UNIVERSITY OF CINCINNATI 14 August 2007 Date:___________________ Brian Christopher Moseley I, _________________________________________________________, hereby submit this work as part of the requirements for the degree of:
Master of Music in:
Music Theory It is entitled: Integrating Analytical Elements through Transpositional Combination
in Two Works By George Crumb
This work and its defense approved by:
Dr. C. Catherine Losada Chair: _______________________________ Dr. David Carson Berry _______________________________ Dr. Steven J. Cahn _______________________________
_______________________________ _______________________________
Integrating Analytical Elements through Transpositional Combination in Two Works by George Crumb A Thesis Submitted to the Division of Graduate Studies and Research of the University of Cincinnati in partial fulfillment of the requirements for the degree of
Master of Music
in the Division of Composition, Musicology, and Theory of the College-Conservatory of Music
2007
by
Brian C. Moseley B.M. Furman University, 2004
Committee Chair: Dr. C. Catherine Losada
ABSTRACT This study investigates ways that transpositional combination (TC) can be used to enhance our understanding of the pitch, form, and extra-musical elements of George Crumb’s Vox Balaenae and Lux Aeterna. Through TC, this thesis develops the idea that Crumb’s music often contains basic pitch material from which larger sets or collections are generated, and that this process is often allied with other musical elements. Chapter one approaches Vox Balaenae through a perspective that appreciates how small pitch-class sets interact with referent pitch collections. This interaction is related to extra-musical ideas that accompany the composition. Chapter two focuses on the details of the TC process. As an analytical cornerstone, this chapter applies new concepts to an analysis of musical form in Lux Aeterna in order to show how the smallest details of the generative process impact the overall form of the piece in a significant way.
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ACKNOWLEDGEMENTS I would like to express my sincere thanks to my advisor, Dr. Catherine Losada, whose guidance, encouragement, and critical mind provided me with the means to undertake this project. She offered her time in copious quantities and was an outstanding mentor. I also owe a great debt to Dr. David Carson Berry and Dr. Steven Cahn for their valuable suggestions. Their courses, our conversations, and their careful reading have shaped this project in innumerable ways. My parents, Will and Danielle Moseley, remained patient and confident during the time I worked on this project; I thank them both for providing unfailing motivation and for reminding me to enjoy my work. Finally, I thank Jessica Barnett, who has been my greatest source of encouragement and support.
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TABLE OF CONTENTS Introduction…………………………..……………………………………………………………1 Chapter One: Transpositional Combination, Collectional Interaction and their Extra-Musical Significance in Vox Balaenae……………………….....………………………………………….………….6 Chapter Two: Transpositional Combination and the Analysis of Form in Lux Aeterna….………………….38 Conclusions...…………………………………………………………………………………….80 Bibliography……………………………………………………………………………………..84
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Introduction
George Crumb’s reputation as a composer has long rested upon his novel use special timbral effects—a characterization that has diverted attention away from other interesting features of his compositions. Recent theoretical studies (canvassed below) have begun to acknowledge this concern by addressing issues of pitch organization and musical form. This study’s contribution to the growing body of Crumb scholarship will consist of an investigation of various ways that transpositional combination can be used as an analytical tool to enhance our understanding of the pitch and formal structure of two of his compositions, Vox Balaenae (1971) and Lux Aeterna (1972). Analytical approaches to Crumb’s music have generally proceeded from either a descriptive viewpoint that catalogs various surface features of a composition, or a more rigorous analytical approach that seeks to discover underlying processes or structural models. The descriptive approach often details the multitude of timbral effects found in a typical composition by Crumb. This approach is appealing because Crumb’s compositional style is so readily identifiable by its exploitation of new sonorities.1 Kenneth Timm’s analysis of Vox Balaenae (Voice of the Whale) is representative of the descriptive approach. In his analysis, Timm devotes considerable attention to the timbral catalog of the piece—where he finds “thirty-eight types of individual timbres”—and these timbres’ relationship to the sound of whales, apparently
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Robert Vernon Shuffett’s analyses of Crumb’s compositions from 1971–75 are perhaps the most comprehensive studies to adopt the descriptive-analytical approach. Robert Vernon Shuffett, “The Music, 1971–75, of George Crumb: A Style Analysis” (D.M.A. thesis, Peabody Conservatory of Music, 1979). See also Stephen Chatman, “George Crumb: Night of the Four Moons—The Element of Sound,” Music & Man 1/3 (June 1974): 215– 224; reprinted in revised form in Don Gillespie, ed., George Crumb; Profile of a Composer (N.Y.: C. F. Peters Corporation, 1986), 1986); and also David Lee Ott, The Role of Texture and Timbre in the Music of George Crumb (Ph.D. diss., University of Kentucky, 1982).
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attempting to link the types of timbres used to the title of the composition.”2 Unfortunately, Timm explores only how these different effects are juxtaposed or superimposed and does not relate his observations to other musical dimensions. While Timm devotes separate sections of his analysis to “Timbral Organization,” “Whale Sounds,” and “Extra-musical and Programmatic Aspects,” he does not incorporates these observations into a format that suggests, for instance, how the different timbres and the whale sounds reflect extra-musical elements of the piece. Rigorous analytical approaches that attempt to integrate the many musical dimensions of a Crumb composition have become more prevalent.3 In his dissertation and a series of articles, Richard Bass discusses both the pitch material and formal structures in Crumb’s Makrokosmos I and II. His analyses elucidate the underlying processes whereby primary pitch materials are assembled into organizational schemes. Through these analyses, Bass is able to make generalizations about not only the types of pitch constructions found in the Makrokosmos, but also the typical ways that these pitch constructions are organized on a larger scale. Symmetry is found to be a facet of both small and large pitch-class sets, as well as the formal structures that organize the pitch material. Bass states that in these pieces, basic pitch materials function as “primary structural units from which larger sets and scale types are generated through
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Kenneth Timm, “A stylistic analysis of George Crumb's Vox Balaenae and an analysis of Trichotomy.” (D.M.A diss., Indiana University, 1977): 17–25. 3
Richard Bass, “Pitch Structure in George Crumb’s Makrokosmos, Volumes 1 and II” (Ph.D. diss., University of Texas, 1987); Bass, “Sets, Scales, and Symmetries: The Pitch-Structural Basis of George Crumb’s “Makrokosmos” I and II.” Music Theory Spectrum 13 (Spring 1991): 1–20; Bass, “Models of Octatonic and WholeTone Interaction: George Crumb and his Predecessors,” Journal of Music Theory 38 (1994): 155–186; see also, Thomas R. de Dobay, “The Evolution of Harmonic Style in the Lorca Works of Crumb,” Journal of Music Theory 28/1 (Spring 1984): 89–111. David Headlam’s approach is interesting in that it attempts to integrate both timbral and pitch-structural approaches using spectrograms to analyze frequency amplitudes. David Headlam, “’Integrating Analytical Approaches to George Crumb’s Madrigals, Book I, no. 3,” in George Crumb and the Alchemy of Sound, Steven Bruns, Ofer Ben-Amots, Michael D. Grace, eds., (Colorado Springs: Colorado College Music Press, 2005), 235–267.
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symmetrically conceived arrangements of pcs.”4 In fact, the creation of large pitch-class sets (many of them being common referential collections like the whole-tone, octatonic, and pentatonic) through the combination of basic pitch material seems to be a common feature of many of Crumb’s compositions. Ciro Scotto notes this feature of Crumb’s Processional as well, though he finds that Bass’s emphasis on symmetry is too constricting.5 Like Bass and Scotto, this thesis develops the notion that Crumb’s music often contains basic pitch material from which larger sets or collections are generated. In order to study the methods of generation and their results, transpositional combination is used extensively as an analytical starting point. Transpositional combination was first formalized by Richard Cohn and has proven to be a powerful tool in the analysis of twentieth-century music.6 Cohn has published a number of articles expanding concepts originating in his dissertation and applying this set of concepts to composers as diverse as Béla Bartók, Alban Berg, and Steve Reich.7 Recently, articles by Mark McFarland and Ciro Scotto have used transpositional combination in novel ways to explore music by Claude Debussy and George Crumb.8 The diversity of approach
4
Bass, Sets, Scales, Symmetries, 3.
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Ciro Scotto, “Transformational Networks, Transpositional Combination, and Aggregate Partitions in Processional by George Crumb,” MTO 8.3 (2003). 6
Richard Cohn, "TC in Twentieth Century Music" (Ph.D. diss, University of Rochester, 1987). Marianne Kielian-Gilbert is an earlier exploration of transpositional combination. Marianne Kielian-Gilbert, “Relationships of Symmetrical Pitch-Class Sets and Stravinsky’s Metaphor of Polarity,” Perspectives of New Music 21 (Autumn 1982): 209–240; and also, Kielian-Gilbert, Pitch-Class Function, Centricity, and Symmetry as Transposition Relations in Two Works by Stravinsky,” Ph.D. diss, University of Michigan, 1981). 7
Richard Cohn, “Inversional Symmetry and Transpositional Combination in Bartók,” Music Theory Spectrum 10 (1988): 19–42; Cohn, Properties and Generability of Transpositionally Invariants Sets,” Journal of Music Theory 41 (Autumn 1991): 1–32; Cohn, “Transpositional Combination of Beat-Class sets in Steve Reich’s Phase-Shifting Music,” Perspectives of New Music 30 (Fall 1992): 146–177. 8
Mark McFarland, “Transpositional Combination and Aggregate Formation in Debussy,” Music Theory Spectrum 27 (Fall 2005): 187–220; Scotto 2003.
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evident in all of these articles indicates that transpositional combination is a highly flexible analytical tool that is able to adapt to different compositional styles and analytical needs. Transpositional combination is an effective tool for studying Crumb’s music because it allows the analyst to explore how basic pitch material is replicated at different transpositional levels to form larger musical units. In addition to this process-oriented approach that “generates upwards” from smaller pitch constructs, transpositional combination allows an analyst to view passages “downwards” from a referential collection and compare the generative paths of sections that make use of different referential collections. While this difference may seem slight—like two sides of the same coin—this thesis takes the viewpoint that the two perspectives can generate different approaches to listening. In a composition with a multitude of referential collections, transpositional combination encourages a hearing that recognizes interaction among these collections. But in a less collectionally interesting environment, transpositional combination allows the listener to attune to the multifarious processes and means by which a single collection is generated. The first chapter of this thesis approaches Vox Balaenae through a perspective that appreciates the role that small pitch-class sets have in creating and facilitating interaction among referent pitch collections. It argues that this interaction is a structural feature of the variation process that occurs in the piece. In addition to distinguishing between types of variation, the use of transpositional combination leads to some conclusions regarding relationships between pitchstructure and extramusical content. Chapter two begins with a discussion of additional concepts allied with transpositional combination that more accurately represent particular realizations on the musical surface. These concepts allow greater attention to be paid to the generative process. The final portion of this
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chapter applies these concepts to an analysis of musical form in George Crumb’s Lux Aeterna, a composition that makes use of a single referential collection. This analysis shows how the smallest details of a generative process can impact the overall form of the piece in a significant way.
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Chapter One: Transpositional Combination, Collectional Interaction and their Extra-Musical Significance in Vox Balaenae
Characterizations of George Crumb’s compositional style frequently invoke referential pitch collections to generalize about passages of music. Some use the terms descriptively to assert a measure of “unity,”9 while others have discussed more intricate features of these collections’ involvement with a composition’s pitch structure.10 This chapter will discuss Crumb’s Vox Balaenae (1971) using transpositional combination to examine the role of referential collections in the composition.11 It will investigate the nature of interaction among referential collections, using transpositional combination as an investigative tool to discuss types of collectional shifting that are important variation techniques in Vox Balaenae. 12 The final portion of this chapter discusses interaction as it relates to quotation and extramusical ideas associated with the composition.
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Shuffett, 1979.
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Richard Bass approaches some portions of the Makrokosmos I and II in this way, though he generally concentrates his efforts on the symmetrical disposition of primary units that ultimately generate these collections. Bass, Sets, Scales and Symmetry, 3–6. 11
Kenneth Timm and Russell Steinberg are the only other published analyses of this work. Kenneth Timm, 1977; Russell Steinberg, “’Meta-Counterpoint’ in George Crumb’s Music: Exploring Surface and Depths in Vox Balaenae (Voice of the Whale), in George Crumb and the Alchemy of Sound, Steven Bruns, Ofer Ben-Amots, Michael D. Grace, eds., (Colorado Springs: Colorado College Music Press, 2005), 211–233. 12
Arthur Berger and Pieter van den Toorn have discussed similar types of interaction in Stravinsky’s music, and Richard Bass has placed interaction between octatonic and whole-tone elements in the context of Crumb’s predecessors. Dmitri Tymoczko has contested van den Toorn’s approach arguing that the octatonic and diatonic interaction is less a feature of Stravinsky’s than proposed. However, rather than discussing these collection’s generative basis in smaller units, Tymoczko discusses Stravinsky’s use of various modes of the minor scale. Arthur Berger, “Problems of Pitch Organization in Stravinsky,” Perspectives of New Music 2 (Autumn– Winter 1963): 11–42; Pieter van den Toorn, The Music Of Igor Stravinsky, (New Haven: Yale University Press, 1983); Dmitri Tymoczko, “Stravinsky and the Octatonic: A Reconsideration,” Music Theory Spectrum 24 (Spring 2002): 68–102. see also Bass, 1994.
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Transpositional combination (TC) was first formalized by Richard Cohn to describe pitch-class sets that may be formed by combining transpositionally related sets.13 Example 1 illustrates a transpositional combination. Here, two (03) dyads are related to one another through T4. Their union produces set-class (0347), which is said to bear the “TC property.”14 Without overcomplicating this preliminary discussion, a note on the standard symbols should suffice. The brackets segment the musical surface into transpositionally related pitch-class sets, and the “*” between the set-class and the interval of transposition is the conventional way of indicating TC. Because transpositional combination is an operation, the transpositionally related set-classes and their transpositional relationship are referred to as operands: 3 and 4 in this case.15 The table underneath the example summarizes the operation. The two operands are represented along the border of the table and their combination is shown in the center. The union is created by adding the two operands to one another. In a TC operation, the transpositionally related set is shown in its normal order, and the level of transposition is represented by a dyad.
13
Cohn, “TC in Twentieth Century Music,” 59–61.
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In “Inversional Symmetry and Transpositional Combination in Bartók,” Cohn compares transpositional combination and inversional symmetry. Specifically, Cohn points out that many set-classes that bear the property of inversional symmetry also bear the TC property. Framing his argument in reference to Bartók scholarship, Cohn notes that it can be fruitful to represent sets as products of transpositional combination than inversional symmetry. Cohn, Inversional Symmetry, 20–27. 15
Dyads are generally abbreviated as single numbers. In Example 1, 3 represents the (03) set-class and 4 represents the (04) set-class. TC operations may have more than two operands. Crumb’s music generally makes use of low cardinality sets, and as a result, I have found that binary operations generally suffice as representations. Cohn, TC in Twentieth Century Music, 77–83.
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Example 1, 3 * 4 TC operation
4 3 6 0 E 2 E 2
Analysis of the variations from Vox Balaenae reveals three variation procedures involving transpositional combinations that result in collectional interaction.16 All of the variation techniques allow similar melodic motives or structural features to exist in different collectional contexts. The first two variation procedures use either of the two TC operands as a convergence point that connects different collections: Type I variation utilizes a set-class operand common to more than one collection as a pivot between collections; Type II variation maintains a common transposition operator to shift collections while varying the other operand. Figure 1 lists the dyadic, trichordal, and tetrachordal subsets of the diatonic, octatonic and whole-tone collections. The overlapping portions of the Venn diagram contain collectionally ambiguous sets that are members of at least two collections, and as a result can be used as convergence points to connect the collections.17 Example 2a is a model of Type I variation, using
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Vox Balaenae is organized formally into a prologue followed by a set of variations and an epilogue. The variations move chronologically through five geological eras. The musical associations with these eras are intended to roughly represent them. For instance, Crumb places a quotation from Strauss’s Also Sprach Zarathustra in the “Cenozoic” variation to coincide with the first appearances of man in that era. George Crumb, Liner Notes from George Crumb: Eleven Echoes of Autumn, Four Nocturnes, Vox Balaenae, Dream Sequence, performed by Hans Peter Frehner, flute, Samuel Brunner, cello, Viktor Müller, piano ([Zürich]: Jecklin Edition, 705, 1996).
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(026) as a convergence point. Here, each line shows two TC realizations with (026) as a common set-class; the first two trichords of each line are identical and the second trichord in each line is related to the first by transposition. In the first line, (026) * 2 forms a nearly complete whole-tone collection, while in the second, (026) * 3 creates a six-note subset of the octatonic collection. In this example, (026) is a literal convergence point that connects two different collections.18 Conversely, Type II variation maintains the transposition value, T6, as a point of convergence while varying the transposed set-class, as shown in Example 2b. This type of variation is generally more abstract because the transpositional relationship is most often a more structural feature that may be less salient on the musical surface. Variation Type III will be discussed extensively in the second half of this chapter. This procedure subjects an overtly diatonic object—such as the perfect fifth, triad, or major–minor seventh chord—to a T6 transposition that creates an octatonic collection. Though diatonic and octatonic collections are similar in terms of subset content—they share all of the major and minor triads, dominant and half-diminished sevenths, and the fully-diminished seventh, among others—this chapter argues that diatonic-octatonic interaction is not simply the result of subset similarity. The significance of diatonic-octatonic interaction is traced to the use of a recurring quotation of the Nature theme from Richard Strauss’s Also Sprach Zarathustra that Crumb slightly alters so as to juxtapose diatonic elements and the octatonic collection.19 The final
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If this hypothesis regarding the importance of collectional interaction in Crumb’s music is sound, it should follow that a survey of Crumb’s music would reveal a predilection for these collectionally ambiguous setclasses. In fact, Richard Bass shows that, other than the prominent octatonic subset (014), Crumb does indeed tend to use these set-classes as motives in many of his compositions. Bass, Sets, Scales and Symmetries, 5–7. 18
In some ways this is similar to common-tone modulation in tonal music, where a triad common to more than one key is used as a common element to modulate from one key to another. I will also show examples where the convergence point is abstract, not literally belonging to either of the two collections. 19 Crumb often uses quotation in his music. Steven Bruns has published extensively on this subject. Steven Bruns, “In stilo Mahleriano: Quotation and Allusion in the Music of George Crumb,” The American Music Research Center Journal 3 (1993): 9–39; Bruns, “Les Adieux: Haydn, Mahler, and George Crumb’s Night of the
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portion of this chapter illustrates that collectional interaction is suggestive in relation to extramusical features of Vox Balaenae.
Figure 1, dyadic, trichordal, and tetrachordal subsets of the diatonic, octatonic, and whole-tone collections
Example 2, two types of TC that use convergence points to connect octatonic and whole-tone collections.
a) Type I
b) Type II
A clear example of Type I variation characterizes the shift from the “Sea Theme” to the first of the variations, “Archezoic.” Both are shown in Example 3 with prominent statements of the collectionally ambiguous pitch-class set (025) indicated.20 In the “Sea Theme,” boundary statements of transpositionally related (025) realizations combine to form a diatonic collection, while in the following “Archezoic” variation the same set-class combines through TC to create Four Moons,” in George Crumb and the Alchemy of Sound, Steven Bruns, Ofer Ben-Amots, Michael D. Grace, eds., (Colorado Springs: Colorado College Music Press, 2005), 101–132. 20
All of the musical examples in this chapter are from George Crumb, Vox Balaenae for Three Masked Players, New York, C.F. Peters Corporation, 1971.
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an octatonic collection. In this process, (025) becomes a literal convergence point allowing variation in collection while maintaining set-class consistency. As Example 3a shows, the “Sea Theme” embeds four realizations of pitch-class set (025). The lower bracket connects the first and last statements—the last related to the first by T2. As the addition table beside the example shows, these two (025) realizations result in a five-note subset of the diatonic collection when combined at this transposition. All but two of the pitches that fall in between these two framing statements are duplications. It is quite clear that in the “Sea Theme,” (025) is functioning as a motivic force that, when subjected to TC at this transposition level, creates the overall diatonic environment. (025) is accentuated as a convergence point at the beginning of the “Archezoic” variation, shown in Example 3b.21 The same [D, E, G] form of (025) that ended the “Sea Theme” begins this variation, as is shown by the arrow connecting the (025) statements. However, this realization of (025) is almost immediately followed by its transposition at T6, which creates a subset specific to the octatonic collection. Example 3b summarizes this interaction in which the [D, E, G] form of (025) is used as a point of convergence to connect two different collections. The collectional contrast between the two sections is a representation of the process of variation, while the common motivic content establishes a link between the sections. A more abstract version of Type I variation is operative in the transition from the end of the “Archezoic” variation to the beginning of the “Proterozoic” variation, shown in Example 4. In this passage, (026) serves as an abstract convergence point linking the octatonic collection at the beginning of “Archezoic” to the whole-tone collection at the end, which prepares the collectional environment of “Proterozoic” where (026) becomes the primary motive in an 21
In Examples 3b and 4a from “Archezoic,” the piano is divided into two staves, of which only the top represents the actual sounding pitches. The bottom staff represents the “chisel-piano” effect and indicates the note on which the chisel is placed.
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entirely whole-tone context. Example 4 reproduces the end of Example 3b on the top staff, where an (025) * 6 realization created the octatonic collection discussed above. The continuation of the passage varies the (025) motive by expanding the boundary interval from (05) to (06), producing
Example 3, (025) used as a convergence point a) “Sea-Theme”
2 3 5 8 0 1 3 6 1 3 6
b) “Archezoic”
Diatonic (02357)
6 9 E 2 0 3 5 8 3 5 8 Octatonic (013679)
c) schematic showing the convergence point that connects the “Sea Theme” and “Archezoic” variation
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Example 4, (026) as a convergence point connecting octatonic and whole-tone a) end of “Archezoic”
2 0
2 4 8 0 2 6 0 2 6 Whole-tone
b) whole-tone beginning of “Proterozoic”
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c) middle of “Proterozoic,” juxtaposition of (025) and (026) reinforce movement back to octatonic
the [F, G, B] (026) set-class that is boxed-in on the left portion of the second staff. Continuity is explicit between the (025) statements at the beginning of “Archezoic” and this (026) statement. Aside from their timbral relationship, they are melodically and rhythmically similar. Furthermore, the octatonicism of the first half of “Archezoic” is somewhat preserved because (026) is an abstract subset of this collection, although it is a subset of a different octatonic collection than that which was established. Example 4a shows through the arrow that this (026) set-class is subsequently projected into the cello where it is a pivot that links the prior octatonicism to whole-tone collections that are established through two iterations of (026) * 2. In the “Proterozoic” variation that follows “Archezoic” (Example 4b), there is a continuation of the whole-tone collection prepared at the end of “Archezoic.” (026), which established itself in the shift from octatonic to whole-tone at the end of “Archezoic” asserts itself as the primary set-class motive in the flute at the beginning of “Proterozoic.” Note that the union of the two (026) motives here forms a larger (0268) that is ambiguous as to which collection it
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belongs, octatonic or whole-tone. Only the whole-tone environment at the end of the previous variation leads to a whole-tone hearing at this moment. After a brief, solo cello statement, the flute reenters (Example 4c) playing a passage clearly linked to the beginning of the variation. As witnessed earlier in “Archezoic,” (025) and (026) are stated in succession here and (026) is emphasized as a convergence point that allows a shift back to the octatonic collection in the succeeding cello passage. This is an unambiguous change as all of these pitches belong to an octatonic collection, even the cello’s grace notes. Underneath the score in Examples 4a–c, a line follows the progress of each collection through “Archezoic” and into “Proterozoic.” To restate these observations, (026) is an abstract convergence point whose membership in both the octatonic and whole-tone collection allows it to serve as a link between the two; the dotted line underneath the [F, G, B] trichord in “Archezoic” (Example 4a) indicates that it belongs to neither the previous octatonic collection nor the following whole-tone collection literally. Instead, this (026) prepares the subsequent whole-tone environment, which is established through (026) * 2 operations. Motivically, this (026) and the previous (025) are related to each other through a simple expansion.22 In “Proterozoic,” (025) reemerged and its juxtaposition with (026) signaled a movement back to the octatonic collection in the succeeding passage (Example 4b–c). (025) is collectionally specific in these passages; in other words, it’s presence always signifies an octatonic environment. On the other hand, (026) prepared the whole-tone environment through its ability to function in either the octatonic or whole-tone environment. As was discussed above, Type II variation makes use of the transposition operator as a convergence point, while varying the transposed set-class. Thus far in the composition, (025) and 22
Miguel Roig-Francoli discusses this type of extension in “A Theory of Pitch-Class-Set Extension in Atonal Music,” College Music Symposium 41 (Fall 2001): 57-90.
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(026) have demonstrated a motivic dominance, with (025) pervading the “Sea Theme” and the beginning of “Archezoic,” and (026) asserting itself at the end of “Archezoic” and the beginning of “Paleozoic.”23 In the “Mesozoic” variation, these two set-classes are juxtaposed as the motivic material in simultaneous, but clearly differentiated textural strands. The convergence point connecting the two is T6, which is involved in TC operations with both set-classes. The TC operations in each strand produce different collections such that the juxtaposition of (025) and (026) is reinforced through collectional contrast as well. In “Mesozoic,” the two strands are demarcated by both texture and instrumentation. The piano, reduced on the lower staff of Example 5a, plays the lower strand in which each hand maintains a different realization of (025) such that there is a T6 relation between hands. (Example 5b shows this relationship as it appears in the score.) Each occurrence of this TC relationship creates a six-note subset of the octatonic collection. Note that an allegiance to a single octatonic collection is not maintained throughout the variation; as the specific pitch realizations of (025) change, the TC relationship remains—causing a regular alternation of two different octatonic collections. The upper strand, shown on the top staff of Example 5a, is a unison melody played by flute and cello. This strand contains four instances of (026)—the other motivic trichord—at both the beginning and the end of the passage. At both moments, (026) is involved in a TC relationship with its T6 partner. As such, each of the two strands articulates different trichords, 23
Some initial findings show that thinking of these variations in different modular contexts is illuminating in regards to the connection between (025) and (026). For instance, the mod-12 set-classes (025) and (026) in this example are identical when thought of as step-class sets in mod-8 (octatonic) and mod-6 (whole-tone) spaces, respectively. Thinking in different modular spaces emphasizes the similarities between the “slightly-different” (mod-12) trichords and seems to show quite effectively how collectional shifting is a variation procedure involving different modular spaces. Matthew Santa, “Defining Modular Transformations,” Music Theory Spectrum 21 (Autumn 1999): 200–229; Santa, “Analysing Post-Tonal Diatonic Music: A Modulo 7 Perspective,” Music Analysis 19 (Spring 2000): 167–201.
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but are related through the T6 convergence point. While the specific trichords are different, the quality of the transformation relating the trichords is the same. Question marks in Example 5c make it clear that an (026) * 6 relationship is not specific enough to articulate a whole-tone collection by itself in the manner that (025) * 6 formed specific octatonic collections in the lower strand. In other words, the result of (026) * 6 is (0268), a setclass common to both the octatonic and whole-tone collections. In order to specify the collectional context of the upper strand, a more remote T2 relationship connects the two (026) * 6 statements, as seen on the upper portion of Example 5a and further clarified in 5c. To summarize briefly, the lower strand articulates the octatonic collection through (025) * 6, while the upper strand articulates the whole-tone collection through two T2 related statements of (026) * 6. T6 serves as the convergence point connecting the strands and reinforcing the collectional interaction. The textural juxtaposition is reinforced by set-class—(025) and (026)—and collectional—octatonic and whole-tone—juxtaposition. Collectional interaction in “Mesozoic” is more intricate than simple juxtaposition, however. Smaller boxes in Example 5a show that the collectional ambiguity of the (026) trichord allows the trichords in the upper strand to function as a member of whichever octatonic collection is being articulated in the lower strand at that time while simultaneously serving in the network of transpositions that create the whole-tone collection in the flute and cello. At the beginning of the passage, the initial, [D, F, A] (026) trichord of the upper strand is not only a member of the (026) * 6 operation that ultimately creates the whole-tone collection, but it is also a constituent of the octatonic collection unfolding in the piano. When B is introduced as a component of the second, [A, B, D] (026) trichord, the piano part rests momentarily because B
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is not a member of the first octatonic collection. When the piano reenters, a new octatonic collection is produced that includes the B and the remaining pitches of this new (026) trichord. This example shows a Type II variation technique where the transposition operator functions as a convergence point, creating a collectionally rich environment where octatonic and whole-tone interact with one another in a very interesting way.24 Both TC operations in the upper and lower strands articulate different collections by using different set-classes, (025) and (026)— which were the same set-classes responsible for collectional shifting in the “Sea Theme,” “Archezoic,” and in “Proterozoic.” The T6 transpositional level serves as a convergence point that relates the collections as does the (026) trichord which maintains membership in whichever octatonic collection is unfolding in the piano at that time.
24
This example does not necessarily posit that these collections exist simultaneously, which would admittedly be difficult to process as a listener. However, I do believe that our collectional perspective evolves over the course of the variation such that the final (026) * 6 relationship, which is of a T2 relationship to the first (026) * 6, could cause us to begin to hear the influence of the whole-tone collection at the end of the passage. In his analyses of Strainvsky’s early ballets, Antokoletz often points out passages in which two or more collections might be operative simultaneously. It is not clear from his discussion how Antokoletz expects listeners to process multiple collections simultaneously, however. Elliot Antokoletz, “Interval Cycles in Stravinsky’s Early Ballets,” Journal of the American Musicological Society 39/3 (Autumn 1986): 587–580, 609–614.
18
Example 5, T6 serving as a convergence point a) reduction of “Mesozoic”
b) lower strand b) lower strand
c) T2 relationship in the upper strand specifices the whole-tone collection T2
19
The remainder of this chapter will explore a final type of variation that uses the referential power of quotation to invoke diatonic tonality. Type III variation maintains T6 as a TC convergence point; the other operand is a diatonic pitch class set with the perfect fourth/fifth (05) as one of its constituents. Such a TC operation will always generate an octatonic collection or one of its collectionally specific subsets as shown in Example 6a, a catalog of Type III variation models. Though the transposition operator is maintained as a convergence point in these examples, this type of variation is fundamentally different from Type I or Type II variation. Type III variation contrasts objects whose associations are diatonic—perfect fifths, triads, and majorminor sevenths—with the octatonic collections that these objects create as a result of T6 transpositional combination. In other words, Type III variation does not juxtapose fully-formed collections like the previous examples of variation did. Instead, the examples of Type III variation that follow contrast the diatonic associations we experience when hearing a triad with the octatonic collection it belongs to as a result of its membership in a TC operation. In other words, diatonic is perceived through referentially. Type III variation does bear an important relationship to the variation techniques explored in the first part of this chapter, however. In these passages, (025) and (026) were primary motives, and they were often juxtaposed and implicated in defined types of interaction between specific collections. Example 6b shows that the transformation of (025) into (026) involves the expansion of (05) into (06). This difference is related to the Type III variation in Example 6b, which shows that that the two central constituents of Type III variation are the transposition operator, T6, and a perfect fourth/fifth, (05). To reinforce the significance of this type of collectional interaction, the remainder of the chapter spends some time relating the conflict between the collections to a recurring quotation of the Nature theme from Also Sprach
20
Zarathustra. The final portions of the chapter discuss how this conflict is resolved in the epilogue of the composition and observes some connections that link the conflict between diatonic and octatonic to a similar conflict in Strauss’s Also Sprach Zarathustra and some extramusical aspects of the Vox Balaenae. Example 6, Type III variation a) Type III variation models
b) relation of Type III variation to (025)/(026) motives
Strauss’s Nature theme from Also Sprach Zarathustra is first heard in the prologue to Vox Balaenae that precedes the set of variations just discussed. At the moment marked “Parody of ‘Also Sprach Zarathustra,’” Strauss’ theme appears in a slightly altered form, as shown in Example 7a. In Also Sprach Zarathustra (Example 7b), the climactic chords that form the center of the theme contrast C major and minor sonorities, which upon repetition become C major and F major. The repetition, in fact, is a transpositional combination of (037 * 5) that defines the
21
diatonic collection. In Crumb’s parody, the Nature theme is stated in B, and maintains the major/minor contrast in the two climactic chords. But, instead of transposition by perfect fourth, the roots instead are related by T6. Substitution of the perfect fourth with a tritone imparts an octatonic quality upon Crumb’s parody. The piano “timpani strokes” that follow add the pitch A so that the complete pitch content of this quotation is a seven-note octatonic subset. To state the
Example 7, contrast between the Nature theme as it appears in Vox Balaenae and its appearance in Also Sprach Zarathustra a) Vox Balaenae, “Vocalise”
b) Also Sprach Zarathustra, mm. 5–12
22
relationship to collection interaction more explicitly, Crumb’s quotation undoubtedly references diatonic tonality through the ascending perfect fifth that begins the passage and the two triads that form the climactic chords; however, the alteration of the root relationship (an expansion of T5 to T6) results in a passage whose pitch content is unambiguously octatonic. This quotation establishes the collectional conflict that occurs repeatedly throughout Vox Balaenae and specifies the conflict as a juxtaposition of diatonic objects that belong to an octatonic collection. Example 8 presents a passage from the “Cenozoic” variation that contains a shortened portion of the quotation at its conclusion.25 In this variation, Crumb combines perfect fifths at T6 to produce small octatonic sets that are further combined at T3 to produce a complete octatonic collection. Immediately following this TC passage, a reappearance of the Nature theme ensues. As the example shows, TC of the first F–C dyad with its T6 transposition results in (0167) and the further combination of (0167) at T3 produces a complete octatonic collection. In the succeeding velocissimo passage, chains of 5 * 6 TC realizations continue, followed by a complete T6 transposition of the entire first half of the passage. After which, Crumb’s parody of the Nature theme reappears as a concluding gesture shown with only the two altered triads and “timpani strokes”—without the ascending perfect fifth that began the quotation in “Vocalise.” The quotation’s appearance here in “Cenozoic” places the 5 * 6 operations occurring before it in the context of the conflict that was first articulated in the prologue (Example 7a). Like the fauxdiatonicism of the triads and fifths of the altered quotation there, the perfect fifths in this passage become members of an octatonic collection when combined with their T6 partners. In other words, in “Cenozoic” and in the quotation from the prologue, the diatonic collection’s fundamental interval–the perfect fifth–is forced into an octatonic context through TC 25
This quotation’s reoccurrence in the “Cenozoic” variation is significant. In the history of Earth, the “Cenezoic” era is believed to have witnessed the appearance of man. Crumb and others have pointed towards this quotation as a representation of man, and as such, it is fitting that it occurs here for the first time in the variations.
23
Example 8, 5 * 6 TC operations in “Cenozoic” followed by an abridged version of the Nature theme
Example 9, 5 * 6 TC operations in “Paleozoic” followed by (0258) * 6 realizations that complete the octatonic collection
24
Example 10, end of the “Cenozoic” variation a) End of “Cenozoic”
b) Network of Transpositions
c) Network that shows “composing-out” of (025)
25
at T6. A similar passage from the “Paleozoic” variation is shown in Example 9. Oscillating quintuplet figures in all three instruments are implicated in 5 * 6 TC operations. As was the case with the 5 * 6 passages in “Cenozoic,” the TC relationship is made explicit here as a contrast between harmonic perfect fifths and melodic tritones. Following the quintuplet passage, an isolated cello statement combines two T6 related major-minor sevenths that are referent to the octatonic collection. This new combination is interesting in that, apart from its inversionally related partner, the major-minor seventh is the only diatonic, triadic construction collection that contains both the perfect fifth and the tritone, the two operands most fundamental to the conflict that has been proposed. 26 The reciprocal relationship between the perfect fifth and the tritone that results in the proposed collectional interaction also occupies the end of “Cenozoic,” the final variation before the concluding epilogue. In this passage, shown as Example 10a, (025) functions as an object persistently transformed at T6. A six-note octatonic referent results from this transformation, which is further transformed at T5. A network showing these transformations is indicated as Example 10b. The examples show the (025) trichords as T5 transformations because the lowest and highest pitch of each trichord are most salient, and these pitches are related through T5. This network is a good model of the reciprocal TC relationship between the perfect fifth and the tritone; both are shown as primary transpositions at different levels of the network. Grouping and ordering contribute to our perception of this relationship. The segmentation in Examples 10a and 10b follows from the regular alternation of (025) trichords with reversed contour. The directionality of the trichords marks the lowest pitch of each ascending trichord and the highest of each descending trichord, creating the path of associations indicated by beams in 26
Also notice that the spelling of the major-minor sevenths, {D–F–A–C} in particular, indicates a conceptual linkage to the diatonic here.
26
Example 10c—a network like Example 10b, but with the content of each node specified. Each beam in the example indicates the composing out of (025), the trichordal object that began the network.27 Interestingly, the tritone is subjugated to a certain degree here. Because T5 is the higher-order operand, it somewhat overwhelms the T6 connections, which are already hidden to some extent by the directional opposition between the T6-related (025) trichords. To summarize briefly, Examples 7–10 demonstrated a specific type of collectional interaction in which diatonic objects were combined with their T6 partners, resulting in octatonic collections. This type of variation is defined as Type III in order to distinguish it from two other types of variation that literally juxtapose fully-formed collections. The origin of the relationship between the perfect fifth/fourth and tritone can be traced to the “Sea Theme” that began the variation set. Example 11 uses various shapes to indicate the role of each in the “Sea Theme.” Two perfect fifths/fourths articulate each boundary of the phrase and three delineate the central portion of the phrase, each of these originating from the high G that initiates the central section. The lone tritone occurs between A and D, marked by a box in the example. Lines and collectional labels under the example indicate possible collectional contexts for each portion of the phrase. Notice that the tritone marks a collectional shift from diatonic to octatonic. A dotted line indicates that this shift gradually unfolds, but is explicit at the arrival on A. The boundary perfect fifths highlight the ability of this interval to function in either of the collectional contexts, a property that is consistently articulated in the “Cenozoic “ and “Paleozoic” variations shown in Examples 8–10.
27
Joseph Straus, “Atonal Composing-Out,” in Order and Disorder: Music–Theoretical Strategies in 20thCentury Music (Leuven University Press, 2004), 31–51.
27
Example 11, reproduction of “Sea Theme” showing the perfect fifths, tritone, and collectional contexts in which each of these occurs
Admittedly, this last point calls attention to a potential criticism of Type III variation; namely, if nearly all of the dyadic, triadic and seventh-chord subsets of the diatonic collection— including all of the dyads, major, minor, and diminished triads, the major-minor seventh, and half-diminished seventh—are also subsets of the octatonic collection, why posit collectional interaction at all? Comparing the subset content of these two collections reveals that, at least abstractly, these sonorities are just as much a part of the octatonic collection as they are part of the diatonic collection.28 The simple answer is that the perfect fifth, major and minor triads, as well as the majorminor seventh immediately evoke diatonic connotations because of their ubiquitous appearance in the centuries of diatonic music that preceded Vox Balaenae and with which most listeners are very familiar. The quotation of the Nature theme (Example 7b) from Also Sprach Zarathustra certainly reinforces the correlation. In its unaltered form, the rising fifth of the theme, along with the tonic pedal, and prominent triads are unambiguously diatonic. Appearances of this theme throughout Vox Balaenae are quite identifiable, and the TC operations in passages from 28
In “Properties and Generability of Transpositionally Invariant Sets,” Richard Cohn uses modular “subuniverses” and interval cycles to show that sets with certain properties will always result in an octatonic collection or one of its subsets through T6 transposition. Essentially, the diatonic objects I have cited all have the perfect fifth in common. Because of their asymmetry, the pitch classes of a perfect fifth also belong to T1 related tritones: C–G is found in the T1 related tritones C–F and C–G. As a result, when any perfect fifth is transposed by T6, the remaining pitches of the T1 related tritones are produced: C–G transformed at T6 produces F–C. This result is (0167), a characteristically octatonic subset. Adding a major or minor third above or below any pitch in the C–G fifth will only increase the cardinality of the octatonic subset. For instance, adding a B above the C–G fifth produces a C–G– B object (026) that when transposed at T6 produces F–C–E and their union creates (013679). Richard Cohn, “Properties,”17–20.
28
“Paleozoic,” “Cenozoic,” and others are heard in relation to the diatonic connotations of these quotations. From this viewpoint, collectional interaction, or conflict, in Vox Balaenae originates in the alteration of the Nature theme. The remainder of this chapter will discuss the resolution of this conflict in Vox Balaenae. In addition, it will discuss some further relationships to Strauss’s Also Sprach Zarathustra. A type of collectional conflict also pervades this composition, and the resolution in the Vox Balaenae’s final passage seems to have a corollary in Also Sprach Zarathustra. Notions of conflict and resolution are of extramusical or programmatic importance in both compositions. The conclusion of this chapter will make some observations about how interaction or conflict in Vox Balaenae and its subsequent resolution in the closing passage relates to issues of mankind, technology, and Nature that were important topics at the time of Vox Balaenae’s composition. A final appearance of the Nature theme quotation appears in Vox Balaenae’s epilogue, entitled “Sea Nocturne” and shown in Example 12.29 In this passage, the Nature theme is further shortened from the version shown in the “Cenozoic” variation (Example 8); it now includes only the final “timpani strokes” without the tritone-related triads. A reprise of the “Sea Theme” quotation (Example 11) follows this shortened quotation and leads into the final piano gestures of the piece. Figure 3a collates the pitch content of these final moments: the right hand quintuplets and B–F pedal of the left hand are on the top line; the central staff’s A and F, which are circled in Example 12, are on the bottom line. This figure shows that the pitch content of these final gestures embeds a pentatonic collection as well as an octatonic collection. The syncopated A and F are the only pitches that prohibit the pitch content at the end from being 29
This entire passage is written with a key signature of 5 sharps. Due to space constraints, I could not reproduce the portion that includes the key signature except on the bottom portion of the example.
29
completely diatonic. In other words, the pitch content of the passage embeds the collectional conflict previously articulated through TC. By stratifying these octatonic pitches against the pentatonic quintuplets and pedal, neither collection is able to fully assert itself. The reemergence of the “Sea Theme” in this passage refers to the conflict in a similar manner, as it has been mentioned numerous times in this chapter as a passage that exemplifies the collectional interaction with which this chapter has been concerned. Like the final moments of the piece, the “Sea Theme’s” pitch content may also be represented by Figure 3a. As a final affirmation of the conflict, two T6 related perfect fifths occur as pedal pitches in the left hand of the closing passage, indicated by the arrow in Example 10. Not only is this 5 * 6 transpositional combination reminiscent Type III interaction that occurred in “Cenozoic” and “Paleozoic” (Examples 8–9), but these exact pitch classes also made up the first instance of tritone related triads in the initial quotation of the Nature theme (Example 7b). Taken together, the occurrences of A and E in the “Sea Theme” and as the stratified accompaniment to the pentatonic quintuplets in the concluding passage, as well as this final occurrence of a 5 * 6 TC operation strongly affirm the conflict between diatonic and octatonic. This conflict is resolved at the end of the composition. The boxed-in portion of Example 11 shows that immediately after the “(dying, dying . . .)” indication, the quintuplet piano gesture returns two final times without the stratified A and E. In other words, only diatonic elements return, without interference from octatonicism. Crumb’s notation and performance directions are suggestive here. Perhaps “(dying, dying . . .)” is written not only as an indication of dynamics and mood, but also as an indication of the final octatonic interference.
30
Example 12, Collectional conflict and resolution at the end of the “Sea Nocturne”
31
Figure 3, various representations of the pitch content at the end of “Sea Nocturne” a)
b)
c)
Visually, the piano staves collapse after the final statement of A and F to fill the space vacated by the two octatonic pitches. Through Type II variation, this chapter has repeatedly mentioned that the composition, the perfect fifth represented diatonicism and the tritone was the single element that impeded diatonicism.30 Two final observations about the conflict and resolution at the end of the composition can be gleaned from this remark. First, the B-major pentatonic that ends the piece (Figure 3b) is not only a common substitute for the diatonic collection, but it is also a large diatonic subset entirely generable through a partial cycle of perfect fifths. As a result of its generation, it does not have the tritone found in a complete diatonic collection. In other words, 30
Of course the perfect fifth is the common element that connects these diatonic objects in the composition. As mentioned in note 26, Cohn has shown that the asymmetry of the perfect fifth allows for its inclusion in the octatonic collection when combined with its T6 relation.
32
the one element responsible for impeding diatonicism in the variations and up until the final moments of the piece is conspicuously absent at the end of the composition. Second, the final octatonic elements, A and E, that formed the stratified accompaniment to the aforementioned pentatonic collection can be regarded as half-step alterations of the B–Major collection’s keydefining tritone, as shown in Figure 3c. Interpreting an alteration here again suggests that these octatonic elements are disruptive forces, and that their disappearance at the end signals a return to diatonicism.
Example 13, Also Sprach Zarathustra, 8 measures after rehearsal 17
This collectional conflict is reminiscent of a similar conflict between C major and B major/minor in Richard Strauss’s Also Sprach Zarathustra.31 For instance, at the climax of the composition shown in Example 13, the ascending fifth of the Nature theme emerges, accompanied by an emphatic C pedal. Immediately following its statement and a pause, a new section suddenly begins in the remote key of B minor, juxtaposing the two tonalities.32
31
This conflict is a salient feature of the piece that has been discussed since the composition’s premier and by Strauss, himself. John Williamson, Strauss, Also Sprach Zarathustra (Cambridge, Cambridge University Press, 1993), 31-38, 66-67.
33
Of more interest to our discussion, the C/B conflict is amplified at the end of Also Sprach Zarathustra, just as the collectional conflict is magnified at the end of Vox Balaenae (Example 14). Like Vox Balaenae, the passage is written with a B-major key signature, which the upper woodwinds, strings, and harp articulate with unambiguous B major triads in much the same way that the right hand of the piano articulated B pentatonic in Vox Balaenae. Every B-major triad, however, is interspersed with the rising perfect fifth motive from the Nature theme, stated in C major. A and F are these fifth’s corollaries in Vox Baleaene. That this passage is representative of the C/B conflict is no more apparent than at the end Also Sprach Zarathustra, where the final B-major chord is followed by three solitary Cs. All of these correlations suggest that Crumb might have had this passage in mind while composing Vox Balaenae. Interestingly however, Strauss’s music seems to end with this conflict firmly intact—neither B major or C major subside at the end of the piece.33 But in Vox Balaenae, the resolution of the collection conflict is apparent as the octatonic elements clearly subside in favor of diatonicism. Example 14, Also Sprach Zarathustra, mm. 979–87
32
The passage after the fermata is based on the Longing motive. This motive generally occurs in the key of B minor. Its juxtaposition with the Nature motive, which usually occurs in C major, is a consistent source of conflict. John Williamson, Strauss, 70–87. 33
The statements of C at the end can be interpreted as the fifth of an altered V7 chord in second inversion, in which case B major is the controlling tonality. However, the final statements of C, even after the final B major chord, indicate that the conflict between the tonalities is somewhat in effect here as well. Ibid., 70.
34
Conflict is compelling in relationship to the extra-musical circumstances surrounding both Vox Balaenae and Also Sprach Zarathustra. Comments by Crumb and others suggest that Vox Balaenae bears some association to the relationship between mankind and Nature—a growing concern in 1971, at the time this piece was written.34 Vox Balaenae, or Voice of the Whale, was composed as Crumb’s reaction to hearing a tape of humpback whales singing. The “Vocalise” portion of the composition emulates the sound of humpback whales as the flautist is required to sing and play at the same time. While the subject matter and many of the geologically named variations suggest a time period before the appearance of man, Crumb’s requirement that performers wear masks indicates an underlying human element. In the preface to the score Crumb states that “the masks by effacing a sense of human projection, will symbolize the powerful impersonal forces of nature".35 Along these lines, one early reviewer suggested that Vox Balaenae’s symbolism “is a condemnation of man's arrogant dominion over the ocean's most mighty dwellers.”36 The Nature theme quotations are evocative in this regard. As mentioned earlier, Crumb himself has stated that “the emergence of man . . . is symbolized by . . . the Zarathustra reference.”37 That the quotation is the source of conflict in both Vox Balaenae and Also Sprach Zarathustra is symptomatic of its symbolism. For Strauss, Also Sprach Zarathustra was a
34
The first Earth Day was held on April 22, 1970, only a year before Vox Balaenae was completed. Roughly basing compositions on world events or current thought is not unlike Crumb. Black Angels, composed a year before Vox Balaenae, has some basis in Crumb’s thoughts about the Vietnam War. 35
Crumb, Vox Balaenae, 2.
36
Alan Segall, George Crumb: Voice of the Whale, liner notes by the author, Zuma Records 102, 1994, compact disc; Michael Walsh, Works by George Crumb, liner notes by the author, New World Records 357–2, 1992, compact disc. 37
George Crumb, George Crumb: Eleven Echoes of Autumn, Four Nocturnes, Vox Balaenae, Dream Sequence, liner notes by the author, Jecklin Edition 705, 1996, compact disc.
35
manifestation of his rejection of Schopenhauerian metaphysics and his acceptance of many of Friedrich Nietzsche’s ideas.38 In particular, Strauss was greatly influenced by Nietzsche’s belief in the power of the individual, the Übermensch, to control his or her own destiny.39 The Nature theme in Also Sprach Zarathustra often interacts with other motives that generally represent human characteristics like longing, desire, and passion. In Vox Balaenae, this Übermensch, as evoked through the quotations of the Nature theme, create an image of man as an oppressive force.40 Throughout the piece, the theme is repeatedly forced into octatonic contexts. The influence of octatonicism only dissolves at the end of the composition. I have mentioned only a few of the many interpretive possibilities that arise from a view of collectional conflict and resolution in the composition.41 A more exhaustive accounting of these possibilities is a possible course for future study. In this study, the goal was to use transpositional combination as a tool to understand this conflict. To summarize, the chapter has shown that collectional interaction is an important variation technique in the set of variations that occupies the central portion of Vox Balaenae. It
38
Bryan Gilliam, Richard Strauss, Grove Music Online ed. L. Macy (Accessed 15 May 2007), . 39
Friedrich Nietzsche, Thus spoke Zarathustra : a book for all and none, trans. Adrian Del Caro (Cambridge: Cambridge University Press, 2006). 40
Interestingly, the opening motif from Also Sprach Zarathustra was widely used in Stanley Kubrik’s 1968 film, 2001: A Space Odyssey, which deals with themes of human evolution and technology, often in relation to nature. 2001: A Space Odyssey, prod. and dir. Stanley Kubrick, 141 min. MGM, 1968, DVD. 41
Some ideas for interpretive study that arise from analysis include the gradual decrease in length of the Nature theme quotation, which seems to exert less influence as the composition continues. In addition, the use of B Major in the “Sea Nocturne” seems to have some relationship to the B Major close of Also Sprach Zarathustra. It is interesting that Crumb used B Major, and not C Major, as an overall tonality for the composition. In Also Sprach Zarathustra, C major accompanies the Nature theme in most passages, but in Vox Balaenae this theme is always in B major. By using B major, Crumb requires the flautist to have a B pedal and forces the cellist to tune the cello scordatura. Finally, a future study might examine the role of electronics in the composition. Such use of technology is certainly suggestive in a piece that is based on the role of mankind and nature.
36
discussed three types of variation, all of them relying to some degree on the notion of a convergence point to connect collections. The first type used an object as the convergence point, while the second viewed the transposition operator as the point of convergence. Using these types of variation, changes in collectional context often represent the variation process, while the common motivic or structural TC operand is the link between variations. The second half of this chapter discussed a more specific type of variation technique that involves the perfect fifth and the tritone, and contrasts diatonic objects that are defined by the perfect fifth with the octatonic collections that are created through T6 transpositional combination. An additional discussion involved the similarities between these types of TC and Crumb’s quotation of the Nature theme from Also Sprach Zarathustra. As shown, this conflict—first articulated in the Nature theme quotation—is not simply a feature of pitch and structural organization, but it may have important implications for an extramusical interpretation of the composition, as well.
37
Chapter Two: Transpositional Combination and the Analysis of Form in Lux Aeterna
The previous chapter discussed how transpositional combination functions in interaction among referential collections, and defined three TC-based variation techniques involving this interaction. Because large pitch collections were the primary concern, that discussion focused mostly on the results of a TC operation, and used transpositional combination only to model how the collections were attained. This chapter will discuss ways to refine the TC apparatus in order to model more closely how operands in a TC operation are represented on the musical surface, and will use these refinements to discuss form in George Crumb’s Lux Aeterna. In this chapter, the referential pitch collections discussed in the previous chapter will be relegated to a subservient role, and operands and specific representations of the TC process will be brought to the fore. Though a lot of Crumb’s music seems susceptible to collectional analysis, such pursuits can easily gloss over important surface details. In Crumb’s music, individual intervals are highly significant. As a composer deeply interested in symbolism, intervallic motives often portray extramusical ideas. In Black Angels for instance, the (016) trichord is generally represented as a tritone [06] with an appended perfect fifth [07] as a symbol of the polarity between the Devil and God. While collectional analysis could refer these sonorities to particular collections, these generalizations might miss the symbolism inherent in the details. The previous analysis of Vox Balaenae relied on an abstract TC matrix that represented most passages as unordered collections of pitch classes without recognizing the register, temporal order, or other surface alterations that may be of significance. The beginning of this chapter discusses some extensions of the TC apparatus that refine the abstract matrices presented
38
in the first chapter to illustrate the various ways that TC may present itself on the musical surface. An analysis of Bartók’s “Fourths” illustrates how the details that these refinements elucidate are influential with respect to musical form. Following this will be a discussion that approaches musical form in Crumb’s Lux Aeterna from this new perspective. Implicit in this study is a movement away from generalizing passages with referential labels such as whole-tone and octatonic, and a refocusing of analytical attention toward the components that create these large pitch-collections (i.e. their TC manifestations).1 The final section of this chapter will discuss this transition and, in particular, TC’s generative function with respect to many of these referential pitch collections.
I
As mentioned, the first chapter relied on an abstract TC operation, one that was neutral regarding temporal and registral attributes of a particular musical passage. This section introduces transformations of TC operations that specify attributes such as register and temporal ordering. In his dissertation, Richard Cohn has explored many of these concepts in extraordinary depth.2 I have chosen to summarize his work in the following pages because many of these concepts are not well known and because this chapter proposes some alterations that are meant to simplify the process and allow the geometry of the TC matrix to model the music visually in a more compelling manner.
1
As my analysis of Vox Balaenae shows, collectional analysis can be significant. However, in pieces that are collectionally unremarkable, attention to the referent collection itself must recede into the background and the details—the way that a collection is generated—should become more important. 2
Cohn, “TC in the 20th Century,” 152–324.
39
The steps discussed below are as follows, proceeding from the most abstract representation to the most specific: (1) The pair of operands are represented in an operation as an object and a transposition; (2) the entire operation is transposed to represent actual pitch-classes and the matrix is created; (3) the matrix is formatted to represent the musical dimensions of a passage; (4) the matrix is permuted to represent temporal and/or registral ordering. Each of these steps represent transformations of an operation by changing the geometry of the matrix. These changes in the matrix’s geometry distinguish the present study’s approach from Cohn’s, which represents format changes and permutations within a standard matrix that does not change.
The TC operation
A TC operation represents the union of two transpositionally-related pitch-class sets. The operands in a TC operation represent an object and a transposition.3 Objects are always represented as pitch-class sets in normal form, transposed to zero,4 and transpositions are directed intervals.5 (As a result, inversionally-related set-classes may not be represented in the same manner; this stipulation occurs because inversionally-related set-classes do not always produce the same larger set when transposed by the same interval.6) In the operation, an object is
3
In this discussion, I consider only binary operations—those with one object and one transformation. This seems to suffice for most of Crumb’s music. However, passages with three or more operands are not uncommon in music, and in these cases, TC operations generally contain one object and two or more transformations; the first transformation acts on the object, creating a new object upon which the subsequent transformation acts. 4
In general, the objects are collections of pitches that undergo a transformation. In my analyses, objects are afforded that status because they seem to belong together as a musical unit. 5
As was the case in the first chapter, dyads are generally abbreviated as single numbers. Cf. p. 4, n. 14.
6
Cohn, TC in 20th century, 70–77 discusses this in more detail in reference to the “Unique Result Condition.”
40
listed first and is followed by its level of transposition. In Example 1a, (035) is serving as an object that is transformed by T6. Example 1b is not as easy to define because it is not clear whether the object is [05] or [06]. Here, the object is given as [06] and each of the [06] dyads is related through a T7 transformation. Had context dictated a different reading, this could be indicated as (5 * 6), choosing the [05] verticalities as objects and a T6 transformation instead.7 Example 1, (035 * 6) and (6 * 7) TC operations a)
b)
Abstract:
(035 * 6)
(6 * 7)
Transposed:
T10(035 * 6)
T6(6 * 7)
The transposed TC operation and the TC matrix
In order to represent the actual pitch-classes present in a passage, the entire TC operation is transposed, as represented by the “Transposed” portion of Example 1. The T-value is equal to the lowest pitch of the first object in the passage, once that object has been put in normal form. In Example 1a, T10 is applied to the entire operation because the lowest pitch of the initial (035)—which is represented in normal form in the passage—is A, or 10. In Example 1b, F is the lowest pitch of the initial object, ic6, so that the T-value is T6. Note that had Example 1b been represented as (5 * 6), the T-value would have been T1.
7
This difficulty generally arises when the TC operation involves temporally adjacent verticalities and both operands are dyads. Context always plays a role in making decisions about which is the object, and which is the transformation.
41
After assigning the transposition value, the resulting pitches can be modeled in a pitchclass–specific matrix like the one shown in Example 2. Notice that the matrix is similar to the addition table used in the previous chapter; the difference being that this TC matrix does not contain the operands along the outer borders. The abstract matrix shows the operation, transposed to begin on zero. The transposed matrix transposes the abstract matrix by the T-value, as determined from the lowest pitch of the initial object in normal form.
Example 2, T10(035 * 6) operation and its matrix representation
T10(035) * 6
Formats8
Formatting an operation further specifies the matrix by assigning a horizontal or vertical dimension to each operand, depending upon whether the operand is realized melodically or
8
Cohn, “TC in the 20th Century,” 180–186. Cohn’s format classes and mine are the same though I have chosen to represent them differently. Cohn seems concerned with representing all reformatting, a change from HV to V2 for instance, as alterations of a basic matrix. As a result, he labels formats on the matrix even if the matrices’ dimensions are not isographic with the actual musical dimensions. This allows Cohn to use complex formats and permutations without changing the basic shape of the matrix. For visual clarity, I have chosen to allow the matrix to mutate so that it isographically represents the musical surface.
42
harmonically. Formatting changes the geometry of the matrix if necessary so as to maintain isography with the temporal and/or registral order of the transpositionally related objects as presented in the music.9 In a simple operation with only two operands, there are three possible formatting situations: (1) the objects are presented melodically and the transpositions occur vertically (HV); (2) the objects are stated vertically, as are the transpositions (V2); and (3), the objects are realized melodically, as is the transposition (H2).10 An HV formatting of the T10(035 * 6) operation is shown in Example 3a. [A#, C#, D#] and [E, G, A]—the objects of the TC operation—are represented as melodic elements and assigned the H dimension in the matrix. T6 transformations connect the two objects vertically. The matrix represents the melodic (035) trichords along the rows and the T6 transformation in the columns. A dimension exchange is shown in Example 3b: here, the (035) trichords are represented as harmonic entities and T6 connects the temporally adjacent structures. When the object of the TC operation is a verticality as it is in Example 3b, it is represented vertically in the columns of the matrix, changing the geometry of the matrix. Cohn does not represent the operation in this way. Instead, he always shows the object on the bottom row, whether or not the object is realized melodically or vertically. Using his method, the matrix in Example 3b would look the same as 3a. To indicate the difference in dimension, Cohn places a V below the row to express its vertical realization. 11
9
As mentioned, this is my own contribution. Cohn is not concerned with preserving isography between the matrix and the score. 10
The temporal ordering of objects, whether they are presented melodically or chordally, will be discussed below as part of the permutation process. 11
Cohn, “TC in the 20th Century,” 182. Cohn uses the terms “congruent” and “non-congruent” to represent Examples 3a and 3b, respectively, because 3a is isographic or congruent with the musical score while 3b is not.
43
Example 3, HV formatting of (035 * 6) a)
b)
A V2 format materializes when the object and transposition both occur vertically. In this format, the operands are labeled V1 and V2 and refer to levels of relation within the sonority: V1 is a local level and V2 a more remote level. Generally speaking, V1 contains the object while V2 represents the transposition value. For instance, in Example 4a the V1 operand is (035)—the two trichordal verticalities—and V2 represents the T6 transposition that relates the two trichords. In 4b, the dimensions are exchanged and V1 now represents the three (06) dyads and V2 is the T3,5 transposition.12
12
When a transposition operation has two or more subscripts, I am indicating a transposition of the original object by both values. In the case of Example 4b, the A–E dyad is transposed at T3 and at T5.
44
Example 4, V2 formatting of T10(035 * 6) a)
b)
An H2 format arises when all of the pitches in a TC operation occur melodically. As in the V2 format, H1 and H2 represent local versus remote levels of melodic structure. The H1 operand is the transposed object, while the H2 operand is the deeper level transpositional relationship between the objects. Examples 5a and b show two realizations of an H2 formatted operation. Example 5, H2 formatting of T10(035 * 6) a)
b)
45
Permutations13
In any TC realization, each operand’s presence is implied, though not always explicitly present. That is to say that the realization itself may not always emphasize the operands in a way that makes them aurally salient. For an operand to be explicit, it must be realized in a way that emphasizes the interval (not just the interval class) or the pitch-class set (not just the set-class). In Example 6 for instance, permuting (a) by retrograding the order of the upper trichord in (b) renders the T6 operand inexplicit; while it may be considered a (hidden) structural element, the realization does not emphasize T6. Example 6b is a “partially explicit” realization where only (035) is aurally salient as a component of the operation. Example 6a, on the other hand, is “fully explicit” because both operands are emphasized in the realization. As a further example, (c) is completely “inexplicit” because, despite being implicitly present, neither (035) or T6 will be easily heard in a transpositional relationship.14 Notice that the H and V values below each passage represent the explicit operands, along with the dimension in which they will be heard
Example 6, three permutations of T10(035 * 6) a) Fully-Explicit b) Partially-Explicit c) Inexplicit
13
Cohn, “TC in the 20th Century,” 174–180. This study follows Cohn’s basic process of permutation.
14
Ibid., 164–170.
46
Permutations are defined as operations performed on a TC matrix that alter the temporal or registral order of the pitches in the operation, and as a result, may affect the explicitness of one or more of the operands in an operation. Some permutations have no effect on the explicitness of a realization: imagine retrograding all of the pitches in Example 6a, for instance. Others cause the operands to become partially explicit, as in Example 6b. A final category, Example 6c, renders all operands completely inexplicit. In a P-type permutation, any change in a column or row of the matrix is accompanied by a corresponding change in all other columns and rows. If this condition is met, the realization of the operation is fully explicit. Example 7b and 7c shows two P-type permutations of T10(035 * 6), as indicated by the arrows in the example. In each of these examples, the P permutation causes a change in the temporal ordering so that the (035) trichords and their T6 relationships to one another are preserved.
Example 7, P-type permutations of T10(035 * 6) a)
b)
c)
H-type permutations represent instances where the horizontal component of a realization remains explicit, and the vertical component loses explicitness. In these cases, the pitch-classes located on the horizontal axis (the rows of the matrix) remain invariant while those in the columns do not. The H-type permutation shown in Example 8b, for instance, retrogrades the
47
temporal ordering of only the upper trichord. The rows of the matrix retain the same pitchclasses, contributing to the explicitness of the H operand, (035). However, the permutation causes the pitch-class content of the columns to change, obscuring the vertical T6 relationship.
Example 8, H-type permutations of T0(035 * 6) a)
b)
c)
The opposite occurs in a V-type permutation. Here, the columns retain pitch-class content while the rows experience some change in content. As a result, only the vertical relationship is explicit. In Example 9, the arrows show that the V-type permutations of T10(035 * 1) in (b) and (c) retain the T1 relationships , while the [035] object is obscured. “V = 1” is shown here to indicate that the vertical operand, [01] is the only explicit operand.
Example 9, V-type permutation of T10(035 * 1) a)
b)
c)
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Finally, X-type permutations hold neither operand explicit, creating completely inexplicit operations. In these passages, neither operand is aurally salient. As a result, we may not hear these passages as exhibiting TC at all. Examples 10(b) and (c) show instances of inexplicit TC realizations in relation to a fully explicit passage shown as Example 10(a).
Example 10, X-type permutations of (035 * 1) a)
b)
c)
Examples 8–10 showed permutations of HV formatted operations.15 Permutations may also be applied to the H2 and V2 formats. Two new permutation types need to be introduced to account for these situations. As shown in Example 11a, within the H2 format a P-type permutation holds both operands explicit, an H1-type permutation emphasizes only the H1 operand, and an H2-type permutation creates a situation in which only the H2 operand is explicitly present. As Example 11b shows, similar permutations apply to the V2 format.16
15
In Cohn’s dissertation he explores multi-dimensional realizations of TC that occur when a passage has more than two operands. Cohn, “TC in the 20th Century,” 265–324. My alterations to Cohn’s system will work with these multi-dimensional passages. I have not included them here because they would take considerable space to discuss and are not especially relevant to the examples I will discuss in Crumb’s Lux Aeterna.
49
Example 11, permutations of the H2 and V2 format
a)
b)
One great benefit of looking at music from this TC perspective is that it allows us to view large sets as embodiments of transpositionally related intervals or small sets. This perspective
16
In discussing formats, I mentioned that within H2 and V2 format types, the H1 and V1 dimensions are generally reserved for objects and the H2 and V2 dimensions are most often transpositions. Conversely, an H1 or V1 permutation will maintain the object as an explicit part of the realization but hide the transposition. Following this logic, it would seem that an H2 or V2 permutation would obscure the object while maintaining the transposition. As this is a conceptually difficult situation to imagine (how can we have transpositions of objects without the objects being present?), these examples are difficult to hear from a TC perspective at all.
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regards a TC set-class as realized in a passage not as a collection of equally important intervals and sets (as may be suggested by an interval vector, for instance) but as a dynamic collection in which certain intervals or set-classes are more prominent than others. As an illustration, Example 12 shows two realizations of set-class (0145). While its interval vector, < 201210>, may suggest that the set contains six equally important intervals, the two realizations in Example 12 clearly do not give equal prominence to all of them. By viewing Example 12a as a fully explicit realization of (1 * 4), these intervals become more important in the makeup of this realization of (0145). As a contrast, 12b retains the ic1 dyads and obscures the transpositional relationship, suggesting that ic1 is the most prominent constituent of this realization. While the two belong to the same set class, they have different presentations that may be used in different ways in a composition. The importance of a permutation lies in its ability to maintain set-class consistency even while the TC constituents of that set-class may intensify and recede in importance.
Example 12, two TC realizations of (0145) a)
b)
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Residual intervals and Transpositionally-Invariant Set Classes
In addition to TC operands, other intervals can be described as byproducts of a TC operation.17 Cohn calls these byproducts residual intervals, and in a composition these intervals can acquire significance, often serving as operands in new TC operations.18 In Example 12a, a residual interval ic3 occurs between C and E. In Example 12b, ic4 is the more salient residual interval, occurring between C and F. In a non-permuted or formatted TC matrix, the residual intervals always occur between diagonal-opposed pitch classes. At this point, an analytical example should clarify how some of the ideas discussed so far can be used in analysis. Example 13 shows the score of Bartók’s “Fourths” from the fifth book of the Mikrokosmos.19 “Fourths” is an exploration of a pedagogical and compositional problem. The pedagogical problem is maintaining perfect fourths in both hands, while moving each hand in contrary motion. Compositionally, Bartók’s piece strives toward the purely quartal sonorities (0257) suggested by the title and represented in the second half of the composition. From mm. 1– 34, Bartók systematically avoids the completely quartal sonority as an explicit vertical harmony. However, beginning in m. 35 the fourths in each unambiguously create quartal chords. This compositional problem and its resolution can be fruitfully explored through transpositional combination, as the perfect fourths in the right and left hand of the piano are explicitly presented as objects related through varying degrees of transposition. The following
17
These other intervals are the remaining, non-operand, intervals in a set-classes interval vector.
18
Cohn, “TC in the 20th Century,” 190.
19
Richard Parks offers a thorough pitch-class set analysis of this composition. Some aspects of his study are represented in my analysis, especially his observations on p. 260 concerning the adjacency interval series of the transpositionally-related fourths. Richard S. Parks, “Harmonic Resources in Bartók’s ‘Fourths’,” Journal of Music Theory 25/2 (Autumn, 1981): 245–274.
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Example 13, Bela Bartók’s “Fourths”
53
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analysis will locate the source of conflict in the piece in the level of transpositional relation connecting the perfect fourths. This transposition value affects the residual intervals that result from each transpositional combination. Ultimately, Bartók satisfies the problem by using a TC operation that contains a perfect fourth as a prominent residual. A format change accompanies the moment of resolution, formally splitting the piece into two large sections. In the first section, from mm. 1–34, the TC relations are all members of the V2 format class where the object is the perfect fourth, and the transpositional level is variable; in other words, this passage contains various statements of 5 * x. Example 14 abstractly represents the TC possibilities of 5 * x. As the score shows, Bartók always insures that no duplicate pitch classes result in a TC operation. For instance, in m. 3 on the last eighth-note, Bartók introduces a slight deviation from the strict contrary motion in order to avoid the duplicates that would have resulted from 5 * 7.20 Had 5 * 7 been the operative TC realization, the right hand pitches would have been D and G, with D being a duplicate of a left hand pitch. As such, we can further define the pitch relationships in the first half of the piece as V2 formatting of 5 * x, where x—the transformational relationships—is any integer but 7(5).21
20
Other passages that show Bartók altering the strict contrary motion to avoid duplicate pitch-classes are found in m.7, mm. 9–10, and mm. 13–14. 21
Because a transpositional relationship and its mod-12 complement will produce the same pitch-classes when transposing any object, they are assumed as equivalent TC representations. For instance, T0(5*8) and T0(5*4) produce the pitches C–F, and A–D. As shorthand, I will indicate this operation as 5 * 8(4) when referring to an abstract situation or stipulation.
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Example 14, TC representations of 5 * x
That Bartók does in fact follow these stipulations is confirmed by the transformational graph of the first phrase, shown in Example 15. The horizontal transformations show the phrase moving in strict contrary motion, while the vertical transformations the level of transpositional relationship between the two hands. Notice that the only point in which Bartók deviates from strict contrary motion occurs at the end of the passage, marked with an asterisk. Here, the T11 transformation in the left hand is accompanied not by T1 in the right hand, but instead by T2, insuring that the penultimate verticality is not formed by 5 * 7, which would have resulted in duplicate pitches.
Example 15, transformational graph of the first phrase, mm. 1–4
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In order to produce a four-note quartal sonority, Bartók’s composition must “find” the transpositional combination that produces a perfect fourth as a residual interval. Example 14 shows that 5 * 10(2) is the only transpositional combination that produces a quartal sonority with ic5 as a prominent residual between the two transpositionally-related fourths. Furthermore, Example 14 shows that the residual intervals of 5 * 9(3) and 5 * 11(1) are ic4 and ic6, interval classes that are situated symmetrically around ic5. With this in mind, the 5 *3 and 5 * 1 TC operations chosen at the outset of the piece (see Example 15) are those whose residuals surround ic5. 5 * 3 and 5 * 1’s symmetrical relationship around the quartal sonority is emphasized at the beginning of the phrase that begins in m. 9, as shown in the score of Example 13. In this passage, the contrary motion established at the beginning continues, but Bartók introduces a canonic component as well, separating the right and left hands by an eighth-note. A 5 * 11 verticality begins the passage, but when the left hand moves to the next sonority an eighth-note ahead of the right hand, a 5 * 10 sonority results that creates the first instance of a quartal sonority in the composition. By introducing the canonic component, this passage foreshadows the purely quartal situation that will arise later in the composition. The resolution to quartal tetrachords occurs in mm. 35–36, which is shown in Example 16a. At this moment, each perfect fifth is transposed at T10, creating the 5 * 10 TC relationship that produces purely quartal tetrachords, with the fourth as the most prominent residual. Example 16b shows that the immediate repetition of mm. 35–36 in mm. 37–38 is a T-5 transformation, creating a larger scale perfect fourth relationship between the two passages. The matrices in Example 20b also show that the resolution to quartal harmony in these measures is accompanied by a format change. Up to this point, the composition has been characterized
57
entirely by V2 formatting, with the TC objects and transformations occuring as verticalities. As such, mm. 35–38 is striking in that the resolution to quartal sonorities is accompanied by a change to the H2 format. In a larger sense, the form of the composition can be represented through the changing geometry of the TC matrices, as is shown abstractly in Example 17. This example views mm. 1– 34 as an attempt to locate the transpositional level that will produce a perfect fourth as a residual interval and form a quartal tetrachord. The resolution of this problem in mm. 35ff is accompanied by a change in format and the dissolution of contrary motion.
Example 16, Bartók’s “Fourths,” mm. 35–38
a)
b)
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Example 17, formal diagram of “Fourths” that shows the changing geometry of the matrices occuring along with a change in prominent residuals that parses the piece into two sections
II
II
Transpositional Combination and Referential Pitch Collections
In “Properties and Generability of Transpositionally Invariant Sets,” Cohn discusses the relative ubiquity of transpositionally-invariant (TINV) sets—like the octatonic and whole-tone collections—in the music of the twentieth-century. He suggests that the group of TINV sets are not common because of any special property these sets have, a priori.22 Following David Lewin,
22
Among their special properties, Cohn points out that TINV always map onto themselves at some nonzero transposition, that they generally have the ability to interact with diatonic collections, and they are susceptible to displaying inversional symmetry. However, Cohn points out that a set’s ability to map onto itself at a non-zero transposition only impoverishes the set’s “fund of resources.” The ability of many of these sets, like the octatonic and whole-tone, to interact with diatonic collections is interesting, but Cohn notes that similarity measures show relatively little similarity between TINV collections and the diatonic. Finally, many other sets display inversional invariance and are capable of exhibiting inversional symmetry; however, TINV sets are much more common. Cohn Properties and Generability, 2–4.
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who is concerned more with transformational processes than set-class identification,23 Cohn instead suggests that TINV are “frequent byproducts of standard transformational routines.”24 Cohn demonstrates this initially in terms of chromatic sequential explorations in nineteenth-century music. In transformational terms, these passages contain diatonic objects that are continuously transformed through chromatic sequence. For instance, Example 18a reproduces Cohn’s Example 1b25, where a three-note diatonic motive [F, E, A] from Wagner’s Die Walküre is sequenced through a series of transpositions by minor third, an (014) * (0369) transpositional combination that produces an octatonic collection (Example 18b). Regarding this and other passages from composers of diatonic, tonal music, Cohn suggests that claiming these composers “strode through the looking-glass into a revolutionary new octatonic universe is to miss the point. It seems much more likely that [these composers were] simply pursuing diatonic routines through chromatic territory, and that the octatonic collection is a result of this evolutionary transformational move, not its revolutionary cause.”26 Such is the case in many twentieth-century compositions, as well. Cohn shows passages from Bartók, Scriabin, and Webern where transpositional combinations sequenced through chromatic territory generate entirely octatonic passages.27 In these cases, the octatonic collection materializes as a result, not as a starting point. While Cohn never suggests that analysts should
23
David Lewin, Generalized Musical Intervals and Transformations (New Haven: Yale University Press,
24
Cohn, Properties and Generability, 7.
25
Ibid., 8.
26
Ibid., Properties and Generability, 7.
27
Ibid., Properties and Generability, 10.
1987).
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abandon collectional thought, his observations acknowledge that the process—the transformational routine—is at least as important as the collection itself.
Example 18, Wagner, Die Walküre, Act II, scene 4 reproduced from Cohn’s Example 1b a)
b)
The emphasis on process and generation as opposed to (or in addition to) referential object or collection has implications in analyses of Crumb’s music. As the first chapter demonstrated, Crumb’s compositions frequently reveal an allegiance to one or another referential pitch collection. However, Cohn’s article suggests that a generative approach may produce an important analytical payoff that might be otherwise ignored or missed through a collectional analysis that ultimately generalizes large spans of music under a single collectional umbrella. Transpositional combination is effective in studying these musical details because it recognizes not only objects and collections as well as the transformative or generative process, but through various formats and permutations, TC appreciates the specific ways that these objects and processes are realized. The remainder of this chapter will use these concepts towards an analysis of form in George Crumb’s Lux Aeterna. Musical form in George Crumb’s compositions has been a
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favored topic among Crumb scholars in recent years. While differing in approach, these studies have all shown that his compositions contain carefully constructed and diverse formal structures.28 Richard Bass has shown that the symmetrical, pitch-structural basis of the Makrokosmos is developed principally through symmetrical organizational (formal) schemes. More recently, both Bass and William Lake have discussed Crumb’s use of simple and complex proportions in a manner not unlike Ernö Lendvai’s study of form in Bartók’s music.29 This study will approach form in Crumb’s Lux Aeterna from a new angle by discussing transpositional combination as a form-defining principle and process.30 From the collectional perspective, it is a rather banal piece; the composition lasts thirteen minutes, most of which are dominated by a single whole-tone collection. The following analysis uses the TC refinements just discussed to explore how this collection is realized. While transpositional combinations do generate whole-tone collections in the composition, this analysis will show how the specific realizations of transpositional combinations are determinants of form in the piece, a formal point
28
Richard Bass, in his article “The Case of the Silent G” applies Golden Section Principles to the theme from Crumb’s Gnomic Variations. He finds that a missing pitch, “G as in ‘Gnome,’” coincides with the golden section. In “Form and Temporal Proportions in Ancient Voices of Children,” Lake discusses proportions including the Golden Section and the ways that these proportions interact with the dramatic trajectory of pieces in the Crumb’s Ancient Voices of Children; Bass, “The Case of the Silent G: Pitch Structure and Proportions in the Theme of George Crumb's Gnomic Variations," in George Crumb and the Alchemy of Sound, Steven Bruns, Ofer Ben-Amots, Michael Grace, eds., (Colorado Springs: Colorado College Music Press, 2005), 157-170; William Lake, “Form and Temporal Proportions in George Crumb’s Ancient Voices of Children,” in George Crumb and the Alchemy of Sound, Steven Bruns, Ofer Ben-Amots, Michael Grace, eds., (Colorado Springs: Colorado College Music Press, 2005), 83– 100. 29
Lendvai’s analyses of Bartók attempt to apply Golden Section and Fibonnaci principles not only to form and temporal proportions, but also melodic construction and harmony. Ernö Lendvai, Béla Bartók: An Analysis of His Music, (London: Kahn and Averill, 1971). 30
Musical form and its relation to TC is also explored by Richard Cohn in his dissertation. Many of my ideas about form in Lux Aeterna originate in Cohn’s TC approach to formal structure in his analysis of Bartók’s 4th String Quartet and Sonata for Two Pianos and Percussion. Cohn, “TC in the 20th Century,” 325–481 In these analyses, Cohn shows how sections are characterized by TC properties and how a sense of musical syntax is created through syntagmatization processes. Overall, my approach differs from Cohn’s in that I am exploring TC’s influence on a larger scale and using the tools to view Lux Aeterna through two different formal “lenses.”
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of view that would be obscured through an entirely collectional lens. In addition, the following analysis aims to use TC as a way to understand the dynamic trajectory of the Lux Aeterna and its interaction with the more obvious formal structure–observations that would also be difficult to explore by viewing the composition simply as a manifestation of a single whole-tone collection. Two initial observations concerning the formal structure of Lux Aeterna are summarized in the two halves of Figure 1. (The length of each section in seconds is indicated as a rough approximation of duration.) First, the composition seems to be divided into sections that undergo a formal liquidation, as shown in Figure 1a. The initial portion of this analysis will discuss how the A sections are defined by certain TC or non-TC properties. This liquidation divides the composition asymmetrically as indicated by the steadily decreasing durations. In consort with the formal liquidation, the composition’s dramatic trajectory reaches a climactic point at roughly the midpoint of the composition, creating a nearly symmetrical parsing as shown in Figure 1b. This climax is reinforced by louder dynamics, greater rhythmic density, and fuller instrumentation. The final portion of this analysis will show that the pitch material that leads to this climax is the result of a TC process initiated at the beginning of the piece.
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Figure 1, Formal diagram of Lux Aeterna reflecting two observations. First (a), sections in Lux Aeterna seem to be steadily liquidated, dividing the composition asymmetrically. Second (b), the composition contains a climatic passage–indicated by the arrow–that divides the composition symmetrically.
a)
b)
Figure 2 gives a detailed view of the formal liquidation, indicating the fact that the A sections are made up of three consistently ordered subsections. While the length of each A section diminishes in length only approximately throughout the piece, each B section decreases in length systematically and is distinguished from the A sections by tempo, instrumentation (these passages use sitar and recorder), and an identifiable rhythmic pulse that A sections lack. The B sections sound improvisatory, as well, though the pitch content of these passages is restricted to the whole-tone collection. Other than the passages labeled non-TC, the pitch content of each A section is made up exclusively of a single whole-tone collection. Using TC as an analytical tool, these subsections can be distinguished by particular TC operations, formats, and permutations, which are collated
64
in Figure 3 for future reference. The following discussion illuminates these details. This generative approach allows not only a better understanding of passages organized by TC, but also sheds some light on the non-TC passages, as well.
Figure 2, While both A and B sections are slowly liquidated, the B sections do so systematically. A sections are further segmented into three consistently ordered subsections identified by specific TC or non-TC properties.
Figure 3, Table showing the TC similarities and differences between TC1, non-TC, and TC2 sections. TC1
non-TC
TC2
TC operation
(2 * 6)
Transpositional objects are borrowed from TCx sections; disruptive ic1 precludes TC parsing
(2 * x), where x is 2, 4, or 6.
Format Type
HV = horizontal + vertical realization
Mostly H2. Vibraphone and Crotale chords are V2.
H2 = completely horizontal realization
Permutation Type
Mostly fully explicit realizations; some partially explicit realizations
N/A
Particular partially explicit permutation
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TC1 sections, shown in Example 19, consist nearly exclusively of (2 * 6) TC operations realized in an HV format, and presented in both fully and partially explicit representations.31 As the TC operations in Example 4 show, the pitch material in nearly every TC1 section can be derived from some transposition of 2 * 6. Musically speaking, this operation signifies ic2 as a motivic force and T6 as its most common transformation; from the standpoint of abstract operations, every TC1 section is identifiable through a web of ic2 dyads subjected to T6 transformations. All of the TC1 sections share the HV format class, as well. In each passage this is manifest not chordally, but canonically. The HV format is represented clearly by the matrices in Examples 19b and 19d, where the fully-explicit realizations equally emphasize both operands. However, because of a permutation in Example 19a and 19c, an alternate matrix is given that better shows the retrograde melodic relationship between the ic2 dyads. Interestingly, this permutation emphasizes only ic2 explicitly and hides the transformational relationship. As a result of the permutation, the ic2 dyads are disposed symmetrically around the pitch F, which is the very next pitch in both passages.32 Shown in Example 20, the four TC2 sections also consistently use ic2 as an operand. The other operand is variable but maintains the whole-tone background established in the TC1 passages. TC2 passages are characterized by entirely horizontal (H2) formatting. H2 formats have two, melodically presented operands: a local operand (H1), and a more remote operand (H2). In all of the TC2 sections, ic2 is the H1 operand and the H2 operand is variable. An H1-type permutation causes only the ic2 operand to be heard explicitly; the variable H2 operand is present structurally, but is not aurally salient. Both Examples 20a and 20d (the first and last TC2 31
In Example 4a and 4b, Crumb asks the percussionist to place a crotale on the top of timpani to produce a timbral effect. This is indicated in these examples on a separate score. In these passages, only the upper staff is audible. All examples are taken from George Crumb , Lux Aeterna, New York, C.F. Peters Corporation, 1972. 32
The F is shown only in Example 4c. Space limitations prevented showing it in Example 4a.
66
Example 19, TC1 sections
67
sections) use this permutation to surround F symmetrically in the same way that the HV format in Examples 19a and 19c disposed the two ic2 dyads symmetrically around F. In the B sections that follow each TC2 section, F is given considerable emphasis as a point of centricity.
Example 20, TC2 sections
To summarize briefly, the TC approach generalizes the generative path taken in each section as well as the specific way that this path is traversed. Both TC1 and TC2 passages pursue
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2 * x TC operations while maintaining the background whole-tone collection. The formatting change from TC1 to TC2 is obvious through a comparison of the geometry of the matrices in Example 20 to those in Example 19. While permutations are common to every TC2 passage, the permutations affect some portions of TC1 as well. Example 21 shows the non-TC sections that lie in between each TC1 and TC2 section and contain the majority of Lux Aeterna’s text. Quite strikingly, non-TC sections do not exhibit the consistent TC parsing that characterizes the other portions of Lux Aeterna, and as a result, they are not analyzable as belonging to a whole-tone collection either. In fact, these passages are the only “non-whole–tone” passages in the composition. However, as the circles and transpositions in the example show, these passages do contain TC operations in which the objects, the circled portions of the example, are ics 2, 4 or 6—those intervals common to the TC1 and TC2 sections. In every one of these passages a disruptive ic1 always prohibits TC from attaining status as the sole organizing force; these disruptive ic1s are shown in the example with grey, slashed noteheads. The disruptive role of ic1 is especially apparent in the tubular bell and crotale chords that initiate the first and last non-TC sections (Examples 21a and d) and come near the end of the second and third non-TC sections (Examples 21b and c). Each of these sonorities can be divided into two trichordal halves, based on instrumentation. Each trichord is written so that the upper and lower pitches form interval 4 and the middle pitch is a half-step from one of these boundary pitches. In Example 21a for instance, the tubular bells have boundary pitches of A and D, and the crotales have boundary pitches of E and A—both interval 4. The middle pitch in both trichords, however, is a half-step above or below one of the boundary pitches, forming a disruptive ic1 that negates any
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transpositional relationship. Furthermore, the tubular bells’s trichord is a half-step above the highest pitch of the crotales trichord. Another representative example is shown in Example 21b where after the initial ascending ic1 from C to B, the soprano sings a steadily descending line of ic2s that follow a clear transpositional path. If not for the initial ascending ic1 this passage would be easily analyzable through both a whole-tone and TC lens. In 6c, each quintuplet figure in the bells and vibraphone contains three unique pitches. The upper two pitches of these figures, circled in the example, all belong to ic4 and are transpositionally related to one another; however, the initial, registrally displaced pitch of each figure forms an ic1 with one of the two upper pitches, obscuring the transpositional relationship. Through the boxes, Example 21 also shows that ic1 is an expressive force in conjunction with the text in each of these non-TC sections. The most striking instance is shown in Example 21b, where the soprano proclaims “Domine” (Lord) in conjunction with an extraordinary ic1 leap two octaves down from B5 to A2. However, ic1 occurs along with important words in all of the non-TC sections. In the first and fourth sections, “Lux” and “aeterna” are highlighted through an ic1 oscillation and a descending ic1, respectively. In the third section, “Requiem” is emphasized in long note values as a leap downward from E5 to D4. The importance of ic1s relationship with the text underscores its important function as a disruptive element among the ics 2, 4, and 6 that accompanied the TC operations in each TC1 and TC2 section. So far this analysis has shown that TC1 and TC2 sections can be grouped together as similar because they share general TC characteristics: both use ic2 as a motivic operand and use transformations that maintain the whole-tone background; however, the analysis has also pointed out ways to distinguish between these sections through formats and permutations. The approach
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Example 21, non-TC sections
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allowed views the non-TC sections in light of transpositional combination and understands these section’s pitch organization as chromatic colorations of the more consistently applied TC technique that characterized surrounding sections. The A sections, which are constructed from this regular ordering of TC1–non-TC–TC2, diminish in length throughout the composition alongside the systematically diminishing B sections. Consequently, the overall impression is of a formal liquidation that parses the piece asymmetrically. In contrast to the asymmetry created by formal liquidation, the dramatic trajectory of Lux Aeterna’s formal structure is much more symmetrical, as was indicated in Figure 1. The following analysis argues that the dramatic trajectory is influence by a TC process called syntagmatization. In the syntagmatization process, interval byproducts, or residuals, of a TC operation become fully explicit TC operands in a later portion of the composition. In this way, the residual intervals attain greater importance as the composition progresses and attain a structural function.
Figure 4, Diagram showing Lux Aeterna’s dramatic trajectory. The climax is reinforced through dynamics, greater rhythmic density, register, and pitch organization.
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Residual intervals of transpositionally-invariant set classes always belong to the same interval class. For instance, as the arrows in Example 22 show, the two residual intervals of the transpositionally-invariant (0268) are C–G and D–F, both members of ic4.
Example 22, residual intervals of the transpositionally-invariant set-class, (0268)
This property creates an interesting situation when a TC operation involving a transpositionallyinvariant set class is permuted. In particular, because the residual intervals are the same, they appear to be explicit when permuted. In Example 23b, a 2 * 6 operation undergoes an H-type permutation, which maintains the explicitness of ic2 in the horizontal dimension but hides the T6 transpositional relationship. When the permutation occurs, the ic4s that were residuals in Example 23a become verticalites, creating a situation in Example 23b where T4 and not T6 may seem to be the transpositional relationship: the impression is of 2 * 4, not 2 * 6. However, this impression is false. Ic4 in this Example 23b is not an explicit interval. In fact, it is not even an operand. (As Example 23c shows, a 2 * 4 operation produces a completely different set-class.33)
33
This interesting result occurs because both ics2 and 4 are interchangeable in their ability to create (0268) when transpositionally combined with ic6. For instance, T0 (2 * 6) = T4 (4 * 6).
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Example 23, full and false explicitness; syntagmatization a) fully-explicit (0268)
b) falsely explicit (0268)
c) fully-explicit (0246)
The relationship between Example 23b and 23c is illustrative of a TC process called syntagmatization.34 In a passage where Example 23c occurs after Example 23b, the false explicitness of ic4 in 23b is realized as full explicitness in 23c. In other words, interval-class explicitness of the vertical operand in 23b becomes interval explicitness in 16c; or, contrary motion created by the permutation in 23b becomes parallel motion in 23c. The illusory 2 * 4 in 23b becomes an actual 2 * 4 in 23c. The relationship is indicated by the question and exclamation marks that appear along with the musical dimensions. A question mark indicates false explicitness, such as occurs with ic4 in Example 23b. An exclamation mark indicates that the operand has become fully explicit through the process of syntagmatization. Alternations between full and false explicitness have important implications that influence the dramatic trajectory of Lux Aetnerna, as shown by the question and exclamation marks in Figure 4. Many passages introduce a falsely explicit ic4 whose implications are realized in the following passage. As mentioned, this ebb and flow often takes the form of contrary versus parallel motion. As I will show, ic4 is ultimately freed from the falsely explicit environment and allowed to function solely as a fully explicit operand. At this moment, shown by the large arrow 34
Cohn, “TC in the 20th Century,” 223.
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in Figure 4, the upward parallel motion that results from the full explicitness stimulates the climax of the composition, where ic4 breaks free from ic2 and the whole-tone collection that has served as the background for most of the piece. Example 24 presents the beginning of the piece, where the first instance of this process occurs. Here, the initial T1(2 * 6) operation is permuted, rendering ic6 inexplicit and causing ic4 to appear to be an explicit operand. These ic4s occur between C and A in the first percussion part and E and G in the second percussion part, the ic2s are present as verticalites.35 The implications generated by the false explicitness of ic4 in the opening are realized as full explicitness in the subsequent passage. As Example 24 illustrates, the TC operations immediately following the opening are T5(2 * 4) and T9(2 * 4), and in both cases, ic4 is an explicit operand along with ic2. Musically speaking, the ic4s in the first part of this example become interval 4s in the second half. This change is emphasized as the instrumentation and format of the operation change along with the explicit intervals. Another way of hearing the passage recognizes that the contrary motion in the initial TC operation becomes parallel motion in the subsequent passage. In fact, alternations between contrary and parallel motion, as a result of changes in partially-explicit and fully-explicit TC realizations, characterize the first half of the composition. The contrary motion, in particular, prohibits any sort of climactic moment produced through registral achievement. As mentioned earlier, in the climatic section of Lux Aeterna, ic4 finally asserts its full potential and completely breaks free of false explicitness, projecting the syntagmatization process onto a larger scale. In the process, ic4 ultimately sheds the ic2 operand
35
The “false explicitness” of ic4 is even more illusory in a live performance. Because each instrument plays an ic4 dyad and the ic2 dyad results only from their relationships to one another, their physical location serves to isolate the two ic4s.
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and the whole-tone background. Fundamental to the climactic character that the passage takes on are fully-explicit (2 * 4) realizations that cause all of the instruments to move in parallel motion. Example 25 shows this portion of Lux Aeterna. Here, the soprano and other instruments form what may be described as a TC canon, each voice continuously stating horizontally formatted, fully explicit versions of (2 * 4). As mentioned, the full explicitness creates parallel motion that causes each member of the ensemble to move uniformly higher in tessitura. Notice the importance that T4 has taken on by this point in the composition. In both the soprano and in the bass flute, the long note values accentuate the T relationships between the oscillating ic2 4
dyads. Shown on the bottom left-hand portion of the score, the vibraphone sheds the motivic ic2 that has been fundamental in each section of the piece thus far and states a complete 4 * 4 transpositional combination over the span of the two measures that lead to the high point of the passage. This moment is significant in that it is the only passage in the entire composition in which ic2 is not a member of a TC operation. Thus, in conjunction with the climax, ic4 has finally attained a sense of fully-explicit TC freedom. To emphasize the point, immediately following the climax the vibraphone plays a descending series of ic4 verticalities that eventually “break free” of the whole-tone background through T5 transformations. The result of this passage in terms of its referent collection is the complete chromatic, an important contrast to the surrounding wholetone passages.
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Example 24, In this passage from the opening of the composition, the falsely-explicit ic4 is subsequently realized as a fully explicit ic4 in the following passage.
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Example 25, Climactic passage of Lux Aeterna
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Unlike the liquidation process that created a formal asymmetry, the dramatic trajectory that results from this TC process splits the composition nearly in half. In effect, the symmetry and asymmetry of these two readings “cut across” one another, as Figure 1 shows. Of interest is the coincidence of the climatic portion of the piece with the second TC2 section. Such a coincidence suggests that the two differing formal perspectives are carefully intertwined, this passage being the fulcrum upon which the two formal perspectives rest. This analysis has shown that transpositional combination can be an effective entryway into a study of musical form. While emphasizing the interesting dual-nature of formal structure in Lux Aeterna, the analysis also accentuated the importance of viewing this music beyond mere collectional labels. In this piece, using the general term whole-tone to describe a passage would have seriously overlooked important compositional details. Rather, TC viewed these passages as moments where single intervals receded and gained in prominence, all against the same harmonic background. In some ways, TC operands and residual intervals act as dynamic interval vectors, pointing out the intervallic ebb and flow contained within any given set-class.
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Conclusions
The objective of this thesis has been to discuss various ways in which transpositional combination can be used as a tool that promotes analysis that integrates many features of a composition. As others have noted, the combination of basic pitch material to generate larger sets or collections is a prominent characteristic of many of Crumb’s compositions.36 In both chapters of this thesis, transpositional combination was used to describe how the interaction of generative material and larger structures function in two compositions by Crumb and how the resulting processes interact with other musical dimensions. Chapter one focused on the results of TC operations—the referential collections—and compared the specific generative paths used to create these referential collections in order to discuss their interaction. Attuned to the set of variations and the variation process that occupies Vox Balaenae’s core, the chapter distinguished between three types of variation involving different manipulations of the two components of a transpositional combination—the object and its transposition. These variation types always involved a change in referential collection—the varied element—while retaining either an object or its transposition as a shared attribute. The final type of variation discussed in chapter one involved a discussion of elements of musical quotation and the extramusical meaning of Vox Balaenae. The concept of variation invoked relied specifically upon a listener’s perception of, or identification with, the diatonic tonality and tonal pulls of familiar tonal quotations. Crumb’s alteration of Richard Straus’s Nature theme from Also Sprach Zarathustra is a type of variation that involves a change in collection (from diatonic to octatonic) while maintaining a diatonic object as shared attribute. A 36
Bass, Sets, Scales, Symmetries, 3; Scotto, Transformational Networks, Transpositional Combination, and Aggregate Partitions, 1–4.
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listener’s perception of these quotations, and the ways in which they are varied, may recognize both the diatonic elements and the transformations that force them into an octatonic context. This chapter integrated these aspects of pitch structure and variation with the extramusical content associated with Vox Balaenae. In particular, both Also Sprach Zarathustra and Vox Balaeane were shown to engage in a dialogue between collections. The diatonic/octatonic interaction in Vox Balaenae is reflective of a similar conflict between C major and B major/minor in Straus’s Also Sprach Zarathustra. Furthermore, conflict as an element of pitch structure in both compositions seems reflective of the intellectual ideas surrounding Vox Balaenae’s composition; namely, the interaction or conflict between man and nature. Chapter two discussed the formal implications of specific features of a TC realization— the ordering and presentation of operands or transformations. In order to discuss these attributes in as specific a manner as possible, chapter two began by discussing refinements to the TC apparatus that better represent the particular way that a TC operation is realized in a passage of music. Unlike chapter one, which was interested primarily in the results of the TC operations, this chapter revolved around the generative process itself and the ways that this generative process is carried out. As an analytical capstone, Lux Aeterna was discussed from two formal perspectives that attempted to appreciate diverse facets of the composition’s organization. Sections were defined based upon specific TC properties. Not only were the types of operation taken into consideration, but also the ways that these operations are realized in the composition. By ascribing TC properties to certain regularly recurring passages, non-TC passages were able to be understood in reference to an overall TC context. Because the non-TC passages contained the majority of Lux
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Aeterna’s text, these passage’s deviations from TC organization were discussed in connection with expressive features of the text. The overall liquidation process that is essential to the formal organization was contrasted with the symmetrical character of Lux Aeterna’s dynamic trajectory. The increased register, dynamic level, rhythmic activity, and instrumentation that result in the climax of the composition were shown to coincide with the realization of a TC process that was initiated at the beginning of the piece. This process involved the residual intervals that result from any TC operation, as well as their increased importance as an operation is permuted. Ic4, which was a prominent residual interval at the opening of the piece, is subsequently elevated to the status of a controlling TC operand at the climax of the composition. This process also involves the disposition of operands at the opening and at the climax. At the beginning of the piece, the permutation of operands that gave rise to the prominent ic4 resulted in contrary motion between the two operands, prohibiting a climax due to any sort of registral achievement. On the other hand, TC operations that distinguished the climax arranged the operands in parallel motion, forcing the register of the music to rise towards the climax. Though the TC approach placed an emphasis on details of pitch organization in both chapters of this thesis, the analytical payoff is significant. By starting with a greater understanding of pitch organization, the analyses were able to approach other musical elements and discuss them in relation to concrete analytical findings. Vox Balaenae’s pitch organization, based on three types of variation produced from slightly different approaches to transpositional combination, was integrated with extramusical ideas surrounding the piece. The small details of Lux Aeterna’s construction were related to larger formal constructions. Because every dimension—musical and otherwise—of Crumb’s compositions is intricately constructed,
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integrative analytical approaches that seek to understand the sum of a composition’s parts may be the most valuable.
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BIBLIOGRAPHY Antokoletz, Elliot. “Interval Cycles in Stravinsky’s Early Ballets.” Journal of the American Musicological Society 39/3 (Autumn 1986): 578–614. Bass, Richard. “Sets, Scales, and Symmetries: The Pitch-Structural Basis of George Crumb’s Makrokosmos I and II.” Music Theory Spectrum 13 (Spring 1991): 1–20. ___________. “Pitch Structure in George Crumb’s Makrokosmos, Volumes 1 and II.” Ph.D. diss., University of Texas, 1987. ___________. “Models of Octatonic and Whole-Tone Interaction: George Crumb and His Predecessors.” Journal of Music Theory 38 (Autumn 1994): 155–186. ___________. “The Case of the Silent G: Pitch Structure and Proportion in the Theme of George Crumb’s Gnomic Variations.” George Crumb and the Alchemy of Sound. Edited by Steven Bruns, Ofer Ben-Amots, Michael Grace. Colorado Springs: Colorado College Music Press, 2005. Berger, Arthur. “Problems of Pitch Organization in Stravinsky.” Perspectives of New Music 2 (Spring 1963): 11–42. Bruns, Steven M. "In stilo Mahleriano: Quotation and Allusion in the Music of George Crumb.” The American Music Research Center Journal 3 (1993): 9–39. ____________. “Les Adieux: Haydn, Mahler, and George Crumb’s Night of the Four Moons.” George Crumb and the Alchemy of Sound. Edited by Steven Bruns, Ofer Ben-Amots, Michael D. Grace. Colorado Springs: Colorado College Music Press, 2005. Chatman, Stephen. “George Crumb: Night of the Four Moons–The Element of Sound.” Music & Man 1/3 (June 1974): 215–224. Crumb, George. Vox Balaenae for Three Masked Players. New York: C.F. Peters Corporation, 1971. _____________. George Crumb: Eleven Echoes of Autumn, Four Nocturnes, Vox Balaenae, Dream Sequence. Liner notes by the author. Jecklin Edition 705, 1996. Compact disc. –––––––––––––. Lux Aeterna, New York: C.F. Peters Corporation, 1972. Cohn, Richard. "Transpositional Combination in Twentieth Century Music." Ph.D. diss., University of Rochester, 1987. ___________. “Inversional Symmetry and Transpositional Combination in Bartók.” Music Theory Spectrum 10 (1988): 19–42.
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. “Properties and Generability of Transpositionally Invariants Sets.” Journal of Music Theory 41 (Autumn 1991): 1–32 . “Transpositional Combination of Beat-Class Sets in Steve Reich’s Phase-Shifting Music,” Perspectives of New Music 30 (Fall 1992): 146–177. DeDobay, Thomas R. “The Evolution of Harmonic Style in the Lorca Works of Crumb.” Journal of Music Theory 28 (1984): 89–111. Gillespe, Don, ed. George Crumb: A Profile of a Composer. New York: C.F. Peters Corporation, 1986. Gilliam, Bryan. “Strauss, Richard,” Grove Music Online. Edited by L. Macy (Accessed 15 May 2007), . Headlam, David. “Integrating Analytical Approaches to George Crumb’s Madrigals, Book I, no. 3.” George Crumb and the Alchemy of Sound. Edited by Steven Bruns, Ofer Ben-Amots, Michael Grace., Colorado Springs: Colorado College Music Press, 2005. Kielian-Gilbert, Marianne. “Relationships of Symmetrical Pitch-Class Sets and Stravinsky’s Metaphor of Polarity.” Perspectives of New Music 21 (Autumn 1982): 209–240. ____________________. “Pitch-Class Function, Centricity, and Symmetry as Transposition Relations in Two Works by Stravinsky.” Ph.D. diss., University of Michigan, 1981. Kubrik, Stanley. 2001: A Space Odyssey. Produced and directed by Stanley Kubrick. 141 min. MGM, 1968. DVD. Lake, William. “Form and Temporal Proportions in George Crumb’s Ancient Voices of Children.” George Crumb and the Alchemy of Sound. Edited by Steven Bruns, Ofer BenAmots, Michael Grace. Colorado Springs: Colorado College Music Press, 2005. Lendvai, Ernö. Béla Bartók: An Analysis of His Music. London: Kahn and Averill, 1971. Lewin David. Generalized Musical Intervals and Transformations. New Have: Yale University Press, 1987. McFarland, Mark. “Transpositional Combination and Aggregate Formation in Debussy.” Music Theory Spectrum 27 (Fall 2005): 187–220. Moseley, Brian. “Transpositional Combination and Collectional Interaction” (paper presented at the Music Theory Midwest, Munice, Indiana, 13 May 2006).
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Nietzsche, Friedrich. Thus Spoke Zarathustra: a book for all and none. Translated by Adrian Del Caro. Cambridge: Cambridge University Press, 2006. Ott, David Lee. “The Role of Texture and Timbre in the Music of George Crumb. Ph.D. diss., University of Kentucky, 1982. Parks, Richard S. “Harmonic Resources in Bartók’s ‘Fourths’.” Journal of Music Theory 25/2. (Autumn 1981): 245–274. Piersall, Edward. “Symmetry and Goal-Directed Motion in Music by Béla Bartók and George Crumb.” Tempo 58/228 (2004) : 32–40. Roig-Francoli, Miguel. “A Theory of Pitch-Class-Set Extension in Atonal Music.” College Music Symposium 41 (Fall 2001): 57–90. Santa, Matthew. “Defining Modular Transformations.” Music Theory Spectrum 21 (Autumn 1999): 200–229. ____________. “Analysing Post-Tonal Diatonic Music: A Modulo 7 Perspective,” Music Analysis 19 (Spring 2000): 167–201. ____________. “Studies In Post-Tonal Diatonicism: A Mod7 Perspective.” Ph.D. diss., City Univ. of New York, 1999. Segall, Alan. George Crumb: Voice of the Whale. Liner notes by the author. Zuma Records 102, 1994. Compact Disc. Schmidt, Tracey. “Borrowing in George Crumb’s An Idyll for the Misbegotten.” D.M.A thesis, University of Colorado-Boulder, 2004. Scotto, Ciro G. “Transformational Networks, Transpositional Combination, and Aggregate Partitions in Processional by George Crumb.” MTO 8.3 (2003). Shuffett, Robert Vernon. “The Music, 1971–1975, of George Crumb: A Style Analysis.” D.M.A. thesis, Peabody Conservatory of Music, 1979. Steinberg, Russell. “’Meta-Counterpoint’” in George Crumb’s Music: Exploring Surface and Depths in Vox Balaenae (Voice of the Whale).” George Crumb and the Alchemy of Sound. Edited by Steven Bruns, Ofer Ben-Amots, Michael Grace. Colorado Springs: Colorado College Music Press, 2005. Straus, Joseph. “Atonal Composing-Out.” In Order and Disorder: Music-Theoretical Strategies in 20th-Century Music, the fourth publication in the series Collected Writings of the Orpheus Institute, edited by Peter Dejans. Leuven, Belgium: Leuven University Press, 2004.
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Timm, Kenneth. “A stylistic analysis of George Crumb's Vox Balaenae and an analysis of Trichotomy.” D.M.A diss., Indiana University, 1977. Tymoczko, Dmitri. “Stravinsky and the Octatonic: A Reconsideration.” Music Theory Spectrum 24 (Spring 2002): 68–102. ______________. “Octatonicism Reconsidered Again.” Music Theory Spectrum 25 (Spring 2003): 185–202. Van den Toorn, Peter. The Music Of Igor Stravinsky. New Haven: Yale University Press, 1983. Walsh, Michael. Works by George Crumb. Liner notes by the author. New Worlds Records 357– 2, 1992. Compact disc. Williamson, John. Strauss, Also Sprach Zarathustra. Cambridge: Cambridge University Press, 1993.
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