Elasticity in three compositions with flute by Boris Blacher

Transcription

1 University of Iowa Iowa Research Online Theses and Dissertations Summer 2009 Elasticity in three compositions with flute by Boris Blacher Cecilia Taher University of Iowa Copyright 2009 Cecilia Taher This dissertation is available at Iowa Research Online: Recommended Citation Taher, Cecilia. "Elasticity in three compositions with flute by Boris Blacher." DMA (Doctor of Musical Arts) thesis, University of Iowa, Follow this and additional works at: Part of the Music Commons

2 ELASTICITY IN THREE COMPOSITIONS WITH FLUTE BY BORIS BLACHER by Cecilia Taher An Abstract Of a thesis submitted in partial fulfillment of the requirements for the Doctor of Musical Arts degree in the Graduate College of The University of Iowa July 2009 Thesis Supervisor: Assistant Professor Robert C. Cook

3 1 ABSTRACT This thesis proposes a new concept of form for Boris Blacher s Divertimento for Woodwinds, Duet for Flute and Piano, and Quintet for Flute, Oboe, Violin, Viola, and Violoncello. In 1950, Blacher began to use systematically varying metric units to provide logic to the apparently arbitrary rhythm of modern music. This practice led the few scholars who have studied his compositions to concentrate on the mathematical organization of metrical units, underestimating other musical elements. This dissertation is based on the idea that it is not the mere disposition of meters, but mostly the interaction between them and other musical elements that makes the peculiar durational scheme audible, thus perceptually relevant. Following this, the technique of expansion and contraction that becomes evident in the organization of the meters is also present in the disposition of durations at other hierarchical levels and in the pitch structure. Furthermore, the mathematical metrical scheme is the foundation for a deeper universe of systematic organization. Blacher s techniques provide a unique sense of movement to his compositions, the aural effect of an elastically developing music. In Duet and Quintet, this idea is also applied in simultaneity to the registral disposition of pitches and textural development, suggesting vertical elasticity. As a result, Blacher s late compositions suggest a replacement of the traditional concepts of form and texture with a new idea of constantly moving, elastic shape. The methodological approach of this thesis is exclusively analytical and technical, with emphasis on durational and pitch organization, form, texture, and the interaction among these aspects. The individual and comparative analysis of the three compositions reveals a consistent conception of the musical space that emphasizes its bi-dimensional quality. In the Divertimento, the unitary conception of the vertical and horizontal construction is mainly reflected in the bi-dimensional treatment of interval classes that provides coherence to the motivic structure. In Duet and Quintet, this mere idea of

4 2 consistent organization of the musical materials in the horizontal and vertical dimensions of the total space becomes a unique principle of formal definition, the elastic development of the musical content itself. Abstract Approved: Thesis Supervisor Title and Department Date

5 ELASTICITY IN THREE COMPOSITIONS WITH FLUTE BY BORIS BLACHER by Cecilia Taher A thesis submitted in partial fulfillment of the requirements for the Doctor of Musical Arts degree in the Graduate College of The University of Iowa July 2009 Thesis Supervisor: Assistant Professor Robert C. Cook

6 Graduate College The University of Iowa Iowa City, Iowa CERTIFICATE OF APPROVAL D.M.A. THESIS This is to certify that the D.M.A. thesis of Cecilia Taher has been approved by the Examining Committee for the thesis requirement for the Doctor of Musical Arts degree at the July 2009 graduation. Thesis Committee: Robert C. Cook, Thesis Supervisor Nicole Biamonte Nicole M. Esposito David Gier Russell V. Lenth

7 ACKNOWLEDGMENTS I would like to express my sincerest and deepest gratitude to my supervisor, Dr. Robert C. Cook, for his wholehearted devotion, his endless patience, and the unconditional offer of his wisdom. It has been an honor to be under the guidance of a master who dedicates the entire spirit and substance of his artistry to education and scholarship. His passion and commitment to music and teaching are, indeed, the inspirational foundation of this thesis, its vital genesis. I acknowledge the members of my graduate committee, Dr. Nicole Biamonte, Prof. Nicole M. Esposito, Dr. David Gier, and Dr. Russell V. Lenth, for their time and invaluable collaboration. I am especially thankful to Dr. Biamonte, whose insightful comments and expert advice have incentivized my thinking and illuminated my ideas. I would like to extend my appreciation to the Graduate College and the School of Music, for providing various kinds of financial support during my doctoral candidacy. My studies at the University of Iowa would have never been possible without their assistance. Finally but not least, I thank my family and friends for their constant support and love. ii

8 TABLE OF CONTENTS LIST OF TABLES... iv LIST OF FIGURES...v INTRODUCTION...1 CHAPTER I DUET FOR FLUTE AND PIANO...7 An Elastic Symmetry...7 Metric Organization...9 Movement I...9 Movements II and IV...12 Metric Connections Among the Movements...16 Pitch Organization...16 The Beginning of the Piece: The Origin of M, CH, and X...24 Vertical Elasticity of M and Further Transformational Networks...28 Associative Function of M at the Largest Level: Inter-movement Motivic Connections and Transformational Networks...37 Elasticity of CH...41 Conclusions...47 CHAPTER II QUINTET FOR FLUTE, OBOE, VIOLIN, VIOLA AND VIOLONCELLO...50 A Symmetrically Framed Elastic Pitch Development...50 A Symmetric Transformational Network...51 Movement I: The Essential Components and Their Operation Within the Network...65 Movement II: The Representation of the Dual Pitch-class Space in a Scale...69 Movement III: The Completion of the Transformational Trajectory Within the Pitch-class Space...80 The Chordal Passages: Their Registral Referential Function and Their Interaction with the Motivic Structure...88 Conclusions...95 CHAPTER III DIVERTIMENTO FOR WOODWINDS, OP Elastic Phrases Within a Traditional Formal Scheme...98 Motivic Development Through the Bi-dimensional Conception of Interval Classes Movement I Movement II Conclusions CONCLUSIONS BIBLIOGRAPHY iii

9 LIST OF TABLES Table 1.1 Main Transformations of M {D,D#,E} Affinity Between Different Versions of M/I Index of Registral Expansion of M and Registral Distribution of Its Components...30 iv

10 LIST OF FIGURES Figure 1.1 Metrical Structure of Movement I Metrical Structure of Movements II and IV Transformation of M in Sub-row Comparison of mm. 2-3 with mm Possible Dispositions of the Components of M Motivic Transformations Through the Movements M-conformation of X Connection Between I4 and I10 at the Symmetry Axis Registral Distribution and Evolution of CH Through the Movements Motivic Transformation of M in the Opening Melody Transformations of Interval Class Inversional Relationship Between the Motives Motivic Transformation in the Theme of Movement III Transformational Network Spatial Distribution of the Pitch Classes in the Quintet Motivic Transformational Paths in the Openings of the Outer Movements Motivic Transformational Path in mm of Movement I Motivic Transformational Path in mm of Movement I Transformational Path in the Cello in mm Distribution of Scale Y in the Pitch-class Space Movement III, mm Registral Distribution and Evolution of CH in the Final Measures Registral Distribution and Evolution of CH Through the Movements Formal Structure...98 v

11 3.2 Metric Row Metric Row and Its Retrograde Interaction Between the Meter and the Form in Movement I First Statement of the Row in Movement II Geometric Inversion of the Metric Row Interaction Between the Meter and the Form in Movement II Formal Interaction in Variation IV Variation IV, mm Movement I, mm Movement I, mm Movement I, mm Intervallic Scheme of the Contracting Melody in mm (Movement I) Sequence of Intervals in the Melodic Fragments of mm Imitative Passage that Opens the Coda in Movement I Movement II, mm Horizontal Expansion of the Pair of Interval Classes {3,4} in Variation I (Clarinet and Bassoon in mm ) Movement II, mm Movement II, mm vi

12 1 INTRODUCTION This thesis proposes a new concept of form for Boris Blacher s Divertimento for Woodwinds, op. 38 (1951), Duet for Flute and Piano (1972), and Quintet for Flute, Oboe, Violin, Viola, and Cello (1973), three representative examples of his late chamber music. A noticeable methodical organization of metric units of different durational values that characterizes many of Boris Blacher s compositions written after 1950 has led the few scholars who studied his music to concentrate on the organization of the metrical scheme, underestimating other musical elements. This dissertation is based on the idea that it is not the mere disposition of meters, but mostly the interaction between them and other musical elements that makes the peculiar durational scheme audible, thus perceptually relevant. Following this, the technique of expansion and contraction that becomes evident in the arrangement of metric units of different length is also present in the disposition of durations at other hierarchical levels and in the pitch structure. Furthermore, the metrical scheme is the foundation for a deeper universe of systematic organization. Blacher s three compositions are presented in this thesis as practical manifestations of a conceptual change from the traditional notion of form and texture to a new idea of constantly moving, elastic shape. This broader conception of the musical structure intends to describe the evolution of the music as a process rather than its objective definition, entailing, thus, analytical aspects and levels that extend beyond the traditional study of form. The first chapter introduces the metric organization of the Duet as the foundational scheme for a musical shape that develops elastically not only in horizontal direction but also vertically. From the three pieces discussed in this dissertation, the Duet is the most illustrative example of vertical elasticity. The second chapter focuses mainly on pitch and registral aspects of the Quintet, finding technical similarities with the Duet and presenting the elastic conception as the frame of a more

13 2 profound transformational motivic network that operates in a symmetrical pitch-class space. Finally, the last chapter studies the notion of elasticity within a more traditional, neoclassical musical word, proposing that the compositional conception of the Divertimento is in many ways consistent with the later pieces. Due to the technical methodology of this thesis, it is highly recommended that the readers have access to the scores. A German composer whose career was interrupted by World War II, Boris Blacher was born in China on January 19, 1903, and died in Berlin on January 30, In 1922, he moved to Berlin, where he began to study architecture and mathematics, switching to music two years later. He was then educated in composition at the Berlin Hochschule für Musik ( ) and in musicology at Berlin University ( ). His teachers included Friedrich Ernst Koch, Schering, Blume, and von Hornbostel. Stravinsky, Milhaud, and Satie influenced his musical style at the same time that incorporated melodic elements from jazz. In the late 1940s, he moved away from tonal harmony and began to experiment with serialism with the intention to find a correspondence between the twelve-tone method and rhythmic organization. In 1950 he started composing with systematically variable meters, which consisted of the methodical disposition of varying metric units, providing logic to the apparently arbitrary rhythm of modern music. 1 The extreme expansion of tonal harmonic development that characterized the music of the late nineteenth century led composers to search for original ways of organizing musical material. In this context, Schoenberg developed a new system of pitch organization while Stravinsky reconstructed tonality from a new rhythmic, textural, and formal approach. According to Francis Burt, one of Blacher s pupils, Blacher chose 1 Josef Häusler, Blacher, Boris, in Grove Music Online, ed. L. Marcy ( ),

14 3 to displace the center of attention from harmony to rhythm. 2 Pitch and rhythm had constituted the most important parameters of music for a long time, causing many modern composers to seek musical coherence in different ways of organizing one or both of these aspects. As a result, rhythm was the focus of interest for many twentieth-century composers. For instance, intricate rhythms and complex metrical schemes are essential features of many compositions by Bartok, Webern, Stravinsky, and Messiaen among many others. 3 Music develops in time and, being the durational organization of the former the most immediate representation of the latter, rhythm and meter offered twenty-century composers a territory for innovation. Following Haüsser, Blacher s technique of variable meters reflects the influence of Stravinsky: it is, in effect, the systematization of the latter s earlier practice of alternating meter signatures. 4 Furthermore, Boris Blacher recognized himself that he derived the technique of variable meters from the Sacrificial Dance, the final movement of the Rite of Spring. 5 As a consequence of Blacher s use of variable meters, the succession of metrical units is frequently systematic. In many of his compositions, the degree of systematization of the metrical structure is strictly methodical to the point of absolute predictability. It is precisely this peculiar way of metric disposition that has led theorists and encyclopedists to concentrate almost exclusively on the systematic organization of the metric units (the meter changes and the logical distribution of the metric units) in Blacher s compositions. Very few analytical works consider his music and they mainly focus on durational aspects. Only one publication discusses in detail one of the pieces to 2 Francis Burt, The Teaching and Ideas of Boris Blacher, The Score 9 (1954): Justin London, Rhythm, in Grove Music Online, ( ), 4 Josef Häusler, Boris Blacher. 5 Christopher Grafschmidt, Variable Metrik, in Boris Blacher, : Dokumente zu Leben und Werk, compiled by Heribert Henrich (Berlin: Henschel Verlag, 1993), 49.

15 4 be studied in this thesis (Divertimento, op. 38). This study (Pape, 1968) limits the analysis to the metrical structure in a superficial relationship to the macro level of the form, mentioning the behavior of the other parameters only roughly. In addition, in his book Boris Blachers Variable Metrik und Ihre Ableitungen, Grafschmidt provides a brief analytical outline for all the works that Blacher wrote after Grafschmidt s technical remarks, which include the Duet, the Quintet, and the Divertimento, are specifically oriented to describe Blacher s use of variable meters, marking considerable differences with the approach of this dissertation. 6 Meter plays a relatively consistent organizational role in many pieces that Boris Blacher composed after As Grafschmidt points out, the technique of variable meters directly influences the delimitation of the form. 7 Furthermore, the composer himself founds his procedure of variable meters in the idea that the changes of meter intensify the formal characteristics of a musical work. 8 Nevertheless, the procedure of expansion and contraction that is evident in the arrangement of meters is also present in the disposition of the durational units of other hierarchical levels (i.e. rhythmic patterns and hypermetrical sequences) and, even more, in the pitch structure. This compositional conception of the parameters of duration and pitch provides a unique sense of movement to Blacher s compositions, the aural effect of an elastic music that seems to be expanding and contracting through time. In the Duet and the Quintet, this concept of elasticity is also applied in simultaneity, creating an elastic effect through space (growing or contracting vertical space) that is mainly accomplished by the disposition of the pitches in register and the behavior of the textural layers. As Hasty points out, expansion, 6 Christopher Grafschmidt, Boris Blachers Variable Metrik und Ihre Ableitungen: Voraussetzungen, Ausprägungen, Folgen (Frankfurt am Main: P. Lang, 1996), , , Christopher Grafschmidt, Variable Metrik, Jürgen Hunkemöller, ed., Boris Blacher: Eigenanalysen und Wekkommentare, International Journal of Musicology 8 (1999): 361.

16 5 contraction, and irregularity of phrases are characteristics of the music of many composers from different periods. 9 However, Blacher s particular way of combining the musical parameters and developing the motivic, durational, and registral structure provides personality to his compositional style. As a result, Blacher s late compositions suggest a replacement of the traditional concepts of form and texture (discrete, relatively separate horizontal units and vertical layers) with a new idea of constantly moving, elastic shape. In this sense, Blacher s compositional technique represents an original alternative to organize the musical elements. intellect. [Blacher s main] pre-occupation is of course form if one must use that term, for a piece of music is surely what it does and form, being a name for what it does, is merely another name for the piece itself. 10 Following Burt, formal conception is a central part of Blacher s compositional Divertimento, op. 38, Duet for Flute and Piano, and Quintet for Flute, Oboe, Violin, Viola, and Violoncello, present the concept of elasticity in three different, although consistent, ways. In the first two pieces, the metric structure works as the skeleton for the rhythmic and pitch organization. The systematic changes of meter and mathematical disposition of the metric units, reinforced by the rhythmic organization, cause expansion and contraction of the durational structure which, supported by the distribution of pitches, conveys a sense of horizontal elasticity. The Quintet, on the other hand, is based on a fixed metrical structure. The rhythmic patterns and organization of the durations independently from the constant meter in direct connection with pitch and texture are the main generators of the elastic effect. In addition, in the Duet and the Quintet, this new concept of elasticity obscures the delimitation of sections, making the application of 9 Christopher F. Hasty, Meter as Rhythm (New York: Grove University Press, 1997), Francis Burt, The Teaching and Ideas of Boris Blacher, 16.

17 6 traditional formal analysis practically impossible, and thus suggesting the idea of an entirely new concept of form. In the Divertimento, on the other hand, the elastic conception, which is mainly applied to the horizontal dimension and only in a lesser degree to the texture, seems to be interacting with a more traditional sectional concept of form. Finally, the idea of a unified bi-dimensional moving elastic shape is the ultimate manifestation of a formal notion that is founded in a unitary conception of the horizontal and vertical development of music, its evolution in succession and simultaneity.

18 7 CHAPTER I DUET FOR FLUTE AND PIANO An Elastic Symmetry Blacher s peculiar techniques for organizing pitch and durations create effects of horizontal and vertical elasticity in the Duet for Flute and Piano (1972). A general noticeable lack of strong articulations between phrases, mainly caused by an interrupted development of the melodic line and the texture, frequently obscures the delimitation of sections and suggests a sense of constant movement. Economy of pitch material in conjunction with Blacher s use of time and register favors the idea of a single constantly stretching or contracting object, creating an elastic shape, over the traditional concepts of successive sections and texture. In general terms, an elastic object possesses the property of being flexible in size without losing the defining features that allow us to identify it as itself. The Duet s elasticity results from the horizontal (durational) and vertical (registral) contraction and expansion of musical elements at different levels (from very small motivic cells to the large-scale scheme). Thus, the notion of elastic shape entails aspects of the musical structure that are beyond its temporal delimitation, intending to convey a comprehensive idea of how the different hierarchical levels of the music evolve in time and pitch space. In the Duet, the elastic shape is supported by symmetrical pitch and metrical organization. A symmetrical design implies similar parts reflected around an axis. Following this idea, a symmetrical shape can be obtained by organizing a succession of elements backwards (i.e. retrograding) or by rotating an object around an axis. In this sense, symmetry conveys, as the idea of elasticity, the image of a moving object. The symmetrical organization of pitch and metrical structures in the four movements that constitute the Duet functions as a framework for the entire piece, playing an essential role in the definition of the form. While movements I and III follow individual, internal symmetrical designs, movements II and IV are conceived as symmetrical counterparts of

19 8 a single larger shape. More specifically, the second halves of movements I and III are retrogrades of their first halves and movement IV is a retrograde of most of movement II. From this point of view, the even-numbered movements operate according to an intermovement symmetry, while the odd-numbered movements delineate an intramovement symmetry. In movement I, a symmetrical disposition of meters is reinforced by the disposition of pitches. An inventory of pitch classes by measure shows that, starting at m. 43, the piano part retrogrades the pitches played by the flute in the previous section of the movement (mm. 1-42). Here, the term retrograde acquires a very particular meaning: with the exception of mm , which are the exact retrograde of mm. 1-4, the operation takes place by unordered pitch-class content of each measure rather than by ordered pitch class. In other words, it is not a retrograde order of the pitches, but a retrograde order of the measures. Accordingly, the notes presented by the flute in m. 42 are the same as the ones introduced by the piano in m. 43, the notes in m. 41 of the flute are reproduced in m. 44 of the piano, and so forth. Since the operation is applied to one of the parts only (the piano retrogrades the flute as the latter introduces new pitches), the retrograde is many times combined with a motivic cell that is not present in the first half of the movement. In other words, the pitch classes stated in the flute in the second half are not necessarily the same as those introduced by the piano in the first half. For instance, in m. 44, the piano plays the same pitches as the flute has in m. 41, but the flute, on the other hand, does not play in m. 44 the same pitch classes as the piano in m. 41. In movement III, the retrograde of the pitchclass and rhythmic structure involves both instruments, becoming more evident. Measures are retrograde of mm with the flute and piano parts interchanged. It is very important to notice that in both movements, the retrograde is applied to pitch classes rather than pitches with certain registral definition. In connection with this, the distribution of the pitches in register provides variety to the retrograded halves of the

20 9 movements, having a direct impact on the overall vertical elasticity. The symmetrical shape that results from combining movements II and IV is the most obvious. Movement IV is a retrograde of mm of movement II in the most traditional sense of the term, involving durations, registrally-defined pitches, and timbre (each instrument is in charge of retrograding its own part). Finally, the organization of the metrical and pitch structures conveys an elastic conception of musical form that is supported by its symmetrical design, rather than simply framed in it. The metrical framework in connection with the organization of the musical figures directly affects the duration of the melodic gestures, creating elasticity in the horizontal sense. The successive and simultaneous distribution of pitches within the pitch-class and registral spaces and the motivic treatment conveys horizontal and vertical stretching. Metric Organization Movement I A symmetrical metric structure works as skeleton for movement I. The progression of meter changes results from methodological applications of principles of order and symmetry. The meter alternates between 4/4 and 3/8. Since the speed of the eighth note remains constant, the meter changes are not only heard as variations in length of the metric units, but also as fluctuations of tempo. Considering the tempo indication for this movement, the 3/8 meter is most likely to be perceived in one, as a single beat that is fifty percent longer than the pulse of the 4/4 bars. Thus, the units do not only contract or expand in length, but they also convey an elastic notion of the speed of the beat. The first six bars are in 4/4; a 3/8 measure follows. The next five measures are in 4/4 and are proceeded by two 3/8 bars. Continuing with this idea, the number of consecutive 4/4 measures decreases by one, while the number of 3/8 measures increases by one. This procedure continues until one 4/4 measure and five 3/8 measures have been

21 10 introduced (m. 36). Then, a symmetrical structure of eleven 3/8 measures framed by one 4/4 measure in each side is presented. This symmetrical disposition of meters, which takes place in m. 36 to 48, represents the climax of the process. Starting at m. 49, the reverse process is applied: the number of 3/8 measures decreases as the number of 4/4 measures increases. Five bars in 3/8 are followed by two bars in 4/4, then four measures in 3/8, three measures in 4/4, and so on. Finally, as shown in Figure 1.1, the metric fragment in mm works as the axis of the symmetrical metric structure on which the entire movement is based. The second half of the movement reproduces the sequence of metric units of the first half but in retrograde. In other words, the metric structure of mm is retrograded in mm Furthermore, the middle group of metric units in mm is symmetric in itself, divisible into two retrograde-related portions. The middle point is situated on the second eighth note of m. 42, one measure before the pitch axis. As a result, the metric and pitch structures of the entire movement are consistently conceived as self-contained retrogrades and the organization of the former anticipates, by one eighth note, the development of the latter. Following this idea and consistently with the serial techniques, we can think of the first 35 measures as a metric row that is retrograded in mm ; the two related forms of the row are separated by the symmetrical fragment of mm It is important to point out here that the symmetrical properties of this segment would justify the delimitation of the metric row on the middle eighth note of m. 42, in which case the original form and its retrograde would be contiguous. Nevertheless, the general musical characteristics of the passage, particularly the pitch and rhythmic patterns, along with the lack of meter change (an essential factor in the delimitation of all the other groups of metric units) do not support an articulation at this point. As a direct consequence of the distribution of the 4/4 and 3/8 meters, the groups of metric units within the row and its retrograde are relatively independent from each

22 11 other, supporting their interpretation as partitions or subsets and their technical definition as sub-rows. The systematic increase or decrease in the number of measures of each meter signature provides the logic for the segmentation of the row. The order of the elements of the row, an essential aspect of its traditional serial conception, becomes the definition of the row itself, suggesting the way of its development. In effect, while the term row here implies a fixed, ordered collection of metrical units that are different from each other, the term sub-row implies a flexible succession of metrical units that is shortened or lengthened in each of its statements. From this point of view, the sub-rows are variations in length of a unique object, different degrees of elastic strain of a single shape: the same succession of meters, 4/4-3/8 or 3/8-4/4, is contracted or expanded in each of its statements. Figure 1.1 shows the metrical structure of the movement and its symmetrical disposition, the row and its retrograde. Each bar represents a metrical subrow. Figure 1.1 Metrical Structure of Movement I

23 12 The progressive process of addition and subtraction of metric units suggests the idea of a single metric entity subject to contraction and expansion rather than a succession of individual metric sequences, supporting the concept of a unique, elastically conceived metric row. 11 During the first half of the movement, the row contracts five eighth notes in each of its sub-rows, symmetrically expanding in the second half. Finally, the resulting disposition of metric units supports the symmetrical form of the movement, with the pitch class axis located in the middle of the central metric sub-row (sub-row 6). Movements II and IV The metrical retrograde of movement II that takes place in movement IV has a significant perceptional corollary. Due to the fact that most measures have attacks on the first beat and do not incorporate rests at the end, the retrograded version of the metrical sequence (i.e. movement IV with respect to mm of movement II) can be clearly perceived. The only exception to this occurs in m. 16 of movement IV, where the 3/4 bar is not attacked on the downbeat. However, the transposition of the right hand of the piano in m. 15, the only register change in the whole retrograde, contributes to outline the 5/8-3/4 metric sequence. Since the retrograding process in movement IV is rhythmically exact, not only the sequence of meters but also their internal grouping is reversed. In other words, the subdivision of asymmetrical meters is retrograded. For instance, the 3+2 in m. 8 of movement II becomes 2+3 in m. 15 of movement IV. The symmetrical disposition of groupings in the only two consecutive measures of equal asymmetrical 11 Pape defines the term row as an ordering of the meters of a composition, generally employed repetitively (Louis W. Pape, Aspects of Meter in Four Selected Compositions of Boris Blacher [MM Thesis, Indiana University, 1969], 25). Following this, he conceives sequences of meters that are based on similar metric patterns as different forms of a single metric row, independently of their degree of variation in length. According to his use of the term through his thesis, the row of this movement would correspond to what it is here considered the first sub-row. The retrograding process applied to the metric structure of the first half of the movement supports the conception of the sub-rows as partitions of a single larger metric entity that I call the row. The analysis of the metric structure of the Divertimento in Chapter 3 supports the conception of the sub-rows as subsets of a larger, indivisible metric unit, providing further justification for the terminology adapted in this thesis.

24 13 meter (two measures of 5/8 in mm of movement II), , makes possible its invariable replication exactly at the middle of the retrograded version (mm in movement IV). Not coincidentally, this point marks the most important phrasal articulation of movement IV, suggesting two balanced 11-measure moments. The sudden absence of the flute in m. 12 along with relative contrast in the register and melodic gestures suggests a formal articulation at this point. Notice, however, that the prolongation of the piano chord over the bar line (insinuated by the unfinished slurs) suggests continuation between m. 11 and m. 12, obscuring the sectional division and contributing to the perception of an uninterrupted form. Figure 1.2 illustrates the metrical structure of movements II and IV. The predominant meter in both movements is 6/8 (3/4). 12 As illustrated in Figure 1.2, meter changes are framed by 6/8 or 3/4. In this sense, the 6/8 meter plays a referential function in the metric structure of both entire movements. As shown, all the meter changes in mm of movement II and mm of movement IV occur symmetrically around 6/8 (3/4). In other words, meter behaves symmetrically between the 6/8 (or 3/4) units. The division of the whole metrical structure in smaller groups of metric units which beginnings and ends are defined by the 6/8 or 3/4 measures results in individual symmetrical patterns. For instance, the first three measures of movement II delineate 3/4 (indicated as 6/8 in the graph), 2/4 (shown as 4/8), and back to 3/4, represented with the triangle outlined by the first three points in Figure 1.2A. The only repetition of the 6/8 meter, which takes place exactly in the middle of movement II (measures 14-15), marks a noticeable change in the overall metrical structure. The second predominant meter is 2/4 (which is durationally equivalent to 4/8) for the first section of movement II and 8/8 for the second part. 4/8 has two beats fewer than 6/8 (represented by two horizontal lines in the graph) and 8/8 has two beats more 12 Even though 6/8 and 3/4 are different metric structures, they have equivalent durations. The meters are equal at the largest grouping level (one strong beat every six eighth notes).

25 14 than 6/8. As shown in Figure 1.2A, the second predominant meter symmetrically moves from 4/8 (located two points below 6/8 in the graph) to 8/8 (two points above 6/8). That is to say, the metric structure symmetrically changes four beats (from 4 to 8) around the axis of 6/8. In a similar way, the shortest metrical unit in the first half is 3/8 (three points below 6/8) and the largest metrical unit in the last part is 9/8 (three points above 6/8). Figure 1.2 Metrical Structure of Movements II and IV A. Metrical Structure of Movement II B. Metrical Structure of Movement IV

26 15 The fact that the last movement retrogrades mm of the second movement has an important effect on the perception of the metrical structure. In movement IV, the first section of the metrical structure equally emphasizes the meters 9/8, 8/8, 7/8 and 6/8 with a total of two measures per meter. The second part exactly replicates the first section of movement II. During the entire piece, the motivic construction and the melodic design contribute to the perception of the sequence of meters. In general, musical gestures coincide with bar lines or groups of measures, favoring the retrograding process, which operates by measure or group of measures rather than by individual contiguous pitch classes ( in blocks rather than lineally). Furthermore, a single motivic unit is often restated and subject to expansion or contraction in subsequent measures, as in mm The metric changes are indeed perceivable. All the meter changes result from a methodic increase or decrease of one eighth note, except for the changes 2/4-3/4 and 3/4-2/4. The 2/4-3/4-2/4 change doubles the relationship to a quarter note. It is important to consider that increasing and decreasing successions of meters have opposite perceptual effects. Increasing meters are perceived as enlarging units (decreasing the frequency of accented events), creating horizontal expansion and a deceleration of the hyper-tempo, 13 while the later creates an acceleration of the hyper-tempo or horizontal contraction. Therefore, at the metrical level (distance between downbeats), progressive acceleration and deceleration of perceived speeds takes place, causing horizontal contraction and expansion. In movements I, II, and IV, the disposition of the metric units creates horizontal elasticity, directly affecting our perception of the way the music unfolds through time. In movement I, the effect is the progressive contraction or expansion of a single concrete object that is subsequently stated: the metric sub-row. Since the sub-row is relatively large even in its most reduced version and it contains only one metrical change (vs. 13 The changes occur at a large durational level. The speed of beats remains constant while the frequency of accented beats changes.

27 16 successions of equal metric units), the horizontal elasticity mainly affects the largest levels of the form. In movements II and IV, the result of the organization of meters conveys the elastic property of time perception itself: the sequence of almost continuously changing meters has the effect of enlargement or diminution of more local durational spans (i.e. single metric units, or measures) that, due to their relatively ephemeral character and the lack of systematic principles of organization at larger levels, cannot be defined as variations of a single object. In other words, the groups that result from associating these durational spans are not related to each other strongly enough to be interpreted as modifications of a definite entity. Metric Connections Among the Movements As mentioned above, the metric structure of movements II and IV is clearly centered on the meters of 3/4 and 6/8, which represent the meter of 3/8 in number of beats per measure (three) and the internal subdivision of the beat (triple) respectively. In movement I, the meter of 3/8, which is nothing but an equivalent version of 3/4 (both meters are simple triple) defines the metric sub-rows: it represents the first metric change that we hear and marks the end of the first sub-row. Movement III is entirely in 6/8. Finally, the metric structure of movements II and IV highlights both the meters of 3/4 and 6/8 as two versions of a single temporal unit, bringing together the very metric definition of movements I (simple triple meter that defines the metric sub-rows) and III (compound simple). Pitch Organization Blacher s procedure of elasticizing musical material becomes evident in certain passages that are constructed by progressively adding pitches to successive statements of a single melodic figure (additive pitch structure). Following this idea, certain melodic

28 17 designs emerge progressively, through consecutive steps, from a small initial gesture. 14 For instance, in movement I, the figure <E-F-A-D-G-G#> 15 in m. 34 is generated by progressively adding notes to the gesture <E-F> in four consecutive steps (one per measure). This idea of expansion of repeated common elements, illustrated at the local level by the previous example, is, in different degrees of explicitness, the main generator of the horizontal elasticity at all levels. It is relevant to point out here that the shared elements, which are pitches in the given example, are often pitch classes. The procedure is consistent with the technique that delineates the sequences of metric units previously described. In movement I, the metric sub-rows are generated by addition or subtraction of measures (metric units), which is the mere expansion or contraction of the common meters 3/4 and 4/4. In a similar way, the single durational unit of 6/8 or 3/4 functions as the invariable value and shared boundary between the successive symmetrical metric groups (durational units that expand and contract around 6/8 or 3/4) in movements II and IV. Finally, the elasticity conveyed by the organization of the pitch and metric structures results from an additive or subtractive procedure based on maintaining common elements, either pitch classes or meters. From this point of view, a unique conceptual notion of elasticity is common to the development of pitch and meter. Similarity and constancy have provided coherence to art works of all times. It is common knowledge that human memory tends to group events according to their similarity in order to remember them. In music, repetition is the maximum possibility of equivalence. Operations such as transposition, inversion, retrograde, and rotation, also 14 Grafschmidt uses the terms Gleitend Wachsende Linien (sliding growing lines) to refer to this type of melodic construction, providing examples in many of his analytical remarks (Christopher Grafschmidt, Boris Blachers Variable Metrik und Ihre Ableitungen, 422). 15 Angle brackets indicate a sequence of elements (i.e. pitch classes or interval classes) in the order that appears in the music. Braces denote a group of elements, independently of the order in which they appear in the music: pitch classes in their most compact ascending arrangement (similar to normal order in set theory terms, but with letters instead of numbers) or interval classes that increase in number. For instance, the form in braces for the motivic figure presented here would be {D,E,F,G,G#,A}. This form represents all the possible ways of combining and arranging these pitch classes.

29 18 imply strong association, since they are variations in the order of the information and not in the quality of its content. 16 Following this idea, the symmetrical correspondence between movements II and IV is probably the most immediate perceptual aspect of the piece for most listeners. The extremely short length of movements II, III, and IV (1:38, 1:05, and 1:14 respectively, according to the recording by Orchester-Akademie des Berliner Philharmonischen Orchesters) 17 in addition to the contrasting characteristics between movements II-IV and movement III, especially in terms of tempo (Andante vs. Presto) and metrical organization, facilitates aural memory, contributing to the connection of the evennumbered movements. In other words, the small quantity of information to be remembered, the short time elapsed between its introduction and its repetition (retrograde in this case), and the clear divergence of the information heard during this time favor the listener s memory. The fact that the last movement retrogrades movement II intuitively suggests a connection among the last three movements (II, III, and IV). In effect, the movements are closely related from the point of view of motivic content and pitch organization. A semitone <A,G#> in the flute, the maximum possibility of registral reduction, the most registrally compressed way of organizing two different pitch classes, opens movement III. A symmetrical (0134) tetrachord compressed in a single register, {C# 4,D 4,E 4,F 4 } in the piano, separates the initial motivic cell (01) in the flute from its variation in mm The variation is a registral and chromatic expansion of the introductory semitone: the 16 According to Christopher Hasty, a structure is defined by the homogeneity of its own domain (associational character of its components) and by the difference of the latter with respect to the domain of other objects. The transposed and the inverted forms of a motive preserve its sequence of interval classes. The retrograded form is, similarly, the sequence of intervals of the original motive in reverse order. Finally, the rotated version preserves the pitch classes and the intervallic content (although not in order). As a consequence, all the forms are perceived as mutations of a single object or structure (Christopher Hasty, Segmentation and Process in Post-Tonal Music, Music Theory Spectrum 3 [1981]: 54-73). 17 Boris Blacher, Boris Blacher Kammermusik, Orchester-Akademie des Berliner Philharmonischen Orchesters (Hannover: Thorofon, 1994).

30 19 initial <A 4,G# 4 > becomes <A 6,G# 5,G 4 >. From this point of view, the figure is vertically (registrally) and horizontally (melodically) elasticized. Notice, in addition, that the process of melodic expansion in the flute is pitch-class symmetrical around G#. This three-note chromatic cell (012), which is here generated as an elastic semitone from {G#,A}, is repeatedly used throughout the entire piece, acquiring different shapes. Its consistent use as an independent figure, its transformational treatment, and the large number of times that it is presented, support the idea of conceiving it as a motive. Its central role in the organization of the pitch structure of the piece, a direct consequence of its consistent use, makes it the main motive (M). As will be discussed later, M in the form of <D-D#-E> is the opening figure of the entire piece. The chord that accompanies M in movement III, mm. 1-3, constitutes the other important element for pitch and registral organization. Marking a difference with M, the identifying characteristic of the chordal structure is not its intervallic content (0134), but its pitch content {C# 4,D 4,E 4,F 4 }. This collection of four notes is subject to pitch-class addition and subtraction that constantly change its intervallic content and total pitch-class definition, questioning its existence as a unique independent unit. In other words, {C#,D,E,F} is only one of the many forms in which this chordal structure (CH) appears. In this particular occasion, C# has been added to a more basic structure {D,E,F}. From this point of view, CH does not indicate a strictly defined set (like M), but a family of sonorities or collections derived from a basic collection of pitches {D 4,E 4,F 4 } that is registrally defined. 18 A chordal structure that contains any of these three notes is a form of CH. Other important defining features of CH are its textural disposition (mostly as a chord) and its consistent lack of two contiguous semitones [CH (012)]. Therefore, CH is considerably different from M. M 18 As will be discussed later, there are only a few exceptions in which CH does not contain D, E or F in register 4. However, the context in which these cases appear suggests a registral change of the same group CH rather than material of different nature.

31 20 and CH are not only the principal motivic configurations of the work but also the generators of the elastic motion. It is important to point out here that CH and M do not literally generate motion: my interpretation of their use articulates the idea of elasticity. Having presented the two most important elements, their uninterrupted presence from the end of movement II to the beginning of movement III becomes evident. The extra measures of movement II (mm ), those that are not duplicated in movement IV, present {G,G#,A} and {D,E,F} in a way that suggests prolongation into one another. Here, prolongation means addition of the components of one pitch structure (either M or CH) to the other one (CH or M). Measure 23 transposes <B-A-G#> into <G-F-E> (T8) and the subsequent measure inverts the latter into <D-E-F> (I E F ). Consequently, the entire passage inverts <B-A-G#> into <D-E-F> (I F# G ) and the three motivic gestures belong to the same set class (013). The disposition of these notes in pitch-space, as descending ninth supports this interpretation of the motivic transformation. The consecutive statements of two different versions of the basic form of CH in m. 23 (i.e. {G#,A,B} or I C C# of {D,E,F} and {E,F,G} or I E F of {D,E,F}) traces M: in effect, <B-A-G#-G-F-E> in m. 23, I C C# {D,E,F} I E F {D,E,F}, includes {G,G#,A}. Even more, the presence of {G,G#,A} in m. 23 is highlighted by the registral disposition of its components ( as opposed to the stepwise motion that dominates the passage). Following this, a motivic figure that contains elements from both M and CH, <B-A-G#-G-F-E> in m. 23, becomes the basic form of CH, {D,E,F} in m. 24. Finally, the <F-E> at the end of the motivic figure in m. 23 (immediately following <A-G#-G>) followed by <D-E-F> in the subsequent measure suggests the prolongation of {G,G#,A} into {D,E,F}. In other words, {G,G#,A} is motivically connected to {D,E,F} through {E,F}, the common tones between m. 23 and m. 24. In m. 24 {D 4, E 4,F 4 }, a form of CH, appears in a melodic rather than chordal setting, thus connected to M from a textural point of view. In a similar way, <A-G#> becomes part of a chordal structure. The chords in mm are {G#,A} {D,E,F}.

32 21 However, the rhythmic disposition of the chord in the last three measures clearly separates {G#,A,D} from {E,F}. {G#,A,D} appears then as a relatively independent chord that has A and G# as outer pitches, anticipating the opening of movement III. Finally, the last measure of movement II presents <A-D-G#> melodically, emphasizing even more the intervallic relationship between A and G# (melodic rather than simultaneous interval in this case) and taking {G#,A} back to their original, horizontal textural distribution. The melodic presentation of <A-D-G#> in the last measure is the culminating point of the process that separates {D,E,F,G#,A} into {G#,A,D} and {E,F} in mm In the original disposition of the chord in mm. 23 and 25, the E is attacked an eighth note later than the other pitches, appearing as a substitute for the F (which is simultaneously attacked with the other pitches). Measure 26 makes this separation even more evident by matching the duration of the F to the rest of the chord (in this measure the F is as long as {G#,A,D}) and placing the E one quarter note after the attack of the chord, immediately after the end of the chord. In the next measure, the separation of the F from {G#,A,D} by one eighth note provides independence to the latter. In m. 28, the duration of the F is lengthened by half of its value, enlarging its temporal distance to the E. Finally, the last measure temporally separates the components of {G#,A,D}, omitting the E and the F. The technique employed is uniform: enlarging the horizontal space between events (i.e. chordal structures or individual pitches). As an aural effect, the components of the initial {D,E,F,G#,A} in mm. 23 and 25 are gradually stretched horizontally in each of the four statements of the structure in mm Paradoxically, this rhythmic expansion is accompanied by a progressive compression of the duration of the successive metric units (7/8, 6/8, 5/8, and 4/8 as shown in Figure 1.2B) that directly shortens the distance between the successive statements of the structure, separated by half-note rest, eighthnote rest, and no rest at the end of mm. 26, 27, and 28 respectively. In terms of time perception, this process of metric compression and the procedure of rhythmic expansion

33 22 previously explained have opposite perceptual effects: as the repetitions of the motive are placed closer to each other (horizontal contraction), the internal components of {D,E,F,G#,A} are separated (i.e. the structure itself is rhythmically expanded). Finally, the repeated statements of {D,E,F,G#,A} at the end of movement II smooth the connection with movement III by means of common tones: these five pitches, along with the C# and later the G, are in charge of opening the Presto. The addition of the C# and the G to the chord and to the melodic cell <A-G#> respectively suggests pitch expansion, vertical in the former case and horizontal in the latter. From this point of view, movement II expands into movement III. The motivic and pitch connection between movements II and III goes further beyond the fragment discussed above. Different versions of M and CH appear throughout the movements, consequently extending the relationships to movement IV: if mm of movement II is related to movement III, then the latter is connected to movement IV. Furthermore, the versions that open movement III, {G,G#,A} and {C#,D,E,F}, are clearly present in mm and 1 of movement II (mm and 14 of movement IV). As a result, movements II and III open with the same version of CH that closes the piece. Versions of M on pitch classes B and C (i.e. original, inverted or rotated forms of {B,C,C#} or {C,C#,D}) appear as independent motivic configurations in mm. 2, 12, 13, and 19 of movement II (and the corresponding mm. of the last movement). As a result, M is stated on G, B, and C. These are precisely the versions of M that are used in movement III. In addition, M is also embedded in larger pitch-class sets that suggest a pitch extension of M into CH, like the one above explained (m. 23). The result of this melodic extension is a collection of pitches that is consistently stated in a scalar manner (stepwise motion, up and/or down) throughout the movements. The total pitch-class collection is X={B,C,C#,D,E,F,G,G#,A}, a chromatic scale that excludes the Eb minor triad. X contains M on G, B, and C and, at least, the forms of

34 23 CH that have been discussed up to this point ({C#,D,E,F} and {D,E,F,G#,A}). It is important to distinguish here between the total pitch collection of the piece, which is the chromatic aggregate, and the most frequently employed selection of notes, which is X. X constitutes the array for the majority of the scalar passages stated after m. 12 in movement I as well as the complete pitch-class domain for movements II, III, and IV. In movement II, X is motivically divided into the tetrachord {D,E,F,G} (m. 8 and m. 10) and the pentachord {A,B,C,C#,D} (mm ) until G# connects both scalar fragments in mm Since movement IV is a retrograde, G# appears at the opening of the passage; thus, X is present from the beginning. This disposition of X highlights D, A, and G#, precisely the three last notes of movement II and the common tones with the opening of movement III. In movement III, X is clearly derived from the opening M and CH. That is to say, the descending scale <G#-G-F-E-D-C#-C-B-A-G#> in mm. 7-9 is the extension of {G,G#,A} into {C#,D,E,F}. During this movement, X is always presented with the same rhythmic contour (eighth notes in compound meter), facilitating its immediate aural identification. The successive statements of X occupy time spans of different durations: four beats in mm. 7-9, three beats in mm , seven beats in mm , nine beats in mm , etc. Consequently, X acquires different lengths through the movements, suggesting horizontal elastic development. In summary, three elements play a decisive role in the organization of the pitch structure of the movements explained above. (i) M=(012), specifically M G, M B, or M C (ii) CH {D 4,E 4,F 4 } and CH (012) (iii) X={B,C,C#,D,E,F,G,G#,A} In general terms, X can be understood as a prolongation of M into CH or vice versa, as follows: X={M G,M B,M C } {D,E,F}

35 24 From this point of view, X is not only a pitch class-collection, but also the operation that combines M and CH. Since the transformation is an addition, its result is nothing but a pitch-class expansion, contributing to the idea of elasticity in terms of successive pitch classes (horizontal stretching). The Beginning of the Piece: The Origin of M, CH, and X The most complex of all movements from a motivic point of view, the first movement presents M in all possible levels of transpositions and inversions. Notice that, since the motive is perfectly symmetric, inversion and retrograde are equivalent and, since M=(012), T n =I n The original version of M is <D-Eb-E>, stated by the solo flute in m. 1; paradoxically, one of the less frequently employed forms of M. Since {D,Eb,E} is the initial form of M, it is logical to define it as M0. The most frequently employed versions of M above mentioned, those embedded in X (M on G, B, and C) are, therefore, M5, M9, and M10 respectively and their inversional equivalents, I11, I3, and I4. As it will be shown in the course of this analysis, M9/I3 plays an essential function framing the entire work. Table 1.1 shows the main operations of M. Table 1.1 Main Transformations of M {D,D#,E} M5 M9 M10 {G,G#,A} {B,C,C#} {C,C#,D} I11 I3 I4 19 All the inversions are labeled around the axis of C. For instance, {Ab,A,Bb} is I(C) {D,D#,E}. Therefore, if {D,D#,E}, the original motive is T0, then {Ab,A,Bb} is I0, because it is the inversion of T0 about C. The axis C is here adopted for mere practical reasons, since it is the most standard axis for labeling inversions and transpositions that do not have contextual connotations. A contextual inversion would not add relevant information to this analysis.

36 25 The initial version of M, {D,Eb,E}, is apparently unrelated to X. However, M0 is pitch-class equivalent to I D of M10, which means the inversion of M10 about its common pitch with M0 and with the initial basic component (i.e. first pitch class of the normal order) of CH. Even more important, during metric sub-row 1, mm. 1-7, M0 (m. 1) is smoothly transformed into M9 (m. 7), one of the most frequently employed versions of M. This transformation is accomplished by means of common tones between the successive versions of M. In effect, the technique of transpositional or inversional invariance is used throughout the entire piece to provide coherence to the motivic organization and the general formal structure. It is therefore significant to explain at this point the commontone relationships between the different forms of M. Table 1.2 shows these affinities organized by number of pitches in common; n indicates the level of transposition or inversion. Following this, n±6 in the first column of the second row indicates that Mn shares three common pitch classes with In±6. For instance, M0 and I6 share all three pitches {D,D#,E}. Table 1.2 Affinity Between Different Versions of M/I Between equal forms of M (M with M or I with I) Between different forms of M (M with I or I with M) 3 common tones 2 common tones 1 common tone n±0 (n=n) n±1 n±2 n±6 n±5 n±4 The common-tone transformation of M0 into M9 that takes place in the first subrow is accomplished through I0 (mm. 5-6) and it is represented in Figure 1.3. The number of arrowheads indicates the number of common tones. Measure numbers are indicated in parentheses.

37 26 Figure 1.3 Transformation of M in Sub-row 1 D works as a common tone between M0 and I4, and {C,C#} between I4 and M9. The emergence of I3 as an extension of I4 in m. 5 highlights the common tones between these two operations. I3 becomes its pitch-class equivalent M9 in m. 7. As a result, the motivic content of the first section is smoothly transformed from M0 to M9 through common tones. Since the versions share common pitches, it is the addition of extra semitone(s) that produces the succession of transformations. In light of this, the internal intervallic structure of the motive recursively provides the pattern for its own transformation. The elaboration of the pitch structure conveys horizontal elasticity from a perceptual point of view, in the sense that a three-note chromatic motive is chromatically transported (i.e. horizontally expanded because of the chromatic nature of both the motive and the process of transformation) from one pitch level to another one (from 0 or D to 9 or B). Once again, the variable elements produce expansion only in connection with the common (invariable) elements. From a perceptual point of view, the musical pitches are added or subtracted rather than changed, providing a sense of gradual motivic transformation that favors continuity over formal contrast.

38 27 The formation of CH and X also takes place during this section. The initial transformation of M0 into I4 is accompanied by a melodic non-chromatic extension of I4 in m. 2. As a result, the total pitch content of this measure is {C,C#,D,E,F,G}. Notice that the incorporation of G as the culminating point of the extension of M10 at the end of m. 2 smoothes the statement of I10 in m. 3, functioning as common tone. Nevertheless, it is important to remember that, as indicated in Figure 1.3, this I10 (i.e. a version of M) is not connected by means of common tones to the forms of M that surround it (I10 does not share common pitches with M10 or I4; the G above mentioned serves as a common tone between I10 and the extension of M10, but not between I10 and M10 itself). As will be explained later, this introduction of I10 anticipates its central role at the symmetry axis where its connection with I4/M10 becomes explicit. In m. 4, {C# 2,E 4,F 4,G 4 }, a form of CH, is the first chord of the piece. From this point of view, the non-chromatic melodic extension of M in m. 2 anticipates the formation of the first version of CH in m. 4. Notice that this original form of CH is simultaneous to D 5 in the flute, which could be interpreted as the anticipation of the D 4 that constitutes most of the forms of CH later in the piece (and therefore a part of CH in this instance); nevertheless, this pitch is registrally and timbrally associated with the first D that the flute plays in the subsequent measure (which clearly prepares the restatement of I4 of M). The next chord, {E 2,F 4,G 3,Bb 3 }, takes place in m. 8. Beyond the registral transposition of E 3, this structure is still a form of CH and it is clearly connected to the previous version of CH. Following the idea of transformation by common tone, the first two chordal structures share the pitch classes {E,F,G}. In addition, pitch class Bb occupies the same position with respect to pitch class G (top pitch class of the set in the most compact ascending arrangement of its elements) that pitch class C# occupies with respect to pitch class E (the bottom pitch class). From this point of view, Bb in m. 8 is the mirror image of C# in m. 2. {C#,E,F,G} and {E,F,G,Bb} are not, of course, literal pitch inversions of each other. The registral change of E anticipates the vertical motion

39 28 of some components of CH that takes place throughout the work at the same time that the registral immobility of F establishes a pitch-space point of reference that is essential during the movement. In this respect, the vertical elasticity in relationship to CH (the registral movement of some of its basic components) will be studied in depth later. I2, another version of M that is not a literal subset of X, appears in the flute in m. 9, while CH in the form of {E 2,F 4,G 3,Bb 3 } is still in the piano. In another instance of invariance, I2 intersects with the last version of M that we heard (M9 in m. 7): {B,C} are the common tones. In addition, the substitution of C# by Bb in CH connects I2 (M) to X by means of the common tone Bb. Measures 9-10 chromatically expand I2 in two consecutive steps, transforming it firstly into I10 (i.e. the version of M in m. 3) and secondly into I6 (=M0 in m. 1). The omission of F# in m. 11 and then Bb in m. 12 comprises the first complete statement of X. As a consequence, X is created by means of expansion and contraction from versions of M and CH that are not contained in X. Nevertheless, as a result of the explanation above, its connection with M and CH is evident. Vertical Elasticity of M and Further Transformational Networks The logical sequence of transformations of M goes beyond the fragments above illustrated, playing an essential connecting function throughout the entire composition and a crucial role in the definition of its formal structure. Since M consists simply of three consecutive semitones, its level of independence (i.e. our ability to hear it as an independent cell) varies throughout the piece. For instance, as mentioned above, many times M appears as a part of X. Its rhythmic contour and registral disposition (either within itself or in respect to its surroundings) contributes to its delimitation as an independent unit. In relationship to this, the three notes of M appear in different registral positions throughout the piece, having a direct impact on our perception of both the motive as an

40 29 independent cell and its registral treatment. Certain registral dispositions of M highlight its presence and suggest vertical elasticity. A comparison of mm. 2-3 with its symmetric counterpart in mm of movement III clearly illustrates this idea. The version of M involved in these passages is I11/M5. The importance of I11 as a connecting element has been suggested at the beginning of the pitch analysis and will be expanded upon later. As shown in Figure 1.4, in mm. 2-3, I11 is presented in the form <A 6 -A 6 -A 6 -G# 5 -G# 5 -G# 5 - G 4 -G 4 -G 4 >. The large registral distance among the three components of M is clearly emphasized at the end of the movement, in the mirror passage (mm ), where the last of the three repetitions of each component (pitch class) is presented in a different register: <G 3 -G 3 -G 4 -G# 4 -G# 4 -G# 5 -A 5 -A 4 -A 4 >. Figure 1.4 Comparison of mm. 2-3 with mm The three components of M can be arranged in three different ways, as shown in Figure 1.5. Each empty circle represents one of the pitches of M and numbers indicate interval classes. Table 1.3 shows the different versions of M employed in the entire

41 30 piece, in terms of the registral setting of its three components. The total level of expansion measures the distance in number of semitones between the lowest and highest pitches of M. The +/- symbols indicate the direction of the interval. Numbers show quantity of semitones per interval. Figure 1.5 Possible Dispositions of the Components of M Table 1.3 Index of Registral Expansion of M and Registral Distribution of Its Components Total level of expansion: Form of M 2 semitones 1-1 <+1,+1> <-1,-1> <-11,+1> <-11,+13> <+11,-13> <-13,+11> <+1,+13> <-1,-13> <-13,-1> <-11,-11> <+11,+11> <+13,+13> <+1,+25> <-13,-13> <-1,-25> 1-2 <-11,+10> <-1,-10> <-11,-2> <-59,+58> 2-1 <+10,+1> <-14,+13> <+46,-11> Note: The forms of M are measured in unordered interval classes while the level of expansion is measured in ordered intervals.

42 31 The main motivic transformations of M in the four movements are represented in Figure 1.6. While all the statements of M are indicated in Figures 1.6B and 1.6C, Figure 1.6A shows only the transformations of M that are relevant from the point of view of location and registral distribution of its three pitch classes, including those appearing at the beginning and end of sub-rows as well as at the beginning, middle (axis), and end of the movement. Since different versions of M are introduced almost constantly in movement I, a graphic reduction represents the most significant transformations in a more understandable and accessible way. Bold letters show registrally expanded versions of M. The subscript indicates the total level of registral expansion according to the values shown in Table 1.3. The absence of subscript indicates the minimum level of registral expansion (i.e. 2). Measure numbers are indicated in parentheses. Slashes show chromatic expansions of M. For instance, I4-3 means <D,C#,C,B>. Square brackets indicate that an occurrence of M is embedded in a larger pitch collection (usually X) or subject to non-chromatic expansion. Movement I has been organized by metric sub-row for methodological purposes. Movements II and IV as well as the two halves of movement III have been overlapped (Figures 1.6B and 1.6C respectively) in order to show their almost perfect symmetric equivalence. Thus, the motivic analysis for movement IV and the second half of movement III must be read from right to left. While in movements II, III, and IV M is narrowly defined in terms of transformational pitch levels and registral setting, the first movement presents the most diversity in this respect. As mentioned above, the last three movements state M on G, B, and C exclusively, only in transposed or inverted forms (no rotations). The level of registral expansion of M in movements II and IV is 2, 13, or 14. In movement III, M appears again on G, B, and C exclusively, in transposed or inverted forms as well, but with a larger level of registral expansion: either 2 or 26.

43 32 As shown in Table 1.3, the registral distribution of M results, in most cases, from combining interval 1, its 8ve-complement 11, or its 8ve-compound 13 (equivalently 2, its complement 10, and its compound 14 in the case of rotations). Figure 1.6 Motivic Transformations Through the Movements A. Motivic Transformations in Movement I Subrows Beginning of the sub-row End of the sub-row 1 M0(m. 1) [I4 M10](2) I10 26 (m. 3) I4-3(m. 5-6) M9(m. 7) 2 I2 I2-10 I2-6 [I2-11](9-11) M9 22 (13) 3 M10 M10 14 I4(21) M11 26 M (15-16) 4 [M9-10](22-23) 5 [I4-3](29) [I11](35) 6 (middle) I4 26 (37-38) RI9 59 (39-40fl/pn)-I1 I10 26 (42) M4 26 (43) I9 I3 14 M9(47) 7 I (48-50) [M5](49) [I4,3,11] [M M5 13 ](55) 8 [M5,9,10](62) 9 [M5,9,10] (63) I3 14 M9 13 (69) 10 M5 26 (70 fl) M RI9 11 -I1(74 total chromatic) [M6-8](75) [M5](76) I3 22 (71 pn) 11 M7(78 fl) M9(77 pn) M10(78) RII9 46 (78-9) I11 M5(79) M10(80) [I11](82-83 fl) [I4 M9](82 pn) I6(82 pn) axis B. Motivic Transformations in Movement II Movement I3 (m. 2) M9 (21) I11 14 (9) M5 13 (14) [I10-8] (11) [M2-4] (12) I4 14 (12) M10 14 (11) [M9-10] (13) [I4-3] (10) [M9 I3] (15-16) [M9 I3] (7-8) [M9-10 I4-3] (17) [M9-10 I4-3] (6) [M9] (18) [I3] (5) I4 (19) M10 (4) [I11] (21) [M5] (2) Movement IV [I11 13 ] (23,25) II C. Motivic Transformations in Movement III Movement III: first half (m.1) (2-3) (8-9) (10) (15-16) (20-22) (24-27) (m. 32) I11 inc. I11 26 [I4-3] I3 13 inc. [M5,9,10] [I4,3,11] [M5,9,10] [I4,3,11] I3 26 M5 26 inc. M5 26 M9 13 inc. [M5,9,10] [I4,3,11] [I4-3] [M9-10] [M5 11 ] [M5] [M5,9,10] I4-3 First note in pn, then fl (63) (61-62) (56-58) (49-50) (45) (44) (43) (39-41) (38-39) I10 inc. M9 26 (37) (m.33) second half

44 33 The four versions that do not belong to this group are stated at structural points in movement I. The largest level of registral expansion of M (59) represents the registral climax (the maximum distance between two simultaneous pitches) in the entire piece. It happens in mm , right before the symmetry axis. Delineated by the C 7 in the flute, an extreme note in the instrument and the highest climax in the piece, and C# 2 in the piano, a pitch only surpassed by B 1 in the third movement, this moment of registral expansion is aurally unmissable. The next expansion level (46) occurs towards the end of the movement, at the beginning of the last metric sub-row (mm ). These two statements of M are the first and second rotational forms of M9 (i.e. M on B) respectively. Finally, the <-1,-25> registral distribution of M is introduced as a clear anticipation of the registral climax (mm ) and it is an I4 form (i.e. M on C). Similarly, <+1,+25> (the ascending version) takes place in m. 70, at the beginning of the penultimate metric sub-row, preparing the appearance of level of expansion 46 at the opening of the final sub-row. Notice that the largest levels of expansion occur on versions of M on G, B, and C (or M5, M9, and M10), directly highlighting them form an aural point of view. Furthermore, all rotational forms of M are presented on G, B, or C. Supporting the idea of vertical stretching towards the middle of the movement, the registral expansion of the three notes of the motive is considerably larger towards the symmetry axis (<-13,-13> in m. 42 and <+13,+13> in m. 43). Not coincidentally, I3 with a registral distribution <-13,-13> and its retrograde (M9 in the form <+13,+13>) shape the symmetry axis in movement III (mm ). Even more, I11 <-13,-13> is the first complete appearance of the motive in that movement. In movement II, I11 and I4, both in the form <-13,-1>, appear in m. 9 and m. 12 respectively. I3, M9 (extended), and I4 are expressed as registrally contiguous semitones in mm. 2, 13, and 19. The retrogrades of these appear in movement IV, in which m. 11 (corresponding to m. 12 of movement II) marks the end of the first perfected balanced half and m. 21 (corresponding to m. 2) is

45 34 the penultimate measure. Consequently, the registral disposition of the motive links movements II, III, and IV: the three movements form a symmetrical structure in terms of the registral disposition of M, with the largest stretching occurring in the middle movement. In a broad sense, this is consistent with the registral disposition of M in the initial movement. However, the final registral expansion of M in this movement causes an additional vertical expansion towards the end. The forms of M at the axes of the symmetric movements are, then, M4/I10 in movement I and I3/M9 in movement III. In addition, M10 marks the most important formal articulation in movement IV, exactly at the middle (movement IV is divided into two equal 11-measure sections). 20 As a result, all the forms of M that are embedded in X play an essential role in defining midpoints except for M5 (or I11) 21 at the same time that the presence of M4/I10 at the axis of the first movement seems to be arbitrary and unrelated to X. Once again, reminding us of the absence of common tones between I10 and the surrounding versions of M (M10 and I4) in mm. 2-4, I10 is the version of M that seems to be out of context. However, the lack of connection of M4/I10 with respect to X is only apparent: M4 is the tacit symmetric counterpart of M10 (with regards to Bb or E) in X as well as the version of M that transforms X into a symmetric scale about C C# or F# G. As shown in Figure 1.7, X is not only an asymmetric pitch collection but also an unbalanced scale in terms of its internal distribution of the subset (012), since it contains an even number of versions of M. M5 is the mirror image of M9 with respect to the axis E or Bb. The missing reflection of M10 with respect to this axis is precisely M4. In addition, M4 contains F#, the pitch that transforms X in a symmetric scale with respect to the axis C C# or F# G. Finally, M5 and M10 are the inversions about E or Bb of M9 and 20 The form of movement II does not imply any kind of axis or midpoint. 21 Notice that I11 (=M5) is also the version of M that is used in the extra measures of movement II, linking it to movement III.

46 35 M4 respectively; in mathematical terms, M4 is to M5 as M9 is to M10 within the domain of X and with respect to a symmetry axis E or Bb: M9 : M5 :: M10 : M4 The version of M used at the axis of movement I is therefore related to the forms of M employed in the axes of movements III and IV. Figure 1.7 M-conformation of X This theoretical justification is musically supported by the transformational networks of M in the initial and middle metric sub-rows. As demonstrated earlier in this chapter, I10 (=M4) appears as an unconnected version of M in the initial sub-row (m. 3). This I10 is not related to the versions of M surrounding it in terms of common tones, nor in terms of registral disposition (it is the only registrally expanded version of M in the first 12 measures). At the same time, its presence is highlighted by means of its unique registral distribution <-25,-1>, the entrance of the piano, and the sudden increase in the

47 36 dynamic level. The subsequent version of M is I4-3 in m It is important to remember at this point that I4 and I3 are pitch-class equivalent to M10 and M9. In addition, M10 appears right before I10 in m. 2. The transformations that take place in the middle group of metric units (mm ) connect these two versions of M (I4 and I10). Reproduced in Figure 1.8, this fragment opens with I4 (m ), which shares the common tone C with the immediately presented I2 (m ). The latter is literally an I2-0, since it is extended by chromatic step (registrally contiguous) to G#. This last G# in m. 42 becomes the first note of I10, which is clearly isolated by means of its contextually different and large registral disposition. The common tone between the two first statements of M in this central sub-row is precisely the highest pitch in the entire work (reached again only in m.74). Figure 1.8 Connection Between I4 and I10 at the Symmetry Axis 22 The first presentation of M in m. 5 is <D-C#-C-B>, i.e. I4 extended to I3. The dash in the notation indicates this extension, representing the four notes as a unit rather than two versions of M intersecting with each other.

48 37 In light of this, a chromatic descent transforms the C (the common tone between I4 and I2) into G# (i.e. I2 is extended to I0), the initial pitch of I10 (and the only common tone between I0 and I10). In this way, I4 (=M10) and its original form I4-3 (=M10-9) are explicitly connected to I10 (=M4) in the symmetry axis. The descending line, the highest uninterrupted disposition of intersecting versions of M in the entire piece, is aurally striking. Associative Function of M at the Largest Level: Intermovement Motivic Connections and Transformational Networks Through transformations of M that take place throughout the entire piece, Movement I anticipates to a certain degree the connection among the movements as well as their order. As shown in Figure 1.6A, M plays an essential role in the delimitation of the metric sub-rows: it is stated at the beginning and ends of all sub-rows except for the opening of sub-row 8 and the end of sub-row 4, reinforcing the motivic connection between the sub-rows and supporting the idea of a continuous rather than sectional form. It is important to notice in this respect that, even though M is not explicitly stated as a complete (012) at the beginning of sub-row 8, it is tacitly present in the first three measures of the sub-row (flute part). The second measure of the sub-row (m. 57) adds C# and F to the <D-E> presented in the previous measure, extending the latter figure to <C#-D-E-F> (notice that the intervallic sequence of this extension is symmetric). Even more, the M5 that the piano plays at the end of sub-row 7 is extended to the beginning of sub-row 8: the last A from sub-row 7 is followed by {G,G#} in the first measure of subrow 8, delineating RII5. The third and fourth measures of the sub-row (mm ) introduce C. As a result, the first pitch class of each of the first three measures of subrow 8 outlines I4. The presence of the six main operations of M from Table 1.1 is evident. As illustrated in Figure 1.6A, M9 plays an important role in defining the metric row through

49 38 the sub-rows of the first half of the movement. I4 and I11 become predominant towards the middle and M5 and M9 play a structural function towards the end. From this point of view, I4 and I11 work as connectors between M9 and M5 at a large scale. M9 and I4 share two common tones and I11=M5 (i.e. they share three common pitch classes). Even though a voice-leading analysis at this level of the form is objectionable because the versions of M are separated by relatively large spans of music, the recurrence of M9, I4, I11, and M5 at the middle ground makes the common-tone connections perceptible to a certain degree. The common pitch class C# links the end of sub-row 2 with M11 at the beginning of sub-row 3 (piano). Presented below M10 (which shares {C,C#} with the last version of M in sub-row 2), M11 is highlighted by the registral distribution of its components <+13,+13> involving C# 2 (one of the lowest notes in the piece and the lowest pitch that marks the climax of vertical expansion). M11 is, together with I10 discussed above, the only version of M that is not contained in X but still obviously emphasized by its registral expansion. M11 in m. 15 anticipates I3 at the end of sub-row 9. These two instances of the motive, which share the first note (i.e. they are inversions about their common tone), are symmetrically distributed within the movement and registrally emphasized through their large levels of registral expansion within their contexts. Even more, C# 2, the first note of M11, reappears right before I3 in m. 69, highlighting the common tone between these two versions of M and suggesting a larger level of registral expansion for I3 (36 instead of 14). 23 As a result, M11 (not contained in X) in the first half of the movement is related to I3 (embedded in X) in the second half. In addition, I3 here is clearly emphasized by a very peculiar symmetrical disposition: it is immediately retrograded in 23 Even though these versions of M are largely separated in time, the C#2 is clearly perceptible throughout the movement due to its extremely low registral disposition. Although the motivic figure starts on beat 3 (C#5), the proximity in time and pitch-class space of the C#2 supports the association of this note with the actual motivic figure (which has a level of registral expansion of 14).

50 39 register and pitch. 24 This symmetrical setting makes evident the relationship between I3 and M9. In addition, I3 in m. 69 is followed by a larger version of itself two measures later. These characteristics along with its relevant location within the form (the end of the sub-row) highlight the I3 operation that, not coincidentally, plays an essential connecting function among movements I, II, and III. It is relevant at this point to discuss the other versions of M that are not literal subsets of X that appear in structural points of the form: I2 at the beginning (sub-row 2) and I9-8 at the end of sub-row 6 and the beginning of sub-row 7. Even though these two instances of M are not highlighted in terms of register, a short explanation illuminates their roles within the transformational network. The importance of I2 as a connecting element and generator of X has already been discussed. The chromatic extension of I9 to I8 in mm is the culminating point of a progressive melodic extension that starts in m. 46. Parting from the two common tones between M4 (at the axis) and I9 and adding a descending semitone per measure (<G-F#> in m. 46, <G-F#-F> in m. 47, and finally <G- F#-F-E> in mm ), a technique of motivic development anticipated in mm , the presence of I9 in this passage clearly emphasizes the G, common tone between M4 (version of M at the axis) and M5 (first version of M in sub-row 7, since I9-8 actually starts at the end of sub-row 6 the first note is in the last measure of sub-row 6). Not coincidentally, M4 and M5 are related by inversion about their common pitch class G. As shown in Figure 1.6A, I3, clearly emphasized in movement I, is the first version of M that appears in movement II (m. 2). The first three forms of M stated in this movement are I3, I11, and I4, which result in M10, M5, and M9 at the end of movement IV. The consequence is a clear motivic independence of the M forms contained in X and their consecutive organization. Despite the relative temporal distance between them, their identical pitch-class structure and the registral disposition of I11 and I4 contributes 24 The first C# is replaced by C at the end, but this does not alter the components of the motive nor the perception of its symmetrical setting.

51 40 to the perception of this progression at the middle ground. In addition, the use of X during the remainder of these two movements makes the presence of these operations evident. It is important to point out here that, even though I10-8 and M2-4 are stated between I11 and I4 and M10 and M5 respectively, they appear as chromatic scales rather than independent motives. Since they are chromatically extended versions of I10 and M4, they have five notes each, being only distantly related to the original three-note motive. Nevertheless, the emphasis on G# and F# is evident in these passages they are the first notes of I10 and M4 respectively. This statement of I10 and M4 is consistent with their position at the axis of movement I. The bold vertical line in Figure 1.6B represents the middle of movement IV, which is located between m. 11 and m. 12. The metric, rhythmic, and motivic organization suggests a formal articulation at this point in both movements (same measure numbers), more precisely, the main formal delimitation in movement IV. In connection with this, I10 appears right after the symmetry axis in m. 37 of movement III. Even though this version of I10 is lacking the F#, it is clearly emphasized by means of the flutter tongue, an immediately perceptible change of timbre. The final measures of movement II that are excluded in the retrograded movement (IV) play an essential connecting function with movement III. The linking function of the extra measures of movement II has already been explained. In movement III, in addition to the almost continuous presence of the motive within scale X, an incomplete version (the first two notes) of I2 and the complete form of I3 are registrally highlighted in m. 10 and the symmetry axis respectively. The registral disposition of I3 recalls, to certain degree, M11 at the beginning of sub-row 3 in movement I (which is associated with I3 at the end of sub-row 9). In addition, I2 marks the opening of sub-row 2 in movement I. It has already been mentioned that I3 and M9 mark the symmetry axis in movement III, with the same registral distribution as I10/M4 at the axis of the initial movement. From a registral point of view, both axes are equal.

52 41 The retrograde of the pitch structure in movement III leads us to M5, connecting with the beginning of the last movement. Finally, the last movement returns to the central operation of movement I, M9, closing the network of motivic transformations to provide coherence to the entire work. Elasticity of CH The common-tone voice leading principle that transforms M throughout the entire work is also applied to CH. Since CH is less clearly defined in terms of its pitch content (its cardinality and intervallic context are modified through the piece), the common tones among its successive forms are essential to its very definition. It is precisely the continuation of common pitches that provides unity to the different versions of CH from a perceptional point of view. In most cases, more than one common note is kept in the same registral position. In addition, as a general rule, the absolute lack of registral continuation between the components of successive versions of CH (i.e. each note of CH is successively presented in different registers) is compensated for by a larger number of notes in common. These factors in addition to the almost continuous presence of either D 4, E 4, or/and F 4, its general chordal texture, and the lack of the set class (012) gives coherence to CH as perceptually independent from M. Figure 1.9 shows the registral distribution of CH throughout the movements. Movements II and IV contain the only case of unconnected (in register and pitch) successive versions of CH, deserving a short explanation at this point. As represented in Figure 1.9B, {B 3,C 4,D 4 } in mm. 6 and 16 of movements II and IV respectively is apparently unconnected in terms of register and pitch class to the subsequent or preceding{f 5,G 5,G# 5,A 5 }. However, the melodic figure in the piano that appears in between {D 4,E 4,F 4,G 4 } provides the common pitch classes F and D between these two versions of CH. Despite its horizontal presentation, this figure is obviously related to CH in terms of pitch content.

53 42 Figure 1.9 Registral Distribution and Evolution of CH Through the Movements A. Registral Distribution and Evolution of CH in Movement I

54 43 Figure 1.9 continued B. Registral Distribution and Evolution of CH in Movements II and IV C. Registral Distribution and Evolution of CH in Movement III

55 44 In effect, this motive contains the three basic elements that are used as connectors between the successive versions of CH in their central and most used registral position. Following this idea, it is logical to consider this instance of CH as the result of a horizontal expansion. Two different aspects directly convey vertical elasticity in CH: (1) registral movement (i.e. transposition by one or more octaves) of one or more of its components between successive versions and (2) addition or subtraction of pitch classes (condensation or decompression of the total pitch collection). Notice that, since CH is a chordal structure, addition or subtraction of pitches generates vertical elasticity. From this point of view, the same process that generates horizontal stretching of the melodic structure creates vertical elasticity of the chordal structure. This marks a significant difference with the elastic treatment of M: while adding or subtracting pitches to or from M affects its melodic development, the same technique applied to CH has harmonic implications. (1) Elasticity by Registral Movement of the Components of CH and the Primordial Space Function of CH In addition to providing internal coherence to CH, the almost mandatory presence of D 4, E 4, or/and F 4 in all its versions gives a spatial reference to the listener. As shown in Figure 1.9, E 4, F 4, and A 4 in movement I, D 4 in movements II and IV, and F 4 in movement III play an essential role in defining the registral reference for each movement as well as connecting the different versions of CH at the middle and large scales (i.e. associating all the versions of CH in each movement beyond the common-tone relationships between consecutive versions). Not coincidentally, these three pitches are the central ones for the very definition of CH (i.e. its basic form ). In the first movement, E 4 is transposed to E 3 at the beginning, immediately returning to its original registral position in m. 22. This registral movement, shown by the arrow in Figure 1.9A, is accompanied by a symmetrical movement of F at the end of

56 45 the same movement. In m , E and F move in opposite directions around the registral axis {E 4,F 4 }. In movement II, on the other hand, {E 4,F 4 } moves to {E 2,F 3 } without coming back to its original register, as illustrated in Figure 1.9B. It is important to point out here that, even though these two versions of {E,F} are widely separated from each other (the former one is at the beginning of the movement and the latter one, at the end), the recurrence of {D 4,E 4,F 4 } or its expanded version {D 4,E 4,F 4,G 4 } suggests continuation of the initial {E 4,F 4 } until the end at a lower scale. The symmetrical counterpart of the registral movement of {E 4,F 4 }, the return to its original register, takes place in effect at the end of movement IV, providing coherence to the entire composition in this respect. Other relevant registral movements of CH occur in mm and 51 of movement I, greatly contributing to the sense of vertical stretching that characterizes the middle of the movement (both registral changes happen in the middle sub-row, before and after the symmetry axis). In these cases, the entire structure CH (each of its components), enclosed in rectangles in Figure 1.9A, is transposed to a different register. The registral change of CH in m is accompanied with an extreme registral movement of C from register 2 to 7, highlighting the two extreme pitches of the movement. Even though C 2 (m. 36) and C 7 (m. 39) are not simultaneous, the incorporation of C# 2 in m. 39 clearly prolongs the register 2 from C in m. 36, delineating the vertical climax of movement I. The registral movement of CH in m. 51 surrounds the central pitches {E 4,F 4 }. In this case, the lower three notes of the first chord are moved two octaves up, therefore conserving their registral order. The D 4, on the other hand, is transposed one octave only. As a consequence, the inversion of the chord about D 4 is transposed up an octave. This registral movement anticipates the transpositions of E and F in the subsequent measures as explained above.

57 46 (2) Elasticity by Pitch-class Addition and Subtraction In addition to the addition and subtraction of pitches between successive versions of CH, clearly shown in Figure 1.9, the relative preponderance of certain pitch classes different from {D,E,F} in each movement illuminates the shift of the registral center among successive movements. The registral centers are determined by the pitches that are consistently repeated in each movement and they have already been suggested at the beginning of the previous section: Movement I: {E 4,F 4,A 4 } Movement II and IV: D 4 Movement III: {E 4, F 4 } In the first movement, CH expands to become a D Dorian hexachord in m. 45 and a D Dorian pentachord at the end. As a result, CH acquires the form {D,E,F,G,A,B} in m. 45. C#, emphasized by its low disposition, and Bb, introduced at the beginning as a clear substitute of C# (this point has been explained at the beginning of the pitch analysis) play the essential role of chromatically compressing CH during the first movement (i.e. they add semitones to its intervallic content). In this sense, C# and Bb are chromatic alterations in CH. In other words, these pitches are the only ones that do not pertain to the collection D Dorian. In movement II, G# becomes the chromatic alteration at the same time that D 4 is established as the new registral center. G# is precisely the inversion of C# about {E,F} as well as the inversion of Bb about A. In this sense, the form CH and registral center in movement I imply the version of CH and registral disposition in movement II. Similarly, D# and G# are the chromatic alterations added to the diatonic form of CH in movement III. Originally introduced as a part of the CH form {D#,F,G,G#} in m. 12, the simultaneous interval {D# 4,G# 4 } is later highlighted by its registral treatment with respect to the rest of the pitches in its original set. As represented in Figure 1.9C, in m , {D#,G#} is obviously separated from {F,G} as the latter is transposed up an

58 47 octave. Finally, in m. 53 {D#,G#} is also transferred to a higher range, resulting in the transposition of the entire set {D#,F,G,G#} to register 5. Following with the connections between successive movements, D#, the chromatic incorporation in movement III (G# is added in movement II) is the inversion of C# about D. Once again, the inversion happens about the pitch that represents the central register in the movement. Even more, D# is also the inversion of G# about {B,C}. {B,C} is obviously emphasized at the axis by the presence of I3/M9, immediately before {D#,G#} becomes relatively independent from CH. Finally, in movement IV, the frequent chromatic alteration of CH is G#, the inversion of C# about the registral center of movement III ({E,F}). The chromatic additions of CH in each movement are created by inversion about the registral center of the previous movement. The inverted pitch class is always C#, a note that is obviously emphasized in movement I, for example the recurrence of C# 2 during the first movement is evident due to its extremely low register. In summary, the registral reference smoothly moves among successive movements, contributing once again to their logical order. In addition, this interpretation of the most important and consistently used pitch components of CH supports its coherence and independence. Conclusions The elastic form of the Duet essentially emerges from the expansion and contraction of the durational and pitch structures at different hierarchical levels vertically and horizontally. The sense of elastic motion is created by adding and subtracting elements from a more or less definite unit (i.e. metric sub-row, durational span, motivic gesture, chordal structure, collection of pitches, or individual pitch classes working as common tones). In other words, the sense of elasticity is conveyed by means of expansion and contraction of an element or group of elements that is always perceptible

59 48 and identifiable as a constant entity. In this way, the idea of elasticity suggests a change in size that does not affect the essential nature of the elasticized object. The addition and subtraction of elements takes place in the vertical and horizontal dimensions. Incorporation and elimination of pitch classes produces elasticity in both directions, depending on their temporal disposition with respect to the fixed element(s) (successive or simultaneous). Addition and subtraction of durational units (i.e. metric units or simple beats) creates elasticity in the temporal dimension (horizontal). Finally, increase or decrease of the registral index of the element(s) clearly conveys vertical stretching. The same compositional procedure that gives the piece its elastic form is locally applied to construct the essential components of the work during the initial measures. The principal pitch-class levels of M (G,B, and C) are derived by expansion by means of common tones. The initial form of CH (m. 4) originates from the vertical disposition of the first melodic expansion of M (m. 2). Finally, X results, both theoretically and in the course of the music, from the addition of M plus CH, representing the essence of the Duet (its fundamental motivic content) in its most defined level of expansion, a concrete collection of pitch classes. The metaphor of elasticity, especially in connection with motivic processes, can aid in preparation for effective expressive performance of the Duet. A brief discussion of mm. 1-7 of movement I with respect to mm will support this idea. The motivic connections between these passages are evident. As discussed in this chapter, the registrally and motivically isolated I10 of m. 3 is linked to I4 in mm Both passages present I4 and M(B) (M9 in one case and RI9 in the other) as two not only consecutive but also clearly connected (by common tones) versions of M. While at the beginning of the movement, I4 and M(B) appear as registrally compressed versions of M stated in contiguous registral positions (i.e. both the internal distribution of M and the disposition of the successive versions of M are registrally contiguous), in mm , I4

60 49 and RI9 are vertically stretched at both the note level (internal distribution of the components of M) and the motive level (position of I4 with respect to RI9). This registral elasticity plays an essential role in the definition of the climax at the same time that it prepares the beginning of the retrograding process. The flute part in mm is not a mere descending chromatic line but the extension of the version of M that precedes it (I4) and the connection between the latter and I10 (m. 42). The C# in the piano in m. 39 plays the role of vertically elasticizing M(B) and it is then, from a motivic point of view, an essential component of the flute line. The C 7 in m. 39 marks not only the highest point in the piece but, even more importantly, its registral climax and the maximum level of vertical expansion of M. The motivic integration of the flute and the piano is essential to understand the motivic coherence and the registral evolution of the movement, its elastic development. Balance of volume and consistency in the articulation between the instruments are crucial performance aspects to be considered in passages of these characteristics. In a similar way, the registral stretching of I10 in m. 3 is a result of the interaction of both instruments. In m. 42, I10 is an exact replicate, from a registral point of view, of its introduction in m. 3. The timbral integration of the components of M in m. 42 makes its statement more obvious. Furthermore, the timbral division of I4 between flute and piano in m. 3 prepares the instrumental change that occurs in mm as the flute part is transferred to the piano to be retrograded. The speed acceleration that I4 acquires at the symmetry axis as M4, immediately following the axis, is presented in sixteenth notes and I4, preceding the axis, is stated in eighth notes, evokes its durational setting in m. 3. As a result, the rhythmic and registral organization of these instances of the motive directly contributes to their perceptual association. Finally, the isolated vertical stretching of M in m. 3 is an anticipation of the unmistakable elastic effect that takes place in mm , in preparation for the symmetry axis. The consideration of this kind of analytical aspects should have a positive impact in the performance of the Duet.

61 50 CHAPTER II QUINTET FOR FLUTE, OBOE, VIOLIN, VIOLA, AND VIOLONCELLO A Symmetrically Framed Elastic Pitch Development The Quintet for flute, oboe, violin, viola, and cello (1973/74) is organized in three movements that can only be completely understood as a continuous whole. Based on a 3/4-4/4-3/4 metric organization, the symmetrical pitch and rhythmic design of the work is supported by clear motivic development. While the middle movement is a retrograde in itself, with the symmetry axis located between mm. 60 and 61, the main melody of the initial movement (mm of the violin) is literally inverted in the final movement to become the theme (mm. 1-17) to be varied. In addition, the variations of movement III follow a symmetrical scheme. Variation 4 retrogrades variation 1 and variation 6 retrogrades variation 2. Variations 3 and 5 present evident motivic similarities without suggesting any kind of systematic relationship. The symmetric design in movement II operates in blocks, by motivic cells rather than individual pitches, and involves only certain motives that are contiguous in time but not in instrumentation. This technique of retrograding, which is consistent with the procedure employed in movements I and III of the Duet, allows motivic, pitch, and rhythmic organization to progress rather than emphasize static balance. The symmetric design is supported by the general motivic organization. As will be discussed later, the creation, development, and transformation of the motivic material follow a balanced trajectory within a symmetrically arranged pitchclass space. In the Quintet, the elasticity is mainly created by presenting intervals in their empty and chromatically-filled versions. For instance, in m. 53 of movement II, the cello presents the interval <D-G>, which is immediately filled in the oboe: <D-D#-E-F- F#-G> in mm Following the same procedure, the <A-D> in the viola in mm is chromatically expanded to <A-Bb-B-C-C#-D> in the flute in the following measure.

62 51 As a result of this technique in connection with the general disposition of the pitches and motives, the chromatic density increases and/or decreases in each movement, suggesting chromatic expansion and/or contraction at different levels of the pitch organization. A Symmetric Transformational Network The piece opens with the cello s melodic presentation of the most frequently used version of the main motive, <D-C#-C>, and closes with its only chordal disposition in the work. We name this motive along with all the possible arrangements of its pitch classes M(D), in which M=(012) and D is the first element of the descending (and original) arrangement of the notes in M. 25 The first and shortest movement presents not only the most important motivic information, but also the basic traces of a symmetric transformational network that organizes the piece and its principles of motivic development. The opening M(D) in the cello is the beginning of a melodic line that dominates the first part of the initial movement (mm. 1-18). This section can be subdivided into two smaller parts, mm. 1-8 and mm The melody in the cello is created by extension in a way that becomes the channel of motivic transformation, transporting M(D), {C,C#,D} 25 Marking a difference with the analysis of the Duet, this terminology will be adopted for the entire chapter, without making reference to numbered levels of transpositions or inversions. In this chapter, all the versions of M are consistently labeled according to the last pitch class of the most compressed ascending order of the three pitch classes that form M. The use of the last pitch class rather than the first one is consistent with the first appearance of M, which does not represent the most common arrangement of its components. In other words, the motivic material of the Quintet is an unordered set (012). Following this, M(D) includes all the possible ways of ordering {C,C#,D}, M(E) represents all the forms of {D,D#,E}, and so forth. This change and simplification of the terminology with respect to the previous chapter favors the analysis of the Quintet. In this piece, M is conceived as a group of three pitch classes which order is not as relevant as in the Duet. The designation of the different versions of the motive according to the pitch classes that contain rather than their transpositional/inversional position with respect to the original version facilitates their identification for the reader (since it is not necessary to translate numbers to pitch classes) at the same time that allows to unify pitch-class equivalent transpositions and inversions (an inherent characteristic of symmetric sets like M) under a single label. Furthermore, this unification is essential to understand the motivic connections in relationship to the organization of the pitch-class space in which they operate. The terminology adopted for each work is, I believe, the one that conveys my verbal interpretation of them in the most comprehensible, contextual, and practical way.

63 52 to M(A), {G,G#,A}. The initial M(D) is expanded melodically, becoming {C#,D,E,F} in mm. 3-4 and {B,C,C#,D} in mm The cello line in mm. 3-4 emphasizes the minor third {D,F} as opposed to the semitone relationship between F and E and D and C#: <F-D> constitutes the only nonchromatic motion and it is thus perceptually highlighted. In addition, the resulting pitch set, {C#,D,E,F} is characterized by a symmetrical intervallic structure, (0134), 26 that clearly highlights the tone distance between D and E. Following the idea of extension by successive presentations of pitch-class sets related by common tone(s), the same principle of motivic organization in the Duet, {C#,D,E,F} in mm. 3-4 shares {C#,D} with the initial cell or M(D). Measures 5-7 extend the original cell to {B,C,C#,D}, having {C#,D} as common pitches with the two previous figures (mm. 1-2 and mm. 3-4). We hear <D-E-F> as melodically consecutive pitches for the first time in m.8. The original arrangement of these pitches is <F-E-F-D> in mm. 3-4, which, as mentioned, reinforces the semitone {E,F}, leaving a pitch space between F and D. In light of this, the motivic gesture <D-E-F> in m. 8 diatonically fills the original interval {D,F} from m. 4 at the same time that it reasserts the tone distance between D and E as opposed to the semitone between E and F. In other words, the apparently separated {E,F} and D, particularly the interval <F-D>, from m. 3-4 are diatonically connected in m. 8. The connection results from a simple reorganization of the pitches presented in mm From this point of view, the <F-D> is retrograded and filled in to become <D- E-F> in m. 8. As a result, the original <D-C#-C> is connected to {E,F} through D (implicitly in mm. 3-4, where {D,E,F} is not contiguously arranged, and explicitly in m. 8, where {D,E,F} follows its most compressed successive order) at the same time that is chromatically extended to {B,C,C#,D} (mm. 5-7). 26 The symmetric design results from the most compressed ascending arrangement of the pitch classes: <C#-D-E-F> delineates the sequence of interval classes <1-2-1>, or, what it is the same, (0134) is a symmetric pitch-class set.

64 53 The diatonic continuation of <D-E-F> from m. 8 leads to the beginning of the second section of the cello melody, which starts with <G-A> in mm The second section of the melody (mm. 9-18) consists of four successive variations of the set {G,G#,A,D}. While the first two statements, <G-A-G#-G-D> in mm and <A-G#- A-G-D> in mm , are different only in terms of order, the last two are subject to extension. Measures add the pitches E and F, extending the set to {D,E,F,G,G#,A}. As a result, the <D-E-F> that links the first phrase with the second one (m. 8) and the characteristic set of the latter are combined into a single motivic unit: {G,G#,A,D} {D,E,F}={D,E,F,G,G#,A}. Finally, mm extend the original set to <A-G#-G-D-C#>, adding it to the initial pitches <D-C#>, components of M(D) and the common tones that establish the motivic connections during the first part: {G,G#,A,D} {D,C#}={G,G#,A,C#,D}. The main melody is then passed to the oboe, which presents <A-G#-G> in mm This entrance of the oboe, which marks the beginning of the second part of the movement, is a transposition of the opening motive in the cello. {G,G#,A} is also a subset of {G,G#,A,D}, the main motivic configuration of the previous cello line. The isolation of {G,G#,A} (i.e. separation from the D) in mm in connection with the melodic parallelism with the opening supports its interpretation as a form of M, specifically M(A), which is later corroborated by other presentations of {G,G#,A} as an independent motive throughout the Quintet. Following this, M(A), which dominates the second part of the cello melody, is connected to M(D), the central form of M in the first part, through {E,F}. The common tones between the sections are {E,F} and {C#,D}. 27 While the former plays the role of diatonically filling the space between M(A) and M(D), the latter is a reduced representation of M(D). Figure 2.1 illustrates the motivic 27 Notice that these common tones refer to the cello part only, which constitutes the main melody (and it is, from this point of view, clearly highlighted and separated from the other components of the texture). The other parts will be considered later in this analysis.

65 54 transformation of M that takes place in mm M(D) and its extended versions are enclosed in rectangles. M(A) is shown in dotted squares and {E,F}, in circles. The asterisks and triangles at the bottom of the score point out the most important common tones. Figure 2.1 Motivic Transformation of M in the Opening Melody Not coincidentally, {E,F} represents the middle point between M(D) and M(A) in pitchclass space. In other words, {G,G#,A} is the inversion of {C,C#,D} about E F: This conception of the operation of inversion has its origin in the theoretical writings of David Lewin, who bases the definition on the mathematical notion of geometric axis the line around which a point or

66 55 M(A)=I E F of M(D) The interval class 5 in the form of {D,G} that is contained in the motivic cells that constitute the second part of the cello melody [i.e. the leap to the D at the end of M(A) or contained in the motive {G,G#,A,D}] is the only instance of interval class 5 in the smallscale motivic gestures of the cello melody in mm Notice especially that even though <C-F> in mm. 2-3 and <D-A> in m. 11 are also interval class 5, they are presented as the connecting element between two successive motives rather than as internal components of independent motivic cells; from this point of view, C is separated from F in mm. 2-3 (C is the final note of the first gesture and F is the initial note of the following motive) and D is separated from A in m. 11 (as explained above, D is the last element of the first presentation of {G,G#,A,D} and A is the initial component of the second statement of that motivic cell). In addition, {D,G} is highlighted for being the largest interval class that is fully contained in the small-scale motivic gestures of the cello line. Furthermore, the motives that constitute the second section of the cello line delineate interval classes 1, 2, and 5 only (being 5 the only non-stepwise distance). The two pitches that define the interval, G and D, are the first and last pitch classes of the most compact ascending order of M(A) or {G,G#,A} and M(D) or {C,C#,D} respectively. The inversion of this interval about D, {D,A}, which connects the first element of {C,C#,D} or M(D) with the last pitch class of {G,G#,A} or M(A), is clearly stated, as an isolated pair of pitches, in the viola in mm As mentioned, {D,A} is also the interval that connects the motivic cells in the second part of the cello line. As a result, the pitch classes G, A, and D are preeminent due to the melodic intervallic context in which they are presented. The outer pitches of M(A) and the first pitch class of the group of points can be rotated in the space. In a circular disposition of the twelve pitch classes, an inversional axis is represented by a line that connects two tritone-related pitch classes or the middle points between them. Since the number of pitch classes that separates M(D) from M(A) is even (either 2, {Bb,B}, or 4, {D#,E,F,F#}), the two points that define the axis are located on middle points between pitches, specifically the points between E and F and B and C, rather than on pitches (David Lewin, Generalized Musical Intervals and Transformations [New Haven: Yale University Press, 1987]: 50-51).

67 56 most compact ascending order of M(D) are therefore represented in the intervals {G,D} and {A,D}. Additionally, G is the inversion of A about D or G#. From this point of view, M(A) is entirely implicit in the largest interval classes of the passage. The chromatic passages in the flute, D# to G# in mm and and A# to D# in m. 16, play the role of chromatically filling the empty space between G and D as well as A and D at the same time they extend the chromatic pitch space to D#. The D# in m. 14 constitutes its first appearance in the piece. Even more, its introduction represents the completion of the chromatic aggregate: all the pitch classes except for D# are introduced in the first six measures. In addition, its initial absence is especially remarked by the motivic cells <F-E-F-D-C#> and <D-E-F> in mm. 3-4 and 8 respectively, which are based on pitch-class sets that are only missing the D# to reach their highest possible level of chromatic density (i.e. {D,E} is the only non-chromatic step between the ordered successive components of the pitch class sets {C#,D,E,F} and {D,E,F}). The introduction of the D# in the flute part in m. 14 is a preparation for its aurally unmissable manifestation in m. 19 in the violin, where it becomes the final arrival point of a relatively long process of descending melodic chromatic extension from G to D# that is clearly emphasized by its extremely high registral disposition (register 7). This process of extension takes place in four motivic steps, each of them starting with G 7 and descending chromatically in half notes one pitch lower than the previous step: <G-F#> in mm. 3-4, <G-F#-F> in mm. 6-7, <G-F#-F-E> in mm , and <G-F#-F-E-D#> in mm , where the D# is finally reached. The appearance of the interval {D,G} in the cello line is simultaneous to the third of these steps and it is highlighted by the latter s inverted anticipation in the viola, <B-C-C#-D> in mm This figure in the viola is an extended version of the retrograde of the original version of M(D) and it is anticipated in the cello line in mm. 5-7: the viola s figure is, in effect, the ascending scalar arrangement of the motivic configuration presented by the cello. The second step of the extending figure in the violin, which coincides with the statement of {B,C,C#,D} in the cello, is a

68 57 version of M, specifically M(G). However, this version of M is used as a part of larger figures (frequently chromatic) rather than as an individual three-note motive. That is to say, the disposition and use of the pitches {F,F#,G} throughout the piece does not generally delineate M as a relatively autonomous motivic figure; therefore, M(G) is understated with respect to M(D) and M(A). The four-step descending line fills the interval {D,G} characteristic of the second part of the cello melody, chromatically completing this empty space, at the same time increasing our expectation for the chromatic motion that governs the overall texture. The first evidence of the absence of D# (as a chromatic skip ) in the cello in mm. 3-4 coincides with the condensation of the rest of the twelve chromatic pitches in mm. 3-5: {F#,G} in the violin, {Bb,B,C} or M(C) in the flute, {G,G#,A} or M(A) in the oboe, and {C#,D,E,F} in the cello. As a result, the arrival of the D# in m. 19 represents both the completion of the chromatic aggregate and the filling-in of the interval {D,E}. It is important to point out here that the presentation of M(A) in the oboe in mm. 3-5 (and not its later introduction in m. 18) constitutes the initial statement of that form of M. However, its homorhythmic combination with the flute part along with its lower registral disposition with respect to that instrument does not favor its perception as an independent motivic cell. Even more, the total pitch-class set of the polyphonic figure that results from combining the oboe and the flute in mm. 3-5 ({G,G#,A,Bb,B,C}) is transposed, as a polyphonic indivisible unit, to generate {G#,A,Bb,B,C,C#} in m. 8, anticipating the filled chromatic space of the interval <G-D> that dominates the cello line in the following passage. In effect, the <D-E-F-G> that connects the first part of the cello melody with the second one occurs in mm In light of this, the M(A) presented in the oboe in mm. 4-5 is part of a larger chromatic figure that functions to complete the chromatic space rather than provide melodic information (the melodic line is, indeed, in the cello during this passage). On the other hand, the motivic and melodic character of the M(A) introduced in the oboe thirteen measures later is its most immediate perceptual aspect.

69 58 The end of the pitch-additive descending process of the violin in m. 19 is confirmed by the viola in mm , which immediately inverts and extends (by one pitch) in imitation the line of the violin, chromatically reaching the D# by ascending motion from A# and consequently filling the ascending chromatic space from A# to G explicitly for the first time the total pitch collection that results from the imitation between the violin and the viola in mm is {A#,B,C,C#,D,D#,E,F,F#,G}. The imitative lines in the violin and the viola share only D#, making its presentation even more imposing. Nevertheless, this arrival on the D# satisfies the empty interval {D,E} only partially, since it is not used as a connecting element between D and E (i.e. the pitches D, D#, and E are not stated adjacently). The out of phase (non-simultaneous) convergence on D# of the imitative process of descending and ascending chromatic motions that takes place in mm recalls mm. 9-13, where the <B-C-C#-D> in the viola is immediately inverted to <G-F#-F-E> in the violin. The original passage (mm. 9-13) emphasizes {D,E} in the final notes of each imitative entrance (D in the viola in m. 10 and E in the violin in m. 12) and {D,G} as its components are contiguously stated to connect the end of the viola line with the beginning of the violin s in m. 10 (while {D,G} is also present, simultaneously, in the cello line). Furthermore, the total collection of pitches in this passage is {B,C,C#,D,E,F,F#,G}, once again a chromatic collection that skips D#. The parallel section in mm chromatically extends the pitch material to {A#,B,C,C#,D,D#,E,F,F#,G}, converting it to a chromatically compressed set that excludes only G# and A from the twelve-tone aggregate. As a result, the converging chromatic lines in the violin and the viola share only one common tone, G, with M(A), contributing to the motivic independence of the latter as it is introduced by the oboe in mm The pitches A and G# lacking in the chromatic figures of the viola and the violin are immediately presented in the viola in the subsequent measures (mm ) within a motivic configuration that contains interval class 5 in the form {E,A}. This figure

70 59 reinforces the connection of the pitches {G,G#,A}, or M(A), at the same time that inverts the interval {A,D} about A. The transformation of interval class 5 is shown in Figure 2.2. Figure 2.2 Transformations of Interval Class 5 In addition, the appearance of {E,A} accompanies {D,G} in the oboe (mm ). The collection resulting from these pairs of notes highlights the outer pitches of M(A), the first pitch class of the original version of M(D), and E, which along with D represents the empty chromatic space that is partially satisfied in m Even more, the cello line in mm emphasizes the pitch classes {G,A,D} (through the repetition of the set {G,G#,A,D} and its development), which is precisely the collection of notes that maps onto itself when it is inverted about D: I D of <G-A-D> is <A-G-D>. The interval {E,A} that characterizes the viola figure in mm is firstly presented in the oboe as part of a larger motivic cell <E-A-B> in m. 22. Finally, {A,B,E} is precisely the inversion of {G,A,D} about A. The axes of inversion D and A are also represented in the viola arpeggios of A 7 (mm. 2 and 7) and D 7 (m. 18). These diatonic collections contain the outer pitches of M(D) and M(A) respectively. The arpeggios are interchanged with respect to the inversional process of interval class 5 and the transformation of M: whereas A 7 dominates the first section, where M(D) is the central motive, D 7 appears simultaneously with the introduction of M(A) in the oboe. From this point of view, the arpeggios acquire a complementary function concerning the motivic plan; even more, their diatonic nature

71 60 clearly contrasts with the chromatic nature of M and the other elements of the pitch structure: if the individual pitches A and D are chromatically represented in M(A) and M(D), they are diatonically symbolized in A 7 and D 7 ; similarly, if A and D are chromatically extended in M(A) and M(D), they are diatonically expanded in A 7 and D 7. The late introduction of the D# in relationship with the non-chromatic {D,E} that is embedded in most of the small-scale sets on which the cello line is based during the first section, the process of chromatic extension in the violin that culminates with the introduction of the missing pitch D#, and the connection among the pitches G, A, D, and E implied by the inversions of interval class 5 about D and A, suggest a version of M on E, or M(E), that would represent the mirror of M(D) about M(A). Inversion about M(A) is equivalent to inversion about D (or G#) because M(A) is a symmetric set with its axis on G# which is a tritone apart from D: I D =I G# because D and G# are a tritone apart AND I M(A) =I {G,G#,A} =I G# because {G,G#,A} is symmetric around G# THEN I M(A) = I D In other words, the D or G# axis of inversion is precisely the one that transforms M(A) into itself and the inversion of M(D) about M(A) is M(E), as shown in Figure 2.3. Figure 2.3 Inversional Relationship Between the Motives In effect, M(E) is suggested in the oboe in mm of movement I. However, in this isolated instance of M(E) in movement I, the three pitches are permuted

72 61 consequently avoiding an explicit statement of the semitone {D,D#}. The importance of M(E) becomes evident in the final movement, which is structured in the form of theme, variations, and coda. A pitch analysis of the theme is relevant at this point of the discussion. Introduced as the initial motive of the movement, M(E) is the cell from which the main theme emerges and, consequently, it is present in all the variations that follow. Stated in the violin in mm. 1-17, the theme is the inversion of the cello melody from the beginning of the piece about D or about M(A). From this point of view, the original missing D# with its consequential skip {D,E} from movement I is completely fulfilled in movement III, where {D,Eb,E} or M(E) appears as two successive iterations of interval-class 1, connecting D and E by chromatic motion through Eb. The operation of inversion that converts the cello melody from movement I into the violin theme of movement III transforms the original skip {D,E} into {C,D} and the missing note D# into C#. Following this, the chromatic skip in the final movement emphasizes the outer pitches of M(D), whereas in the first movement it delineates the boundaries of M(E), anticipating its posterior motivic relevance. That is to say, at the large scale, the first movement presents M(D) as it increases our expectation for the upcoming M(E) in the final movement: in effect, the missing pitch from the beginning of the piece is precisely the pitch center of M(E), the very symmetry axis of the motive that opens the final movement. Since M(A) maps onto itself when it is inverted about D or G#, its original presentation in the cello melody in mm of movement I becomes a mere permutation of the order of its components in the violin in mm of movement III, <A-G-G#> as opposed to the original <G-A-G#>. The diatonic motion that connects the two versions of M involved in the melody becomes, then, {B,C,D}, being {B,C} the pitches that connect M(E) with M(A). {B,C} is, consequently, the inversion of {E,F}, the linking pitches between M(D) and M(A) in movement I, about D. M(A)=I B C of M(E)

73 62 The motivic transformation that takes place in the theme of movement III is represented in Figure 2.4, which is to movement III as Figure 2.1 is to movement I. In the final movement, M(A) is generated from M(E). Eb and E work as the common tones between M(E) and its extensions, {B,C,D,Eb} in mm. 3-4 and {D,Eb,E,F} in mm The first of these pitch sets associates, although only implicitly because the pitches do not appear in contiguous order (as {D,E,F} in mm. 3-4 of movement I), M(E) (stated in its complete form in the previous measures) with {B,C}. Finally, the reorganization of <B-C-B-D> from mm. 3-4 results in <D-C-B> in m. 8, directly and explicitly connecting, melodically (and non-chromatically), D, the lowest pitch of M(E), with M(A) (m. 9). Corresponding with the function of {C#,D} and {E,F} in movement I, {D,Eb} and {B,C} work as the common pitches at the small and middle levels in movement III. As a result, the symmetric conception of the motivic development in the first twenty measures of movement I allows the re-generation of M(A) in movement III through the symmetrical counterpart of the original motivic trajectory. As will be expanded later in this chapter, the last two measures of the theme present {D,Eb} and {B,C} in simultaneity, preparing the return of M(E) through {B,C} (the reverse process) in the first variation. Figure 2.4 Motivic Transformation in the Theme of Movement III

74 63 As a conclusion, M(A), M(D), and M(E) are the main motivic configurations of the piece. Their importance is confirmed by their consistent appearance throughout the movements. In movement I, M(A) is initially generated by inversion of M(D) about {E,F} at the same time that the central pitch of M(E) is created by consecutive chromatic motion. In the final movement, the inversion of M(E) about {B,C} re-generates M(A). Finally, M(A) constitutes the axis of inversion of M(D) into M(E) and {E,F} into {B,C} and vice versa. In addition, the chromatic lines that converge on D# in mm of movement I, <G-F#-F-E-D#> in the violin and <Bb-B-C-C#-D-D#> in the viola, complete the chromatic space between M(A) and M(E) and M(A) and M(D) respectively. As a result, two different types of relationships rule the motivic connections, defining two diverse possibilities of connecting M(A) with each of the other two versions of M. (i) The first type is by inversion about M(A), D, or G#. This kind implies a tone/semitone dichotomy that results from transforming the motive through the axes {E,F} or {B,C}: {C,C#,D,E,F,G,G#,A} or {G,G#,A,B,C,D,D#,E}. Even though the axes are of chromatic nature (they are, in effect, a semitone), they are not chromatically contiguous to M; thus, the transformation of M through these axes implies non-chromatic motion. (ii) The second type of relationship is by continuous chromatic motion and interchanges the pitch classes that play the role of axes in the first type: M(A) connects with M(D) through {Bb,B,C} and M(E) links to M(A) through {E,F,F#}. In other words, while the transformation of M(D) into M(A) through inversion implies ascending motion in the pitch-class space, its transformation through chromatic motion results from descending direction. The transformation of M(E) into M(A) operates reversely. Finally, the combination of Figures 2.1 and 2.4 along with the previous explanation results in the transformational network that guides the development of the motivic material throughout the entire composition. Illustrated in Figure 2.5, this network operates in the pitch-class

75 64 space as represented in Figure 2.6. The symmetrical conception of the transformational network generates a symmetrical arrangement of its components in the pitch-class space. Figure 2.5 Transformational Network Figure 2.6 shows the disposition of the main versions of M and the inversional axes that transform them into each other in the pitch-class space. M is shown in rectangles and the inversional axes, in circles. To facilitate the discussion of the motivic structure, let us define Z as the vertical pitch space that connects M(A) with M(E) and Y as the vertical pitch space that links M(A) with M(D). Figure 2.6 Spatial Distribution of the Pitch Classes in the Quintet

76 65 Following the previous discussion, movement I begins at the bottom right of the pitch space represented in Figure 2.6 and diagonally moves to {E,F} to generate the central form of M or M(A). Movement III travels across the symmetrical counterpart in pitchclass space, departing from M(E) to re-generate M(A) through the diagonal {B,C}. This is illustrated in Figure 2.7. In addition, as will be expanded upon later, the axis of inversion, M(A), G#, or D is clearly exploited in the middle movement. Figure 2.7 Motivic Transformational Paths in the Openings of the Outer Movements Movement I: The Essential Components and Their Operation Within the Network The components of the entire network are suggested in the final section of movement I. As a continuation of the rotated form of M(E) in mm , the oboe begins a descending line that culminates with Bb in the final measure of the movement. This line, which shares its initial pitch D with the previous M(E), is interrupted with rests in m. 34, suggesting a subdivision into two parts. 29 The first part (mm ) descends 29 Even though the oboe melody starts in m. 18 with the presentation of M(A), the general characteristics of the music (pitch and durational organization, articulation styles, and texture) suggest a subdivision in mm The D in m. 29 marks the beginning of a continuous descending line that clearly differs from the previous ascending/descending fluctuations. In addition, the tenuto markings and the triplet rhythms that start on the D in m. 29 continue until the end of the movement, functioning as important factors of

77 66 from D to G by chromatic motion with the exception of the Bb (skipped in m. 32). Nevertheless, Bb is present in the violin (mm ) and also anticipated in the initial section of the oboe melody (mm ). As a result, M(E) is connected to M(D) through D and the latter is linked with M(A) through {B,C} (via chromatic, intra-y-domain motion), as shown in Figure 2.8. The melodic connection of M(D) with M(A) through the inversional axis that belongs to M(E), i.e. {B,C}, suggests motivic transformation within the Y domain (i.e. vertical rather than diagonal network movement). This idea is supported by the {B,C,C#} presented in the cello in mm : {B,C,C#} is the chromatic extension of {B,C} that links it with M(D), that is to say, the reverse path of the arrow 2 in Figure 2.8. Even more, in mm , the cello part traces <D-C#-C>: {C,C#,D} and {B,C}, the two essential elements of the Y domain, are clearly connected without interfering with the Z domain. Figure 2.8 Motivic Transformational Path in mm of Movement I The second part of the oboe melody (mm ) moves down from F# to Bb by chromatic steps except for the skipped Eb in m. 37, suggesting both the cross-domain delimitation. Notice that the phrasal articulation in the oboe anticipates the posterior articulation in the cello (in the cello part, the formal subdivision falls between mm. 29 and 30).

78 67 inversional relationship between M(D) and M(A) and their vertical chromatic connection within the Y domain. <F#-F-E> in mm connects the M(A) stated in the two previous measures with the presentation of M(D) that follows (mm ), linking M(A) with M(D) through {E,F} (the crossed movement within the pitch space represented in Figure 2.6). <F#-F-E> can then be interpreted as the chromatic connection between M(A) and the axis {E,F}, since it implies purely chromatic motion (a pure chromatic path between M(A) and {E,F}). The chromatic extension of M(D) that finishes the movement, <D-C#-C-B-Bb> in mm , represents the chromatic, intra-y path that links M(D) with M(A) (the vertical motion in Figure 2.6). In addition, the interpolation of the flute line <C-B-Bb-A-G#-G-F#> in between the two parts of the oboe melody highlights the chromatic connection between them. As a result of the skipped Eb, <F-E> is followed by M(D) (mm ), which is extended downwards to Bb, filling the chromatic space that belongs to the Y domain. Finally, Bb leads to A, the opening pitch of the following movement. In this way, M(D), the initial version of M in movement I, is extended to A and, consequently, connected to the highest pitch of the most compressed ascending arrangement of M(A) through chromatic movement within the Y domain. This is illustrated in Figure 2.9. The <F-E-D#-D> in the cello in mm connects M(E) with the inversional symmetry axis that belongs to M(D) (i.e. {E,F}), making even more explicit the similar process that takes place in mm In these passages, the cello line emphasizes the chromatic connection between M(D) or M(E) and the inversional axis located in their respective domains, favoring linear (vertical) rather than cross (diagonal) relationships within the space represented in Figure 2.6: M(D) with {B,C} and M(E) with {E,F}. The incorporation of the D at the end of the final measures of the cello highlights the intra-domain connection between {F,E} and M(E), since the passage is precisely a descending chromatic line from F to D.

79 68 Figure 2.9 Motivic Transformational Path in mm of Movement I In a similar way, the cello line in mm connects M(D) with {B,C} by means of stating {B,C,C#,D} contiguously in the pitch-class space. In light of this, the cello plays the role of outlining the linear relationship between M(D) and {B,C} as well as M(E) and {E,F}. Illustrated in Figure 2.10, this connection, which is chromatically founded, partially represents the chromatic nature that governs the motivic transformations within Y and Z (vertical movement in the pitch-class space). Figure 2.10 Transformational Path in the Cello in mm

80 69 The portions of the chromatic paths that complete the vertical, intra-domain connection of M(D) and M(E) with M(A) through {B,C} and {E,F} respectively (i.e. the pitch classes Bb and F#) are stated in the oboe line (Bb in the final measure and F# in m. 35). Furthermore, as mentioned above, mm in the oboe represents precisely the chromatic connection of M(D) and M(A) within the Y domain. Notice that the intra-zdomain connection of M(E) and M(A) never becomes explicit (i.e. the descending chromatic line in the oboe is interrupted without the complete statement of M(E)), being then subordinated to the chromatic intra-y-domain transformation. Finally, the initial and final pitches of each part of the oboe melody are important points in the transformational path. G and D in the first part represent the three versions of the motive: G is the lowest pitch of the most compressed ascending form of M(A) and D is the common tone or symmetry axis between M(D) and M(E). The second part delineates Bb and F# or, what is the same, Bb and its inversion about M(A). These two notes signify the chromatic filling of the Z and Y domains. Movement II: The Representation of the Dual Pitch-class Space in a Scale To summarize, movement I principally exploits the inversional, cross-domain relationship and the chromatic, vertical intra-y-domain path between M(D) and M(A) or, in other words, the connection of M(D) with M(A) through {E,F} and {B,C}, at the same time that it highlights the pitch classes D, E, G and A through interval class 5 and the empty versions of M(E) and M(A). These components and motivic procedures that define the network and the organization of the pitch-class space are combined to become the pitch-class collection that dominates movement II. Similar to the emergence of scale X in the Duet, the basic motives and the essential features of the transformational network generate a scalar pitch collection, musical material in its most rudimentary combinatorial form.

81 70 The <B-C-C#> that the cello presents in mm and inverts in mm of movement I, connecting M(D) with {B,C} and {E,F} with M(E) respectively, becomes the initial motivic cell of the middle movement. Presented in the oboe in mm. 1-2 of movement II, {B,C,C#} or M(C#) is immediately extended to {B,C,C#,D}, emphasizing the chromatic intra-y-domain path between M(D) and {B,C} suggested in movement I as well as strengthening the continuation between the two movements implied by the descending line in the oboe at the end of movement I that leads to the opening A of movement II. The oboe line at the beginning of the middle movement consists, then, in two adjacent versions of (012): {B,C,C#} or M(C#) in mm. 1-2 and {C,C#,D} or M(D) in mm. 2-3, with {C,C#} working as the common tones. The closeness in time of these two statements of M and the chromatically contiguous (uninterrupted) arrangement of the four pitches suggests a unified motivic configuration rather than two separate versions of M. From this point of view, the opening oboe line is an extended version of M or, more precisely, a retrograded expansion of M(D) evoking that form of M introduced in the viola in mm of movement I. In effect, the subsequent measures confirm the hierarchical importance of M(D) over M(C#). In measures 4-5, the oboe extends {C,C#,D} to {C,C#,D,E}. This motivic configuration, which is reinforced by the registral distribution of its components and its counterpointing duplication in the flute (which presents a rhythmic elaboration of it), is the combination of M(D) with the empty version of M(E) and, therefore, it highlights the whole tone {D,E} characteristic of movement I. The following motivic cell in the oboe is, once more, M(D) in mm. 8-9, which is, in this instance, counterpointed by {D,F} in the flute. The homorhythmic character of this passage suggests the integration of the oboe and the flute generating the set {C,C#,D,F}, the combination of M(D) with one of the pitches of the inversional axis that transforms it into M(A). In light of this, the motivic figures in mm. 4-5 and 8-9 are both non-chromatic extensions of M(D) and they imply, from this point of view, a non-

82 71 chromatic/chromatic contrast. While in the first passage the extension is horizontal, in the second one, is vertical. The dimension of the extension, the sense in which it is developed, is determined by the spatial relationship of the added pitches with respect to M(D), successive in the first case and simultaneous in the second one: in mm. 4-5 the flute and the oboe melodically and independently present {C,C#,D,E}, whereas in mm. 8-9 the flute harmonically extends the M(D) stated in the oboe by means of the common tone D i.e. the extension results from the simultaneous incorporation of {D,F} with M(D), or, what is the same, from combining the flute and the oboe. These two non-chromatic extensions of M(D) are separated by an imitative passage created with the collection {E,F,G,A,Bb,B,C,C#,D} in the strings. This selection of pitches connects by stepwise motion M(D) with {E,F} not only in an upward direction in the pitch-class space (through <C,C#,D,E,F>), as occurs with the motivic figures in the woodwinds, but also by downward movement (through <D,C#,C,B,Bb,A,G,F,E>), adding more pitches to the collection of notes presented by the oboe and the flute in the previous measures. The motive that is imitated in the strings contains the entire chromatic Y domain, the axis located in the Z domain that transforms M(D) into M(A) through inversion, and the outer pitches of M(A); in other words, it represents the two essential paths that link M(D) with M(A) introduced in movement I. A similar imitative passage in mm leads to the presentation of the same pitch collection in the woodwinds in mm In this instance, the continuous ascending arrangement of adjacent pitches that characterizes this pitch collection from its introduction expands through more than two octaves in individual parts (i.e. the collection is completely and independently stated in the flute and the oboe), defining the immediately perceivable scalar character that this selection of notes assumes throughout the movement. With respect to the transformational network of the piece, this group of pitch classes is the ordered combination of the chromatic pitches that belong to the Y domain, the inversional axis that transforms M(D) into M(A), {E,F}, and the outer pitches of

83 72 M(A) and M(E). We define Scale Y as {D,E,F,G,A,Bb,B,C,C#} and its distribution in the pitch-class space is illustrated in Figure Scale Y amalgamates the most chromatic manifestation of the Y domain with the least chromatic representation of the Z domain that is possible while preserving its essential elements. The Z domain is defined by three main elements: the first pitch class of the most compressed ascending form of M(A), the axis {E,F}, and M(E). Figure 2.11 Distribution of Scale Y in the Pitch-class Space It is important to point out here that, since {E,F} represents an inversional axis, the totality of its components is essential to its definition: the axis is located exactly in the middle of both pitches. On the other hand, M(E) (like any version of M) is a symmetrical three-component structure that can be subject to reduction without losing its external shape. 30 In light of this, it is logical to interpret {D,E} as the reduced version of M(E), especially considering the treatment of these notes in movement I. 30 Intuitively, two-point figures have the especial quality of irreducibility: omitting one point of a form that is defined only by two points results in a completely different shape rather than in a reduced version of the

84 73 The non-chromatic/chromatic dichotomy suggested by the chromatic and nonchromatic expansion of M(D) at the beginning of the movement and reinforced by the internal structure of scale Y (i.e. chromatic Y domain vs. non-chromatic X domain) plays an important role in the overall pitch organization of the middle movement. As mentioned above, the second half of movement II, mm , is a retrograde of the first half, mm However, the retrograde principally involves the strings, while the woodwinds present motivic material that is different from the first half. Many of the modifications in the pitch structure of the woodwinds generate changes in chromatic density. The two halves defined by the symmetry axis (mm and 61-end) start using a less-chromatic pitch-class collection, mostly scale Y, and finish employing the complete chromatic selection, suggesting the chromatic filling of the Z domain or, what is the same, the transformation of scale Y into the chromatic aggregate. This increase in chromatic density is supported by the pitch collections presented in scalar manner: whereas the beginning of each half is characterized by scalar passages based on scale Y, the end favors chromatic scalar configurations. Once scale Y has been fully presented in mm , G#, Eb, and F# are the necessary pitch classes to complete the chromatic aggregate. A scale-y-based ascent in the flute in mm culminates in the highest point of the entire movement: G# 6 in m. 18. The arrival of the G# is, then, aurally evident. Its unmissable appearance is prepared by the motivic figure presented in the oboe in the previous measures. In mm the melody in the oboe is based on the set {G,A,C,C#,D}, which is equivalent to the combination of M(D) and the empty version of M(A). Recalling the process that leads to Eb in the first movement, the late arrival of G# in connection with its chromatic skip ({G,A} in the oboe) increases our expectation of it during the second movement, original form (the original form is indeed not identifiable anymore). On the other hand, eliminating the middle point of a form that is defined by three points located in the same line alters only the internal elements of the shape but not the form itself.

85 74 strengthening its already registrally prominent appearance in m. 18. G# transforms scale Y into a symmetric pitch collection around E F, precisely the inversional axis that transforms M(D), the motive of the initial section of movement II, into M(A), the central motive of the network. Even more, E and F are also emphasized in the initial nonchromatic extensions of M(D) explained above. The appearance of G 6 in the violin five measures later (m. 23) suggests a connection between these two pitches (G and G#) that are registrally differentiated within the overall texture. In addition, the G in m. 23 rhythmically evokes the initial A in mm. 1-4: both pitches are repeatedly stated in sixteenth notes. As a result, M(A), the motivic center of the network, is delineated across mm (at the middle ground). A noticeable process of registral transfer of the note G that takes place in the following measures prepares the manifestation of M(A) at the small scale. Starting in m. 23, G 6 is gradually transferred through the subsequent five measures, reaching its lowest position, G 2, in m. 28 (first in the violin, then in the cello), precisely where Eb is introduced. The incorporation of Eb in m. 28, its first appearance in this movement, preserves the symmetrical quality that the pitch collection acquires through the previous addition of G#, at the same time that moves the axis from E F to F#, now the only missing pitch of the twelve-tone domain. The registrally descending G leads to the first explicit presentation of M(A) in this movement, which appears in m. 31 as a part of a larger set, {G,G#,A,D,F}, in the cello. This motive evokes the second part of the cello melody (i.e. the repeatedly stated motivic cell {G,G#,A,D}) that originates M(A) in movement I. In addition, {G,G#,A,D,F} resembles {C,C#,D,F} from mm. 8-9 of movement II: in effect, both figures are the combination of a form of M with {D,F}: M(A) {D,F} in one case and M(D) {D,F} in the other. The relatively gradual incorporation of pitch classes, which is suggested from the beginning during the formation of scale Y, is interrupted, before the introduction of F#, by the only section in the movement that is retrograded in its complete texture (involving

86 75 all five instruments): mm , which is retrograded in mm The formal and textural development of these passages, their most immediate perceptual effect, consists in the horizontal and vertical elastic manifestation of scale Y. From this point of view, they represent the bi-dimensional display of scale Y. In these passages, the strings present the collection in a way that delineates parallel moving chordal structures. This textural layer is rhythmically differentiated from the woodwinds (sixteenth notes in the strings vs. triplets in the flute and the oboe), which also play scale Y in parallel motion but in the opposite direction with respect to the strings. In other words, the two strata, strings on one hand and woodwinds on the other, present scale Y as parallel moving chordal structures, but they move by contrary motion with respect to each other, aurally conveying vertical expansion and contraction of the overall texture. The diverse lengths of the scalar passages create horizontal elasticity while the imitative passages and the alternation and contrary motion between strings and woodwinds suggest vertical stretching. The momentarily interrupted completion of the chromatic aggregate caused by the interpolation of the original form of the scale-y-based section is resumed in m , where the flute presents the first complete chromatic scale in the movement, after the G# and the Eb have been reintroduced, in that order (i.e. following the same order in which they are originally introduced in the movement), in the oboe and the cello in mm The presentation of the F# in m. 52 is accompanied by {G,G#,A,D,E}, a motivic configuration that not only contains M(A) but also resembles the {G,G#,A,D,F} presented in the cello in m. 31. In effect, {G,G#,A,D,E} is derived from the restatement of {G,G#,A,D,F} in the oboe in m. 49. Restated in the cello in mm and reaffirmed in the viola in mm , {G,G#,A,D,E} is simultaneous to the flute s and the oboe s confirmation of the chromatic scale. Already discussed at the beginning of this chapter, the introduction of the chromatic scale in this section plays the role of chromatically filling the larger intervals outlined by the motivic configurations of the cello and the

87 76 viola. The subtraction of the {Eb,F#,G#} at the end of this section gives back to the scalar passages their original scale-y form in m. 56, decompressing the pitch-class space in preparation for the symmetry axis of the movement four measures later. This reduction of the pitch-class content prepares the increase of chromatic density that takes place in the second half. The axis (mm ) is highlighted with a registrally stretched version of the repeated A that opens the movement, now homorhythmically stated in different registers (i.e. vertically stretched) by the three string instruments. The parallel increase of chromatic density between the two halves of the movement implies changes in the overall pitch development of the retrograded section. In effect, the part that is fully chromatic during the first half of the movement (the passage following the elastic display of scale Y) is purely based on scale Y in the second half. This is possible because the retrograde involves the strings only, while the woodwinds present different motivic material that evolves within the pitch-class space in a parallel (rather than reverse) manner with respect to the elements introduced during the first half. The result of the non-coincident pitch development of the strings (retrograde) and the woodwinds (parallel increase of chromatic density) that takes place in the second half with respect to the first one results in the pitch-class decompression (at first) and densification (later) of the retrograded music heard before the axis, at the same time that it allows for the simultaneous presentation of scale Y and the chromatic scale at the end of the movement. A comparison of certain passages from the first half with respect to their respective retrograded versions illustrates this point and illuminates the comprehension of the operation of the transformational network and the pitch-class space in this movement. The passage that presents F# for first time in movement II is retrograded in mm The most preponderant motivic material of this section is, consistently with the first half, based on {G,G#,A,D,E} (viola in mm and cello in mm ). The only scalar passage in this section, presented in the flute in m. 68, is based on scale Y (as

88 77 opposed to the corresponding chromatic scalar passage in the first half), clearly decreasing the chromatic density with respect to the first half. In a similar way, most of the motivic configurations that are based on scale Y during the initial section (the fragment prior to the scale-y-based passage of the first half) become more chromatic during the retrograde version (in the final part of the second half). For instance, the Scale-Y pattern in the flute in mm is replaced with a chromatic passage in the oboe in mm Even more, the oboe continues playing portions of the chromatic scale until m. 93. Other evidence of increase of chromatic density in passages that retrograde the pitch structure of the strings with respect to their original versions occurs in mm (retrograde of mm ), 109 (corresponding to mm ), (mm. 7-9), and (mm. 4-5). The first of these sections has already been mentioned in connection with the introduction of the G#: mm are based on the set {C,C#,D,G,A} in the oboe. It is relevant to point out here that this set is related to {G,G#,A,D,E}, one of the forms of the main figure of the final and initial sections of the two halves respectively (oboe in mm , cello in mm , and viola in mm ), by both transposition (the second set is T7 of the first one) and inversion (the second set is I9 of the first one). As a corollary of this particular transformational affinity, these sets highlight M(A) and M(D): whereas {C,C#,D,G,A} is the combination of M(D) with the empty version of M(A), {G,G#,A,D,E} is exactly the opposite: the fusion of M(A) with the empty version of M(D). Consequently, the figures based on these two sets represent the empty, nonchromatic version and the complete, chromatic form of the central motive of the network, M(A). In effect, {C,C#,D,G,A} appears precisely before the introduction of the pitch G#, suggesting motivic transformation within the Y domain of the network and preparing the presentation of M(A). Measures (retrograde of mm ) preserve M(D), this time in the flute, clearly emphasizing it motivically and rhythmically, replacing the empty M(A) with {Bb,B,C} or M(C) (oboe). As a result, the Y domain is chromatically filled in

89 78 a explicit way. 31 A similar situation takes place in mm , where the original {C,C#,D,E} from mm. 4-5 is replaced by the complete chromatic Y domain. A noticeable symmetric disposition of the chromatic scale, created by the contrary motion delineated between the flute and the oboe in m. 109, makes the chromatic character of the final section even more prominent. The peculiar distribution of pitches results in the compression of the progressive process of incorporation of the pitches G#, Eb, and F# that takes place during the first half of the movement (in mm. 18, 28, and 52 respectively). The gradual increase of pitch classes that characterizes the beginning of the piece is then summarized in one measure (m. 109). This measure is the retrograde of m which coincides with the first explicit presentation of scale Y (i.e. the pitchordered statement of the total collection in a single part through more than an octave). The absolute equivalence of the chords in the strings of m. 11 and 12 allows the reduction of the passage to only one measure in the retrograde version (m. 109). Since m. 12 repeats exactly the same sequence of chords presented in m. 11, the omission of one of these measures in the retrograde does not alter the total pitch definition of the chordal passage: m. 109 is simply the abbreviated version of mm (i.e. a progression of two chords <n-m> instead of a repeated progression of two chords <n-m-n-m> ). The sequence of chords in mm accompanies the presentation of scale Y, which extends over two measures, while the chordal progression in m. 109 supports the condensation of the chromatic scale in a single measure. The result (with respect to mm ) is the chromatic extension of scale Y distributed in a more compressed, compact form. Consistently, the contrary motion between the flute and the oboe in m. 109 suggests vertical contraction towards the middle of the measure as opposed to the parallel ascending movement (i.e. non-contracting texture) of mm Although Bb is also present in m. 13, the contiguous arrangement of the chromatic scale in mm makes the chromatic connection evident. In m. 13 <B-Bb> is not linked to M(D) (not even to C).

90 79 Finally, measures represent the chromatic version of mm As mentioned above, mm. 8-9 clearly separate M(D) from the interval {D,F}, contrasting the chromatic form of the Y domain with the non-chromatic manifestation of the Z domain by means of timbral qualities. Measures complete the twelve-tone aggregate, explicitly filling the entire chromatic space that belongs to the Z domain (i.e. <G-F#-F-E- D#-D> in the oboe) simultaneously with a chromatic extension of M(D) in the flute. The coincidence of the final note in the oboe (D in m. 113) with the first one in the flute (D in m. 112) as well as the chromatic extension of M(D) into the Z domain (an extension that covers all the downward chromatic space between D and F#) suggests the confluence of both domains into a single, indivisible chromatic universe. Finally, the retrograde of the imitative passage that generates scale Y (mm. 6-7) engenders the vertical coexistence of scale Y, in the strings, and the chromatic domain, in the woodwinds. Measures deserve special consideration with respect to Figure 2.6. Based on motivic material that is evidently related to that presented in the flute and oboe in mm. 8-9, this section emphasizes the contrast between the chromatic Y domain and the nonchromatic Z domain that defines scale Y. The initial set in the flute in mm. 8-9, {D,F} above M(D) in the oboe, is extended to {D,E,F} in m. 27. The retrograde of this measure is m. 94, where {D,E,F} and M(D) are emphasized by means of registral expansion. Furthermore, these two motivic figures that operate in simultaneity as part of a twofold vertical motive are repeated in mm in a registral and timbral setting that resembles mm. 8-9 and m. 27, clearly contributing to the perceptual association of the passages. The sixteenth notes in the viola and the cello in m. 95 and 96 (and 26-25) highlight the contrast between the non-chromatic Z domain and the chromatic Y domain respectively: the diatonic part of scale Y, represented in the strings in m. 95, is juxtaposed with the chromatic part of the same scale, stated in m. 96. Finally, Movement II suggests the symmetrical quality of the pitch space represented in Figure 2.6 as it exploits its axis in connection with the dividing function

91 80 that precisely separates the chromatic path from the non-chromatic trajectory, providing the essence of the definition of the network, its dual-domain feature. In addition to the frequent use of M(A) as a part of larger motivic figures and the delineation of M(A) at the large scale in mm. 1-23, scale Y, the essence of movement II, is precisely conceived with respect to the symmetry axis, since it contains the two sides of the space, Z and Y, as two separate domains: it is the convergence of the most chromatic manifestation of the Y domain with the less chromatic representation of the Z domain. Even more, the procedure of completing the pitch-class space by increasing its chromatic density is accomplished in a symmetrical manner: once scale Y has been formed, the addition of G# transforms it into a collection with symmetrical properties that are preserved with the subsequent incorporation of the D#. The later addition of the F# results in the twelvetone aggregate, which is one of the most perfect representations of pitch-class symmetry: a domain that is symmetric around any of its components. The exploitation of the symmetric axis of the pitch-class space suggested by the repetition of the pitch A at the beginning and the pitch axis of the movement becomes evident in the final three measures of the movement, where M(A) acquires a polyphonic, vertical setting as its components are presented almost in simultaneity: G# in the oboe, A in the flute, the viola, and the cello, and G in the violin. Movement III: The Completion of the Transformational Trajectory Within the Pitch-class Space The transformational path of the piece is completed in the final movement, where the main theme is an inversion of the melody that opens the first movement. Consequently, in the last movement, the inversion of M(E) about {B,C} re-generates the original M(A) from movement I. It is only in movement III that the pitch trajectory within the space acquires a sense of coherence and closure. As demonstrated above, movement II exploits the symmetry axis. However, the symmetric property of the space is only anticipated in movement II and it does not become completely evident until the

92 81 end of the work. If movement I mainly explores the diagonals that connect the bottom of the Y domain with the center of the network through the top of the Z domain, movement III is principally centered on the complementary path: the diagonals that connect the pitch components located at the bottom of the Z domain with those in the center of the network through the pitch elements in the top of the opposite domain. Furthermore, M(D), the point of departure of the piece, is verticalized to become the final chord, a process that is partially anticipated with the vertical (although not completely coincident) disposition of the components of M(A) at the end of movement II. From this point of view, the entire work is the musical manifestation of a symmetric and closed theoretical conception. The formal organization of movement III consists of a theme followed by six variations and a coda. Variation 4 is a retrograde of variation 1 and variation 6 is a retrograde of variation 2. Variations 3 and 5 are motivically related in a non-systematic way. The movement is essentially based on M(E) and its connection, both inversional and chromatic (i.e. inter and intra-domain), with M(A). In addition, in variations 1 and 6, M(A) is stated only once, as part of a larger motivic figure. M(D) appears only in variations 3 and 5 and at the end of the movement (coda). The elements of the transformation of M(E) into M(A) through {B,C} that delineates the theme in movement III are compressed in the last three measures of the theme: M(A) in the violin in m. 15 and the first two pitches of M(E) in the violin above the axis {B,C} in the cello in mm {D,Eb} and {B,C} in the flute and the oboe open the first variation in m. 18. The appearance of the E followed by {B,C} in the cello in mm not only satisfies the incomplete version of M(E) stated in the flute in the previous measure but it also connects M(E) with {B,C} from a melodic point of view: the motivic line is <E-B-C-B>. In mm , M(E) is again timbrally and temporally divided into {D,Eb} (violin in m. 20) and {D,E} (flute in mm ), evoking the whole-tone distance between D and E that leads to the Eb in movement I. The pitch collection {B,C,D,Eb,E}, the combination of M(E) with {B,C}, becomes explicit in the

93 82 cello in mm , precisely before the introduction of the only M(A) in the variation. In this passage, M(A) is contained in the figure <D-C-B-A-Ab-G-F> in the violin in mm This figure is continued in the next measure by the viola, which presents the scale <E-Eb-D-C-B-A-G> [i.e. the combination of M(E), the axis {B,C}, and the empty version of M(A)], connecting M(E) with M(A) through {B,C}. The last two measures of the variation chromatically extend M(E), linking it with M(A) through the Y domain (<Eb-D-C-B-Bb-A> in the flute) and emphasizing its chromatic relationship with the inversional axis located in Z, {E,F} (<F-E-Eb-D> in the oboe). Variation 2 diagonally connects M(E) with M(A) through the inversional axis {B,C} (m. 36). The repetition of M(A) in m. 37 leads to the succession <A-D-G-D-A> in m. 38, highlighting not only the pitches that represent M(D) and M(A) but also their symmetrical relationship. Furthermore, D is the common tone between M(D) and M(E) and, from this perspective, it is the representation of both forms of M in a single pitch. From this point of view, this passage symbolically embeds M(D) and M(E) in M(A). The perceptual preeminence of A, G, D as relatively individual pitches (i.e. separated from, although not unrelated to M) is consistently created throughout the work by means of repetition and intervallic setting, particularly the use of interval-class 5. The imitative passage that follows the symmetrical arrangement of the three pitches, <A-B-C-D-Eb> in mm , leads M(A) back to M(E) through the original cross path {B,C}. In measures 43-46, the flute presents M(E) in a horizontally and vertically stretched form, repeated a number times with different registral dispositions. The motive is supported by the empty vertical statement of M(A), G in the violin and A in the cello. The variation finishes with an imitative passage that melodically extends M(E) to the two inversional axes: <F-E-Eb- D-C-B>. As discussed at the beginning of this chapter (during the explanation of the transformational network and the pitch-class space), {B,C} represents the non-chromatic connector between M(A) and M(E) in the sense that it does not involve complete

94 83 chromatic motion: {G,G#,A,B,C,D,D#,E} is not a fully chromatic set since it has whole tones between A and B and between C and D. 32 This function of {B,C} as a nonchromatic connector between M(E) and M(A) is gradually weakened as the C# and Bb are incorporated in the next variation. The initial five measures of variation 3 highlight, once more, M(E), M(A) and the two inversional axes, {B,C} in mm and {F,E} in m. 54 both in the violin. The variation opens with a trilling melodic setting of M(E) in the cello in which {B,C} is interpolated in mm M(A) is introduced in the oboe in mm In mm , M(A) takes the trilling effect, first as {G,A}, or empty M(A), in m. 58, then as {Ab,G} in m. 60. The middle measure (m. 59) links the F# from the Z domain with the representative components of the Y domain, {B,C} and M(D), through the outer pitches of M(A). The chromatic lines in the woodwinds in the last seven measures of this variation fill the chromatic space that belongs to the Z domain, at the same time extending it to the Y domain by presenting M(D) and <E-D#-D-C#-C-B> in the flute in mm. 63 and respectively. Simultaneously, the violin completes the chromatic space within the Y domain, a process confirmed by the <C-B-Bb> in the cello in the final measure. Even more, the three notes in the cello along with the <G#-A-D- C#> that precedes them tacitly (not in order) imply the vertical-y-domain chromatic transformation of M(A) into M(D). Simultaneously, the viola converts M(A) in the empty version of M(D) through {B,C} (first) and in the complete version of M(D) through {E,F} (last) by means of the initial trilling motivic shape (mm ). Variation 5 resembles variation 3 in the trilling motives and the general chromatic character. The variation starts with M(E) (m. 85) and M(D) (mm ) in the violin and M(A) in the viola (mm ). The next motivic figures fill the chromatic space, in both directions, between the two initial motives. As a result, the Z and Y domains are 32 It is important to distinguish here between the chromatic conformation of {B,C} (which is obviously a semitone) and the non-chromatic path that lies behind its function of inversional axis for the motivic transformation.

95 84 here unified in a single chromatic universe. The entire variation is based in chromatic cells. Not coincidentally, during the first five measures the chromatic structure is segmented into different versions of (012). Following this idea, the main motivic configuration, M, recursively becomes the path of its own transformation. That is to say, the successive transpositions of M progressively move within the chromatic space, aurally suggesting that M is being chromatically transported (through its same internal component the semitone) to different pitch levels. Notice, however, that this is only a unique characteristic of variation 5 and not a procedure outlined at the large level of the entire work. The rest of the variation chromatically extends all these versions of M, creating patterns of progressively larger number of pitch components, thus conveying horizontal elasticity. In m. 98, the cello introduces an A 7 arpeggio that contrasts with the general chromatic character of the variation. The diatonic quality of this set, the internal registral distribution of its components and its timbre (in harmonics) immediately contributes to its perceptual association with the A 7 and D 7 from movement I. Finally, while variations 3 and 5 are highly chromatic, the other variations are mostly based on M(E), M(A) and their non-chromatic melodic connection through {B,C}. As a result, the chromatic/non-chromatic dichotomy suggested in the pitch organization of the entire work contributes to the delineation of the large-level formal structure in the final movement. Starting with a counterpoint of the chromatic (i.e. filled) form of M(C#), {B,C,C#}, with the non-chromatic set {A,B,C}, the coda insistently states M(E) and M(A), the central motives of movement III, presenting M(D) as an independent motive in only two, although unmissable, instances: mm and m. 164 (final chord). 33 Notice especially the repeated M(A) and M(E) with different registral dispositions in 33 M(D) is also stated in m. 126 in the violin. However, in this instance, the motive is part of a larger figure that ascends chromatically from C to F#.

96 85 mm and the scale that connects these two motives through both {B,C} and {E,F} in m The first incorporation of M(D) in the coda, the most perceptible instance of this version of M(D) as an independent figure in the entire movement, occurs in the flute in mm , marking the culminating moment of the only passage in the whole piece that presents the three principal forms of M as evident transpositions of a single motivic figure in a consecutive manner. In mm , M(E) and M(D) are literally connected through M(A), which appears as an inverted mirror of the other two, precisely in between them. The rhythmic features of these three converging instances of M are here not only identical but also noticeably similar to the original version of M in the piece. This section coincides with the beginning of a process that gradually extends the chromatic scale in several steps, melodically stretching it. The horizontally and vertically elastic texture of measures evokes the passages based on scale Y from the middle movement at the same time that it completes the total chromatic space as an indivisible whole: the initial and final pitches of each motivic figure and their total number of pitches varies, denying any kind of subdivision of the chromatic space. Consequently, the convergence of the separate domains is ultimately achieved. The similarity between this section and the scale-y-based passages from movement II is evident also from the point of view of the polyrhythmic structure: they are the only sections in the work that polyrhythmically combine sixteenth notes with triplets (mm ). The Quintet ends with an imitatively developing chordal passage (mm ) that comprises the totality of the twelve-tone aggregate, leading to the only vertical distribution of M(D) in the piece: the final measure is the M(D) chord. As shown in Figure 2.12, the first set of imitations delineates a diatonic pentachord {A,B,C,D,E}, the second one, the set {F,G,G#,Bb,C#}, and the third one, {B,C#,D#,F#,G#}.

97 86 Figure 2.12 Movement III, mm The distribution of these apparently unrelated chords horizontally delineates the essential components of the network and the pitch-class space. According to the voiceleading interpretation shown in Figure 2.13, this passage presents, for the last time, M(D) and M(A), the inversional axis {E,F} that connects those forms of M, and the movement from B to Bb that fills the chromatic space in the Y domain (an essential component of scale Y). The outer components of M(D), C in the violin and D in the flute in m. 159 (first set of entrances), move to the C# 5 that appears in the oboe in m. 161 (second set of entrances), which is transposed up an octave in the last imitative group (flute in m. 163). The middle pitch of M(D) is then dropped three octaves to become the registral center of the only chordal disposition of M(D) in the final measure.

98 Figure 2.13 Registral Distribution and Evolution of CH in the Final Measures 87

99 88 The registral movement and development of M(D) is shown with the darker arrows in the figure. In a similar way, the dotted arrows point the delineation of M(A) and the lighter arrows illustrate the movement from B to Bb. The ascending motion in the registral space is evident, preparing for the final presentation of M(D) in its lowest, although not vertically compact (each pitch is presented in a different, contiguous register), disposition. In addition, D# in the cello and F# in the viola in the final imitative chordal structure represent the apparently absent Z domain: in effect, D#, the missing note in movement I, is the note that defines M(E), its central element, and F# is the pitch that fills the chromatic space in the Z side of the network. The Chordal Passages: Their Registral Referential Function and Their Interaction with the Motivic Structure The horizontal trace of the main motivic cells as internal lines hidden in a succession of chords illustrated above is a consistent technique throughout the work. Marking a similarity with the Duet, the chordal passages in the Quintet are conceived as mutations of a single and relatively definite structure. Resemblances in the conception and characteristics of this configuration in both pieces support its homonymic definition as CH, a chordal structure that contains almost without exception D, E, or F in register 4. The only exception to this condition occurs at the symmetry axis of movement II, where E is transported to register 5. In addition, as in the Duet, CH is here a set that explicitly excludes any version of (012). The exception to this is, of course, the vertical form of M(D) in the final measure mentioned above. As a result the definition of CH is, indeed, the same one given for the Duet: CH {D 4,E 4,F 4 } and CH (012) Finally, the special property that distinguishes CH in the Quintet from the same kind of structure in the Duet is the horizontal conception of the individual lines delineated by the successive presentations of its different forms. Figure 2.14 illustrates the chordal passages throughout the piece.

100 89 As shown in Figure 2.14A, the sequence of chords in movement I clearly establishes the vertical position of D, E, and F in register 4 supporting M(D) and the chromatic path between M(D) and M(A) that belongs to the Y domain (<C-B-Bb-A>). Notice that the chromatic descent from C to Bb becomes an explicit melodic line at the end of the movement, leading to movement II. In addition, the initial G in the viola in connection with the final A in the violin represents the empty version of M(A) and the outer components of M(D) are vertically aligned at the beginning of the passage. In the initial chordal section of movement II (mm ), E is transposed to register 5, up an octave in regard to the previous chordal passage in movement I, while D and F are alternated as two different representations of register 4. More importantly, the succession of pitches {B,Bb} is the unifying element of the entire passage from a horizontal point of view: the uninterrupted alternation of B and Bb between consecutive chords becomes a defining factor of the chord progression. As illustrated by the arrows in the first graph of Figure 2.14B, this melodic succession is vertically highlighted by the registral movement. The registral disposition (vertical elastic treatment) of the pitches B and Bb implicitly delineates a symmetric figure. Following the bi-dimensional conception of the space that dominates the work, the symmetric design does not result from a single axis but rather from two different lineal centers of reflection: the chords in mm are the vertical mirror image of those in mm with respect to the Bb 3 in m. 15, while mm are the horizontal reflection of mm Whereas the first axis of symmetry can only be delimited temporally, the second one is defined in the pitch space. Not coincidentally, the pitch axes around which B and Bb are horizontally reflected are F 4 and E 4 respectively, the two central pitches of the chordal passage in movement I (see Figure 2.14A). F 4 is, in effect, the most invariant pitch during this manifestation of CH in the second movement, while E 4 is precisely the note that is registrally transferred.

101 90 Figure 2.14 Registral Distribution and Evolution of CH Through the Movements A. Registral Distribution of CH in Movement I B. Registral Distribution of CH in Movement II

102 Figure 2.14 continued 91

103 92 Figure 2.14 continued C. Registral Distribution of CH in Movement III This symmetrical design is extended (becoming asymmetric) in the last measure of the passage (Bb and B in m. 20). The chromatic step between B and Bb represents the chromatic filling of the Y domain, the essential defining factor of scale Y, precisely the pitch collection that dominates the movement, playing a central role in its formal organization. The consistent presence of G in register 4 prepares the noticeable registral transference of this pitch in the following measures, an aspect discussed earlier in this chapter. This passage is retrograded in mm

104 93 The horizontal registral movement of G that takes place for the first time in mm (explained during the pitch analysis of movement II) links, at the middle ground, the section discussed in the previous paragraph with the next chordal passage in the movement (mm ). This passage precedes and follows the two sections based on the elastic display of scale Y. As shown with the rectangular geometric enclosures in the second graph of Figure 2.14B, the succession of chords in this section delineates horizontally and also vertically to a lesser degree, complete and empty versions of M(A), M(D), and M(E). The registral movement of the pitches involved in the motivic sets suggests vertical elasticity. Notice, in addition, the harmonic statement of the inversional axis {B,C} in m. 34 (violin and viola) and the appearance of {Bb,B}, the chromatic path that connects M(A) with {B,C}, at the middle of the passage (cello, mm ). {B,C} and {B,Bb} are illustrated with the transparent enclosures in the figure. The passage is retrograded in mm The first section based on scale Y is followed by another chordal passage (mm , represented in the third graph of Figure 2.14B). Reversed in mm , this section horizontally delineates {Bb,B,C}, {Eb,E,F}, and {C,C#,D}, the versions of M that highlight the chromatic Y domain, the intra-z-domain relationship between M(E) and {E,F}, and M(D) respectively. This is illustrated with the geometric enclosures in the figure. The other succession of chords in movement II is located before and after the symmetry axis (mm and mm ). Surrounded by the repetition of the pitch A in the strings (cello in mm and all strings in mm ), this passage, illustrated in the last graph of Figure 2.14B, is the chordal representation of the two main transformational paths between M(D) and M(A) or, what is the same, the vertical manifestation of scale Y. The axis {E,F} that converts M(D) into M(A) through inversion results from the horizontal interpretation of the top pitches of the first two chords. In a similar way, {Bb,B}, the most essential portion of the chromatic path that

105 94 connects M(D) and M(A) through the Y domain, is horizontally traced in the second voice of the last two chords of the passage. While M(A) is represented only by its highest pitch class (that appears in the strings before and after the chordal passage and it is not included in the figure), precisely the note that is adjacent to the Y domain, M(D) is horizontally stated both as an incomplete and complete three-note chromatic cell in the previous figure in the cello (mm ). As a result, the chordal expression that highlights the symmetry axis contains the essential motivic information of the movement. Finally, illustrated in Figure 2.14C, the succession of chords that finishes the work (mm of movement III) brings the totality of the defining pitches of CH, {D,E,F}, to the central registral position (register 4) at the same time that it vertically transports the central element of M(D), C#, through three different registers (5, 6, and finally 3) in the short time span of three measures even though the first C# appears four measures before the end, it is prolonged until the next measure. In addition to the horizontal statement of the components of the network explained in the previous section, the registral disposition of the succession of pitches {B-Bb-B} in this passage evokes the chordal sequences of the middle movement. Finally, the registral evolution of this passage creates the largest level of vertical expansion in the entire piece: the ascending line delineated by the successive entrances of pitches suddenly drops to register 2 to present the lowest version of M(D). The remaining chordal structures that appear in movement III are represented in the first part of Figure 2.14C. It is important to point out here the relatively large separation between the chords that are illustrated in this figure: mm , mm , mm , and mm As shown by the geometric designs in the graph, a large-scale analysis reveals the alternation of the empty version of M(A) with a vertical structure that is directly derived from the essential components of the inversional axes. Furthermore, F and B are precisely the pitches that are not contained in any of the main versions of M.

106 95 Conclusions The symmetrical conception of the Quintet that becomes evident in its formal organization is strengthened with a perfectly balanced theoretical model. In effect, the transformational network that rules the pitch organization is not only conceived but also exploited in a symmetrical way. The chromatic/non-chromatic dichotomy constantly present throughout the piece plays an essential role in the definition of the transformational network and at the same time conveys fluctuations in pitch-class density. This particular conception of the pitchclass space enables Blacher to change the chromatic density of the pitch material, conveying pitch-class compression and decompression, and to create motivically related figures of different lengths, suggesting horizontal stretching. The empty and complete successive statements of M and interval class 5 in connection with the idea of filling pitch-class spaces also contribute to the sense of elasticity. Finally, the conceptual notion of the network and its operational method imply motivic transformations that are the result of stepwise melodic extensions of two different kinds, chromatic and nonchromatic or inversional, contributing to the sense of horizontal elastic evolution. The compression and expansion of the pitch-class space are significant factors in the musical effect of the Quintet. As a general rule, chromatic density tends to increase the need for some kind of less chromatic resolution. For instance, the obvious chromatic condensation that precedes the final chords is an evident preparation for the end, a basic, symmetrically distributed-in-register version of M. In this passage, careful management of the dynamic levels along with a sense of increasing intensity for the growing-in-length and expanding-in-register musical phrases should effectively build the audience s expectation for the final resolution. In connection with this, balance of volume, length, stress, and articulation in the very last chord in the strings will contribute to the communication of its motivic function and its ultimate resolving role. A similar performance perspective should have a powerful perceptual effect in the gradual

107 96 chromatic compression towards the symmetry axis in movement II. In addition, tension and relaxation relationships result, at different levels, from the empty versions of the motives in direct connection with the pitch completion that frequently follows them. For example, the G# 6 in m. 18 of movement II is not the mere final pitch of an ascending scale: it is the fulfillment of the empty version of M(A) stated in the oboe in mm and the resolution of a gradual process of increasing tension that starts at the beginning of the movement. A meticulous performance of the crescendo indicated in the flute in mm along with interpretative emphasis (motivic, even dynamic prominence) on the <A- G> presented in mm (even with respect to the rest of the oboe melodic figure, which is precisely a combination of the empty version of M(A) and M(D)) should have a positive impact in this respect. Precise interpretation of the relatively long duration of the G# in the flute is another important aspect to be considered. At a larger scale, the G# in m. 18 is the continuation of the initial A, delineating a version of M(A) that is not complete until the violin presents G in m. 23. In light of this, equal performance stress on the A in the violoncello in mm. 1-2, the G# in the flute in m. 18, and the G in the violin in m. 23 should contribute to the delineation of M(A) at this level. The performers awareness of the pitch structure of the overall texture is, therefore, essential. Likewise, the D# in m. 19 of movement I satisfies the empty versions of M(D) and the gradual chromatic compression that characterize the opening of the piece. Finally, the comprehension of the motivic transformations through the entire texture in relationship with the completion of the chromatic pitch-class space should contribute to the performance quality of passages of these features. Strong similarities in the overall formal, textural, and registral conception of the Duet and the Quintet in conjunction with evident resemblances in the specific motives employed and the way they are developed and transformed suggest that the systematic organization present in the two pieces is the result of a more general theoretical model.

108 97 However, further analyses of Blacher s compositions are crucial to describe a generic systematic model and its universal operational principles.

109 98 CHAPTER III DIVERTIMENTO FOR WOODWINDS, OP. 38 Elastic Phrases Within a Traditional Formal Scheme Consistent with its earlier date of composition, Divertimento, op. 38 (1951) is a three-movement work that belongs to a more traditional universe of formal conception and pitch organization. As shown in Figure 3.1, movement I is based on a ternary form that concludes with a coda and Movement II follows a classical theme-and-variations scheme. In addition, the motivic material of the piece is frequently based on centric diatonic collections. Figure 3.1 Formal Structure Movement I A B A Coda a * aa aaa b bb a aa aaa bbb bb m Movement II Theme Var. I Var. II Var. III a b b a m (Movement II) Var. IV Var. V Coda I I I II I III m * Lowercase letters are here intended to compare the motivic content of the internal sections across the larger parts. The letter denomination (i.e. a or b ) of the internal sections has been chosen accordingly to the major part to which they belong. Following this, a quantitative distinction in the label implies differences in the motivic content. Identity of motivic content is thus reflected in labels that are qualitatively and quantitatively equal. Following this, aa is, for instance, different from aaa. To provide the most illustrative example, the opening melody is in the key of D minor accompanied by a C# that moves to D at the end of the first phrase, causing the

110 99 convergence of the four instruments on the pitch D (m. 11). This idea of pitch centricity is still present in the Duet in a less traditional way. In the Duet, the center is not merely based on pitch classes but, more specifically, in registrally established pitches: {D 4,E 4,F 4 }, which operates in opposition to the chromaticism that results from the development of M, (012). The traditional general outline of the Divertimento is only the external frame of an original conception of form. The elastic notion that shapes the Duet and the Quintet is also conveyed, to a lesser degree, in the Divertimento. While in the two later pieces the elastic conception is the defining process of the form, in the Divertimento the stretching of phrases and textural layers interacts with a stricter sectional formal design. A clear traditional form frames phrases that evolve elastically. This sense of elastic development is mostly a consequence of the metric organization reinforced by the pitch and rhythmic structure. The systematically evolving metric units that Blacher used in the first movement of the Duet is in the Divertimento the basis for the formal structure at the lower levels. The pitch and rhythmic organization of the individual measures directly contributes to the perception of the metric units: all the sequences of metric shifts are clearly audible. The motivic material outlines the metric structure, which is almost without exception reinforced by downbeat attacks. The metric structure of both movements is based on a metric row composed of systematically related sub-rows. The master row occupies mm Each sub-row is constituted of measures that progressively increase in length by an eighth note: each measure is one eighth note longer than the previous one. Following this additive procedure, subsequent sub-rows systematically extend by one measure. As a result, the first sub-row, 2/8 3/8, is followed by the sub-row 2/8 3/8 4/8, and so forth. The row in its original form (mm of movement I) is represented in Figure While in the 34 Pape uses in his thesis the term row to refer to what it is here defined as sub-row. As will be discussed later, future statements of the sub-rows as a group and transformational operations applied to the entire

111 100 Duet the meter changes are notated within the music, in the Divertimento the metric organization is explicitly indicated by Blacher only once at the top of the score. Figure 3.2 Metric Row 2/8 3/8 2/8 3/8 4/8 2/8 3/8 4/8 5/8 2/8 3/8 4/8 5/8 6/8 2/8 3/8 4/8 5/8 6/8 7/8 2/8 3/8 4/8 5/8 6/8 7/8 8/8 2/8 3/8 4/8 5/8 6/8 7/8 8/8 9/8 This is possible only because in the Divertimento the partitions of the row (sub-rows) are not defined in terms of the systematic increase or decrease in the number of measures in each meter as in the Duet, but according to a succession of metric changes without repetition. This procedure marks the central difference with regards to the internal conception of the metric row in the two compositions. In both pieces the elastic effect is created by the progressive increase and/or decrease in the length of the sub-rows; in the Duet, this is a consequence of the addition and subtraction of measures in a single meter (3/8 or 4/4) while in the Divertimento, a result of the addition (and also subtraction, as will be discussed below) of a measure that is the sole representation of its metric configuration. Another essential difference in the metric organization of both pieces is that in the Divertimento the row is stated more than twice and certainly more than once in its original form. In both pieces the row is retrograded. However, the axes that generate the reflected forms of the rows are conceived slightly differently in the two compositions. In sequence of sub-rows in connection with traditional serial terminology support the classification adopted in this dissertation. The representation of the metric structures in Figures 3.2 and 3.3 has been inspired by Pape s work (Louis W. Pape, Aspects of Meter, 74-75). It is also relevant to point out here that Boris Blacher recognized himself that his idea for the metric organization was inspired by Schoenberg s melodic row (Christopher Grafschmidt, Variable Metrik, 42).

112 101 the Divertimento the last sub-row is treated as the central point between the original form and the retrograde. In other words, the last sub-row of the original form works also as the initial sub-row of the retrograde. In the Duet, on the other hand, the middle sub-row is, in fact, the combination of the last sub-row of the original form with the first sub-row of the retrograde, which become an inseparable group of metric units because of the lack of meter change at the middle point. As shown in Figure 3.3, the last sub-row of the original row in the Quintet is its own retrograde, functioning as the axis of the larger symmetrical structure that results from combining the original form of the row with its reverse statement. Figure 3.3 Metric Row and Its Retrograde 2/8 3/8 2/8 3/8 4/8 2/8 3/8 4/8 5/8 2/8 3/8 4/8 5/8 6/8 2/8 3/8 4/8 5/8 6/8 7/8 2/8 3/8 4/8 5/8 6/8 7/8 8/8 2/8 3/8 4/8 5/8 6/8 7/8 8/8 9/8 2/8 3/8 4/8 5/8 6/8 7/8 8/8 2/8 3/8 4/8 5/8 6/8 7/8 2/8 3/8 4/8 5/8 6/8 2/8 3/8 4/8 5/8 2/8 3/8 4/8 2/8 3/8 Just as the central eight-measure sub-row represents the axis of the symmetrical metric figure, in a similar way a single measure in 2/8 separates the successive statements of the latter. This symmetric scheme, which is stated four times during movement I, delineates the large scale of the formal structure. The original form of the row by itself (Figure 3.2) does not coincide with the internal formal sections. As shown in Figure 3.1, the section aa begins in m. 27, where a textural change and a whole note E that works as a pedal lead to new motivic material that is centered on A rather than on D (flute in m ). The next important formal articulation falls in m. 43, after a weaker division

113 102 between mm. 38 and 39. The metric row, on the other hand, is not completed until m. 35 and the last sub-row, which is precisely the one that is its own retrograde (the larger indivisible unit), begins in m. 28. Notice, however, that a motivic progression elaborated by pitch addition on the pitch center A starts in the flute in m. 28. In effect, the E pedal in m. 27 prepares the presentation of the flute melody in m. 28. In other words, m. 27 functions as the upbeat of m. 28. The graph in Figure 3.4 shows the interaction between the extending and contracting metrical structure and the sections of the movement. The continuous and discontinuous lines show the subdivisions of the form at the middle level. 35 Figure 3.4 Interaction Between the Meter and the Form in Movement I 35 Specific measure numbers are shown in Figure 3.1.

114 103 The entire cycle of extension and contraction, i.e. the symmetric design resulting from the row and its retrograde as represented in Figure 3.3, coincides with the articulation of the form at the macro level. In other words, as Pape points out, the formal and metrical structures are coincident at a large scale. 36 However, the middle-level articulations of the formal structure are out of phase with respect to the metric organization. While the general shape of the musical phrases, the textural changes, and the overall pitch organization suggest strong articulations at the middle ground that do not coincide with the metric sub-rows, the rhythmic and pitch organization of the individual measures contributes to the perception of the evolution of the metric structure. The elastic development suggested by the expansion and contraction of the metric sub-rows is therefore audible. Finally, the overall shape of the Divertimento results from the interaction of a sectional, traditional idea of form with an original notion of elastic development. The delimitation resulting from both conceptions is coincident only at the largest (ternary form) and smallest (measure) levels of the total shape. It is precisely the lack of coincidence at the middle scale that supports the idea of an original elastically evolving shape framed in a static, sectional formal design. The metric row of movement II is clearly derived from the master row represented in Figure 3.2 and it expands through the first twenty-seven measures. As shown in Figure 3.5, the first sub-row comprises a sequence of measures decreasing in length from 8/8 7/8. Figure 3.5 First Statement of the Row in Movement II 8/8 7/8 8/8 7/8 6/8 8/8 7/8 6/8 5/8 8/8 7/8 6/8 5/8 4/8 8/8 7/8 6/8 5/8 4/8 3/8 8/8 7/8 6/8 5/8 4/8 3/8 2/8 36 Louis W. Pape, Aspects of Meter, 137.

115 104 In arithmetic terms, the difference between 3/8 and 2/8 is equivalent to the distance between 8/8 and 7/8: 3/8-2/8=8/8-7/8=1/8. From a musical point of view, the pair of meters 2/8-3/8 and 7/8-8/8 are equivalent in terms of the distribution and nature of their individual components: in both cases, the second meter is the first meter increased by an eighth note (i.e. 8/8 and 3/8 are an eighth note larger than 7/8 and 2/8 respectively). Therefore, 8/8-7/8 is related to 2/8-3/8 by operational inversion (the subtraction of an eighth note) in a similar way than it is to 7/8-8/8. The arithmetic mean of 3 and 7, the lineal middle point, is 5. In effect, 8/8-7/8 is the inversion of 2/8-3/8 about 5/8. However, this satisfies the idea of inversion in a lineal plane only, while the notion of musical inversion in its most complete sense (and especially in the way that it has been treated in this thesis) cannot be entirely conceived separately from the geometrical space. The eight different components that constitute the original row in movement I (2/8, 3/8, 4/8, 5/8, 6/8, 7/8, 8/8, and 9/8) are geometrically represented in Figure 3.6. In this context, 8/8-7/8 is the inversion of 2/8-3/8 about 5/8 and 9/8. Since the meters are now points in a bi-dimensional space, two elements are necessary to define an axis. Figure 3.6 Geometric Inversion of the Metric Row

116 105 The first sub-row that forms the metric row of movement II is, then, the inversion of the original sub-row of the piece about 5/8 and 9/8. Even more, the entire initial row of movement II is an inversion of the first six sub-rows presented in movement I about 5/8 and 9/8. In light of this, the only sub-row from the original row of movement I that is not inverted in the final movement is the last one, precisely the one that functions as its own retrograde in the initial movement. The sub-row that is not inverted is the most complete symbolization of the inversional axis, since it is the sole metric segment that contains both the meters of 5/8 and 9/8. Finally, the metric row acquires a similar meaning to a traditional serial row in the sense that it is subject to different transformational operations throughout the piece. Movement II presents this inversion of the row (as represented in Figure 3.5) six times, coinciding with the delimitation of the theme and each of the variations. Nevertheless, as will be expanded below, the formal divisions at the lower levels are not always concurrent with the internal partitions of the inverted metric row (inverted subrows). Figure 3.7 shows the interaction of the metric and formal structures at different levels. As illustrated by the design delineated by the horizontal lines in the graph, the principle of retrograding applied in movement I returns to provide metric symmetry to the second coda. However, the repetition of the sub-row 7/8-8/8 (retrograde of the inverted sub-row 1) in m. 217 and the omission of the sub-row 4/8-5/8-6/8-7/8 (retrograde of the inverted sub-row 4) at the end of the last section make the symmetric design only apparent. 37 As shown in Figure 3.7, the metric and formal subdivisions are coincident in the theme and variations III and V (see the consistency between the boundaries of the horizontal lines and the change in their continuous or discontinuous design). In variation 37 Both Pape and Grafschmidt point out the omission of the sub-row at the end of the last section, but they miss the repetition of the 2-measure sub-row in the first section (Ibid., 80-81; Grafschmidt, Boris Blacher Variable Metrik Ihre Ableitungen, 136).

117 106 I, the second important formal articulation is displaced one measure with respect to the metric partitions. This is illustrated in the graph by the difference in format of the second horizontal line (which corresponds to the second inverted sub-row) of that variation. In effect, m. 37 is texturally, rhythmically, and timbrally associated with the previous passage. As will be explained in the final section of this chapter, the chordal figure <{D,F}-{Db,F}> plays in this variation, in connection with the presentation of the theme, a central role in highlighting the interaction between the metric structure and the sectional form. Variation II is characterized by a noticeable lack of coincidence between the articulations of the form and the sub-rows (they are never aligned). Figure 3.7 Interaction Between the Meter and the Form in Movement II

118 107 The displacement of the parts with respect to the metric sub-rows is mainly a result of the motivic parallelism between the sections: mm and are motivically related to the opening section (mm ) and mm is connected to the final part of the variation (mm ). Furthermore, the five sections are parallel in terms of their textural outline, which strongly reinforces the formal divisions. Illustrated in Figure 3.8, variation IV represents the paradigm of the notion of interacting shape as each instrument delineates a different formal scheme. Figure 3.8 Formal Interaction in Variation IV The motivic organization of the clarinet highlights the elements of the metric row: the same motive starts at the beginning of every sub-row and it is expanded according to the organization of the metric row. The metric structure is reinforced by the bassoon, which plays a continuously expanding figure that is situated at the end of each metric sub-row. Consequently, the progression of meters is indeed perceivable. Measures , corresponding to the beginning of this variation, are shown in Figure 3.9. In the first inverted sub-row (mm ), the bassoon figure is in its most compressed form and it coincides with the last measure (the second measure in this case) of the sub-row and/or

119 108 clarinet motive. As a result of the process of melodic and metric expansion that takes place in both instruments, the bassoon s motive always starts one measure after the clarinet s. The elastically evolving shape is thus highlighted in the lower parts. At the same time, based on expanding motives that extend beyond the articulation of the lower parts, the flute and the oboe contradict the formal definition of the clarinet and the bassoon. The first entrances in each of the top parts (m. 113 in the flute and m. 117 in the oboe) emphasize, as the bassoon does on m. 110, the final measure of the metric sub-row. However, the second entrance of the flute and the oboe (mm. 116 and 121 respectively) melodically extends across the articulation of the metric structure: m. 118 of the flute and m. 123 of the oboe are motivically connected to the previous passages. Figure 3.9 Variation IV, mm

120 109 In addition, the same motivic cell is stated in these two measures, reinforcing the relative pitch dependence of the upper parts, which frequently move in parallel thirds. The last two phrases of this pair of instruments (mm and mm ) proceed in a similar way to the clarinet and the bassoon in the sense that the flute and the oboe have coincident ends of phrases (mm. 130 and 135) that are displaced by one measure at the initial point (mm and ). Nevertheless, except for the end of the variation (m. 135), the upper parts articulate the form in a different way than the lower instruments: mm. 128 and 129 are strongly connected by articulation (the slur across the bar line) and their similar rhythmic and chromatic motion. Finally, the relatively long periods of rests between the successive phrases of the flute and the oboe as well as the bassoon along with the staggered entrances of the instruments convey the perceptual effect of a continuously moving space in which musical figures appear and disappear rather than a form sectioned in parts. The first twenty-seven measures of each section of the Coda are equal in terms of melodic and rhythmic content, with a change in instrumentation and a registral transposition of the flute part in mm (parallel to mm ). 38 Transposed up an octave, the flute part creates vertical expansion in this passage. The registral transference is anticipated by the vertical displacement of the oboe line in m. 220, where the original E 5 (from m. 163) is transposed to E 4, delineating ascending motion to the next statement of that note (E 5 in mm. 222). The only formal articulation that does not coincide with the partitions of the metric row in these two parallel sections is emphasized with the registral transposition of the flute part in m The only remaining point in the entire coda in which the metric and formal structures operate independently takes place in m. 198, where the flute introduces a progressively expanding arpeggiated figure (motivic development by pitch addition) exactly one measure after the beginning of the 38 Louis W. Pape, Aspects of Meter, 81.

121 110 retrograded inversion of the second sub-row. This is the last point in the piece in which the metric and formal structures are out of phase. Not coincidentally, the articulation (the slurs across the bar lines) and the general unfolding of the motivic lines in mm obscure the segmentation of the metric row (the articulation between the retrograded inversion of the first metric sub-row and the second one, which falls precisely between those measures) for the only time in the work. 39 Blacher s idea of form implied by the interaction of the elastically developing shape with the more traditional notion of successive, relatively proportional sections is accompanied by a single conception of horizontal and vertical musical space. This notion, which is later developed into the principle that generates the very elastic shape in the Duet and the Quintet, is here mainly present in the motivic development in direct relationship with the harmonic and melodic presentations of interval classes. As will be discussed, the simultaneous and successive statement of interval classes establishes motivic connections at different levels of the formal structure. This idea, which is the same that Schoenberg introduces in his essays Composition with Twelve Tones, provides a great sense of coherence and continuation to the entire work. 40 From this point of view, the organization of the components of the Divertimento operates consistently with Schoenberg s method. Nevertheless, the essence of the elements and the pitch structure are closer to the tonal world than they are to the atonal universe. After 1948 he [Blacher] began to come to terms with the 12-note method, but in this he was attracted more by its possibilities of interval ordering than by its atonal features This is the only moment in which an indivisible motivic cell extends across the boundary of the metrical structure. In the remainder points in which metrical and formal structures are not coincident, the lowest level of the motivic structure is still concurrent with both delimitations. 40 Arnold Schoenberg, Style and Idea: Selected Writtings of Arnold Schoenberg, ed. Leonard Stein (London: Faber, 1975), Josef Häusler, Blacher, Boris, in Grove Music Online.

122 111 The Divertimento constitutes a representative example of Haüsler s point. It is a neoclassical work based on the intervallic conception of Schoenberg s method. Motivic Development Through the Bi-dimensional Conception of Interval Classes Movement I As shown in Figure 3.10, the piece opens with an octave-doubled melody in the flute and the clarinet that is, as mentioned at the beginning of this chapter, not only based on the diatonic collection of D minor but also centered on D. Even though the melody opens with scale degree ^b7, it clearly moves to and away from D. The initial descent from ^b7 to ^1 in mm. 1-5 and the line from ^1 to ^5 and back down to ^1 in mm emphasize the relative diatonic position of D with respect to the other notes. Figure 3.10 Movement I, mm. 1-11

123 112 In addition, the C# held in the oboe and the bassoon contributes to the D-centered effect as it resolves to D in support of the melodic cadence (m. 11). As a result, the four instruments converge on D. The initial measure presents, then, the simultaneous version of {C,C#}, illustrated by the vertical rectangle in Figure As shown in Figure 3.11, this collection of pitches is horizontally stretched to become the main motivic configuration of the opening of section B. In mm , {C,C#} is repeatedly stated in successive eighth notes in the bassoon line. Figure 3.11 Movement I, mm The motivic development of interval 1 continues until m. 88 through successive transpositions of the original {C,C#}, prolonging the idea of motivic extension to higher levels of the formal organization. In addition, if we consider pitches instead of pitch classes, the {C,C#} in mm signifies also a registral contraction of the original disposition of {C,C#} in m. 1: the beginning of section B states {C,C#} in a single register whereas the opening of the work presents the pitches in four different registral versions: C# 3, C 4, C# 5, and C 6. Finally, the figure that starts section B is the horizontal

124 113 expansion (motivic development) and vertical contraction of the initial measure. The opening chord of section A (and the entire work) becomes the primary motive of the contrasting section B. An analysis of the simultaneous interval classes between the melody and the C# pedal in the first two measures reveals the sequence <1-3-4>. As shown in Figure 3.10, interval classes 3 and 4 are the most frequent harmonic sonorities in the passage. In addition, the melodic versions of the three initial harmonic intervals along with interval class 2 constitute important successive components of the flute melody. As shown in Figure 3.10, interval class 3 is delineated by the initial and final notes of the first two subrows, which coincide with the internal phrasing of the melody: {C,A} in mm. 1-2 and {G,E} in mm Melodic interval class 1 (illustrated in circles in Figure 3.10) is mainly emphasized by the isolated slurs of Bb to A in m. 2 and F to E in m. 7. In addition, interval classes 3 and 1 are conjunctly highlighted in the motivic figures <F-E- F-D-E> and <F-E-F-D> in mm. 4-5 and 6-8 respectively, which are not only two forms of (013), more precisely {D,E,F}, but also two figures that directly contribute to the perception of the D-minor center. Interval classes 3 and 4 are especially emphasized in the second part of the melody (mm. 6-11): the initial F in m. 6 is four semitones apart from the highest pitch of the section, A, and three semitones away from the lowest pitch, D. The preponderance of interval class 2 is an obvious consequence of the diatonic stepwise melodic motion. Finally, the entire passage highlights interval classes 1, 2, 3, and 4, especially emphasizing the thirds, firstly in its minor kind (interval class 3) and secondly in its major quality (interval class 4), by means of their role in the definition of the shape of the melody and the frequent appearance of their simultaneous forms. Furthermore, the only non-stepwise motion in the melody is {D,F} or interval class 3. The same interval classes presented in the initial measures acquire a peculiar disposition in the following passage. In mm , the homorhythmic figure in the flute and the oboe delineates the sequence of simultaneous intervals <3-3-2> as well as the

125 114 successive version of intervals 1 and 2. As a result, the original bi-dimensional disposition of interval class 1 becomes purely horizontal, transferring its bi-dimensional manifestation to interval class 2. Notice that, as a direct consequence of the superposition of the D minor natural collection with C#, interval class 2 appears only in its successive version during the presentation of the opening melody (as shown in Figure 3.10, it is never stated in simultaneity). The sequence of simultaneous intervals outlined by the homorhythmic figure (mm ) is retrograded and expanded two measures later, becoming <2-3-5>, at the same time that the successive intervals are reduced exclusively to 1 (mm in the flute and the oboe). Altogether, these homorhythmic figures emphasize the vertical quality of interval classes 2 and 3, two dominant successive intervals of the opening melody. These motivic configurations are the accompaniment of a melody in the clarinet that is still centered on D minor. The clarinet line is inverted about G# in the oboe part in mm. 21, with a variation in m. 25 (that corresponds to m. 19). In addition to the alteration of m. 19, the inversion is not exact in the sense that it does not preserve the quality of all the original intervals: the inversion is conceived in a way that allows the preservation of the D minor collection (it is diatonic rather than chromatic). The successive interval that is modified through the inversional process is the second, which is major in the clarinet s line and minor in the oboe s. Following this, while the line in mm (clarinet) emphasizes interval 2, its inversion (oboe in mm ) highlights interval 1. In mm , the inverted melody is accompanied with the homorhythmic figure of the previous passage, now in the form of simultaneous interval class 3s. As a result, the importance of interval class 2 vanishes in both the melodic line and the homorhythmic figures that accompany it. In a similar way, harmonic interval class 3 gives place to simultaneous interval class 4 in m. 27, where the bassoon and the clarinet play the sequence of vertical intervals < > to arrive at a homorhythmic figure that evokes, in parallel motion by

126 115 thirds, that from mm Presented in mm , this polyphonic motive delineates simultaneous interval class 4, returning to 3 in mm It is relevant to point out here that harmonic interval class 4 in m. 27 is, in general lines, gradually derived, by vertical expansion, from interval class 2: the sequence of vertical interval classes <3-3-2> in the homorhythmic figure from mm becomes <2-3-(5)> in mm , <3-3-3> in mm , and finally < > in m. 27. Even more, this process of intervallic vertical expansion has its origin in the simultaneous presentation of interval class 1 from m. 1. The flute line in m. 27 moves in contrary motion with respect to the bassoon and the clarinet. Contrary motion and pitch (as opposed to pitch-class) inversion are two similar kinds of reflecting processes: they both operate with respect to a horizontal axis. They differ only in the degree of exactness of the reflected image that they create with respect to the original object: while contrary motion reflects only the general contour (upward or downward motion) of a motivic line, inversion horizontally reverses its particular intervals. In the case of the section being discussed here, as it is in most of the inverted passages in the three pieces analyzed in this thesis, the procedure of inversion is applied melodically (rather than harmonically). The oboe line in mm represents, then, the successive (i.e. horizontal) unfolding of the horizontal reflection of the clarinet line in mm The temporal arrangement of the horizontal reflection of the clarinet part that takes place in mm is reinforced by a textural change that also suggests an inversional relationship (in a broad sense) with respect to the previous passage, being the horizontal axis located between the oboe and the clarinet parts (lines 2 and 3): in m. 21 the main melodic material passes from clarinet (line 3) to oboe (line 2) at the same time that the homorhythmic figures that accompany it move from flute and oboe (lines 1 and 2) to clarinet and bassoon (lines 3 and 4). In sum, the textural change that occurs in m. 21 is based on a horizontal axis of symmetry, supporting the motivic relationship between the phrase that finishes at that point (clarinet melody) and the one that begins (oboe line). In m. 27, on the other hand, the clarinet-and-bassoon line and the flute part are

127 116 simultaneous reflections of each other with respect to horizontal axes of symmetry. Since the reflection develops harmonically, a distinction between original and mirror image is irrelevant. Finally, the difference between the symmetrical processes of mm and m. 27 is the temporal arrangement of the mirror image with respect to the object being reflected, successive (i.e. horizontal) in one case (as the oboe inverts the clarinet in mm ) and simultaneous (i.e. vertical) in the other (as the lower parts and the flute move in contrary motion in m. 27). The restatement of the homorhythmic figures in mm , in the forms of harmonic interval classes 4 and 3, continues highlighting successive interval 1. Furthermore, by this point the bi-dimensional expression of interval class 1 that opens the piece has become almost exclusively horizontal. The homorhythmic figures serve as the accompaniment for a new melody that is, like the initial passage, mostly diatonic. The pitch center is now A. Marking a difference with the opening, the flute line in mm is not based unequivocally in one mode but rather in the bi-modal center of A major/minor, since it incorporates both C and C#, as well as G and G#. The idea of horizontally presenting two parallel pitch-class collections, A major and A minor, is vertically applied in the middle of section B (mm ) and its equivalent passage in the coda (mm ), where the D 7 and d 7 arpeggios are simultaneously presented. The horizontal statement of pitch collections in the flute in mm (which is restated in mm ) anticipates, then, the vertical arrangement of the two different modal qualities of the clarinet and the bassoon parts in mm (and the flute and the clarinet lines in mm ). All the passages discussed here contain the complete sub-row 7, which is the one that works as its own retrograde. The connection between these passages is also supported by the motivic evolution of the lines, which is based on successive presentations of a single, repeated cell that expands melodically in every measure, strengthening the metric progression and consequently causing the aural effect of horizontal stretching. Notice, in addition, that the arpeggios

128 117 are the major and minor representations of the original pitch center. Measures are illustrated in Figure Also present in the Duet and the Quintet, and applied in later passages of the first movement of the Divertimento, the procedure of additive pitch structure is an effective generator of the sense of horizontal elastic motion. It is relevant to point out here that this technique is also employed in mm and of movement I of the Divertimento. Not coincidentally, these passages develop from the cell {D,F}. Figure 3.12 Movement I, mm In terms of pitch content, the only difference between the arpeggios is the F# and the F, highlighting intervals 4 and 3 with respect to the root. Following this, while the

129 118 bassoon emphasizes interval class 4, the clarinet highlights interval class 3. This pair of interval classes is precisely the one that creates the homorhythmic figures in mm Finally, while the horizontal idea of combining parallel keys in mm (and mm ) is verticalized in mm (and mm ), the spatial conception of interval classes 4 and 3 operates reversely, first harmonically and then successively. The textural layer of the flute and the oboe in mm , which is clearly differentiated, rhythmically and in terms of pitch content, from the lower parts, horizontally unfolds the set (013) through a melodic shape that emphasizes interval 3: the first and last note of each of the two small sections in which the flute-and-oboe melody can be articulated (mm and mm ), <Eb,C> and <C,A>, are three semitones apart. Furthermore, the minor third is explicitly stated in the two last contiguous notes of the passage in each instrument. This is illustrated in Figure These intervallic characteristics evoke the motivic elaboration of the initial measures. At the same time, the simultaneous intervals that result from the superposition of the flute and the oboe delineate a sequence of interval class 3s, not only recalling the homorhythmic figures of mm , but also suggesting the convergence of the melodic and harmonic forms of interval class 3: in mm , the flute and the oboe highlight the bi-dimensional quality of interval class 3. Finally, the passage in mm brings together the elements that are gradually introduced during the first thirty-eight measures. This passage leads to a section that is mainly constructed in parallel interval class 3s that horizontally trace intervals 3 and 1 as well as 2 to a lesser degree. In this section, mm , the texture is stratified into two parts that are not only different in their rhythmic and melodic shape but also non-coincident in their formal articulations. The flute and the oboe on one hand and the clarinet and the bassoon on the other move in parallel thirds, which are almost without exception of minor quality. The passage contains four sub-rows: length-decreasing units of 7, 6, 5, and 4 enlarging measures (it corresponds to the beginning of the retrograded form of the metric row). In the section

130 119 that coincides with the first sub-row (mm ), the oscillating interval class 1 from the beginning of section B is alternated with interval class 2 (flute and oboe). Simultaneously, a slower-moving figure highlights melodic intervals 3, 1, and 2 in the lower parts. Even more, the rhythmic shape of the melody in the clarinet and the bassoon delineates three closely related pitch sets, (0,3) in mm , (0,1,3) in mm , and (0,1) in mm , that especially highlight interval classes 1 and 3, clearly recalling the previous parallel line in the upper parts (mm ). The successive variations of this figure strongly reinforce the segmentation of the metric row from a perceptual point of view. The way in which this figure is contracted clearly emphasizes the melodic intervals it contains without explicitly repeating the original intervallic sequence, suggesting the idea that the motive is intervallically conceived. The form generated by the horizontal contraction of this motivic configuration interacts with a relatively non-coincident shape resulting from the pitch and rhythmic organization of the upper parts. In the flute and the oboe parts, m. 106 is the continuation of m. 105: the dynamic indication in these two measures (crescendo at the end of m. 105 and diminuendo in the following measure), the closeness in pitch (A to Bb in the flute and F to G in the oboe), and the long durational value of the Bb and G supports the idea of interpreting these two notes as the ending point of the preceding passage, contradicting an articulation at the end of m The next sections of the upper stratum are delineated in mm. 113 and 116, one measure later and one measure earlier than the metric row (and the clarinet-and-bassoon stratum) respectively. This lack of coincidence in the formal articulation of the different simultaneous strata (the flute and the oboe on one hand and the clarinet and the bassoon on the other) anticipates the process of shape development in the fourth variation of the second movement explained at the beginning of this chapter. The contraction of the musical phrases of the lower stratum (clarinet and bassoon) results from a systematic organization of successive intervals that is clearly perceptible without being directly exposed. The sequence of horizontal intervals in each of the four

131 120 contracting versions of the melody is systematically derived from the series of intervals < > as represented in Figure Figure 3.13 Intervallic Scheme of the Contracting Melody in mm (Movement I) Each melodic fragment and sub-row, except for the last one, follows the circle of intervals clockwise, parting from the penultimate interval of the previous segment, i.e. receding one step in the circuit. The first fragment starts at the top of the circle. The number of steps traveled systematically decreases by one: mm follows a sequence of six successive intervals, mm , a series of five intervals, and so forth. The last fragment (mm ) begins in the final position of the previous segment, without receding in the circle, and travels the circle in the opposite direction (counter clockwise) to finish at the departure point of the process, providing a great sense of coherence. Consequently, the last segment (mm ) retrogrades and inverts the first four measures of the initial segment (mm ). The resultant sequence of intervals outlined by each fragment is represented in Figure Figure 3.14 Sequence of Intervals in the Melodic Fragments of mm < > < > < > <1-1-3> m. 99 m. 106 m. 112 m. 117

132 121 The sequence of parallel minor thirds continues until the beginning of the recapitulation in m Even more, the individual lines are based on the pitch-class sets (01) (mm ) and (013) (mm ), connecting, at the middle ground, not only the preceding sequential passage discussed in the previous paragraph but also the motivic material from mm (flute and oboe) with the recapitulation of the opening melody. The analysis above has demonstrated the importance of interval classes 3, 2, and 1, without making reference to interval class 4 further than its brief introduction at the beginning of the intervallic discussion in connection with the evolution of the harmonic intervals during the opening section (the progressive incorporation of interval classes 2, 3, and finally 4 in m. 27). The vertical setting of interval class 4 that is emphasized in m. 27 acquires an essential function in the initial section of the coda (mm ). This passage, which is reproduced in Figure 3.15, is based on the imitation of melodic descending octaves. The interval classes that separate the entrances of each pair of imitative entrances are 4 (oboe and clarinet in mm ), 3 (flute and bassoon in mm ), 4 (oboe and clarinet in mm ), 2 (flute and bassoon in mm ), 4 (oboe and clarinet in mm ), and 6 (flute and bassoon in mm ), 4 being the returning interval class that separates the rest, framing the entire sequence. Consequently, interval class 4 is highlighted both in succession, as a result of the imitative process, and simultaneity, as the pairs of instruments overlap. This is illustrated with the continued arrows in Figure In addition, the second entrance (C in the flute in m. 194) overlaps with the end of the imitating voice of the first one (E in the clarinet), suggesting, at a lower formal level, a group of four entrances instead of two groups of two at the same time as it highlights interval class 4. In a similar way, the end of the imitating voice in m. 202 outlines interval class 4 against the A in the flute. The last overlapping of this kind in m. 210 generates interval class 1, evoking the beginning of the piece. These interval classes are shown with dashed lines in Figure In addition, the balanced disposition of interval classes (centered on interval class 4) is reinforced by a

133 122 symmetrical sequence of temporal distances between the parts: two eighth notes, four eighth notes, five eighth notes, and its retrograde (five eighth notes, four eighth notes, two eighth notes). The boxes at the bottom of the staves in Figure 3.15 illustrate the horizontal distance between the parts of each pair of imitative entrances measured in eighth notes. Figure 3.15 Imitative Passage that Opens the Coda in Movement I As mentioned at the beginning of this chapter, the second part of the coda (mm ) is very similar to mm : it is the same passage with the parts interchanged and a generally higher registral disposition. It is important to point out here that the restatement does not start in m. 215 but in m. 214, which is the repetition of m. 88. This idea suggests a formal articulation in mm. 88 and 214 rather than in the