Harmony in Elliott Carter’s Late Music: Core, Non-Core, and Derived Core Harmonies
Harmony in Elliott Carter’s Late Music
During his middle period, Elliott Carter typically divided the landscape of set classes among the distinct layers of his split-ensemble works. Then, in the mid-1990s, he focused his sights on three set classes that had been essential to his music for decades: the two all-interval tetrachords and the all-trichord hexachord. These three set classes are what I call the “core harmonies.” Carter himself frequently pointed to their central importance, and they have served as the foundation for analyses of his music since the 1950s. Yet, as many analysts have observed, it is not always straightforward to see how these core harmonies account for the abundance of intricate harmonic relationships in Carter’s late works.
I propose adding a third category—“derived core”—to the existing classifications of “core” and “non-core.” Derived core harmonies emerge when pairs of core harmonies are combined, with or without shared pitch classes. David I. H. Harvey first suggested this idea, noting pentachords and hexachords formed by two all-interval tetrachords (AITs) that share two or three notes. Tiina Koivisto’s analysis of the String Quartet No. 2 also draws on this approach. Here I broaden and generalize this concept, briefly connect it to other areas of Carter scholarship, and demonstrate several analytical uses for it in Carter’s late compositions.
One strength of this derivation is that it condenses some larger collections containing core harmonies as subsets. For example, Carter’s hexachord 9 (013579) contains both AITs as subsets, but its two all-interval tetrachords together yield only five pitch classes, thereby generating an instance of derived core pentachord 36 (02368). Because it is not a core harmony, the chord symbol for hexachord 9 appears in parentheses. The result of this approach is a tight yet wide-ranging harmonic vocabulary of five-, six-, seven-, and eight-element set classes, each listed alongside its derivations. Crucially, many derived core harmonies have more than one possible derivation.
A further benefit is that derived core harmonies mesh seamlessly with much existing analytical writing on Carter. For instance, they can be understood as collections of transformations implied by their derivations. Derived core hexachord 22 (013479) has three possible origins from pairs of AITs sharing two common tones. Guy Capuzzo and John Roeder have done valuable work on transformational networks of core harmonies in Carter’s music, often drawing on the complement union property. In Joshua B. Mailman’s analysis of Carter’s Scrivo in Vento, hexachord 20 (012357) serves as the “arbiter” of a dramatic confrontation between the two AITs. Among the derived core harmonies are the seven octachords formed by non-intersecting AITs, a specific type in which the number of common tones is zero. These octachords have a long history in Carter’s music; he published a table of them, and in recent years they have been the target of much analysis, including work by Koivisto and Laura Emmery on the String Quartet No. 2, J. Daniel Jenkins on the String Quartet No. 5, and Guy Capuzzo on What Next?, where Capuzzo coined the term “combination sets.”
Andrew W. Mead provided an elegant informal proof of the combination sets’ shared structure: each is formed by removing two discrete dyads of interval class 3 or 6 from the aggregate. The complement of each combination set can therefore be divided into pairs of the same interval class in at least two distinct ways—a feature that Carter frequently exploits. In a brief passage from Carter’s Oboe Quartet, the violin and viola, having already begun their duet, come to the foreground with a vigorous marcato statement that leads to sustained double-stops. Together they produce an instance of non-core tetrachord 2 (0167), arranged to project interval class 1 by instrument and interval class 5 by register. The oboe and cello respond with the final gestures of their own duet—a pair of AITs that together create derived core combination set 2, thereby completing the aggregate. Carter mentioned in interviews that he returned to using combination sets, and they appear especially prominently in a run of works from the mid-to-late 1990s, including the String Quartet No. 5, Luimen, the Quintet for Piano and String Quartet, and What Next?.
Yet Carter also investigated a variety of other derived core harmonies in his late period. In August 2012, he sent me a transcription—later annotated—of what he referred to as his “seven-note chord system.” The sketch displays two instances of derived core septachord 34 that share two common tones and thus also form an aggregate as a union. On the right side of the sketch, Carter illustrates how to build the chords practically from a segment of the circle of fourths combined with an interval-class-5 dyad.
Little more than a month after sending that sketch, Carter completed his final composition, Epigrams. An excerpt from the first movement—also the last movement he wrote—reveals three pairs of sustained string dyads framed by contrasting gestures. The first two pairs both yield instances of AIT 18 (0146), the first arranged as i7+i4 and the second as their inversions, i5+i8. The framing gestures sustain the derived core harmony.
The initial four-note violin melody forms non-core tetrachord 9 (0134), but the violin’s next utterance—the dyad G-B—completes not only the first AIT 18 together with the cello, but also two instances of AIT 23, interlocked in derived core hexachord 27 and derived core septachord 33. In measures 5–6, the piano answers with a three-note figure that completes derived core septachord 34 (Carter’s sketch version) and anticipates the subsequent pitches in the strings. The piano’s three staccato notes in measure 7 similarly complete the same derived core septachord. Recurring AITs in the strings establish a pattern; when the violin plays B –A in measure 8, one expects an interval-class-1 dyad to round out a third instance of (0146). Instead, the cello surprises us with a new register and an i5 dyad that completes non-core tetrachord 22 (0126). However, this chord together with the preceding tetrachord in the strings forms the all-trichord hexachord, and it is quickly absorbed into another derived core septachord 34 in the piano.
The technique of recontextualizing non-core sonorities within larger derived core harmonies is extremely valuable for highlighting both harmonic variety and consistency across Carter’s late music—especially in passages where the three core harmonies themselves vary greatly in salience. The opening bass clarinet melody from the Adagio of Carter’s Concertino for Bass Clarinet and Chamber Orchestra illustrates this. As in many of Carter’s late concertos, the eleven ordered pitch-class intervals are divided between soloist and ensemble: the soloist plays intervals 2, 4, 7, 9, and 11, while the ensemble takes their inversions (1, 3, 5, 8, and 10; the interval 6 is shared). Because nearly all of the bass clarinet’s smaller intervals are even, its melody tends to stay confined to a whole-tone scale segment when its range is narrow. The first eight notes form non-core, whole-tone pentachord 6 (02468). When the bass clarinet eventually plays an odd interval—i7—the melody breaks free of the whole-tone collection, adding G♭ and D so that the entire phrase becomes derived core septachord 37 (shown with its derivations). In the context of the bass clarinet melody, neither derivation is especially evident. That is one of the key advantages of the derived core harmonies: they reveal a high degree of harmonic consistency even when the sonic fingerprints of the three core harmonies are explicit (as in Epigrams I) or well-hidden (as in the present case).
The derived core harmonies are immensely helpful for untangling passages in which neither the core harmonies nor the combination sets are easy to discern. Among the derived core octachords are four collections formed by several all-trichord hexachords projecting four common tones (see Example 11). Carter deploys these collections to striking effect in his setting of Wallace Stevens’s poem “Re-Statement of Romance,” from the song cycle In the Distances of Sleep. The poem’s speaker addresses the beloved: “Only we two may interchange / Each in the other what each has to give.” In Carter’s setting, the vocal line at first articulates two AITs that share pitch classes 10 and 3 as common tones, creating derived core hexachord 31. The larger derived core harmony is relatively conspicuous here. The vocal line can also be heard as a pair of overlapping, inversionally related (01367)s—Carter’s pentachord 31. This set class takes a leading role among Carter’s late harmonies as one of only two derived core pentachords and as the only one that is also a subset of the all-trichord hexachord. That property comes to the fore in this excerpt. The first pentachord 31 combined with its accompanying G4 in the viola yields the all-trichord hexachord at T8; the second pentachord 31 does the same with the viola’s B3 at TeI. The resulting two all-trichord hexachords together compress into derived core octachord 8.
The full excerpt illustrates a common pattern in Carter’s harmony: a clear presentation of a harmonic relationship is followed by a more ambiguous one. The second part of the vocal line brings harmonic variety in the form of two non-intersecting transpositions of non-core tetrachord 16 (0248). Together, the two (0248) tetrachords assemble into octachord 15—another derived core harmony this time linking two all-trichord hexachords interchanging “Each in the other” the four pitch classes they share. Especially compelling in this passage is the way a musical relationship between the melody and its accompaniment is later “internalized” inside the melody alone. This process mirrors the speaker’s gradual shift of attention in the poem: first absorbed in the romantic story being told, the speaker begins to grow self-conscious about the very act of telling that story. The speaker’s slowly rising self-awareness leads to considerable uncertainty about the nature of romantic love.
A comparably challenging passage in Carter’s late harmonic idiom is the opening of the Oboe Quartet. Three times the oboe answers a dramatic string outburst with a heavily accented G4. Eventually all four instruments play a simultaneous attack, which launches into an oboe solo (not shown). The four string gestures are heard as long-short-long-very-short, while the oboe’s notes grow progressively longer, building toward the culminating rhythmic unison.
Here, the derived core harmonies offer a valuable analytical lens. Each of the oboe’s repeated G4s adds a seventh pitch class to the preceding string hexachord, transforming it into a derived core septachord. The sense of rising energy is strengthened by a widening registral wedge in the strings and increased harmonic diversification moving from the composer’s core framework (the all-trichord hexachord, number 35), to derived hexachord 10, and on to non-core hexachord 17. In each two bar oscillating call from the middle register of the oboe then draws the harmony back to bring forward again the now grown surface-patterned shape of passage shaping across analogous down beat structures in page positions prior within the instrumental families ongoing roles and a set of core resolution frameworks with a periodic schema that listeners encode easily enough when defined as separate phrases now with derived core rest points—then on the culminating downbeat of measure 5, the strings at last mimic the oboe’s gesture, producing a derived core septachord on their own, as the violin prominently asserts the bright surprise dynamics in a reemerged G4 relationship with its aligned whole texture—absorbing cadential goal. But, simultaneously, the oboe slips away to an eighth pitch to form, in partnership with the strings composite texture, derived core octachord 19—the same set class generated by the ensemble block of the number and two sets resolution gestured earlier with parallel expansion with opposing tether so thus gluing together one against ante-arch chain extended arc of string groupings rounding number repetitions balanced still.
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Several useful conclusions emerge from this analysis. First, the derived core harmonies establish intermediate referential collections positioned between the ATH and the concluding combination set. As the register expands and dynamics increase, derived core harmonies become more prominent, expanding in size until they reach the concluding combination set. Consequently, we can interpret the passage as one that strengthens harmonic resolution, since derived core harmonies replace non-core ones. The harmony undergoes two internal morphs, producing non-core hexachords 9 and 12 in the process. Moreover, these two internal morphs—moving from referential to non-referential harmonies and back again—can be understood as motion within two larger derived core septachords (numbers 36 and 6), effectively functioning as a type of arpeggiation. This multi-layered functionality proves particularly striking, as it enables harmonic change and voice leading to operate differently across varied time scales. For example, the C⁴ in hexachord 9—played by the second violin—may be perceived on one level as a quasi-dissonance; it creates a non-core hexachord by anticipating derived core pentachord 36. On a somewhat larger time scale, that same move can be heard as an arpeggiation within the larger derived core septachord.
Distinguishing among core, derived core, and non-core collections offers a clear foundation for this kind of multi-layered interpretation of harmonic events and aids in modeling the sense of harmonic inevitability—or surprise—that listeners often experience in Carter’s music. The approach presented here possesses broad analytical applicability and complements existing work in the field. It can account for harmony in nearly any passage from Carter’s late music, while leaving ample room for multiple interpretations and methodologies to illuminate each of Carter’s singular compositions in a distinctive way. It also suggests a productive approach to thinking about Carter’s creative process, his sketch materials, and the interaction of harmony and voice leading in non-tonal music more generally.