37? /set Mo, 7S7i
PREFERENTIAL STRATEGIES IN ELLIOTT CARTER'S
STRING QUARTET NO. 2
THESIS
Presented to the Graduate Council of the
University of North Texas in Partial
Fulfillment of the Requirements
For the Degree of
MASTER OF MUSIC
By
Elizabeth B. Crafton, B.Mus.
Denton, Texas
May, 1999 Crafton, Elizabeth B., Preferential Strategies in Elliott Carter's Strine Quartet
No^2. Master of Music (Theory), May, 1999, 55 pp., 8 examples, 5 figures, 7 tables,
references, 48 titles.
For the purposes of expressive intent, Carter developed compositional strategies
that possess qualities congruent with the musical language in his Second Quartet (1959).
He employed strategies including tempo modulation, triple groupings, and large-scale
ratios to assemble the musical discourse and to guide the listener's perception of large-
scale continuity. I label these devices collectively as "preferential strategies" because it is
Carter who selects certain pre-compositional ideas that organize musical material and demarcate structural locations. Tempo modulations that organize dual meters and triple groupings that interact in transitional and transformational ways demonstrate his concern with controlling the overall time continuity through local level organization. Large-scale ratio relations between nine interlocking sections of this four movement work illustrate how Carter employs a local strategy that projects a large-scale structure. Recognizing that Carter s ultimate compositional goal prioritizes temporal processes, these proposed preferred strategies articulate a convergence of musical elements. 37? /set Mo, 7S7i
PREFERENTIAL STRATEGIES IN ELLIOTT CARTER'S
STRING QUARTET NO. 2
THESIS
Presented to the Graduate Council of the
University of North Texas in Partial
Fulfillment of the Requirements
For the Degree of
MASTER OF MUSIC
By
Elizabeth B. Crafton, B.Mus.
Denton, Texas
May, 1999 Copyright by
Elizabeth B. Crafton
1999
111 TABLE OF CONTENTS
Page
LIST OF EXAMPLES
LIST OF FIGURES vi
LIST OF TABLES vii
Chapter
1. INTRODUCTION
Concern with Large-Scale Structure Concern with Discourse Current Research Criteria for Preferential Strategies Design of the Second Quartet
2. TEMPO MODULATIONS
Tempo Modulation as a Term Tempo Modulation as a Strategy
3. TRIPLE GROUPINGS 29
Grouping Divisions Triple Grouping Strategy Triple Groupings as Transitions Triple Groupings in Transformations
4. LARGE-SCALE RATIOS
Local Level Ratios Large-Scale Ratios
5. CONCLUSION
BIBLIOGRAPHY 52
IV LIST OF EXAMPLES
Example page
1. Notated Tempo and Projected Speed as Simultaneous Stream 20
2. Transformed Successive Streams 23
3. Tempo Modulation that Organizes Dual Meters, measures 171-186 27
4. Tempo Modulation that Organizes Dual Meters, measures 228-237 28
5. Rounded Grouping Division; Adjusted Grouping Division, measure 75 30
6. Triple Groupings as a Transition, measures 72-75 37
7. Triple Groupings in Transformations, measures 75-78 39
8. Polyrhythms as Vertical Ratio Relations 43 LIST OF FIGURES
Figure Page
1. Formal Diagram of the Second Quartet 15
2. Grouping Division Diagram, measures 75-87 33
3. Triple Groupings as a Transition, measures 73-75 37
4. Triple Groupings in Transformations, measures 75-78 40
5. Tempo Relations in the Second Quartet 44
VI LIST OF TABLES
Table DPag e
1. Formal Structure of the Second Quartet 15
2. Character-Patterns in the Second Quartet 15
3. Grouping Divisions, measures 75-87 32
4. Triple Grouping Transformations, measures 76-84 39
5. Measure Analysis of Ratio Relations 47
6. Pulse Analysis of Ratio Relations 47
7. Large-Scale Ratio Relations 49
Vll CHAPTER 1
INTRODUCTION
Concern with Large-Scale Structure
Very early in his career, American composer Elliott Carter (b. 1908, New York) became interested with issues concerning large-scale structure in contemporary composition. Even as a student at Harvard and later in Paris, Carter felt that few composers had novel conceptions of large-scale continuity in their works. Part of the reason for this related to their preoccupation with writing works that were publicly accessible. Carter, on the other hand, was not so concerned with writing works simple enough for an audience to grasp on first hearing. Rather, he noted that he was interested
more "in the speed of presentation.. .how rapid and dense [the music] might be, and so
on." In a 1971 published conversation with Allen Edwards, he stated: "I began to think
about all sorts of different processes and ways of making music hang together which I
would probably not otherwise have brought into conscious focus and made into a primary
field of thought and musical action."1 Carter felt that his concern with the flow of music
as primary goal set him apart from other composers writing after the Second World War.
1 Allen Edwards, Flawed Words and Stubborn Sounds: A Conversation with Elliott Carter (New York: W.W. Norton, 1971), 36.
2lbid., 37. About 1944, Carter became concerned that composers were only consciously interested with "this or that peculiar local rhythmic combination or sound-texture or novel harmony and had forgotten that the really interesting thing about music is the time of it—the way it all goes along."1 He believed that while contemporary composers had explored "every imaginable kind of harmonic and timbral combination" there had been only a degree of rhythmic innovation on the local level in the music of Stravinsky,
Bart6k, Varese, and Ives. Carter further stated that "the way all this went together at the next higher and succeeding higher rhythmic levels remained in the orbit of what had begun to seem to [him] the rather limited rhythmic routine of previous Western music."2
Bothered with composer's lack of concern in "larger-scale time continuities," Carter reacted by exploring aspects of tempo modulation and found this technique, particularly when combined with controlled musical elements (e.g., prescribed intervals, character- patterns), created a convincing "musical logic" that was new and appropriate to the richness of the modern musical vocabulary. Through a union of rhythmic innovation paired with certain controlled other musical elements, Carter's musical syntax was
created. Though his degree of manipulation and the controlled elements themselves have
changed throughout his compositional career, Carter's rhythmic vocabulary appeared in
its first representative state in the Second Quartet (1959). This quartet, written at age
fifty, serves as the beginning of his truly representative works.
lIbid., 90. The vocabulary employed in the Second Quartet allowed Carter to experiment with musical elements that enabled local patterns of activity to contribute to large-scale
continuity. As a result of this experimentation, I propose that Carter adopted strategies
that propel musical motion and influence the listener's perception of large-scale
continuity. These strategies include tempo modulations, triple groupings, and large-scale
ratios. I label these collectively as "preferential strategies" because it is Carter who
selects strategies to organize musical material on a local level and to demarcate structural
locations within the overall continuity. Recognizing that Carter's ultimate compositional
goal prioritizes temporal processes, these proposed preferred strategies articulate a
perspective of a convergence rather than an isolation of musical elements.
Concern with Discourse
A primary feature of the Second Quartet is its "musical flow," an element Carter
considers to be the ultimate goal of each work that is characterized in this composition by
seemingly non-cooperative elements coexisting in a new way. For Carter, musical flow
represents his treatment of musical time, an aspect he considers to be more important
than the particulars or the novelty of the overall vocabulary. In turn, the morphological
elements of any Carter composition owe their musical effect largely to their specific
placing in the musical time continuity. This idea parallels structures of verbal languages,
systems in which verbal expressions of anything, including time, require words and
letters for contextual meaning. Verbal expressions or idioms depend on specific word
ordering for their meaning no matter how unconventional the ordering may be. He feels
that each piece of music sets up its own frame of norms and deviants, and that the only question about this is the 'linguistic' level on which the identity of the piece and of its norms and deviants is established....In post-tonal music it's simply that each composer, every time he writes a piece, has the opportunity of 'making up his own language,' so to speak, conditioned only by the requirement that it be a language, i.e., that from the point of view of the imagined listener the morphological elements have a recognizable identity in each case and that their status as mutually relative "norms' and 'deviants'.. .be clearly established in the work. It's perfectly obvious that the liveliness of the operation of musical composition has always been just this~the composer's finding or inventing a linguistic 'frame' for his piece that will make all the details come to life in a new way.3
Internal conversational elements are recognizable within the frame of a Carter composition. Discourse becomes a means to facilitate interaction between instrumental parts, to which behavioral gestures of specific expressive characteristics, intervals, and rhythmic styles are assigned. Carter's prominent use of discourse as a significant compositional element may be traced to his Piano Sonata of 1945-46. In successive compositions written after 1945, Carter pursues the dramatic possibilities of contrast and interplay offered by the individual personalities of instruments. He juxtaposes non- idiomatic behavioral characters between contrasting instruments to exploit their dramatic possibilities. This method allows each instrument to play sonorities and gestures that go against its traditionally assigned nature. Carter's musical gestures and combinations exist in independent time-worlds and refrain from creating mere sound effects (e.g., col legno, tapping on wood, playing on the bridge). To him, sound effects impress an elementary past or future, neither of which can express nor manipulate varied past and future relationships.
3Ibid., 85-86. A notable influence upon Carter's evolution of musical discourse was his contact with mathematician, philosopher, and cosmologist, Alfred North Whitehead. During his student years at Harvard in the late 1920's, Carter took a course under Whitehead about the same time. Whitehead published Process and Reality. Carter admits that what he could understand of this work and Whitehead's other writings, with their stress on organic patterns, molded his thinking, not only about music.4
Carter's understanding of Whitehead's ideology evolved into an aesthetic awareness of time. Carter's manipulation of musical time embodies Whitehead's idea of
"process" and exploits its inherently consequential nature. For Carter, time exists in music as independent streams of rhythmic patterns that often interact polyphonically.
These patterns, juxtaposed and facilitated by tempo modulations, project a story with dialogue between characters rather than portray a collection of isolated rhythmic moments. Rhythmic tension-release patterns combine with specifically assigned pitch collections to motivate without interruption the musical phenomena of each instrumental part and to constitute the composition as a whole. A piece results as a singular teleology that occasionally comes to a climax or makes a point. Carter explains that his teleological approach is not analogous to the familiar "story-telling" kind. Following
Whitehead, he describes his approach to time-awareness as "believing that existence of
4Daniel Breedon applies Whitehead's philosophy as a methodology through which he explores Carter's compositional process. See Breedon, An Investigation of the Influence of the Metaphysics of Alfred North Whitehead upon the Formal-Dramatic Compositional Procedures of Elliott Carter, DMA dissertation, University of Washington, UMI No. 7617414, (Ann Arbor, Michigan: University Microfilms International, 1975). any kind is a teleological process, in which various kinds of concrescences attain and then later lose integrated patterns of feeling. 'The ultimate creative purpose' is 'that each unification shall achieve some maximum depth of intensity of feeling, subject to the conditions of its concrescence.'"5
Each of Carter's compositions represents a synthesis of organic elements and expresses an interest in dramatic discourse, process, and transformation in some form.
Explorations of polyphony-one of Carter's trademarks-are not haphazard, isolated moments of sound frozen in time. Rather, these moments are combinations of sounds that dually represent former musical events and anticipate future ones. Carter's attempt then is to reflect the encounters one experiences daily, chaos and order. Foreground and background elements regularly lend contextual meaning to whatever component is perceived in focus at a certain time. As a result, numerous divergent yet simultaneous feelings and perceptions synthesize into meaningful units.
Carter's musical vocabulary exhibits qualities of discourse through his rhythmic innovations. Today critics widely recognize Carter for his approach in creating large patterns of rhythmic activity. His approach exhaustively employs tempo modulations, a technique first incorporated in the Sonata for Violoncello and Piano (1948), to facilitate these patterns. Though tempo modulation plays an important role in the evolution of
Carter's musical style, it is not the only means through which he achieves large-scale continuity in his works. His ongoing interest in large-scale continuity results in diverse compositional methods. These include, for example, the structural connection of
5Edwards, 96-97. rhythmic contrast to the domain of pitch, the exploitation of juxtaposing behavioral characters between contrasting instruments, and the replacement of a traditional thematic exposition, development, and recapitulation with organic structure and transformation.
Current Research
Beginning with Martin Boykan's 1964 article, "Elliott Carter and Postwar
Composers," analysts have concerned themselves with Carter's compositional techniques.6 Current methodologies vary widely: from set-class considerations by
Bernard,7 Harvey,8 and Mead,9 to structural polyrhythm and phraseology design analyses by Link10 and Nelson,11 to spatial design and composite attack pattern approaches by
6Martin Boykan, "Elliott Carter and Postwar Composers," Perspectives of New Music II/2 (Spring-Summer 1964): 125-128. Reprinted in: Perspectives on American Composers. Edited by Benjamin Boretz and Edward T. Cone (New York: W.W. Norton, 1971): 213-216.
'Jonathan Bernard, "Problems of Pitch Structure in Elliott Carter's First and Second String Quartets," Journal of Music Theory 37/2 (Fall 1993): 231-266.
8David I.H. Harvey, The Later Music of Elliott Carter: A Study in Music Theory and Analysis (New York: Garland Publishing, Inc. 1989).
9Andrew Mead, "Pitch Structure in Elliott Carter's String Quartet No. 3," Perspectives of New Music 22/1-2 (Fall-Winter 1983/Spring-Summer 1984): 31-60.
10John F. Link, Long-Range Polyrhythms in Elliott Carter's Recent Music, Ph.D. dissertation, City University of New York, UMI No. 9417488 (Ann Arbor, Michigan: University Microfilms International, 1994).
11 Jon Christopher Nelson, Compositional Technique in Elliott Carter's Penthode: A Study in Phraseology and Formal Design, Ph.D. dissertation, Brandeis University, UMI No. 9118711 (Ann Arbor, Michigan: University Microfilms International, 1991). Cogan and Escot12 and Roeder,13 to literary and cultural analyses by Bernard14 and
Schreffler,15 to philosophical and phenomenological discussions by Breedon16 and
Lochhead,17 and to Lewin's generalized interval transformation approach.18 This listing
is only a sampling of the diverse types of analyses that make up the present theoretical
literature that discusses Carter's music. Each methodology explores one or more topics
compacted into Carter's compositional process that signals the expansiveness of his
musical language.
Since the first analytical presentation of a Carter work, it has taken nearly forty
years for analysts to standardize the musical terminology that adequately defines and
12Robert Cogan and Pozzi Escot. Sonic Design: The Nature of Sound and Music. (Englewood Cliffs, N.J.: Prentice-Hall, 1976).
13John Roeder, "A Processive Model of Rhythmic Groups in Atonal Polyphony," paper presented at the Society for Music Theory Annual Meeting, Baton Rouge, Louisiana, 1996.
14Jonathan W. Bernard, "Elliott Carter and the Modern Meaning of Time," The Musical Quarterly (Winter 1995): 644-682.
15Anne Schreffler, "Elliott Carter and his America," Sonus 14/2 (Spring 1994): 38-66.
16Breedon, Investigation.
17Judith Irene Lochhead, The Temporal Structure of Recent Music: A Phenomenological Investigation, Ph.D. dissertation, State University of New York-Stony Brook, UMI No. 8307398 (Ann Arbor, Michigan: University Microfilms International, 1982).
18David Lewin, Generalized Musical Intervals and Transformations, (New Haven: Yale University Press, 1987). explains the processes Carter employs. In the second edition of his book, The Music of
Elliott Carter, David Schiffs inclusion of a "Technical Glossary" represents solidified terms presently used in the theoretical literature.19 Prior to this book, Jonathan Bernard s article, "The Evolution of Elliott Carter's Rhythmic Practice," as well as Schiffs first edition, made positive contributions to term standardization. Future analysts benefit greatly from their efforts and may study Carter's compositional processes from diverse analytic approaches without spending enormous efforts redefining the basic conceptions of his style. At the same time, performers and readers of theoretical literature on Carter's music now have a resource to consult that is representative of the scholarship within the last decade. In the same way that Carter's compositional practices evolved over time, so has its representative vocabulary emerged.
Nearly all the above approaches address rhythmic complexity in Carter's works.
It is nearly impossible for an analyst to discuss components of Carter's compositional process without considering his approach to rhythm. Its value and presence cannot be ignored because the rhythm is intimately connected to each musical element of the piece.
A detailed consideration of Carter's rhythmic practice is vital to formulating an understanding of a particular work since a listener, even during a first hearing, is affected on some level by its profound influence upon the character and unity of a work.
Carter scholars recognize that tempo modulation plays an indispensable role in the overall rhythmic organization of each work even to the point at times that the
19David Schiff, The Music of Elliott Carter, 2nd ed. (Ithaca, N.Y.: Cornell University Press, 1998). See the Technical Glossary provided on pages 34-50. 10
"continuous journey is more important than its temporal destination." This observation by Schiff comes from a composer/analyst who intimately understands Carter's intention and values the importance of placing a rhythmic study of Carter's music in perspective.
Carter's conception of a work does not imply hierarchical levels of motion in terms of pitch or rhythm that are a simple augmentation of local events. His primary conception of a work revolves around elements that make the music cohesive: its speed of presentation, the density of the material, and, the flow of the music. To Carter, "the flow of the music [is] the most important thing; the 'now' of any given point... is only as significant as how it came to be the 'now' and what happens afterward.
At this stage in the current literature, the challenge to the analyst is to present musical observations that, while they may focus on a particular musical aspect, elucidate the meaning of this element only in consideration of its surrounding musical elements.
Bernard notes the difficulty of separating Carter's dense musical layers in order to examine rhythmic elements suppressed within the overall texture of a piece. He explains that
Carter's rhythmic organization can be just as subtle and intricate as pitch organization and is thereby capable of entering into a reciprocal, essentially equal relationship with pitch. The mosaic-like qualities of both pitch and rhythmic organization suggest also that 'structural levels' in Carter's music are quite a different matter from both those of tonal music as construed by Schenkerian theory and those of theories of 'transitional'
20Schiff, The Music of Elliott Carter (London: Eulenburg, 1983), 48.
21Edwards, 37. 11
or post-tonal music based, however generally or partially, on Schenkerian models.22
These suppressed rhythmic elements are often presented in a simple manner possessing an independent existence, when actually their role is complex and dependent on other musical elements of the work.
Criteria for Preferential Strategies
Carter is somewhat unique among contemporary composers in that the reader is fortunate to have access to many of Carter's own writings about his music 23 His level of activity has only increased as his life and works have matured. This wealth of materials is accessible not only due to the technology of the day, but also because Carter established himself very early in his career as one who enjoys discussing his works. He thinks of them not necessarily as works in and of themselves but as creative artworks that embody expressive intent. His view on analysis differentiates analysis as an exercise from analysis associated with discovering aesthetic value. On this subject, Carter feels that
the only way to analyze a piece meaningfully is to begin not with anything that 'exists' on paper, but with the detailed impression that the piece makes on an intelligent listener after many hearings. Any analysis of music has to be the analysis of the means by which a piece makes its expressive point and produces the impression one has of it. Thus any analysis
22Jonathan Bernard, "Review of The Later Music of Elliott Carter: A Study in Music Theory and Analysis," by David I.H. Harvey, in Music Analysis 9/3 (1990): 354.
23Bernard's edition of Elliott Carter: Collected Essays and Lectures, 1937-1995 (1997) is the most recently collected work. There are many other writings and interviews included in the Bibliographies of Doering and Schiff, Music, 2nd ed. 12
presupposes that the piece to be analyzed is worth analyzing, in that it does in fact communicate something esthetically before one studies the ,24 printed score.
With his words in mind, suppose the Second Quartet communicates to the listener in a meaningful way and that it possesses qualities worth understanding more fully. Then the question remains: how does one understand his means to communicate expressively?
In his early years, Carter worked out compositional strategies that possess qualities congruent with the musical language he desired through which to communicate.
He found strategies such as tempo modulation, triple groupings, and large-scale ratios possess individual qualities that guide temporal processes and assemble diverse musical events such as discourse and transformation. In his writings, Carter mentions the topic of tempo modulation often and explains it as "a way of evolving rhythms and
continuities."25 He does not hint at using any sort of triple groupings to facilitate
passages, but he does suggest that polyrhythms play a structural role. Carter prefers to
use these strategies and realizes ways their properties help him achieve the musical
expressive goals he desires. In my opinion, features of these strategies are present aurally
and constitute the musical motion of the piece. In the present study, I highlight two
preferential strategies employed in the Second Quartet that demonstrate his concern with
controlling the overall time continuity through local level organization. The third
strategy examines a method Carter employs to project the large-scale structure.
24Edwards, 118.
25Ibid., 91. 13
Design of the Second Quartet
The technical rigor of the Second Quartet served a dramatic end. Carter simplified the game-rules of his music in order to create a complex, compact totality.26
Carter designed the Second Quartet as a through-composed nine-section work and organized into four large sections that are assigned movement status with tempo indications. The only traditional movement title assigned by Carter was to the last section, Allegro. To the other three large sections, Carter assigned tempo indications of
Allegro Fantastico, Presto Scherzo, and Andante espressivo to what are commonly labeled movements I, II, and ID. As Table 1 shows, cadenzas for three of the four instruments occur between each designated movement. Violin II never plays its own cadenza though this absence of instrumental showcasing is not surprising. The nature of violin II's role as time-keeper throughout the work does not limit it from expressing itself in a free manner; rather, the idea of a cadenza for violin II goes against its inherent character. The relationship terms and instrument assignments assigned by Carter
27 appropriately characterize the nature of each section.
The organization of this quartet is presented in diagram form by Schiff, who characterizes it as a "double series of movements [that] creates a twofold progression towards order and unity, and at the same time towards disorder and non-cooperation. The double sequence also allows much of the material of the Quartet to be heard twice, in
26Schiff, Music, 2nd ed., 73.
27"Section" is used for analysis organization in the present study. 14 changed circumstances."28 (See Figure 1.) This plurality of goals pursued by four singular voices creates a remarkable listening experience, one guided by their individual character-patterns, shown in Table 2 and described by Schiff in his definition of
"character-patterns."29 Because Carter carefully assigns intervallic, rhythmic, and expressive material to each instrumental part, Schiff suggests that these character patterns take the place of thematic structures in Carter's music beginning with this work. His observation is only referential to the present study. However, this analysis is interested in exploring the expressive capability of Carter's preferential strategies in terms of process, not in terms of development. Through large-scale pre-compositional planning, Carter projects the "compact totality" of the Second Quartet. These strategies contribute to important structural points or sectional landmarks within the piece. That he conveniently aligns these arrival locations at the notated rehearsal numbers in the score is a welcome sign to the performer and analyst.
2&Ibid., 78. See Schiff s description with examples on pages 71-78.
29Ibid., 36. 15
Table 1. Formal Structure of the Second Quartet
Section Movement Designations Relationship Lead Instrument
1 Introduction Companionship 2 I. Allegro Fantastico Discipleship Violin I 3 Viola Cadenza Confrontation Viola 4 II. Presto Scherzo Discipleship Violin II 5 Cello Cadenza Confrontation Cello 6 III. Andante espressivo Discipleship Viola 7 Violin I Cadenza Confrontation Violin I 8 IV. Allegro Discipleship Cello 9 Conclusion Companionship
Table 2. Character-Patterns in the Second Quartet
Instrument Intervals Rhythmic Type Expressive character
Violin I m3, P5, M9, M10 free bravura Violin II M3, M6, M7 pulse laconic Viola Aug4, ml, m9 rubato espressivo Cello P4, m6, mlO accelerando-ritardando. impetuous
(427-598) (286-373) Allegro ' Andante^ (171-242) -espressivo Presto- (V»h) (35 - 134) -schexzaMo Allegro' (v»ii*n) (1-34) "fantastico Introduction" Conclusion (Violin.I) ^Vi ^Viola (959-633)
(135-170) „ Cello caitaca •a (2*3-285) Violin I ' eafaaza (373 -438)
Figure 1. Formal Diagram of the Second Quartet 30
30This diagram is adapted from Schiff s chart. See Ibid., 74. CHAPTER 2
TEMPO MODULATIONS
Tempo Modulation as a Term
Tempo modulation provides a way of moving from one tempo to another by means of time signature changes and redivisions of the beat. While some basic pulse remains the same, the grouping of the pulses creates a new metric orientation. Although this technique is commonly referred to as metric modulation, Carter prefers the term
"tempo modulation" because it is ultimately tempo, not the meter that changes. Tempo modulations are easily recognizable in Carter's scores and serve practically as a visual aid
to the performer and govern the alignment of rhythmic streams.
Carter's rhythmic techniques evolved from those he used in the Sonata for
Violoncello and Piano (1948). Each subsequent rhythmic development opened up new
possibilities with which he could facilitate musical continuity. Carter scholars recognize
that tempo modulation plays a vital role in the overall organization of each work even to
the point at times that the "continuous journey is more important than its temporal
destination."2 This idea accurately follows Carter's sentiment in his writings. However,
Carter scholars have yet to acknowledge specific ways tempo modulation is important to
'For an overview on the development of Carter's rhythmic practice, see Jonathan Bernard, "The Evolution of Elliott Carter's Rhythmic Practice," Perspectives of New Music 26/2 (Summer 1988): 164-203.
2Schiff, Music, 2nd ed., 48.
16 17
Carter as an expressive means, to offer an explanation why Carter incorporated it as a stylistic component, and to examine the multiple ways Carter employs tempo modulation as a process. Before examining tempo modulation as a preferential strategy that organizes dual meters, it is necessary to clarify a few definitions that are not included by Schiff in his
"Technical Glossary." Some of the terms are borrowed and adapted from Bernard, most of which first appeared in his "Evolution" article. A "rhythmic stream" (hereafter
"stream") is a series of equally-spaced note values that may or may not project a
metronomic speed (hereafter "projected speed"). A projected speed may be slower than,
faster than, or the same as the notated tempo (metronome marking) in the score. Whereas
Schiff defines the same term as "metronomic speed,"11 prefer the term "projected speed,"
because it more clearly identifies the musical character of the line. Although they are
intimately related to one another, notated tempo and projected speed rarely sound
identical.2 Projected speed usually sounds faster than the notated tempo.
Shown as Example 1, each instrumental part throughout measures 208-214
demonstrates a different projected speed. This is an uncommon passage for this work
because as the four parts interact, there is a steady pulse emphasizing 5/4~usually there is
no steady pulse present. Following the tempo modulation in measure 207, the cello and
Schiff defines metronomic speed as "the speed of any notated duration figured in terms of beats per minute as shown on a metronome." Ibid., 43.
2Bernard defines projected speed as "figurational speed." For example, the speed of a notated figuration is a multiple of some basic pulse (metronome marking). See Bernard, "Evolution," 173. Schiff defines this as "metronomic speed." 18 viola establish MM 175. The viola projects a speed of MM 175, the notated tempo.
Even when the viola displaces the beat (measures 211-212), MM 175 remains the projected speed. The cello projects a speed of MM 350 (175 x 2) while violin I projects a speed of MM 525 with its triplet division of each pulse. The violin I stream begins to break down in measure 213 but does so only briefly before playing one last triplet division to emphasize the completion of its line (measures 214-215). Visually, it appears that violin II does not project a stream of its own like the others. Aurally, it projects a steady 4/4 pulse that is layered between the 5/4 accompanimental voices. Since violin II is the leader of Movement II it is expected that this instrument may present an unfamiliar idea to the other instrumental parts. This passage projects diverse streams that coexist and confront the lead instrument.
The previous example shows streams that occur simultaneously. Streams may also occur successively. Bernard's definition of successive streams is "the arrangement of two or more different rhythmic patterns is followed by a second, the second followed by a third, and so on. [They] may be superseded by one another or may undergo transformations or a series of transformations as in the tempo modulation process that gradually produces the second rhythmic stream from the first."1 Simultaneous streams may occur as separate strands or parts of the musical texture and differentiate themselves in terms of projected speed, timbre, dynamics, rhythmic gesture, articulation, characteristic behavior, and intervallic range. Example 1 demonstrates notated tempo
'H, 166. 19 and projected speeds that occur as simultaneous streams. Example 2 shows a successive stream that is transformed through tempo modulation into a second rhythmic stream.
The transformed successive stream occurs between multiple parts. (See Example
2.) The first stream begins in the cello part on the last beat of measure 54. Its forte attack on beat 3.5 completes the violin I forte attack on beat 3; therefore, the cello line gives the impression of articulating a duple division (projected speed MM 224). Next the cello plays a triplet division in measure 55 before the viola completes the gesture. Both project speeds of MM 336. Then the cello enters in measure 56 with a quadruple division of the beat (projected speed MM 448) that is continued by violin I in measure 57. The quadruple division is further broken into quintuplet divisions and projects a speed of MM
560. Violin I continues this line through two tempo modulations in measures 59-60 while assisting the transformation of a second stream occurring in the cello part.
Through redivisions of the beat, the first stream is transformed on a lower level in that, in its process of transformation, it projects various speeds (MM 224, 336,448, and 560).
This first stream is also transformed on a higher level because no stream truly exists independently when multiple parts are present.
The interest of successive stream transformation occurs when a multi-voice stream shifts the coordinated pulse between parts to a new notated tempo.2 For example,
2Neither Schiff nor Bernard note this type of stream transformation. Their examples of stream transformation consider only successive changes in grouping divisions within a single part. 20
rr-i p p ^Tf A—= m , (j . - "if -a^L • *[,/ ,,f i>^—fir m ^mf-pp nf-ppti ^ if piu f) 3® fV> r b Q , 175) />*«/ Pespr. ™P'PP Example 1. Notated Tempo and Projected Speed as Simultaneous Stream beginning in measure 57.3, the cello plays a line in a rubato manner that helps prepare the tempo modulations in measures 59-60 that establish dotted eighth note = MM 186.7.3 Considering articulated beats from 57.3 through 60.1, the cello plays pitches that receive six divisions, five, five, four, three, and three. By measure 60, the cello articulates stream two, the newly transformed pulse of dotted eighth note = MM 186.7. Throughout this process, violin I plays quintuplet divisions of the former notated tempo, quarter note = MM 112. The simultaneity of these two streams enables a smooth transition into the new 3 Hereafter I will indicate beats within a measure as one numeral. For example, 'measure 57, beat 3" will be written as "measure 57.3." 21 notated tempo in measure 60, a tempo that continues with a displaced pulse for the next fifteen measures. "Stratification" or "stratified streams" is another way Carter scholars describe the simultaneity of streams. Schiff defines stratification as "the division of the musical texture into separate layers with contrasting harmonies, tone-colors, rhythms, and expressive characters."4 The terms are not always used interchangeably. Stratification characterizes Carter's counterpoint more descriptively because it considers the additional musical elements present that create his diverse polyphony. Example 2 certainly demonstrates stratified streams because the instrumental intervallic quality, expressive behavior, and rhythmic characters are unique to each instrumental part. Tempo Modulation as a Strategy Tempo modulation is recognizable in Carter's scores by the change of time signature and notated equivalent note values above the staff where each modulation occurs. An examination of tempo modulation as a preferential strategy aims to demonstrate how Carter controls large-scale structure beginning on a local level. The following two examples from Movement II exhibit how tempo modulation, when employed as a preferential strategy, organizes conflicting dual meters. In this Presto scherzando movement, there is a metrical conflict between the sounding referential 4/4 meter and the additionally-notated 5/4 meter. Violin II leads this movement as time-keeper. The remaining instruments playfully imitate its character 4Schiff, Music., 2nd ed., 46. 22 while retaining their individual personalities. Beginning in measure 171, Carter notates a 5/4 meter written within brackets and he indicates that the tempo is quarter note = MM 175. In sections where the dual tempos exist, Carter inserts an alternate rhythmic notation in 4/4 meter on an inner stave that indicates the resulting sound of the second violin part. The alternate notation indicates how long each note should sound; its notated tempo is dotted quarter note = MM 140. Through the employment of conflicting notated meters, Carter contrasts a sounding referential meter with a notated meter. Dual meters exist but the violin EE's MM 140 tempo prevails as the sounding referential line within the 5/4 meter. The added fifth stave serves as a notational device for the referential meter and coordinates the four instrumental parts. Carter instructs the performers' interpretation with written indications of "with rhythmic precision in all parts," ma ben marcato (violin II), and leggiero sempre (violin I, viola and cello). Example 3 demonstrates how tempo modulation organizes conflicting dual meters in measures 171-186. The notated tempo in measure 171 is quarter note = MM 175. Carter establishes both 4/4 and 5/4 meters in measures 171-176 before the first tempo modulation occurs in measures 176-179 that establishes quarter note = MM 140 as notated tempo in measure 179. In measure 175, violin I continues its triplet division stream projecting a speed of MM 525. Throughout measures 175-177, violin II does not participate in its designated role of time-keeper; it plays only on beat three of measure 175. The 5/4 meter is preferenced by the other three parts during these two measures. 23 —J««n.«n2—. (£= 186.7) 4f -f ff ubjaidi i5E«r • S i l- S i E—-1 fn, ^ ! "•'-r r •cherundo !l"'" JSS = j ; .. P—' pp PP = Example 2. Transformed Successive Streams Prior to the tempo modulation in measures 176-177, the rhythmic interaction between instruments becomes more sophisticated in measure 175 and emphasizes duple and triple polyrhythms. Carter discloses that he subdivides pre-compositional patterns or cycles in several degrees down to the smallest level of rhythmic structures. This subdivision limits the output of possible polyrhythmic combinations to simple ones, so 24 that "no polyrhythm is higher than what is playable and distinguishable for the listener."5 The interacting polyrhythms also project two speeds (violin I projects MM 525; the cello projects MM 350). These two speeds were foreshadowed by single polyrhythmic occurrences beginning in measure 171. The following list shows another way Carter prepares for a tempo modulation to occur. Examples of duple versus triple division that foreshadow the interaction in measures 175-176 are found in • measure 171.3, violin I, violin II and cello • measures 172.4, violin I and viola • measures 173.2,173.5,173.6, violin I and cello • • measure 175.2, violin I and viola • measure 175.3, violin I, violin II and viola • measures 175.4 and 175.5, violin I and cello • measure 176, violin I and cello. In measure 177 the meter changes to 3/4 (quarter note = MM 175) and violin II reenters with two notes (not a stream but a beginning of one) that articulate beats three and four of dual tempo 4/4 re-notated in measure 176. Violin II continues its stream of quarter note = MM 140 through measure 178 and realigns its referential tempo of 3/4 with the new 3/4 meter. The strategy employed in this passage establishes dual meters and tempos (measures 171-174), transforms them to a single meter and tempo (measures 175-176), Edwards, 112. 25 then transforms them through polyrhythms to reestablish violin II in its time-keeper tempo (quarter note = MM 140) as the leader of this movement (measure 178). Violin II emerges as leader with its solo in measures 177-182. Through a second series of tempo modulations (measures 182-186), the notated tempo returns to its previous tempo, quarter note = MM 175. For notational convenience, the meter changes to 2/4 in measure 182. This one measure transition aids the shift from the basic pulse of quarter = MM 140 to half note = MM 105 in 3/2 meter (measure 183). During the following 2/4 measures, violin II maintains the pulse while violin I prepares the tempo modulation to half note = MM 105. Violin I briefly establishes the pulse for the tempo modulation in measure 186. Throughout measures 183-186, violin I projects a speed of MM 525. Its quintuplet division of the half note (measure 183) is equivalent to its eighth note figure in the 3/8 meter (measure 184), in the 6/8 meter (measure 185) and in the 3/4 meter (measure 186) when the dotted eighth note becomes equivalent to a quarter note. A second example where tempo modulation organizes the 4/4 and 5/4 meters occurs in measures 228-236. (See Example 4.) Measure 228 continues a 5/4 meter established in measure 208 where the quarter note = MM 175. A fifth stave was re- notated in measure 211, so the violin II's MM 140 pulse continues. No tempo modulations occur since measure 208. One might characterize measures 228-236 as the climax of this movement due to its dramatic nature. A series of tempo modulations occur between measures 231 and 236 that result in the quarter note being re-established as MM = 140 in measure 234. While violin II maintains its constant MM 140 pulse, beginning in 26 measure 229, violin I and cello emphasize 5/4. Violin I projects a speed of MM 525 and cello, MM 350. In measure 232, violin I continues its stream of MM 525 although the meter shifts to 15/8 and the grouping division per pulse shifts to quintuples from triples. To make a clearer notational transition for the performers, Carter writes a triple pulse measure as 3/2 (measure 233). Violin I continues its quintuple division. At the same time, violin II emerges with a solo gesture that re-establishes 4/4 as the meter for all parts by measure 236. Tempo modulations that organize dual meters as a preferential strategy occur less frequently than Carter's customary type of tempo modulation employment in the Second Quartet. This perspective demonstrates Carter's creative employment of this strategy and his care to guide the listener through complex passages. A seemingly complex method appears simple when one notices that streams commonly function between parts rather than remain confined to their own boundaries. By stretching assigned character-patterns, instrumental parts experience musical freedom. An examination of their subtle interrelationships would like lead to further aspects of musical spontaneity. Whether organizing dual meters or facilitating a swift change in time signature, tempo modulations direct the musical flow throughout the Second Quartet. 27 Presto scherzando JJ U » 175) (titA rAythmie jtrttition iw *il ftrt») [11 SOLO- U"^> nfefwwA *«woj- ' • ^ F v'frtf H •* /P—V K h / V—• rn r * n r T IU. «tlN. •v - r • - i i ML-==— % (fit*) U§fitr* mrmfrt *(---<-) JS105~ 1 if f "fT> /SSSs"''-I* VTaST* ' wm T / fifif (PP) mf.p f f F ffr f»tri i ^vJ a— Al. - f'f . Lr'». A vgwi-i—l • A •• - =f7 > liM r r i. '1 i. - = .I i 1 ^ ^ ^ i % v W V *IP Sijr ^ j Biiri:i.L 15 = * T " V^i 1 a i' 'j"'' no —•—- - - »y-/> /i.rr. — jiiwrr ;J i i Example 3. Tempo Modulation that Organizes Dual Meters, measures 171-186 28 — Jw ^i (J J. , 10S) -J -105 Example 4. Tempo Modulation that Organizes Dual Meters, measures 228-236 CHAPTER 3 TRIPLE GROUPINGS Grouping Divisions Carter employs a preferential strategy that propels the musical motion, directs the listener's hearing, and facilitates all his musical innovations. This device does not exist for its own sake but for the benefit of the composition's overall structure. Throughout the Second Quartet, a logical organization of "grouping divisions" (my term) exists in which notes are arranged according to how many span one pulse. Though it is possible for many notes to occur within a single pulse, Carter's commonly uses grouping divisions range from 2's, two notes per pulse, to 5's, five notes per pulse. He primarily employs grouping divisions of three notes per pulse (3's), four notes per pulse (4's), and five notes per pulse (5's).1 Measures 75-87 of Movement I demonstrate Carter's statement and transformation of grouping divisions. (See Example 5.) I believe that Carter purposely employs and manipulates a larger strategy that I label Triple Grouping Strategy and define more fully later in the chapter. This strategy obscures a regular pulse within a passage that has a relatively strict notated tempo (3/4 and 4/4 time signatures), contains no metric modulation, and involves an average amount of instrumental stratification. Consisting of these characteristics, one might expect this lrrhis grouping criterion excludes single notes occurring per pulse. Single notes are not considered as a grouping division and are referred to in Figure 1 by the number '1" and in the text as "l's." 29 30 passage to project metric regularity. While the pulse constantly shifts, violin II maintains its characteristic time-keeper role. Grouping divisions may not be entirely exact at times. Some grouping divisions round to a higher number; I label these "rounded grouping divisions." For example, see the violin I part in measure 75, beat two, shown as Example 5. A sixteenth note rest and sixteenth note follow two sixteenth notes. This division on beat one is grouped as a 4 because the rest, while contributing to the skipping motion of the violin I part, propels the musical motion of the line by progressing to beat three. Therefore, three literal divisions round to four. .JvEStfJ * uo) Example 5. Rounded Grouping Division; Adjusted Grouping Division, measure 75 The cello part in measure 75 provides an example of an "adjusted grouping division." Also shown in Example 5,1 label beats one (75.1), two (75.2), and four (75.4) each as a 1 in this instance. Beat one is the adjusted pulse because the attacks on the first and fourth divisions fulfill two different functional roles. The downbeat attack aligns with violin I and violin II entries. The attack on the fourth division propels the combined triplet figure that occurs on beats one and two. Within beat two there is a tied sixteenth 31 note on the first division of the beat followed by a sixteenth-note rest. An articulated eighth note occurs on the second half of the beat. Although there are two pitches notated on beat two, only one is articulated in sound. Therefore, the grouping division of beat two remains a 1. An alternate way of determining the cello groupings on beats one and two in measure 75 could be as a 4 followed by a 2. This reading is more literal than the one given above in that each division of the pulse is accounted for; however, this reading does not account for how the piece sounds. Because the tempo indication is Allegro fantastico (quarter note = MM 140) with leggiero added in measure 75, one might suspect that a larger gesture or grouping exists between individual notes. This appears to be the case as the first three notes in the cello part in measure 75 form a relationship that requires a grouping division adjustment. This passage demonstrates how a numeric grouping division, in this case 3's, serves to facilitate interaction and fluidity between the instrumental parts. Figure 2 provides a diagram of grouping divisions in measures 75-87. Single digits written in each instrumental part indicate grouping divisions. Brackets indicate functional roles of grouping divisions. Straight line brackets represent an important secondary line indicated in the score by Carter. Long dash brackets represent primary melodic material. Widely spaced dotted line brackets indicate gestures, and closely spaced dotted line brackets indicate accompanimental figures. The number of regularly- and irregularly-notated grouping division occurrences for l's-5's is shown in Table 3. Besides l's, which serve as the primary pulse in this passage, 3's occur more frequently 32 than 2's, 4's, and 5's. The 3's or "triple groupings" shown in this passage occur primarily in the viola and cello parts. They are notated either as eighth- note triples that span one beat or as quarter-note triples that span two beats. A closer look at the number of 3's employed in this passage reveals that not all 3's are treated equally. In fact, Carter employs and manipulates four types of triple grouping divisions in straightforward and hidden ways. These grouping divisions are present on both local and higher structural levels and ultimately promote the overall motion of individual movements and the composition. When grouping divisions exist on a local level then interact to form a significant relationship on a higher structural level, I label this activity as a "grouping strategy." Table 3. Grouping Divisions, measures 75-87 Grouping Division Number of Occurrences (Is) (Primary pulse) 2s 9 3s 18 4s 5 5s 3 33 mi 1 Yin. 2 Vln. ! ! L t i i Via. , 1 ' 3 3 2 ti\ 3 i L Vc '» I t ' l 2 2 ; -J -J • I """ _ (3) Ui ' ! ' u 78J 79 |§_ 1 Vln. 2 •' — 2 Vln.- I I 1 I J I e ra | 4 i i i i i j i__ Vc. 7 t F3 F3 pf f F3 S3 81 h 1 Vln. J 1 t I B 2 Vln. -= ! = ! = i = !—= ! ! ' * ? 1 Via. S3 S3 S3 F3 F3 Vc. i _ i » J J 5 2 Vln.— 7 i 7 f Via. Vc. I ! - I 3 4 Figure 2. Grouping Division Diagram, measures 75-87 34 Triple Grouping Strategy I categorize four types of triple groupings collectively as a Triple Grouping Strategy (TGS) because each type involves some arrangement of triple-note figures. The types include: eighth-note triples, quarter-note triples, and two kinds of irregularly- notated triples. I classify them in the following ways—their order reflects how commonly they appear: • eighth-note triples occurring over the span of one pulse are labeled as Fast Three's (F3) • quarter-note triples occurring over the span of two pulses are labeled as Slow Three's (S3) • irregularly-notated triples with equal grouping divisions that span multiples pulses are labeled as Hidden Three's (H3) • irregularly-notated triple with unequal grouping divisions that span multiple pulses and provide an aural illusion of a triple grouping are labeled as Loose Three's (L3). It is necessary to differentiate between these types of 3's because each arrangement functions differently than the others. Carter distinguishes types of 3's then transforms one type of triple grouping into another to preserve the musical motion. Fast and Slow 3's contribute to the notated secondary line in measures 77-83, are identifiable to the ear, and are easy to locate in the score. Fast 3's occur primarily in the cello part (measures 76, 79, 80.2) and in the viola part (measures 78 and 82). Slow 3's occur most clearly in the viola part beginning in measure 80. One feature of the Fast and 35 Slow 3's is their tendency to appear adjacent to another triple grouping. This pairing of 3's demonstrates Carter's concern to achieve smooth rhythmic and melodic connections without any sudden breaks in the overall texture. Hidden 3's are aurally recognizable but more difficult to locate in the score. Their aural influence results as triple groupings exchange between instrumental parts, between viola and cello most often. Acknowledging the possibility that each type of triple grouping may transform into another type, instrumental exchange aids the transformation process into regularly- or irregularly-notated triple groupings. The transformations emphasize both the primary and secondary lines, reinforcing Carter's interest in polyphonic discourse. Carter rarely indicates a Hauptstimme, (I have done so with straight line brackets in Figure 2), although his notation often suggests that a figure receive preference within the musical texture.1 The closest Carter comes to acknowledging a primary line is the wichtige Nebenstimme shown in Figure 2 by a long dash dotted line. Where one expects a single projected line, Carter indicates rhythmic gestures or pitch groupings to direct the listener's attention through a passage. His care to include an instruction to the listener through the performer supports my suggestion that Carter purposely guides the listener directly through aural density. While violin II maintains the steadiest notated pulse throughout measures 75-83, grouping divisions 1 As an example, see the cello line in measures 73-74. Here the cello articulates four equally-spaced eighth notes. Although there are four equally-spaced eighth notes present in this series, I include only the last three eighth notes in my grouping based on the criteria that the figures ascend and repeat sequentially in the cello line (measures 75 and 76). 36 facilitate the combined instrumental relations and guide the listener's perception of the material. Triple Groupings as Transitions Carter employs triple groupings as transitions and in transformations to guide the listener through a passage that is absent of any regular pulse. Example 6 and Figure 3 demonstrate how 3's facilitate a notated tempo modulation between measures 72-75. Before examining how triple groupings advance this tempo modulation, I will describe some basic features of this passage in detail (See Example 6). Beginning in measure 72, the melodic gesture of violin I propels the motion for all parts into measure 75. Throughout measures 73 and 74, the dotted eighth note represents the unit of beat (MM 186.7). Sixteenth-note triple groupings in violin I represent Fast 3's, a figure that articulates the division of the beat and projects a speed of MM 560.1 (MM 186.7 x 3). Violin I plays a dramatic ascending line that climaxes on the downbeat of measure 75. Throughout this gesture the sixteenth-note stream continues to project a speed of MM 560. This constant projected speed aids the time signature shifts from 12/16 (measure 73) to 15/16 (measure 74) then facilitates the tempo modulation to 4/4 (measure 75). Beneath the MM 560 projected stream in violin I lie three instrumental parts with contrasting rhythmic patterns that work to displace the metric orientation (See Figure 3). In measure 74, the cello begins an ascending Hidden 3 that is followed by a second Hidden 3 in the viola (measure 75.4-5) then is completed by a third Hidden 3 in the cello line (measure 75.1-2). These three instrumental parts complement each other and highlight the violin I climax by articulating their greatest intervallic distance between the 37 «'s:eI£MZB!_ "•»/ <•*) P .J*®J * MO) ppltggi Example 6. Triple Groupings as a Transition, measures 72-75 73 74 75 / i i i i / i i i i i / i I I F3 F3 F3 F3 F3 F3 F3 F3 F3 1 Vln. 2 Vln. '222 H3 Via. 2 2 2 3 3 3 H3 H3 Vic. Figure 3. Triple Groupings as a Transition, measures 73-75 38 viola's "A-flat2" and the cello's "C" on the downbeat of measure 75.2 As shown above, the projected speed of violin I operates through Fast 3's and the cello/viola line functions through Slow 3's. These two types of triple groupings interact to facilitate the tempo modulation in measures 74-75 and more importantly, to emphasize the structural importance of the downbeat in measure 75. This arrival provides a climax between the violin I and viola/cello lines, as well as a structural alignment location for all instrumental parts.3 The basic pulse shifts in measures 74-75 from dotted eighth note = MM 186.7 to quarter note = MM 140; this shift is led by violin I. Once the tempo modulation occurs, the character of the individual instrumental parts changes immediately. The cello articulates its second Hidden 3 while violin I completes its melodic line by shifting from three to four divisions per beat. Triple Groupings in Transformations Example 7 and Figure 4 demonstrate how triple groupings transform from one type to another and prepare time signature changes in measures 75-78. Throughout these measures the cello and viola play a gesture that begins with the cello in measure 76.2 and ends in the viola in measure 78.2. (See Example 7). Violin II maintains its role as time- keeper and articulates a steady pulse throughout the shift from 4/4 to 3/4 to 4/4 (measures 76-78). The interest of this passage lies in Carter's usage of triple groupings through 2An alternate reading of this sequence would label the triple sequence as a series of three Slow 3's. While each part plays a deliberate triple that spans two beats, these Slow 3's are not notated as quarter notes. 39 transformations. Table 4 lists all transformations throughout measures 76-84. Referring to Example 7, Figure 4, and Table 4,1 will highlight three examples that demonstrate how triple groupings in the viola and cello parts conceal and reveal the time signature changes. Table 4. Triple Grouping Transformations, measures 76-84 Instrument Measure TGS TGS Measure Instrument Cello 76.2-3 Fast 3 Loose 3 77.1-3 Cello Viola 76.3-77.1 Loose 3 -» Loose 3 77.2-3 Viola Viola 77.2-3 Loose 3 Fast 3 78.1-2 Viola Viola 78.4-79.2 Loose 3 Fast 3 79.2-4 Cello Cello 80.2 Fast 3 -» Slow 3 80.3-4 Cello Cello 80.3-4 Slow 3 Slow 3 81 Cello Cello 81.1-4 Slow 3 Loose 3 82.1-4 Cello Viola 82.1-3 Slow 3 Fast 3 82.4 Viola Violin I 83.1-3 Slow 3 Fast 3 83.4 Violin I Violin I 84.1-2 Slow 3 Loose 3 84.2-85.2 Violin I .J\|§!(J • 140) Example 7. Triple Groupings in Transformations, measures 75-78 3Although the viola part does not articulate 75.1, it assists in preparing the downbeat with its three-note gesture on the previous two beats. 40 i 5 * 76 77 78 7i i i i / i i i i / i ' 1 / i i i i 1 Vln. 2 Vln. - , , ^ 3 ' (. T )' J 3 2 ' 13 L9 > F3 F3 Via. 1 1 •3 3 3' « 'X' . < 3 3" H3 F3 F3 LS Vic. Figure 4. Triple Groupings in Transformations, measures 75-78 While violin II reveals the pulse in these measures, viola and cello groupings conceal the pulse. In measure 76.2-3 the cello plays two Fast 3's that are transformed in measure 77 into a Loose 3 that spans the entire measure.4 The Loose 3 in measure 77 is not exact in that the values are not all equal though they are very close. The "d#" and "G" last for four divisions while the "b-flat" receives five divisions. There are two reasons, however, to support the equality of these three notes: first, Carter notates each pitch as a dotted eighth note; this seems to indicate an intention of equality. Second, one could suggest that the "b-flat" borrows two divisions from the succeeding measure to complete its five division total. The cello's "d" in measure 78 is not part of any triple grouping and completes the cello gesture. 4In Figure 3, the arrow represents transformations between triple groupings. 41 The second and third triple grouping transformations occur in the viola part beginning in measure 77.1. The Loose 3 in the viola has four notes and consists of "b- flat1" (3 divisions), "c1" (3 divisions), "c#1" (2 divisions, and "g" (3 divisions). I label this a Loose 3 because even though there are four notes, a hidden three exists within it (depending on which notes are chosen to represent it). Regardless, the figure is nearly equally spaced and leads into the Loose 3 in measure 77.2-3. This Loose 3 is not exact either because the three pitches do not receive an equal number of divisions: the "f' and "f#" receive three while the "e" receives only 2. This Loose 3 is similar to the one described above in the cello part, measure 77.3. Here however, I maintain that this grouping remains a Loose 3 because the difference of missing divisions between the notes is minute and because the figure is transformed into Fast 3's in measure 78.1-2 to complete its gesture. The nature of the viola's rubato character-pattern is to play slight accelerando and ritardando within a line. This feature also supports that this figure could be read as a Hidden 3. Table 4 shows all the 3's transformations that occur throughout measures 76-84. It is striking that Carter carefully selects occasions when 3's will transform within the same instrumental part or when they transform between two parts. These choices suggest that he carefully employs 3's groupings for the expressive purpose of the line. As shown in the above examples from the Second Quartet, Carter employs grouping divisions in order to guide the listener through passages of polyphonic complexity or through successive time signature changes. He preferences grouping divisions of 3's and employs them in more unique ways than he does 2's, 4's, or 5's. 42 Using 3's in transition areas-usually in a series of tempo modulations, Carter employs various kinds of 3's groupings: fast, slow, loose, and hidden. Having several kinds of 3 s groupings to interchange propel the musical flow while coordinating a steady stream throughout a passage that initially sounds absent of a referential line. Transformed 3 s often occur in simultaneously yet still maintain musical continuity. Rather than rely on a single line to guide the listener, Carter achieves a new means through which to direct one through complex passages. CHAPTER4 LARGE-SCALE RATIOS Local Level Ratios Local level ratio relations exist in Carter's music in linear, successive ratios and in vertical, stratified ratios. Linear ratios manipulate successive grouping divisions that are related locally. Vertical ratios occur when each part plays a different polyrhythm and the lines are stratified. As noted above and shown below in Example 5, polyrhythms are a familiar component of Carter's compositional style. They are familiar in terms that while grouping divisions occur during the pulse, there is often some sort of occasional alignment between the parts that signifies this type of coordination. One may not be able to determine precise polyrhythmic relationships but their general factors are recognizable. Example 8 shows a passage with more complex linear ratio relations than presented in Chapter 3. In measure 589 Carter combines polyrhythms of 8:7:6:5. These polyrhythms exist on a local level of organization and, at the simplest level, constitute breaking the pulse into different grouping divisions (8's, 7's, 6's, 5's). UMl Tempo giusto * J} * 168— nJ - y t»-1*-•-^ f ffmntv. ™ f Example 8. Polyrhythms as Vertical Ratio Relations 43 44 Linear ratio relationships also exist on a higher level. Schiff has compiled a list of tempo relations in the Second Quartet that summarize ratios employed by Carter.1 (See Figure 5.) This Chart also represents Carter's idea of a "time-screen," a field of related notated speeds.2 MttA? j iZtnf mil/Ji jack? = 105 2:3 J. * 70 J J = 71 J = 14B 1:2 = 112 5:8 J = 70 J 5:8 = 112 J «7§ O Ji = m.i 3:8 3:7 J.. = 70 JT = 163.3 I = 7® J = 93.3 3:4 L — 2:5 J J> = 70 J = 175 5:6 | = 7® J = 84 7:6 SaST = 70 J. = 60 Figure 5. Tempo Relations in the Second Quartet Schiff, Music., 2nd ed., 48. "Carter discusses this idea in, "Music and the Time Screen," in Elliott Carter: Collected Essays and Lectures, 1937-1995 ed. Jonathan Bernard (Rochester: University of Rochester Press, 1997): 262-281. 45 These linear ratios present relations that coordinate all notated tempos in the quartet. The chart lacks detail because it does not cite all instances of recurring notated tempos through the work. Perhaps that study would reveal another level of linear ratio relations. These types of vertical and linear relations assist our understanding of another component of Carter's style: structural polyrhythms that serve as the background rhythmic structure of a piece. Both Schiff and Link discuss how Carter pre-plans large- scale structure through polyrhythmic alignment and misalignment~the latter is preferable to Carter. Knowing Carter made such a plan incorporating structural polyrhythms during the formative stages of composition suggests that he may have made a similar plan involving large-scale ratios. Schiff notes that Carter did not employ the technique of structural polyrhythms until 1961 in his Double Concerto for Harpsichord and Piano with Two Chamber Orchestras. In works following the Second Quartet, Schiff remarks that Carter desired to develop a systematic way of approaching all tempo relations in a piece and to develop the structural equivalent of the slow phrase rhythms found in tonal music. He adds that "the calculation of a structural polyrhythm on graph paper is usually the first step in composition, a clear indication, if one were needed, that for Carter rhythm always comes first, albeit in a highly abstract form."3 Because rhythm is one component of large-scale organization and there is supportive evidence that Carter related ratios between tempos in the Second Quartet, it seems likely that Carter planned large-scale ratios between sections as well. 3Schiff, Music., 46. 46 Large-Scale Ratios The strategy of large-scale ratios exists among the four movements that contain nine through-composed sections, when examined either through measure or pulse analyses. With the exception of the Introduction (Section 1) and Conclusion (Section 9), which serve as the outer framework within which the piece is composed, the large-scale plan reveals that the entire work is organized around a series of interlocking 2.1 ratios. The seven internal sections reflect this plan in terms of the number of measures assigned to each section and of the number of pulses projected within each section.4 As shown in Tables 5 and 6,2:1 ratio relations exist between sections in an interlocking order rather than in a successive one, although two pairs of relations occur between adjacent sections (Section 2: Section 1 and Section 6: Section 5). Non-adjacent related sections occur between Sections 2 and 7 and Sections 8 and 6. Section 2 is twice the size of Section 7. While Section 2 contains 99 measures and 355 pulses, Section 7 contains 52 measures and 174 pulses. Section 8 exists in an almost 2:1 relation with Section 6: Section 8's 171 measures and 455 pulses are twice the range of Section 6's 87 measures and 267 pulses. When considered through pulse analysis, it is not surprising that the ratio relations for Section 8 do not align as clearly because the major feature employed in Section 8 is a large-scale accelerando. Due to this fluctuation in number of 4tr"Numbe* r of pulses" is analogous to the term "unit of beat" as a fundamental musical term. For example, in a piece written in Common Time with a moving eighth- note figure, the quarter note functions as the "unit of beat," whereas the eighth-note functions as the "division." 47 discernable pulses, the pulse analysis is greatly affected. Still the two sections exist in almost a 2:1 relation; their relationship equals 1.77, which rounds to 2. Table 5. Measure Analysis of Ratio Relations Ratio Section Measure No. Measure Length Measure Length per Section 35 3 2 35 99 2.83 1 3 135 35 1 U- 2 4 171 71 2.03 J I 1.25 5 243 42 1.2 U- 2.5 6 286 87 2.49 I 1.5 7 374 52 1.49 1 8 427 171 4.89 5 Table 6. Pulse Analysis of Ratio Relations Section Number of Pulses Number of Pulses -f-120 Ratio 2 355 2.99 3 3 151 1.27 JL 1.25 4 295 2.49 L 2.5 5 172 1.45 "] 1.5 6 267 2.25 2.25 7 174 1.47 V, 1.5 8 455 3.84 4 For the measure analysis, I divided the measure length per section by 35. The number "35" is significant and serves as a common divisor for each section due to its meaningful role in the Introduction and Conclusion sections.5 Another signifier that "35" might somehow be important to the structure of the entire piece became evident through 5There are thirty-five measures in Sections 1 and 9. Other similarities between the two sections that frame the work include similar pitch relationships, number and types of phrase groupings, and formal layout. 48 measure analysis. Almost every measure number beginning each section when divided by three results in a whole number. Dividing each section's measure length by 35 creates the large-scale ratios noted in Table 5. Similarly, in the pulse analysis, I averaged the number of pulses in Sections 1 (133) and Section 9 (104). Using 118.5 as a divisor for each section's number of pulses, I achieved nearly the exact same ratio relations as found in the measure analysis. Table 7 shows the significant relationships between the sections. Compared with Tables 5-6, it is striking that the relationship between Section 2 and 7 is so clearly presented. The ratios in those sections require the least adjustment and maintain consistency throughout each type of analysis. This signifies Carter's plan for some kind of large-scale control between sections. This formal structure is represented in the linear and vertical relationships established on the local level. Further study could reveal that relationships of 3:1.5,2:1, 2.5:1.25, and 5:2.5 exist in the selection of notational values employed in grouping divisions, in ratios between tempo modulations, and also on the large-scale formal structure. 49 Table 7. Large-Scale Ratio Relations Section Ratio Significant Relationships Movement Lead Instrument 2 3 3 I Viola I 3 1 1 Viola 4 2 2 II Violin II \r 5 1.25 1.25 Cello 6 2.5 2.5 in Viola 7 1.5 1.5 ^ y Violin I J 8 5 5 IV Cello CHAPTER 5 CONCLUSION Preferential strategies of tempo modulations, triple groupings, and large-scale ratios enable Carter to impress and manipulate future and past effects associated with each musical event. Even if Carter does not recognize these strategies, their existence is supported in the music and through the complex relationships it possesses. These strategies demonstrate that simple relationships govern his Second Quartet. With tempo modulations (shown in Chapter 2), single relationships facilitate rhythmic streams on a local level and coordinate rhythmic streams on an advanced level. Triple groupings discussed in Chapter 3 are effective both through transitional and transformational relationships. Duple relationships are also essential to the large-scale 2:1 ratio relations that govern its overall structure. The present study suggests that such strategies exist as a result of Carter's concern for the process and transformation of independent musical events. Further exploration of these strategies could determine if they exist on both local and advanced levels and in multiple works composed over several decades. We know that Carter worked out these strategies over the early part of his composing career and that they were perhaps first fully employed in the Second Quartet. Because Carter was comfortable working with these strategies, he possibly internalized their processes and effects. These strategies then achieved a presence of their own that did not demand his awareness and have recently been discovered by analysts. Meanwhile, Carter's attention turned elsewhere; primarily, 50 51 to devising musical devices that would achieve the expressive focus of each subsequent composition. Mote important than the effect of these compositional strategies in structuring the work, of course, is Carter's expressive intent in this quartet. These strategies do not present themselves on the surface level of listening but affect the listener's curiosity in order that the questioning process begins. 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