Ensemble E – Melbourne, Australia

253

85. 86. 87. 88. 89. 90. 91. 92. 93. 94. 95. 96. 97. 98. 99. 100. 101. 102. 103. 104. 105. 106. 107. 108. 109. 110. 111. 112. 113. 114. 115. 116. 117. 118. 119. 120. 121. 122. 123. 124.

18-03-08 – Daniel Phillips – Ensemble E – Melbourne, Australia 06-04-08 – Franck Ollu – KammerensembleN – Stockholm, Sweden 22-04-08 – Daniel Kawka – Ensemble Orchestral Contemporain – Marseille, France 10-05-08 – Sandro Gorli – Divertimento Ensemble – Monza, Italy 11-05-08 – Sandro Gorli – Divertimento Ensemble – Milano, Italy 31-05-08 – Susanna Mälkki – EIC – Essen, Germany 26-06-08 – Daniel Kawka – Ensemble Orchestral Contemporain – Eglise de la Bénisson-Dieu, France 23-08-08 – Franck Ollu – Royal Stockhom Philharmonic Orchestra – Stockholm, Sweden 30-09-08 – Michel Swierczewski – Prager Kammerphilharmonie – Prague, Czech Republic 28-10-08 – Normunds Sne – Sinfonietta Riga – Riga, Latvia 11-11-08 – Chan Tze Law – Student Ensemble – Singapore 11-12-08 – Boulez – EIC – London, UK 13-12-08 – Boulez – EIC – Grenoble, France 16-12-08 – Daniel Kawka – Ensemble Orchestral Contemporain – Opéra National de Lyon, Place de la Comédie, Lyon, France 29-01-09 – Boulez – EIC – Mozerteum Großer Saal, Salzburg, Austria (FOURTH version - UE 32528) 26-04-09 – Otto Tausk – MusikFabrik – Berlin, Germany (45 minutes and 39 seconds) 13-08-09 – Boulez – EIC – Helsingin Juhlaviikot, Helsinki, Finland 21-09-09 – Franck Ollu – Ensemble Modern – Mozart-Saal, Alten Opera, Frankfurt am Main, Frankfurt, Germany (44 minutes and 11 seconds) 01-03-10 – Ed Spanjaard – Nieuw Ensemble Amsterdam – Arnhiem, Netherlands 04-03-10 – Ed Spanjaard – Nieuw Ensemble Amsterdam – Muziekgebouw aan’t IJ, Amsterdam Netherlands (44 minutes and 25 seconds) 12-03-10 – Boulez – EIC – Cité de la musique, Paris, France (FINAL Version, 44 minutes and 37 seconds) 13-03-10 – Boulez – EIC – Köln, Germany 01-04-10 – Geoffrey Paterson – Theseus Ensemble – London, UK (51 minutes and 10 seconds) no violin grace notes (fourth version) 20-05-10 – Peter Rundel – EIC – Rotterdam, Netherlands 21-06-10 – Hans-Christian Euler – Ensemble Musica Viva Hannover – Hannover, Germany 26-06-10 – Boulez – EIC – Aldeburgh Festival, Aldeburgh, UK (44 minutes and 42 seconds) final version 03-09-10 – Jean Déroyer – Lucerne Festival Academy Ensemble – Luzern, Switzerland (abbreviated version, 27 minutes and 26 seconds) 06-09-10 – Omer Meir Welber – Jerusalem 20-09-10 – Daniel Barenboim – Staatskapelle Berlin – Berlin, Germany 21-09-10 – Daniel Barenboim – Staatskapelle Berlin – Berlin, Germany 06-11-10 – Daniel Kawka – Ensemble Orchestral Contemporain – Auditorium National de Musique Madrid, Spain 14-11-10 – Eric Hewitt – The Boston Conservatory Boulez Festival players – Boston, US (49 minutes and 42 seconds) no violin grace notes (fourth version) 06-12-10 – Jim Baker – Talea Ensemble – Miller Theater, New York, US (51 minutes and 37 seconds) no violin grace notes (fourth version) 04-01-11 – Bill Linwood – Aventa Ensemble – Toronto, Canada 06-01-11 – Bill Linwood – Aventz Ensemble – Montreal, Canada 07-01-11 – Bill Linwood – Aventz Ensemble – Edmonton, Canada 08-01-11 – Bill Linwood – Aventz Ensemble – Calgary, Canada 21-01-11 – Jean Deroyer – Ensemble Contrechamps – Genève, Switzerland (48 minutes and 31 seconds) no violin grace notes (fourth version) 14-03-11 – David Milnes – ECO Ensemble – University of Berkeley, Berkeley, US 12-05-11 – Arie van Beek – Nieuw Ensemble Amsterdam – Köln, Germany

254

125. 126. 127. 128. 129. 130. 131. 132. 133. 134. 135. 136. 137. 138.

17-06-11 – Daniel Kawka – Ensemble Orchestral Contemporain – Room Guy Poirieux, Montbrison, France 18-06-11 – Daniel Kawka – Ensemble Orchestral Contemporain – Room Guy Poirieux, Montbrison, France (50 minutes and 44 seconds): final version with violin grace-notes, but ending is the third version 03-07-11 – Daniel Barenboim – Staatskapelle Berlin – Berlin, Germany 25-10-11 – Andrea Pestalozza – Brandenburg, Germany 27-10-11 – Andrea Pestalozza – Bologna, Italy 09-12-11 – Pavel Šnajdr – Studentenorchester – Janácek-Musikakademie, Brno, Czech Republic 14-12-11 – Michael Wendeberg – Collegium Novum Zurich – Zürich, Switzerland (7 minutes and 36 seconds) - Version 1988 14-12-11 – Michael Wendeberg – Collegium Novum Zurich – Zürich, Switzerland (47 minutes and 30 seconds) - Version 2006 no violin grace notes (fourth version) 09-02-12 – Ernest Martínez-Izquierdo – BCN216 – Barcelona, Spain 10-12-12 – Ernest Martínez-Izquierdo – BCN216 – Barcelona, Spain 01-03-12 – Andrea Pestalozza – Divertimento Ensemble – Milan, Italy 18-05-12 – Pierre-André Valade – Athelas Sinfonietta – Copenhagen, Denmark (44 minutes and 12 seconds) no violin grace notes (fourth version) 20-07-12 – Daniel Barenboim – West-Eastern Divan Orchestra – London, UK (49 minutes) no violin grace notes (fourth version) 28-05-13 – Orchester-Akademie der Berliner Philharmoniker – Kammermusiksaal, Berlin

255

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Music Scores Boulez, Pierre. Anthèmes 1 pour violon seule (1991/1992). Vienna: Universal Edition (UE 19992): 1992. ______. Anthèmes 2 pour violon et dispositive électronique (1997). Vienna: Universal Edition (UE 31160): 1997. ______. Dérive pour six instruments (1984, version 1986). Vienna: Universal Edition (UE 18103): 1986. ______. Dérive 2 pour 11 instruments (1988/2006). Vienna: Universal Edition (UE 31940): 2006. ______. Dérive 2 pour 11 instruments (1988/2006). Vienna: Universal Edition (UE 32528): 2006. ______. Incises pour piano (1994). Vienna: Universal Edition (UE 30200): 1994. ______. Incises pour piano (version 2001). Vienna: Universal Edition (UE 31966): 2002. ______. Le Marteau sans maître. London: Philharmonia Partituren in der Universal Edition (UE12450): 1957. ______. Messagesquisse pour 7 violoncelles. Vienna: Universal Edition (UE 16678): 2000. ______. Memoriale (…explosante-fixe…originel) pour flute solo et huit instruments. Vienna: Universal Edition (UE 18657). ______. Messagesquisse pour 7 violoncelles (1976). Vienna: Universal Edition (UE 16678): 2000. ______. Rituel in memoriam Bruno Maderna für orchester in acht gruppen. Vienna: Universal Edition (UE 15941): 1975. ______. sur Incises pour trios pianos, trois harpes et trois percussions-claviers (1996/1998). Vienna: Universal Edition (UE 30939): 2006. ______. sur Incises pour trios pianos, trois harpes et trois percussions-claviers (1996/1998). Vienna: Universal Edition (UE 31514): 2006. ______. sur Incises pour trios pianos, trois harpes et trois percussions-claviers (1996/1998). Vienna: Universal Edition (UE 34302): 2006. Discography Boulez, Pierre. Le Marteau sans Maître, Dérive 1 & 2. Deutsche Grammophon CD DDD 0289 477 5327 8GH (20/21 Series). 2005. Program Notes Universal Edition. “Pierre Boulez, Dérive 2 for 11 instruments – Work Introduction,” Universal Edition. http://www.universaledition.com/ Pierre-Boulez/composersand-works/composer/88/work/9777/ work_introduction (accessed October 18, 2011).

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