“Waves” in Debussy's Jeux
A notable feature of Debussy’s Jeux is the cyclic musical process. This process is characterized by motivic particles recycled within a series of waves, and is part of a duality of continuity and discontinuity found within the piece. Moreover, although there are discontinuous elements within Jeux, there is an underlying continuity, largely due to the wave process. The cyclic process in Jeux results from the use of motivic-thematic regeneration, melodic waves, and statistical waves working together, at different formal and hierarchic levels, and there is a large-scale structure and temporality which results from the process.
In Jeux, arabesques and wave processes co-exist with fragmentation and mosaic structure. The dialectic of continuity and discontinuity is also highlighted through change, interruption, inhibition, and reversal of process. The duality of discontinuity and continuity parallels other general dualities, some of which are expressed in Jeux, including that of object and process, particle and wave, nonlinear and linear, heterogenous and homogenous, and others (Figure 1).
particle, motive, mosaic
form, object, duration
process, flow, change, motion, tension, release
heterogenous, difference, variety
homogenous, similarity, unity
heaven, permanence, eternity
earth, change, temporality, beginnings and endings
Figure 1: Dualistic Elements
In addition, Jeux expresses other dualistic elements, including Ruwet’s “play of symmetries and asymmetries,” “the dialectic of the repeated and the non-repeated,” a dichotomy of diatonic and chromatic materials, and the synthesis of East-West ideas.
The idea of a continuity of waves within a modular structure parallels Asian concepts of time. Concerning the duality of continuity and discontinuity, ancient Chinese thought emphasizes continuity, waves, and cyclically recurring change. On the side of discontinuity, time was divided into parts or phases. Needham says, “An important question here is how far the individual cycles, or particular parts of cycles, were compartmentalized, separated off from one another into discrete units,” that “time in ancient Chinese conception was always divided into separate spans,” and that “it was essentially discontinuous ‘packaged’ time,” but that ultimately “Chinese physical thinking was dominated throughout by the concept of waves rather than of atoms.”
Debussy said “I would like to make something inorganic in appearance and yet well‑ordered at its core.” Boulez notes of Jeux, that “the general organization of the work is as changeable instant by instant as it is homogenous in its development.” Debussy also says of the contrasting sections in Jeux, that “the link between them may be subtle but it exists” and Eimert notices the juxtaposition of fragments, similar to that of mosaic and ‘miniature-art,’ while emphasizing the continuous aspects of the piece, the arabesque-like melodic and statistical waves, motivic regeneration, and “timbres and flowing tempi.” Others have also indicated, on the one hand, the discontinuities and incipient moment-form qualities of Jeux, and on the other hand, various qualities of continuity.
Cinematography also integrates continuity and discontinuity, through frame-to-frame changes, similar to Langer and Zuckerkandl’s concept of virtual time, wherein a succession of notes creates the illusion of motion. Cinematography influenced Debussy to create musical fades, slow motion, freeze-frame, cuts generating musical blocks, creating “a linked series of images, musical ideas,” sometimes approaching moment-form, and cross-cutting stratification. Debussy’s mosaic technique itself contains many elements of cinematography, with which it shares a focus on the moment. Pasler relates Bergson’s durée to the focus on the moment in Jeux, process versus object, the duality of heterogenous elements within a homogenous flow, and “the dialectic of continuity and discontinuity,” stating that “Jeux embraces both continuity and discontinuity,” and that “Jeux lies at the crossroads of a change in aesthetic values from the need for continuity to the desire to create discontinuity.”
Levels of formal discontinuity in Jeux occur as fragmentation, contrast, and stratification of materials, which depending on the degree of contrast may or may not produce functional discontinuity, while functional discontinuity results from the separation created by the troughs between successive waves, moment-form tendencies, and through process inhibition, interruption, change, deflection, and reversal, wherein discontinuity is presented as a form of inhibited continuity. Levels of formal continuity in Jeux occur in the form of repetition, periodicity, metric regularity, gradation, consistent timbres, the linear writing, and motivic connection and recurrence, while functional continuity results from open-ended and directed wave processes. The dialectic of continuity and discontinuity is sometimes highlighted, as for example through the refraction of continuous wave motion between contrasting modules of the mosaic. The general effect is one of connected blocks and contrasts wherein successive modules are, to different degrees, “set off” from one another through the timbral, textural, rhythmic, and motivic coherence within modules, and the different degrees of change or discontinuity between modules. The contrasts are always connected through motivic-thematic means and the underlying wave process runs through the entire piece like a “mysterious thread” giving it flow and continuity.
Levels of Design
Debussy’s composed-out musical waves have three general levels of design: (1) a micro level which corresponds to the motivic-thematic particles, processes, and melodic waves; (2) a middle level, which corresponds to the metric grid, metric waves, and the mosaic structure; and (3) a macro-level, which corresponds to the statistical waves, wave groups, and larger sections. With the wave processes and mosaic structure being synchronized by the metric grid, these three levels of design work together, and integrate continuous and discontinuous elements. The basic relationship between these levels can be illustrated as follows (Figure 2). In Jeux this basic idea is developed into elaborate patterns.
Figure 2: Structural Levels of Wave Process
These three levels are general levels of design. Each level can be further subdivided and there is some overlap between them. However, the general order of micro- to macro- applies. There are minimal building blocks at each level which are combined to form larger units. The smallest building block at the micro level is the single note, which combines to form motivic fragments and motives. The smallest building block of the middle level of mosaic structure is the cell, which combines to form modules and sections, and the smallest building block of the macro-level is the single wave, which combines to form wave groups, wave trains, and large-scale wave processes. A motive is often the size of a mosaic cell, while a statistical wave is often composed of two or more mosaic modules. Even though the elements of each of these levels may be analyzed separately, in practice each higher level subsumes the lower levels, i.e., the mosaic structure includes the motivic particles, and the statistical waves include the mosaic structure and motivic particles.
Micro Level: Motivic-Thematic Particles and Processes, Melodic Waves
The motivic-thematic particles of Jeux are small, relatively self-contained, and open-ended. These motives and fragments thereof form the building blocks of the larger motivic-thematic process and structures, and are generated by an additive and combinatorial process. The process is non-developmental and ‘inorganic’ in the traditional sense of gradual, linear transformation and incorporates elements of continuity and discontinuity, process and object. As process, the motives are continuous, step-wise, and wave-like, while as objects they form the particles within the statistical waves and cyclic processes.
Debussy’s motivic particles are characterized by a kind of invariance under transformation. Meyer says that “Debussy, too, chooses motivic constancy.” Eimert refers to the “associative and motivic coherence of the melodic shapes (Gestalten, figures) in the piece” and creates a “table of associations” to show the connections between the different motivic-thematic shapes and a general “wave pattern.” Trezise presents the similar idea of “motivic constellations,” as does Wenk with his concept of global form. This coherence is facilitated by Debussy’s use of simple, universal, and ubiquitous shapes, e.g., simple wave shapes, ascending and descending scale fragments. Pasler refers to the idea of “a short, clearly recognizable motive that is flexible and easy to manipulate.” Bergson’s “mean image,” and Barraqué’s “thèmes-objets” similarly do not develop but recur, divide, and recombine in ever-new ways and Trezise comments on the same yet different quality of the motivic shapes, referring to the “quest for motivic diversity as for motivic homogeneity” and “the sensation that an idea is familiar, even when its surface bespeaks disparity and incongruity.” Debussy thus also creates less easily recognizable variations of motivic-thematic ideas, hence the idea of transformation and so, in keeping with Debussy’s aesthetics, the similarities between such motivic particles are often ‘subtle’ or subconsciously perceived.
Within the self-similarity of materials there is free variation of motivic design. Debussy flexibly uses a number of variation techniques including inversion, augmentation, diminution, reversion, interversion, change of tempo, rhythm, or accent, thinning and filling of thematic shapes, and techniques similar to liquidation, dissolution, condensation, and reduction. Sometimes melodies are “varied” by simply doubling them with intervals or chords, similar to organum but Debussy’s variations are largely based on arabesque techniques, including those from Baroque ornamentation changes of articulation, simple embellishments such as grace notes and trills, “free ornamentation” and melodic elaboration using neighbor notes or intervals, filling of intervals, alternation with common tones, micro-rhythmic variations, syncopation, etc. Debussy’s also generates “families” of related motives through the use of secondary variations - variations becoming the basis of further variations, and so on. He goes further by creating a more radical motivic‑thematic transformation through change of “character,” for example, the transformation of a legato motive into a motoric rhythmic figure (m. 342 clarinet, m. 387 bassoon). Motivic transformation also appears as an extended form of traditional change of mode between diatonic and chromatic. This often involves semitonal transformations from major to minor 2nd, major to minor 3rd, and perfect 4th or 5th to tritone (or vice versa). The motivic particles and arabesques form part of an overall alternation and synthesis of diatonic and chromatic materials wherein motivic-thematic shapes are composed from diatonic, modal, pentatonic, overtone (or acoustic), chromatic, whole-tone, octatonic, and other symmetrical scales, or from an intersection and combination of these, resulting in a more exotic “chromatic modality.”
Even though Debussy mentions Bach and Palestrina as examples of musical arabesque and the “principle of ornament,” his method and aesthetic is also similar to that of arabesque in Middle-Eastern music, from where the term ‘arabesque’ derives, and of Art Nouveau, both of which interested Debussy. These styles share an emphasis on melodic linearity and embellishment. The use of embellishment and ornament also provide an improvisation-like element of free play within a structure. Al-Faruqi, in her description of arabesque in Arabian music gives a description that perfectly fits Debussy’s technique:
... a conjunct string of short motif or motif conglomerates in varied repetitions. Each one of these chains of repeated or symmetrical combinations creating an aural arabesque, a labyrinth journey of melodic and rhythmic details... towards a musical resting place. ...Each phrase is a new excursion into a tonal or durational unknown with its aesthetic tension, followed by an eventual relaxation... to a point of musical stability.
The arabesque is “open-ended,” “capable of accepting the addition of further motifs and modules. The arabesque is experienced in parts, for “the significance of each motif and module must be appreciated one by one, ...with its many mini-climaxes.”
The open-ended quality, alternations of tension and relaxation, “mini-climaxes,” and “point of musical stability” noted above are characteristics of the ‘continuous’ aspects of wave processes. At the same time, the combination of motives, “motive conglomerates,” and chains of varied repetitions relate to the additive or ‘discontinuous’ aspect of these designs. Debussy uses such additive and combinatorial techniques, including repetition, transposition, division, fragmentation, addition, subtraction, and permutation. His generative process also connects the motivic particles at the micro level, creating moment-to-moment continuity, ensuring that even contrasts are connected. The connective elements or ‘subtle links’ consist of single pitches, intervals, chords, fragments, or shapes. Other connective devices include immediate transformation, embellishment, overlapping, and change of function, or kinesthetic shift. Connection is also achieved at the phrase level and sectional levels through what Ruwet calls duplication, which involves using the same or varied beginning, or ‘head’ of an idea as the basis for a successive variant, but with a different ‘tail’ or ending. Debussy also uses the opposite technique, that is, using the ‘tail’ of an idea as the basis for the ‘head’ of a following idea, producing a ‘propagation’ or transference of motives between modules or waves (e.g. mm. 27-31). The use of repetition in the form of reiteration provides both cohesion and segregation, both defining and dividing events, producing fragmentation. Reiteration (and variation) also contributes to a focus on the moment through an emphasis on differences.
The qualities of continuity, flexibility, fluidity, and “curve” of melody which Debussy defines as part of musical arabesque, give the arabesques of Jeux the quality of melodic waves with motivic particles arranged so as to create wave patterns, circularity, and open-endedness. Eimert makes numerous references to the “motivic particles,” “linear writing,” and “ornamental stepwise waves.” Pomeroy notes that Debussy’s arabesques generally tend to fill registral space with a “combination of stepwise and relatively undulating disjunct motion,” and generally many of Debussy’s melodies have shapes described as undulating, wavering, oscillating, etc. Even though the motivic particles are small and frequently shift from one instrument or group to another, the continuity of arabesque and wave patterns are still at work.
The motivic particles and melodic waves thus shape the larger textural and statistical waves of Jeux. The melodic waves take the form of either the more frequent, ornamental, and often rhythmic motivic waves, or of simpler and more lyrical thematic waves. These two kinds of waves are sometimes expressed as a play of rhythmic-motivic and lyrical ideas. Trezise quotes Barraqué on a similar play of “rhythmically-oriented” and “melodically-oriented” ideas in Jeux de vagues. In contrast to the more common fragmentary and ornamental motivic waves, there are few extended thematic statements or thematic waves in Jeux, and these are more prominent in the second part of the piece. However, lyrical fragments are found throughout the piece, often as secondary or background parts. There is nearly always a melodic element in the music, whether it is arabesque‑like or lyrical, and figuration is nearly ever‑present. Motives and themes are composed through the same generative process and from the same pitch cells, providing unity and allowing motives to be transformed into themes. Trezise refers to “the transformation of arabesque into motif,’ and Eimert gives the example of the “alpha” motive from m. 49 augmented into climactic theme at m. 677. Another example is the transformation of the descending, chromatic, “beta” motive from m. 10 into the thematic lines at mm. 475, 481, and 489. Motivic and thematic ideas are also presented together, including the simultaneous, or heterophonic, presentation of the same idea (e.g., m. 585 horns and m. 587 oboes) where an idea emerges in both thematic and ornamental form, thus combining the ideas of arabesque and heterophony (e.g., mm. 100-117, the flute part presented ornamentally in the clarinet part). Moreover, Debussy often uses a more generalized form of “heterophonic orchestration” through slightly varied doublings of parts. The longer thematic waves of Jeux have a joyful and flowing contour. The ‘waltz theme’ of Jeux (mm. 566-604) is parallel in function to the “smooth sailing” theme of Jeux de vagues.
In addition to the moment-to-moment connection between the motivic particles of the arabesques and melodic waves produced by the generative process, there is also a regeneration or recycling of particles that occurs between waves in the form of returns and ‘return effects’ of motivic-thematic materials which recur, usually transformed. Debussy wrote of the search for “forms in constant renewal.” and renewal suggests cyclicity, as distinct from “constantly new forms.” Some of these recurrences are found at apparently random positions within waves and sections, leading some to refer to this as a “separation of theme from structure.” The cyclic nature of the wave process itself encourages such recycling, and provides a format for Debussy’s tendency to reiterate and transform motivic-thematic material. The constant recycling of numerous particles of motivic-thematic materials also produces a static and nonlinear sense of multiplicity and simultaneity. Eimert says, “in the vegetative circulation of the form there is no development”:
The ornamental formations based on close intervals move in simple waves, which circulate or are spun out, are repeated and give rise to fresh waves, a circulation which is always at its goal and therefore never ‘going’ anywhere, never building up thematic figures, with no motivic ‘working-out’. Instead of this we find motivic association, which produces inexhaustible variants, not in fixed thematic form, psychologically bracketed off, but in a freely growing process of breeding.
Eimert’s comments are somewhat overstated, his point being to emphasize an important aspect of the motivic process. The incessant recycling of the motivic particles creates an ever-presence of motivic-thematic materials submerging and emerging from a static ‘background.’ This emphasis through reiteration and recurrence on what Pasler calls the “essence” of a motivic idea creates a focus on the moment similar to that of Eastern music, which is, as Koelreutter says, to “circle round a central idea.” However, Jeux expands this idea by using two basic germinal ideas which are recycled, a modal “alpha” motive-theme (m. 49) and a chromatic “beta” motive-theme (mm. 1-2). Most of the motivic-thematic designs are derived from these basic ideas and their combination or synthesis. In addition, a number of secondary motivic and rhythmic motives are recycled throughout the piece. Secondary materials are also transformed into primary materials. Fragments of primary materials are also often transformed into secondary and accompaniment material, creating arabesque‑like backgrounds and figurations. Because of the fragmentary nature of the materials, the distinction between figure and ground, or between primary and secondary parts is sometimes weak or even non‑existent, resulting in passages of figuration or accompaniment which are purely “textural” in effect. The textural and background elements, figurations, secondary parts, and accompaniment patterns are also subject to recycling. Purely textural passages sometimes provide a lull or break from the focus of the motivic-thematic process (e.g., mm. 130-137, at the end of the first section). Eimert’s discussion of “ornaments, motives and flocculi, which have a secret associative power” suggests that these background and secondary parts also follow the “principle of ornament.” The manner in which Debussy transforms the motivic-thematic identities themselves does not differ from traditional music. Rather, it is the ornamental design of the motivic particles and the manner in which they are combined which differs.
The recycled materials have different degrees of connection between waves. There are, generally speaking, three basic levels of recurrence of motivic‑thematic materials in Jeux. The first is within sections, that is, between successive waves or through returns after contrasting materials, e.g., AA, AAA, ABA, ABAB, and involves materials returning in literal or varied form. The second level is between sections, e.g., ABCB ADED BFGF BHIH, with materials returning at the beginning of each section, and involves a return of materials usually transformed, e.g., the “alpha” and “beta” themes return transformed marking the beginning of different sections. This technique resembles that used between movements in cyclic forms, except that here the returns occur between sections and more than one theme is used, although the “alpha” theme is more central. The use of two basic themes instead of one vastly expands the formal possibilities. Out of the twelve sections into which Jeux is analyzed below, seven begin with some form of the “alpha” theme, while five begin with some form of the “beta” theme. In addition, the “alpha” and “beta” materials are freely developed and alternated within these sections. The third level of returns is large-scale, between the large parts of the piece, as in the reprise and epilog (postlude) of Jeux, and are usually literal. More literal returns create a greater sense of stability and closure, as with sectional and large-scale returns, which often form ABA patterns. Short-scale recurrences define sections while large-scale recurrences define the entire piece.
There is a kind of cyclic development or “working out” of motivic-thematic material within the piece which consists of a gradual departure from the original form of the “alpha” motive during the first two sections of the piece, where it appears in more literal and varied forms, followed by a greater use of the “beta” theme and more derivative forms of the “alpha” theme during the following sections of the first part of the piece. This helps give the literal return of the “alpha” motive at the reprise (m. 455) a strong sense of return, pointing back to the beginning of the piece, and creating large-scale structure. The second part of the piece beginning at the reprise forms a further cycle of motivic-thematic development which further integrates and brings together elements from the first part. This involves the increased use of literal rather than transformed versions of the “alpha” theme, helping to integrate and resolve the motivic process toward the end of the piece. Finally, the literal return of material in the epilog (postlude) points back to the very beginning and closes the piece. The effect of “departing” from original motivic forms is due to the anchoring effect of their initial presentations along with the different degrees of transformation these undergo within the larger processes. Structurally, the original forms and their recurrences are generally more ‘stable,’ while their transformed versions are more ‘unstable’. The original presentation of the chromatic or “beta” motive is also less rhythmically defined than that of the “alpha” motive, and lends itself to more fluid transformations and adaptations.
In addition to the cyclic regeneration and development of materials, there are several other developmental motivic-thematic processes in Jeux which differ from traditional procedures. For example, the piece begins with fragmentary motivic ‘seeds’ or germinal materials that combine and develop as the piece unfolds, giving a sense of growth. As the piece unfolds, original, transformed, and new ideas mix together. Trezise refers to Debussy’s technique as “a procedure of development in which the notions of exposition and development co-exist in an uninterrupted stream, permitting the work to be propelled along by itself without recourse to any pre-established model.” Leading up to the final climax of the piece there is a use of shorter and shorter alternation, interaction, and integration of motivic materials (mm. 627-676). Moreover, as Kinariwala says of Debussy, “The succession of ideas does not always lead to an inevitable goal but is often subjected to interruptions and circular patterns,” and that “Debussy’s structures at higher levels progresses not through a linear development but through one made up of starts and stops, returns and contrasts, seemingly without order or direction but always connected through the continual process of generation.” In Jeux, these motivic-thematic processes work within the larger statistical wave process.
Middle Level: Metric Grid, Metric Waves, and Mosaic Structure
The flow of waves in Jeux is “rhythmicized.” This occurs of course through the rational rhythms, and through the mosaic structure itself. The nearly constant 3/8 meter provides measure, continuity, and periodicity, and forms a ground upon which a play of rhythmic patterns can unfold, i.e., recurring 3/8 rhythms, cross-rhythmic patterns, hemiolas, waltz patterns, two-measure patterns producing a 6/8 feel, etc. The regular meter forms a metric grid, and provides a foundation and underlying continuity for the mosaic structure and larger wave processes. The piece is also governed by a “reference pulse” which is part of the temporal foundation of the piece. This is the Tempo initial of m. 9 (dotted quarter note = 72 M.M.), which returns regularly with the marking Mouvement initial. Pasler considers this an important organizational aspect, as well as Boulez, who believes that “a single basic tempo is need in order to regulate the evolution of thematic ideas.” Boulez thus sees continuity as prevailing in Jeux. Moreover, the pulse is not mechanical. Eimert likens the fluctuating tempo, rubato, and statistical wave process of Jeux to breathing, as a “respiratory flux of movements.” Pasler likewise compares the tempo fluctuations to a heartbeat, “accelerating and decelerating depending on the context.” These tempo fluctuations are coordinated with the statistical waves and even though the mosaic structure is composed of discrete parts, these features provide it with temporal regularity and an elastic quality.
The regularity of pulse and meter create periodicity. This is, of course, felt more or less depending on the strength of its musical articulation. Natural waves are also characterized by periodicity. In Jeux, the periodicity of pulse and meter creates metric waves, with the strongest being at the basic metric level, which Zuckerkandl calls the ‘principle wave.’ The metric wave has an inverse climax function in that the downbeat or trough is stronger than the crest or upbeat portion of the metric pattern or wave. The principal or basic metric wave interlocks with the metric grid and the mosaic cells, and combines to form larger nested metric waves corresponding to mosaic modules and statistical waves. Zuckerkandl states that “the meter in a piece of music does not beat simply in a single wave but in a complex involving superordinate and subordinate waves, of which at least the two closest to the principal wave can be felt distinctly,” making the periodicity most immediate at the metric and middle levels. The intensification phase of a wave can be seen to form a large ‘upbeat’ in relation to the ‘downbeat’ at the climax of the wave, or vice versa. In Jeux, this gives rise to interplays between the downbeats of wave climaxes and the structural downbeats of sections. Debussy also uses syncopation, cross-rhythms, and hemiolas to increase forward-drive towards the downbeat at the crest or climax of the wave, followed by rhythmic regularity (or vice versa) and de-intensification. Small repeated wave shapes and ostinati are often coordinated with the metric wave. Examples are the undulating strings of the opening of La Mer and later ostinati (mm. 31-42, 105-121).
Another feature of the wave process is momentum. Momentum may vary in strength, and results from periodicity, the irregular 3/8 meter, which is self-propelling, and rhythmic and wave processes which produce intensification, forward drive, and a structural accumulation of energy. Zuckerkandl makes reference to momentum and the cumulative effect of periodicity while discussing the polarity and intensification of wave processes, saying that “The swing that makes the wave reach its crest at the same time carries it beyond its crest, toward a new cycle, a new crest” and that “the repetition of the same metrical wave now produces intensification. Every new wave, in comparison to the similar wave that preceded it, is experienced as an increase.” The periodicity works at the middle level of the metric wave and the macro-level of the statistical wave and momentum increases as the wave crest is approached. Together, the basic tempo, metric grid, and associated features of periodicity and momentum form the temporal basis of the mosaic structure and wave process.
Figure 3 Coordination of Tempo Fluctuations, Metric Wave, and Statistical Wave
The mosaic structure is made up of three kinds of units: cells, modules,and sections. Cells form the smallest units of 1-2 measures in length, often coinciding with motives, and interlocking with the metric grid. The concept of a cell is different from that of a motive in that a cell consists of a measured unit which is interlocked with the metric grid. In distinction, a motive is usually a smaller, metrically independent, and monophonic part. In addition, mosaic cells originate at the middle level and form the building blocks of the mosaic structure, whereas motives originate at the micro level from single notes. Cells thus bring motives into the metric context and mosaic structure. The motivic-thematic material is subsumed within the mosaic cells, and gives each cell or module its identity.
The timbral-textural definition of the mosaic units is also important, and because Jeux uses an orchestra, a cell consists of a textural unit of several interlocking parts. These cells usually consist of a main part and a secondary part or parts which are in simple homo-rhythmic, contra-rhythmic, or hetero-rhythmic relationship to each other. Modules are made up of cells, usually two to eight or more, correspond in size with waves or parts thereof, and form the basic functional units of the mosaic. There may be several levels of modules (see Figure 7). Sections are made up of modules, forming ABAB, etc. patterns which are relatively self-contained. The modules which make up sections are often defined through the juxtaposition and alternation of diatonic and chromatic materials which they contain. At the highest level the mosaic corresponds to the complete structure, with its sections and constituent modules and cells. In addition to the linear, additive aspect of mosaic units, there is also an occasional superimposition or layering of parts within cells and modules. Wenk notes that in Jeux “Debussy frequently splits the individual unit into two or more layers, producing a horizontal as well as a vertical fragmentation of the musical texture.” This occurs to different degrees, ranging from the simple superimposition of rhythmically regular and interlocking parts, to a heterophonic or polyrhythmic layering of regular parts (e.g., mm. 130-137), to the layering of more irregular parts. Because of the gamelan influence on Debussy, the parts sometimes contain heterophonic, interlocking, or hocket-like elements.
The mosaic structure is produced through an additive and sequential process. Cells are generally repeated to form modules while modules are usually contrasted and have a greater sense of juxtaposition. The repetition of cells has a form-building function which provides coherence and structural redundancy and allows cells to group into modules through similarity. Pomeroy gives a description of Debussy’s use of mosaic structure, its integration of motivic-thematic ideas, and its construction:
Debussy integrates such thematic ideas into larger formal sections via a characteristic technique whereby arabesque-like units, typically two bars in length, are combined through a chain-like process. This constructive technique is so prevalent as to constitute one of his most readily identifiable stylistic traits. The units’ identity as such is established by symmetrically balanced contrast and juxtaposition in factors such as motive, texture, and harmonic rhythm (also, of course, secondary parameters of instrumentation, dynamics and so forth); larger form is generated by the hierarchical chaining of units and their compounds.
The contrast between modules is a matter of degree and ranges from mild to relatively strong. Debussy increases the sense of juxtaposition through the use of interpolation, stratification, and moment-form tendencies. Moreover, the juxtaposition of homogenous elements is more common than that of heterogenous elements. Shattuck states that the juxtaposition of homogenous or similar elements creates an “illusion of smoothness and rhetorical progression” which however “rapidly disintegrates.” Shattuck adds that “By their very repetitiveness the parts establish their independence” and that “style here becomes circularity, a distortion of linear development and direction in the traditional sense.” On the other hand, the juxtaposition of heterogenous elements interrupts linear development. Furthermore, the additive and combinatorial aspect of mosaic structure is similar to that of arabesque, consisting of small units which are repeated and combined (or “woven”) into larger patterns. Motives are thus to melodic waves what modules are to statistical waves. However, musical arabesque emphasizes the elements of similarity and continuity, while mosaic structure, as inspired by visual mosaic, emphasizes the spatial, discontinuous, and open-ended elements. This difference forms part of the duality of continuity and discontinuity in Jeux, i.e., the micro and macro-levels are continuous while the middle level is discontinuous.
Mosaic modules are combined according to the law of symmetry, which is applied as a sense of balance, complementation, and equilibrium. This results in techniques including repetition, binary constructs (antecedent - consequent, statement - counterstatement), and complementation (intensification and de-intensification, contrast, Al-Faruqi’s ‘symmetry’). An example is Debussy’s common practice of repeating once and moving on, e.g., (AA) (BB) (CC) (DD) etc. Debussy takes this process a step further by repeating groups of modules, e.g., (AB AB) (CD CD) or (ABC ABC) (DEF DEF) or (ABCD ABCD) (EFGH EFGH) and even (ABCDE ABCDE) (i.e., mm. 224-244, 245-263). This method is used commonly in Jeux, and results in a chain of relatively self-contained sections. A group of modules is repeated, forming a section, followed by a different group of modules, forming a new section, and so on. Although the sections may seem loosely connected, they form part of a large-scale process in the piece. However, mosaic structure works mostly at middle levels of the formal hierarchy in the form of modules and sections. Sections do not tend to combine to form larger sections, except for some contiguous sections that form free duplications (or replications) of each other. This aspect of the mosaic structure relates to Boulez’s comment on Jeux having a ‘woven form.’ Moreover, even though the mosaic structure is focused on the middle level, there are large-scale proportions, including the use of the golden section.
The contrasts and alternations of modules in the above forms of repetition, e.g., ABAB, produce returns, interpolation, and stratification of materials. The return or recurrence of materials after an intervening contrast emphasizes similarity and provides structural definition to the mosaic structure. The law of symmetry also expresses itself here as the law of return, in both repeated (ABAB) or ternary (ABA) organizations, and contributes to a cyclic effect and closure. Interpolation involves the temporary interruption and alternation with a secondary process, whereas Cone’s sense of stratification involves a more sustained alternation of processes. Cone’s concept of ‘interlock’ of strata, results from the resumption of an interrupted or suspended process after an intervening contrast corresponding to the cinematographic technique of “cross-cutting.” Debussy employs stratification to some extent in Jeux, in the form of overlapping stratification (or dovetailing) between sections (e.g., mm. 284-308), but more commonly in the simpler and short-scale form which Kinariwala calls “interlock through interpolation.” An example of duplication within mosaic structure is the open repetition pattern of ABAB’, where B’ is open, or AABA’, which creates a functional open-endedness and passive continuation. Debussy also uses closed repetition, e.g., AA or ABAB, which, due to its formal closure and self-containment, leads to a focus on the moment. Modules are also combined according to Ruwet’s “dialectic of the repeated and the non-repeated” or of the “expected and unexpected,” e.g., AAAB or ABAC. In addition, the interaction of duple and ternary groupings sometimes produces a blurring of structural boundaries and alternate interpretations and as to modular groupings, e.g., (ABA BCBC) vs (ABAB BCBC) or (ABAB CBCB).
Debussy’s mosaic structure evolved from simple block-like modules with steady intensity levels to modules with increasing and decreasing intensity levels corresponding to the parts of a wave. Figure 4 shows the basic idea of modular structure, from simple blocks to parts of waves, represented by the triangular modules, e.g., intensifications only, intensifications and de-intensifications in stages. Block modules can also represent standing waves, which are of relatively steady intensity. Other modules within the mosaic structure serve as transitions or bridges.
Through the complementation and symmetry of intensification and de-intensification phases of waves, Debussy discovered a new and powerful method of coherence and continuity between successive modules of the mosaic structure. The alternating phases of waves produce a corresponding open-closed functionality which links modules into pairs through complementation and symmetry, whereas returns of modules help define sections. The mosaic structure incorporates further rhythmic elements which relate to the wave process, including static and dynamic symmetries, a “play of symmetries and asymmetries,” irregular groupings, imbalance, and forward drive. Debussy’s basic repeat patterns are duple and square, 2+2 = 4, 4+4 = 8 measures, etc. This is made more interesting in several ways. One method is the setting up of regular repeated patterns of cells or modules which are shortened, e.g., (4+4)+(4+2), 4+3, or 4+4+3+3, 4+2+2+1+1+1+1, a process which reinforces the intensification phases of waves. Often the successive shortening or truncation of a repetition is used to reinforce the intensification phase of a wave, the build-up of energy, momentum, and forward drive towards the climax or release (e.g., mm. 639-644 form a (2+2)+2 or 4+2 pattern). As a way of shifting the duple predominance, sometimes there is a play of 2s and 3s, that is, the repetitions in the modular mosaic become part of rhythmic play of patterns which are set up and varied. These apply to the cellular and modular construction and in addition to the techniques already mentioned of syncopation, cross-rhythm and hemiola used within the metric wave.
Figure 4: Evolution of Mosaic Structure
The mosaic structure has an open-ended quality which is produced through different means. Its combinatorial and permutational construction has appealed to the ‘serial’ and open-form sensibilities of later composers, wherein mosaic units can be re-combined and added indefinitely and the modular juxtaposition and fragmentation of materials produces a sense of discontinuity and inconclusiveness.  Other means of producing open-endedness are the use of passive continuation and non-functional harmony and symmetrical scales. Howat also refers to the “enigmatic mixture of closed and open-ended formal characteristics” in relation to the fact that the mosaic units are formally relatively self-contained yet functionally open. Together with elements of discontinuity and lack of hierarchy, the non-functional and static elements often give the mosaic structure qualities of moment-form. Kramer notes how Messiaen took the mosaic technique of Debussy and Stravinsky closer to moment-form and Stockhausen’s further development of the idea. Kramer uses the term sectionalized collages as synonymous with moment-forms. However, Debussy’s mosaic modules are smaller than collage blocks, and are connected through his ‘subtle links,’ and form a mosaic of connected lines rather than a collage of contrasts.
The additive and combinatorial aspect of the mosaic structure lacks organic development, and instead produces a static sense of juxtaposition and simultaneity. Evans says of mosaic structure that “the impression of simultaneity of the whole is achieved through a collection of seemingly constantly present shapes that may intensify or vary independently without affecting an overall progression towards an end.” She also suggests that even though mosaic structure consists of an inorganic and “non-progressive development,” other kinds of processes and “development” are possible within the mosaic structure, e.g. processes of repetition, permutation, variation, ornamentation, fragmentation, extension, expansion, interruption, abbreviation, recombination, stratification, different rates of change, and intensification and de-intensification of materials. The “non-progressive development” of the mosaic structure leads to a sense of simultaneity and a focus on the moment. “Lack of overall development seems to be an essential feature of the mosaic model; it is perhaps the most crucial way of depicting a sense of’ ‘now’.”
Debussy combines the elements of discontinuity and open-endedness of the mosaic structure with the continuity of the motivic-thematic and statistical wave processes. There is a kind of development within the mosaic structure of Jeux whereby Debussy first uses larger modules, then parts of modules, and then sub-parts of modules, these corresponding to complete waves, then the intensification and de-intensification phases within waves, and then stages within these phases. This all becomes part of an increasingly elaborate play of repetition, variation, and alternation of modules which is particularly notable within the first part of the piece. Moreover, the wave process not only provides an underlying continuity but also shapes the entire piece. The wave process provides a continuity and flow which underlies and unifies the successive modules, including repetitions, contrasts, alternations, returns, and which can be distinguished from the block-like contrasts exploited by Stravinsky. Debussy’s use of mosaic structure in Jeux is unique. His modular waves may well have been his response to Stravinsky’s block-like structures, and an ingenious implementation of the “dialectic of continuity and discontinuity.”
Macro-Level: Statistical Waves
The principles of flexibility, fluidity, ornament, and the natural curve which Debussy mentions are also present in the textural waves of Jeux. Debussy’s statement that musical arabesque is “subject to laws of beauty inscribed in the movements of Nature herself” makes an implicit connection between musical waves and natural waves, of “sinuous arabesque based on natural forms.” This description includes Eimerts’s “vegetative forms”, the sinuous vines of Art Nouveau arabesques, with their leaves of different sizes and similar shapes, and what he calls“vegetative inexactness” and the “ornamental-vegetative formal principle” in Jeux. However, although the ballet story certainly suggests back and forth motions, Debussy was not trying to portray natural waves in Jeux. Instead, he plays with a more abstract sense or “essence” of wave behavior. Debussy says “I wanted music to have a freedom that she perhaps has more than any other art, as it is not restricted to a more or less exact reproduction of nature, but instead deals with the mysterious correspondences between Nature and the Imagination.” Debussy expresses his admiration of “the music of nature,” and states that “It is the musicians alone who have the privilege of being able to convey all the poetry of night and day, of earth and sky. Only they can re-create Nature’s atmosphere and give rhythm to her heaving breast.” However, Debussy stresses that “there is no attempt at direct imitation, but rather at capturing the invisible sentiments of nature.” Trezise refers to this as a “synthesis of the natural world and human emotion.” A reference to “capturing the invisible sentiments of nature” or the synthesis of nature and human emotion is seen in Debussy’s comment that the orchestration of La Mer “ is as stormy and varied... as the sea!.” Potter also writes of musical onomatopoeia in reference to “a multitude of water figurations” that “evoke the sensation of the swaying movement of waves and suggest the pitter-patter of falling droplets of spray” in La Mer. The elements of process, flow, and the “free play of sound” enter in the temporal dimension of these comparisons with nature.
Debussy’s “correspondences between Nature and the Imagination” are a form of analogy. The word analogy refers to the comparison of things that are different but which show similarity in some respect, though not necessarily in others. The idea of representation is similar though more specific, what Debussy called “reproduction of nature.” For example, a painting of an object is not the object, a graph of a wave is not a wave, music notation is not music, but these are understood as representations or analogues. In the same way, composed-out musical “waves” can represent or, more abstractly, be analogous to natural waves. Trezise states that “Debussy did not believe in absolute music, but he also mistrusted portraiture, so his works lie midway between, encouraging neither abstract analysis nor straightforward story telling.” Waves are universal and natural shapes and processes which have been scientifically studied and codified, and relate to what Debussy called “the laws of beauty inscribed on the movements of Nature.” It is therefore useful to apply the language of waves, including descriptions and terms from physics, oceanography, and other fields, in order to draw “correspondences” or analogies with the waves of Jeux. Debussy may not have used such technical terms, but we do know of his interest in the natural subjects of water, rain, and the sea and these terms have a great power in their ability to illuminate the connection between natural waves and musical waves.
A wave is generally defined as a disturbance that is propagated through a medium, as an increase and decrease, a rising and falling, back and forth, or oscillatory motion around a point of equilibrium, as an undulating curve, sinuous motion, etc., which is generally characterized by continuity, gradation, and flow. In addition, waves are described in terms of their structural features, or according to their type, for example different types of ocean waves. Different kinds of waves behave similarly no matter what the medium of propagation. By analogy, the medium of sound is air, while the medium of music is sound. In nature, physical energies shape material mediums such as air and water, while in music human creative energy shapes the medium of sound. The resulting shapes and forms, in this case composed-out waves, are in turn transmitted to the listener through the medium of air, in the form of sound waves. The comparison is figurative. Sound as a medium for music is a virtual medium, not physically manifest as is air or water, but projected through the music itself. In addition, natural mediums are composed of material particles such as atoms, molecules, water drops, etc., while in music, the particles consist of notes and motives. Of course, these musical particles can be further divided into basic dimensions of timbre, pitch, duration, etc.
The musical particles in Jeux form part of larger textures and statistical waves, though not all statistical waves form “statistical textures.” It is also possible to distinguish between statistical waves which form “statistical textures” or which are characterized by figuration, from those that are basically “arranged” forms of melodic waves. Furthermore, the texture moves between these different kinds of waves. All of these waves are perceived as larger forms, or gestalts, made up of parts and particles. Although the particles make up the waves, the waves behave like energy fields within which the particles are organized. This is according to the gestalt field concept, which states that a whole and its parts mutually determining each other’s characteristics. Eimert notes the textural waves of Jeux, pointing to the “quasi-statistical accumulation of sound” and “the build-up and alternating play of piled-up groups of sound.” Eimert adds that these statistical waves “result from the closest interweaving of all categories of form,” and that “the categories of form take part all with equal weight in the wave-movement.” Pasler refers to the play of balance, counterbalance, and equilibrium which is part of the wave process, stating that “this succession of impulses and tensions- which can be conjunct or disjunct- keeps the form fluid. Balance or equilibrium is constantly being recreated.” She quotes Grisey on “music as the organization of “tensions” moving in waves of contraction and expansion” and Stravinsky on “the drawing together and separation of poles of attraction.” Pasler also suggests that “In Jeux, “arabesque” might apply to the play between the various sections of music. Each section develops its own vector, its own force of contrasting shape and direction, which needs resolution or balance.” These statements sound like those by Al-Faruqi above, and further imply, as Eimert does, that in Jeux the wave-like principle of arabesque is extended to form textural arabesques. Eimert refers to these forms simply as waves, or as “waves of time,” “envelope-curves,” and “movement-curves.”
In contrast to the surface motivic-thematic and timbral level, statistical processes function at an underlying or more ‘subtle’ level, and can be thought of as independent from the surface level. Eimert refers to the underlying continuity produced by this ‘invisible’ process, stating that “ever-different elements appear within the same unitary wave-movement,” and that “It is as if charged by a current, with constant tension in its elastic brilliance, whose most wonderful quality is an unchanging vibration. The hidden impetus of the current creates a new organic coherence, that of flowing form.” The “hidden impetus” that Eimert mentions forms one of Debussy’s ‘subtle links’ at a statistical level of form and creates a new kind of continuity. Eimert mentions how the wave energy gives continuity to the mosaic structure, describing it as “an ornamental kinetic form which makes the cellular plan and line-by-line structure of the four-bar phrases so supple that they can freely follow the vibration of the form and can themselves become flexible form.” Howat similarly says that there is “a very definable system of block construction underlying the apparent fluidity of Debussy’s forms.”
The energy of these waves is a quantitative and statistical factor corresponding to the overall density, dynamics, or intensity of the waves. This overall intensity dimension is composed of statistical parameters including attack density, vertical density, loudness, and pitch height in the case of melodic waves. In Debussy’s waves the separate parameters thus form parametrical waves which move in a synchronized manner to create cycles of intensification followed by de-intensification. The intensification and de-intensification phases also occur in stages. Eimert says, “Dynamic evolutions and gradations in Jeux are indicated not only by crescendo and decrescendo, but are equally manifest in the changing intensity of the sound-mass resulting from addition or subtraction of groups of sound - a process which is developed to its highest degree of subtlety in the course of the work”. The wave phenomena thus largely result from a sophisticated interplay of intensification and de-intensification, or disturbing and restoring forces which are governed by the larger principle of balance and equilibrium. Pasler refers to this process as the “constant creation of a new balance.” Waves thus include features of symmetry and asymmetry, ying-yang, polarities, balance of opposites, growth and decay, antecedent-consequent, and thesis-antithesis. The following table summarizes common means through which the intensity dimension works (Figure 5).
Increase in loudness
vertical or horizontal density
decrease in loudness
decrease in horizontal or
Figure 2: Intensification/De-intensification Parameters in Wave Processes
The intensification and de-intensification processes give waves their characteristic overall shape or envelope, consisting of a rise and fall, or increase and decrease, the prototypical shape being that of a sine wave, though there are other shapes (triangle, square, sawtooth, etc.). In natural waves, the shape results from the displacement of particles within the medium. In a transverse wave, particles are displaced perpendicularly to the plane of motion. In music, this corresponds to the pitch contour of melodic waves, where the particles or notes move up and down in pitch, perpendicular to the plane of motion of time. In a longitudinal wave, particles are displaced parallel to the plane of motion, in a compression and rarefaction of particles. In music, this would correspond to the statistical aspect of the waves, wherein the average density of particles or notes, or intensity, increases and decreases over time, producing an overall shape. Surface waves (or complex waves) combine qualities of both transverse and longitudinal waves, the particles moving both perpendicular and parallel to the plane of motion. The textural waves of Jeux achieve this by combining melodic and statistical elements. Moreover, it is the “curve” or envelope of these waves that matters (Eimert’s“ envelope curves”). These “curves” are characterized by flow, continuity, cyclicity, and an ornamental and playful quality or “jeu,” a continuous flux which is connected at micro- and macro-levels through corresponding motivic-thematic and statistical wave processes. The ‘invisible’ energy of the statistical waves behaves as though independent of the motivic-thematic material or particles, as in Eimert’s description that “ever-different elements appear within the same unitary wave-movement.” Debussy seems to be aware of this, not only through his use of motivic recycling at the micro level, independence of particle and wave and of theme from structure, but in its most obvious form as refraction. Moreover, according to the field concept, the energy of the waves dominates the particles. The rise and fall or intensification and de-intensification phases of a wave also complement each other as two symmetrical parts of a larger unit and give waves coherence and closure. There is also the gestalt principle that states that the parts of a form, in this case the intensification and de-intensification phases of a wave, have different structural values, and that some parts of the whole are indispensable if wholeness is to be retained while others may be relatively unnecessary. This explains why a single rise or a single fall may by itself qualify as wave-like without actually completing a wave.
The intensification phase of a wave leads to the crest or high-point of the wave, which usually coincides with the climax and the beginning of the de-intensification phase. In the case of a plunging breaker the climax occurs at the low-point, forming an inverse climax, which reverses the functional polarity of the wave and the position of the climax in relation to the crest. This typically occurs when a melodic line descends from a crest, or de-intensifies, while other (statistical) musical parameters, e.g., tempo, vertical density, and loudness intensify, leading to a climax at a low-point of the wave. This also explains Debussy’s use of contrary motion of melodic lines in approach to the climax of a wave. A climax can also be delayed to some point after the crest yet still within the de-intensification. Climaxes can be of different intensities (amplitudes), or they can be attenuated (inhibited, damped, anti-climactic) or non-attenuated. Waves can also release energy gradually or suddenly, as in non-breaking and breaking waves. Breakers are classified as spilling breakers, which break up gradually and over a distance, plunging breakers, which curl over and break with a crash, the plunge point being the point at which the wave breaks, and surging breakers, which instead of spilling or plunging, surge up onto and break on the beach face. Another classification is collapsing breakers. In practice, these different types overlap.
Toch mentions the wind-up as a means for gathering momentum at the beginning of a wave, and initiating the build-up toward the crest, stating that, melodically, this character is “produced by a group of short, quick notes, anything from a turn (mordent) to an independent, characteristic motif” and that “the “winding up” motion is then followed by the “throw” or “jump,” which in turn is followed by a stepwise retrocession.” This applies to the “alpha” motive (Example 1). The idea of a wind-up is also accomplished through the statistical waves, where it is often present as the first stage of two or more in the intensification phase of a wave, but which is most noticeable in those instances of functional waves that set a new wave or section in motion, notably at the beginning of wave trains or wave groups (e.g. mm. 357-358, 395-396).
Example 1 “Alpha” motive from Jeux m. 49 in clarinet
The trough is the low-point between successive waves, and ordinarily a point of relative cadence and repose. Both the climax and the trough may consist of an instantaneous point, including an elision point where the end of one process coincides with the beginning of the next process, or they may be extended. The crests and troughs of waves are points at which there is a process reversal between intensification and de-intensification, or progression and recession, and vice versa, and hence a change of direction which is manifested as the cyclic back and forth wave motion. The trough is sometimes also used as an ambiguous transitional area in the shift or change of direction of the musical process. Moreover, both crests and troughs relate to expectations as to whether or not, or how a wave will climax or continue. Pasler says that “The anticipation aroused by this music is not one of what will recur - what melody or harmony - but a sense of what quality of sound and rhythm will provide counterbalance.”
The intensification and de-intensification phases form complementary parts of a wave. Schweitzer defines a motion-completion unit as made of two parts consisting of motion, as drive, tension, and directional sense, followed by completion, as goal, relaxation, and completion relative to its preceding material. This corresponds to Al-Faruqi’s concept of symmetry between modules, as “a pair of design entities which are in some aspect balanced opposites or reciprocals of each other.” The resulting wave shape and complementation creates a self-containment which is in contrast to traditional functional and linear processes, and contributes to the moment-form qualities of many of the waves and sections. Composed-out waves found in traditional music are usually part of a linear and functional process. A functional wave has a defining function in that an intensification phase leads to a new section, with the crest of the wave coinciding with the structural downbeat of the section, and serving to set the new section in motion (similar to the idea of a wind up). The new section often stands apart from the wave leading up to it and consists of sustained activity rather than a de-intensification. In contrast, Debussy’s non-functional waves are relatively self-defined and self-contained, with the beginning of a non-functional wave usually coinciding with the beginning of a section. Figure 6 illustrates the difference between these two types of waves.
Figure 6: Functional and non-functional waves
Most of the waves in Jeux are non-functional. There are a few functional waves whose function is similar to those of traditional music, that is, of leading to a new section, sometimes in the form of a transition or of a wind-up. Because the non-functional waves are relatively closed or self-contained and succeed each other in an additive fashion, the formal process is also relatively open-ended, like that of mosaic structure, wherein waves and modules can be added indefinitely. A further level of interplay between functional and non-functional waves is introduced through the use of the inverse climax function. This allows the momentum and release at the inverse climax point to set a new module or section in motion (e.g., mm. 469-472 leading to m. 473) or, alternatively, for the non-climactic crest of a wave to form the beginning of a new section (e.g., m. 224). Moreover, the predominance of non-functional waves places emphasis on statistical rather than syntactic factors as formative.
Standing waves, or stationary waves, are characterized by a relatively stable and symmetrical shape, an oscillatory motion around a fixed reference point without achieving a progressive movement or shape, of which there are a number of examples in Jeux. The non-progressive character of these waves also contributes a non-functional quality. Standing waves may also result from interference patterns between waves, e.g., as with independent parametrical waves. This occurs to an extent in Jeux through the inhibition of waves and inverse climaxes. A standing wave may also gradually change into a progressive way or vice versa. Progressive waves are characterized by an asymmetrical shape, having a longer intensification than a de-intensification, mounting size and/or increasing speed and momentum, forward motion, directionality, and may or may not break. Progressive waves are also often associated with wave trains leading to a cumulative climax. Deep water waves contrast to the typical breakers of the relatively shallow surf. The effect of these waves is created through a slower undulating motion and a thicker texture with greater mass (e.g., mm. 535-550).
A wave group is a series of waves of similar proportions. This term can also be used in the general sense of a series of waves that group to form a section within the mosaic structure, e.g. ABAB, or (ABC ABC), etc. The mosaic structure, with its reliance on well-defined and organized modules, plays a central part in this grouping of waves into sections. A wave train is a series of waves coming from the same direction. Waves in a wave train are part of the same process, line, or “train” of thought. In Jeux, this is most noticeable at the motivic-thematic level, where motivic particles are ‘propagated’ and ‘regenerated’ from wave to wave, forming a continuous motivic thread, and thus coming from the “same direction.” The continuity of wave trains is often enhanced through a gradual intensification of waves within the wave train which helps group the waves into a larger wave or section. Waves can also combine to form a choppy sea, a state of the sea characterized by short, rough waves tumbling with an abrupt and quick motion. A cross sea is a more confused and irregular state the of the sea due to groups of waves coming from different directions, the musical analogy of which would involve combining wave trains and phasing, possibly with other features such as refraction and breaking. A series of waves may also be related by systematic processes of phasing, damping, amplitude modulation or frequency modulation, or constructive and destructive wave interference, though this is not common in Jeux.
Textural or statistical waves can appear in simple or more “ornamented” or “embellished” forms. Whereas a basic wave may consist of a simple rise and fall, an embellished wave might rise in several stages, or “smaller partial waves” where “the successive climaxes add up, as it were, to one big wave.” A wave may crest, climax, or break in different ways, and may recede in several stages A wind-up can also function as a form of embellishment. Zuckerkandl says “the wave will display contours now soft and rounded, now sharp and jagged; and will beat softly and calmly or with ever-increasing impact; will heave, topple, break against resistances.” One form of the wave “embellishment” analogy which Debussy favors and uses numerous times in Jeux is the uprush (or swash), the rush of water up the foreshore following the break of a wave, usually of less intensity than the other parts of a wave, and which should be distinguished from the surge of a breaking wave. Debussy also uses the opposite effects of a backrush (or backwash), of water receding back into the sea, and of undertow, an underwater current caused by backwash, which produces an attenuation and retards the flow or reverses the process. Other naturally inspired forms of the ornamental play of waves include reflection, in the form of echo-like repetition or variation of a wave or fragments thereof, usually from the tail end, and refraction, the transfer of wave motion between two different mediums or between immiscible states of the same medium. Debussy creates the effect of refraction by transferring wave energy between contiguous and differing textural and/or motivic-thematic organizations. This transfer of wave energy often occurs between the intensification to the de-intensification phase of a wave, or, in more elaborate form, between the stages of the intensification phase of a wave. The modular aspect of the music is highlighted with this technique. However, the underlying continuity, the “hidden impetus,” or invisible “current” of the wave process is always present.
A sense of flow is always present in the “ornamental” play of waves in the form of varied ebbs, lulls, surges (upwellings), eddies, swirls, and the mostly laminar though sometimes turbulent flow of particles. Debussy’s typical harp glissandi produce the effect of a smooth and laminar flow of particles. A more delicate wave pattern occurs in the form of ripples (or capillary waves), which have a musical analogue in the form of trills, vibrato, and tremolo (e.g., mm. 690-693). In addition, there are the soft, “luminous,” and shimmering textures of Jeux, with their dynamic nuances and attention to detail, which are analogous to the ‘sparkle’ on the surface of the ocean as it reflects the sunlight, along with the spray, mist, water droplets, and foam, and which is projected through the coloristic orchestration, trills, tremolos, glissandi, “musical pointillism,” “sound textures,” and general ornamentation.
In summary, there are waves of many shapes, heights, lengths, speeds, rates, slope angles, etc. The play of waves in Jeux expresses the gamut from soft to loud, slow to fast, simple to elaborate, and delicate and playful to powerful, and can be “stormy and varied.” There are waves of many types, including breakers and non‑breakers, different length climax moments, from attenuated to short, to fuller and longer releases, or energetic breaks, progressive waves, standing waves, slow motion waves, reflected waves, refraction versus no refraction, wave phases in stages, uprush, undertow, deep water waves, wave groups, wave trains, and large‑scale wave processes. There are rhythmic‑motivic waves (e.g., mm. 627‑648) and lyrical‑thematic waves (e.g., mm. 264‑283). The “game of waves” in Jeux includes “waves of every color and mood.” These composed‑out waves can be extremely varied and flexible in their behavior, even capricious and dramatic. Moreover, the statistical waves are part of the general cyclic processes of Jeux related to the global form, or ‘vegetative’ organic regeneration, of the motivic‑thematic process. The wave process thus operates at different levels, from local to global (e.g., propagation is contiguous, regeneration is non-contiguous), including the micro-level of motivic particles, the layering of metric waves of the mosaic structure at the middle level, and the statistical waves, wave groups, and larger scale wave processes of the macro-level. Again, parallel cyclic elements are found in Middle‑Eastern, Indian, and Balinese music and cosmologies, in some examples of Western music, and are ultimately universal in nature. The idea of musical waves, or wave‑like shapes in music, both melodic and statistical, is not necessarily new, though not as unique and developed as those of Jeux. Debussy applies wave and cyclic ideas at many levels, which together with their non‑functional qualities, incorporation of discontinuous elements, modern harmonic materials, “textural” orchestration, etc., sets them apart as unique and innovative.
Analyses of Waves in Jeux
The dialectic of continuity and discontinuity in Jeux is expressed not only through the duality of mosaic structure and waves, but also through the relative continuity and discontinuity of the wave process itself. This occurs both within individual waves and series of waves, and is in addition to the process reversal of intensification followed by de-intensification which is characteristic of all wave motion. At the level of individual waves, the intensification phase builds energy which seeks release during the de-intensification phase. The inhibition of this process creates a play between two kinds of waves: waves that release, either gradually, or suddenly, as with a breakers, and waves that do not release, which are inhibited, suppressed, attenuated, deflected, anti-climactic, or cut-off. This process is likely inspired by the play between ocean waves that break and do not break.
At the next level of a series of waves, or between waves, there is a play between two kinds of wave processes: waves that continue, through release, momentum, periodicity, wave trains, development, and waves that do not continue, through interruption, inhibition, attenuation, or damping. The terms release and continue are similar and are used in a relative sense, these being a matter of context. The difference here is that one is applied within a wave while the other is applied between waves. Furthermore, the process that begins at the individual wave level and can have a cumulative effect at the level of a series of waves. This explains why a wave train can have a cumulative climax in the last wave of the train, or conversely, why the cumulative inhibition of wave energy release can lead to the structural accumulation of unreleased energy. Moreover, release and continuity correlate with tendency realization (or implication realization). The tendency inhibition, or process inhibition, of release and continuity directly correlates with the ‘hesitation’ between the dancers which occurs in the ballet. This play or ‘dialectic’ between waves that release and those that do not stems from Debussy’s mosaic techniques, and is parallel to the “the dialectic of the repeated and the non-repeated,” or of the expected and unexpected.
The piece will be analyzed in two parts and twelve sections. First part: mm. 1-46 (prelude), 47-141, 142-223, 224-308, 309-356, 357-434, 435-454 (middle section), and second part: mm. 455-514, 515-564, 565-626, 627-688, 689-709 (postlude). The play of continuity and discontinuity of the wave process starts from the beginning of Jeux, and becomes gradually more complex through the introduction of stronger contrasts, and a changing pace through tempo fluctuations, metric changes, faster waves, slower waves, etc. Moreover, the first part of the piece is characterized by the general use of process inhibition, e.g., slowing towards the crest, anti-climaxes, etc., while the second part is characterized by continuity and release. The analysis will focus on the statistical waves and wave process, with references to important related features of the motivic-thematic and mosaic structure. The table in the appendix summarizes the waves of Jeux.
First Part: mm. 1-434
The prelude (intro, prologue) music of mm. 1-8 conveys the stillness of the empty park with the static quality created through the held strings, while an element of mystery is provided by the slow chromatic tones of the horn and harp (“beta” motive). This static module forms a backdrop to the wave process which follows immediately with the first wave of the piece, spanning mm. 9-42. This wave consists of a gradual intensification towards a sharp cut-off just before the crest at m. 38, followed by an immediate return to the previous static level from mm. 39-42. This gives the wave an overall wedge, or sawtooth shape, which has no de-intensification. The cut-off interrupts the wave process and creates a discontinuity, suppressing the release of energy. The envelope of this wave is shown in Figure 7.
Figure 7: Wave Process Schematic and Mosaic Structure of mm. 9-38
The intensification phase occurs in three stages, each of which can be further subdivided. This is shown in the mosaic analysis also in Figure 7. The second modular level (i.e., the 8:2:6:4:5:5 pattern) is the first level at which motivic-thematic contrasts occur. Below that level, repetition and similarity of materials are used. The 3/8 meter combines into two-measure patterns to produce a 6/8 feel during the first two stages of the wave (mm. 1-28), after which the pattern breaks. This is easily seen at the cell level of the mosaic analysis in Figure 7. The breakup of the hypermetric groupings is enhanced through the hemiola patterns of mm. 34-35 and mm. 37-38. This metric-rhythmic process prepares the cut-off at the crest of the wave, giving it a more natural flow. The wave is followed by a return of the static opening music (mm. 43-46), elaborated with a undulating line in the strings. These three modules, the static opening (mm. 1-8), first wave (mm. 9-42), and the return of the static material (mm. 43-46), give the introductory section an ABA pattern.
The curtain opens at m. 47 with a second wave (mm. 47-83) which marks the beginning of the next section (mm. 47-141). This wave is a variation of the first wave. However, instead of being interrupted, this wave follows through to a climax, which coincides with the fall of the tennis ball onto the stage, and has the form of a plunging breaker (Figure 8).
Figure 8: Wave Process Schematic of mm. 47-83
The intensification phase, which also occurs in three stages, is created through increase in vertical density and the use of cross-rhythms (violas, cells, perc., mm. 63-66) which resolve to a regular rhythm at the crest (m. 67). There is a superimposition or layering of parts beginning at m. 47 which coalesce by m. 63, bringing together the different components and energy of the wave, and leading to the crest, from which all instruments descend together to the climactic plunge point (or break point) at m.70. The climax point is thus emphasized, in a sense everything from the beginning of the piece having lead to it, and is of structural importance. The break releases energy which creates a splash, expressed as a rapid change of timbres and a return to a high register at m. 70, which then falls to a release at m. 74, and further releases in two small waves (mm. 75-76, and 77-83). The second of these slows at the end (Retenu), helping to resolve and close the entire wave while, at the same time, the rising string and harp figures of mm. 80-83 provide a transition to the next module.
In a structural sense, the release of the second wave at m. 70 can be said to also complete the unresolved process or “gap” of the first wave of the piece (i.e., mm. 9-42). Even though the second wave forms the beginning of a new section, it is connected to the first wave. This can give the modules so far discussed an alternate grouping of ABAB’ (mm. 1-83), where B and B’ are the first and second waves. This alternate analysis is possible due to the blurring of structural boundaries previously explained. However, the sectional boundary chosen here is based on the choreography and similarities between the second and third sections. Moreover, it is clear by the end of the second wave at m. 84 that Debussy is playing with modules and interrupted processes. This play of continuity and discontinuity, and of interrupted or inhibited waves followed by continuous waves, is a pattern which shapes Jeux in its entirety.
At the micro-level, these first two waves share common elements. For example, the descending chromatic “beta” motive introduced in m.10 of the first wave returns in the second wave, along with the opening pedal tone, tambourine rhythm, and string pizzicato figures. Also, the motives in the first wave all exploit the chromatic “beta” motive which figures prominently in Jeux, and which first appears in the harp and horn in mm. 1-2. In the second wave, diatonic-modal material introduced in the clarinet at mm. 49-50, or “alpha” motive of Jeux, and is combined with the chromatic material. Moreover, these two waves have a “seeding” function wherein many motivic-thematic ideas and fragments which are later developed are presented. These two waves also largely form statistical textures, although toward the end of the second wave the separate textural components join to become more of a doubled melodic wave. The many small ornamental motives are clearly perceived as particles within the larger textures. In addition, the motivic-thematic material of the intensification phase (clarinet, oboe, and flute at mm. 63-66) carries through to the de-intensification phase (in mm. 67-69), giving the wave homogeneity. The descending chromatic figures in the woodwinds in mm. 63-66 produce the effect of the wave starting to spill, as with a spilling breaker, before the plunge from the crest to the inverse climax. The descending chromatic figures “resolve” to the more consonant diatonic (pentatonic) material at the climax.
The second wave is followed by a standing wave module (mm. 84-99). The narrow, undulating chromatic line of the violins and harp is synchronized with the metric wave and has a standing wave quality which is emphasized through the repeated accompaniment of woodwinds, brass, and pizzicato and tremolo strings. The chromatic line is an ornamental variation of the opening “beta” cell from mm. 1-6, with chromatic elements filling a perfect 4th frame (also mm. 57-60, 2nd violins). The major third and half-step fragment (m. 86, harp and 2nd violins) within the cell is also later reused. This standing wave module can be divided into an ABAC ABAC pattern, where each module is two measures long and made of two one-measure cells. After the initial AB pattern, AB is expected again, but instead, A is followed by C, producing ABAC, which is then repeated, producing ABAC ABAC.
This pattern of expectation and deviation repeats in the following wave (mm. 100-109). The intensification phase of this wave (mm. 100-105) uses a variation of the motivic-thematic material from the preceding standing waves, here presented heterophonically as theme accompanied by an ornamented version of itself (m. 100, clarinets). It forms a free duplication (replication) or spin-off of the previous standing wave pattern, in that the beginning or ‘head’ of the wave is the same while what follows, or the ‘tail,’ is in this case unexpectedly different. Mm. 100-105 give the impression that the standing waves will repeat as in mm. 84-99. However, mm. 104-105 turn out to be an intensification leading to contrasting motivic-thematic material in the de-intensification phase of mm.106-109. The de-intensification is signaled melodically by the descending lyrical fragment in the oboe and violins which is based on the third with half-step motive, in this case a minor third and a half-step. The secondary horn parts fill in with a chromatic line, a contrapuntal procedure common in Jeux. This is followed by a second wave (mm. 110-117) which immediately introduces further contrasting material in the intensification phase (mm. 110-113), which occurs in two stages of two measures each, the second stage being an upward transposition of the first. Inversions of the third with half-step motive are combined with the hemiola bass pattern first introduced in m. 23 (mm. 110 and 112), while the fast rising figures in the woodwinds and harp can be traced to the flute in m. 52. The de-intensification (mm. 114-117) consists of an upward transposition of the de-intensification of the previous wave (i.e., of mm. 106-109), which also creates a localized stratification (alternative continuity, or interlock through interpolation). The standing waves and the two progressive waves that follow are thus grouped together by means of shared motivic-thematic materials, that is, the first wave begins like the prior standing waves, and the second wave ends (de-intensifies) like the first wave. Debussy thus takes the wave process a step further by making these waves shorter and more complex than the first two waves of the piece in that the intensification and de-intensification phases have contrasting motivic-thematic and textural organizations. This creates a refraction of the wave energy which also leads to the above mentioned stratification. The contrasts introduce further elements of discontinuity while the refraction highlights the duality of contrasts within the “unitary” wave movement.
The final part of the section contains a return of motivic-thematic materials from the wave of mm. 47-83, with an unexpected recurrence of the “alpha” theme of mm. 49-50 (clarinet) combined with the descending chromatic motive of m. 57 (English horn) at mm. 118-119, 120, 123, 127. These motives are combined to form two essentially melodic waves which are followed by a “textural” closing wave. The first of these waves melodically rises (mm. 118-119), then falls (m. 120), and then rises again (mm. 121-122) to higher climax (m. 123) which is followed by a small echo in the flute and uprush in the trumpet (mm. 124-125). This is followed by the second wave (mm. 126-129), which consists of a repeat of the last four measures, or the tail end, of the first wave, thus forming an echo or a reflection of the first wave. A final closing wave grows out of the reflected wave. This wave is “textural” in that there is no main part. The wave is created and intensifies through the repetition and cross-rhythms of the layered parts (mm. 130-137), to a release of tension at the crest (m. 138) through the held brass and woodwinds. The syncopated pizzicato cellos and violins delay full release until m. 141. However, the cross-rhythm in the cello and double bass also gives the music a subtle “shove” toward the next wave (section). Note the smooth rhythmic transition from the previous wave, decelerating from sixteenths to quintuplets, to quadruplets, and the tied-over trumpet, all of which provide continuity and connection at a micro-level. The dominant progression from B7 to E (mm. 134-138) provides extra resolution and closure and prepares the next section. These waves have no contrasts or refractions between the intensification and de-intensification phases, and are thus simpler and motivico-thematically more homogenous than those in the second part of the B segment. The three large modules which shape this section form another ABA pattern: A (mm. 47-83), B (mm. 84-117), and A (mm. 118-141). The return of materials from the opening of the section provides the section with an element of closure while the closing textural wave provides a form of “liquidation” of the motivic-thematic process, enhancing closure and creating a temporary rest from the focus of the motivic-thematic process.
The next section (mm. 142-223) forms a replication (or free duplication) of the previous section (i.e., mm. 47-141), producing yet another ABA pattern. The following table lists these correspondences (Figure 9).
wave, modal “a” theme
standing wave-like, chromatic “b” theme, and more complex smaller waves
wave, modal “a” theme return, build-up, climaxes, reflected wave, closing wave
Figure 9: Wave Process Correspondences mm. 47-141 and mm. 142-223
The first part of this section (mm. 142-173) contains a wave train of two waves since the second wave forms a development of the first. These waves are light and rhythmic, with delicate orchestration. The “alpha” theme returns in a transformed modal-chromatic form in the bassoons and flutes at mm. 142-144. The intensification occurs in two stages (mm. 142-145, 146-149), propelled largely by the syncopated pizzicato strings. These string parts and the viola accompaniment figures carry through from the end of the last wave (mm. 138-141), providing continuity to the wave process, and are an expression of Debussy’s generative process. The crest occurs at m. 150, de-intensifying with the release consisting of a steady stream of descending pentatonic notes in the strings and flutes. The wave is given a final ornamental touch with an uprush in mm. 155-156, followed by a holding back of tempo (Retenu) closing the wave. The second wave (mm. 157-167) begins with a return to tempo and the “alpha” theme repeated in more intensified form in the English horn, bassoon, and oboe at mm. 157-159. The crest and de-intensification are also more intense than in the preceding wave, which ties these two waves together as a wave train. The crest and climax occurs at m. 164 and is repeated at m. 166. However, the de-intensification is attenuated at m. 167 with a slowing or yielding of tempo (Cédez), immediately leading into a transition from mm. 168-173, with the “alpha” theme in the violins, which has the effect of a backwash or undertow (mm. 168-171). The slowing of tempo and activity counteracts the momentum of the previous wave and pulls it back, creating a dampening of the wave process and a sense of reversal of process through the slower, lower-pitched texture, as if the waves had receded. The Rubato tempo emphasizes the flux of this transition. The process turns forward again in mm. 172-173 (Serrez), which lead in to the next segment. This transition also forms a functional intensification phase in that it leads to the structural downbeat of the next segment. The fragment of the “alpha” theme in the English horn, oboe, and clarinet at mm. 171-173 supports this interpretation, since the “alpha” theme (or its variants) usually signals the beginning of a wave.
This slight intensification leads to the standing wave-like module of mm. 174-177, with its narrow undulating chromatic line (a duplication or variation of mm. 84-87), which also complements and can be interpreted as a de-intensification to mm. 171-173. This is followed by a series of shorter and more complex waves. The first is a purely melodic wave (lyrical fragment) in the strings, rising in mm. 178-179 and falling in mm. 180-181. The next wave intensifies in two stages, beginning with a return of the “alpha” theme in the bass clarinet and flute in Dorian mode, at mm. 182-183, leading to a more intense repetition by the bassoon and oboe in the second stage at mm. 184-185, now in the original Locrian mode, and leading to the de-intensification and return of the standing wave idea at mm. 186-189. Note the heterophonic relationship between flute and 2nd violins in mm. 186-189. Two small waves grow out of this process, as if tagged onto the end of the wave. These cover mm. 190-191 and 192-193. The intensification and de-intensification phases of these are each one measure in length. These waves are followed by a wave and its reflection (mm. 194-203 and 204-209), which are a free duplication of the wave and its reflection of mm.118-129 (as noted in Figure 9). The intensification also introduces a variant of the “alpha” motive in the oboe at m. 196, which is repeated and intensifies, as does the entire texture, toward the climax at m. 202. The climax is inhibited such that it forms more of an anti-climax. The reflection of this wave follows (mm. 204-207), with the addition of two smaller echoes (mm. 208-209). The de-intensification of the reflection (mm. 206-207) forms a varied return and stratification with the de-intensification of the original wave (mm. 202-203).
The variant of the “alpha” theme introduced in m.196 returns again at m. 210, at the beginning of the last wave of the section, which has a very smooth and rounded overall shape. This is one of the best examples in Jeux of a wave with a “statistical texture,” which is relatively self-enclosed, forming a ‘quasi-moment.’ The intensification proceeds in mm. 212-215 as it did in mm. 204-205, beginning with the rising pizzicato figures of the double bass and cellos, varied and reinforced by strings and woodwinds in mm. 214-215, and reaching an energetic though fleeting climax at m. 216, identifying this wave as a breaker. Even though the wave reaches a break it is quickly attenuated in mm. 217-219. Mm. 220-222 begin with another presentation of the variant of the “alpha” theme, now in the oboe at m. 220, and forms a functional transition, lead-in, and gentle intensification toward the next section, the crest coinciding with the structural downbeat of the next section.
The next section (mm.224-308) can be divided into two parts (mm. 224-263, 264-308). The first part consists of a series of slow motion waves, produced through a change of meter and a slowing of rhythm and tempo. Even though the waves of the first part of this section move slowly, their wave shape is clearly shown by their melodic contour. These waves also have a more lyrical quality. The first part consists of a series of five modules that are repeated, producing an ABCDE ABCDE pattern. Module A (mm. 224-225) forms the de-intensification of mm. 220-222. The melody, a variant of the chromatic “beta” motive, descends slowly from the crest at m. 224, as part of a parallel chord progression. The change from 3/8 to 3/4 also slows the momentum. The next wave begins at m. 226, where the 3/8 returns with reduced tempo. This wave is made of two contrasting modules, B (mm. 226-229) and C (mm. 230-233), corresponding to the intensification and de-intensification phases. The violin melody in mm. 226-229 is rhythmically derived, in the form of diminution of the cross rhythms from the variant of the “alpha” theme at m. 142, here also presented with chromatic modality. The descending melodic line of the de-intensification (mm. 230-233, strings) consists of an open repetition of a lyrical form of the major second cell of mm. 208-209, 202-203, 33, and 16. Module D (mm. 234-237) consists of small once-repeated (or reflected) uprush from the preceding wave. The second uprush leads into and overlaps with the beginning of the next wave at m. 237, corresponding to module E. The motivic material of the intensification of this wave is derived from that of the uprush. The intensification builds up significantly but is stalled before the climax through the restrained tempo (Retenu) and thinning of texture.
The momentum is further slowed by a return to 3/4 meter at m. 245, which forms the crest of the wave and the beginning of the second ABCDE pattern. The pattern repeats nearly the same way as the original, slightly reinforced, and with the notable difference that the ending E module is more abbreviated and not as inhibited in its flow as the original occurrence. Mm. 262-263 form a transition and functional lead-in to the next module and are still a part (albeit attenuated) of the intensification begun in m. 258 and stalled in mm. 261 and 262 (Très retenu). What is interesting about this connection is that the intensification leads to the beginning of the intensification of a new and more active wave (marked au Mouvt). This helps increase the temporal flow after the slower waves, as if the wave process had shifted gears. The high point of the slower wave forms the low point of the more active wave. Nevertheless, due to the overlapping of the two waves, the wave beginning at m. 264 also releases energy from the intensification of mm. 258-263.
The next part of this section (mm. 264-308) can be further subdivided into two sub-parts, ABAB and CBCD. The first sub-part (mm. 264-283) forms a wave train of two waves, the second of these waves forming a repetition of the first wave at a higher pitch level and with a stronger climax than the first wave (marked Passionnément). Each of these waves has a contrasting intensification and de-intensification phase (with each phase occurring in two stages) forming an AB pattern for each wave and the ABAB pattern between both waves. The de-intensification of the second wave is extended by six measures (mm. 278-283), slowing the wave further through a lessening of intensity and a slowing of tempo (Retenu, then Plus Retenu). The motivic materials consist of a variation and further development of that of the first sub-part. For example, the fast figuration of the woodwinds and harp at mm. 234-236 and 256 are brought to the fore at mm. 264 and 272 in the harp and descending chromatic flute part. It is also interesting that related variants of this chromatic idea up to now are presented in successively quicker rhythmic forms (mm. 1, 84, (100), 174 and 186, 264 and 272). Two different forms of the “alpha” theme are presented, one lyrical and the other motivic. The lyrical theme of the de-intensification phases of these waves (mm. 268-271, 276-277) combines transposed fragments of the “alpha” motive with the rhythm of the violin part of mm. 237-238 and 258-259, while the arabesque-like variant of the “alpha” motive which last appeared in the flute at m. 220 is re-introduced in the oboe at m. 278, now in a cadential context.
The second sub-part, the CBCD pattern, spans mm. 284-308, and features sharper contrasts and an overlapping block-like stratification (alternative continuity, process interpolation) which serves to continue and end the section while also preparing the next section. Module C (mm. 284-289) introduces a change of meter to 2/4, new and more rhythmic material, more dissonance through chromatic material and the harmonic use of the major second cell, and appears to mark the beginning of a new section. Despite the contrasts, the descending flute line at m. 287 has a connection with the preceding flute line at m. 274, both chromatically filling a tritone, while the horn part of mm. 284-286 prepares the oboe and English horn theme of m. 309. The de-intensification, which corresponds to module B (mm. 290-293), consists of an unexpected recurrence of the lyrical material of the last B module (mm. 276-279), with which it interlocks and forms an alternative continuity. Nevertheless, because of the ‘unitary’ motion of the statistical wave process and the complementation of wave phases, the two contrasting modules group to form a unit (CB). The material of the C module returns again at mm. 294-300, intensifying toward the crest at m. 301.There is a slight release at the climax at this point, but it is interrupted through rhythmic discontinuity and a sudden reduction of textural density. Mm. 305-308 provide further de-intensification, a return to the rhythmic continuity, and form a transition to the next section. The use of stratification in the section just analyzed is part of the process of increasing complexity of modular contrasts and waves in Jeux.
The next section (mm. 309-356) can be divided into two parts. The first part consists of a single large wave (mm. 309-345). The intensification continues to build on the rhythmic ideas of mm. 284 and 294, now using a regular rhythm and meter (the same returning 2/4), with a gradual quickening of tempo (Animez peu à peu at m. 315). Debussy takes the process of contrasting modules and refraction within waves a step further by using contrasts between the stages of intensification of the wave, in this case four stages, ABCA’ (Figure 10). The chromatic theme in the A modules (mm. 309, trumpet and woodwinds, and 325) was prepared in mm. 284 and 294 and consists of a further variation of the theme at m. 100, which goes back to m. 84 and the first six measures of the piece. The rhythm of the modal motivic idea introduced in the bassoons in module B (mm. 315-320) is combined with the chromatic theme in the A’ module (mm. 325-330). The interlocking rhythms begun in module B, which are similar to that of gamelan music, become more regular and textural in module C (mm. 321-324) and continue varied through module A’.
Figure 10: Modules and Refraction in Wave Process mm. 309-345
The wave crests at the beginning of module D (mm. 331-335), coinciding with a change of texture. The wave energy is slowed and delayed through a shift in meter back to 3/8, which breaks up the rhythmic momentum, and leads to a lesser release of energy toward the end of the module at m. 335, which can be considered the actual climax, that is, the climax occurs after the crest because it is delayed. The second stage of the de-intensification, module D’, (mm. 336-345) also serves as a transition to the next part of the section. The strings continue the cross-rhythms of the preceding module in simplified form. The modal clarinet melody (mm. 335-343), a derivative of the “alpha” motive, has a standing wave shape, circling symmetrically above and below the axis pitch of F (mm. 340-343). This melodic shape can be traced back to the pentatonic form in mm. 150-152 (celesta and harps).
The second part of the section (mm. 346-356) consists of a succession of two standing wave modules, the second one being more intense than the first, a pattern found in several earlier wave trains of two waves each. The pizzicato string pattern brings the rhythm back to the 3/8 waltz pattern. The solo melodies (flute at mm. 346 and 354, horns at m. 352) continue the standing wave shapes around the same pitch axis, while the rising hemiola string notes (mm. 348-351 and 354-356) produce an intensification towards the end of each of these modules, adding forward drive. This gives the modules qualities of both standing waves and of progressive waves. The intensification at the end of the second module serves to functionally lead to the structural downbeat of the next section, transferring momentum and giving the beginning of the next wave a “push” similar to a wind-up. In this case, the beginning of the next section beginning at (mm. 357-359) has a dual function (and thus a dual interpretation) serving both as de-intensification for some of the energy from the last module, and as the beginning of the new section.
The next section (mm. 357-434) can also be divided into two parts (mm. 357-395 and 396-434). These two parts exhibit the pattern of inhibited waves followed by more continuous waves. The first part begins with a burst of energy and directionality which is soon subdued, while the second part consists of a return of the original impetus in a more uninterrupted manner, leading to the structural climax at m. 429. The first part (mm. 357-395) consists of five waves. The first wave (Joyeux) spans mm. 357-366. As mentioned above, the beginning of this wave releases energy carried over from the previous section. This occurs in mm. 357-361, beginning with the sforzando string and harp chord, and leading into the modal woodwind figuration and horn theme of m. 358, which is based on the Locrian mode as is the “alpha” theme. This energy is carried over into a quick rise to the crest in m. 363, followed by an equally quick a descent in mm. 363-366 which subdues the wave momentum through the 3:2 hemiola pattern of the violins and woodwinds.
The next wave (mm. 367-378) begins with an ambiguous area in 367-370 which extends the trough of the preceding wave, and produces a lull through a re-introduction of the earlier circular, standing wave-like melodic pattern in the English horn and oboe. Mm. 371-376 continue the intensification toward the crest at m. 377, with the modal melody pivoting into a rising chromatic line. However, the wave motion is inhibited once more just before the crest through a holding back of tempo (Cédez) at m. 376. The de-intensification (mm. 377-378), gently releases the little energy left in the wave, with the motion being slowed further by halving the meter from 3/8 to 3/4 at the beginning of the de-intensification. The two phases of the wave also consist of contrasting motivic material. However, the orchestration is similar, which lessens the contrast between them. The wave thus has an AB modular pattern (i.e., mm. 367-376, 377-378).
Two more waves continue at the same slow tempo, forming a series of slow motion waves. The first wave introduces further new material within the intensification (mm. 379-380), although again, the orchestration is similar to that of the preceding module and there is a continuity of voice-leading between the modules. However, the de-intensification (mm. 381-382) consists of a return of the de-intensification of the previous wave (mm. 377-378), transposed up. The second of these waves consists of an intensification only, and brings back the material from the intensification of the first wave, also transposed up and extended. A stratification of materials is produced in mm. 377-385, forming a BCBC pattern, though it is subtle due to the homogenous orchestration and reduced tempo of the waves. The energy of the last intensification-only wave is restrained through a holding back of tempo (Retenu, Serrez un peu, Plus retenu), and is carried over to the beginning of the next wave by the woodwind and horn figures at the end of m. 386. Even though restrained, the wave is functional in that it leads toward the beginning of the next wave. The next wave (mm. 387-395) ends the first part of this section. It also consists of an intensification only, which leads into the second part of the section, and is therefore also functional. Having two functional waves in a row gives added emphasis to the structural downbeat at m. 396. At the motivic-thematic level, this section has a consistency of material, the material of mm. 357, 367, and 387 being derived from the melodic standing waves of m. 335, which lead back to the modal “alpha” theme. The rhythmic articulation of the bassoon figure in m. 387 is derived from the bassoon figures in mm. 315 and 325.
The second part of this section (mm. 396-434) consists of a wave train, with each successive wave increasing in intensity, and leading to the first structural climax of the piece at m. 429. As noted above, the process begun and subdued in the first part of the section is resumed and given more energy the second time around. The first wave (mm. 396-402) thus consists of a reinforced return of the first wave of the section (mm. 357-366) and begins the same way as the section, prepared by a functional wave, with an effect similar to a wind-up. The modal figuration of the wind-up wave (i.e., mm. 387-395) is carried over into the first wave so this wave can be considered part of the wave train, though m. 396 marks the structural downbeat of the section. The hemiola string pattern in mm. 401-402 slightly stalls the rhythmic flow. The next wave in the wave train (mm. 403-420) returns to the rhythmic regularity and a gradual increase of tempo (En animant progressivement). The intensification builds steadily from mm. 403-410, through an intensified repetition of the modal-chromatic theme in the oboe and English horn. The process continues in mm. 411-412, which begin another stage of intensification. However, mm. 413-414 produce a slight sense of reversal of process and restrained release or de-intensification through the downward turn of the melody, similar to the effect of a spilling breaker. Mm. 415-416 repeat mm. 411-412 and seem to put the intensification back on track, but the reversal of process and de-intensification are repeated and extended in mm. 417-420.
retrospect, mm. 411-412 and 415-416 form repeated crests (similar to the later
repeated climaxes) of the same wave with the second crest followed by the final
de-intensification. These crests are climactic but are dampened and do not
break or fully release. The accumulated and unreleased energy carries forward
to the last wave of the wave train (mm. 421-434). The intensification of this
wave occurs in two connected stages (mm. 421-424, 425-428). There is a return
of the modal material of the prior waves in the section. The cross-rhythmic and
“textural” string figures at m. 425 form a return of those at m. 130. The
intensification reaches the crest and climax at m. 429. However, the climax is
muffled by a sudden reduction of instrumentation. The melodic component carries
the wave while the statistical component drops out. The climactic high B-flat
descends through a modal-chromatic scale in the violins, alone forming the
de-intensification phase of the wave (mm. 429-434). There is also a reversal of
the usual chromatic to diatonic “resolution” of pitch materials in moving from
the intensification to the de-intensification phase of a wave. However, this
“anti-climax” has a potent structural function, as part of the drama of the
ballet and of Debussy’s play with the continuity and discontinuity of the wave
Motivically this last wave connects back to the first wave of the wave train
and the section and to the diatonic standing wave melodies of mm. 335-356.
These examples illustrate Debussy’s free recycling and interconnection of
motivic particles. The envelope of the wave train is shown in Figure 11,
beginning with the wind-up wave of mm. 387-395, which gives the wave train four
Figure 11: Wave Process Schematic mm. 387-434
Middle section: mm. 435-454
Mm. 435-454 form the middle, relatively static and inactive section which separates the two large parts of the piece. The meter shifts to 3/4 and the tempo is slowed to a moderate pace (Trés modéré). A series of four gradually intensifying slow motion waves is followed by a transition to the next section. These start at mm. 435, 439, 441, and 445, with the transition starting at m. 450. Even though these waves are slow, their rising and falling melodic shapes are clear. Each wave begins with a transformation of the “alpha” motive, while the de-intensifications introduce the descending clarinet figures of m. 440 and a return of the descending violin figures from the climax of m. 429 at mm. 443 and 449. The strings and clarinet of mm. 453-454 form a small connective melodic wave leading to m. 455, and releasing with the harp glissando.
Second Part: mm. 455-709
Measure 455 marks the beginning of the second large part of Jeux and the beginning of the next section (mm. 455-514). This section forms a reprise, consisting of an abbreviated return and free duplication (or replication) of basic ideas presented at the beginning of the piece. A wave train of two waves begins the section (mm. 455-464, 465-472). The “alpha” theme returns in its original form at mm. 457 and 467 in the English horn, combined with the cross-rhythms of m. 142. These waves combine melodic elements from the wave at m. 47 with the transparent orchestration of the wave at m. 142. The de-intensification of the first wave (mm. 461-464) descends similarly to that of the plunging breaker of mm. 47-83, releasing with a harp glissando as in m. 455. The release coincides with the beginning of the next wave, to which it transfers momentum.
The second wave is a repetition of the first, with a reinforced de-intensification which generates greater drive and momentum toward the beginning of the next wave, to which the energy is transferred, again, similar to a plunging or surging breaker. However, instead of breaking or releasing, the transferred energy combines with the release of the next wave. The descending line in the flutes of mm. 469-472, heterophonic variant in the oboe, and contrary (ascending) motion in the 2nd violins and violas, also reinforce the convergence toward m. 473. After the establishment of the reprise and return of the “alpha” theme in the first two waves, this point at m. 473 marks the beginning of the second wave train, and corresponds to a change in the large-scale wave process. Howat calls this the turning-point at which there is a reversal of the inhibition of continuity of the first part of the piece. Debussy verifies this, stating that it marks “un nouvel état de choses” (“a new state of things”). This point corresponds to the plunge point (break point) of the first plunging breaker of the piece at m. 70, with which it forms what is comparable to an inverse harmonic point (or node). In other words, m. 70 forms a point after which the wave process becomes inhibited and discontinuous, while m. 473 forms the complementary point after which the wave process becomes continuous. The following wave train bears out this change through a series of two short yet powerful waves (mm. 473-478, 479-488), the second of which is transposed upward and more powerful than the first. These waves begin with a permuted fragment of the “alpha” theme in the bass and cellos. The two-measure intensifications of these waves quickly surge to climaxes. The climax of the first wave at m. 475 is reinforced with the crash cymbal, a distinguishing feature of climactic and breaking waves from this point on in the piece, along with reflections and repeats of climaxes. The climactic figures, the descending chromatic motives of m. 475, are reflected in m. 476, followed by an uprush in mm. 477-478 made from a rhythmically augmented fragment of the “alpha” theme in the horns. The second wave proceeds similarly, except that it has a longer de-intensification phase consisting of three stages, and reaches greater repose.
The chromatic motive is ‘propagated’ to the beginning of the next and final wave of this section which spans mm. 489-510. From the relative repose of m. 489, the chromatic theme in the English horn is intensified in four stages, creating the effect of waves ‘stirring up,’ reaching the crest climax at m. 503, which consists of a return of the lyrical melodic ideas of mm. 106, 178, 230, and 268, and which is formed by the descending sequence of the three-note fragment from the beginning of the “alpha” theme. This is also accompanied by a “resolution” from chromatic to modal materials. An uprush follows in mm. 507-510, now with the chromatic motive. Moreover, the waves within this section form a free duplication and summary of the ideas of the beginning of the piece, with the modal “alpha” theme being followed by the chromatic theme, shorter waves, and lyrical melodic waves. Mm. 511-514 form another surging intensification and functional lead-in to the next section.
The next section (mm. 515-564) can be divided into two parts, consisting of a wave train of five short and powerful waves plus a transition, followed by the pas de trois theme and a closing wave. The surging intensification of mm. 511-514 leads to the climax of the first wave of the wave train at the beginning and structural downbeat of the section at m. 515 (marked Mouvt initial). The climactic figure in the violins, violas, and trumpets uses a retrograde form of the opening fragment of the “alpha” theme. This wave has characteristics of a surging breaker with the syncopation of the climax at the end of m. 515 having the same rhythm as the wave break at m. 70. As is characteristic of the powerful breakers of the second part of the piece, the climax is reinforced with crash cymbals.
The next wave (mm. 517-520) consists of a repeat of the first wave, with a shorter intensification. The close succession of the climaxes of these first two waves produces the effect of overlapping waves and climaxes, as when a breaking wave is overtaken by a another wave which also breaks, which in ocean waves is due to the shoreline halting the wave movement. The intensification of the third wave (mm. 521-522) is a repetition of that of the second (i.e., mm. 517-518), leading to the climax at m. 523. This climax is slightly less intense than the previous two, with no crash cymbal reinforcement, giving the impression that the process is diminishing at that point. However, this lasts for only two beats before it is suspended and overtaken by a more powerful wave.
The intensification of this next wave consists of the fast rising violin figures at m. 523 beat 3. These reach the breaking climax at m. 524, which is reinforced with crash cymbals, and which is immediately followed by a softer echo or reflection of the climax at m. 525. The final wave in the wave train (mm. 525 beat 3-526) is a repetition of the preceding wave, except that the echo of the climax is softer than the first previous echo at m. 525. This occurs at mm. 527-530, which coincides with the beginning of the transition, and forms a further de-intensification of the wave. The next four measures (mm. 531-534) form the actual transition, and produce the effect of an undertow created through, in addition to the softening of dynamics, the retarding of tempo (En retenant), and the cross-rhythms of the harp figures. This wave train is powerful in the brevity and intensity of its component waves and their overlapping and repeated climaxes. Debussy also plays with the timing of the climaxes. The climaxes of the first two waves, which are reinforced by crash cymbals, occur on off-beats (third beats of mm. 515 and 519), while those of the last two waves occur on downbeats (mm. 524 and 526), a play of open followed by closed rhythm which makes them more emphatic and culminating, and which is part of the process of climax and release in the second part of Jeux. The first part of this section can be alternately analyzed as forming part of the previous section (mm. 473-534).
The undertow at the end of the first part (i.e., mm. 531-534) transitions the music to the beginning of the second part of the section which consists of the pas de trois (mm. 535-550). This segment has an ‘oceanic’ effect of standing waves in deep water and the undertow leading up to it has the effect of pulling back to deeper water. The standing wave quality is achieved through the pas de trois theme in the bass clarinet, bassoon, and horn, which circles around the A-flat axis (D Locrian, modal-chromatic), and the periodicity of the metric wave and accompaniment, while the effect of deep water is achieved through the full, thick, though relatively soft and “luminous” orchestral texture. The pas de trois theme is stated twice, the second time of which is left open, like an antecedent without a consequent, a process which will come to the fore in the next section. This is also the first appearance in the piece of a longer, more sustained theme.
Mm. 551-564 contain the closing wave of this section. The intensification (mm. 551-560) is achieved through the repetition in the oboe, English horn, clarinet, and violins, of melodic fragments derived from the pas de trois theme. The de-intensification (mm. 561-564) is expressed primarily through the held notes of the horn theme and the harmonic cadence in m. 563. Moreover, there is a rhythmic and metric continuity underlying the second part of this section, in this case created through the repeated accompaniment patterns, which continues through to the next section and to the end of the piece.
The first part of this section (mm. 565-604) consists of a series of variations on the ‘waltz theme.’ These thematic variations or thematic waves form a wave train of four waves (mm. 565-578, 579-584, 585-592, 593-600), and which are rhythmically and melodically more flowing than the pas de trois. The ‘waltz theme’ forms a rhythmically augmented variation on the version of the “alpha” theme presented in the previous waltz section (mm. 340-343, clarinet), transformed here from arabesque into lyrical theme with a melodic wave shape, and given varied accompaniment and orchestration from wave to wave. This also illustrates the progressive elaboration of the “alpha” theme from more rhythmic and arabesque-like to lyrical and thematic. There is also a rhythmic play in the mosaic structure whereby the usually regularly repeated patterns are shortened towards the end of the wave. This occurs in the first two waves, creating a 4+4+4+2 = 14 measure pattern beginning at m. 565, shortened to a 4+2 = 6 pattern beginning at m. 579, creating a sense of truncation and a subtle element of discontinuity within the otherwise continuous process. In addition, the first three waves are characterized by an intensification-only (wedge shape) which corresponds to the ‘antecedent’ portion of both waves and thematic statements, referred to by Eimert as a “chain of varied ‘antecedents’.”
These melodic and statistical features also keep the formal process open. The intensification-only waves have a functionality and forward drive which is partially transferred from wave to wave even though there is a cut-off before each climax, while the rest of the energy forms part of a structural accumulation in preparation for the final climax of the piece in the next section. Intensification is achieved at the end of the first wave (mm. 575-578) through the reinforcement of the horn and English horn theme in the violas and cellos and the repeated harp glissando. The second wave (mm. 579-584) provides further intensification through an upward transposition and a reinforced presentation of the theme in the cellos, violas, and violins. Because of this progressive intensification, the first two waves can be heard as forming a single larger intensification-only wave (i.e., mm. 565-587). The third wave (mm. 585-592) displays an independence between melodic and statistical aspects. While the horn theme remains relatively fixed, the textural density builds toward the cut-off at the end of m. 592, similar to the combination of melodic standing waves with a textural intensification in mm. 346-356. The “alpha” theme is also presented in its original motivic form as a secondary part in the oboes at m. 587, and in counterpoint with the slower waltz theme in the horns. The fourth wave (mm. 593-600) is not progressive but consists of a repeated standing wave motion in the waltz theme. The theme undulates around the axis pitch of C in a repeated four measure pattern. The standing wave pattern plus the return to the original key as the beginning of the section create a return and close to the thematic variations. Mm. 601-604 has a dual function, serving as a transition or bridge, in which the theme is reduced to a repeated two-measure pattern, connecting to the restatement of the “alpha” theme in the next wave at the beginning of the second part of the section, for which m. 601 can also be interpreted as the beginning. This segment can also be interpreted as the beginning of the first wave of the next part of the section.
The second part of the section (mm. 605-626) consists of two wave trains of two waves each (mm. 605-610, 611-626). The first wave is complete in itself. However, as stated above, in context it connects with mm. 601-604, which can be heard as a first stage in the intensification of the wave. The intensification of mm. 605-606 features the return of the “alpha” theme in its original form (Mouvement initial), which occurs in a higher register, and is accompanied by a rhythmic shift from a predominance of eight notes back to sixteenth notes of the increasingly scherzando waves. The climax at m. 607 is reached quickly and uninterruptedly and is followed by a quick descent, with a brief uprush in the trumpet and harp at m. 608. The second wave immediately overtakes and overlaps with it (mm. 608-610), reaching a quicker and higher climax (m. 609) than the first wave, and with a brief uprush in m. 610, these two waves forming a quick succession of climaxes. The next wave train (mm. 611-626) continues to build on the flow (En animant progressivement) and rounds off the section with two waves (mm. 611-618, 619-626), the second wave consisting of a higher transposition of the first. The intensification phases of these waves (mm. 611-614, 619-622) occur in two stages of two measures each, with the second stage presenting the “alpha” theme. This reinforces the interpretation of the transition of mm. 601-604 as the first stage of a wave. The irst stage of each intensification begins with a melodic wave (cellos and basses) whose crest coincides with entry of the “alpha” theme at the beginning of the second stage (mm. 613, 621), after which the lines move in contrary motion and converge toward the main (statistical) climax. In essence these form two overlapping melodic waves within the larger statistical wave, the intensification of the first melodic wave functioning as a wind-up to the second wave.
The climaxes of these waves are sustained through repetition, forming a climax area (each four measures long) rather than a climax point, and release the energy of the waves without further de-intensification. Taken together, the two wave trains of the second part work to further increase the “joyful” quality of this section, and grow out of and culminate the waltz of the first part of the section. They also prepare the next section through the re-introduction of the “alpha” theme (beginning in m. 587) and the more rhythmic-motivic scherzando waves.
This section (mm. 627-688) forms the ‘grand finale’ of Jeux, and consists of two continuous and connected wave trains (mm. 627-656, 657-688). Together these form an even larger wave train, with the second wave train being more intense than the first. The overall increase builds on the marking En animant progressivement of m. 611, leading to the climaxes of the first wave train, marked Violent at mm. 645 and 653, and continuing relentlessly, through the marking Toujours animé et expressif at mm. 665, to the final climax and release at m. 677, marked Très modéré. These waves are very rhythmic and, as mentioned, involve the return of the motivic scherzando waves. These wave trains consist of a chain of motivic variations which bring together, integrate, and synthesize previous rhythmic and melodic ideas. Debussy’s technique is here expressed more directly, including the use of motivic combination, compression, and intensification. These wave trains are more complex than the ones leading to the climax of the first part of the piece in that they also have stages within the waves of the wave train. Moreover, the entire section is climactic, consisting of repeated and sustained climaxes (versus climax points), exhausting the accumulated structural energy, with the final climax signaling final release. The two consecutive waves trains are represented in Figure12. The overall outline indicates the wave envelope, with the vertical divisions indicating the wave stages and sub-stages, and enclosing the dynamic shapes of these and the letters which indicate motivic-thematic relations between modules.
Figure 12: Wave Process Schematic mm. 627-688
The first wave train (mm. 627-656) contains two waves. The first wave (mm. 627-648) has an intensification phase that builds in three stages. The first stage presents a variation of the chromatic motive from the beginning of the piece (m. 63, oboe, with the minor third intervals expanded to major thirds here), combined with a shortening of the rhythmic and articulation ideas from the transformation of the “alpha” motive in the bassoons at m. 387. The second and third stages, labeled B in Figure 12, introduce a different variation of the chromatic motivic-thematic ideas, now combined with diminished seventh (minor third or octatonic) fragments, and characterized by a hemiola rhythmic pattern. The background sixteenth-note pattern in the oboes in m. 635 and then in the violins at m. 639 is rhythmically related to the opening figures of the section. The first viola part at m. 639 also combines rhythmic elements derived from the “alpha” theme with chromatic elements of the “beta” theme. The continuous and regular repetition of four-measure modules is shortened by two measures in the third phase of the wave, producing a six-measure module (mm. 639-644). This pushes forward to the repeated climax of mm. 645 and 647, bringing the background rhythmic-motivic pattern to the foreground in the violins, emphasizing the axis pitches of A-flat, F, and G-flat of the violin motive of m. 635, and continuing the 3:2 hemiola pattern (marked in lower part of Figure 12). The second wave (mm. 649-656) has a shorter intensification (mm. 649-652) which returns to the regular metric accentuation and reintroduces a variation of the modal “alpha” theme at m. 649, while the repeated climax (mm. 653 and 655) is a repeat of that of the previous wave. The climaxes of these two waves are strong (marked Violent), being repeated as they are and reinforced with crash cymbals, a clear suggestion of the ‘crash’ of a breaking wave. Even though these climaxes are powerful, they do not present full resolution, largely due to the chromatic materials and reiterated hemiola patterns.
The second wave train (mm. 657-688) continues the intensification process of the first wave train. It also consists of two waves which join into a single larger wave, each of these being divided into several stages and sub-stages. The first wave (mm. 657-664) forms an extended version of the intensification phase of the previous wave (mm. 649-652). At m. 657, two versions of the “alpha” motive are presented simultaneously, in original form in the horns (and English horn at m. 659), and in varied form in the clarinets and bassoons. The second wave (mm. 665-688) has an intensification consisting of three stages each of which is sub-divided into two sub-stages. The first stage (mm. 665-668) resumes motivic variation of the chromatic violin motive of m. 635 of the first wave train, thus connecting the two wave trains. The motive recurs in the violas, but without the hemiola rhythm, which makes the waves more continuous. This is repeated, producing the two two-measure sub-stages. The second stage (mm. 669-672) upward transposes, rhythmically compresses and adds to the violin motive of the first stage, which is also repeated to produce two sub-stages. The figures in the violas and cellos at mm. 665 and 669 can be traced back to the violins at m. 87, though now in a very different context. Each of these stages and sub-stages has an intensification-only wedge shape emphasized by the cymbal rolls at the end, in the second measure, of each sub-stage. These wedge-shaped component waves can also be interpreted as successive freeze-frames of waves, depicting waves in successive stages of development, like a successive unfolding combined with replays, a technique which is also used in La Mer (see the < < < < notation in Figure 12). The third stage (mm. 673-676) slightly varies the motivic-thematic material, which becomes more fragmented and intensified through increased event density, and with a variation of the viola and cello motive of the first two stages now in the flutes and oboes. This also occurs in repeated two-measure sub-stages each emphasized at their ends with harp chords. A 2:3 hemiola pattern is also used in all parts as a means of producing the final push towards the climax at m. 677. The block-like alternation of materials gives the impression of alternating wave trains coming from different “directions,” as in the analogy of a cross sea.
The climax consists of a rhythmically augmented version of the beginning of the modal “alpha” theme presented in the woodwinds, brass, and violins, which is drawn out and made more expansive through the marking Très modéré. This rhythmically augmented form of the “alpha” theme can be said to be prepared by the previous version of the “alpha” theme presented at m. 477, while the placement of the theme at the crest of the wave rather than its usual position at the beginning of a wave is prefigured in the overlapping melodic waves at mm. 613 and 621. However, taking the two wave trains into consideration, the sustained climaxes of the first wave train at mm. 645 and 653 are stronger than the culminating release of m. 677. Howat notes this, saying that this makes “the last part of the approach to the climax even more intense than the climax itself at figure 78.” The sense of resolution is also increased by the change from chromatic to diatonic-modal materials. The second stage of the de-intensification consists of lingering repetitions of the augmented “alpha” motive in the flute at m. 680, oboe at m. 682, and flute again at m. 683, counterpointed with brief reminders of the of the viola and later flute and oboe motives of the intensification phase, now in a diatonic setting. The final stage (mm. 684-688) has a dual function both as further de-intensification (mm. 684-685) and as transition. The transitional portion (mm. 685-688) can also be heard as the intensification phase of a slow moving and subtle wave, ethereal in its orchestration, with the de-intensification beginning at m. 689, the beginning of the last section. Moreover, even though the previous waves in these wave trains have strong climaxes, they do not have further de-intensification, so that the drawn out de-intensification of the last wave also serves for the entire section.
The last section (mm. 689-709) forms a postlude (epilog) to the entire piece. It consists of two parts, a transition (mm. 689‑701) and static ending (mm. 702‑709). The first part (mm. 689-701, marked Mouvt initial) begins with the fast and delicate descending notes and glissandi. As mentioned above, these can be heard as the de-intensification of mm. 685-688, and have the shape of a plunging breaker, even though delicate in orchestration. These lead (like an inverse climax) to a texture of trills with a standing wave quality, and which resemble ripples (mm. 690-693). Mm. 694-701 are a faint reminder of the end of the first wave of the piece (mm. 35-38), with the short isolated pizzicati and open quality. This is followed by the second part of the section (mm. 702-709). Just as the pizzicati of mm. 35-38 was followed at mm. 43 by a return of the static prelude music, so also is m. 701 followed by a return of static prelude music (marked Mouvt du Prélude) in mm. 702-705. Mm. 706-709 form the last reminder of the beginning (marked Mouvt initial), with the beginning rising chromatic (“beta”) pitch cell of the harp and horns (mm. 1-2) returning in the strings (mm. 708-709), appearing now in inverted or descending form.
The first part of Jeux is longer than the second, with the first climax at m. 429 occurring near the golden section of the piece. Howat suggests that Debussy most likely consciously utilized such proportions in Jeux and other works. Debussy thus coordinates the additive, ‘bottom-up’ approach of mosaic structure construction with the ‘top-down’ method of golden section division. Moreover, the existence of a large-scale proportional structure and wave process reveals Debussy’s structural thinking. Figure 13 diagrams the dynamic envelope and structure of the entire piece.
The diagram shows the twelve sections analyzed above, indicated by the measure numbers on the bottom, and the climaxes of the two large parts, indicated by the numbers above the diagram. The audio waveform representation is from the recording of Jeux by the New Philharmonia Orchestra conducted by Pierre Boulez (CBS Odyssey MB2K 45620). The dynamic shape of the sections, the two large parts and their climaxes, and of the entire piece is clear. It can be seen that the second climax at m. 677 is stronger than the first at m. 429, and how the second part of the piece displays a greater density of activity than the first. In addition, the outlines of many waves within the sections are also visible.
This analysis has revealed that in the first part the wave process is generally discontinuous and inhibited, while in the second part it is continuous and free-flowing. The first part can be seen as a series of ‘games’ of different kinds of waves, all of which are generally inhibited, and which becomes progressively more varied and elaborate. The motivically homogenous waves of the beginning are followed by a breakdown of wave modules into contrasting phases, and then stages, producing refractions and stratification. These contrasts and discontinuities highlight the block-like modules and sections of the mosaic structure. Shorter waves are also introduced, along with increasing tempo fluctuations and meter changes. Waves move at different paces, speeding up, slowing down, leading to relatively slow motion waves, speeding up again, and so on. Even though the tempo fluctuations coincide with the wave motion, they inhibit the continuity and flow between waves. Sometimes the wave process is almost halted through pauses preceding the crest of a wave, that is, within waves (e.g. mm. 224, 265). The cyclic returns, “new starts,” and alternations of material also create a back and forth process, stalling and inhibiting continuous and linear development. It isn’t until m. 309 that the first clear signs of linear development return, though soon inhibited, followed by another try at m. 357, also soon inhibited, and followed by the final forward thrust of the wave train at the end of the first part, leading to the inhibited climax or “anti-climax” of m. 429.
In the second part of the piece, following the turning-point at m. 473, the wave process becomes notably more continuous, connected, and flowing. The waves become more progressive and climactic, with repeated and extended climaxes, more powerful climaxes, “breakers,” and “wave trains.” More continuous and linear thematic waves are introduced along with the rhythmic-motivic waves. In this part of the work there is a greater rate of recycling of motivic ideas, and greater motivic-thematic interrelatedness, interworking, and bringing together. Pasler points out the synthesis of rhythmic and metric ideas in the second part of the piece. The meter remains unchanged from the reprise up to the main climax, and tempo and rhythm becomes much more continuous and regular, almost like a perpetuum mobile in mm. 605-676, where there is a bringing together of previous elements in a kind of synthesis leading to the final climax. The transformation through augmentation of the “alpha” motive into theme at the main climax (m. 677), along with its movement from its usual initiating position at the beginning of a wave to that of the crest at the climax of the piece, symbolizes the change from rhythmic to flowing, and can be called the “thematic resolution” of the piece. It coincides with the “dramatic resolution” of the piece, the ‘triple kiss,’ and the movement of the young man, often symbolized by the modal “alpha” theme, from an ‘initiating’ position to a ‘culminating’ position. Moreover, the two parts of the piece complement each other. The ‘inhibition’ and ‘hesitation’ of the waves of the first part, together with the “anti-climax” (m. 429) of the first part creates a tension which is resolved or ‘fulfilled’ in the second part, which follows the dramatic structure of the ballet, and provides closure. These factors form the structural expression of the two complementary parts of a wave, of tension followed by release.
Howat writes about this process in several works of Debussy, including Jeux and the earlier Jeux de vagues from La Mer. In discussing the two parts of this process, Howat points to the “avoidance of sustained dramatic accumulation in the earlier part of each work” and the “making way decisively for two sustained build-ups to large climaxes in the later part of the score,” and how “a first sustained crescendo leads the music to a semi-final climax, which then subsides to set off the final, and longest build-up.” He also refers to these as “initially inconsequential games that gradually become channeled towards an inescapable dénouement.” Howat characterizes the second part of Jeux as a “sequence of small-scale dynamic undulations within the overall accumulation between  and , these subsidiary surges contributing to, but never obscuring, the large-scale surge.” Howat never uses wave terms, but gets his meaning across with words like dynamic sequences, dynamic undulations, dynamic ups and downs, overall accumulation, and surges. Howat also says that “The aspect of dramatic accumulation and this type of overall dramatic shape is a facet of the music that has received less attention from recent composers and analysts than the intricate intervallic aspects of these scores.” Trezise also notes the pattern in Jeux de vagues, mentioning “the gradual transformation from a state of discontinuity... epitomised by a fragmented, edgy succession of events, to an expansive, continuous flow...” Trezise references Berman, also on Jeux de vagues, on how events on the way to the climax “set up an alternation of impediments and forward impulses, before the final forward sweep,” and on the parallel of the waltz theme in Jeux to the ‘smooth-sailing’ theme of Jeux de vagues, which is associated with “a flow of developing material in place of interruptions and deflections.” Evans also notices a related though different process in “Sirénes” from Nocturnes, stating that “The shapes follow a clear pattern of being very brief in the beginning of the piece and, for the most part, become longer as the piece progresses.”
Howat refers to the above pattern as an “archetypal dramatic shape” in Debussy, which is rooted in drama and literature. Classical and romantic ‘architectured’ music has a similar pattern inspired from the same dramatic sources. Muns calls this pattern the climax cycle, while Toch notes that “the long ascending line, ascending as a whole, will show curves, notches, retarding moments, similar to those of the melodic [wave] line,” and that “In musical form these little contrasts within the main trend are not felt as impairing the drive; on the contrary, they set it off to better advantage, giving each section a new impetus.” However, there are differences between the traditional musical application of this pattern and Debussy’s application. The traditional climax cycle relies primarily on syntactic processes for its definition, essentially functional harmony and ‘organic’ or linear motivic-thematic processes. The gradual increase in intensity is relatively smooth and consistent with the predominantly linear quality of the music. The resulting structure forms an arched hierarchy with a major climax towards the end of piece. Debussy’s climax cycle differs in that it relies more on statistical processes for its definition, i.e., the statistical wave process, along with the motivic-thematic process. The gradual increase is achieved in cycles and through the inhibition of the wave process, resulting in the accumulation of unreleased energy which is then released at the structural climaxes.
The large-scale harmonic structure of Jeux combines tonal and non-tonal elements, and results from Debussy’s merger of diatonic and chromatic materials. This produces an expanded form of tonality which is relatively static, and a structure which is hierarchically relatively flat, and which is more organizational than functional. Wenk notes that “A framework of symmetrical tonal centers, corresponding to details of the stage action, provides a coherent basis for Debussy’s excursions into non-tonal procedures.” These symmetrical tonal axes are centered on the main axis of A and form the overall pattern of A-F#-D#-C-(A). Pomeroy also notes of Debussy that “Areas of harmonic inactivity often coexist with a highly chromatic musical surface, often of a whole-tone or octatonic orientation. Such collectional focus in the foreground attenuates harmonic-functional sense, which nevertheless still emerges at a higher level.” Within this structure, non-tonal passages thus result from the use of chromatic, octatonic, whole-tone, diminished seventh, and other symmetrical formations, chromatic transposition in parallel or contrary motion, the superimposition and juxtaposition of chords by semitone, tritone, or other intervals, and “chords used for coloristic rather than harmonic function.” Tonal passages result from the use of diatonic and modal materials (melodic tonality), occasional dominant preparation of a key area, or “deceptive” resolution from V7 to vi, and the reiteration of a tonic, including the use of ostinati used to establish “tonality without progression.” Pomeroy notes Debussy’s “non-functional diatonicism.” Wenk also points out that “Debussy’s sharpest dissonances, even if considerably prolonged, eventually resolve, usually to a simple triad,” and that “While Debussy’s score displays various techniques associated with non-tonal composition, these passages function as embellishments to an essentially tonal framework.” In addition, Pomeroy notes that in Debussy “Harmonic progression often appears as a slower-moving underlay to motivic arabesque-activity on the surface,” and that “synchronization of this harmonic underlay with the motivic symmetry of the ‘two-bar block’ technique results in a metrical (and hypermetric) regularity.” Pomeroy further adds that mosaic structure “enabled the construction of expansive thematic-presentational areas incorporating sufficient internal contrast without (necessarily) the component of harmonic progression,” and that “When the block technique serves an extended area of harmonic inactivity, harmonic progression sometimes takes the form of a distant undercurrent far beneath the surface.” Moreover, both tonal and non-tonal passages are primarily determined and connected by linear factors, i.e., melodic tonality, pitch axes, voice-leading, directional tones, etc. Wenk concludes that in Jeux “Debussy accommodates atonality within a tonal context.”
Debussy’s weakening of functional harmonic elements leads to a greater reliance on motivic-thematic and statistical processes to define structure. Meyer points out that with the advent of atonality, “motivic structure (together with the organizing capabilities of the secondary parameters) had to bear the main burden of musical process and form.” Because of the weakened functional processes and relatively self-contained sections of the mosaic structure, the structure of the piece differs from the arched hierarchies of traditional music and forms a relatively flat or an additive hierarchic structure. Meyer points out that “the weaker and flatter syntactic structure is, the more readily a statistical plan can combine with it,” and that such statistical shapes are additive, continuous, and form an emergent hierarchy. Boulez, on Debussy’s late works, says that “One must experience the whole work to have a grasp of its form, which is no longer architected, but braided; in other words, there is no distributive hierarchy in the organization of “sections” (static sections; themes, dynamic sections; developments) but successive distributions in the course of which various constituent elements take on a greater or lesser functional importance.” As an example of this in Jeux, the first large climax is not prepared for in a straightforward way by the preceding sections, its effect being due primarily to its relative intensity and structural placement. The kind of ‘braided’ form Boulez mentions is also suggestive of a ‘string of pearls,’ which also resembles a wave series. The meaning of the word “wave” as back and forth motion is similar to that of weaving and ‘braided’ or ‘woven’ form. However, even though the hierarchic structure in Jeux is relatively flat compared to classical and romantic music, there is a large-scale and process at work incorporating elements of both functional and hierarchic European music and of relatively non-functional and non-hierarchic Eastern music. Boulez says “it is, above all, the poetics of the Far Eastern musics which imposes its corrosive influence.” This allows Jeux to have a large-scale structure, with a corresponding kind of ‘structural hearing,’ while also having elements of a ‘focus on the moment.’
Focus on the Moment
Boulez states that even though the piece is no longer “architectured,” that “one must experience the whole work to have a grasp of its form.” However, Boulez also states that “Debussy rejects every hierarchy not found implicated in the musical instant,” and that “Jeux announced the achievement of a musical form that, in renewing itself instantaneously, implies a no less instantaneous way of hearing.” Boulez thus implies that even though there is a focus on the moment-to-moment process, there is also a large-scale process and structure in Jeux. The focus on the moment is thus a matter of degree. Debussy certainly introduced this element into his own music, though it was later composers who took these ideas to their furthest reaches through the further breakdown of hierarchy, increased nonlinearity, moment-form, ‘spectral composition,’ etc. Jeux lies “at the crossroads” and is thus seen as a precursor of these later developments.
The duality between ‘structural listening’ and a ‘focus on the moment’ is parallel to that of object versus process, discontinuity versus continuity, i.e., objects are discontinuous and processes are continuous, and that of particle versus wave. Cone distinguishes between these two polarities or “esthetic modes of perception” as “synoptic comprehension” versus “immediate apprehension.” Within this duality, Debussy emphasizes the moment-to-moment process, becomingness (devenir), flow, and continuity, creating a heterogenous flow, or flux of connected differences. Parks states that Debussy “proceeds from a concept of musical form as dichotomous, in its opposing aspects of morphological and kinetic form” and that “kinetic form is a potent organizing force in Debussy’s music.” Furthermore, hierarchic structural listening is Western and ‘rational-discursive’ whereas the focus on the moment involves a listening mode which is ‘intuitive-peripheral,’ showing the connection of durée to Eastern modes of temporal experience. Koelreutter says, “Time in Indian music is “experienced” time, not rationally “measured” time.” This also relates to what Pasler calls the exploration of the “essence” of a musical idea, and Koelreutter’s description of Indian music, which is to “circle round a central idea, whose reality is always present in the raag, and whose sphere of validity is the present.” Wheeldon also says, “Melodic circularity- whether realized or inferred- focuses attention within the boundaries of the moment and attenuates connection to the next moment.” The definition of ‘moment’ in this context ranges from an instant in time, to an indefinitely short amount of time, anywhere from a few seconds to over a minute, sometimes called the specious present.
The primary means through which Debussy achieves a focus on the moment is through the weakening of hierarchic syntactic and directional processes. Debussy spoke of “the discovery of harmonic ‘moments’ to caress the ear.” He also said that “...We’re still in the age of ‘harmonic progressions’ and people who are happy just with beauty of sound are hard to find.” Meyer says of Debussy, “by explicitly weakening the processive (i.e., goal-oriented) aspects of the experience can we be forced, as it were, to attend primarily to the qualities of sense experience - the sound of a particular instrument the specifically sensuous shape of a motive, the changing qualities of such sounds as they follow one another.” Pomeroy points out that Debussy’s “ornamental conception finds its most characteristic form in harmonic inactivity.” Meyer also states that “Nonhierarchic music- that of John Cage, for instance - moves, like the ocean, in undulating or sporadic waves of activity in which we attend to, but can scarcely remember, the particular events.” Meyer’s comparison with ocean waves refers to how the lack of hierarchy maintains a focus on the moment. Even though Meyer refers to Cage, the same description could be applied to Reich, who also achieves a focus on the moment, not through change, juxtaposition, and simultaneity, which can lead to a ‘verticalization’ of the moment, but through reiteration, producing a ‘dilation’ of the moment. These two kinds of focus on the moment correspond to the ‘continuous present,’ through constant change, and the ‘extended present,’ through constant sameness. However, Debussy’s hierarchic structures are not as flat as that of aleatoric or minimalist music. Meyer says that “Both complete uniformity and total heterogeneity preclude syntactic organization, and hence establish no stability-instability relationships.” Syntactic organization, in this reference, presumably creates hierarchy. The non-functional aspects of the music thus contribute to a focus on the moment. Debussy’s motivic-thematic process creates a flux through forms which are always the ‘same yet different.’ The cyclicity and circularity of the motivic-thematic and statistical wave processes diminish linearity and hierarchy, and focus attention on the moment. These elements work together, e.g. heterophonic layering creates a focus on the moment due to elements of simultaneity, reiteration, closed repetition, nonfunctional harmony, and sometimes cyclicity (as in gamelan music). The nonlinear and spatial effect of simultaneity of diverse motivic elements emerging and submerging from a static ‘background’ also creates a focus on the moment. The mosaic structure, which has few hierarchic levels, keeps attention at the middle levels of modules and sections, with durations within the specious present. The qualities of non-linearity, “non-progressive development,” open-endedness, and simultaneity of the mosaic structure all contribute to a focus on the moment. In addition, the ‘encapsulated,’ relatively closed, non-functional waves and sections, the relative discontinuity of the wave troughs (and nodes), and incipient moment-form, create a focus on the moment.
A focus on the moment is also provided by the changing surface of the music, in the form of intricate arabesques, figuration, ornamentation, particle-constituted textures, and changing timbral and textural effects, “sound textures,” and “orchestration-invention.” As Eimert states, “the entire form is in a state of flux.” Embellishment itself is a form of focus on the moment. The use of contrasts, juxtapositions, and discontinuities also draw attention to the moment, to the fact of change or surprise. Since motivic-thematic connections between the contrasting segments are often lacking or tenuous, attention becomes less focused on making forward or backward comparisons between successive contrasts. Kramer relates the use of blocks, contrasts, and stasis to the focus on the moment and Stockhausen’s definition of moment-form. The constant change in different parameters relates to what Stockhausen calls degree of information and density of alteration, and is demanding on the listener’s processing resources, keeping attention focused on the moment, and making the psychological speed of passage of time seem quicker (i.e., the 17 or so minutes of the piece seem closer to 10 minutes). Eimert says that “incomprehensibility increases in so far as the material repeated is constantly new”, yet on the other hand, reiteration is also a means of focusing on the moment, by focusing on micro-level differences, changes, or variations, while also providing coherence and continuity. Furthermore, the rate of change of materials in Jeux is variable, e.g., longer more continuous sections or segments versus choppier and quicker contrasts. In combination with the weakened hierarchic, syntactic, and goal-oriented processes, the element of sameness often appears in the form of passive continuation and open repetition, and contributes to the flow and continuity of the moment, another important aspect of durée.
Debussy uses a number of methods to achieve the flow of durée. The ideas of arabesque and waves embody the elements of continuity and flow, through the conjunct, stepwise, undulating melodic movement, intensity gradations, and curves. “Gradual, rather than abrupt change is what gives Jeux its unique, seamless quality.” Eimert mentions the “highly developed art of transitions and shadings,” “dynamic evolutions and gradations,” and that “What sounds at any moment is constantly adapted to what follows, often in a gentle shift of movement.” This includes the smoothly connected voice-leading, the soft, fuzzy, and blended timbres and orchestration, and the speeding and slowing of tempo and use of tempo rubato. The underlying tempo and metrical structure is also part of the continuity and coherence, the ‘periodicity’ underlying Jeux which Boulez and Pasler point out. The motivic-thematic generative process also plays a large role, providing ‘subtle links” through motivic associations and moment-to-moment connection at the micro-level. Toch mentions two general means of achieving movement and flow, through periodicity and through constant renewal of motivic-thematic material, and talks about ‘the art of joining’ and creating an ‘unbroken flow,’ using sequences, pre-announcing of motivic fragments, preparation of the following melody, propagation of motivic fragments, transitions, bridges, connecting links, introductory links, organic connections, overlaps, voice leading, elisions, etc., all of which Debussy uses. In reference to the ‘constant renewal’ of motivic-thematic materials, Pasler mentions the “constant becoming” on the level of form, and “forms entirely based on fluid relationships of textures and timbres.” The music stays on the leading edge of time, the devenir, “always starting” and not always finishing, and is hence open-ended. This is reflected in the < < < < shapes of some wave stages. Boulez says of Jeux that “the structure is rich in inventions and flowing complexity,” and that “Motion, the instant, erupt into his music, not merely an impression of the instant, of the fugitive to which it has been reduced, but really a relative and irreversible conception of musical time and, more generally, of the musical universe.” Notwithstanding the constant transformation, heterogeneity, contrasts, and discontinuities in Jeux, the process is always smoothly connected, even if in ‘subtle’ ways. Moreover, the focus on the moment, flow, and irreversible time are part of Debussy’s larger sense of musical time.
Jeux is characterized by an evolved sense of multiple-time which combines cyclic, linear, and nonlinear elements. Pasler says that Debussy was “inspired by the time of nature and of the universe, which he found multiple and characterized by a different quality from moment to moment.” The “time of nature” is also cyclic, with events recurring in cycles. Due to the dialectic of continuity and discontinuity within the wave process of Jeux, the cyclic process itself is sometimes more linear and other times more nonlinear. As stated before, Debussy’s use of cyclic processes includes the influence of Non-Western (Balinese, Javanese, Indian, Middle Eastern, Chinese, Japanese, e.g., Zen and durée) music and time concepts. Wenk emphasizes the fact that Debussy created an ‘alternative’ form of musical process, that “Debussy opened a new approach to the problem of form by offering alternatives to linear, architectural structure,” and that Debussy was working with “models for replacing a strictly linear conception of music with a circular conception in which the moment might take precedence over the relationships among moments, in which time might take the form of succession rather than progression, and in which the notion of goals might be greatly diminished if not altogether abolished.”
The cyclic element in Jeux results from the combination of motivic-thematic regeneration, melodic waves, and statistical waves. Cyclicity begins at the wave level through the periodicity of the metric wave, the reversal of process between wave phases, and is periodic at the level of a series of waves. A single cycle of a wave involves a linear motion away from the trough and towards the crest, and then away from the crest and towards the trough. The “constant renewal” of motivic-thematic materials is not only moment-to-moment but also cyclic. The cyclic use of a number of different motivic-thematic particles and secondary materials creates the sense of a multiplicity of events coming and going within the statistical wave process. The regular cycles of the statistical waves are combined with more independent patterns of motivic-thematic recurrence. The beginning of each new wave is a point of expectation as to what motivic-thematic materials might appear or recur. Depending on context, this may be more or less predictable, e.g., a wave train is more continuous and predictable. This combined use of motivic-thematic recycling and statistical waves produces a strong cyclic sense. In addition, when a series of waves is considered, the cyclic element can combine with linear or nonlinear elements, giving the cyclic process an overall linear or nonlinear orientation. The returns toward the end of the piece, including the reprise and epilog, create a further large-scale cyclic return and close the circle of Jeux.
The nonlinear elements in Jeux are varied, and are created through both continuous and discontinuous means. At times, nonlinearity is achieved through continuous though static processes, as in the prologue and epilog of the piece. The use of passive continuation also provides nonlinear continuity. Nonlinear, non-directional processes can thus have flow and continuity, e.g. minimalist, Indian, and some gamelan music. Kramer calls this kind of process non-directed linear time. The ornamentation, figuration, and prolongation of static pitches and chords, including the use of ostinati, also contributes a sense of “motion without progression.” This is often achieved through embellishment using both diatonic and chromatic neighbor tones and chords, e.g., back and forth motion or trills, and can occur in compound form. Discontinuity in the form of juxtapositions, interruptions, irregular and apparently random returns, “new starts,” and other devices to different degrees also diminish directional tendencies and create a nonlinear effect. The sense of simultaneity of multiple motivic-thematic materials that are recycled throughout the piece and seem to ‘co-exist’ also produces a spatial and nonlinear effect, a stratified simultaneity of multiple strands of cyclically recurring ideas. Cone comments on the ‘presence’ of stratified materials, saying that “although heard in alternation, each line continues to exert its influence when silent.” Wenk proposes the concept of global form, in which “a musical idea extends its influence over the work as a whole,” and which “allows a single musical idea to organize a composition into a unified, though nonlinear, entity,” and Hedges states that “what stands out is not a long-range goal, but cycles of musical sound which arise as manifestations of abstract ideas in a continuously changing state.” Other non-functional aspects of the piece contribute to nonlinearity, including the non-functional harmonies and the relatively self-enclosed non-functional waves. Moreover, the nonlinear element appears prominently as the inhibition of the linear wave process.
When cyclic and nonlinear elements are combined, a nonlinear cyclic process is produced, one which is cyclic with no overall sense of direction. Standing waves, whether melodic or statistical, are a prime example of a wave process without an overall sense of direction. This is what Eimert meant by “a circulation which is always at its goal and therefore never ‘going’ anywhere.” Wheeldon also says, “A melody or motive is deemed circular if its implication for repetition - indeed, endless repetition - outweighs its connections to the following melodic idea.”
Finally, the relatively self-enclosed and non-functional waves and sections, in combination with elements of discontinuity such as fragmentation, juxtapositions, random recurrences, and open-endedness produce nonlinear moment-form tendencies. Wheeldon points out that the more “random” recurrences of modules have greater moment-form qualities due to their greater sense of disconnection. The troughs (or nodes) between waves, with their relative resolution and closure-defining properties, help segregate waves into ‘quasi-moments.’ In addition, due to the greater elements of discontinuity in the first part of the piece, there are more examples of ‘quasi-moments’ there. Modules, waves, and sections are relatively self-enclosed and independent from each other, contributing to a sense of multiplicity and nonlinearity. However, Jeux is not a moment-form piece, although it has elements of moment-form. This is due to the ‘subtle’ links between the modules. Wheeldon states that “if successive melodic ideas are linked, it often seems incidental or secondary in nature,” but that “the connecting function of the motivic links departs from moment-form aesthetics.” The dramatic curve of the piece and the structural returns at the reprise and epilog also depart from moment-form ideas. Again, many of the above cyclic and nonlinear elements are found in Eastern musics and are strongly related to the focus on the moment. In essence, anything which produces a focus on the moment contributes to a nonlinear effect.
The are a number of linear or directional elements in Jeux. Any process that implies a goal of motion is directional. The crests, climaxes, and troughs of waves are goals of motion or points of arrival of a number of syntactic and statistical processes working together, e.g., voice-leading, intensification, momentum, cross-rhythms, de-intensification, cadential processes. Wave motion incorporates linearity within waves in the form of motion away from and towards each crest and trough. Progressive waves have a sense of momentum, forward drive, and directionality which often points beyond the wave itself. This is especially true of progressive waves that do not break, since the unreleased energy tends to be propagated. A series of waves can also form part of a larger directional process, as with wave trains. These linear, functional processes are supported through the accumulation of structural energy and the propagation of motivic shapes from wave to wave. Furthermore, the motivic‑thematic and statistical processes work closely together to create the sense of development. There are clear strands of motivic development within the piece, i.e., the “alpha” and “beta” motives mixed in the first sections with the “alpha” motive predominating, followed by the development of “beta” related ideas, the varied return of the “alpha” motive at m. 335, prepared at m. 315, and leading to the climax at m. 429, the variation of the “alpha” theme in the middle section, the literal and varied return of “alpha” and “beta” motives in the reprise, the pas de trois section, the long development of waltz theme, leading to the intensified chromatic motivic development running through mm. 627, 635, 665, 669, to the main climax at m. 677.
The large-scale wave process in Jeux is directional, moving as it does through the two structural climaxes of the piece. Within this overall direction, the process fluctuates between being more or less linear and nonlinear. Even though the process of modular and wave development in the first part is often inhibited, it still provides linearity, a kind of ‘intensification’ through increasing variation and elaboration. The process then becomes notably more linear in the second part of the piece. Because the overall process occurs in cycles or waves, the temporal orientation of the piece is cyclic with an overall sense of direction, which might compare to the overall rise and fall of waves due to the tide or a spiral. The process forms a cyclic development as opposed to a purely linear development, but a development and resolution nonetheless. Notwithstanding the duality and dialectic of continuity and discontinuity, which often significantly inhibits the process, there is an underlying though sometimes subtle directionality due to the motivic-thematic and statistical wave processes. Moreover, the linear and nonlinear elements work together to create a sense of what Kramer calls multiply-directed linear time, that is, the juxtapositions, refractions, and stratification create relatively self-contained modules, often open-ended, and with varied goal implications. In addition, the piece contains a multiplicity of linear elements, or strands, many of which gradually evolve. Kramer states that “Jeux exists in a complex and fascinating time world of multiply-directed time that anticipates the still more radical “moment time” of Stravinsky, Messiaen, and others.”
The multiple-time aspect of the process brings all of the above elements together, i.e., the cyclic, nonlinear, and linear. Of the several definitions which Kramer gives of multiple-time, one is that “the nature of time is multiple,” which corresponds to Pasler’s reference to Debussy’s multiple “time of nature.” Kramer also quotes Langer on the idea that multiple-time is made up of several concurrent strands of processes. Langer states, “If we could experience only single, successive organic strains, perhaps subjective time would be one-dimensional, like the time ticked off by clocks. But life is always a dense fabric of concurrent tensions, and as each of them is a measure of time, the measurements themselves do not coincide.” This illustrates the elements of multiplicity and simultaneity of multiple-time. Kramer distinguishes the idea of “intertwined strands of continuity” and “separate directional continuities,” from clock-time as an abstraction, as a unidimensional and linear conception of time (the influence of Newtonian time). Bergson’s durée is also multiple in that it consists of a heterogenous flow, a multiplicity comparable to a stream of consciousness. Kramer further relates multiple-time to the sense of “re-ordering” of events, a “time scrambled” view, which he compares to traditional “well-ordered time-experiences,” and in which cinematography has an influence. The motivic recycling, including the mixing of new and old ideas, along with the additive and permutational aspect of the mosaic structure, with its modular alternations, contrasts, interruptions, regular and irregular returns, and discontinuities, lead to a “re-ordering” of events. The breaking up of linearity relates to Kramer’s concept of multiply-directed linear time, i.e., “pieces where the direction of motion is frequently interrupted by discontinuities in which the music goes so often to unexpected places that the linearity, though still a potent structural force, seems reordered.” As Kinariwala’s notes, that “The succession of ideas does not always lead to an inevitable goal.” Kramer says the same, that some sections are in “motion toward goals (or from sources) that do not appear in adjacent sections and may not even appear.” This means that the “good continuation” of the wave process towards goals of motion is often inhibited, interrupted, suspended, or discontinued such that only some of these goals are subsequently reached or processes continued or resumed. In addition, the regeneration of motivic materials is often irregular, as if the wave particles were being reshuffled. The independence of particle and wave, and to a lesser extent between “theme and structure,” allows a multiplicity of motivic-thematic materials to return at any point within the larger statistical wave process. Together with the nonlinear stratified simultaneity, this produces the effect of multiple interpolated cycles (of recurrences of materials). Moreover, because the process has a large-scale sense of direction, it can be said that Jeux has a temporal orientation of multiple-time which is cyclic with an overall sense of direction, as in the “time of nature.”
Debussy’s musical language developed into culminating form in Jeux. Debussy developed an alternative kind of music, incorporating a focus on the moment, continuity and discontinuity, arabesque, mosaic structure, and cyclic processes. The waves and cyclic processes of Jeux are one of his most advanced developments. Debussy’s compositional techniques reveal his “search for a world of sensations and forms in constant renewal.” Even though Debussy’s innovations in harmony and orchestration are often noted, many of the ideas he developed and which inspired other composers are in the area of musical process, or “kinetic form,” including his exploration of music as process, statistical waves, and cyclical time. These contributions have been heralded by later composers such as Stockhausen and Eimert to be the precursors of later twentieth-century developments such as statistical-form and moment-form. As the analyses have shown, the largely unexplored wealth of wave characteristics displayed by Debussy’s composed-out waves is truly remarkable and consistent.
The analogy between natural waves and composed-out waves developed and applied herein is simple, useful, and valid, and explains what more traditional approaches might not. Eimert’s states that “traditional music theory is helpless in the face of this work,” Pasler says that “they have failed to find the key to its form,” and other authors make similar comments about the difficulties of analyzing Jeux. The wave concept emphasized by Eimert has been an important key to understanding Jeux. The wave concept is a great unifying concept because it clarifies, explains, and uncovers many kinds of natural behavior, and produces an insight which makes it possible to understand something not as understandable without the concept. The wave concept can be usefully applied, by way of analogy, to the analysis and composition of musical composed-out “waves” and cyclic processes.
References and Bibliography:
Al-Faruqi, Lois Ibsen. “Ornamentation in Arabian Improvisational Music: A Study of
Interrelatedness in the Arts,” The World of Music, vol. XX, no. 1, 1978.
Berry, Wallace. “Rhythmic Accelerations in Beethoven,” Journal of Music Theory, vol. 22, no.
2, Fall, 1978.
Benjamin, William. “‘Pour les Sixtes’: An Analysis,” Journal of Music Theory, vol. 22, no. 2, Fall, 1978.
Bloom, Victor. Jeux by Claude Debussy: Some Recent Reviews, Ph.D. Publishable Paper,
University of California at San Diego, 1990.
Carpenter, Patricia. “The Musical Object,” Current Musicology, no. 5, 1967.
Chimenes, Myriam. “The Definition of Timbre in the Process of Composition of Jeux,” in
Debussy Studies, ed. by Richard Langham Smith, Cambridge University Press,
Cambridge, New York, 1997.
Cone, Edward T.. “Stravinsky: The Progress of a Method,” Perspectives on Schoenberg and
Stravinsky, Benjamin Boretz and Edward T. Cone, eds., W.W. Norton & Co., Inc., New
DeVoto, “The Debussy Sound: Colour, Texture, Gesture,” in Trezise, Simon (ed.), The Cambridge
Companion to Debussy, Cambridge University Press, Cambridge, New York, 2003.
Eimert, Herbert. “Debussy’s ‘Jeux’,” Die Reihe, vol. 5, 1959 and 1961, Theodore Presser Co.,
Gabor, Dennis. “Acoustical Quanta and the Theory of Hearing,” Nature, vol. 159, no. 4044,
Gelleny, Sharon. “Cyclic Form and Debussy’s Nocturnes,” Cahiers Debussy, no. 20, 1996.
Howat, Roy. “Dramatic Shape and Form in ‘Jeux de Vagues’, and its Relationship to Pelléas,
Jeux and Other Scores,” Cahiers Debussy, Series 2, vol. 7, 1983b.
Koelreutter, H. J.. “Indian and Western Music as the Expression of Different Attitudes of
Consciousness,” in Music East and West, edited by Roger Ashton, Indian Council for
Cultural Relations, Bhatkal Books International, 1966.
Kramer, Jonathan D.. “Multiple and Non-linear Time in Beethoven’s Opus 135,” Perspectives of
New Music, vol. 11, no. 2, Spring-Summer, 1973.
_________________. “Moment Form in Twentieth-Century Music,” The Musical Quarterly, vol.
LXIV, no. 2, April, 1978.
Lendvai, Ernö. “Duality and Synthesis in the Music of Béla Bartók,” in Module, Proportion,
Symmetry, Rhythm, Gyorgy Kepes (ed.), George Braziller, Inc., New York, 1966.
Reprinted from The New Hungarian Quarterly, vol. 3, no. 7, 1962.
Leydon, Rebecca. “Debussy’s Late Style and the Devices of the Early Silent Cinema,” Music
Theory Spectrum, vol. 23, no. 2, Fall, 2001.
Lidov, David. "Structure and Function in Musical Repetition,” Canadian Journal of Research in
Ligeti, György. “Metamorphoses of Musical Form,” Die Reihe, vol. 7, 1965, Theodore Presser
& Co., Philadelphia, Pennsylvania.
Marcel, Gabriel. “Bergsonism and Music,” in Reflections on Art, edited by Susanne Langer,
Oxford University Press, London, 1958, pp. 142‑51.
McFarland, Mark. “Debussy and Stravinsky: Another Look into Their Musical Relationship,”
Cahiers Debussy, no. 24, 2000.
Parks, Richard, S.. “Music’s inner dance: form, pacing anc complexity in Debussy’s music,” in
Trezise, Simon (ed.), The Cambridge Companion to Debussy, Cambridge University
Press, Cambridge, New York, 2003.
Pasler, Jann. “Debussy, Jeux: Playing with Time and Form,” 19th Century Music, vol.6, no. 1,
__________. “Resituating the Spectral Revolution: French Antecedents and the Dialectic of
Discontinuity and Continuity in Debussy’s Jeux,” in Aspects du Temps dans la Création
Musicale, Irène Deliège, ed., ESCOM, Lìege, Belgium, 2004.
Pike, Kenneth L.. “Practical Phonetics of Rhythm Waves,” Phonetica, vol. 8, 1962.
Pomeroy, “Debussy’s Tonality: a Formal Perspective,” in Trezise, Simon (ed.), The Cambridge
Companion to Debussy, Cambridge University Press, Cambridge, New York, 2003.
Potter, “Debussy and Nature,” in Trezise, Simon (ed.), The Cambridge Companion to Debussy,
Cambridge University Press, Cambridge, New York, 2003.
Pousseur, Henri. “Pour une Périodicité Généralisée,” Fragments Théoriques I sur la Musique
Exérimentale, Henri Pousseur, Éditions de l’Institut de Sociologie Université Libre de
Ruwet, Nicolas. “Note sur les duplications dans l’oeuvre de Claude Debussy,” in Langage,
musique, poésie, aux Éditions du Seuil, 1972.
Schneider, Mark. “Measured Diversities: Transformation Structures In Music, Art, And
Architecture 1880‑1935,” Unpublished paper, presented at On Growth and Form: the
Engineering of Nature, ACSA East Central Conference, University of Waterloo, 2001.
Simon, Herbert, and Newell, Allen.. “Models: Their Uses and Limitations,” The State of the
Social Sciences, Leonard D. White, ed., The Univ. of Chicago Press, Chicago, 1956.
Sommer, Avo. Musical Syntax in the Sonatas of Debussy: Phrase Structure and Formal
Function, Music Theory Spectrum, vol. 27, no. 1, Spring 2005.
Stockhausen, Karlheinz. “Structure and Experiential Time,” Die Reihe, vol 2, 1958, Theodore
Presser Co., Philadelphia, Pennsylvania.
__________________. “Von Webern zu Debussy- Bemerkungen zur Statistichen Form,” Texte
zurElektronischen und Instrumentalen Musik, Karheinz Stockhausen, Verlag M. DuMont
Schauberg, Köln, 1963.
Wichmann, Siegfried. “The ‘Wave’ as Picture and Ornamental Form in Japanese Art and its
Influence on European Art,” World Cultures and Modern Art, Bruckmann Publishers,
Zenck, Claudia Maurer. “Form- und Farbenspiel: Debussy’s ‘Jeux’,” Archiv Für
Musikwissenschaft, v. 33, 1976.
Beck, Stephen David. Structural Impact of Binary Phrase Constructs in Debussy's Jeux, Ph.D.
Dissertation. UCLA, 1988.
Bersano, James Richard. Formalized Aspect Analysis of Sound Texture, Ph.D. Dissertation,
Indiana University, 1980.
Biggs, George Browning Jr.. The Return Effect in Works for Orchestra by Stravinsky, Bartok,
and Schoenberg, as Determined by Factors other than Theme and Key, Ph.D.
Dissertation, Indiana University, 1968.
Brooks, Richard. Structural Functions of ‘Musical Gesture’ as Heard in Selected Instrumental
Compositions of the Twentieth Century: A Graphic Analytic Method, Ph.D. Dissertation,
New York University, 1981.
Evans, Laura Jean. Mosaic Structure in Music, Ph.D. Dissertation, New York University, 2003.
Friedman, Edward Arthur. Texture and Ornament in the Music of Claude Debussy, Ph.D.
Dissertation, University of Connecticut, 1987.
Gallaher, Christopher Summers. Density in Twentieth-Century Music, Ph.D. Dissertation,
Indiana University, 1975.
Hartman, James Barclay. A Gestalt Theory of Musical Perception, Ph.D. Dissertation,
Northwestern University, June, 1959.
Hedges, Bonnie Lois. The Structural Significance of Duration and Concepts of Linear and
Cyclical Development in Two Chamber Works of Claude Debussy, Ph.D. Dissertation,
University of Texas at Austin, 1976.
Jenkins, Susan Elaine. Representationalism in Selected Twentieth Century Compositions
about the Sea, Ph.D. Dissertation, The Ohio State University, 1984.
Kam, Dennis. Repetition and the Drift Towards Constant Focus in the Pattern-Pulse Works of
Terry Riley and Steve Reich, Ph. D. Dissertation, University of Illinois at Urbana-
Kinariwala, Neela Delia. Debussy and Musical coherence: A Study of Succession and
Continuity in the Preludes (Repetition), Ph.D. Dissertation, The University of Texas at
Lipkis, Laurence (Larry) Alan. Aspects of Temporality in Debussy's Jeux and Ives' Symphony
Symphony No. 4, Fourth Movement, Ph.D. Dissertation, University of California at Santa Barbara, 1984.
McFarland, Mark. Claude Debussy and the Concept of Chromatic Harmony in His Late Works
(1911‑1915), Ph.D. Dissertation, University of California at Santa Barbara, 1997.
McGinness, John Randolph. Playing with Debussy's Jeux: Music and Modernism, Ph.D.
Dissertation, University of California at Santa Barbara, 1996.
Muns, George E., Jr.. Climax in Music, Ph.D. Dissertation, University of North Carolina, 1955.
Pasler, Jann. Debussy, Stravinsky, and the Ballets Russes: The Emergence of a New Logic,
Ph.D. Dissertation, University of Chicago, 1981.
Ringgold, John Robert. The Linearity of Debussy's Music and Its Correspondences with the
Symbolist Esthetic: Developments Before 1908, Ph.D. Dissertation, University of
Southern California, 1972.
Schweitzer, Eugene William. Generation in the String Quartets of Carter, Kirchner, and
Schuller: A Concept of Forward Thrust and its Relationship to Structure in Aurally
Complex Styles, Ph.D. Dissertation, The University of Rochester, Eastman School of
Sepe, Randolph Neal. Large‑Scale Structure and the Compositional Idea in the Music of
ClaudeDebussy, Ph.D. Dissertation, Yale University, 1993.
Silver, Sheila. Some Aspects of Harmonic Organization in the Music of Debussy, Ph.D.
Dissertation, Brandeis University, 1979.
Watson, Linda Lee. Debussy: A Programmatic Approach to Form, Ph.D. Dissertation,
University of Texas at Austin, 1978.
Wennerstrom, Mary Hannah. Parametric Analysis of Contemporary Musical Form, Ph.D.
Dissertation, Indiana University, 1967.
Wheeldon, Marianne. Interpreting Discontinuity in the Late Works of Claude Debussy, Ph.D.
Dissertation, Yale University, 1997.
Barraqué, Jean. Debussy, Éditions du Seuil, Paris, 1962.
Bascom, Willard. Waves and Beaches, revised, updated, and enlarged edition, Anchor Books,
Anchor Press/Doubleday & Co., Garden City, New York, 1980.
Berry, Wallace. Structural Functions in Music, Prentice-Hall, Inc., Englewood Cliffs, New
Besancon, Robert M.. The Encyclopedia of Physics, second edition, Van Nostrand Reinhold
Company, New York, 1974.
Boulez, Pierre. Notes of an Apprenticeship, A. A. Knopf, New York, 1968.
Capra, Fritjof. The Tao of Physics, Shambala Publications, Inc., Boulder, Colorado, 1976.
Christ, William, et. al., Materials and Structure of Music, vol. 1, third edition, Prentice-Hall, Inc.,
Englewood Cliffs, New Jersey, 1980.
Cogan, Robert, and Escot, Pozzi. Sonic Design, Prentice-Hall, Inc., Englewood Cliffs, New
Cone, Edward T.. Musical Form and Musical Performance, W.W. Norton & Co., Inc., New York,
Cox, David. Debussy Orchestral Music, University of Washington Press, Seattle,
DeVoto, Mark. Debussy and the Veil of Tonality: Essays on His Music, Pendragon Press,
Edwards, Arthur C.. The Art of Melody, Philosophical Library, New York, 1956.
Eitan, Zohar. Highpoints: A Study of Melodic Peaks, University of Pennsylvania Press,
Erickson, Robert. Sound Structure in Music, University of California Press, Berkeley, 1975.
French, A.P. Vibrations and Waves, W.W. Norton & Co., Inc., New York, 1971.
Glasford, Irene A.. Rhythm, Reason, and Response, Exposition Press, Jericho, New York, 1970.
Howat, Roy. Debussy in Proportion: A Musical Analysis, Cambridge University Press,
Cambridge, New York, 1983a.
Katz, David. Gestalt Psychology, its Nature and Significance, Ronald Press Co., New York, 1950.
Koffka, Kurt. Principles of Gestalt Psychology, Harcourt, Brace, and World, Inc., New York, 1935.
Köhler, Wolfgang. Gestalt Psychology, Liveright Publishing Corp., New York, 1947.
Kramer, Jonathan. The Time of Music, Schirmer Books, New York, London, 1988.
Langer, Susanne K.. Feeling and Form, Scribner, New York, 1953.
Lapedes, Daniel N.. McGraw-Hill Dictionary of Scientific and Technical Terms, second edition,
McGraw-Hill, Inc., New York, 1978.
_______________. McGraw-Hill Dictionary of Physics and Mathematics, McGraw-Hill, Inc.,
New York, 1978.
La Rue, Jan. Guidelines for Style Analysis, W.W. Norton & Co., Inc., New York, 1970.
Leichtentritt, Hugo. Musical Form, Harvard Univ. Press, Cambridge, 1961.
Meyer, Leonard. Emotion and Meaning in Music, The University Of Chicago Press, Chicago
and London, 1956.
_____________. Music, the Arts, and Ideas, The University of Chicago Press, Chicago and
_____________. Explaining Music, The University of Chicago Press, Chicago and London, 1973.
_____________. Style and Music: Theory, History, Ideology, University of Pennsylvania Press, Philadelphia, 1989.
Needham, Joseph. Science and Civilisation in China, vol. 4, pt. 1, Cambridge Univ. Press,
______________. Time and Eastern Man, Royal Anthropological Institute. Occasional paper, no. 21, Royal Anthropological Institute of Great Britain & Ireland, London, 1965.
Nichols, Roger. Debussy, Oxford University Press, London, 1972.
____________. Debussy Letters, Faber and Faber, London, Boston, 1987.
Orledge, Robert. Debussy and the Theatre, Cambridge University Press, Cambridge, New
Parks, Richard S.. The Music of Claude Debussy, Yale University Press, New Haven, 1989.
Pike, Alfred. A Phenomenological Analysis of Musical Experience and other Related Essays,
St. John's University Press, New York, 1970.
Read, Gardner. Style and Orchestration, Schirmer Books, New York, 1979.
Reti, Rudolph. The Thematic Process in Music, The Macmillan Co., New York, 1951.
Roads, Curtis. Microsound, MIT Press, Cambridge, Massachusetts, 2002.
Salzman, Eric. Twentieth Century Music- An Introduction, second edition, Prentice-Hall, Inc.,
Englewood Cliffs, New Jersey, 1967, 1974.
Schillinger, Joseph. The Schillinger System of Musical Composition, Da Capo Press, New
York, (1941), 1978.
Shattuck, Roger. The Banquet Years, revised edition, Vintage Books, New York, 1968.
Smith, Richard Langham. Debussy on Music, A. A. Knopf, New York, 1977.
Tenney, James. Meta(+)Hodos- A Phenomenology of Twentieth-Century Musical Materials and
an Approach to the Study of Form, The Inter-American Institute for Musical Research,
Tulane Univ., New Orleans, 1964.
Thompson, D’Arcy Wentworth. On Growth and Form, Cambridge University Press, Cambridge, 1942.
Toch, Ernst. The Shaping Forces in Music, Dover Publications, Inc., New York, 1948, 1975, 1975.
Trezise, Simon. Debussy: La Mer, Cambridge University Press, Cambridge, New York, 1994.
____________ (ed.). The Cambridge Companion to Debussy, Cambridge University Press,
Cambridge, New York, 2003.
Vallas, Leon. The Theories of Claude Debussy, Dover Publications, New York 1979.
Walker, Alan. A Study in Musical Analysis, The Free Press of Glencoe, New York, 1962.
Wenk, Arthur B. Claude Debussy and Twentieth‑Century Music, Twayne Publishers, Boston, 1983.
Wiegel, Robert L.. Waves, Tides, Currents and Beaches: Glossary of Terms and List of Standard
Symbols, Council on Wave Research, The Engineering Foundation, July, 1953.
Zonis, Ella. Classical Persian Music- An Introduction, Harvard Univ. Press, Cambridge,
Zuckerkandl, Victor. Sound and Symbol, Pantheon books, New York, 1956.
Debussy, Claude. Jeux; poème dansé, miniature orchestral score, Éditions Durand & Cie.,
______________. Jeux, poème dansé, Dover Publications, New York, 1992.
______________. Jeux, piano music score, Première Representation le 15 Mai 1913, Éditions Durand & Cie., Paris, 1912.
______________. Jeux, poème dansé, piano arrangement, 4 hands, transcribed by Léon Roques, Éditions Durand & Cie., Paris, 1914.
Appendix A: Schematic Representation of Wave Structure of Jeux
 Ruwet, pp. 73, 75. Pasler, 1982, p. 68, and 2004, p. 136, also points out timbral and metric dualities between the dance characters in Jeux. See also Barraqué, p. 169, Howat, 1983a, p. 13, Lendvai.
 Needham, 1954, pp. 1, 3, 10, 14.
 Needham, 1965, pp. 6-7, 1954, p. 3.
 Trans. by Pasler, 1982, p. 69, quoting from Barraqué, p. 159, from Debussy letter to Louis Laloy (unknown date).
 Boulez, p. 354.
 Nichols, 1987, p.288, from Debussy letter to composer/conductor Gabriel Pierné (who conducted first performance of Jeux), 5 March 1914.
 Eimert’s translation of Debussy’s phrase that music is of “couleurs et de temps rhythmés,” Eimert, p. 20, in Nichols, 1987, p. 184, from Debussy letter to Jacques Durand, Tuesday 3 September 1907. Pasler, 1982, p. 72, and 2004, p. 133, goes more into what Debussy may have meant by this phrase, i.e., as “rhythmicized color (timbres) and time.” Debussy’s remarks also imply that the nature of time is a non-measured flow (durée) which is given rational measure or “rhythmicized” through music. Debussy’s use of speeding and slowing of tempo and tempo rubato is a means of preserving the sense of flow of natural time, here used within the context of wave processes.
 Cox, p. 50, Kramer,1988, pp. 48-49, Orledge, p. 170, Salzman, p. 22, Wenk pp. 73, 121, Wheeldon, p. 9. To some extent, there is a wave-particle duality in analytical approaches to Jeux, with some authors stressing continuity, others discontinuity, and some incorporating both. The complexities of the dualities involved in Jeux has lead some authors to describe it as difficult to analyze, e.g., Eimert, p. 20, Pasler, 1982, p. 60, Kinariwala, p. 198, Bloom, p. 1.
 Langer, pp. 108-109, Zuckerkandl, pp. 83-84.
 Smith, p. 261, quoting Vincent d’Indy, originally in d’Indy’s article in SIM, February 1913, see also p. 298. With matched cuts, common elements between the successive shots make the changes seem smooth, whereas jump cuts abruptly change to a different scene and are used to boldly change or terminate action. Other cinematographic techniques that can be applied to music, but which Debussy does not use in any obvious way in Jeux, are accelerated motion, compressed time, flashbacks and flashforwards, replays (often in slow motion), reverse (retrograde) motion, different cutting rates and rhythms, insert shots, montage editing, wipes, and dissolves (involving mixing or cross-fading).
 Cone, 1972, p. 156, see also Leydon, Kinariwala, pp. 207-209, Toch, p. 191.
 Pasler, 1982, p. 75.
 Meyer, 1956, p. 93, 1989, p. 304.
 Smith, p. 21, from Debussy (Monsieur Croche) article “The Nursery, Poem and Music by M. Moussorgsky” in La Revue blanche, 15 April 1901.
 The terms ‘organic’ and ‘organicism’ are used in a specific sense in classical-romantic music, referring to motivic homogeneity, coherence, and linear development. However, the term organic in a more general sense has also been applied to aspects of Debussy’s music, e.g., Eimert’s mention of “organic growth” and the “ornamental-vegetative formal principle.” The term organic relates to similarity with organic forms and processes, homogeneity, coherence, connectedness, and gradual development from motivic “seeds,” while inorganic relates to heterogeneity, diversity, and juxtaposition. In this sense, Debussy’s music has elements of both the organic and inorganic. Debussy’s use of organic elements differs from traditional methods. The terms analytic versus synthetic clarify this difference in that classical-romantic music is analytic because it generally develops themes by breaking them down into motivic fragments, whereas Debussy’s method is synthetic because it develops motivic fragments by combining them into larger patterns. See Lipkis, p. 10. This latter method is also similar to Baroque motivic development processes, and further shows Debussy’s affinity for Bach’s music.
 Meyer, 1989, p. 269, also p. 332. These function as motivic ideas or sometimes similar to “nuclear themes” such as used in Middle-Eastern and gamelan music.
 Eimert, pp. 9, 15. Eimert also states that “Debussy was richly able to transpose shapes” in ”other gestalt dimensions,” p. 13. It is interesting to compare Eimert’s “vegetative inexactness,” p. 3, to Thompson’s morphological transformations, pp. 1026-1095, which involve deforming, shearing, skewing, stretching, and warping of coordinate systems of natural shapes. The emphasis on contour rather than specific details facilitates the transformation of shapes into expanded or compressed versions. Schneider indicates how Thompson’s work influenced the arts during Debussy’s time. Nichols, 1972, pp. 54-55, writes about the “organic development and the assimilation of all themes into a wave pattern. This new-found predominance of shape over interval is crucial to his later style, culminating in the use of similar wave patterns in Jeux.”
 Trezise, 1994, p. 84, Wenk, p. 71. Meyer, 1973, p. 44, calls such similarity relationships conformant relationships.
 Pasler, 2004, p. 134, 1982, pp. 69-70. Gelleny, p. 26, quotes Lionel de La Laurencie on Debussy’s “monodic cells” that “incessantly germinate new forms.” Wenk, p. 121, mentions “Fleeting motives, capable of being combined in a number of ways, take the place of self-sufficient themes.” Motives are easier to manipulate and interject into the musical process than themes.
 Trezise, 1994, pp. 82, 84. Reti, p. 244, also writes of “thematic identity beneath outer variety” as a general ideal of classical-romantic music.
 This is related to Bergson’s ‘unconscious’ perception of time. See also Hedges, p. viii, Walker on “background unity,” pp. 91, 107-108, Reti, pp. 243-244, and Kramer on “nonlinear listening,” 1988, pp. 20-22.
 Reti, pp. 66-105, 194-206, Sommer, p. 72.
 See Gelleny‘s article on similar procedures in Nocturnes.
 Gelleny, p. 37, also notes that in Sirènes motives are “augmented both rhythmically and melodically, i.e., half steps become whole steps.”
 See Pomeroy, pp. 156-159. Debussy also practiced the combination of “primitive music with all the modern conveniences” as he said of Stravinsky. McFarland, 2000, p. 95, also notes that “In Jeux, the young man’s diatonic music contrasts with the girls’ chromatic music.” Parks, 1989, pp. 114-124 further analyzes chromatic and diatonic pitch materials in Jeux. Wenk, pp. 73-82 also discusses the tonal framework and advanced harmonic ideas used in Jeux, some of which were influenced by Stravinsky. In Jeux, the chromatic materials very often but not always “resolve” to diatonic material. Dissonance and consonance are thus related to the tension and relaxation, or intensification and de-intensification, of the wave process. There is generally also a ‘brighter’ sense of pentatonic and diatonic versus chromatic as part of the play of diatonic and chromatic.
 Al-Faruqi, pp. 26-27. See also Zonis, pp. 105-120, and Potter quoting Arguelles on arabesque, p. 145. Reti, pp. 11-12, also says “We call motif any musical element, be it a melodic phrase or a fragment or even only a rhythmical or dynamic feature which, by being constantly repeated and varied throughout a work or a section, assumes a role in the compositional design somewhat similar to that of a motif in the fine arts.”
 Kinariwala, p .48, Pasler, 2004, p. 137, and Trezise, 1994, p. 53 on Barraqué.
 Parks, 1989, p. 99, uses this term for change of function through a common pivoting element.
 Ruwet derived this term from from André Schaeffner. Barraqué, Benjamin, Kinariwala (p. 47), and others later refined and diversified the use of this and related terms, including free duplication and replication. Further terms include correlate, analogue, model, and different levels of reproduction of ideas. The rondo-like recurrence of an idea in early and non-western music, e.g., raga, forms an elementary form of duplication, i.e., same head with a different tail. In Jeux, the “alpha” theme is often used as head, sometimes varied, followed by different tails (e.g., diatonic, pentatonic, chromatic, or mixed), thus often signaling the beginning of a new wave, or “excursion into a tonal or durational unknown” (Al-Faruqi). See also Beck, pp. 20-21. Treize, 1994, pp. 76-86, and 2003, p. 244, mentions further motivic-thematic methods used by Debussy.
 Carpenter, p. 80, Kinariwala, p. 24, also Meyer, 1956, p. 152, on the principle of successive comparison.
 Smith, p. 27, from Debussy article “Good Friday” in La Revue blanche, 1 May 1901. See also p. 84. This also shows an emphasis on overall melodic shape or contour as an element in itself.
 Eimert, pp. 13, 9, 12, 10.
 Pomeroy, p. 159, Gelleny, pp. 28-33.
 Pasler, 1982, p. 62, notes the “passing the motive among instruments” and the timbral variety created. See also Chimenes, Parks, 1989, pp. 261-263.
 Trezise, 1994, p. 69.
 Trezise, 1994, p. 85, Eimert, p. 10. Reti, pp. 11-12, 193, 351-352, treats motives and themes as essentially the same as far as their structural function within the thematic process, and shows, pp. 193-206, how Debussy achieves “structural consistency” and “evolution” within the motivic realm in La Cathédrale engloutie.
 See also Trezise, 1994, pp. 62-63, DeVoto, 2003, p. 181, and Erickson, p. 155.
 See also DeVoto, in Tresize, 2003, pp. 180-185.
 Berman quoted by Trezise, 1994, p. 64.
 Biggs, p. 4. The ‘return effect’ involves the more subtle and partial return of motivic-thematic or other traits. The returns and ‘return effects’ relate to the above idea of the motivic-thematic particles being ‘same yet different.’
 Howat, 1983a, pp. 9, 12, from Debussy articles “About a Few Superstitions of Ours, and an Opera” in La Revue Blanche, 15 November 1901, and “The Opera and Its Connections to Music” in Gil Blas, 9 March 1903.
 Wenk, pp. 71, 121.
 Shattuck, pp. 318, 345.
 Eimert, p. 11. See also Wenk on global form, p. 71, and Hedges, p. viii.
 Pasler, 1982, p. 63, Koelreutter, pp. 45-46.
 This also forms part of the interplay of diatonic and chromatic materials. Gelleny, p. 38, refers to the similar idea of a “core of thematic material” that recurs in the Nocturnes, as well as to “thematic hybrids” and “composite themes” made from combinations of the basic thematic ideas which leads to further integration, pp. 33-34. Zonis, p. 109, refers to a similar technique used in Persian music as “centonization.”
 Examples include octave chromatic runs in seconds, upward and downward (mm. 18, 30), a half-step combined with a major or minor third (m. 86 harp, mm. 106-107 violins, mm. 264-265 oboe, m. 358 horns), a major second used with pizzicato or melodically (mm. 15-17, mm. 230-233 violins), a tritone (and other intervals) often “filled in” chromatically, a repeated note pattern (mm. 9, 315 viola). Fragments from the “alpha” and “beta” cells are shortened or lengthened, e.g., three to four consecutive notes, up to an octave. Recycled rhythmic patterns include a pizzicato accompaniment pattern (mm. 17-18 violas, mm. 182-183 cellos, mm. 212-213, mm. 635-636 bass and cellos). Characteristic hemiola patterns are also used throughout the piece, along with a few generic 3/8 rhythms (e.g., dotted-eighth, sixteenth, eighth). These rhythmic patterns are varied as independent “motifs” and help unite diatonic and chromatic elements by combining freely with these.
 DeVoto, 2004, p. 150, describes this as “where a structurally significant motive first appears unobtrusively, only to be brought to the fore at a later point.”
 For example, mm. 110‑113, 155‑156, 212‑219, 226‑229, 264‑267, 387‑395, 396‑402 (fragmentary theme followed by texture), 421‑428, 511‑514, 611‑618 (motive mixed with figuration), 615‑618 (pure figuration), and 689‑700.
 Eimert, p. 4. 85. Trezise, 1994, pp. 84-85, mentions two similar kinds of motivic types in La Mer: cyclic themes and motifs, like leitmotifs, “emblematic” in character and subject to development and recapitulation, and more diffuse motifs, malleable and undergoing metamorphoses, and “liable to end up in the delicate tracery of motivically insignificant arabesque.”
 See Reti, p. 276, on a similar technique used by Bach. See also Gelleny on similar procedures in Nocturnes. Reti, p. 205, also says of Debussy , “a theme is that shape around which a section is centered. But from the point of view of thematic development, a theme is that basic musical thought from which the further utterances of the work are derived in constant transformation and evolution, possibly leading to a resolution.” See also Reti, p. 309, on a similar method in Chopin.
 First part: mm. 1-46 (prelude) “beta”, 47-141 “alpha”, 142-223 “alpha”, 224-308 “beta”, 309-356 “beta”, 357-434 “alpha”, 435-454 (middle) “alpha”, Second part: mm. 455-514 “alpha”, 515-564 “alpha”, 565-626 “alpha”, 627-688 “beta”, 689-709 (postlude) “beta”.
 Also Kinariwala, pp. 66, 199-203.
 Trezise, 1994, p. 53. See also Pasler, 1982, p. 70. Wenk, p. 120, on the mixture of “both simple recurrence and continuous development of motives,” and Wheeldon, p. 33, on the breaking up of a pattern of “constant newness” through the “jumbling together old and new ideas.”
 Kinariwala, pp. 66-67.
 Pasler, 1982, p. 71, Boulez, p. 35, see also Eimert, p. 17.
 Eimert, p.14, Pasler, 2004, p. 137. It is interesting to note that a regular heartbeat is 72 beats per minute.
 Zuckerkandl, pp. 168-169, p. 187.
 Ibid, p. 187.
 Zuckerkandl, p. 175. In physics, momentum is defined as mass x velocity, which could lead to a musical analogy.
 This is an example of a progressive wave. There are, of course, many examples in Jeux where the tempo is steady and the wave shapes are non-progressive.
 The mosaic terms used here were derived from the most applicable terms presented by Wenk, p. 119, Evans, pp. 14-16, and Al-Faruqi, p. 19.
 Heterophony occurs in several forms, including the simultaneous presentation of the inverse or retrograde version of the same pattern, the “switching” of pitches of melodic parts, and the simultaneous presentation of an idea and its ornamentation using arabesque ornamentation principles. Heterophony also contributes to a focus on the moment. See Kinariwala, p. 92.
 Parks, 1989, p. 253 notes that pitch-class sets are coordinated with the mosaic structure, and on p. 124, that ‘subtle links’ are created through pitch-class invariance between modules.
 Wenk, p. 73.
 Beck, pp. 6-20, Kam, pp. 55-59, see also Lidov.
 Pomeroy, p. 159.
 Shattuck, pp. 337-338. This technique is similar to that of matched cutting versus jump cutting in cinematography.
 However, visual arabesque in Arab art is also appreciated spatially.
 This method also has a similarity to block-like and open-ended medieval musical forms.
 See also Evans, p. 14.
 Boulez, p. 201.
 Howat, 1983a, pp. 156-158.
 Carpenter, p. 80.
 Meyer, 1956, pp. 151-156.
 Cone, 1972, p. 156. Another term for stratification is alternative continuity, Barraqué quoted by Pasler, 1982, p. 65. A more exact term might be process interpolation, which may occur in temporary or sustained fashion as in true stratification.
 Kinariwala, pp. 117-118, says that in distinction to stratification, which is more sustained, interlock through interpolation is “an interruption pattern in which the main texture of the piece is temporarily disrupted by the interpolation of bits and pieces of some opposing idea.” She further distinguishes that “While the duplication pattern brings opposing strata into a single event, the interlock through interpolation maintains a separation of the interlocking ideas into distinct units.”
 Lipkis, pp. 7-10, Beck, p. 20.
 Eimert, p. 11.
 Trezise, 1994, p. 78, defines passive continuation as that which “occurs when the second phrase in a pair (‘unit’) fails to react to the implication of the first,” and that “repetition without development is a prime example.”
 Wenk, p. 58, Kinariwala, p. 117.
 Parks, 1989, p. 211, notes of Debussy, “The standard four-bar length is often modified by pre-, inter-, and post-extensions. Phrases also may be truncated or elided. The result is a predominantly regular musical surface for which four-bar phrases are the norm, but which are rendered flexible and unpredictable by the liberal use of extension, truncation, and elision.” See also Kramer, 1988, pp. 102-107.
 Evans notes that “The symmetry and repetition of shapes makes possible their fragmentation, and in turn, an open-ended aura.” p. 19.
 Howat, 1983b, p. 7, 1983a, p. 111, Lipkis, p. 10.
 Kramer, 1988, pp. 189-190. Wheeldon, p. 18, also emphasizes the esthetic and developmental relationship between Debussy’s mosaic structure and Stockhausen’s moment-form. Howat 1983b, pp. 19-21, says “The carefully measured (though not obvious) structural ‘blocks’... that underlie Debussy’s form-building, eventually tear off their mask and come boldly to the fore in Stravinsky’s music, more and more from the time of his acquaintance with Debussy onwards.”
 Evans, pp. 20, 24-25, 78, 23-27. See also Shattuck, p. 345.
 See also Trezise, 1994, pp. 61 and 77.
 See McFarland on mutual influences and some competition between Debussy and Stravinsky. Stravinsky’s use of mosaic structure also uses slowing of tempo of modules, gradual processes within modules, and a motivic-thematic generative process similar to that of Debussy. Cone mentions connective elements between successive blocks in Stravinsky’s use of stratification. Influences go back to Mussorgsky, who used such connections, the "mysterious thread" mentioned by Debussy. Even though Jeux shows influences of Stravinsky through sharper contrasts, and certain harmonic and orchestrational devices, these combine with Debussy’s own stylistic developments and softer and more flowing tendencies.
 Smith, p. 84, from Debussy article “The Orientation of Music” in Musica, October 1902.
 Ibid, p. 31, Smith made this comment referencing Maurice Denis.
 Eimert, pp. 3, 6, 12, 14.
 Potter, p. 13. Also in Smith, p. 74, from Debussy article “Why I Wrote Pelléas” in Gil Blas, April 1902.
 Smith, pp. 48, from Debussy article “Conversation with M. Croche” in La Revue blanche, 1 July 1901.
 Ibid, p.295, from Debussy article in SIM, 1 November 1913.
 Trezise, 1994, p. 38.
 Wenk, p. 67, from Debussy letter to Jacques Durand, 24 September 1904. See also Pasler, 2004, p. 133.
 Potter, p. 149.
 Trezise, 1994, p. 43. See also Jenkins’ dissertation.
 Pousseur has developed a more contemporary and generalized application of wave analogies to musical composition and Schillinger has developed related ideas.
 Debussy expresses himself in poetic and mystical terms, saying “Music is a mysterious mathematical process whose elements are a part of Infinity. It is allied to the movement of the waters, to the play of curves described by the changing breezes.” Smith, p. 199, from Debussy article “A Consideration of the Prix de Rome from a Musical Point of View” in Musica, May 1903. In part this quote alludes to the laws of physics. See also Potter, pp. 148-149, and Trezise, 1994, pp. 40-41.
 These particles are the smallest that Debussy had to work with. Computers have allowed the production of truly micro-sound particles (Roads). See also Gabor on the wave-particle duality at the sound level. An example of musical particles in Debussy’s music can be heard in the piano piece Jardins sous la pluie, where the fast repeated notes simulate rain drops.
 The term statistical is applied here in several ways. First, these textures move in an overall wave-like manner, thus producing statistical waves. Second, some of these statistical waves appear in the form of statistical textures composed of more independent particles and perceived in terms of their overall (emergent) qualities, which led Stockhausen, 1963, to consider Jeux a precursor of statistical-form, and Boulez, p. 201, to refer to this technique as “musical pointillisim.” Finally, the statistical waves are shaped by statistical parameters, which often coincide with what Meyer calls secondary parameters as opposed to syntactic parameters. Meyer, 1989, pp. 340-342, i.e., secondary parameters are those other than harmonic, e.g., timbre, texture, dynamics, while statistical parameters are those that move continuously rather than in discrete steps such as pitch steps or scale degrees. Meyer, 1989. p. 330, also says “Motivic unity, statistical form, and secondary parameters, then, share an important characteristic: all tend to change incrementally - in degree rather than kind.”
 Examples of “arranged” or “doubled” melodic waves are at mm. 67-69, 230-233, 276-277, 371-376, 429-434, 489-508.
 In particular, the field concept states that although the organization of the parts affects the organization and determines the structural coherence of the whole, in a perceptual sense, the parts are dependently related to the whole. In addition, the nature of the whole cannot be apprehended by examining its parts in isolation, it can only be understood when taken in its entirety. Thus, the total form quality of a gestalt perceptually dominates the qualities of the parts, just as an energy field dominates the particles within a medium. See also Zuckerkandl, p. 205.
 Eimert, pp. 18-19, 16.
 Pasler, 2004, p. 135, 1982, p. 64. Berry, 1976, p. 6, describes music generally “as dialectically in balance between intensifying and resolving tendencies, and involving complementary or counteractive relations.”
 Eimert, pp. 11, 16-17.
 Ibid, pp. 13, 10-11.
 Ibid, p. 11.
 Howat, 1983b, p. 23.
 Attack density, composite density, or successive density refers to the number of notes per time unit, e.g., per beat or second, while vertical density or simultaneous density refers to the number of simultaneously sounding notes per time unit. Other authors have proposed an intensity function of music similar to that presented here, including Brooks, p. 57, and Muns, pp. 4-5, although Muns includes the more general idea of complexity, and Meyer, 1989, p. 303, on statistical plans.
 Eimert, p.16. This comment also implies that the intensification and de-intensification processes can be quantized and occur in incremental steps.
 Pasler, 1982, p. 64.
 See Toch, Muns, Schillinger.
 See Meyer on statistical climax, 1989, p. 311.
 Muns, pp. 99-105. Meyer, 1989, p. 304, makes a distinction between a syntactic climax, corresponding to the structural point of resolution of syntactic and functional processes, and a statistical climax, and that “though often congruent with a statistical high point, a syntactic climax essentially involves a change in function. It is an action in which the tensions of instability are resolved to the relaxation of regularity. This being so, a syntactic climax can occur at a low point in the statistical/dynamic curve shaped by the secondary parameters.” Meyer, 1989, p. 204, makes a similar distinction between statistical climaxes and apotheoses.
 Smith, p. 84, Debussy article “The Orientation of Music” in Musica, October 1902, where Debussy mentions contrapuntal arabesque lines in parallel or contrary motion. The contrary motion also creates a convergence or divergence of the lines towards the climax.
 Wiegel, p. 10.
 Toch, p. 95.
 Technically, in physics, the node between waves is the point of repose or equilibrium of the medium. The musical composed-out waves described here move between zero and some positive value. There are no negative values. These musical waves thus involve full wave rectification and the node, or point of equilibrium or disconnection between waves, usually corresponds to the trough, though not every trough represents full resolution. An exception is the inverse climax function, where the functional polarity is reversed. Intensification, de-intensification, and the point of rest, equilibrium, or closure are thus relative, depending on contributing factors and context.
 See Muns, pp. 106-108, on climax moment, of which he distinguishes two types: extended, and instantaneous or point climax moments.
 Meyer, p. 93, 1989, p. 304.
 Berry, 1976, pp. 4-13.
 Pasler, 2004, p. 135.
 Schweitzer, pp. 5-6.
 Al-Faruqi, p. 21, i.e., as balance of opposites, antecedent-consequent, and thesis-antithesis.
 This can happen melodically and/or texturally and is reinforced through functional harmonic means.
 Toch, pp. 79-80, also Muns.
 Zuckerkandl, p. 172.
 Trezise, 1994, p. 40, quoting Oscar Thompson.
 Wenk, p. 71, Eimert, pp.3, 6, 11.
 Gelleny, p. 27, discusses the early impact of the “cyclic form” of Franck and d’Indy on Debussy and, pp. 38-39, how Debussy took French cyclic forms further, incorporating his “subtle and innovative approach to thematic transformation.” See also DeVoto, 2004, pp. 1-23.
 See Zuckerkandl’s, p. 175, quote above on the cumulative effect of waves, also Toch, and Muns on climax cycle.
 Meyer, 1956, p. 93, 1989, p. 304.
 All references herein are to the orchestral score of Jeux. Debussy made some additions and modifications to the original 1912 piano score.
 See Pasler, 1982, pp. 61-62, and Kinariwala, pp. 199-203 for further motivic-thematic analyses of the first wave.
 Meyer, 1956, pp. 130-135.
 Eimert, p. 10, also calls it the ‘rondo theme.’
 The layering of parts in mm. 47-66 also has elements similar to that of the introduction to The Rite of Spring, including the statistical texture, superimposition and build up of independent “strata” of melodic fragments, arabesque-like figures, chromatic modality, repeated note pattens, scale fragments, etc.
 This passage is similar to that of mm. 36-43 of Jeux de vagues. DeVoto, 2004, pp. 24-65, discusses Debussy’s recycling of ideas from previous works, or works of others, across works, what DeVoto calls “aural images.” DeVoto’s concept can also be related to cinematography, moment-form, and mosaic structure. It also reflects the experimentation and evolution of Debussy’s compositional ideas.
 This term is used descriptively and does not necessarily imply the influence of cinematographic ‘slow motion’ on Debussy in this case.
 The procedure used here is parallel to that of mm. 131-138 of Jeux de vagues. See footnote 139.
 Nichols, 1972, p. 72, calls this “the premature termination of a climax” and indicates the technique is foreshadowed in Reflets dans l’eau. See also Leichtentritt on anti-climax as a structural device, p. 239.
 Howat, 1983b, p. 23.
 Nichols, 1987, p. 289, from Debussy letter to composer/conductor Gabriel Pierné, 5 March 1914.
 The melodic pitch axis does not necessarily equal the key or harmonic center.
 Eimert, p. 10.
 Wenk, p. 81.
 Eimert, p. 10.
 Lipkis, p. 13-14, and Trezise, 1994 , p. 81, mention the increased motivic interrelatedness.
 Zenck, p. 47, also uses this notation to represent wave stages.
 Howat, 1983b, p. 11.
 Howat, 1983a, pp. 157-158, and 1983b, pp. 21-22.
 This sectional division compares to Pasler’s sectional division, 1982, p. 73, and Howat’s, 1983a, p. 157.
 Lipkis, p. 14.
 Pasler, 1982, pp. 72-73.
 Reti, p. 157, defines “thematic resolution” as “those specific transformations through which a theme in the course of its structural evolution becomes the decisive, the culminating expression into which the idea of the work finally “resolves” and adds that “we include in it the whole sphere of phenomena which produces the effect in question.”
 The second part of the piece could be interpreted to form a kind of “second development” which works together with the change from inhibited to uninhibited wave process in the piece, i.e., a second development with a different course. Meyer, 1989, p. 307, says “second developments make it possible to place the statistical climax (and in some cases the syntactic climax as well) late in the movement,” which is true of Jeux.
 Howat, 1983a, p. 155-156.
 Howat, 1983b, p. 11.
 Ibid, p. 15.
 Howat, 1983a, p. 158. References are to rehearsal numbers  at m. 455 and  at m. 677.
 Howat, 1983b, p. 19, 1983a, p. 12. This is a reference to the particle-wave duality in approaches to the piece.
 Trezise, 1994, p. 77, also p. 82.
 Ibid, pp. 61, 64.
 Evans, p. 74.
 Howat, 1983b, p. 16.
 Toch, pp. 160-161.
 Meyer, 1967, p. 309.
 Wenk, pp. 74-75. Within this diminished-seventh formation, the tritone relationships of A-D# and C-F# are important in the piece. The use of such symmetrical tonal axes is also common to Stravinsky and Bartók, and is an outgrowth and combination of traditional chromatic third relations, major-minor chords, substitute chords, and symmetrical divisions of the octave. See also Pomeroy, p. 157, and DeVoto, 2004, pp. 126-143.
 Pomeroy, p. 161.
 Wenk, p. 118.
 Pomeroy, pp. 157-158.
 Ibid, pp. 159-160.
 Wenk, pp. 118, 79, 75. See also DeVoto, 2004, Silver, McFarland, 1997.
 Meyer, 1989, p. 335.
 Ibid, pp. 310- 312.
 Meyer, 1989, p. 311, says “statistical form is itself additive. Each event - each dynamic curve - in such an emergent hierarchy consists of three phases: an intensification, a climax (high point or apotheosis), and an abatement. The dynamic curves making up the larger form generally follow one another with increasing intensity,” and that “emergent hierarchies, which give rise to statistical climaxes, are continuous. All of the constituent curves, and the largest, overall curve as well, have similar shapes and are shaped in similar ways.”
 Boulez, p. 201.
 Ibid, p. 33.
 A large-scale structure does not have to be hierarchic in order to have a shape and be comprehensible, it simply forms a different kind of structure, process, or gestalt.
 Boulez, p. 31.
 Ibid, p. 353.
 The duality of object and process begins at the note level and up to the level of waves, modules, and sections. This results in an alternation of continuous and discontinuous levels of form due to musical parameters alternating between cohesive and formative levels. A process, when closed, becomes an object on the next higher level, and determines form. See Meyer, 1973, pp. 90-91, 1989, pp. 303-304, Tenney, pp. 64-65.
 Cone, 1968, pp. 89-90.
 Parks, 1989, pp. 203, 234. Also Reti, p. 109, on the two form-building forces in music.
 Koelreutter, pp.43, 45.
 Pasler, 2004, p. 134.
 Koelreutter, pp. 45-46.
 Wheeldon, p. 55.
 Smith, p. 255, from Debussy article “Jean Philippe Rameau,” November 1912, Smith says it was for an American magazine but does not specify which.
 Nichols, 1987, p.303, from Debussy letter to Bernardo Molinari, 6 October 1915.
 Meyer, 1989, p. 269. See also Pasler on Casella and Asian music “exploiting the resources of substance,” 2004, p. 138.
 Pomeroy, p. 158.
 Meyer, 1973, p. 80. The French word for wave is vague, from the Latin vagus, meaning wandering, rambling, vacillating, vagrant, roving, and relates to nonhierarchic and static processes.
 See Kam, p.82 , also Pasler, 2004, pp. 130, 134.
 Shattuck, p. 348, quotes Donald Sutherland, from his book on Gertrude Stein, on these terms, i.e., “a prolonged present assumes a situation or a theme and dwells on it and develops it and keeps it recurring” whereas “The continuous present would take each successive moment or passage as a completely new thing essentially,” from Gertrude Stein, “Composition as Explanation,” (1926) “Continuous present is one thing and beginning again and again is another thing."
 Meyer, 1973, p. 83.
 Hedges, pp. viii, 144.
 Erickson, pp. 45-46, 139.
 Boulez, p. 353. Stravinsky also influenced and was consulted by Debussy on the orchestration in Jeux.
 Eimert, p. 16.
 Kramer, 1988, p. 201.
 Stockhausen, 1958.
 Eimert, p. 6.
 Carpenter, Kam, see footnote 30.
 See Parks, 2003, p. 198, on pacing and complexity as important compositional variables in Debussy’s music.
 See footnotes 78 and 80.
 Lipkis, p. 11.
 Eimert, pp. 14, 16.
 Kinariwala on generative process.
 Toch, pp. 195-196. These correspond to the’ extended present’ and the ‘continuous present’ mentioned above.
 Pasler, 2002, p. 133. Also Eimert’s “timbres and flowing tempi.”
 Boulez, pp. 355-356.
 Pasler, 1982, p. 63.
 See Koelreutter article.
 Wenk, p. 120.
 Ibid, p. 57.
 Also an element of symmetry, Wenk, p. 61.
 Kramer, 1988, p. 39. It is therefore important to distinguish continuity and flow from directional or linear qualities. Continuity and flow are formal qualities while linearity and directionality are functional qualities.
 Kinariwala, pp. 66-67. See quote to footnote 55.
 Cone, 1972, p. 156.
 Wenk, p. 71, Hedges, p. viii, and Kramer on nonlinear listening, see footnote 21.
 Eimert, p. 11.
 Wheeldon, p. 54.
 Ibid, pp. 14, and 49, where Wheeldon says “the repetition of musical ideas is not antithetical to moment time, if the return of previous moments is not anticipated and does not instill any formal expectations.”
 Ibid, pp. 55, 53.
 Kramer, 1988, p. 46, p. 49.
 Kramer, 1973, p. 132.
 Langer, pp. 111-113, Kramer, 1973, p. 124.
 Kramer, 1973, p. 126, p. 130.
 Ibid, 1973, pp. 123-124. See also Shattuck, pp. 333-334, Howat, 1983a, pp. 112-113.
 Kramer, 1988, p. 46.
 Kinariwala, p. 66-67. See quote to footnote 55.
 Kramer, 1988, p. 49.
 Howat, 1983a, pp. 9, 12. See footnote 41. Also Smith, pp. 54, 137.
 Eimert, p.20.
 Pasler, 1982, p. 60.