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
Ex. 3.1.10. (cont’d)
Violoncelle
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
Ex. 3.1.11. (cont’d)
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
Clarinette
Vibraphone
Marimba
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
Ex. 3.1.12. (cont’d)
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).
Clarinette
Violoncelle
Ex. 3.1.13. A realization of the 6-element collections from R9-3 to R9-4
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
Ex. 3.1.14. (cont’d)
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
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
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).
Ex. 3.1.15. A realization of the 9-element collections from R14 to R16
Assez rapide (q = 92) Tempo variable en'sassordant á la dynamique. Les indications métronomiques sont à relier avec souplesse
Ex. 3.1.15. (cont’d)
Clarinette en
Violoncelle
Vibraphone Marimba
Ex. 3.1.15. (cont’d)
Ex. 3.1.15. (cont’d)
Clarinette en
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.
Ex. 3.1.16. The opening all-trichordal hexachord
[012478] - All-trichord Hexachord
Ex. 3.1.17. A reduction of the simultaneities in the opening measures
Ex. 3.1.17. (cont’d)
Ex. 3.1.17. (cont’d)
Ex. 3.1.17. (cont’d)
Ex. 3.1.17. (cont’d)
Pierre Boulez „Dérive 2|für 11 Instrumente“ © Copyright 2006 by Universal Edition A.G., Wien/UE 32528
Clarinette en la Basson
Vibraphone
Marimba
Ex. 3.1.17. (cont’d)
Clarinette en la
Ex. 3.1.17. (cont’d)
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
10 Philippe Albéra, and Pierre Boulez, “Pierre Boulez in Interview (2): On Elliott Carter, ‘A Composer Who Spurs Me On’,” Tempo 217 (July 2001): 3.
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
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
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 – a labyrinth out of a labyrinth.
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
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
Ex. 3.2.2 (cont’d).
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
Ex. 3.2.3. (cont’d)
Ex. 3.2.3. (cont’d)
Ex. 3.2.3 (cont’d)
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.
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
Ex. 3.2.4. (cont’d)
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.
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.
CA., Cl., Bsn. N N N N N N Vla.
3 cycles Vln.
E E E E E Vln., Vla., Vc N N N N N N Cl.
3 cycles Bsn.
E E E E E CA., Bsn., Vla N N N N N N Cl.
3 cycles Vln.
E E E E E CA., Vln., Vc. N N N N N N Vla. 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 Rhythmic pattern
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 schemes 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-n 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.
Ex. 3.2.7. Opening measures re-barred according to the instrumental partitioning of E-nn-n-E-E-n-E-E-n-n
pattern
Cor Anglais
Cor
