Meaningfulness in Bach's Cyclical Works

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Site 1996 Timothy A. Smith

Author BACH Courses

Intentionality and Meaningfulnessin 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. The musical examples are designed to be played from Trevor

Pinnock's (Archiv 415 130-2) recording of the "Goldberg Variations" BWV 988. This method for

obtaining sound was intended for my students who have access to this particular disk. If you have

obtained the FREE helper application from Voyager (and configured your computer to receive

these files), you should be able to insert the Archiv disc in the CD drive of your computer and hear

the examples.

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 necessarilywarrant inference that a large-scale cycle ought to be performed in set order or withoutinterruption. But, in situations where arise uncertainties about completeness, or whenextraneous factors call into question the "belonging" of elements to a set, evidence ofintentionality can be enlightening. Internal coherence of the B-Minor Mass, forexample, indicates that Bach conceived of that work as a set, whereas historicalincidents culminating in the composition of the Mass are sometimes used to urge apoint 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 firstprelude and fugue are in C-major, second in c-minor, third in C#-major, fourth in


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c#-minor, and so on. This pattern infers that the set is completed after Fuga XXIV. Tocontinue beyond this point would be to commence a repetition of the cycle, which thecomposer does in Volume II. Further, if parts of the cycle were missing, apprehensionof the pattern would enable one to determine not only how many preludes and fugueshad been lost, but also in what keys.

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

As the name implies, the Well-Tempered Clavier was also written to demonstrate the

range ofAffektenlehre that could be achieved by a "well-tempered" scale. These affectsaccrue, writes Harnoncourt, from "various strong tensions, created by intonation, whichincrease with distance from the C major center and which are also felt as a kind oflonging for the beautiful, relaxed keys (F major, C major, G major)." The set structureof the Well-Tempered Clavier can be seen, in such a light, as an artistic manifestation ofphysical and metaphysical presuppositions, peculiar to the eighteenth century. Thepresupposition is one ofAffekt, the most important aesthetic concept of that era, and themanifestation seeks to exploit affect to the fullest extent possible. Obviously the cardcatalogue organization of the set is not a product of affect, but the systematic structureimplies 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, notmerely with respect to order, but wherein the order itself is born out of higher patternsof thought or thing (see Pike). To J. S. Bach, for whom music was not only the physicalrepresentation of affection, but subject to the same laws governing the cosmos,meaningfulness would have been exceedingly important. After all, music was, in thewords of his student, Mizler, "sounding mathematics." Nicolaus Harnoncourtcomments, of the 18th-century preoccupation with proportions in music, art, andarchitecture, 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)


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Because of their position within the large-scale context of the Clavier-bung, Bach'sGoldberg Variations hint of meaningful possibilities to wit. That the Goldbergscomprise an intentional pattern is self-evident; every third variation is canonic, with theinterval between leader and follower expanding by one scale step in each successivecanon. The apparent intentionality of this set gives pause to wonder if it was generatedby 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. Weshall begin by examining Bach's model (Kuhnau's two volumes by the same name), andthe preceding three volumes of Bach's own Clavier-bung.

The Goldberg Variations comprise the fourth, and last, volume of the Clavier-bung.The composer produced volume I of that cycle in 1731, a year when he surely could nothave envisioned writing the Goldbergs nearly a decade later. Bach had moved toLeipzig only five years earlier to assume the post of predecessor, Johann Kuhnau. Thefirst composer to use the term Clavier-bung ("Keyboard Practice"), Kuhnau titled twovolumes 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-bung,shall help us to understand Bach's complementary titles, especially his GoldbergVariations.

Whereas the first volume of Kuhnau's model consists of seven partitas in ascendingmajor keys (C, D, E, F, G, A, B-flat), the second consists of a similar sequence, inminor (c, d, e, f, g, a, b). The intentionality of this set is found in its ordering andgrouping of tonic and mode. Curiously, the order incorporates intonationalimpediments that render both volumes impractical in performance. These impedimentsare difficult to justify without recourse to symbolic associations that the Lutheran

culture would have traditionally attached to the numbers six and seven. From antiquitythe number six has represented perfection. In Judeo-Christian cosmogony six stands forthe creation, while seven represents the perfection, fullness, and completeness of thatcreation. Kuhnau's conclusion of both volumes with a seventh partita probably stoodfor such completeness, while his order may have represented the form and structurethat God renders, out of void and chaos, into that which is visible, and, in the case ofmusic, invisible:factorem coelli et terrae omnium visibilium et invisibilium.

Like Kuhnau's, the first volume of Bach's Keyboard Practice may have been conceivedas 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. Thispattern allows us to predict that the key of the unpublished seventh partita, would havebeen a seventh below the key of the sixth: i.e. F major.

the 2nd partita is at the interval of a 2nd above the 1st

the 3rd partita is at the interval of a 3rd below the 2nd

the 4th partita is at the interval of a 4th above the 3rd

the 5th partita is at the interval of a 5th below the 4th

the 6th partita is at the interval of a 6th above the 5th

The tonal plan of the first volume of Bach's Keyboard Practice is immediatelyrecognizable as an intentional set comparable to those of his Well-Tempered Clavier,


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Inventionen, & Sinfonien. While there can be little doubt that the order of the partitas isintentional, the meaning of the order is a matter of conjecture. Whether or not Bachplanned to include a seventh partita, the six comprising his set echo Kuhnau's allusionto completeness.

Volume II of the Clavier-bung contains two titles: the Italian Concerto in F-major and

French Suite in b-minor comprising a total of fourteen movements. In them Bachexplores the two national idioms of his day, the French manier and Italian gusto. Notsurprisingly, the Italian Concerto is in the key implied by set characteristics of thepreceding six partitas. If there is intentionality in volume II it has to do with contrastbetween two national styles in juxtaposition of unrelated keys (a tritone apart), flatsignature versus sharps, opposing modes (major vs. minor), and genres (concerto vs.suite). Thus the composer's perfect "creation" of the preceding volume, is marred bydualistic juxtaposition of unrelated, if not opposing, elements in volume II. Theseopposing elements await resolution until 1739.

Resolution (and symbolic redemption) comes in the form of the so-called Organ Mass.Volume III of the Clavier-bung was published eight years after the first folio in theseries and two years before the Goldberg Variations. Unlike the other three, Volume IIIwas composed for organ, the grandest instrument of the keyboard family. The book iscomprised primarily of chorale preludes to hymns of the Lutheran liturgy. The "OrganMass" contains numerous set characteristics that convey meaning. The most superficialof these is its abundance of ternary patterns representing the triune God. While there ismuch more to this cycle than triads, they do contribute to our understanding of a higherpattern 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 cyclebegins 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 threeproportional meters. In the second exposition of this fugue Bach divides the beat intothree equal parts, and in the third exposition he subdivides the divisions into three equalparts. Such parsing of meters is highly unusual for Bach and underscores the extent towhich 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 followingis immediate, but in this instance the fugue is reserved for the conclusion of the Masstwenty-seven movements later. Such framing Bach uses rarely in his instrumentalworks but routinely in his choral, where it always has theological significance. Thesignificance can be determined by looking at what falls between the framingmovements...what the Germans call theHerzstck. The twenty-five sectionssandwiched between this framing prelude and fugue have three functions within theLutheran liturgy: Mass, catechism, and communion. The functions of specificmovements are as follows:

Nine (3+3+3) chorale preludes based upon the German equivalents of the Mass1.


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texts Kyrie & GloriaTwelve chorale preludes on texts related to Luther's Short Catechism: (12 beingthe number of ecclesiastical order--12 tribes of Israel, 12 apostles, etc.)

2.

Four duets for use during communion: It is possible that the four duets allude tothe four application precepts with which Luther prefaces his Catechism.

3.

Whereas the catechistic preludes of the Organ Mass are twelve in number, because theyexist in pairs, they speak to but six texts. These six poems represent the heart ofReformation dogma in that they address, in order, the six components of Luther's ShortCatechism: 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 musicaltechniques (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-bung III

As with the nine chorale preludes on Mass texts from the Clavier-bung III, the canonsof the Goldberg Variations conform to an extended ternary arrangement. But unlike thepreludes, where the symmetry is built of texts and cantus firmi, the Goldberg builds itssymmetry upon metrical relationships. In that sense the canonic set of the GoldbergVariations is, quite literally, symmetrical. In 1966 Jander discovered that, in thecreation 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) tocreate nine meters, each with a unique architectonic structure. Two's, three's and four'scomprise the groupings for beats per measure and motoric underlayment. Theintentional aspect of this set is represented as in Fig. 2 and in the appendix. (Click thegrid to play each canon)

Figure 2: metrical matrix of the Goldberg canons

Stop


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In the figure to the right there appears to be no relationshipbetween 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 inchronological order. Immediately above this, a second series represents correspondingMOTOR 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 canonslater: motor group of canon 1 becoming measure group of canon 4, motor group ofcanon 2 becoming measure group of canon 5, and so on. This arrangement makes for arecursive cycle in which the motor group of canon 7 becomes measure group of canon1, 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 thatshould have appeared in canons five and nine (represented by the red arrows of Figure3a). Without the exchange, the metrical set would have comprised an invariant andclosed system with each of its metrical properties in correspondence with every other.This correspondence is represented by the diagonal lines of Figure 3a. With theexchange of motor groups from canons five and nine, the mimetic characteristics of theset are nevertheless recognizable as intentional. Not only so, but the pattern becomes allthe more interesting for its subsequent disruption (represented by the Greek letter infigure 3b).

Whereas the maintains linkage between corresponding measure and motor groups incanons three, five, eight, and nine, the transposition of motor elements effects a patterndistortion in which those links cross. The distortion draws attention to itself because itdoes not participate in the set. Indeed, the distortion could not exist without the patternagainst which it has been superimposed. Inasmuch as it is impossible to disruptsomething 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 thedisruption, not in the pattern.

It is significant that this marring is consummated by a transposition of elements fromcanons five and nine. Between these two lie the only three where motor and measuregroups 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, witheach beat divided into 4 parts. The eighth canon has 3 beats per measure, each beatdivided into 3 parts. In the anagogic parlance of the eighteenth century, this simplestand noblest of numerical ratios (1:1) was the point ofUnitas, and symbolized God.


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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 theUnitas) is 14. Not only so, but the sum of transposed motor units, plus theircorresponding measure units (3+4+3+4), is also 14. That there is an intentional set tothe Goldberg canons, there can be no doubt. That the set has been disrupted, there canbe 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 hewishes from the following discussion.

The symmetry of the Goldberg set devolves from a sequence of motor groups that isechoed, three canons later, by measured groups. This sequence reverberates motoricallywith two anomalies: products of the transposition of one pair of numbers. If onecomprehends evidence of design in the undisrupted pattern, one might suspect that theanomaly itself might have been intentional. If not, then surely the composer has made amistake. There are twenty-seven such "mistakes" possible of which two-thirds wouldhave 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 (reviewFigure 2). If the switched integers were a mistake, there would have been a .67probability 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 thiswould tend to have destroyed a pattern that could not have existed without carefulforethought, and which the composer seems to have been intent upon creating. If not amistake, then the disruption itself must have been conceived and executed by thecomposer.

Figure 4:

likely product of a random transposition of motor groups

Too, from what we know of this particular composer, erratic componentsseem out of character. Because his music is so tightly structured, theproposition that Johann Sebastian Bach made a mistake seems farcical. If the anomalywas a mistake, how did it mark something as significant as Unitas, and how so usingdisparate elements adding to 14 twice? There exist too many coincidences for thenotion of an accidental disruption to be credible. I have defended this view byappealing 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 conveyingmeaning. 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, andhimself in relation to his art. As for what is said, I shall attempt to show that this patterndistortion is driven by soteriologic and catechistic themes evident in similar structuresof the "Organ" and B-Minor Masses. To demonstrate this it shall be necessary, first, toconsider 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


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fascination for Bach. TheArt of the Fugue contains, for example, two mirror fuguesthat can be played right side up or upside down. This mirroring extends not only to theinversion of melodies, but also to the entry of voices, key relationships, tonal functions,sequence directions, and cadences. Without speculating as to why Bach held suchprocesses in high regard, it should be enough, when he employs them, for us to sit upand take notice. They often point to something else.

But what? In the literature of Bach's day similar transpositions demarcated boundarystructures lending emphasis and coherence to the whole. The model for this practicemay have been the Holy Scriptures, especially its Hebrew poetry. Whereas, forexample, the book of Ruth refers to her sons, at first, as "Mahlon and Kilion," towardthe end they become "Kilion and Mahlon," thereby marking the boundaries of thatnarrative. While chiasmus was well known as a rhetorical device, it is unclear theextent to which this may have been a conscious imitation of Scripture. Regardless of itssource, 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 anotherwise symmetrical sequence, we do well, therefore, to ask: what might he bemarking? The answer can be found in what lies between the transposed elements:canons six, seven, and eight--the only canons where the proportion between measureand motor groupings is in a one-to-one relationship. It is more than a coincidence thatthese three are contiguous. It is also significant that the composer uses not one canon,but three, in proportional Unitas. The connotation ofTri-Unitas is eloquent, and theTrinitarian conception of God as "three persons in one and one in three persons" comesimmediately 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. Theproportional relationship between measure and motor groups of canon six is 2:2, ofcanon seven is 4:4, and of canon eight is 3:3. This order is extremely meaningful, for itintersects with Martin Luther's division of the Nicene creed. Chafe informs us thatLuther interpreted that creed as having articles related to creation, redemption, andsanctification. 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, Lutherdevotes 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 isdivided according to an asymmetrical 2+4+3 and not the symmetrical 3+3+3 of theKyrie and Gloria.

Was Bach aware of Luther's asymmetrical division? The answer to that question isfound in his Symbolum Nicenum of the B-Minor Mass. There he devotes 2 movementsto the Father, 4 to the Son, and 3 to the Holy Spirit. Chafe emphasizes how this is amusical demarcation as well as textual...each of these sections is marked by a dramaticcrescendo involving instrumentation and voices. In view of the fact that Luther'sdivision 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 theFather: Patrem omnipotentem, factorem coeli et terrae. Likewise, it uses canon seven


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(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 inminor mode. Canon seven is, therefore,Herzstckof two chiastic formulations. Thefirst of these is the Triunitas itself, while the second is comprised of three variations inminor 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 arecanons. Canon No. 7 (fullness and completeness) is positioned 7 movements afterCanon No. 5 (Christ's passion), and 5 movements before Variation 25 (5x5). This lastminor 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 Jesuspassion, which is not the passion of an ordinary man but of God (three variations inminor). Of these minor mode variations, two are canons and one is not. Whereas it wasGod 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 ofthat passion as well as Christ's dual nature: God and man. Finally, Variation 25, beingthe final act (so to speak) in minor mode, represents the "It is finished" that Jesus spokefrom the cross. Wanda Landowska's designation of Variation 25 as a "Crown ofThorns," was probably suggested by these implied associations as well as a certainresemblance 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 onethat is minor (No. 7) is the Christ canon, as represented by its position in the middle ofthe Triunitas, and by its 4:4 proportion (four clauses of the Creed). The other twocanons of the Triunitas (No's 6 and 8) are not minor. Thus, between the three variationsin minor and the three canons in Triunitas, there exists a distinction of functions, but acommingling of essence in which one member is set apart from the other two. Yet, overthe course of the two chiastic formulations, each member is represented once as acanon. This double chiastic separation of one as belonging apart from the other two is, Ibelieve, a representation of the incarnation.

Whereas the disruption of the canonic set serves to mark, chiastically, the unity ofproportions in canons six, seven, and eight, visualizing the links between disruptedmeasure and motor units also produces the Christ sign: . This sign is superimposedupon the Triunitas, symbolizing not only Christ's deity but also the participation of theGodhead in redemption. It is intriguing to think that the composer may have chosen toeffect this out of units suggesting his own name. Motor and measure units 3+4+3+4comprising the extremities of this cross add to 14, the same sum as B+A+C+H, and thesame sum as canons 5+9. Did the composer use these elements to suggest hiswillingness 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 artistpaints himselfinto his own picture?


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We have focused our attention, so far, upon the canons of the Triunitas. But thetransposition 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 beviewed, like a reversible vest, from the inside out. This view is justified inasmuch asthe pattern is circular. The set can be seen, then, as an eternal and self-generatingcontinuum 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 circlerepresents motor groups, the middle circle represents measure groups, and the outercircle represents canon order. Notice that when the innermost circle is rotated clockwisethree degrees (Figure 5b) the result is a convergence of values in two groups separatedby a non-convergence in one group. This anomaly, the result of pattern distortion incanons five and nine, engulfs two variations. Convergence enfolds a group of fourcanons 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 of2half steps separating a group of4 pitches from another group of3 pitches, the circlebecomes an analogue of the diatonic system! Inasmuch as Luther's division of the creedwas suggested by the order of creation, is it possible that Bach meant, here, to representmusic as another object of God's creation generated by the Divine proportions of theTriunitas itself!

Consider the musical implications of proportions found within this Triunitas. Becauseof the disruption framing the Triunitas, canons 3,5,8,9 become unwilling participants inthe 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, theasymmetrical proportions between 3:5:8:9 (inclusive of each number) represent yetanother analogue, in retrograde, to Luther's creedal series 2-4-3. Second, and of musicalsignificance, 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 proportionyields the six aliquot divisions of the monochord that Zarlino (later Kirnberger &Rameau) used as acoustical generator for all voices, consonances, and species ofharmony. 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 chromaticsystem, and the pitches of an all-interval tetrachord!

Returning to the circular characteristics of the set (Figure 5b), notice that the pattern


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disruption that defines the canons of the Triunitas as belonging together also defines theremaining six as belonging apart. And, if the Triunitas can be expressed in a singleproportion, the six can be expressed in a variety of proportions, all factors of one. Therelationship 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 thedisruption 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 HolyTrinity. While three pairs of reciprocals are inevitable in the closed system Bach hasconstructed, the symmetrical pairing of them is a direct consequence of his patterndistortion. This symmetrical pairing (Figure 7, below) is eminently recognizable andfull of significance. It is the same pattern of three interlocking and elided frames thatthe composer used in his "Organ Mass" to emphasize the structure of Luther's shortcatechism (figure 1). But here the commingling of mathematical elements in thesereciprocals 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 intersectswith the 2-4-3 symbol of the B-Minor Mass, it also intersectswith the chiastic structure of the "Organ Mass." It is important to understand that theBachian symmetry of these interlocking frames is made possible only after the motorgroups of canons five and nine have been transposed. Without transposition thereciprocating 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 arrangementtypical of Bach, and exactly the same frame employed in the catechistic preludes of theClavier-bung 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 foundnot in its pattern, but in its disruption. This disruption creates numerous significances ofwhich 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


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of the three interlocking reciprocals are individually ONE, and, when multiplied bythemselves (1x1x1) STILL one, the canons are mystical expressions of the creed: Wirglauben all an einen Gott, Vater, Sohn, und heilign Geist . . . Der durch seine grosseKrafft Alles wurcket, thut und schafft. This interpretation is not only congruent withTrinitarian connotations traditionally attached to the "Organ Mass" preceding theGoldberg 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 thecreed, Bach eloquently affirms that this is not just any creed, but his own. In this aspectof the disruption we detect a soteriological theme, for the transposed elements producethe familiar --the symbol of Christ, his cross, and the believer's salvation, since thefirst century A.D. Thus the meaning transcends theism to embrace a theology bothredemptive, explicitly Christian, and highly personal. As if to show that it was his ownsin that marred God's perfect creation, Bach switches motor groups in canons five andnine, 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 offourteen.

While Trinitarian apologists delight in analogies from nature and the constitution ofman, we must politely reject the notion that the mathematical unity of Bach's canonicset was intended to be an argument for the triunity of God. While analogies areinteresting, they do not prove anything, nor is the canonic set of the Goldberg canonssuch a proof. The Lutherans were very clear on this point. Johann Muller'sJudaismus, acopy of which Bach owned, identifies two types of numerological symbols of whichonly the first was permissible. Cabbala Speculativa involved the use of numbers to

allude to Scripture in an ingenious manner, while Cabbala Practica used numbers tointerpret Scripture. Bach could not possibly have had in mind a defense of Lutheranbelief and practice (he would have been preaching to the converted), but, rather, anexpression of it by means of mathematical, and musical, processes.

Like a string of pearls adorning the neck of a lady, nine canons grace Bach'sAria withThirty Variations, giving them mystery as well as form and substance. "Prepared for theenjoyment of music lovers" they have indeed made joyful generations of thankfulmusicians. Yet through these pleasurable tones the thoughtful ear faintly hears echoesof 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 wasespecially 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 bung--oneorder--that which was decreed by the Spirit of God. In the esoteric symbols of his setJohann Sebastian participates not only in the spirit of his age, but also the spirit ofcreative ingenuity, and in the Spirit of his God.

Hearing this masterpiece of "autonomous" music in harmony with Bach's faith andliturgical oeuvre cannot help but raise, for some, as many questions as they might havehoped 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 theone who scratched it, at least, this reading of his Variations would not have seemed at


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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 metricalsymmetry of the Goldberg canons. Jander apparently did not understand the plenaryrelationship between measure and motor groups itself to be nearly canonic. Hespecifically avoided speculation of the sort that might have gotten him into polemicalhot water. Jander's analysis, first published in 1966, and republished in the "MusicalQuarterly" ("Rhythmic Symmetry in the Goldberg Variations," Winter 1991, pp.188-193) was crucial to this writer's understanding of the sequential disruption andconsequent proportional/reciprocal meanings. Jander's original observation is reiteratedhere for the reader's convenience.

Of the canons with TWO beats per measure:

No. 6 divides each beat into TWO

Stop Play Canon No. 6

No. 9 divides each beat into THREE


No. 5 divides each beat into FOUR


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Of the canons with THREE beats per measure:







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Of the canons with FOUR beats per measure:







Notes

A "set" is a collection of things that belong together as a group. Music theorists use pitch

class sets, for example, to demonstrate organic relationships between chords and melodies inatonal music. In the context of this article these things are large-scale cycles, or, in the case


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of the Goldberg Variations, a subset (the nine canons) of a larger cycle. This author uses

"set" in the same sense as Michael Marissen's, "J. S. Bach's Brandenburg Concertos as a

Meaningful Set," Musical Quarterly 77:2 (Summer 1993): 193-236. Unlike Marissen's, this

essay distinguishes between "intentional" and "meaningful" sets. The latter are collections

where order represents a reality extrinsic to itself. Not all intentional sets are meaningful,

but all meaningful sets are intentional.

In the Mass in B-Minor, Bach constructs a twenty-one-movement symmetry in which the

Crucifixus is placed precisely between the Gratias and theDona Nobis Pacem. Other thanthe repeat of theHosanna (an essential component of this symmetry), the Gratias andDona

Nobis are the only movements in the B-Minor where a musical idea is repeated. Thetwenty-one movements begin in the Gloria and continue through the Credo, Sanctus,

Benedictus, andAgnus Dei to the end of the Mass. Corresponding movements on either sideof the Crucifixus mirror key relationships, performance forces, use ofcantus firmus, andcompositional styles. The cruciform plan of the Mass is evidence that Bach conceived that

work as a whole in spite of the fact that pieces of it were composed for different patrons and

occasions, much of it borrowed from earlier works, and the completed Mass never

performed during the composer's lifetime.

Contrary to popular understanding, the Well-Tempered Clavier was not written todemonstrate that the same piece could be played in any key. If anything, it showed the

opposite. To be sure, the cycle did exploit an expanded range of key relationships that would

have excited the imagination of composers before this date restricted, by mean-tone, to

exploring closely related keys. Unlike modern equal temperament, Bach's Well-TemperedClavier retained sonorities unique to each key that the skilled composer could exploit toexpressive advantage. Such a composer would have united each piece to the key that created

the desired affect. The eighteenth-century ear, accustomed to hearing temperamental

imperfections as expressive of longing for perfection, would have heard an E major preludeas embodying a different affect when transposed to the "softer and more relaxed" key of F

(Harnoncourt p. 66). For an excellent discussion of this subject see "From 'Mean-Tone' to

'Well-Tempered' Tuning" in Harnoncourt's Baroque Music Today: Music As Speech

(Portland: Amadeus Press, 1982).

Tuning to an F# mean-tone it is possible to play Kuhnau's major partitas in B-flat, C, D, F,

G, and A, and minor partitas in g, d, and a. The remaining works (E, b, c, e, f) would have

required retuning of the instrument. So, the scalar ordering imposes an intonational obstacle

that makes Kuhnau's cycle impractical in performance at one sitting. We surmise,

accordingly, that Kuhnau's order exists to "mean" a theoretical perfection that before the

Enlightenment was always rooted in theological presupposition.

The following ideas on the number six are synthesized from Marissen. Most Europeans of

the 18th century continued to regard the number six as a sign of perfection (signumperfectionis). The basis for this association had been, since ancient times, mathematical,musical, and theological. As the first number to equal the sum of its proper divisors 1+2+3,

six was, according to Pythagoras, the first "perfect number."

But to musicians, six was even more significant. When, in 1558 (Institutioni Harmoniche),Gioseffo Zarlino increased the possible divisions of the monochord (a vibrating string) from

four to six, he was able, for the first time, to generate every consonant interval. From thisdivision Zarlino theorized that there are six species of voices (Unisone, Equisone, Consone,


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Emelle, Dissone, & Ecmele), six consonances (Diapason, Diapente, Diatessaron, Ditino,Semiditino, & Unisono), and six species of harmony (Doria, Frigia, Lidia, Mistalidia o

Lochrense, Eolia & Ionica).

Zarlino's senario became the acoustical foundation for all of music theory through theeighteenth century. J. S. Bach's relative, Walther devoted a chapter of hisMusicalisches

Lexicon (1732) to the properties of theNumerus perfectus, while Bach's pupil, Kirnberger,inDie Kunst des reinen Satzes in der Musik(1771-79), explained scales, temperament, andintervals, in terms of acoustical proportions derived from Zarlino's senario. Before helearned of the harmonic series, Rameau also based his ideas upon the division of the

monochord (Traite de l'harmonie, 1722). It was theorized by musicians of the day that thereason God created the cosmos in six days was because of the strongly sonorous

characteristics of that number.

See Genesis 1:31. Biblical connotations were not only common in eighteenth-century art,

they were expected. Without allusion to the symbols of mathematical or theological

perfection, works of art were in jeopardy of being seen as less than perfect. Marissen points

out that J. S. Bach did not hesitate to incorporate symbolic allusions to the completeness orperfections of his own works...even in jest, as when he refers to the "imperfections" of his

six Brandenburg concerti.

See Genesis 2:1-2.

We suspect that Bach conceived of a seventh partita because an announcement in the

Leipzig Post-Zeitung, May 1730, indicates that after completion of the fifth partita Bachcontinued to work on two more.

Bernard Greenberg pointed out to me that the symbolic connotations of six and seven may

have represented, to the native German speaker, a distinction without a difference. The term

Vollkommenheitexpresses both the domains of perfection (whether it be mathematical orcreative) as well as completeness and fullness. The speculation that necessarily attends such

interpretations is cause for no small controversy. Individuals not inclined to theorize in like

manner see, in those who do, a simplistic following whose hermeneutics enable them

always to win. "Regardless of the status of the missing seventh partita," they might well

object, "the interpretive Simons will have their symbol." To be sure, allegorical

interpretations of the music of Bach have been taken to the extreme, but there does swim,

within the mainstream of Bach scholarship, a species of interpretive literature, with which

there is general agreement, and which is cognizant of intentionality and meaningfulness,

particularly in large-scale sets that are the subjects of this investigation.

English organists began calling this the "St. Anne" fugue for its resemblance to William

Croft's "O God Our Help in Ages Past." While they were contemporaries, it is unlikely that

Johann Sebastian had knowledge of Croft's tune.

"Chiastic" structure (from Greek "Chi") involves the creation of analogous units on either

side of a structural "heart" (German Herzstck). Also known as "cruciform structure," suchpatterns are considered to be, in the context of Christian liturgy, Christological symbols.

The first group, consisting of a Kyrie, Christe, Kyrie. is united by similarity ofcantus firmustechnique in stile antico. A second group ofKyrie, Christe, Kyrie is united by verset-like


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structure and proportional meters. The three settings ofAllein Gott in der Hh are arrangedtwo for manuals framing a setting for full organ.

David Humphreys, The Esoteric Structure of Bach's Clavier-bung III (Cardiff: Universityof Cardiff Press, 1983), 8.

Chi, the first letter in the Greek spelling ofChristus, and a Christological symbol sinceancient times, appears in Bach's manuscripts as a substitute for the written words Christus(Christ) or Kreuz (cross). See, for example, the Handexemplar of the aria, Gerne will ichmich bequemen und Becher anzunehmen, from the St. Matthew Passion. Whether Bachconceived of metrical pattern distortion in the Goldberg canons as such a symbol, it is

impossible to determine. I do, however, see the distortion as symbolic, and appeal to

Dorothy Sayers:

The function of imaginative speech is not to prove, but to create--to discover

new similarities, and to arrange them to form new entities, to build new

self-consistent worlds out of the universe of undifferentiated mind-stuff.

("Christian Letters to a Post-Christian World" Grand Rapids: Eerdmans, 1969,p. xiii)

Meaning can be found wherever one chooses to find it. The composer's intentions, while

they are useful, and interesting, are not a prerequisite for meaningfulness to be attached to a

musical work of art. Two lovers listening to "our song" on the radio should be proof enough.

Any theory of meaning must therefore allow humans to follow one or both of two

directions: what the set may have meant to the artist (and I emphasize "may"), and what the

set means to the beholder. While determination of the former may well influence the latter, it

may well not. And any meaning attached by the beholder, whether or not it converges with

the intentions of the artist, is valid. I cannot say, "No, this cannot mean thus and such toyou," when you know full well that it does. By the same token you cannot say that what is

meaningful to me is not, when I know it is. Finally, none of us can say with any degree of

certainty that any of this meant anything to Bach...he is not here to tell us what he meant.

Given our understanding of eighteenth-centuryZeitgeist, Bach's own writings, and theimmediate context of his Goldberg Variations, it is possible, however, to assert that Bach

"may" have meant certain things by his canonic set, and that, regardless of what he meant

(or didn't mean), we are justified in apprehending meaning, in structure, nonetheless. If

nothing else, structure indicates intelligence and purposefulness, which itself is a powerful

meaning.

Accordingly, the purpose of the remainder of this essay shall be to create intersections

between the intentional patterns of the Goldberg set, and a higher pattern of thought, a

Christian world view, with which we know Bach was acquainted, would have understood,

and would have agreed. These intersections are intended to suggest meanings that the

composer may have intended, or, if he did not intend, with which he might have identified at

least. In the process I shall describe what these structures mean to me (a valid form of

criticism). All of this, in turn, may or may not mean anything to the reader (both valid forms

of reader response). I appeal, again, to Sayers and Pike.

Whereas most of Bach's music is procedural (generated by contrapuntal processes) rather

than formalistic (conforming to closed templates such as sonata-allegro, rondo, etc.), it is


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significant that one form that Bach did employ, often, was cruciform structure. His sacred

music is full of it. I have alluded to the chiastic plan of the B-Minor Mass. The interested

reader may wish to study Cantata BWV 4, MotetJesu Meine Freude, and the St. JohnPassion for other fine examples.

Don O. Franklin, ed.,Bach Studies (Cambridge University Press, 1989), p. 92 (from chapter

on the St. John Passion by Eric T. Chafe).

I [the author] am unable to verify the basis upon which Chafe makes this conclusion.

See "The Book of Concord," translated and edited by Theodore Tappert (Philadelphia, 1959)

pp. 344-5, 411-20. Project Wittenberg has also published the full text of Luther's Large and

Small Catechisms on the Web.

"God the Father almighty, maker of heaven and earth" (from the first clause of the Nicene

Creed).

See Colossians 1:19 and Colossians 2:9

"God of God, light of light" (from the second clause of the Nicene Creed). Consider also

Hebrews 1:3.

See Matthew 16:24-26

On numerous occasions Bach puts himself into his music. In Contrapunctus XIV of theArtof Fugue, the third subject spells BACH. Two of canonic variations on vom Himmel hochcontain the BACH motive. The Kreuz und Becher aria from the St. Matthew Passioncontains an amazing juxtaposition of the composer's name with the word Kreuz, in a context

where textual associations make it clear that the composer himself is willing to pick up thatcross and carry as the Savior did. Bernard Greenberg has observed that the theme of the bass

aria BWV 87.1Bisher habt ihr nichts gebeten in meinem Namen also contains the musicalrepresentation of Bach'sNamen.

"We all believe in one God, Father, Son and Holy Spirit . . . who controls, creates and does

everything by his great power"; from the first stanza of Clausnitzer's strophic rendition of

the Apostle's creed, Wir Glauben All. Bach composed two settings of this chorale as organprelude in the catechistic portion of his Clavier-bung book III.

Saying that J. S. Bach was a religious man is like saying Wagner was German. This is not

been my purpose. I have attempted to show, instead how large-scale structures have been

driven by higher patterns of thought. Bach's musical affirmations of faith represent

confirmations, however strong, of what we know from other sources such as the recently

discovered Calov Bible Commentary.

The most famous of these is Augustine's argument that for the divine perfections to exist

they must have expressed themselves before creation. Thus, if God is love, and if love

cannot exist without an object, God must be Triune.

Bach wrote these words in the margin of his recently discovered Bible Commentary by

Lutheran theologian Abraham Calov. The marginal notation refers to portions of thetwenty-fifth chapter of I Chronicles.


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Meaningfulness in Bach's Cyclical Works

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