Brahmsfo(u)rmations or: Transformations within Brahms 4

You're probably familiar with the opening theme of Brahms's fourth symphony. It's the one that descends by thirds and then ascends by thirds. No, not that one. It's this one. Big deal, right? Well I had the audacity to look a little closer at it and let me tell you, the results are quite illuminating. Take a gander at them, won't you?

Let me preface my so-called analysis of the piece by saying that this is but a part of a greater study of mine on this work. It's also heavily inspired by Steven Rings's book Tonality and Transformation, which was a pretty decent read. I use a lot of its terminology, but I won't cite the pages I used because I can't be bothered. 

Somewhere in that book of his he mentions the concept of "tonal intention." This more or less describes the phenomenon of our attention being directed to the tonic when encountering some subordinate tonal element. A basic example would be the momentum created by the dominant sonority; it generates a strong pull towards the tonic within the context of a tonal work. I think that's what it means anyway. I will refer to this concept again in a minute, so don't forget it.

Let's get down to brass tacks. Take a look at the score. Here's the first page of a piano reduction for the work:

Figure 0 Johannes Brahms, Symphony No. 4 in E minor, Op. 98, i

Notice there are brackets labelled X and Y. They demarcate the aforementioned chains of descending and ascending thirds, respectively. That's all there is to it, right? Wrong. See below:

Figure 1 Network of the Main Theme in the first violins, mm. 1-9

Now to explain this. X and Y, the labels that bracketed a part of the theme within Figure 0, are modeled by Figure 1, the transformational network above. In it, nodes contain pitches accompanied by the number of their octave (e.g. middle C = C4); they are arranged approximately in pitch-space, where the highest node represents the highest registral pitch, and vice versa. Solid arrows connect the majority of nodes, and these are accompanied by an ordered pair consisting of scale-degree intervals and pitch intervals. Henceforth I'm going to refer to scale-degree intervals and pitch intervals as sdints and pints, respectively, so don't get confused. Each sdint represents an interval between sds; for example, the interval from sd 1 to sd 3 is a 3rd (or its inverse, a 6th-1). When an octave or unison is encountered, that is, when there is no change in sd, the symbol e, for identity, is used. The pints conform to pitch-space, where negative and positive integers represent intervals. The solid arrows trace the literal path of the theme's melody found in the score.

Take a deep breath and continue to persevere through this molasses-like prose.

The other two types of arrows are dotted and dashed. Dotted arrows trace the top layer of the theme; a path determined by the network's upper registral notes. This path does not happen as explicitly as the one traced by the solid arrows due to the many intermediary pitches within the score of the work. The sole dashed arrow is essentially a combination of the dotted and solid arrows; it literally appears in the score where B5 moves to E6 (in m. 4), but it also traces that upper line based on register. The rightmost node of Figure 1 has a B5 enclosed by a dotted circle. This special type of node implies that the pitch contained inside of it has not been realized within the measures pertaining to this network (i.e. mm. 1-9).

Before the significance of the dotted arrows can be explained, I'm going to talk about the significance of the arrival and emphasis of pc C on the downbeat to m. 9 until m. 13. Note that most of these measures are not included in Figure 1 (read: go look at the score to see them). Within them, C, as a pitch, alternates repeatedly between C5 and C6. Although three eighth-notes are found at the end of each measure which include pcs other than C, these deviations, don't disrupt the emphasis of pc C found on the downbeat of each of those measures. This area marks a point of contrast to the preceding measures.

So Brahms has decided to play around on this emphasis of pc C. I'd like to ask why he has decided to do so, but he's dead. Therefore, speculating on composer intention is little bit futile. Instead, I'd like to take a closer look at what the consequences of the prolonged C might be.

Looking to the score shows that before C5 arrives on the downbeat of m. 9, each of the first eight measures of the movement has two thematic notes (i.e. melodic notes within the theme) per full measure. These unfold supported by basic triadic harmonies built on diatonic scale degrees. There is a tonic pedal within mm. 1-4, but afterwards until m. 8 the theme is supported by root-position harmony. In contrast, the chordal harmonies within mm. 9-13 are far more dissonant; they include fully-diminished-seventh applied chords based on chromatic scale degrees (mm. 9, 12), as well as a second-inversion chord (m. 10). Therefore, in addition to emphasizing primarily the melodic pc C, as was discussed in the above paragraph, these measures also feature prominent harmonic instability.

As a pitch, chord, and key, C is an element that binds the symphony together. As a chord it plays an important role in the recapitulation of the first movement. As a pitch and key area it is prominent in the second movement as an inflection within the key of E-major. The third movement is fully in the key of C-major, and in the fourth movement C-major emerges as a key area despite this movement being in E-minor.

Returning to Figure 1; the dotted arrows trace the paths of the line formed by the highest registral notes. Within X, the pitches B5-C6-B5 are highlighted by these arrows. This idea, reminiscent of a neighbour-tone, sets up an expectation: a departure from B5 is followed by a return to it. In this sense, the motion away from B5 is intentionally directed back towards it. Another departure from the B5 initiates Y. This motion is a leap to E6, which then descends stepwise to D6, and C6, again demonstrated by the dotted arrows in this bracketed area. It appears that this is fulfilling a similar idea to what X worked out; a departure from B5 begins to return to its origin through stepwise motion, which is anticipated in Y because of the expectation introduced by X. By the time the C6 in Y is reached, however, the path to B5 is heavily delayed. As was mentioned before, there is an emphasis on pc C within mm. 9-13 which is supported by dissonant harmonies. Based on the intentionally directed dotted arrows of Figure 1, that emphasis on C builds up energy in its anticipation towards the B5 in the dotted circle node; a pitch that remains unrealized within the measures that this transformational network models. This B5 in the dotted node is eventually encountered in the score, but not until the anacrusis to m. 18. Here the expected B5 dramatically returns with the leaping of first violins and oboes through two of its lower octaves in m. 17. At this moment there also appears to be an allusion to the B5-C6-B5 traced by X within the top voice of mm. 17-18. This more literally outlines the neighbour-tone dotted arrow motion X first introduced. 

Figure 2 models a similar kind of reading of the theme but from a different perspective. This network uses a pitch-class space. Beginning with the upper-left node and then following the path directed by the arrows will trace the theme, not unlike to the solid/dashed arrows of Figure 1. Although the bottom row is labelled Y, it is slightly different from the Y in Figure 1: here in Figure 2, any repetitions of pitch-classes are omitted. The purpose of this network is to articulate that, unlike X, which begins and ends on pc B through a chain of descending thirds, Y begins on pc E and stops prematurely at pc C. This is one ascending third away from returning to pc E and the completion of a chain that returns to its origin. This missing pc E is articulated in the network by the dotted node, which implies that it is not realized in these measures, similar to the dotted node within Figure 1. Beyond the measures Figure 2 represents, however, are no immediately discernible candidates to satisfy this unrealized node. One might speculate that the pc E in the dotted node is in fact realized in m. 13, immediately following pc C. But because pc E here is merely a lone eighth note, supported by a dissonant harmony, and in a formal area where the theme becomes fragmented, it is quite alien to the material from mm. 1-9. Therefore, in my hearing, it does not grant a convincing relationship to the way each pc contained within the other nodes of this figure are presented in the score. The potential chain of thirds beginning and ending on pc E within Y remains incomplete.

Figure 2 Another network modeling mm. 1-9.

The horizontal dotted arrow in Figure 2 traces the unrealized path leading from pcs C to E; it is also consistent intervallically with the solid arrows preceding it on the bottom row, as was discussed above. The relevance of the dotted arrow leading vertically from pcs E to B, however, is a little more interpretive. Its addition unifies the entire network and enables it to be closed in a logical manner; but the music itself does not follow this path literally, nor does it convincingly imply an intentionally directed motion from pc E to B as strongly as the preceding pc C to E does. X and Y, however, are essentially repeated in the transition following the main theme (see Figure 0). This action restarts the network, but only after having skipped over that pc E in the dotted node. The two dotted arrows and dotted node of Figure 2 therefore represent the portion of the network that the music itself had ignored, and provide what I consider to be a more predictable and logical way for the network to return to the beginning of X and initiate the transition. In addition, this vertical dotted arrow creates a kind of symmetry with the solid arrow from pcs B to E on the right-hand side of the network, which alludes to the inversional qualities of X and Y.

The avoidance of Y's return to the originating pc E, represented by the dotted node of Figure 2, is metaphorically similar to a stepwise ascent of a major scale that begins on its tonic and halts motion once reaching the leading-tone. In this example, there tends to be an acoustic desire for a resolution to the tonic from that leading-tone pitch. In the case of Brahms's symphony, I would argue that the cessation of ascending thirds on pc C in Y impels a similar anticipation for a resolution to the tonic, pc E. Given that Brahms has already offered a complete cycle in X beginning and ending on the same pc, this only serves to add to the expectation that the first pc which initiates Y should also be its closer. Rings's concept of tonal intention is applicable here as well because it involves the process of subordinate tonal elements directing our attention to the tonic. The corner nodes of the bottom row, Y, in Figure 2, would be the tonic in this case, and the subordinate elements can be understood as the intermediary pcs between them. Although the corner nodes of X are not the tonic, in the sense that pc B is not the tonic of the symphony's key of E-minor, within this localized context of the network it does not seem to me a stretch of the imagination to conceive of our attention being intentionally directed in some way back towards pc B once the subordinate intermediary pcs of X are encountered. This skipping over of pc E in Y parallels of the avoidance of resolution to the tonic throughout the entire movement (a typical Romantic aesthetic), and its prolonged acoustic absence heightens intentional energy towards it. 

Both Figures 1 and 2 demonstrate that pc C plays a role in anticipating and delaying a resolution that is expected, represented by the dotted nodes. It is intentionally directed towards both pcs B and E, depending on which of these two networks one is engaged with. In a listening experience of the movement, however, one could claim that pc C is directed towards both of these pcs simultaneously because of its dualistic role within the first two figures.

The End.