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
51 Gérard Grisey, “Tempus ex Machina: A Composer’s Reflections on Musical T ime,” 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 œ œ œ œ 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
56 "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 te mporal 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
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
60 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
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
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
62 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
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
Ex. 1.4.10. Out-of-phase periodicities where the pulses do not converge
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
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
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
64 14th-century isorhythmic motets can be considered as a form of periodicity between pitch and rhythm.
65 “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.
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.
67 “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 em ployed 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.”
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
68 Gérard Grisey, "Tempus ex Machina: A Composer’s Reflections on Musical Time,” 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.
70 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.
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 Désordre 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 György 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
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
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
75 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.
78 Allen Edwards, Flawed Words and Stubborn Sounds: A Conversation with Elliott Carter, 111.
79 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.
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
82 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 c omes 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
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 Dérive 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 – 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.
1 Boulez, Penser la musique aujourd’hui, 28.
2 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 structure s) 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.
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 flu te 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 avant l’Artisanat furieux
Flute, Vib., Guitar, Viola
2 Commentaire I de Bourreaux de Solitude Flute, Xylo., Perc., Viola
3 l’Artisanat furieux
Voice, Flute
4 Commentaire II de Bourreaux de Solitude Xylo., Vib., Perc., Guitar, Viola
5 Bel édifice et les pressentiments version première
6 Bourreaux de solitude
7 après l’Artisanat furieux
Voice, Flute, Guitar, Viola
Voice, Flute, Perc., Guitar, Viola
Flute, Vib., Guitar
8 Commentaire III de Bourreaux de Solitude Flute, Xylo., Vib., Perc.
9 Bel édifice et les pressentiments double 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
7 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
13 “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.
15 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.
Ex. 3.1.1. The opening viola line up to the first double barline
Rapide (q = 152)
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
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
1 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 – (024), (027), (036), (037), and (048),2 while putting an emphasis on trichords that involve the tritone – (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
2 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
4 Throughout this document, all the transposing instruments will be notated in and referred to in concert pitch (in C).
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
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
5 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
6 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
Ex. 3.1.7. Trichords are combined to form 12-element collections & 1+2+3
&
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
&
&
&
&
&
&
&
&
&
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.
9 Boulez’s method of trichordal combination in creating polyph onic structure resembles that of Milton Babbitt. However, Babbitt’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
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