Brendan P. McConville - Society of Composers P. McConville ... taken from John Link’s 1999...

REFINEMENT THROUGH REVISION OF COMPOSITIONAL METHODS: THE RECENT MUSIC OF ELLIOTT CARTER AS A PARADIGM FOR

COMPOSERS IN THE 21ST CENTURY

Brendan P. McConville

There was at a certain point, certainly within the last eight or ten years, that I decided to use only a certain very small vocabulary of chords. And I used the chords that I had discovered in the course of searching through the whole system-the two 4-note chords that have all the intervals and the 6-note chord that contains all the 3-note chords. And then I went back to previous work which I used in my Double Concerto combining 4-note chords, either no. 18 or no. 23, and so I made many 8 note chords out of those. It’s something I was aware of when I first wrote the Double Concerto. I simply reverted to what I had been doing long ago. I discovered no. 18 entirely by chance when I was writing my First String Quartet. And then when I came to write the Double Concerto I discovered that there was also no. 23, but I never made a list until much later.1

This quote, taken from John Link’s 1999 interview with Elliott Carter, which constitutes part of the prefatory material to the composer’s Harmony Book, speaks volumes about the maturation process of Carter’s highly personalized harmonic language. Moreover, the recent works Carter first mentions represent a methodology of administering pitch materials that has undergone a forty-year gestation period towards ultimate codification. In this article we will consider an example of Carter’s recent compositional activities in connection with his life’s work, not only to better understand his intriguing approaches to “harmony”, but as composers, to acutely observe an exemplary career which included self-examination, self-discovery, and perhaps most significantly, self-discipline. His recent works reveal a pitch language which distills the composer’s immense vocabulary of materials, as evident in his Harmony Book, to the crucial harmonic elements which may represent the composer’s total compositional output. By studying the harmonic evolution in Carter’s works we may affirm to ourselves that our constant efforts to define, refine, and efficiently re-approach our compositional principles and methodologies do provide successful results. Furthermore, Carter’s recent music also demonstrates that if we carefully organize our procedures while maintaining loyalty to our innate compositional instincts, perhaps ultimately the music that may best represent ourselves is yet to come.

Before satisfactorily arriving at these observations, we must accurately deconstruct Carter’s career approaches to pitch organization.2 We will start by examining the layout of his Harmony Book, which categorizes the growth of his “chords,” or pitch sets (Carter preferred the term chord over prime form pitch set, as used in Forte analysis), from about 1965 into the 1980s.3 Next, we will see three brief examples which may demonstrate Carter’s self-examination of harmonic materials resulting in chord expansion: String Quartet No. 1 (1951), Double Concerto (1961), and A Symphony of Three Orchestras (1976). Finally, we will consider how Carter’s Lauds (completed in 2001), for solo violin, may serve not only as a refined continuation of techniques associated with

iSCI: The Composer’s Perspective 1/1

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previously favored harmonic materials, but as a representative of his harmonic categorization project, the Harmony Book.

Perhaps no better example of Carter’s commitment to his craft exists than his recently published Harmony Book (2002). The book may serve as a collection of clues in defining a consistent and concise approach to elucidating Carter’s complex harmonic language. Jonathan Bernard has noted that the inordinate amount of analytical attention that has been paid to Carter’s pitch structure may be attributed to those seeking to interpret the composer’s comments with his music.4

Congruent with Carter’s explanation above, many analyses of the works prior to his “middle period” (John Link mentions this period begins with his Piano Concerto in 1965 and extends into the late 1970s) uncover a harmonic language built on a foundation of 3 and 4-note chords. Thereafter, Link explains that the increasing sizes and complexity of the chords Carter was investigating aligns roughly with the chronology of his works, as he used 3-note chords for the Piano Concerto, 4-note and 5-note chords for the Concerto for Orchestra (1969), 2-note to 5-note chords for the Duo for Violin and Piano (1974) and finally 3-note and 6-note chords for A Symphony of Three Orchestras (1976).5 Carter’s investigation of chords ultimately led to the Harmony Book, which not only catalogs all of his chords, but designates two large sections, “Synthesis” and “Analysis”, as varying approaches to using these chords. “Synthesis” divides 3-note through 7-note chords into potential additive possibilities, allowing the user to combine the various sub-sets into larger chords. “Analysis” presents the abundant divisions of 3-note to 8-note chords in possible sub-sets. Therefore, by the time he composed Night Fantasies (completed in 1980; now using the all-interval 12-note chords), Carter’s extraordinarily extensive chord and chord-partitioning vocabulary began to reach its technical climax.6

One could perhaps argue that the subsequent reduction in Carter’s applied harmonic vocabulary may be considered a reactionary measure to this technical climax. Yet, we must not forget that embedded in his reduced harmonic parameters remained the experiences the composer had cultivated during his period of expansion. Carter’s examination of his own works, combined with the arsenal of his Harmony Book, has more recently allowed him to discover a language which may most accurately represent his compositional style.

As mentioned in the opening Carter quotation, the composer began consistent use of the (0146) all-interval tetrachord, or his 4-note chord 18, as early as his String Quartet No. 1.7 The cello solo that initiates the work may serve as an early example of Carter’s music relying heavily on the 4-note chord 18. (Fig. 1 – 0146 in green8) Yet as both Bernard and David Schiff have articulated, it is impossible to discount the importance other 4-note chords in String Quartet No. 1, including the only other all-interval 4-note chord, number 23. But how could this 4-note chord have been intentionally integrated into the work, as Carter has mentioned he did not discover this second all-interval 4-note chord until he composed the Double Concerto?

Figure 1 highlights three other 4-note chords: number 23 (0137) in red, number 13 (0347) in blue and number 9 (0134) in yellow. Schiff argues that these other tetrachords are abstractions from chord 18.9 Bernard on the other hand, believes the composer was still in the process of exploring 4-note chords.10

Here, we may consider a slightly different approach to understanding the frequently varied tetrachords in String Quartet No. 1. Both Schiff and Bernard have interesting positions: we will consider both, plus move one step further. Schiff’s theory (abstracting


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other 4-note chords, especially 23) and Bernard’s theory (exploring intervals of various 4-note chords) may combine to present a third possible theory: that Carter was administering a harmonic process that he didn’t fully realize until many years later, as he used a primary chord (4-note chord 18), used an abstraction (4-note chord 23), and then combined them to discover that a symmetrical tetrachord always remained.

Carter explained this process in the 1999 interview with Link, and Figure 2 reproduces Link’s chart from the Harmony Book, demonstrating this phenomenon.11 Although this technique was not mentally codified by the composer until he had undergone the process of compiling the Harmony Book, uses of other favored tetrachords in String Quartet No. 1 may serve as primitive examples of his combinatorial exploration. After all, as Link has suggested, it was for this reason that Carter decided to assemble the Harmony Book.12

In Carter’s search for combinatoriality, it is certainly plausible that he may have employed two of his exploratory 4-note chords (18 + 18, 18 + 23 or 23 + 23) and subsequently “filled out” the chromatic aggregate with one of the resultant tetrachords. As shown in Figure 1, the resultant, symmetrical 4-note chords from Figure 2 appear frequently in String Quartet No. 1. But at the time it was simply part of an investigation, as he did not consciously execute this harmonic process until composing the music of his “reduced vocabulary.” Essentially, this timeline of events in Carter’s exploration for a pitch language may be quite analogous to many of our own experiences as composers: from a thorough examination of our intuitively composed music we may ascertain useful information about our approach to composition.


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FIGURE 1: STRING QUARTET NO. 1, OPENING BARS STRING QUARTET NO. 1

Music by Elliott Carter Copyright © 1962 (Renewed) by Associated Music Publishers, Inc. (BMI)

International Copyright Secured. All Rights Reserved. Used by permission.


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FIGURE 2: HARMONY BOOK P. 32, “RESULTANT” 4-NOTE CHORDS Used by permission from Carl Fischer, LLC.

As he mentions in the quote opening this essay, by the time he wrote Double Concerto he was now fully aware of the relationships projected from, and between, both all-interval 4-note chords 18 and 23. Here, the two ensembles clearly articulate individual affinities with each of the 4-note chords 18 and 23: the harpsichord ensemble (orchestra 1) with chord 23 and the piano ensemble (orchestra 2) with chord 18.

With the designation of his two favored tetrachords firmly established in this work, it is now critical to see how he uses them in the music. First, Carter frequently alters their pitch content while maintaining their “movable” pitch class anatomies. He achieves this by common tone (invariant pitch) re-voicings of the 4-note chords. The circled groups of pitches in Figures 3, and again in reduction in Figure 4, exhibit this practice.13 In orchestra 1, the G and Ab that represent the movable “0” and “1” pitch classes, respectively, of tetrachord 23 in bar 13, turn into movable pitch classes “1” and “0,” respectively, of a new iteration of tetrachord 23 in bar 14 (chord 23 in bar 14 = T3I of chord 23 in bar 13). Similarly, the Bb and C played in the violin and cello of orchestra 2, which represent pitch classes “4” and “6,” respectively, of tetrachord 18 in bar 12, turn into classes “6” and “4” in bar 13 (chord 18 in bar 13 = T10I of chord 18 in bar 12). Common tone retention, while altering other pitch classes of a tetrachord, was an intuitive approach to pitch organization for Cater which proliferated through the examination of his own processes. We will see how this technique reappears in his late works.


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FIGURE 3: DOUBLE CONCERTO, MM. 10–14 DOUBLE CONCERTO FOR HARPSICHORD AND PIANO




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FIGURE 3 (CONTINUTED)


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FIGURE 4: MM. 11–14, REDUCTION; DOUBLE CONCERTO (DOUBLINGS EXCLUDED)


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The second manner in which Carter employs his all-interval tetrachords in the Double Concerto also recalls the quote which opened this essay. This method involves combining two of the 4-note chords to produce 8-note chords. He mentions only that he made 8-note chords out of them, so the possibility of any of the three combinations of these two 4-note chords (18+18, 18+23, 23+23) remains open. A clear example of a harmonic sonority employing this method may be found in bar 111 during the harpsichord cadenza. Reproduced in Figure 5, we see that on the downbeat of this bar the sonority is an amalgamation of two 4-note chord 18s or two 4-note 23s.14 Measure 111 of the Double Concerto provides an example of early employment of an 8-note chord, albeit without then complementing this with the “resultant” 4-note chord 2, or set (0167), as shown in Link’s musical chart (Fig. 2). So far, we have seen an example from String Quartet No. 1 exploring these resultant chords, and now an example from the Double Concerto leaving these chords unrealized. But as we will see, these resultant chords will play a much stronger role in Carter’s late works.

FIGURE 5: DOUBLE CONCERTO, MM. 109–111 DOUBLE CONCERTO FOR HARPSICHORD AND PIANO



Our final work to examine before turning to Lauds will be the Symphony for Three Orchestras, where, as mentioned, Carter now expands his harmonic foundation to 6-note chords and their 3-note subsets. By the late 1970s, his 6-note chord 35, or set (012478), became as fundamental to his music as the all-interval tetrachords were in the String Quartet No. 1 and Double Concerto.15 Like the two 4-note chords’ ability to project any interval onto the musical surface, 6-note chord 35 may harmonically or melodically present any of the twelve trichords.16 Furthermore, the hexachord retains the two all-interval tetrachords as subsets, so his intervallic and trichordal palettes remain extremely flexible. Nancy Rao has demonstrated the manner in which this 6-note chord presents itself in the opening bars of A Symphony. She explains how the introductory hexachords in the high strings, like the tetrachords seen in our previous discussion of Double Concerto, morph in pitch content yet retain their basic identities of (012478). Her chart, as reproduced here in Figure 6, demonstrates that if we consider all of the hexachords produced over the course of the opening 5 bars, we find that Carter retains invariant symmetrical tetrachords (those pitches not in boxes) between groups of these morphing 6-note chord 35s.17 Therefore, the harmonic background retains a consistent musical


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syntax while constantly interpolating new expressions of the hexachord. From a purely hexachordal perspective, this example reveals how Carter has now evolved his common-tone retention of chords to involve invariant tetrachords instead of merely dyads, as we saw in the Double Concerto.

FIGURE 6: HEXACHORDS MM. 1–5; A SYMPHONY OF THREE ORCHESTRAS Musical example reproduced from Rao, Nancy Yunhwa, “A Comparison of Trichordal Relations in Milton Babbitt’s String Quartet No. 2 and Elliott Carter’s A Symphony of Three Orchestras,” Ewha Music Journal

Vol. 6 (2002): 164-165.

FOUR LAUDS, FOR SOLO VIOLIN

We shall now see that even though Carter has reduced his primary chords in Lauds to three (4-note chords 18 and 23 and 6-note chord 35), his use of these chords has become affected by his “synthesis” and “analysis” explorations of the Harmony Book. His stylistic employment of these chords has not radically changed, but has been modified by virtue of his dedication to meticulous systemization. We have chosen two movements for specific analysis, “Statement – Remembering Aaron (1999),” and “Rhapsodic Musings (2000),” as they best present passages maintaining a connective tissue with the past. This score study will be broken into two primary sections. First, we consider familiar uses of the three Carter “reduced vocabulary” chords in Lauds, in both situations similar to what we have seen in previous music, but also in more evolved, post-Harmony Book, post self-examination influenced moments. Second, we will see how both the “Synthesis” and “Analysis” sections of Harmony Book may work together to elucidate the complex yet efficient construction of musical passages of this work.

The opening bars of “Statement” (first movement) immediately recall Carter’s early use of 4-note chords 18 and 23. Figure 7 exhibits the clear constructions of tetrachords 18 (circled in green) and 23 (in red) as they permeate these measures.18 Notice in bars 2-3, two representations of tetrachord 23 are projected: one, as a vertical sonority sounding on the downbeat of measure 3, and two, the bass (E) and soprano (Bb) voices of this chord combining with the two pitches flanking it (F and D) to create yet another tetrachord 23. We may consider structure a classic illustration of Carter’s common-tone (E and Bb) bonding of two all-interval tetrachords. In fact, we saw a similar example of this technique earlier in the Double Concerto (see Fig. 3). However, if we continue to bars 3-4, we see Carter systematically coalesce his three favored chords together into one brief gesture. Here, the two 4-note chords 18 and 23 are not tied together by a common-tone, but rather enveloped by 6-note chord 35 (circled in purple) emanating in the


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registrally higher pitches of this gesture. In this way, Carter succinctly presents his favored chords while evolving the practice of connecting his all-interval tetrachords.

FIGURE 7: “STATEMENT” FROM LAUDS; MM. 1–4 FOUR LAUDS by Elliott Carter

© Copyright by Hendon Music, Inc. A Boosey & Hawkes Company

Reprinted by permission.

Another example of common-tone association between chords comes in measures 5-15 of “Statement.” After an exclamatory introduction in bars 1-5, the music subdues itself to transition into this “tranquillo” section. Here, Carter facilitates the peaceful atmosphere with a consistent harmonic agenda of smoothly connecting all-interval tetrachords. In Figure 8, we see this connective proliferation through 4-note chord 18s, again circled in green, and chord 23s, circled in red. Furthermore, invariant pitches between subsequent tetrachords are graphically connected by arrows. Interesting as well, are the varying methods in which Carter connects these tetrachords, done either by three common-tones (denoted by black letter A) two common-tones (letter B) or one (C). Moreover, if we then move to “Rhapsodic Musings,” (third movement) we may see continuation of this technique, yet now on the level of connecting 6-note chord 35s. In Figure 9a (mm. 19-21), hexachord 35s (again circled in purple) shift their pitch contents while maintaining their core anatomies in a manner similar to what we saw in A Symphony for Three Orchestras (Fig. 6). Furthermore, as accomplished in Figure 8, these shifting hexachords alternate the amount of invariant pitches between them. Figure 9b reproduces the nine hexachords from this triple-stop based gesture. Notice how invariant content (circled in black) fluctuates from five pitches (between H1 and H2), to three pitches (H3 to H4; H5 to H6) and finally to two pitches (H7 to H8). Ultimately, Figures 9a and 9b not only present Carter’s recollection of a scheme used in A Symphony, but demonstrate how he has now evolved this scheme to increase the invariance level of four pitches connecting hexachords, as we saw in Figure 6, to five pitches (between H1 and H2 in 9b) in “Rhapsodic Musings.” Naturally, changing one pitch to maintain a consistent hexachord is limiting; therefore, this increased level of invariance provides evidence of Carter’s further exploration in musical hexachordal relationships beyond his investigations in 1976.


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FIGURE 8: “STATEMENT” MM. 5–15 FOUR LAUDS by Elliott Carter



FIGURE 9A: “RHAPSODIC MUSINGS” MM. 19–21 FOUR LAUDS by Elliott Carter




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FIGURE 9B: “RHAPSODIC MUSINGS” MM. 19–21, INVARIANT PITCHES

In addition to these classic and mature examples of his three favored chords in Lauds, Carter also isolates the “resultant” tetrachords that were alluded to earlier in this essay. Recall that in Figure 1, we saw these resultant tetrachords (as explained by Link in Fig. 2) appear in the opening bars of String Quartet No. 1, however, not yet isolated with gestural emphasis. In Lauds, a resultant tetrachord may now have elevated importance to accentuate climactic moments by complementing the incessant amalgamations of 4-note chords (two tetrachord 18s, 23s or both combined). For example, one such moment occurs in bars 12-14 of “Rhapsodic,” as shown in Figure 10. Here, we find an energetic and rapid ascension to a high G, played double forte, concluding the dolce music in bars 9-12. Although 6-note chord 35s do manifest themselves in bar 14, it is important to recognize the combination of pitches agogically and/or registrally emphasized here: C, G, C# and F#, or 4-note chord 2 (0167), circled in orange. This resultant tetrachord sonically dominates the measure and serves as a climactic signpost for the movement. Embedded within this climax lies the evidence of Carter’s compositional self-study.


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FIGURE 10: “RHAPSODIC MUSINGS” MM. 12–14 FOUR LAUDS by Elliott Carter



Yet, how does Lauds demonstrate the joint contribution of the “Synthesis” and “Analysis” volumes of Harmony Book? Recall that his “Synthesis” volume essentially combines other chords, while the “Analysis” section simply divides chords.

An example of Carter’s use of these two volumes may be found in the final bars (mm. 23-24) of “Rhapsodic.” In Figure 11a, we see a segmentation of 4-note chord 23 (in red) heard near the end of the bar, commencing on F#6, and continuing into the next measure.19 We also find 4-note chord 18 (in green) interlocked with tetrachord 23, sounding in a contrasting lower register. The manner in which these pitches unveil themselves, layered 4+4, reflects the segmentation Carter suggests of 8-note chord number 12 (0124678t) on page 335 of the “Analysis” volume in the Harmony Book (see Figure 11b; pitches in “Rhapsodic” transposed down a major second from those present here). However, this combination of tetrachords only presents two layers of the gesture, and in a period of composition where his harmonic materials have evolved by coalescing his three favored chords, we must also recognize Carter’s third layer: the projection of 6-note chord 35. True, 6-note chord 35 is a subset of 8-note chord 12, but if Carter wished to generate a 6-note chord as a natural extension of one of his 4-note chords, he could move to the “Synthesis” section of his notes to see what needed to be added to either of his tetrachords in order to realize hexachord 35. If we turn to page 162 of the Harmony Book, we see his notes clearly construct 6-note chord 35 out of 4-note chord 18 plus interval 1 (see Figure 11c). In Carter’s “Synthesis” language, this operation is 4+2=6, but, we have arrived at this operation by virtue of dividing the 8-note chord and then combining its 4-note subset with a dyad. If we look back to Figure 11a, we see this process in a musical context, as a resulting 6-note chord 35 appears in the first six notes of bar 24 (circled in purple). This hexachord clearly represents the synthesis as created on page 162 in the Harmony Book. Therefore, his efficiency and organization allow all three of his favored chords to materialize in one brief gesture.


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FIGURE 11A: “RHAPSODIC” MM. 23–24 FOUR LAUDS by Elliott Carter



FIGURE 11B: “ANALYSIS” VOLUME, HARMONY BOOK; P. 335 Used by permission from Carl Fischer, LLC.


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FIGURE 11C: “SYNTHESIS” VOLUME, HARMONY BOOK; P. 162 Used by permission from Carl Fischer, LLC.

Finally, now suppose Carter wished to project two statements of 6-note chord 35 from the 8-note chord number 12, discussed in the previous paragraph. (Incidentally, 8-note chord 12 is one of importance in his relatively recent music, due to its construction) To do this, he would need to extract both a 4-note chord 18 and a 23 from this 8-note chord, then turn to the “Synthesis” volume to see what needed to be added to each to create 6-note chord 35s. As mentioned, 4-note chord 18 would need a minor second added to it, as shown in Figure 11c. Four-note chord 23, as seen on page 164 of the Harmony Book and reproduced here in figure 12a, needs i5, which may be written as a perfect fourth or fifth. Then, Carter only needs to write down the pitches associated with the transformation of either of these tetrachords, and keep them adjacent to the pitches of the 4-note source chords. For example, in bars 5-6 of “Rhapsodic,” as shown in Figure 12b, we find a linear unveiling of 8-note chord 12 commencing on C4 and ending on a G#5/A#5 dyad. Furthermore, the two abstractions of 6-note chord 35 are beamed in purple. Notice how these two are inversions of each other. But for Carter to sonically portray these two simultaneously as inverted versions of 6-note chord 35, he needed to isolate the only pitch of the original 8-note chord 12 that does not contribute to these inversional hexachords, in this case C. Or we could look at this in another way. As per the “Synthesis” volume, we could isolate two tetrachords of this 12-note chord (18 again circled in green, 23 in red – on 12b) and then remember what needs to be added to each. Chord 18s, either set I: [D,D#,F#,G#] or II: [A#,A,F#,E], would need to add dyads [A,A#] or [D,D#], respectively, to complete hexachord 35. However, each of these tetrachords, with their dyad partners, also need to be added to a tetrachord 23 with its dyad partner in order to supply all seven pitches required for the desired inversional hexachordal relationship. Chord 23s, either set I: [A,G#,F#,D] or II: [D#,E,F#,A#], would also need appropriate dyads [D#,A#] or [D,A], respectively, to complete 6-note chord 35. As shown in Figure 12c, 18-I would need to be added to 23-II, in order to achieve this effect (as circled in 13b). Likewise, 18-II could be added to 23-I. Either approach would work, as long as Carter isolated the seven pitches from the non-


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contributing pitch, C, as realized musically in 12b.20 In this way, Carter has taken an 8-note chord and partitioned it (1+7) in a manner in which the second partition could project inversional, all-trichord hexachords.

FIGURE 12A: EXCERPT FROM HARMONY BOOK, P. 164

FOUR LAUDS by Elliott Carter © Copyright by Hendon Music, Inc.

A Boosey & Hawkes Company Reprinted by permission.

FIGURE 12B: “RHAPSODIC” MM. 5–6


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FIGURE 12C: 7 + 1, WHERE 7 PITCHES = 2 SYMMETRICAL 6-NOTE CHORD 35’S

Although examining the compositional method of a composer like Carter may be a convoluted exercise at times, the examples his music provide may give today’s composers revived energy in taking time to study their own music. In Carter the result is clear: a refined harmonic agenda codified and enriched by assembling the chords of the Harmony Book. The efforts of our own compositional self-study may not one day yield a published catalog of our organizational methods, and our approaches to composition may not ever be revealed to anyone but ourselves. But then again, perhaps the most important person to “discover” our compositional techniques is ourselves.


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SELECTED BIBLIOGRAPHY

Bernard, Jonathan. “Problems of Pitch Structure in Elliott Carter’s First and Second String Quartets,” Journal of Music Theory, Vol. 37, No. 2 (1993): 231-266. Carter, Elliott. Double Concerto for Harpsichord and Piano. New York: Associated

Music Publishers, Inc., 1962. Carter, Elliott. Four Lauds: for solo violin. New York: Boosey & Hawkes, 2001. Carter, Elliott. String Quartet No. 1. New York: Associated Music Publishers, 1962. Carter, Elliott. A Symphony of Three Orchestras. New York: Associated Music Publishers, 1978. Carter, Elliott. Ed. Link, John F. and Hopkins, Nicholas. Harmony Book. New York: Carl

Fischer, LLC, 2002. Carter, Elliott and Link, John. “Elliot Carter Talks About His Harmony Book,” in

Harmony Book. New York: Carl Fischer, LLC, 2002. Koivisto, Tiina. “Aspects of Motion in Elliott Carter’s Second String Quartet,” Integral, Vol. 10, (1996): 22. Rao, Nancy Yunhwa. "A Comparison of Trichordal Relations in Milton Babbitt's String Quartet No. 2 and Elliott Carter's A Symphony of Three Orchestras,” Ewha Music Journal Vol. 6 (2002): 164-165. Schiff, David. “Carter in the Seventies,” Tempo, Vol. 130 (September, 1979): 2-10. Straus, Joseph N. Introduction to Post-Tonal Theory, Second Edition. Upper Saddle

River, NJ: Prentice Hall, 2000.


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ENDNOTES

1. Carter, Elliott and Link, John, “Elliot Carter Talks About His Harmony Book,” in Harmony Book (New York: Carl Fischer, LLC, 2002), 31. 2. Although Carter’s music possesses a highly organized and evolutionary approach to rhythm, this article confines our examinations to pitch materials only. 3. Straus, Joseph N. Introduction to Post-Tonal Theory, Second Edition. (Upper Saddle River, NJ: Prentice Hall, 2000), 49. In this article we will use a notation system for describing prime form sets as done by Straus in Introduction to Post-Tonal Theory. 4. Bernard, Jonathan, “Problems of Pitch Structure in Elliott Carter’s First and Second String Quartets,” Journal of Music Theory, Vol. 37, No. 2 (1993): 232. 5. Carter and Link, 9. 6. Ibid, 15. 7. In Harmony Book, Carter organizes his chord charts by assigning a number to each of his 3-note through 8-note chords (sets). The advantage of this is that it “illustrates that the number of available forms of a chord matches the number of available forms of its complement.” This information may be found in Carter, Elliott. Ed. Link, John F. and Hopkins, Nicholas, Harmony Book (New York: Carl Fischer, LLC, 2002), 40. 8. Carter, Elliott. String Quartet No. 1. (New York: Associated Music Publishers, 1956), 1. 9. Bernard, 233. 10. Ibid, 242. 11. Figure 2 reproduced from: Carter, Elliott, and Link, John, 32. 12. Carter and Link, 7. 13. Carter, Elliott, Double Concerto for Harpsichord and Piano with Two Chamber Orchestras. (New York: Associated Music Publishers, Inc., 1962), 2-3. 14. Ibid, 33. 15. Schiff, David. “Carter in the Seventies,” Tempo, Vol. 130 (September, 1979): 5. 16. Koivisto, Tiina. “Aspects of Motion in Elliott Carter’s Second String Quartet,” Integral, Vol. 10, (1996): 22. 17. Rao, Nancy Yunhwa. "A Comparison of Trichordal Relations in Milton Babbitt's String Quartet No. 2 and Elliott Carter's A Symphony of Three Orchestras,” Ewha Music Journal Vol. 6 (2002): 164-165. 18. Musical examples 7, 8, 9a, 9b, 10, 11a, 12b and 12c all derive or are taken from Carter, Elliott. Four Lauds: for solo violin. (New York: Boosey & Hawkes, 2001). 19. This article refers to a system of pitch identification where the lowest A on the piano is A0; all other pitches are numbered accordingly. 20. Naturally, there are other possible segmentations of tetrachords 18 and 23 in this 8-note chord, but since C is musically isolated, we must consider the possibilities within the seven adjacent pitches.

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