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date: 18 November 2017

Sequence

Abstract and Keywords

This article examines the musical sequence as a nexus of melodic, linear, and harmonic processes commonly found in tonal music from the seventeenth to the early twentieth centuries. It locates sequential repetition as a problematic concept within the history of tonal music, considering the various controversies it has sparked in both its critical reception and the competing models advanced throughout the history of tonal theory. The article also explores how the sequence is both prototypical for tonality more broadly and an exception to its norms. Besides acting as an agent of tonal, formal, and stylistic disruption, the sequence also provides a distinctive way of structuring the experience of musical temporality between diachrony and synchrony. The article advances the provocative idea of the sequence as a bipolar machine for transforming identity into difference and difference into identity.

Keywords: circle of fifths, diachrony, linear intervallic pattern, rosalia, sequence, sequential repetition, synchrony, temporality, tonality, tonal theory

The Sequence between Identity and Difference

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Example 1: Schubert, Gretchen am Spinnrade, mm. 112–18.

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Figure 1: Reduction of Schubert, Gretchen am Spinnrade, mm. 112–18.

A sequence is a bipolar machine for transforming identity into difference and difference into identity. At first blush, this seems a somewhat baffling, and perhaps unnecessarily complicated, way to describe a common harmonic-melodic phenomenon of tonal music. An example from Schubert’s Gretchen am Spinnrade (see Example 1) indicates why it might be useful. A traditional definition would focus on two constitutive moments of the sequence: (1) a fragment of musical material that is to be repeated at different pitch levels and (2) a pitch-based schema that determines the relationship between successive statements of the material. The material to be repeated is readily identifiable as a rising, stepwise dyad, embellished with an upper neighbor to the second note, harmonized by an applied dominant seventh resolving to the root-position triad it tonicizes. The schema dictating the trajectory of transposition is less easy to pinpoint, because the quality of the intervals within and between each melodic statement varies so as to adapt to the diatonic context. In this way, intervallic differences are needed to achieve a diatonic sameness, at least melodically. Preserving identity between the melodic material repeated, however, precipitates a discrepancy between the melodic and harmonic dimensions. The rising melodic tetrachord A–B♭–C–D, which is entirely diatonic in F major, could have been harmonized by exclusively diatonic focal harmonies with A minor supporting the C, but this would have entailed the insertion of a melodic chromatic passing note B♮ to be harmonized by the applied dominant. Schubert’s solution is to harmonize the melodic C with the nondiatonic A♭ major, thus privileging melodic over harmonic diatonicism, while allowing the harmonic dimension to determine the irregular projection with its mixture of half and whole steps together with the modal inflection of the second step (see the reduction in Figure 1). This passage exemplifies the way in which sequences commonly contend with negotiating between identity and difference across various parameters and structural levels.

There are, of course, other phenomena in tonal music taking place at various structural levels, from local detail to global form, which could be said to convert identity into difference or difference into identity. For the most part, however, these various processes have as their primary task the subordination of one to the other, thereby privileging either identity or difference. The hallmark of the sequence, by comparison, is that it is traversed by a double movement such that it produces both identity and difference at one and the same time and holds them in tension. That is not to say that particular uses of this musical process in the context of specific pieces do not tend toward one pole or the other (which they almost always do). Rather, not only can the trace of the countermovement not be entirely eliminated, but this double traversal is constitutive of the sequence: the sequence depends for its musical effect upon two simultaneous processes of transformation that cut across one another. As this essay explores, the play of identity and difference manifests itself in a number of dimensions: as a tension rhetorically between parataxis and hypothaxis and stylistically between baroque and classical, and as a function of temporality, between diachrony and synchrony. Finally, as the last section of this essay demonstrates, it is both different from, even disruptive of, other harmonic, melodic, and formal principles in tonal music and a prototype for the operation of tonality more widely.

In this way, my description of the sequence seeks to capture not only its technical internal workings, but also the position it occupies within theories of tonality: the sequence tends both toward coinciding with the definition of tonality (most obviously in the theory of Jean-Philippe Rameau, for example) and toward diverging from normative patterns of tonality’s functioning, even to the point of enjoying an exceptional status (as in the theories of François-Joseph Fétis and Hugo Riemann). This interplay of identity and difference within tonal theories can be seen to replicate itself at a higher level if one surveys the body of theoretical discourse on this topic from the seventeenth to twentieth centuries. One of the difficulties in providing a definition of the sequence is the sheer proliferation of terminologies and theoretical models generated by thinking about this seemingly straightforward little device, and with it the risk that the definition will be so diluted in order to cover all these positions that it would extend to any vi—ii—V—I progression. For this reason, the sequence provides a fruitful vantage point from which to grasp the differentiation of various theoretical traditions, from which to divide figured-bass from fundamental bass models, French from German schools, melodic from harmonic conceptions.1

The sequence provides an exemplary lens, for instance, through which to highlight a particular fracture in the history of theory: in a certain strand of nineteenth-century harmonic thought, the idea that figured-bass theories do not adequately acknowledge the hierarchical nature of relationships within the tonal system gains currency. By leveling out the differences between the scale degrees or harmonic functions upon which the hierarchy of the tonal system is constructed, the sequence effects a temporary suspension of this system. Fétis describes, for example, how “the mind, absorbed in the contemplation of the progressive series, momentarily loses the feeling of tonality and regains it only at the final cadence, where the normal order is reestablished.”2 Riemann adopts much of this line of thinking, crediting Fétis with realizing that the sequence is an essentially melodic formation that consists in the suspension of harmonic movement throughout its duration.3 Riemann, though, effectively misrepresents Fétis in attempting to assimilate Fétis’s assessment of the sequence to his own functional-harmonic perspective. Riemann’s theory eschews any of Fétis’s arguments about the relative instability of the diatonic scale and instead consolidates chords into three primary perceptual categories of tonic, dominant, and subdominant. Unlike Rameau, who sees the diatonic sequence as paradigmatic for tonality as a whole insofar as it projects the cadential movement from tonic to dominant across a descending-fifth progression, Riemann denies the sequence the status of harmonic prototype because the D–T relation of the model is not strictly replicated at each step. Riemann in fact requires a more substantial T–S–D–T to express tonality fully.

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Example 2:Rosalia, mia cara.”

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Example 3: Beethoven, Theme from “Diabelli,” Variations for Piano in C major op. 120, mm. 1–32.

Nothwithstanding these divergences, it is possible to discern the emergence of a common thread among these various theoretical traditions: the multiple discussions of the sequence converge upon a single anxiety that sequences are too static, too monotonous; in short, they present too much of the same. Hence the advice found in modern and historical texts to limit the number of repetitions, often to no more than the optimal three. Heinrich Christian Koch’s dismissal of the melodic sequence is typical of concerns in Germany during the later eighteenth century: he claims that these types of transposition of a phrase segment are “obsolete” and “are to be avoided … unless they appear in a new form.”4 Koch’s worry that the sequence exhibits insufficient innovation is echoed by what Jairo Moreno describes as “an oddly assorted, transhistoric jury loudly proclaiming the lackluster qualities of sequential repetition in a variety of contexts.”5 Moreno’s jury includes figures as diverse as Beethoven, Wagner, Charles Burney, Heinrich Schenker, Theodor W. Adorno, and Richard Taruskin. Particular condemnation is reserved for what came to be known as rosalia after the Italian popular song “Rosalia, mia cara” (see Example 2). Christian Friedrich Daniel Schubart, for example, disparages these stepwise rising melodic sequences for sounding out-of-date by the late eighteenth century; they are best suited to the incessant repetitions of Trinklieder in the beer halls or to empty virtuosic displays, provided that the musician’s hands are on display so that the dizzying visual display of the approach toward the end of the fingerboard might compensate for the boredom induced by the sonic repetition.6 Burney had lamented the “tediousness” of the rosalia and, in words penned by Adorno, Thomas Mann’s Doktor Faustus condemns them as “cheap.” Beethoven was even said to have poked fun at the “Schusterfleck” or cobbler’s patch in the theme he later used by Diabelli (see Example 3).7 In other words, theoretical discourse moves toward identity precisely in its anxiety about the sequence’s tendency toward identity.

As if to underscore the point that the sequence—in both its internal workings and its theoretical elaboration—is always marked by a double movement of identity toward difference and difference toward identity, there is, as Moreno notes, an important exception to this otherwise univocal condemnation.8 Chief among the dissenting minority are Adolf Bernhard Marx and Anton Reicha, whose theories detach the sequence from notions of monotony and stasis by associating sequential repetition with a dynamic process of generating melodic content. As Moreno argues, this change in view can be situated within the context of a broader shift from the rhetoric-inspired Satzlehre of the later eighteenth century to the Melodielehre of the early nineteenth century; while the former considers how preformed and self-contained melodic units are repeated and added in a block-like fashion to preexisting phrases for the purposes of expansion, the latter thinks of these melodic units themselves as both capable of being decomposed into smaller components and also containing an intrinsic demand for transformation and expansion through repetition. From the perspective of Satzlehre, musical material is subjected to an external process of repetition, while for Melodielehre the material itself necessitates the repetition and thereby generates further melodic content. This emphasis on a mutable dynamic process as opposed to the subsequent spatial layout of preexisting blocks firmly inserts the sequence into the temporal realm. In turn it becomes possible, as discussed in the third part of this essay, to think of the sequence as a means for producing different representations of time.

Types of Sequence

Perhaps the strongest argument, however, against the notion that the sequence is a means of producing identity and boredom is the fact that as a category, the sequence resists reduction to a single, self-identical musical phenomenon. Instead, any attempt to pin down the sequence with a narrow definition is met by a wealth of self-differentiation. Examining these differences provides a summary of the various issues and debates in the history of music theory’s engagement with the sequence. These differences fall on either side of the binary division inherent within the standard definition of the sequence, between the material to be repeated and the logic governing the trajectory of its repetition. One immediate source of differentiation in the category of the sequence lies in the variety of musical materials and combinations of materials that may be subject to sequential repetition. This has led to a lack of terminological consistency in describing the primary material of a sequence. In his recent theory of formal functions in the classical style, William Caplin uses the term “model-sequence,”9 which no doubt derives from Reicha’s terminology, but Moreno prefers the more processual implications that come with “repetend.”10 More frequent is the term “pattern,”11 although the potential difficulty here is that pattern can connote not only a model designed for imitation, but also a regular design itself formed through repetition.

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Example 4 Mozart, Piano Sonata in G, K. 283, i, mm. 112–18.

This terminological inconsistency reflects the difficulty in defining what type of material is capable of being repeated sequentially. The use of the word pattern suggests that the primary unit of material itself has a recognizable form of its own; indeed, some theorists exclude the possibility of a sequence where the repeated component has fewer than two states, so that there is always an internal relation within the unit to be repeated (two different inversions or a triad and a seventh chord would be admissible, but not a chain of 63 chords, for example).12 This makes sense if the unit of repetition is considered in terms of vertical simultaneities, because the transposed repetition of a single chord requires careful revoicing to avoid excessive parallel motion, if not actually parallel fifths or octaves. Moreno is content to consider a series of 63 simultaneities as a sequence mainly because it allows him to forge connections between different theorizations of scale-degree steps, but notes that the requirements of good voice-leading often yield pairs of voice-leading patterns in any case.13 One such example is the embellishment of the successive 63 chords with local suspensions (see Example 4). The term pattern thus foregrounds the idea that a sequence consists of the repetition of a particular contrapuntal fragment; this notion has its roots in the figured-bass tradition, for which the sequence is the repetition of an intervallic progression above the bass, and is taken up in more recent Schenkerian theorizations with the notion of a linear intervallic pattern.14 An alternative thread of theoretical discourse sees a melodic or motivic fragment as the object of sequential repetition.

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Example 5: Mozart, Piano Concerto no. 21 in C, K. 467, ii, mm. 45–50.

Daniel Harrison, for instance, conceives of the rosalia as a primarily melodic procedure of transposition either up or down a step, but his interest in this process lies not simply in the fact that it presents a counterpoint to harmonic or intervallic sequences, but, more significantly, in the way in which the appropriation of this device enables him to trace the fusion of melodic and intervallic elements in the practice of Arcangelo Corelli, his contemporaries, and his descendants.15 Specifically, Corelli’s sequences exploit a kinship between serial transpositions of thematic patterns and the chains of consonant syncopes and dissonant suspensions in fourth- species counterpoint. Conversely, the plurality of basic materials capable of being subjected to sequential repetition also opens up the possibility of disconnecting melodic, harmonic, and intervallic components such that only one is subject to strict transposition, while others are reworked more freely. Example 5 shows a purely melodic sequence in which the harmony does not follow the same transpositional scheme as the melody.

This separation of melodic from harmonic or intervallic material is just one of the ways in which the sequence may escape uniformity in its repetition phase. Besides the possibility of subjecting different components of the original material to different processes, there is considerable diversity of theoretical opinion on what kind of musical object or principle may govern these trajectories of repetition. Variously described as a “projection”16 or a “vector” by modern theorists,17 the tradition largely agrees that the logic of repetition is pitch based. It is far from clear, however, that this will always be conceived primarily as a transpositional schema (rising or falling by a certain interval with each restatement). In many cases, sequential repetition is determined not by replicating a fixed interval of transposition, but by motion through scale-degree steps. Other theoretical models dispense with the requirement for a linear-melodic schema to privilege instead the underlying harmonic progression or cycle as generative of the sequence: harmonic sequences frequently elaborate the circle of fifths (a descending-fifth progression) or other common root motions such as ascending or descending seconds or thirds.

That the vector of sequential repetition may be governed by scalic or harmonic arrays has led to one of the important and frequent distinctions in theories of the sequence, between those that are based on exact transpositional schemas and those that retain the number, but not consistently the quality, of the interval of transposition. The first type is typically described as real and the latter as tonal, but a further distinction between modulating and nonmodulating sequences has led to a degree of terminological inconsistency. The two systems of classification do not coincide exactly, especially when analyzing sequences in nineteenth-century repertoires. Real sequences are often taken to be modulating, in contrast to tonal sequences, in which the interval is modified precisely for the purpose of maintaining the prevailing tonic.

It is precisely the capacity of the real sequence to resist confinement within a hierarchical framework of the scale that grants it a certain utility within tonal music. The real sequence produces this tonal difference at the level of the entire progression, though, only by reproducing the very same intervallic relation between each of its individual steps. In other words, the real sequence can be said to be a machine for producing global difference out of local identity. The local relations between each step of a sequence governed by a circle of perfect fifths, for instance, coincide with one another, while the gap between the tonal centers at the beginning and end of the sequence marks a higher-order noncoincidence. The tonal sequence, by contrast, produces a higher-order identity out of local difference; a slight modification to the vector (e.g., contracting one of the steps into a diminished fifth) allows what would have been a modulating progression to function instead as a prolongation. The local disjunction is thus subordinated to global sameness.

It is equally possible, however, that a tonal sequence might modulate and that its intervallic modifications might be performed with this goal in view. A further distinction then suggests itself: sequences might be tonal without being strictly diatonic. Richard Bass uses the term “tonal anchor” to describe an overall tonal framework through which local tonicizations at various junctures of the sequence might be united.18 In his model, the tonal sequence’s subordination of local difference to global identity is maintained. These tonal anchors also operate on various structural levels such that local anchors may be subordinated to a larger-scale movement in a modulating sequence from an initial to a closing tonal anchor. Here, it seems as if the equivalence of local tonicizations (each is the same insofar as it has the status of a tonic) is secondary to the higher-order difference produced through the modulation as a whole. The securing of identity may simply be deferred here by subordinating the sequence to a tonic prolongation on a yet larger scale. But what is interesting about the case of the modulating tonal sequence is how it demonstrates that otherness is not simply to be found in the real sequence’s apparent suspension of tonality, but is fundamentally constitutive of tonality’s own elaboration and articulation. This disruptive potential of the sequence within the tonal framework is discussed in greater detail in the final section of this essay.

Whereas the typical usage defines tonal sequences as those that subordinate their local harmonic progressions to the larger-scale prolongation that sustains the prevailing key, Bass groups together under this category all progressions whose transpositional schemas are modified by the presence of tonal anchors, or what might therefore be described as tonal magnets to explain how their attractive force is able to make the transpositional vector veer off course. To this extent, the tonal sequence could be said to be defined by its local difference in contrast to the real sequence’s local identity. This discussion shows, however, that there is little consensus about whether a sequential vector should be classified from a local or global perspective: while the real sequence is typically defined by its local consistency rather than the global effect of modulation, the tonal sequence is constituted by the interaction between a set of tonal forces that operate across a spectrum from local to global tonicizations. If the theoretical tradition is right to express some anxiety about the tendency of the sequence to produce sameness, it is significant that this identity manifests itself in a diversity of ways—local and global, melodic and harmonic—whose combination resists straightforward classification.

If real sequences are, by contrast, not subject to any tonal influence, Bass nonetheless argues that this category is not selfsame either: “Nineteenth-century harmonic practice … admits alterations to the patterns and projections of real sequences.”19 At the same time, apparently tonal sequences in nineteenth-century music may be modified for reasons other than the influence of a tonal anchor. In this way nineteenth-century examples of sequential repetition dismantle the binary opposition between tonal and real sequences. It is not simply that those sequences that are not real are tonal and those that are not tonal are real, for within the category of real sequences there are those that are not-real (which Bass calls “unreal”) and within the category of tonal sequences those that are not-tonal (insofar as the modifications are not exclusively determined by tonal anchors). This is to say that, while real sequences may frequently be modified at their end to provide a more coherent transition to the subsequent harmonic progression, others, especially in nineteenth-century practice, contain modifications that are not attributable to tonal concerns, but rather to larger-scale motivic or harmonic processes. For instance, a local modification within the sequence may serve to ensure an exact echo of a thematic reference point at an earlier juncture of the overall form, thereby sacrificing local difference for higher-order identity. This kind of “unreal” sequence reverses the common assumption that it is the sequence’s real element that harbors a disruptive impulse, capable of unraveling tonal articulations; rather, it is sometimes tonal considerations at the level of the overall form that disrupt the exact transpositional schemas of the local sequence. The next two sections of this essay consider interactions between the sequence’s disruptive potential and questions of larger-scale form.

It is possible to analyze many sequences as containing a mixture of real and tonal elements, but only on the condition that one also recognizes that each of these categories is marked by internal difference.20 Recall how the example from Gretchen am Spinnrade (Example 1) illustrates that it is not always possible to preserve diatonicism within melodic and harmonic dimensions at the same time. Such examples act as a prism, which refracts the classification of “tonal” sequences along parametic grounds, distinguishing between sequences that are tonal by one criterion but not by another; the effect is to introduce a subset of sequences that are neither wholly tonal nor straightforwardly not-tonal (i.e., real), but might be described as “not-not-tonal.” In certain situations it becomes difficult to discern between this category and the corresponding one of the “not-not-real.” This tendency of sequences to produce identity or difference at one structural level or within one parameter, while producing the other in another dimension, is what motivates the description of a double traversal or bipolar operation, in which one movement is never fully subsumed into the other.

The Temporality of the Sequence

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Example 6: Beethoven, Piano Sonata op. 111, ii, mm. 106–30.

This double movement is also what explains the sequence’s distinctive and fascinating capacity to produce an experience of music’s temporal unfolding. The theoretical tradition has long grasped intuitively that sequences participate in the construction of musical time, as both anxieties about their monotony and alternative theories of their role in melodic development testify. The exact mechanism by which sequences build a representation of time, however, has not yet been rigorously theorized. Across the body of theoretical writings on the sequence there nonetheless exists a certain ambivalence about whether the sequence tends toward stasis or dynamism, toward space or time. Similarly, it is unclear whether to attribute the spatial impulse to the mechanical repetition of melodic fragments, as opposed to the inherently dynamic logic of dissonance-resolution in the suspension chain (as in Harrison’s example of the cross-fertilization of the rosalia tradition with fourth-species counterpoint), or the harmonic progression that governs the transposition can itself be the agent of stasis. The sequential episode that interrupts the flow of variations in the second movement of Beethoven’s Piano Sonata op. 111 illustrates these difficulties (see Example 6). For Charles Rosen, the “greatest master of musical time” here succeeds in freezing time altogether.21 It is not so much that Beethoven suspends the temporality of harmonic succession from without through the imposition of a rhythmic or textural brake, but rather that the descending-fifth progression realizes its potential to suspend its own movement when it appears as a diatonic sequence:

The mastery lies in Beethoven’s understanding that a sequence does not move, that a diatonic circle of descending fifths within classical tonality does not exist on a plane of real action, so that the long series of tiny harmonic movements that prolong this immense inner expansion serve only as a harmonic pulse and in no sense as a gesture.22

This suspension of time—a treadmill-like display of pseudo-labor without a goal—is possible precisely because the circle of fifths that governs this sequence contains within it a double temporal potential.23 The overall effect in this example is the absolute subjection of time to space. And yet it is upon this same harmonic trajectory that the dynamic linear propulsion of Corelli’s suspension chains would be founded. This then raises the question of how the sequence is simultaneously capable of freezing time and of being the source of forward momentum. This duality lies at the heart of Rameau’s theory of the sequence. If the cadential movement from dominant to tonic is paradigmatic of harmonic progression more broadly, the sequence becomes an ideal expression of the tonal system by projecting a cadential progression onto every degree of the scale via the fundamental bass’s motion through descending fifth. The succession from one chord to another in the sequence, as in the cadence, is motivated by the presence, sounding or implied (sous-entendu), of a dissonant seventh above the bass that compels the dominant to resolve to the tonic. In a descending-fifth sequence, “the progressions of harmony are nothing but a chain of tonic notes and dominants, and we should know the derivatives of these notes well, so as to make sure that a chord always dominates the chord which follows it.”24

As Moreno notes, however, all the chords in the sequence, with the exception of the cadential goal, enjoy a certain sameness by virtue of the fact that none can claim a hierarchical superiority. The result is that, cut off from its closing progression, the sequence “reduces harmonic content, particularly function, to pure motion.”25 Without any relation to an origin or end, the motion itself appears static. The absence of hierarchy between scale degrees in Rousseau’s system at this point and the preference for a single local relation between tonique and dominante-tonique remove the differential on account of which one might perceive movement. To this extent, it is possible to think of the interplay of identity and difference in expressly temporal terms: the diachronic or moment-to-moment perception of each local progression is leveled out as each of these moments is collapsed into a single moment of synchronous perception. The real sequence, by contrast, starts from a local synchrony (the identity of each local progression to the next), out of which it produces a global diachrony (the opposed initial and closing tonics).

What makes the sequence a representation of time, however, is that it resists the twin possibilities of pure synchrony or pure diachrony. While the effect of the nonmodulating tonal sequence is to subordinate the sequential progression to a second-order tonic prolongation, it is unable to produce this synchrony without there being a trace of residual diachrony at the level of local relations between steps of the cycle. This stain of first-order difference consists of the introduction of the diminished fifth into the progression of otherwise perfect fifths, in the trace of the Pythagorean comma that prevents the tonal system from coinciding with itself. This residue of noncoincidence is what permits an experience of time. If there were pure synchrony, there would be no sense of time unfolding from moment to moment, but rather the collapse into an eternal present in which every moment has always-already taken place. Pure diachrony would equally destroy all sense of time’s passing, because each new moment would be completely unrelated to any prior moment and would always be experienced as a never-before. Only with a differential margin between diachrony and synchrony—only with a residue of one forestalling the other’s totalization—does it become possible to grasp the passage of time.

A fascinating example of the way sequences participate in small- and larger-scale temporal representations occurs in the slow movement of Beethoven’s String Quartet op. 131. Here the sequence interacts with a formal process, with which it shares a repetitive impulse. Like the sequence, variation has often been maligned by comparison with thematic Entwicklung for its propensity to repeat the basic substance of its melodic and harmonic materials. More sophisticated analyses of variation forms, however, tend to recognize that the process is marked by a double traversal similar to that of the sequence. The process of variation seeks difference in repetition; each successive variation repeats the theme only insofar as it marks its own distance and transformation from its original statement. In the slow movement of op. 131, the fulfillment of a real sequence in the final variation produces a pair of unusually striking interruptions of the form, marked by two outbreaks of trills, thus highlighting the sequence’s potential for formal disruption.

The local origin of the trills can be traced back to the introduction of the ornament in the second half of the third variation, Andante moderato e lusinghiero. Here, the trill functions in a fairly conventional way, as the decoration proper to a cadential flourish. It is then taken up in imitation across the four parts (mm. 113 onward), but what is more important is that the imitation actually cuts across the division of the template binary form. With the exception of the second half of the fifth variation, this set contains written-out repeats throughout, but instead of simply repeating each half of the form with supplementary ornamentation, Beethoven redoubles and accelerates the momentum of variation by internalizing the process within each variation. Here the cello’s entry in the second half of m. 113 is an echo of the cadential figure with which the repeat of the A section culminated. In this way, the trill introduces a zone of indistinction between A and B sections, between inside and outside.

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Example 7a: Beethoven, String Quartet op. 131, iv, mm. 225–35.

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Example 7b: Beethoven, String Quartet op. 131, iv, mm. 250–64.

This structure gives rise, in the final variation, to two outbreaks of trills where one would expect the written-out repeat: the first at m. 228 and the second at m. 250, an octave higher (see Example 7a and Example 7b, respectively). In both cases, the trill moves up from B to C♯ underpinned by a V53  64 progression, only to fall unexpectedly with a modal inflection to C♮ so as to produce an A-minor chord before the bass rises to reharmonize the C♮ with IV in C major. At this point the two passages part ways. The first time, in m. 231, the bass rises again by a step to G to form a V64  53 progression, which resolves in m. 235 before it is deflected back to A major via A minor. When the trills return for the second time, the F major chord is instead reinterpreted as a tonic in m. 254, and the common tone 3^ in A provides the glue to bind this interlude to an abridged reworking of the theme’s second half at m. 264.

In this way, this variation set thematicizes the way in which return necessarily coincides with transformation, identity with difference. A significant digression becomes the occasion for structural return. While the expected return to the tonic is denied at the end of the first eight measures of the final variation, the thematic return is not entirely absent, even if it is held back by the trills for four measures. When it does arrive, this is a return, not simply within the context of the variation, but across the entire set, for the material at mm. 231 and 250 reprises the theme in its original, rather than varied, form, albeit in the wrong key. A tonal and thematic return of the variation’s opening phrase is postponed until m. 243, where the trills from the interruption are absorbed into a varied repeat.

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Figure 2: Linear intervallic pattern in Beethoven, String Quartet op. 131, iv, mm. 136–37.

The eruption of trills in the final variation of op. 131 appears as a moment of synchrony when it is seen, not as the production of tonal identity, but as the completion of a real sequence. This large-scale sequential process can be traced in reverse: a search for a precedent for the C♮ trill yields a chromatic upper neighbor to B in m. 137 in the fourth variation, which scarcely seems significant enough or early enough in the movement to provoke the later disruption. Here it functions as part of a 7–3 linear intervallic pattern (see Figure 2) and is to that extent perfectly predictable: the C♮ corresponds exactly to the E and D♮ earlier in the sequence. It is the D♯ in the bass that breaks with the real sequence in order to maintain the local tonic by means of a half cadence (an exact sequence would have demanded a D♮). In the final chord of the sequence in m. 137, the B alone is synchronous.

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Example 8: Beethoven, String Quartet op. 131, iv, mm. 5–9.

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Figure 3: Linear intervallic pattern in Beethoven, String Quartet op. 131, iv, mm. 6–8.

The sequence originates in mm. 6–8 of the theme (see Example 8), where it again begins as a nondiatonic sequence until the fourth harmony, which introduces a D♯ in the viola (see Figure 3). In both the theme and the fourth variation, this D♯ moves to a G♯ harmonized as E major, thus ending the progression of fifths. The theme, however, disguises the projected course of the sequence, misconstruing the relation between the final two chords and the sequence that precedes them by misaligning the melodic and harmonic sequences. The point of imitation between the violins implies that the falling third from B to G♯ in the second violin is a sequential repetition of the first violin’s fall from E to C♯ and, by extension, of the slightly more embellished falling third F♯ to D♯ earlier in the second violin. It might seem that the sequence has skipped two steps, but the viola’s D♯ suggests otherwise. It is this note, in fact, and not the second violin’s imitative figure, that continues the 6–3 linear intervallic pattern, but the expected B is absent on the downbeat of m. 8.

The turn back to the repeat of the phrase seeks to iron out this wrinkle: the first violin takes up the point of imitation again, beginning from the top with a descent from F♯ to D♯, but in order to effect the return to the tonic, the second violin counters with a falling dyad (D♮ to B). This tiny detail “corrects” two “errors” in the measures beforehand that disrupt the sequence’s local synchrony: the B completes the foreshortened melodic sequence, while the D♮, the real continuation of the harmonic sequence, contradicts the erroneous D♯. In the 7–3 pattern in the fourth variation, real and diatonic components of the sequence coincide. Whereas the continuation of the melodic sequence is already distorted into a D♯, the penultimate step of the sequence in m. 137 is both synchronous and diachronous, insofar as the C♮ allows one to hear the continuation of the pattern, while the D♯ subverts it. Further, the doubled third in the final E-major chord with a doubled third hints that it should have been a chord on G♯.

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Figure 4: Linear intervallic pattern in Beethoven, String Quartet op. 131, iv, mm. 225–57.

The unexpected intrusion in this final variation dramatizes the indissoluble residue that inheres in the tonal sequence by enabling the sequence finally to coincide with itself through an exact transpositional schema only in a moment of seemingly absolute diachrony. The final variation as it were “corrects” the theme by providing a model version of the real sequence at whose possibility the earlier incarnations only hint (see Figure 4). This final variation adopts the 7–3 linear intervallic pattern from the fourth variation, but where one now expects a D♯ at the end of m. 226, the cello moves down a perfect fifth from A to D. But this is only the first inkling of what is to come. The C♮ that is part of the real sequence does not come immediately, but is approached via a pair of lower and upper neighbor notes, the rise to C♯ surely playing with expectations about the continuation. The bass persists with the continuation of the real sequence, rising to a G with the return of the original thematic material. The C♮ is reinterpreted as a fourth above the bass and then resolves to B to form the third of V in C major. The return of the theme thus completes the hitherto unfulfilled step of the sequence.

The intrusion that breaks through in m. 250 then advances the real sequence by another step. This time the C♮, transposed up an octave, is not reharmonized when the theme returns, but continues to be supported by F major. When it does fall, it moves to a Bb, picking up where the previous descending chromatic line left off in m. 232. This in turn steps down to A to complete the descent. The F in the bass thus reveals itself as a continuation of the descending-fifth progression passing through the D in m. 226, the G in m. 231, and the C in m. 235. In this way the sequence finally becomes real, seemingly purifying itself of the diachronous residue that had haunted its previous occurrences, precisely at the moment at which it abandons the synchrony of variation procedure, introducing a higher-order formal diachrony.

Sequence as Disruption

The realization of the sequence in op. 131 exemplifies in nuce the disruptive potential of sequences, especially real ones. In other cases, the disruption is contained as a local interruption of the form, but has much more wide-ranging ramifications for the overall form and for the attainment of global tonal goals. The theoretical tradition has variously conceptualized the sequence as an agent of tonal and formal disturbance. 26 I first look in more detail at those nineteenth-century harmonic theories that see the sequence as an obstacle to tonal articulation and its theorization. Unlike Rameau, for whom the sequence is prototypical of tonal harmony in general, both Fétis and Riemann find that the sequence sits at odds with their models of harmonic progression and tonality. I then consider how the sequence’s association with harmonic instability begets the idea that this type of progression fulfills its proper function within certain looser parts of formal constructions, both within the phrase and as larger-scale agents of formal expansion.

For Fétis, the sequence’s repetition and sameness lead to a temporary loss of the sense of tonality: “The mind suspends any idea of tonality and conclusion until the final cadence, so that the degrees of the scale lose their tonal character, the ear being preoccupied only with the similarity of movement.”27 In contrast with Rameau’s theory of implied or supplementary dissonances, which give the sequence’s harmonic progression its forward momentum, 28 Fétis maintains that not all triads can be thought of as unstable and in need of resolution. This is because Fétis argues for a correlation between notes of the scale and the kinds of chords that may be built upon each scale degree. If the scale itself is hierarchically structured around the pair of tendency tones 7^ (resolving to 8^) and 4^ (resolving to 3^), the chords built upon each degree must reflect the varying degrees of stability of the scale, such that only triads may be constructed on scale degrees 1^, 4^, 5^ and 6^, while 2^, 3^, and 7^ support inversions only.

Sequences threaten this hierarchy of stability by putting triads over every degree of the scale and in this way forget the way in which the law of tonalité, grounded in the scale’s relations of attraction, depends on differentiation. The suspension of tonality thus comes about precisely through an inattention to differences between degrees of the scale, which are instead supplanted by a notion of pure identity. Fétis in this way objects to the idea that every step in the sequence is a local temporary tonic. Sequences undermine the set of relations between scale degrees through which Fétis conceptualizes the system of tonality, because they are too symmetrical.

Riemann’s analysis of the sequence, discussed above, also questions whether the sequence is not, in fact, better explained as a melodic rather than harmonic formation.29 He argues that the sequence as a whole has no functional-harmonic significance, but only makes sense if we hear each pair of simultaneities as an autonomous fragment with a caesura between each fragment. In much the same way that the mind of Fétis’s listener is absorbed in the similarity of the movement, it is only the process of repetition that forges a connection between these pairs of harmonies and that gives the sequence a measure of coherence.

More recently, espousing a mixture of Stufen- and Funkionstheorie, William Caplin’s theory of formal functions in late eighteenth-century music retains the idea that the sequence is a locus, if not of outright tonal suspension, then of heightened harmonic instability. Whereas Riemann sees melodic process as taking priority, Caplin argues that “although some sequential progressions exhibit a degree of harmonic functionality among their constituent chords, this aspect of the progression is secondary to the fundamental purposes they are meant to serve.”30 While this is mainly for the purpose of modulation, sequences are also “especially suitable for destabilizing harmonic activity in a given key.” Caplin then categorizes sequences not only according to their underlying linear intervallic pattern, but also by the degree of harmonic functionality they express; even the descending-fifth progression, in which this functionality is most prominent, “nevertheless promotes a weakening of the harmonic-tonal environment.”

This observation leads Caplin to develop a tripartite model that maps harmonic progressions onto formal functions: if cadential progressions form natural endings and prolongations tend to open up beginnings, model-sequence technique is most closely associated with middle-type functions and specifically with the continuation phrase of a sentence, which works in combination with other destabilizing effects such as fragmentation of structural units and increased rhythmic activity. In a broader view, sequential repetition finds its proper place in those thematic constructions and formal regions that Caplin defines as “loose” as opposed to “tight-knit.” Loose insofar as “the individual links in the sequential chain are harmonically nonfunctional,” the sequence sits alongside other modulatory processes, asymmetrical grouping structures, motivic diversity, and unconventional formal types as a means of destabilizing formal organization. Hence, looser constructions and sequences in particular are associated with transitions, developments, and, within the second half of a sonata exposition, expansion strategies aimed at postponing cadential closure in the new key.

The capacity for sequential repetition to disrupt the attainment of cadential goals in sonata form is exemplified in a remarkable fashion in Beethoven’s Grosse Fuge op. 133, originally composed as a finale for the String Quartet op. 130. After an introduction in which a forceful opening statement of the fugue subject is followed by a snippet of a contrasting piano Meno mosso, a lengthy opening fugue in the tonic of B♭ gets underway before collapsing into an expansive version of the Meno mosso material in G♭ to give a brief period of calm. Roughly midway through an ensuing Allegro molto e con brio in <6/8> that fragments the fugue subject, the extraordinary descending-fifth sequence begins. Taking the tail of the subject as its motivic material, the sequence initially passes its model imitatively between the two violins before descending into the lower parts. The exchange produces a series of half-step dyads aligned with the descending-fifth progression so that the focal pitches form a 10–10 linear intervallic pattern with the bass. The repetition of the melodic patterning at four-measure intervals gradually dissolves into fragments of the fugue subject, scattered across a drawn-out extension of the harmonic sequence.

SequenceClick to view larger

Figure 5: Reduction of Beethoven, Grosse Fuge op. 133, mm. 325–453.

What is most striking, however, is how this sequence takes the logic of repetition and sameness to extreme lengths, with the final result that it produces a tonal rupture in complete violation of normative formal expectations. The sequence holds to an exact transpositional schema throughout its dizzying six cycles of repetition, plummeting flatward through the circle of fifths without any tonal correction, and the real harmonic sequence even extends beyond where the melodic pattern of repetition dissolves, until it reaches what is technically B♭♭♭ (see Figure 5). The enharmonic reinterpretation from m. 331, staggered across the four parts, seems like a matter of purely notational expediency rather than a decisive shift across this harmonic seam. The result of this extended sequence, then, is that in holding to an exact (real) transpositional schema, the closing tonic is technically not B♭, but C♭♭: from a harmonic-functional perspective, not a tonic, but the end of a lengthy chain of nested subdominant functions. As David Lewin has demonstrated elegantly, such harmonic sequences illuminate the conflict between Stufen and Riemannian space and show how the sequence assumes an important role in mediating between these two theoretical constructions of tonality.31

This outcome is all the more remarkable when one considers that the fugue is, if not exactly a sonata structure itself, a rewriting of the first movement’s own peculiar sonata deformation. The extended sequence is a retracing of a similar descending-fifth progression in the first movement’s development, confirming the association of model-sequence technique with loosening impulses. That the fugue repeats the unusual choice of ♭VI for a second contrasting thematic area cements the connection to the first movement; any sonata background structure that may be implied in the fugue is filtered through this relation. In the first movement, however, the enharmonic reinterpretation takes place before, rather than midway through, the sequence at the juncture between exposition and development and is given greater rhetorical weight, taking place across a gap of silence. The enharmonic shift here also seems less like a matter of notational convenience, because it straddles an important thematic contrast from the movement’s opening, separating material from the Adagio introduction from the motivic content of the main Allegro body of the exposition and thereby suggesting a repositioning from outside the frame to inside. Unlike in the fugue, the real sequence descends from D major down a whole step to C and then a further whole step to B♭ and thus provides a more convincing return to the tonic.

In any case, the first movement’s sequence is normalizing, correcting an unexpected sharpward turn, rather than disruptive, because the unsettling sleight of hand of enharmonic equivalence lies outside its scope. The effect of the fugue’s sequence, though, is to destabilize the large-scale tonal resolutions that articulate the form. The first movement’s recapitulation transposes the second group down a fifth to D♭ major, thereby maintaining the typical transpositional relation if not the strict tonic reprise. In the fugue, by contrast, the sequence yields a reprise of its Meno mosso in A♭ (notated) or B♭♭♭ (by diachronic listening), eschewing the usual mechanics of recapitulatory transposition while also aurally projecting a type of 1^ precisely at the moment when the tonic is expected. In this way, Beethoven’s sequence forms a quasi-essay on the sequence’s capacity to drive wedges between theoretical systems and even to unsettle their internal construction, but also at the same time to provide a passage or a means of transitioning between one mode of tonal hearing to another. This example also demonstrates how the sequence may precipitate formal destabilization: the fugue’s sequence is instrumental in derailing a crucial juncture in the articulation of sonata form, not only displacing tonal expectations for the reprise of the second group, but also suppressing the return of first-group material by allowing the overgrown sequence to extend beyond and blur the essential structural boundary between development and recapitulation.

This example also suggests another facet to the sequence’s disruptive potential. If the theorists of the late eighteenth century found sequential repetition stylistically outdated, this reflects the possibility that sequences may generate interstylistic conflict. In its fusion of fugal and sonata processes, the Grosse Fuge presents a way of dramatizing the tension between baroque Fortspinnung and the classical style’s rhythm-punctuation model. While the baroque aesthetic is premised on the interminable forward momentum of the cycle of fifths and relegates cadences to the status of momentary deflections, the classical style elevates the cadence into the governing form of harmonic progression and the primary determinant for the comprehension of form. In the latter style, the music is divided by a series of endpoints of varying degrees of closure into a gridlike structure; the idea of periodicity allows the hierarchy of cadences to project the local metrical patterns onto increasingly higher hierarchical levels, such that a global cadence may subsume an entire span of music under its concept. This idea is central to sonata form, whose structure is generated through this projection of closing function from local to global. It is also this mechanism that permits a certain predictability, insofar as the listener is able to form a synoptic view of the whole and thereby to foresee what might happen next.

Beethoven’s Grosse Fuge stages a confrontation between these two stylistic worlds, attempting to contain within the boundaries of sonata form the untrammeled momentum of the baroque fugue with its abundance of sequential processes. Just as Johann Georg Sulzer observed a shift in rhetoric from the old-fashioned, list-type construction of parataxis to the contemporary hypotactic practice that groups clauses together under a conceptual unity, music undergoes a similar change. Baroque Fortspinnung, exemplified above all by sequential repetition, exhibits the successive quality of paratactic construction, while the classical sonata form, with its strongly articulated formal divisions and cadential goals, typifies the synoptic character of hypotaxis. This contrast replicates and separates out the two strands of the sequence’s double traversal: the classical style tends toward collapsing local difference into global identity through its hierarchy of rhythmic grouping and punctuation, while the baroque Fortspinnung works by producing differences out of identical or similar musical materials, be they motivic units, linear patterns, or harmonic progressions. The interstylistic tension thus plays out the double movement toward synchrony and diachrony inherent in the sequence. At the same time, through this stylistic transformation from baroque to classical, the status of the sequence also shifts from being prototypical of tonal operations in general to becoming a disruptive exception to tonal and formal norms. Far from being an inert mechanism at risk of inducing boredom, it is in fact the sequence’s repetitive character that enables it to become an agent of stylistic change and a driving force in the transformation of tonal processes and their theorization across the common practice era.

Bibliography

Aldwell, Edward, and Carl Schachter, with Allen Cadwallader. Harmony and Voice Leading. New York: Schirmer/Cengage Learning, 2010.Find this resource:

Bass, Richard. “From Gretchen to Tristan: The Changing Role of Harmonic Sequences in the Nineteenth Century.” 19th-Century Music 19, no. 3 (1996): 263–85.Find this resource:

Berger, Karol. Bach’s Cycle, Mozart’s Arrow: An Essay on the Origins of Musical Modernity. Berkeley: University of California Press, 2008.Find this resource:

Caplin, William. Classical Form: A Theory of Formal Functions for the Instrumental Music of Haydn, Mozart, and Beethoven. Oxford: Oxford University Press, 1998.Find this resource:

Fétis, Joseph-François. Complete Treatise on the Theory and Practice of Harmony. Translated by Peter M. Lanley. Hillsdale, NY: Pendragon, 2008.Find this resource:

Harrison, Daniel. “Rosalia, Aloysius, and Arcangelo: A Genealogy of the Sequence.” Journal of Music Theory 47, no. 2 (2003): 225–72.Find this resource:

Hook, Julian. “Generic Sequences and the Generic Tonnetz.” In Oxford Handbooks Online (October 2014). doi: 10.1093/oxfordhb/9780199935321.013.003.Find this resource:

Hyer, Brian. “‘Sighing Branches’: Prosopopoeia in Rameau’s Pigmalion.” Music Analysis 13, no. 1 (1994): 7–50.Find this resource:

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Notes:

(1) This is the premise of Jairo Alberto Moreno Rojas’s dissertation, “Theoretical Reception of the Sequence and Its Conceptual Implications” (Yale University, 1996), which uses shifting conceptions of the sequence to parse the theoretical traditions into a series of broad paradigms. Within this framework, Moreno nonetheless notes a considerable degree of cross-fertilization between and even fusion of paradigms that threaten any paradigm’s claim to self-identity.

(2) Joseph-François Fétis, Complete Treatise on the Theory and Practice of Harmony, trans. Peter M. Lanley (Hillsdale, NY: Pendragon, 2008), 27.

(3) Hugo Riemann, Harmony Simplified or the Theory of the Harmonic Functions of Chords, trans. H. Bewerunge (London: Augener, 1896), 122.

(4) Heinrich Christian Koch, Introductory Essay in Composition, trans. Nancy Baker (New Haven, CT: Yale University Press, 1983), 45.

(5) Moreno, “Challenging Views of Sequential Repetition: From ‘Satzlehre’ to ‘Melodielehre,’” Journal of Music Theory 44, no. 1 (2000): 127–28.

(6) Christian Friedrich Daniel Schubart, “Von der Rosalien,” in Vermischte Schriften (Zurich, 1812), 1:220–26.

(7) Charles Burney, The Present State of Music in Germany, the Netherlands, and the United Provinces (London, 1775), 2:329; Thomas Mann, Doctor Faustus, trans. H. T. Lowe-Porter (New York: Vintage Books, 1948), 60. Anton Schindler reports Beethoven’s words in Elliot Forbes, ed. Thayer’s Life of Beethoven (Princeton, NJ: Princeton University Press, 1967), 856n.

(9) William Caplin, Classical Form: A Theory of Formal Functions for the Instrumental Music of Haydn, Mozart, and Beethoven (Oxford: Oxford University Press, 1998), 77.

(11) Examples include Richard Bass, “From Gretchen to Tristan: The Changing Role of Harmonic Sequences in the Nineteenth Century,” 19th-Century Music, 19, no. 3 (1996): 263–85; Daniel Harrison, “Rosalia, Aloysius, and Arcangelo: A Genealogy of the Sequence,” Journal of Music Theory 47, no. 2 (2003): 225–72; and Edward Aldwell and Carl Schachter with Allen Cadwallader, Harmony and Voice Leading (New York: Schirmer/Cengage Learning, 2010).

(12) See, for example, Arnold Schoenberg, Theory of Harmony, trans. Roy E. Carter (Berkeley: University of California Press, 1978), 283.

(14) See Allen Forte and Steven Gilbert, Introduction to Schenkerian Analysis (New York: Norton, 1982), 83.

(20) An alternative approach, which instead emphasizes the common origin of real and diatonic sequences in patterns in generic pitch space (i.e., indifferent to exact interval sizes and qualities), is found in Julian Hook and Adam Ricci’s use of a combination of diatonic set theory and transformation theory (Hook, “Generic Sequences and the Generic Tonnetz,” Oxford Handbooks Online [October 2014] and Ricci, “A Theory of the Harmonic Sequence” [PhD diss., University of Rochester, 2004]).

(21) Charles Rosen, The Classical Style: Haydn, Mozart, Beethoven (London: Faber and Faber, 1997), 445–46.

(23) For a discussion of this twofold temporal character, see Karol Berger, Bach’s Cycle, Mozart’s Arrow: An Essay on the Origins of Musical Modernity (Berkeley: University of California Press, 2008), 10.

(24) Jean-Philippe Rameau, Treatise on Harmony, trans. Philip Gossett (New York: Dover, 1971), 288.

(25) Moreno, Musical Representations, Subjects, and Objects: The Construction of Musical Thought in Zarlino, Descartes, Rameau, and Weber (Bloomington: Indiana University Press, 2004), 118.

(26) Two recent articles on sequences have focused on this disruptive possibility: Bass, “From Gretchen to Tristan,” which looks at the tension between real sequences and their tonal context, and Adam Ricci, “Non-Coinciding Sequences,” Music Theory Spectrum 33, no. 2 (2011): 124–45, which examines simultaneous melodic sequences with different intervals of transposition.

(28) The translation of sous-entendu as “implied,” with the psychological intentionality this suggests, is contentious and arguably reflects an anachronistic preoccupation of more reflective Anglophone theory. The nuance of the French is perhaps better captured in the idea of “hearing-as-understanding.” Rameau also describes this dissonances as ajoutées, whence Brian Hyer’s bid to capture the deconstructive impulse here with his notion of “supplementary dissonances” in “‘Sighing Branches:’ Prosopopoeia in Rameau’s Pigmalion,” Music Analysis 13, no. 1 (1994): 7–50.

(30) Caplin, Classical Form, 29–30. Caplin explicitly rejects a melodic definition of sequence, insisting that the presence of a harmonic sequential progression is essential.

(31) David Lewin, “Amfortas’s Prayer to Titurel and the Role of D in Parsifal: The Tonal Spaces of the Drama and the Enharmonic C<flat>/B,” 19th-Century Music 7, no. 3 (1984): 336–49.