A Source of Richness in the Main Themes of Samuel Barber's Violin Concerto and Piano Concerto: Hierarchical Conflict and the I

Figure 1: Interaction of Learning and Cognitive Constants

Three cognitive constants affect our perception of the three following parameters: pitch, intervallic motion, and registral direction.² The constants are applied only to one parameter at a time. The first of the three is one of similarity based on Gestalt laws of common direction, similarity, and proximity. Thus, if A + A, then a further A is implied, which means that sameness or similarity (A +A) causes the subconscious expectation of more sameness or similarity (A), all things being equal. The second constant is one of differentiation, a hypothetical construct symmetrical to the Gestalt rules of similarity. Thus, if A + B, then C is implied, which means that differentiation (A + B) causes the expectation of more differentiation (C), all things being equal.³

The definitions of the first constants, (A + A → A and A + B → C), raise two important questions. First, how does the analyst define "similarity" and "differentiation" in musical items. And second, how does the analyst evaluate the meaning of such distinctions when the data are necessarily relative? The solution to these questions lies within the third theoretical constant, the syntactic parametric scale, which defines the relative terms "similarity" and "differentiation." The syntactic parametric scale hypothesizes that intervals from the unison through the P4 are small intervals that imply continuation, or similarity, of intervallic size (I) and registral direction (V) (up, down, or lateral). Intervals from the P5 through the M7 are larger intervals that imply reversal or differentiation, of intervallic size and registral direction (i.e. a small interval and a change of registral direction is expected after a large interval) (Figures 2 and 3). Given the relative nature of intervallic size, the analyst must consider that an interval's degree of implication is directly proportional to its size.

Figure 2: Syntactic Parametric Scale

Five melodic archetypes derived from the theoretical constants form the basis of the system of melodic structures (Figure 4). The first two archetypes, process [P] and reversal [R] are direct analogues of the first two constants (similarity and differentiation, respectively.) A process [P] is a melodic structure with at least three pitches that have relations of similarity of intervallic size and registral direction between all the pitches. A reversal [R] is a melodic structure with three pitches that have relations of differentiation of intervallic size am:1 registral direction.

Figure 4: Synthetic example of five melodic archetypes derived from the theoretical constants

The third melodic archetype is the dyad, a two-pitch structure symbolized by the arabic numeral that corresponds to the size of the interval. This structure occurs when closure on the second pitch denies further implication, due to metric emphasis, harmonic resolution of dissonance, directional cumulation or some combination of these.

The fourth melodic archetype is the monad [M], a single-pitch structure that has neither implication nor realization since implication is created by two pitches and realization by three pitches. Monads are isolated pitches usually because they are followed by a silence (i.e. the last pitch of a piece.)

The fifth and final melodic archetype is exact or near registral return [a b a; a b a¹]. These symbols indicate instances where a melody departs from a previously established pitch [a], then returns, either to the same pitch [a] or one very close by [a¹]. This archetype does not involve implication since the pitches so labeled need not be successive.

The implication-realization model also proposes six melodic structures derived from the archetypes (Figures 5a + b). Duplications [ID] are derived from processes and consist of three or more repeated notes, where the implicative interval is the unison and the registral direction lateral. Sometimes only one parameter, interval or register is realized while the other is denied. For example, when the implicative interval implies continuation but the realization denies the expected intervallic size or registral direction, the resulting structures are called registral process [VP] and intervallic process [IP]. Just as duplications [D] are derived from processes [P], intervallic duplications [ID] are derived from intervallic processes [IP]. Intervallic duplication refers to the case where the intervallic differentiation is nil and registral implication is denied (i.e. the first and third tones of the structure are the same pitch, and the second is a different pitch.)

Figure 5a: A list of archetypal melodic structures and their derivatives

Figure 5b: Synthetic examples of the melodic structures derived from the archetypes

When the implicative interval implies differentiation but the realization denies registral direction or intervallic motion, the resulting structures are called intervallic reversal [IR] and registral reversal [VR] respectively. Thus far, each of the implicative archetypes and their derivatives have been discussed in their prospective form. A structure is said to be prospective, for instance, when the implicative interval is small and the terminal pitch reaffirms the resulting structure to be of a the implicative interval is small but the final structure turns out to be of the reversal type (and vice versa). These structures are said to be retrospective, since the listener comprehends their type only in retrospect upon hearing the terminal pitch of the structure (Figure 6). Note that retrospective structures retain the same pattern of intervallic motion and registral direction as the prospective structures (A + A or A + B) yet often have a different set of denials and realizations of the implicative interval, thus creating a different cognitive effect. Retrospective structures are symbolized by the use of parentheses around the particular structural symbol [(P), (R), (VR).]

Rhythm and meter play a major role in the segmentation or parsing of melody into structures, since an interval in a melody implies metric and rhythmic placement as well as intervallic motion and registral direction. Factors such as harmonic resolution of dissonance, placement on a weak beat, and a counter-cumulative rhythmic position can interfere with the closure of a structure and cause it to combine or chain with other structures (Figure 8). A combinational structure is formed by two dovetailed structures which share the middle interval (in other words, the realization interval of the first structure is the implicative interval of the second). A chain is simply a structure containing three or more such dovetailed structures.

Figure 8: Synthetic examples of harmonic resolution of dissonance (x), placement on a weak
beat, and counter-cumulative (long-short) rhythmic position causing compound structures

Finally, the analyst must address the issue of transformation, the creation of hierarchical levels (conceptually comparable to the foreground, middle-ground, and background of a Schenkerian analylsis.) The initial tone and the terminal tone in a structure are those that transform to the next level and form their own implication and realization on the next level. Registral and intervallic activity on one level is in some sense embodied in the next higher level. These interactions of level contribute to the richness and subtlety of melodic cognition.

We began this discussion of the implication-realization model, with the hypothesis that our cognitive constants interact with prior learning when we listen to music. Hopefully the relevance of this hypothesis to musical analysis has become clear to the reader. It forces the analyst to clearly separate expectations created from style knowledge from basic human pattern processing and further to provide empirical evidence supporting any claim of style impingement.

Now let us turn to the specific issues involved with the application of the model to the melodies of Samuel Barber's piano and violin concerti.

Introduction

The melodies analyzed in this critical study are presented in order of increasing complexity and aesthetic affect.⁴ In this paper, complexity corresponds to the degree of cognitive conflict between embodiment of suppressed implication from the low level and larger-scale patterning on the high level. For instance, I find that the opening theme of the first movement of Samuel Barber's Piano Concerto is less complex than that of the first movement of his Violin Concerto because the latter displays greater conflict between high and low-level patterns. I will argue that it is this hierarchical conflict (i.e. level interference) that contributes to melodic richness.This is not to suggest that the melody of the Piano Concerto is neither complex nor rich. Indeed, it is both. However, its complexity and richness are found more in the relationships between adjacent pitches within a phrase than in high-level patterning between adjacent phrases, such as in the opening theme of the Violin Concerto.

The main themes from the slow movements of these works will serve as intermediary stages in the discussion of melodic complexity and aesthetic affect. In both, clearly separated phrases develop subtle motivic variations that create relatively simple conflicts between high and low levels. Melodic dissonance in the slow movement of the Piano Concerto creates a richness and complexity greater than that found in the slow movement of the Violin Concerto. In the most complex melody to be discussed in this study, that of the first movement of the Violin Concerto, all structural hierarchies engage in an elaborate intra- and inter-level dialogue, interlocking with one another to seam an exquisite eleven-bar phrase.

Piano Concerto, First Movement, measures 19 -27.

Over the course of this melody, hierarchical levels diminish in number and complexity (Example 1). The initial angular leaps, contramodal dissonances on stressed beats, and the unusual syncopation of the structural tones create intense excitement in the first four bars (mm. 19-22). The second part of the melody (mm. 23-24) is calmer because the melodic structures are simpler. Measures 25-27, the third section of the melody, are nearly "flat" (i.e. they have very few transformational tones and create only two levels).

The main motive (a descending arpeggio) fights the meter with syncopation, thus shifting the transformational tones to weak beats (Example 2). Note the fortissimo entrance of the violins off the beat in measure 19, the metrically enveloping process [P]⁵, the sudden reversal [R] on the downbeat of measure 22 (see middle level), and the resolutions of the dyads [7] off the beat in measures 20 and 22 (see again Example 1).

Example 2: Middle-level analysis

This phrase syncopates on a lower rhythmic level as well. Note the [bl a b² a] that groups the falling arpeggios into three eighths. This conflict between motive and meter and the ensuing deformation of the congruence between structural tones and meter are aesthetically pleasing for their unpredictability and freshness. The motive/meter conflict continues through measures 21 and 22, disappears in the second part of the melody mm. 23-24), then reappears briefly in the third and final part (mm. 25-27). In this respect, and in many others to be discussed below, the three parts are dissimilar.

This comparison illustrates the compositional plan of the theme: the gradual abatement of complexity. Both low and high level structures and registral differentiation underscore the disparity between the three parts of this melody. Note that the first part consists of a four-bar phrase (or two two-bar subphrases connected by an underlapping a¹b a¹ ..., the second part is a two-bar phrase united by an upper-level process [P]; the final part contains three short one-bar phrases unconnected on the high level. Thus the phrases grow progressively shorter and more independent.

On the low level, Barber's strategy in handling complexity is different. The opening string (mm. 19-20), P-RP(VR)-7, indicates that intricacy lies in the middle of the phrase. The simplicity of the opening process [P] and the closing dyad [7] is integral. These structures facilitate the motion into and out of the chain [RP(VR)], by initiating and terminating the underlapping a b a b^l a b² structure. In fact, the high B is also the initial and terminal note of the high-level I. ID (another a b a, see again Example 2), join the three lower-level structures on a single higher level. Thus the B that initiates the opening process and the B that initiates the closing dyad anchor the first phrase.

The same pitch, B, becomes the point of departure for the second phrase. As in the first phrase, this structure - [embedded VP-P]-7 - is simple in the beginning and end, and complex (i.e. multi-leveled) in the middle. The dyad [7] is marked with (i, os) because: 1) the B is the resolution of the dissonant A# , 2) it duplicates the structure Barber uses to close the previous phrase, and 3) it is the completion of an underlapping process that spans measures 19-22 (D# D C# C B).⁶

The subtle difference between the descending resolutions of the dyads [7] illustrates the aesthetically affective conflict between embodied yet suppressed low-level implication and larger-scale patterning on the high level. The dyad [7] in measure 20 is stylistically prospective, thus of little surprise, because of the a b a ... patterning which "teaches" the listener to expect descent from the B. In contrast, the dyad [7] in measure 22 is both prospective and retrospective. As the terminal pitch in a registral process [VP] initiated by D# and A# on the highest level, the B is prospective in terms of registral direction and partially prospective in terms of intervallic size. The denial of interval - large instead of small - is made partially prospective by the underlapping process from measure 19 that implies the specific pitch, B, as well as its register.

The conflicting retrospective nature of the downward resolution of the dyad in measure 22 arises because of low-level interference with the high level. The ascending process on the quarter-note level, C F A, in measure 22 embodies ascent and implies a small interval, specifically, a minor second up to B, because of the harmony. This low-level implication, the ascent to B, persuasively fights those of the high level, descent to B, caused by the above mentioned registral process [VP] on the high level and the underlapping a b a... both in measures 19-23 and in measures 21-22. Thus the registral process [VP] is marked with a symbol (e) that indicates the conflicting presence of low-level embodiment. Had the A# in measure 22 resolved upward to B, the transformational tones, D , A, B, would have formed a reversal [R] with conflicting prospective (low level) and retrospective (high level) qualities.⁷

The second part of this melody (mm. 23-24), while simpler than the first in many ways, also displays hierarchical conflict.⁸ In measures 23-4, the low level interferes with the middle level causing the middle-level registral reversals [VR] to seem prospective, rather than retrospective. The ascent embodied in the low-level process [P, C# E# B, in measures 2- 4 makes the first of the two middle-level registral reversals [VR], G C#^jB, prospective. The registral and intervallic differentiation (A + B) in the middle-level structures [VR] come as no surprise because of this low-level interference. Also, the initial interval of the registral reversal [VR] is fairly large, a diminished fifth. As it is a threshold interval, in between being small or large, it is not surprising that it reverses direction (Example 3).

Example 3: Low-level interference

In contrast, the initial interval of the second middle-level registral reversal [VR] (m. 24), B G D, is smaller, a major third; thus the registral reversal is less expected, all things being equal. However, all things are not equal here. For the ascent embodied in the low-level process [P], E G B D, of the IDP combination influences the upper level. In effect, the descent implied in beats one and two on the middle level is challenged and overthrown by the low-level ascending process in the second half of beat two. The low level makes the change of direction (down - up) and the intervallic differentiation (small-large) in the middle level [VR] prospective. Also, the first two notes of the high level process [P, G B D, imply similarity of interval (small-small) and register (up-up), enhancing the prospective nature of the registral reversal [VR] on the second level at the arrival of the D in measure 24. Thus the interference from both high and low levels virtually drains the aesthetic power of surprise from these usually highly implicative and open structures.

A cursory glance at the structural symbols of the third part of this melody (mm. 25-7) will reveal its simplicity relative to the two preceeding parts. Note that there are fewer levels, predominantly small (less implicative) intervals, repeated intervals and pitches, and shorter, less dynamic phrases. We see, then, that Barber's basic compositional plan is one of complexity moving toward simplicity. Let us now focus our attention on the subtler hierarchical conflicts that arise from relatively simple melodies constructed of fewer and smaller intervals.

Violin Concerto (1940), Second Movement, measures 3-14

The conformance of the intervals, rhythms, harmonies, and phrasing in this melody create an easily recognizable coherence. Example 4) Unlike the raw energy of the fragmented main theme of the first movement of the Piano Concerto, this melody gathers its aesthetic power from motivic variation. The lower level consists primarily of four-note PIP structures (eight of the sixteen structures are PIP) in which the IP, with one exception, always moves from larger intervals to smaller intervals, or motion-left. Thus, the most common structure of this melody is closural, and not strongly implicative.

That such simplicity and conformity yield an interesting melody is remarkable. The subtle alterations these PIP structures give this melody its distinct, attractive profile. The richness of this motivic variation is manifest in the hierarchical conflicts between the low and high level, as well as in the changes in low-level structures from those implied to those that are unexpected.

Barber's subtle variation technique is best displayed in his handling of the two dyads [7] which are placed in the middle of the first and third phrases (mm. 3-4 and 8-9). Although the G# in measure 3 is the consonant tone of the two that create the dyad [2], the mildly dissonant [(x)] A transforms to the next level because it has a cumulation of 1:3 over the G . This affective G# A dyad is implicative, marked with (i), because the listener expects the beautifully suspended A, an upper neighbor note, to fall to the G# once it becomes clear that it is not going to ascend to B, (i.e. after two beats of the A).

The dyad in measures 8-9 is analogous, motivically, but the dissonance and consonance are reversed. Here, the initial note of the dyad, B, is dissonant and articulative, while the second note, C#, is consonant and transformational. This dyad is also implicative. The listener expects the CO to fall to B because of intraopus style [(os)] (i.e. since the tone following the dyad [2] in m. 4, G# tt, descends from A, the listener expects the same treatment here.) However, the dyad [2] in measures 8-9, B C#, does not immediately imply a downward resolution, as did the dyad in measure 4, because the C# is initially consonant with the harmony. Thus there is a conflict between intraopus style (implied descent) and bottom-up constants (implied ascent). However, with the harmonic change in measure 9, the C# becomes a suspension and the listener expects it to resolve downward, 6-5, to B. Note that the harmonic rhythm in the second half of this melody is faster than that of measures 3-7. The harmonic change on the downbeat of measure 9 gives the C new meaning. Had the harmony been repeated over the bar line, as it was in measure 4, the consonant C# would have seemed lifeless.⁹

The PIPs that terminate the third phrase and initiate the fourth phrase recall tleh motivic link between the first and second phrases. However, the two PIPstructures are, in both cases, only superficially alike. In the latter case, the second PIP (mm. 10-11) begins with a long note (four beats) and moves, at first, toward closure, motion-left (B GO Fti) like the PIP in measures 9-10. Then, however, it becomes more open, moving motion-right (G# F# A).

The only chain [R(R)IP] of this melody follows (mm. 11-12).¹⁰ On the low-level, the largely stepwise, intervallically conformant melody gradually shifts from small intervals to a mixture of small and large intervals. In the stylistic context of this melody, fourths and fifths are large intervals; seconds and thirds are small. Thus, Barber saves the larger intervals, which initiate the fresh, contrasting reversal structures, for the final phrases of the melody, measures 11-14. This, too, reveals Barber's strategy of altering the patterns set up over the course of the melody. Also note that in measure 12 we hear the only articulative retardation on the downbeat of a measure. The F# is a particularly "good" note because all three previous phrases end on a consonance.¹¹

The R(R) portion of the chain (A D# F# F#, mm. 11-12) is closural as it moves motion-left, circling in on, and anticipating, the dissonant F#.The dissonance resolves in the final structure of the chain, IP (F# F# G#). Had the F# fallen to E, as the listener might expect it to, the phrase would have closed more decisively. The resolution up to G# is mildly surprising. It keeps the IP more open and implicative, and prepares the relatively large leap into the following reversal.

The final PID-(VR)(R) combinations over the subdominant-dominant progression (mm. 13-14) imitate a stock cadential patterrn. The high level influences the final reversal structure, making it seem partially retrospective. The structural tones in measure 13 are C# above the subdominant and D# above the dominant. Naturally, the listener expects E to follow over the tonic. This would create the analysis (VR) (IP). However Barber closes melodically on B, thus creating a reversal that is neither prospective nor retropective because the B is anticipated. The 'celli play the anticipated E an octave lower as they take the melodic line. Thus the phrases dovetail elegantly.

If the listener hears measures 10-14 as a single extended phrase, as I do, then the formal relationship of the four phrases should be identified as A (mm. 3-5) B (mm. 5-7) A¹ (mm. 8-10) C (mm. 10-14). This formal organization is readily apparent in the structures formed on the second level. Remember that the second level IPP (mm. 3-5) is very much like the RP (mm. 8-10) since the IP of the IPP moves motion-left closely resembling a reversal (Example 6).

Example 6: Second-level analysis

Also displayed in the high level analysis are the level interferences that subtly add to the aesthetic affect of this deceptively simple melody. One change of direction occurs in the high-level IPP structure of the first phrase. Relative to the first, the second structure, (R)VR, which changes direction twice, is more closural. Although the first change in registral direction is unexpected, (R)], the second, [VF1], is expected. Thus, the usually prospective reversal structure and the usually retrospective registral reversal structure are altered by low-level interference (Example 7). The reversal is influenced by the low-level intervallic process [IP] because the denial of register, B A C#, is unexpected. Note that this is the first motion-right structure in the melody. Thus, the implied high-level process, E B C#, reverses unexpectedly. The usually retrospective C# A G#, of the PIP on the low level implies descent. Such dynamic interaction between levels denies the listener the expected high-level structural tones while providing the comforting familiarity of the low level structures (i.e. the PIPs).

Example 7: Low-level interference

As I mentioned above, the high-level RP structure of the third phrase (Example 6) is very much like the IPP of the first. These similar structures reveal the formal similarities between the phrases while indicating that there is a fundamental difference between them. The difference stems from low-level variation. The ID half of the low-level PID combination (m. 9), denies registral direction by ascending and returning to B. On the high level this creates a reversal [R], G# C# B, rather than, say, an intervallic duplication, G # C# G#i or G# C# A, an intervallic process.

A look at the high-level symbols of measures 10-13, the first half of the final phrase, might indicate that this phrase is highly open. Yet the PVP, which spans a seventh, B C#, is largely the result of the necessary melodic return from the dominant scale degree to the closure of the final cadence. In fact, the closural structures on the lower level (i.e. reversals) conflict with the openness of the upper level structure, thus making this the most dynamic of the phrases, for this hierarchical conflict masks the approach of the ensuing quasi-stock cadential gesture, discussed above, that follows in measures 13-14.

One final observation, on the parameter of meter, supports the argument that the key to this melody's aesthetic effect lies in subtle variation. The ambiguous nature of the meter in the opening measures (i.e., the variation between six-four and three-two) brought out by the interplay between horns and first violins allows the melody to seemingly float, within a flexible meter, through the first measure. In fact, the opening ascent to A can be parsed in two ways, one in each meter. Though the violins eventually clarify the meter in measure 4, the same flexibility can be heard again in measure 8. These moments of metric ambiguity breathe life into this melody.

The subtle variation techniques that individuate the delicate phrases of the slow movement of the Violin Concerto are also found in the slow movement of the Piano Concerto. The two melodies are alike in that they both are deceptively simple, constructed of few motives and highly similar phrases. The latter is, however, somewhat more complex, because it is not rhythmically homogeneous, as is the former, nor do the phrases resemble each other as closely .

Piano Concerto, Second Movement, measures 1-9.

The immediately apparent formal repetition of phrases is, in the Piano Concerto as it was in the Violin Concerto, deceptively simple. This melody plays with the listener's expectations of formal repetition because Barber's strategy centers on the modification of intraopus style structures. Although high-level structures indicate the readily apparent similarities between phrases (e.g. measures 2 and 3, 4 and 8, 6 and 7, see Example 8), close inspection of the low levels of this melody supports the idea that such formal relationships between phrases can be reductive. The aesthetic pleasure derived from the subtle variations from phrase to phrase can be best understood from the low-level structures.

Each phrase borrows and/or alters some aspect of the phrase it follows. The phrases can be labeled as A-A¹-AB-B¹-1- BA. Each letter or pair of letters accounts for one of the six phrases. The meaning of these symbols will become clearer after a detailed discussion of each phrase. The second phrase (m. 3) is much like the first (m. 2), but the closing dyad is deformed to create a dependant relationship between the second and third phrases. The third phrase (m.4-5) is similar to the first and second in rhythm and the use of the dyadic beginning and ending. Its ending more closely resembles that of the first phrase because the dissonance resolves. As a result of the altered treatment of the opening dyad, the third phrase has an entirely new high-level structure, P. The fourth phrase (m. 5-6) grows out of the new melodic shape in the middle of the third phrase and maintains the use of dyadic beginnings and endings. The fifth phrase (m. 7) subtly varies the fourth. Finally, the sixth phrase (mm. 8-9) recalls the second and third, combining their materials into a new structure (Example 9.)

Example 9: Second-level analysis

The first phrase, 3-ID-ID - P-2, moves from simplicity, the dyad [3], to slight complexity, dyad [2] (see again Example 8). The opening dyad, E C#, is the germinal cell of this melody, as it is at first integral to the two embedded processes [P], and then initiates the low-level process (C# E G#). The G# transform. and becomes a structural dissonance because of its strong metric position and cumulation. It implies F a harmonically weak resolution. Thus the dyad is implicative and is marked with an (h). The G# is a particularly "good" note because intraopus style (os) and harmony make the F#, instead of the G , the expected note. The leap (E G#) to a cumulative moderate dissonance [x] on a strong beat is affective.

Note Barber's consistent use of dyadic structures to open and close the phrases of this melody. The opening dyads do not function, like many opening dyads, as anacruses. The treatment of the dissonance, consonance, duration, and transformational power of both opening and closing dyads changes throughout this melody. Such subtle variation in dissonance, consonance, and hierarchy in the repetition of these structures adds to the delicacy of this melody.

For example, although the symbols on the low level of the second phrase, 3-ID-ID-P-2, duplicate those of the first phrase, marked differences exist between the meanings of these symbols. Both opening and closing dyads of the second phrase (m. 3) are treated differently from those of the first phrase (m. 2) with regards to dissonance and transformation. In the case of the opening dyad [3] in measure 3 the initial note is a weak dissonance, [(x)]. In measure 2, both notes of the opening dyad [3] are chord tones. In the case of the closing dyad in measure 3, both notes are dissonant. The first is a moderate dissonancd [x], falling on the strong beat; the second is a strong contramodal dissonance [(x)]. Thus, the latter is articulative while the former is transformational. In contrast, only the first note of the closing dyad in measure 2 is dissonant.

The D# and the contramodal C in measure 3 are strongly implicative. They deny the closure implied by intraopus style learning in measure 2 where the dissonant [x] first note (G#) of the dyad [2] resolves to the second note (F#). The dissonances in measure 3 differ because they require further resolution. This conflict between melody and harmony is shown with the (h) above the dyad [2]. The expected resolution of the D# comes in the oboe's initial E (m. 4). Thus the variation in the treatment of the closing dyad in measure 3 joins measures 3 and 4 on the low level, shown by the horizontal dotted line. This linkage between phrases two and three (mm. 3-4) is unexepected because phrases one and two (mm. 2-3) are separate on the low level. The a b a¹ D# C E, and the steady crescendo through the end of the flute phrase also helps dovetail phrases two and three.

The third phrase offers yet another treatment of the dissonance, consonance and transformation of the pair of dyadic structures. At first, the new twist seems to be the metric displacement of the first dyad. But that is only part of the scheme. The interval of this dyad is a second [2], rather than a third [3], and in this case the first note (E) is consonant while the second (DO) is a weak dissonance [(x)], which yields the opposite placement of the dissonance in the opal... .g dyad in the preceding phrase (m.3). Also the second note is counter-cumulative in this measure, whereas it is cumulative in the first two. These variations affect the high-level structure as well as the low-level perception of this phrase. Note that it is the initial note of the dyad in this phrase that transforms to the second level. This is not the case in the first two measures.¹²

In the third phrase (mm. 4-5) the melody adds new structures to alterations of the familiar ones discussed above. From intraopus style (os) learning we might expect in measure 4 an intervallic duplication to follow the opening dyad. The harmonic change to an augmented chord B# E G# on beat three strengthens this expectation. The first IP (D# G E) of the IPPIP chain is marked with (h) because the harmony leads the listener to expect the change in registral direction and perhaps even the specific pitch, E, as the resolution of the D#. The a b al and the x (!) (o) highlight this relationship. The leap up to the G#, not the nearest resolution of the DO, but a chord tone nonetheless, is a small surprise. The ensuing chain, IPPIP, is a welcome change because it introduces a new structure to the meiody.¹³

Closure is strengthened by strong motion-left through the ensuing IP-2. This symbol (mL) indicates the propulsion of the melody toward G. The initial note (A) of the dyad [2] is a "good" note because the Fx embodies reversal to G#.Thus the dyad [2] is marked with (h). Barber delays the implied resolution with a cumulative dissonance, A, much as he did at the end of the first phrase. Note the further similarity between this phrase and the first. They begin simply, with dyads, move toward complexity (embedded structures), and end simply, with implicative dyads.

Further transformation of this melody leads off the fourth phrase, IR [embedded PIP-IP]ID-2 (mm. 5-6). The embedded PIP-IP (m. 5, beats 3 and 4) imitates the chain of the previous phrase (IPPIP, m. 4) and functions as an upbeat to the passage with the most embedded melodic structures (m. 6).¹⁴ The complex embedding in measure 6, which resembles an augmented turn, results from various influences of harmony and voice leading. The B, a lower neighbor note to C, is dissonant against the A in the harp and violins. The E is dissonant against the F in the accompaniment. Barber uses a dyad [2] to reach closure. Note that t^his dyad differs from those that come before in that it has an embedded trill.

A subtle variation on the complex chain in measure 6 follows in measure 7. The IP structure embedded under the IP in measure 7 is an aesthetically pleasing surprise because the large leap up to G# is unexpected, particularly due to this phrase's similarity with measure 6. The closural dyad is implicative, a resolution from a half-diminished seventh chord to a fully-diminished seventh. This is made clear by the flat trill. In contrast, either C or B can be considered a chord tone in measure 6. From intraopus style learning, the listener might hear the former as an appoggiatura.

Measure 8, the sixth and final phrase, combines phrases two and three (mm. 3-4) motivically. The underlapping dyad, [2] rather than [3], resembles measure 4. The dissonance and the lack of syncopation of the initial note of the opening dyad and the embedded processes [P] and intervallic duplications [ID] recall measure 3.

It is interesting to note the conflicting degrees of closure from level to level. The final phrase closes with non-motion on the low level, motion-left on the middle level, and motion-right on the higher level. The closing low-level intervallic duplication [(ID)] (mm.9) is retrospective because of intraopus style interference. That is, because the closing low-level structures in measures 2-7 are all dyads, the intervallic duplication in the final phrase is at first, unexpected, and thus retrospectively closural.

It may be helpful to restate the motivic relationships between phrases described previously as A-A¹-AB-B-B¹-BA. The second phrase (m. 3) is much like the first (m. 2), but the closing dyad is deformed to create a dependent relationship between the second and third phrases. The third phrase (mm. 4-5) is similar to the first and second in rhythm and the use of the dyadic beginning and ending. Its ending more closely resembles that of the first phrase because the dissonance resolves. As a result of the altered treatment of the opening dyad, the third phrase has an entirely new high-level structure, P. The fourth phrase (m. 5-6) grows out of the new melodic shape in the middle of the third phrase and maintains the use of dyadic beginnings and endings. The fifth phrase (m. 7) subtly varies the fourth. Finally, the sixth phrase (mm. 8-9) recalls the second and third, combining their materials into a new structure.

Violin Concerto, First Movement, measures 1-11

Deceptively simple, this melody is, perhaps, the most Elegant of the four discussed in this critical study (Example 10). Its sub-phrases are woven into an extraordinarily seamless eleven-bar phrase. The key to this melody's beautiful sense of forward propulsion lies in the interlocking of denied, abandoned, and/or deformed implications in the measure-to-measure level structures which finally blossom (i.e., realization is achieved) when the high B is reached in measure 8. This having been said, the low-level structures will not be given quite the same exhaustive treatment as have those in the previous analyses. Still, it is necessary to address issues of smaller importance first to lay some of the groundwork for the closing discussion.

One of the problems encountered in analyzing this melody is that the nature of dissonance and consonance varies from measure to measure. This has a direct effect on how the melody is parsed. For example, the A in measure 2 is a mild dissonance [(x)] in a downward enveloping process (E C A F0). Thus the melody does not break on that A. However, in measure 4, A is a chord-tone dissonance because the horns sustain the seventh and the eleventh above the root, blurring the subdominant. For this reason, and due to the influence of intraopus style learning from measure 3, the melody breaks on the A, and parsing occurs on the eighth-note level in measure 4. Such distinctions are important because they directly influence .one dynamics and meanings of the analytic symbols.

Intraopus style further effects the low-level parsing of this melody in measure 5. With the subdominant harmony, the D on beat two is mildly dissonant [(x)]. It subsequently resolves to C. Therefore, ;t seems the dissonant note (D) should be enveloped in a combination. However, this D occupies the metric position of an important implied structural tone, namely C, due to an intraopus style pattern found on the second beats of measures 3 and 4, A (D) C. Thus, this measure-to-measure patterning, along with the effect of cumulation, causes the melody to break on the mildly dissonant D.

Barber's interest in subtle motivic variation is as amply displayed in this melody as it is in the slow movements of his piano and violin concerti. For example, measures 3 and 4 are very much alike formally, but the analytical symbols reflect their subtle dissimilarities. Measure 3 is constructed of two separate structures, P and IP, on the low level. They are not linked because of the cumulation on A (beat 2) and the stress on the first note of the three-note upbeat to measure 4. The triplet figure helps separate the two structures (P and IP). In contrast, the augmentation of the analogous three-note upbeat figure in measure 4 stresses the second note (A), thus breaking the melody on beat four of that measure. This creates three adjacent structures, P-IP-IP, that flow smoothly from one to another because they link on the low-level.

Another example of motivic variation in measures 8-10 helps abate the climactic tension through formal repetition of the descending thirds figure of measure 2 (see again Example 10). These echoes of melody break into smaller chunks, beat by beat, because they lack dissonance. In this manner, they become more closural and less implicative, which seems fitting as they are essentially cadential. Even so, Barber makes this figure elastic, and therefore interesting, by recycling the different groupings of the same notes through a repetitive rhythmic pattern. For example, the triplet figure changes from an intervallic process [IP] in measures 9 and 10 to a combination [PIP] in measure 10.

The leap up a tenth to D (m. 10) also derives its beauty from subtle variation. After numerous repetitions of the descending arpeggio pattern (mm. 8-10) followed by an upward octave leap, to B, the leap from B to D is fresh and pleasantly surprising. Although this melodic fragment's structure [VR] is, at first, highly open D B D¹, it is immediately followed by a strongly closural reversal, B D¹ D¹. The closure of the reversal [R] is, however, exquisitely suppressed by the dovetailed arrival of the opening melody, supported by a seventh chord in third inversion. Thus as the melody closes, the harmony becomes implicative.

Similar to the compositional plans of previously discussed melodies, the middle-level symbols indicate patterns of simplicity moving toward some of considerable complexity, followed by a return to simplicity. For instance, complexity begins in measure 3, where the reversal [(R)] of the IP(R) structure seems retrospective because the listener expects the Ft on beat 4 to resolve upward to G when the subdominant arrives in measure 4. However, the F# is reiterated and becomes part of the chord, C 13, and therefore a structural tone on the next level.

Simultaneous with the high-level process is an a b al b¹a² b² pattern, the b-parts of which create an underlapping process on the second beat of each measure. The first two b-parts of this underlapping process, A B, imply the specific pitch, C, and metric position, second beat, of the third b-part. The D in measure 5 is an effective point of arrival, because it satisfies the implication of metric location, yet it denies the specific pitch. Thus it does not realize the ascending underlapped process. Note that it differs from the initial and medial notes of this process in that it is a weak dissonance. Therefore, it only temporarily suppresses the realization of this underlapping process, A B C, until it resolves to the implied C two beats later.

The underlapping dyads [4] in measures 3-4 are the combination of the a and b of the a b a¹... pattern (Example 15). This motive, initiated by the sixteenth notes on the lowest embedded level, implies continuation in measure 5. However, the pattern is broken by the arrival of a new rhythmic and intervallic structure, the dyadic fifth, shown underlapping the reversal on the downbeat of measure 5. Even as Barber denies the processive implication of the dyadic fourth motive, he initiates a new one, for another dyadic fifth motive arrives on the fourth beat of measure 5. This, in turn implies another motivic process. The reversals [IR] in measure 5 (beats 1 and 4) introduce a new structure and larger intervals. The following pair of structures, IP- P in measure 6, abandon the pitch aspect of the opening chain in measure 7, denies the rhythmic process begun in measure 5, and leads to the climax of the phrase, a restatement of measure 2 transposed up a fifth to the tonic (Example 1 6.) Fj ascends only to A (mm. 6-7.) But the beauty of this A is its double function as a denial of the motivic implication and as the medial note in the final process up to the climactic B. The breaking of the ascending fifth pattern (mm. 6-7) suggests that the line has completed its rise on the A. This is why the B is such a beautiful arrival point.

²Keep in mind that implications and realizations take place on many levels. The interaction of levels either reinforces or interferes with our on-going understanding of the melody.

³The variables in the formulas (A,B,C) can stand for both small-scale and large-scale elements in music. They apply to single elements in a parameter, such as a pitch or an isolated duration, and to relationships within a parameter, such as interval (I), registral direction (V), or rhythmic patterns, as well as to form on any level.

⁴ Consequently, their ability to manifest the strength and depth of Eugene Narmour's implication-realization theory of melody.

⁵Ascending and descending processes EP] tend to envelop additive and counter-cumulative rhythmic relationships, regardless of metric accent. Thus, I do not parse the melody on the downbeat of measure 20 because the G is the medial tone of a metrically enveloping process (B G D), which is the middle structure in the RP(VR) chain.

⁶The second half of this underlapping process is marked with a dashed line because the C comes so much later than the C#. While this descending process might not be immediately apparent, I believe the formal similarity of the closing dyads makes it noticeable. Also, this helps account for the resolution of the A# to the B a major seventh below. That this resolution is an octave lower than expected makes the B a particularly "good" note.)

⁷ Not only does the hierarchical conflict of the dyad [7] provide the listener with a rich aesthetic experience, it also makes apparent one of Barber's most effective compositional strategies: the inventive and unconventional use of the dyad. Dyads as anacruses can be found in almost all of the musical literature, for the notion of upbeat-downbeat is widespread. However, the use of the dyad as a closural structure also became somewhat of a cliché in tonal western music, when aesthetically pleasing appoggiaturas became cadential style structures. As we can see in this melody, Barber took this style structure and "modernized" it, for the closural dyads here are of such large intervals that they hardly resemble those of, say, the eighteenth and nineteenth centuries. Yet they function in much the same way. The second movement of this concerto makes this clear, for the dyads are smaller, seconds and thirds, and seemingly much more conventional. This is not to say that all of Barber's closural dyads are either conventional or modernized conventional style structures of the eighteenth and nineteenth centuries. The dyads in measures 20 and 24-26 imitate that style structure, but do not have the resolution from dissonance to consonance necessary to make them function as such. Thus, a third kind of closural dyad is that of stylistic imitation.

⁸ The relative simplicity of these measures lies in the use of smaller intervals, lack of syncopation, the predominance of low and high-level realization. The low-level structures, l-DP-(R)-IDP-4, and the highest level structure, P, indicate the prevalence of realizations.

⁹Because the second tone of the dyad in measure 3 is dissonant, the listener may initially perceive the second tone of the dyad in measure 8, C#, to be dissonant as well. That this tone does eventually become dissonant is somehow pleasing. Barber has his cake (i.e., the illusionary, intraopus style-influenced, implicative dyad in m. 8) and eats it too (i.e., the actual implicativeness of the dyad at measure 9). Such subtle variation in the treatment of dissonance in this melody is truly elegant, for it not only affects the low-level structures, but also alters the implications on the higher level, to be discussed later.)

¹⁰ Unlike the compositional plan of the first movement of the Piano Concerto, that of the second movement of the Violin Concerto positions the most complex phrase at the end. The reversal structures on the lower level seem well-suited to this final phrase (mm. 10-14) for they are more closural than the IPs which end the first three phrases.

¹¹The F# in measure 12 occurs over a change of harmony. This makes it stronger than the B in measure 6, where the harmony does not change.)

¹²Of the two tones that form the opening dyad in the first, second, and sixth phrases, the second is the structural tone. In all three instances, the second tone is cumulative. In the second and sixth phrases, the first note of the dyad is deformed, though by different degrees, by dissonance. Note again that it is the initial note of the opening dyad of the third phrase (m. 4) that transforms. Of the two tones that form the closing dyad in the first, third, fourth, and fifth phrases, the second is the structural tone. The deformational dissonance on the terminal note of the dyad in the second phrase (m. 3) prevents closure on both high and low-levels. Thus, as I indicated earlier, Barber's varied treatment of dissonance in the dyads on the low level alters the structural tones on the upper hierarchical level. The high-level symbols reflect the subtle variations from phrase to phrase.