Site ©1996 Timothy A. Smith

Author BACH Courses

Intentionality and Meaningfulness
in Bach's Cyclical Works


This paper was read at the Third Conference of the Rocky Mountain Society for Music Theory in Tucson, Arizona, April 19-20, 1996.
A personal search for knowledge involves the search for patterns within patterns in a holistic context. No pattern can occur in isolation, autonomous from a larger context or set of assumptions, and still be meaningful to human beings. Patterns require larger contexts, with relevance to more inclusive patterns, if they are themselves to be meaningful to us. The total autonomy of parts of knowledge does not exist. Conviction 5.1: A description of units within patterns within larger intersecting patterns is a kind of knowledge and a component of truth.

Kenneth L. Pike from "Talk, Thought, and Thing: the emic road toward conscious knowledge" (Dallas: Summer Institute of Linguistics, 1993) p. 55

A set is a collection of objects. In music, these objects may be movements, keys, meters, pitches, etc. An "intentional" set is a collection of like objects that achieves its "likeness" by means of patterns indicating that the order or attribute of events comprising the set, is purposeful. The presence of a pattern does not necessarily warrant inference that a large-scale cycle ought to be performed in set order or without interruption. But, in situations where arise uncertainties about completeness, or when extraneous factors call into question the "belonging" of elements to a set, evidence of intentionality can be enlightening. Internal coherence of the B-Minor Mass, for example, indicates that Bach conceived of that work as a set, whereas historical incidents culminating in the composition of the Mass are sometimes used to urge a point of view that he did not.

Johann Sebastian Bach often shows intentionality by means of musical-theoretical constructs such as key or mode. His Well-Tempered Clavier is a case in point. The first prelude and fugue are in C-major, second in c-minor, third in C#-major, fourth in c#-minor, and so on. This pattern infers that the set is completed after Fuga XXIV. To continue beyond this point would be to commence a repetition of the cycle, which the composer does in Volume II. Further, if parts of the cycle were missing, apprehension of the pattern would enable one to determine not only how many preludes and fugues had been lost, but also in what keys.

Whereas set characteristics of the Well-Tempered Clavier constitute a referential device allowing one to locate preludes and fugues quickly, in most instances such utilitarian intentionalities, like the alphabetics of a dictionary, "mean" nothing. But we know that the Well-Tempered Clavier did mean something. The cycle was written first to demonstrate the feasibility of an intonational system that the composer himself had helped to invent. This system was predicated upon an emerging acoustical science as well as practical constraints that had puzzled (and plagued) musicians for a hundred years.

As the name implies, the Well-Tempered Clavier was also written to demonstrate the range of Affektenlehre that could be achieved by a "well-tempered" scale. These affects accrue, writes Harnoncourt, from "various strong tensions, created by intonation, which increase with distance from the C major center and which are also felt as a kind of longing for the beautiful, relaxed keys (F major, C major, G major)." The set structure of the Well-Tempered Clavier can be seen, in such a light, as an artistic manifestation of physical and metaphysical presuppositions, peculiar to the eighteenth century. The presupposition is one of Affekt, the most important aesthetic concept of that era, and the manifestation seeks to exploit affect to the fullest extent possible. Obviously the card catalogue organization of the set is not a product of affect, but the systematic structure implies that each affect has been fully explored. The set is therefore rationally distinguished as more than utilitarian, but meaningful.

Meaningful sets, then, are intentional formulations that indicate purposefulness, not merely with respect to order, but wherein the order itself is born out of higher patterns of thought or thing (see Pike). To J. S. Bach, for whom music was not only the physical representation of affection, but subject to the same laws governing the cosmos, meaningfulness would have been exceedingly important. After all, music was, in the words of his student, Mizler, "sounding mathematics." Nicolaus Harnoncourt comments, of the 18th-century preoccupation with proportions in music, art, and architecture, as follows:

...the perfection of sounds is revealed by numbers. And vice versa, all simple numeric ratios could be imagined as sounds. Kepler's harmony of the spheres, as well as harmonically "resounding" architecture, are based on this notion: if the visible proportions of a building could be expressed in simple numeric ratios, then these relationships could be seen and heard as "chords." In many ways, Palladio "composed" the ground plans for his structures as a kind of petrified music. According to theory, harmony in music rests on a principle similar to the golden section in architecture. Both impose order on the hearts and minds of men by virtue of their simple, natural relationships. The Baroque idea that music was a reflection or a likeness of the divine order was applied to all music, sacred as well as secular. (Baroque Music Today: Music As Speech pp. 61-62)
Because of their position within the large-scale context of the Clavier-Übung, Bach's Goldberg Variations hint of meaningful possibilities to wit. That the Goldbergs comprise an intentional pattern is self-evident; every third variation is canonic, with the interval between leader and follower expanding by one scale step in each successive canon. The apparent intentionality of this set gives pause to wonder if it was generated by a higher pattern of thought giving meaning to the whole. I believe that it was. But to explain the meaning it shall be necessary to view the Variations in a larger context. We shall begin by examining Bach's model (Kuhnau's two volumes by the same name), and the preceding three volumes of Bach's own Clavier-Übung.

The Goldberg Variations comprise the fourth, and last, volume of the Clavier-Übung. The composer produced volume I of that cycle in 1731, a year when he surely could not have envisioned writing the Goldbergs nearly a decade later. Bach had moved to Leipzig only five years earlier to assume the post of predecessor, Johann Kuhnau. The first composer to use the term Clavier-Übung ("Keyboard Practice"), Kuhnau titled two volumes by that name. With little doubt, Kuhnau's is the model that inspired Bach to attempt a similar collection. Knowledge of the structure of Kuhnau's Clavier-Übung, shall help us to understand Bach's complementary titles, especially his Goldberg Variations.

Whereas the first volume of Kuhnau's model consists of seven partitas in ascending major keys (C, D, E, F, G, A, B-flat), the second consists of a similar sequence, in minor (c, d, e, f, g, a, b). The intentionality of this set is found in its ordering and grouping of tonic and mode. Curiously, the order incorporates intonational impediments that render both volumes impractical in performance. These impediments are difficult to justify without recourse to symbolic associations that the Lutheran culture would have traditionally attached to the numbers six and seven. From antiquity the number six has represented perfection. In Judeo-Christian cosmogony six stands for the creation, while seven represents the perfection, fullness, and completeness of that creation. Kuhnau's conclusion of both volumes with a seventh partita probably stood for such completeness, while his order may have represented the form and structure that God renders, out of void and chaos, into that which is visible, and, in the case of music, invisible: factorem coelli et terrae omnium visibilium et invisibilium.

Like Kuhnau's, the first volume of Bach's Keyboard Practice may have been conceived as seven partitas of which only the first six were published. Bach's partitas are in the keys of B-flat, c, a, D, G, e: a pattern of expanding intervals in alternate directions. This pattern allows us to predict that the key of the unpublished seventh partita, would have been a seventh below the key of the sixth: i.e. F major.

The tonal plan of the first volume of Bach's Keyboard Practice is immediately recognizable as an intentional set comparable to those of his Well-Tempered Clavier, Inventionen, & Sinfonien. While there can be little doubt that the order of the partitas is intentional, the meaning of the order is a matter of conjecture. Whether or not Bach planned to include a seventh partita, the six comprising his set echo Kuhnau's allusion to completeness.

Volume II of the Clavier-Übung contains two titles: the Italian Concerto in F-major and French Suite in b-minor comprising a total of fourteen movements. In them Bach explores the two national idioms of his day, the French manier and Italian gusto. Not surprisingly, the Italian Concerto is in the key implied by set characteristics of the preceding six partitas. If there is intentionality in volume II it has to do with contrast between two national styles in juxtaposition of unrelated keys (a tritone apart), flat signature versus sharps, opposing modes (major vs. minor), and genres (concerto vs. suite). Thus the composer's perfect "creation" of the preceding volume, is marred by dualistic juxtaposition of unrelated, if not opposing, elements in volume II. These opposing elements await resolution until 1739.

Resolution (and symbolic redemption) comes in the form of the so-called Organ Mass. Volume III of the Clavier-Übung was published eight years after the first folio in the series and two years before the Goldberg Variations. Unlike the other three, Volume III was composed for organ, the grandest instrument of the keyboard family. The book is comprised primarily of chorale preludes to hymns of the Lutheran liturgy. The "Organ Mass" contains numerous set characteristics that convey meaning. The most superficial of these is its abundance of ternary patterns representing the triune God. While there is much more to this cycle than triads, they do contribute to our understanding of a higher pattern that its composer appears to have invoked.

The "Organ Mass" consists of twenty-seven (3x3x3) movements grouped as follows: prelude and triple-fugue, twenty-one (3x7) chorale preludes, and 4 duets. The cycle begins with a prelude in E-flat major (3 flats) on the chorale Was mein Gott will, das. The prelude has three themes arranged in an enlarged ternary structure of nine parts: ABACABACA. The fugue that concludes the cycle has three subjects in three proportional meters. In the second exposition of this fugue Bach divides the beat into three equal parts, and in the third exposition he subdivides the divisions into three equal parts. Such parsing of meters is highly unusual for Bach and underscores the extent to which the composer appears to employ small patterns to intersect with a larger pattern (or, in this case, Being).

Ordinarily, when Bach follows a prelude with fugue on the same subject, the following is immediate, but in this instance the fugue is reserved for the conclusion of the Mass twenty-seven movements later. Such framing Bach uses rarely in his instrumental works but routinely in his choral, where it always has theological significance. The significance can be determined by looking at what falls between the framing movements...what the Germans call the Herzstück. The twenty-five sections sandwiched between this framing prelude and fugue have three functions within the Lutheran liturgy: Mass, catechism, and communion. The functions of specific movements are as follows:

  1. Nine (3+3+3) chorale preludes based upon the German equivalents of the Mass texts Kyrie & Gloria
  2. Twelve chorale preludes on texts related to Luther's Short Catechism: (12 being the number of ecclesiastical order--12 tribes of Israel, 12 apostles, etc.)
  3. Four duets for use during communion: It is possible that the four duets allude to the four application precepts with which Luther prefaces his Catechism.

Whereas the catechistic preludes of the Organ Mass are twelve in number, because they exist in pairs, they speak to but six texts. These six poems represent the heart of Reformation dogma in that they address, in order, the six components of Luther's Short Catechism: the Ten Commandments, the Creed, the Lord's Prayer, Baptism, Penitence, and the Lord's Supper. While the order is Martin Luther's, Bach employs three musical techniques (canon, pedal, and full organ) to correlate the whole within a threefold-elided frame, below.


Figure 1: musical frames of the catechistic
preludes from the Clavier-Übung III

As with the nine chorale preludes on Mass texts from the Clavier-Übung III, the canons of the Goldberg Variations conform to an extended ternary arrangement. But unlike the preludes, where the symmetry is built of texts and cantus firmi, the Goldberg builds its symmetry upon metrical relationships. In that sense the canonic set of the Goldberg Variations is, quite literally, symmetrical. In 1966 Jander discovered that, in the creation of this set, Bach used every possible combination of three measure groups (beats per measure) with three motor groups (underlying motor rhythm of each beat) to create nine meters, each with a unique architectonic structure. Two's, three's and four's comprise the groupings for beats per measure and motoric underlayment. The intentional aspect of this set is represented as in Fig. 2 and in the appendix. (Click the grid to play each canon)

Figure 2: metrical matrix of the Goldberg canons
Stop

In the figure to the right there appears to be no relationship between the meter of each canon and its chronological position within the Variations. But when the canons are returned to chronological order, a pattern does indeed emerge. Study Figure 3a, below. The bottom series of numbers represents the canons in chronological order. Immediately above this, a second series represents corresponding MOTOR groups, and above that, the MEASURE groups, in each canon. Figure 3a shows how the sequence of motor groups is repeated in measure groups three canons later: motor group of canon 1 becoming measure group of canon 4, motor group of canon 2 becoming measure group of canon 5, and so on. This arrangement makes for a recursive cycle in which the motor group of canon 7 becomes measure group of canon 1, motor group of canon 8 becomes measure group of canon 2, ad infinitum.



Figure 3: measure groups mimic motor groups three canons later
Curiously, the composer disrupts this pattern by a transposition of motor groups that should have appeared in canons five and nine (represented by the red arrows of Figure 3a). Without the exchange, the metrical set would have comprised an invariant and closed system with each of its metrical properties in correspondence with every other. This correspondence is represented by the diagonal lines of Figure 3a. With the exchange of motor groups from canons five and nine, the mimetic characteristics of the set are nevertheless recognizable as intentional. Not only so, but the pattern becomes all the more interesting for its subsequent disruption (represented by the Greek letter in figure 3b).

Whereas the maintains linkage between corresponding measure and motor groups in canons three, five, eight, and nine, the transposition of motor elements effects a pattern distortion in which those links cross. The distortion draws attention to itself because it does not participate in the set. Indeed, the distortion could not exist without the pattern against which it has been superimposed. Inasmuch as it is impossible to disrupt something that does not exist, I conclude that the pattern exists to permit its disruption. If this is true, and there is meaning to be found in this set, it should be found in the disruption, not in the pattern.

It is significant that this marring is consummated by a transposition of elements from canons five and nine. Between these two lie the only three where motor and measure groups exist in proportions of one-to-one. The sixth canon has 2 beats per measure, with each beat divided into 2 parts. The seventh canon has 4 beats per measure, with each beat divided into 4 parts. The eighth canon has 3 beats per measure, each beat divided into 3 parts. In the anagogic parlance of the eighteenth century, this simplest and noblest of numerical ratios (1:1) was the point of Unitas, and symbolized God.

I shall develop the significance of this proportional unity in greater detail momentarily. But, for now, it bears observation that the sum of five and nine (the canons framing the Unitas) is 14. Not only so, but the sum of transposed motor units, plus their corresponding measure units (3+4+3+4), is also 14. That there is an intentional set to the Goldberg canons, there can be no doubt. That the set has been disrupted, there can be little doubt. That the composer deliberately marred the set, is debatable. But that the composer stamped the marring with his number, the reader shall be left to infer what he wishes from the following discussion.

The symmetry of the Goldberg set devolves from a sequence of motor groups that is echoed, three canons later, by measured groups. This sequence reverberates motorically with two anomalies: products of the transposition of one pair of numbers. If one comprehends evidence of design in the undisrupted pattern, one might suspect that the anomaly itself might have been intentional. If not, then surely the composer has made a mistake. There are twenty-seven such "mistakes" possible of which two-thirds would have undone, as this one undoes, not only the sequence of measure and motor groups but also the architectonic structure of the nine canons, which this one does not (review Figure 2). If the switched integers were a mistake, there would have been a .67 probability that the canonic matrix would have looked like Figure 4 instead of Figure 2. The principle of entropy argues, therefore, against Bach having made a mistake as this would tend to have destroyed a pattern that could not have existed without careful forethought, and which the composer seems to have been intent upon creating. If not a mistake, then the disruption itself must have been conceived and executed by the composer.


Figure 4:
likely product of a random transposition of motor groups

Too, from what we know of this particular composer, erratic components seem out of character. Because his music is so tightly structured, the proposition that Johann Sebastian Bach made a mistake seems farcical. If the anomaly was a mistake, how did it mark something as significant as Unitas, and how so using disparate elements adding to 14 twice? There exist too many coincidences for the notion of an accidental disruption to be credible. I have defended this view by appealing to internal coherence of the set, logic, and the principle of entropy.

I propose, therefore, to develop the alternate view, that the marring was purposeful, itself part of the plan, and designed to intersect with a larger pattern conveying meaning. Having created something perfect, the composer appears to have sacrificed it. And in sacrificing it he may have said something profound about himself, his art, and himself in relation to his art. As for what is said, I shall attempt to show that this pattern distortion is driven by soteriologic and catechistic themes evident in similar structures of the "Organ" and B-Minor Masses. To demonstrate this it shall be necessary, first, to consider the eighteenth-century rhetorical device known as chiasmus.

Chiasmus involved the transposition of words at the beginnings and endings of sections. Transposition of elements, inversion, and mirroring held an unusual fascination for Bach. The Art of the Fugue contains, for example, two mirror fugues that can be played right side up or upside down. This mirroring extends not only to the inversion of melodies, but also to the entry of voices, key relationships, tonal functions, sequence directions, and cadences. Without speculating as to why Bach held such processes in high regard, it should be enough, when he employs them, for us to sit up and take notice. They often point to something else.

But what? In the literature of Bach's day similar transpositions demarcated boundary structures lending emphasis and coherence to the whole. The model for this practice may have been the Holy Scriptures, especially its Hebrew poetry. Whereas, for example, the book of Ruth refers to her sons, at first, as "Mahlon and Kilion," toward the end they become "Kilion and Mahlon," thereby marking the boundaries of that narrative. While chiasmus was well known as a rhetorical device, it is unclear the extent to which this may have been a conscious imitation of Scripture. Regardless of its source, there is no doubt that Bach used chiasmus, extensively, as a symbolic construct.

When Bach transposes the motor groupings three and four in his repetition of an otherwise symmetrical sequence, we do well, therefore, to ask: what might he be marking? The answer can be found in what lies between the transposed elements: canons six, seven, and eight--the only canons where the proportion between measure and motor groupings is in a one-to-one relationship. It is more than a coincidence that these three are contiguous. It is also significant that the composer uses not one canon, but three, in proportional Unitas. The connotation of Tri-Unitas is eloquent, and the Trinitarian conception of God as "three persons in one and one in three persons" comes immediately to mind. Accordingly I shall refer to these three canons as the "Triunitas" of the Goldberg set. Inasmuch as the triadic formulations of the prior "Organ Mass" also symbolized the triunity of God, I suspect that this Triunitas does the same.

Consider now how the composer assembles the Triunitas in his Goldberg set. The proportional relationship between measure and motor groups of canon six is 2:2, of canon seven is 4:4, and of canon eight is 3:3. This order is extremely meaningful, for it intersects with Martin Luther's division of the Nicene creed. Chafe informs us that Luther interpreted that creed as having articles related to creation, redemption, and sanctification. These articles corresponded to the three persons of the Trinity: Father, Son, and Holy Spirit. Chafe asserts that, in both his small and large catechisms, Luther devotes 2 clauses to the Father and creation, 4 clauses to the Son and redemption, and 3 clauses to the Holy Spirit and sanctification [note]. Thus, the total number of clauses is divided according to an asymmetrical 2+4+3 and not the symmetrical 3+3+3 of the Kyrie and Gloria.

Was Bach aware of Luther's asymmetrical division? The answer to that question is found in his Symbolum Nicenum of the B-Minor Mass. There he devotes 2 movements to the Father, 4 to the Son, and 3 to the Holy Spirit. Chafe emphasizes how this is a musical demarcation as well as textual...each of these sections is marked by a dramatic crescendo involving instrumentation and voices. In view of the fact that Luther's division of the creed was dictated by the order of creation, it is significant, perhaps, that the Goldberg's Triunitas employs canon six (the number of creation) to designate the Father: Patrem omnipotentem, factorem coeli et terrae. Likewise, it uses canon seven (fullness) to designate the Son: Deum de Deo, lumen de lumine.

Canon seven, at the center of the Triunitas, is also the second of three variations in minor mode. Canon seven is, therefore, Herzstück of two chiastic formulations. The first of these is the Triunitas itself, while the second is comprised of three variations in minor mode. This second of three places is, anagogically, the position occupied by Jesus, the second person of the Trinity. Only two of the minor mode variations are canons. Canon No. 7 (fullness and completeness) is positioned 7 movements after Canon No. 5 (Christ's passion), and 5 movements before Variation 25 (5x5). This last minor mode variation is the only one of the three that is not a canon.

The minor mode itself may be seen, from this perspective, as an allusion to Jesus passion, which is not the passion of an ordinary man but of God (three variations in minor). Of these minor mode variations, two are canons and one is not. Whereas it was God the Son who hung upon the cross (Variation 25 not a canon), it was God the father, and God the Holy Spirit (Canons 5 and 7) who participated in this passion (minor mode) with the Son. In this view Christ is linked to the one variation that is not canonic (his humanity), while the Father and Holy Spirit are linked to the two that are canonic. The non-canonic Variation 25, being the product of 5 squared, represents the fullness of that passion as well as Christ's dual nature: God and man. Finally, Variation 25, being the final act (so to speak) in minor mode, represents the "It is finished" that Jesus spoke from the cross. Wanda Landowska's designation of Variation 25 as a "Crown of Thorns," was probably suggested by these implied associations as well as a certain resemblance to the double mirror canon Christus Coronabit Crucigeros.

But if one looks at the chiastic of the Triunitas, one sees the inverse. Of the three canons in proportional unity, one is minor, while two are not. In this instance the one that is minor (No. 7) is the Christ canon, as represented by its position in the middle of the Triunitas, and by its 4:4 proportion (four clauses of the Creed). The other two canons of the Triunitas (No's 6 and 8) are not minor. Thus, between the three variations in minor and the three canons in Triunitas, there exists a distinction of functions, but a commingling of essence in which one member is set apart from the other two. Yet, over the course of the two chiastic formulations, each member is represented once as a canon. This double chiastic separation of one as belonging apart from the other two is, I believe, a representation of the incarnation.

Whereas the disruption of the canonic set serves to mark, chiastically, the unity of proportions in canons six, seven, and eight, visualizing the links between disrupted measure and motor units also produces the Christ sign: . This sign is superimposed upon the Triunitas, symbolizing not only Christ's deity but also the participation of the Godhead in redemption. It is intriguing to think that the composer may have chosen to effect this out of units suggesting his own name. Motor and measure units 3+4+3+4 comprising the extremities of this cross add to 14, the same sum as B+A+C+H, and the same sum as canons 5+9. Did the composer use these elements to suggest his willingness to participate in Christ's passion? In "suspending himself," so to speak, from the , was the composer creating a metamorphic representation of vicarious atonement? Or was he simply perpetuating a venerable tradition in which the artist paints himself into his own picture?

We have focused our attention, so far, upon the canons of the Triunitas. But the transposition of elements that mark this structure also forms an exclusive, or reverse, frame for the remaining six canons. I am suggesting, here, that the canonic sequence be viewed, like a reversible vest, from the inside out. This view is justified inasmuch as the pattern is circular. The set can be seen, then, as an eternal and self-generating continuum as in Figure 5.


Figure 5: circular characteristic of the Goldberg set
(a) Goldberg canons in chronological order
(b) innermost circle rotated clockwise three degrees

Figure 5a presents this continuum as a clock diagram in which the innermost circle represents motor groups, the middle circle represents measure groups, and the outer circle represents canon order. Notice that when the innermost circle is rotated clockwise three degrees (Figure 5b) the result is a convergence of values in two groups separated by a non-convergence in one group. This anomaly, the result of pattern distortion in canons five and nine, engulfs two variations. Convergence enfolds a group of four canons separated from a group of three. As a consequence, the circle can be seen as an eternal reiteration of the Credo sequence 2-4-3. But look again! With its sequence of 2 half steps separating a group of 4 pitches from another group of 3 pitches, the circle becomes an analogue of the diatonic system! Inasmuch as Luther's division of the creed was suggested by the order of creation, is it possible that Bach meant, here, to represent music as another object of God's creation generated by the Divine proportions of the Triunitas itself!

Consider the musical implications of proportions found within this Triunitas. Because of the disruption framing the Triunitas, canons 3,5,8,9 become unwilling participants in the mimetic parallels of their counterparts. Not only are they rendered oblique to the pattern, they become oblique to each other in a very interesting way. First, the asymmetrical proportions between 3:5:8:9 (inclusive of each number) represent yet another analogue, in retrograde, to Luther's creedal series 2-4-3. Second, and of musical significance, the differences between adjacent pairs of this eccentric proportion are 2, 3, and 1--proper divisors of that most sonorous number, six. Third, the 3:5:8:9 proportion yields the six aliquot divisions of the monochord that Zarlino (later Kirnberger & Rameau) used as acoustical generator for all voices, consonances, and species of harmony. Finally, to the modern musician, an interval vector of this set yields [1,1,1,1,1,1] showing that it is capable of producing each interval in the chromatic system, and the pitches of an all-interval tetrachord!

Returning to the circular characteristics of the set (Figure 5b), notice that the pattern disruption that defines the canons of the Triunitas as belonging together also defines the remaining six as belonging apart. And, if the Triunitas can be expressed in a single proportion, the six can be expressed in a variety of proportions, all factors of one. The relationship between the six canons of the reverse frame is one of reciprocating pairs (Figure 6). The integers comprising each of these pairs involve an exchange like the disruption that marked the Triunitas itself.


Figure 6: reciprocal pairs in the reverse framed canons
(canons 6, 7, and 8 representing "Triunitas")

A reciprocal is a proportion that, when multiplied, yields a product of one: Unitas. So, in these THREE reciprocals, Bach constructs yet another representation of the Holy Trinity. While three pairs of reciprocals are inevitable in the closed system Bach has constructed, the symmetrical pairing of them is a direct consequence of his pattern distortion. This symmetrical pairing (Figure 7, below) is eminently recognizable and full of significance. It is the same pattern of three interlocking and elided frames that the composer used in his "Organ Mass" to emphasize the structure of Luther's short catechism (figure 1). But here the commingling of mathematical elements in these reciprocals yields a powerful symbol of the unity of the Godhead. Each reciprocal is comprised of the mathematical "essence" of the other two.
Figure 7: triple elided frames within the reversely framed canons
So we come full circle. The Goldberg set not only intersects with the 2-4-3 symbol of the B-Minor Mass, it also intersects with the chiastic structure of the "Organ Mass." It is important to understand that the Bachian symmetry of these interlocking frames is made possible only after the motor groups of canons five and nine have been transposed. Without transposition the reciprocating pairs would have elided as in Figure 8a--an arrangement atypical for Bach. But, after the transposition, the pairs elide as in Figure 8b--an arrangement typical of Bach, and exactly the same frame employed in the catechistic preludes of the Clavier-Übung III (please review Figure 1).
Figure 8:
(a) atypical frame (before disruption of set)
(b) typical frame (after disruption)

It should be obvious by now that the meaning of the Goldberg set is primarily found not in its pattern, but in its disruption. This disruption creates numerous significances of which I have touched upon several. The preeminent meaning is, of course, theological--specifically Trinitarian in substance and in scope. In view of the fact that the products of the three interlocking reciprocals are individually ONE, and, when multiplied by themselves (1x1x1) STILL one, the canons are mystical expressions of the creed: Wir glauben all an einen Gott, Vater, Sohn, und heilign Geist . . . Der durch seine grosse Krafft Alles wurcket, thut und schafft. This interpretation is not only congruent with Trinitarian connotations traditionally attached to the "Organ Mass" preceding the Goldberg Variations, it is also consistent with Bach's understanding of the Credo structure exemplified in his B-Minor Mass.

By using the symbols of his own name to disrupt the set, thereby symbolizing the creed, Bach eloquently affirms that this is not just any creed, but his own. In this aspect of the disruption we detect a soteriological theme, for the transposed elements produce the familiar --the symbol of Christ, his cross, and the believer's salvation, since the first century A.D. Thus the meaning transcends theism to embrace a theology both redemptive, explicitly Christian, and highly personal. As if to show that it was his own sin that marred God's perfect creation, Bach switches motor groups in canons five and nine, the sum of which is fourteen--his signature number--while the reciprocating pairs of integers (3/4 & 4/3) marking the four points of this cross are also addends of fourteen.

While Trinitarian apologists delight in analogies from nature and the constitution of man, we must politely reject the notion that the mathematical unity of Bach's canonic set was intended to be an argument for the triunity of God. While analogies are interesting, they do not prove anything, nor is the canonic set of the Goldberg canons such a proof. The Lutherans were very clear on this point. Johann Muller's Judaismus, a copy of which Bach owned, identifies two types of numerological symbols of which only the first was permissible. Cabbala Speculativa involved the use of numbers to allude to Scripture in an ingenious manner, while Cabbala Practica used numbers to interpret Scripture. Bach could not possibly have had in mind a defense of Lutheran belief and practice (he would have been preaching to the converted), but, rather, an expression of it by means of mathematical, and musical, processes.

Like a string of pearls adorning the neck of a lady, nine canons grace Bach's Aria with Thirty Variations, giving them mystery as well as form and substance. "Prepared for the enjoyment of music lovers" they have indeed made joyful generations of thankful musicians. Yet through these pleasurable tones the thoughtful ear faintly hears echoes of an earlier note humbly scored in his Bible and with his hand: "splendid proof," wrote Bach, "that besides other arrangements of the service of worship, music too was especially ordered by God's spirit through David." All music--choral, orchestral, sacred, secular, vocal or Clavier--for Bach there appears to have been but one Übung--one order--that which was decreed by the Spirit of God. In the esoteric symbols of his set Johann Sebastian participates not only in the spirit of his age, but also the spirit of creative ingenuity, and in the Spirit of his God.

Hearing this masterpiece of "autonomous" music in harmony with Bach's faith and liturgical oeuvre cannot help but raise, for some, as many questions as they might have hoped to have had answered. But, least we forget, two-thirds of the ink that flowed from the composer's quill was scratched upon the parchment of Lutheran praxis. To the one who scratched it, at least, this reading of his Variations would not have seemed at all strange. Book IV of his "Keyboard Practice" is best heard after Book III.

Appendix

Owen Jander is, to this writer's knowledge, the first to have noticed the metrical symmetry of the Goldberg canons. Jander apparently did not understand the plenary relationship between measure and motor groups itself to be nearly canonic. He specifically avoided speculation of the sort that might have gotten him into polemical hot water. Jander's analysis, first published in 1966, and republished in the "Musical Quarterly" ("Rhythmic Symmetry in the Goldberg Variations," Winter 1991, pp. 188-193) was crucial to this writer's understanding of the sequential disruption and consequent proportional/reciprocal meanings. Jander's original observation is reiterated here for the reader's convenience.

Of the canons with TWO beats per measure:

Of the canons with THREE beats per measure: Of the canons with FOUR beats per measure: