Bach's mastery of large-scale organization is readily apparent in works such as the Goldberg Variations, BWV 988, and The Art of Fugue, BWV 1080. The Variations are cyclic in nature and are organized around a succession of canons at increasing intervals. Erich Schwebsch's account of The Art of Fugue reveals a "symmetry of the most subtle kind" (Ulrich 110), one whose "plan of thematic transformation [is] unequaled in the history of music" (Ulrich 110). Both the Variations and The Art of Fugue were written late in Bach's life, the former in 1742 and the latter left unfinished at the time of the composer's death in 1750. Conversely, the Passacaglia and Fugue in C minor, BWV 582 was composed relatively early in Bach's career. There is some uncertainty about the exact date of composition, but it is believed to have been written before or during the early part of Bach's tenure at Weimar (Crist 78; Grout and Palisca 405; Marshall 6). How developed was Bach's sense of large-scale organization when he wrote the Passacaglia? In other words, what is the overarching mechanism used to organize the twenty variations?
Both Christoph Wolff and Siegfried Vogelsänger have proposed symmetric organizations of the twenty variations of the Passacaglia, summarized in figures 1 and 2 (Kobayashi 63). As Yoshitake Kobayashi points out, though, "they are so beautiful — almost too beautiful — that [one] cannot escape the impression that in research up to now, attractive analytical goals have first been established, and a path to them searched out after the fact" (64). Kobayashi's criticism of symmetric models is followed by an attempt to show that "there is an ever more complex rhythm in [the Passacaglia], manifested as well in gradually decreasing note values" (Kobayashi, 65). While it is true that the half- and quarter-notes of the ostinato theme give way to eighth- and then sixteenth-notes, sixteenth-notes are introduced in the fourth variation and are the predominate rhythmic value from then on! Sixteenth-note triplets are used extensively in variation 17, but regular sixteenth-notes quickly regain their status as the baseline note value in the three remaining variations.
Though Kobayashi criticizes Wolff and Vogelsänger's attempts at painting a perfectly symmetric picture of the variations, it is impossible to ignore the fact that elements of symmetry do exist. During his tenure at Weimar, Bach learned from Italian examples how to "develop subjects into grandly proportioned formal structures" (Grout and Palisca 405). Since the Passacaglia was written before or shortly after his appointment in 1708 it does not exhibit the heightened awareness of large-scale structure displayed by later works such as the Goldberg Variations or The Art of Fugue, but the composer's natural instinct for long-term organization is clear. It must also be remembered that the Baroque passacaglia was a variation composition (Hudson 268), one that presented an accomplished organist such as Bach the chance to show off his improvisational skills. It is possible that the Passacaglia is a realization of an improvisation over the ostinato theme. Based on these observations I suggest that the organization of the Passacaglia is very organic and self-referential, that it is based on the ostinato theme, shown in example 1.
The ostinato theme is unusual in that it is eight measures long rather than the traditional four (Geiringer 249). The first four measures, borrowed from André Raison's Trio en Passacaille (Geiringer 249), move from i to V while the second four measures move from V back to i. This suggests an arc of increasing and decreasing tension and mimics the melodic shape of the ostinato theme. Its fundamental contour is very symmetric and can be described as two low points with a high point in the middle. The melodic importance of the ostinato theme is underscored by the fact that it begins the piece unaccompanied; surely if Bach had used a simpler basso ostinato (such as the ever-popular descending tetrachord) it would not have received the same treatment.
Bach employs the arc as an organizational tool throughout the Passacaglia. In all cases, symmetry is reflected around the two central variations, ten and eleven. On a very broad level the arc is mirrored by the tessitura of the ostinato theme. From its stately solo entrance in the pedal through the tenth variation there is no change in register. The theme's sudden leap from bass to soprano in the eleventh variation may seem impulsive and unprepared, but the shift is in fact hinted at in the fifth and ninth variations. In the fifth variation the familiar rhythm of the ostinato is modified, resulting in a more lively pattern. The new rhythm is present in the upper voices, thus providing a subtle psychological connection between the theme and a higher tessitura. Similarly, the ostinato theme of the ninth variation is given an enlivened rhythm () which is imitated, this time more clearly, by the upper voices. A stronger connection is made between the theme and a higher tessitura.
Variation eleven is the climax of the arc. The isolated soprano register of the ostinato theme and sparse accompaniment make it stand out. In the twelfth variation the addition of two voices significantly narrows the registral gap between the theme and the accompaniment. Additionally the increased contrapuntal activity takes attention away from the ostinato theme. In variations thirteen through fifteen Bach hides the theme by placing highly ornamented versions of it in the middle of the texture. As each variation progresses, the theme sinks down through the tangle of inner voices and back to the bass register, which it finally reaches in variation sixteen.
Bach uses the arc on a more local level to plot out the texture of variations six through fifteen. In this scheme the texture grows increasingly dense until the comparatively sparse tenth and eleventh variations are reached. With the twelfth variation comes a return of the dense texture and subsequent mirror imaging of the preceding buildup; the texture thins out considerably by the time the fifteenth variation is complete.
In addition to range and texture, Bach uses an inverted arc (two high points framing a single low point) to explore the tension between melody and accompaniment. He does this using the first, middle, and last pairs of variations. For purposes of this analysis the ostinato theme, usually the accompaniment, will be called the melody and everything else will be referred to as the accompaniment. Variations one and two and nineteen and twenty are characterized by an extremely static accompaniment, both rhythmically and harmonically. At first the syncopation of the first two variations does not seem to be particularly static, but its constant repetition effectively discharges any initial effect it might have had. Harmonically the accompaniment is a large prolongation of tonic harmony. The nineteenth and twentieth variations are an elevated form of the first two. They remain rock steady both rhythmically and harmonically, changing only at the end of the last variation in order to cadence. It is also interesting to note that the succession of accompaniment figures forms a hemiola that plays off of the syncopation seen in the first pair of variations. The tenth and eleventh variations exhibit the opposite structure. The accompaniment figure, a single line of running sixteenth-notes, is much more active and interesting than the ostinato theme.
While the range and texture arcs seem to single out the middle pair of variations for special attention, the tension arc does just the opposite. The tenth and eleventh variations are treated normally with the ostinato theme serving as accompaniment while the framing pairs of variations are given such static upper voices that the ostinato takes over a melodic function. This juxtaposition is part of what keeps the listener's interest. It is worth observing that variations sixteen through eighteen present a conflated version of the tension arc, with variation sixteen representing the first two variations; variation seventeen representing the central pair; and variation eighteen representing the nineteenth and twentieth variations.
Bach's use of the arc as an organizational tool lends a certain coherence and symmetry to the Passacaglia, even though the overall form of the work is not as neat and systematic as the Goldberg Variations or The Art of Fugue. It seems that Bach's awareness of large-scale structure was not yet fully developed at the time of composition, but his natural instinct for organization was quite good. His use of a single concept to coordinate otherwise unrelated material is the seed that flourished under the guidance of Italian examples, and the possibility that it was all improvised makes the work all the more impressive.