The Symmetrical Binary Principle
This lesson ©1997 Arizona Board of Regents and Timothy A. Smith

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Please begin this lesson by listening to the Trevor Pinnock recording of the Bach Goldberg Variations (BWV 988) (Archiv 415-130-2).

The turn of the 17th century saw a philosophical shift of great importance to music theory and composition. Whereas, the stile antico of the Renaissance had treated words as subservient to harmony and counterpoint, the stile moderno would attempt to subjugate music to text. One consequence of the new paradigm would be a period of general discontinuity which Bukofzer describes as having characterized nearly all forms of baroque music. He observes ambiguity of form, or lack thereof, in the monadic madrigal, recitative, toccata, ricercar, fantasia, canzona and motet, noting that, in spite of attempts to incorporate continuous elements, such as refrain sections, strophic variation and ritornelli, "nearly all early baroque forms had one formal trait in common: multisectional structure." These sections were "discontinuous" because they typically lacked formal or motivic symmetries to unite them. What united them, was text. Bukofzer concludes: "The nervous and erratic flow of early baroque music merely reflects its intensely affective nature."

Yet the philosophy promulgating discontinuity of form in the early baroque contained a constructive seed which, if not in Bach's generation, certainly that of his sons, would flower in the most highly evolved forms of the western tradition not to mention a renewal of music for music's sake. Bach's generation is responsible, you see, for the ascendancy of motive--the musical representation of text-associated feelings, moods and emotions--in what baroque theorists called the Affekt. This "doctrine of affections" was a rationalization of motivic developments and transformations appreciable apart from the Affect they existed to represent. While Bach's generation would not have sensed the irony, by means of their word-dependent model they scribed a different rubric in which motive would ultimately attain identity altogether autonomous from text. Thus, however paradoxical, a philosophy intended to associate music with text laid the foundation for the dissociation of music from text culminating in the absolute music of "beyond"--e.g. Brahms and Schoenberg. Autonomous music of the 19th and 20th centuries, especially the practice of motivic development, can be seen in such a light as having sprouted from cuttings rooted in associative genres of the early baroque.

Whereas the aforementioned motive has comprised the focus of our study this far, we turn now from motive to form. What, if anything, did the baroque contribute to complex homophonic forms that would evolve in the half century following Bach's death? To the general characteristic of discontinuity in early baroque music the ostinato forms--passacaglia and chaconne--had provided moments of relief. To these were added theme and variation, but more importantly, music of the dance. Unlike the tongue which moves more comfortably at a non-proportioned pace, the feet keep the body in proper orientation to the ground when they move in symmetry with each other (gravity does interesting things to art). It is in the music of the dance, moreover, that the baroque saw its first crystallization of nationalistic styles which coincided with the appearance of binary precursors of classic sonata-allegro, rondo, and song form with trio. Because these precursors emanated from music associated with ballet, they, too, tended to be symmetrical.

In retrospect we can see the middle baroque rebellion against discontinuity of form as having been led by Lully, whose patron king Louis was, himself, a dancer. The Lullists exported French idioms to places like Celle, in Saxony, where they were absorbed by young Sebastian Bach into his own Vermischte Geschmack. A synthesis of German, Italian and French idioms, this "eclectic style" featured symmetrical binary form in its suites and ternary in its overtures.

Like the symmetrical gardens at Cöthen where Bach was employed from 1717-1723, Leblanc extolled the symmetrical qualities of French music when he wrote, ten years before Sebastian's birth:
The imitation of the dance in the arrangement of tones, in which phrases are joined together like intertwining steps in the formal dance, the symmetrical arrangement of measures, all this has shaped that creation known as the pièce, which represents poetry in music. The entire ambition of the French nation is directed toward....that subtle symmetrical division, which shapes the musical figures in a pièce, like those ornamental shapes of boxwood hedges which comprise a garden in the parterre of the Tuileries."

Bred on the asymmetrical chorale variations, canzona and ricercare of Pachelbel, Froberger and Frescobaldi, it is likely that Sebastian Bach first encountered the symmetrical style when he visited the Ducal Court of Celle in his teens. There he would have had opportunity to hear the courtly clavier music of Marchand, Dieupart, Anglebert, Chambonnières, Clairembault and, most importantly, "le Grand" François Couperin.

What Johann Sebastian would have heard in Celle would have represented the first compound form to have reached maturity in the 18th century. The inner movements--allemande, courante, sarabanda, and gigue--would have subscribed to a rigid binary structure--more often than not, symmetrical. Today we describe this arrangement as "continuous" because part "A" modulates to the key of the dominant, requiring continuation to part "B" and eventual return to the tonic. Thus by 1700 key relationships hint of what they would shortly become--one of the most form-defining elements in western music. This particular pattern--two sections repeated, producing the expanded binary form AABB--would later be recognized as precursor of the Viennese classic sonata-allegro. But, unlike its descendent, the baroque binary was monothematic--developing the same motive in the second section as it did in the first. The two-part structure was achieved by statement of the motive in related keys, an innovation that would eventually function, more so than multiple themes, as the distinguishing characteristic of sonata-allegro. Repeats served, accordingly, to reinforce tonal development, but also to render the sections of baroque binary as easily recognizable.

Goldberg Variations (BWV 988)

In its stylized dances and symmetrical outlines, Bach's Goldberg Variations illustrate salient characteristics of the French repertoire. As in Canon #2, both halves of each variation are of the same length--usually 16 bars, sometimes 8--while the thematic content is, likewise, the same. (Examples in the following discussion will be taken from the notebook to the left.)

When the first section of a piece implies continuation to a second (continuous binary), the two sections have an antecedent-consequent" relationship. Whereas parts "A" and "B" of each Goldberg Variation are so related, both "A" and "B" are themselves divided into antecedent and consequent phrases. Variation #17, for example, contains four phrases, each one terminated by an authentic cadence in a related key. In many of the variations it is possible to detect further subdivision into "semi-phrases" or "phrase members," but this is a subjective hearing, at best. While some may hear a semi-phrase as supported by a cadence, others may hear the same segment as having no such support, or, perhaps, a cadential inflection. The Goldberg cycle projects structural semi-phrases fairly consistently, as the following attest: Variation #4 and Canon #6.

In addition to symmetry of section, phrase and semi-phrase, a peculiarly Bachian characteristic is found in "Back 'n Forth" technique where analogous measures in sections "A" and "B" employ the same motives but in textural or melodic reversals. While "Back 'n Forth" is not a requirement for binary form, it represents a sophisticated spin on symmetry that appears to have been Bach's singular contribution to the form. It is a technique to which we shall return in the Ludus Tonalis of Hindemith and, the beyond portion of this study, Drei kleine Stücke of Webern.

Anton Webern: Drei kleine Stücke Op. 11 No. 1
(Little Piece for Cello & Piano)

Please continue the lesson by inserting the Webern Complete Works disk No. 1 (Sony SM3K 45845), and listening to the Little Piece for Cello and Piano.

"To live is to defend a form."
Anton Webern (quoting Hölderlin)

Like his own life, cut short in 1945 by a nervous American GI, Webern's music is concise and to the point. His 1914 Three Little Pieces for Cello and Piano represent the most concentrated of these, prompting teacher Schoenberg to remark: "One has to realize what restraint is needed to express oneself with such brevity."

My interest in the Three Little Pieces was sparked by Robert Clifford's paper, "Textural Contour and the Medial Structural Level in Webern's Op. 11, No. 1" delivered at the RMSMT Annual Conference last March. Anticipating the writing of this unit, it occurred to me that Webern may have used textural contour to create symmetry, "atonal" development and "Back 'n Forth." The notion of Webern's Little Piece as abstraction of an 18th-century model is certainly debatable...this, of course, is not Clifford's point. But Clifford does at least make a compelling argument for symmetry of contour which I see as not unlike Bach-style "Back 'n Forth." As a matter of fact, I hear three theoretical likenesses which make it reasonable to imagine Webern's piece as "beyond" Bach and the Baroque. Rather than make these connections myself I shall leave you to ponder them as guided by the following questions. Please begin by studying the Score & Analysis.


Questions pertaining to binary structure of the Goldberg Variations

Begin by listening to the nine canons of the Goldberg cycle while following the score. When you've become thoroughly familiar with the work, answer the following questions.

  1. Is every variation symmetrical in terms of the number of measures devoted to the two parts? Write a paragraph discussing anomalies to the symmetrical binary principle.

  2. Write a paragraph comparing elements contributing to unity (continuity) in the Goldberg Variations with elements contributing to variety (discontinuity). What elements tend to be the same in each variation and what elements tend to be different?

Questions pertaining to the Webern Little Piece for Cello and Piano

  1. Clifford hypothesizes that Webern has employed eight vertical sonorities to define the highs and lows of his textural contour. These chords are highlighted in the animated score. To Clifford's hypothesis I have added the possibility that the first four chords may comprise part "A" of a two-part structure with the second group comprising part "B". In a Bach-style binary, part "A" would have ended, of course, with a dominant...which would have required continuation and return to a tonic. Inasmuch as there is no perception of dominant function in Webern's work (nor tonic for that matter), do you sense anyway that the first section "wants" to be continued to the second? If so, to what might you attribute this perception? Perhaps you don't hear sections at all...feel free to disagree.

  2. One of the cardinal binary principles of the 18th century is tonal development. But Webern's music is atonal. What does it appear that Webern has done with texture to emulate the |: I-V :|: V-I :| of a tonal binary in an atonal context?

  3. How does Webern's textural contour resemble Bach-style "Back 'n Forth?"
Make sure that your response is organized as pertaining to the Goldberg Variations (answers numbered 1-2) and the Webern Little Piece for Cello and Piano (answers numbered 1-3). Briefly restate the question before you answer it using complete sentences, correct grammar and spelling.



  1. Manfred F Bukofzer, Music in the Baroque Era (New York: W. W. Norton, 1947), pp. 351 ff.

  2. as quoted in Nikolaus Harnoncourt, Baroque Music today: Music as Speech (Portland: Amadeus Press, 1982), pp. 182-183.

  3. Today these repeats are left to the discretion of the performer who must consider not only the technical limitations of recorded media (approximately 75 minutes) but also the modern distaste for extended performances.

  4. From the liner notes to Sony disk SM3K 45845, p. 21.

  5. Ibid, p. 17.
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