Towards An Understanding of Structural/Rhetorical Development In J. S. Bach’s ‘Art of Fugue’: a companion to the score

Rama Gottfried – Manhattan School of Music, May 2007 Email: rama.gottfried@gmail.com

Towards an Understanding of Structural/Rhetorical Developmentin J. S. Bach’s ‘Art of Fugue’: a companion to the score.

Rama Gottfried

One of the keys to an intimate understanding of Bach’s fugal writing is his use of

a rhetorical model serving as a structural basis for contrapuntal development. The

progressive structure of rhetoric forms an underlying semantic architecture for the fugue

– playing out the drama of a musical idea, put forth and submitted to the process of

introduction, discussion, argument – culminating in the reiteration of the idea as a truth.

In The Art of Fugue, Bach’s great treatise on the subject, we find an exponential

confirmation of just this – that the work as a whole is a telescoped form of the fugue

itself. Through its design of rhetorical progression, the large-scale shape mirrors the

small – imbuing the fugal forms of the interior with new semantic layers. Subject and

counter-subject, through a fugue of approximately 90 minutes in length, form the two

layers of rhetorical debate. They are set against each other, fragmented and re-

constructed in a meditation on the nature of the fugue, and through the fugue to rhetoric,

and in the end through the form of rhetoric, The Art of Fugue become a mirror outlining

Bach’s contrapuntal dualistic-monody. This is one of the great encoded statements of the

composer on the topics of music, but quite possibly it is also an deeper expression of his

inner dialog on the issues of religion and experience of life, surely on the surface of

Bach’s mind as he worked to the end of his illness to complete this last set of fugues.

Left to us in its nearly, but un-completed form, some aspects of the 250 year old work are

still being uncovered, each stepping closer towards a greater perception of Bach’s

intentions. Through the course of this paper I hope to guide the reader through an outline

of the dialogs in play, in order that a connection of object and intent might emerge.

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Beginning with the macro-scale, we now turn to the total shape of the Art of Fugue, at

which point one immediately has to deal with the much-debated issue of the order of

pieces.

The number and chronology of fugues (and canons) went through several changes

from the Berlin Autograph to the first printed version. The plans for the first printed

edition were overseen by J. S. Bach before his death, and then taken over by C. P. E.

Bach and Johann Friedrich Agricola afterwards in its incomplete state. Musicologist,

Hans Heinrich Eggebrecht, in his insightful perspective on the work, suggests that C. P.

E. and Agricola must have been responsible for the mis-ordering of the printed version, in

part with the decision to add the final chorale, which is clearly disconnected formally

from the rest the score. This debate is mostly outside of the framework of this paper, but

in brief I will refer the reader to the research on this topic by Gregory Butler, who in his

1983 article appears to have conclusively determined the original order of pieces as

intended by J. S. Bach before his death. If his study of the engraving plates and

preparation techniques of the time is accurate, it confirms, as most scholars have

suspected, that the canon in augmentation and contrary motion was meant to follow the

other three canons, continuing the design of increasing complexity. In addition, Butler

has determined that most likely the unfinished quadruple fugue, Contrapunctus XIV, was

meant to immediately follow the two mirror fugues, thus placing the four canons at the

end of the large-scale form. This changes things considerably, and places the quadruple

fugue in a rhetorical position of culminating final argument, rather than a conclusion as

previously understood. Without being able to directly determine the authenticity of

Butler’s research, I propose that this re-organization is logical based on the structure of

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fugal form. This placement of the canons as conclusion brings a critical balance and

symmetry to the work that was previously dependent on an imagined version of what the

final fugue would be in its completed form. If the works are placed as suggested in

Butler’s article (Example 1),

flexible presentation of subject/counter-subjects

Example 1: Organization of movements as suggested by G. Butler's research

I. simple fugue (suspension)

II. " (dotted)

III. " (chromatics)

IV. " (thirds)

V. stretto fugueVI. "

VII. "

VIII. triple (double) fugueIX. double fugue

X. "XI. triple (double) fugue

XII. (a,b) mirror fugue

XIII. (a,b) "

XIV. quadruple fugue

canon at the octavecanon at the twelfth

canon at the tenthcanon in aug./contrary motion

contrapuntal solidifying of subject/counter-subject

two subdivisions emerge, separating the 18 works into two sets of 9, which alternate by

group. The first collection of movements contains the four simple fugues (I – IV), double

(or triple) fugues (VIII – XI), and final quadruple fugue (XIV). These 9 are alternated

with the works dealing with more ‘strict’ fugal procedures: the stretto fugues (V – VII),

mirror fugues (XII a, b – XIII a, b), and concluding with the four cannons (at the octave,

twelfth, tenth and augmentation/contrary motion). As I will show below, this not only

creates a clear rhetorical flow to the unfolding of ideas, but also sheds light on Bach’s

own understanding of fugal structure.

As is well known by all students who have ever taken a class in counterpoint, a

fugue is based on a subject, usually coupled with a counter-subject and follows the

pattern of statements, answers (real or tonal), and episodes; frequently utilizing stretto or

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canonic imitation. Bach, in the Art of Fugue, has over of 57 strettos using only the main

subject in 14 fugues, including the two mirror fugues (rectus and inversus), which are

completely invertible – yet the subject that allows him this flexibility seems plain at first.

Perhaps it is this simplicity that lends itself to such successful uses.

Examination of the subject in its simplest form reveals that it is symmetrical in

shape, and is contained within the span of a perfect 5th. The subject expands evenly from

the center point of C# in both directions, the notes D, F and D falling equal distantly from

the center (Example 2). The subject itself implies the rhetorical structure of a fugue: the

opening two notes tonic and dominant – a statement and answer; measure 2 could be

interpreted as a slow return to tonic, now placed on the weaker third beat; in measure 3,

the center of the subject, a pressure point chromaticism is introduced forming a

symmetrical neighbor-tone cadence on D leading from the previous measure – arguing

the tonic, and lifting away, with the introduction of an ascending stepwise tetrachord,

breaking the half-note pulse now to quarters (a form of diminution) and an increased

sense of motion; the concluding eighth-note gesture, to the down beat of measure 5,

increases in fluidity and reiterates the return to tonic through a descending tetrachord

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bringing the subject to a total of 12 notes and 17 beats. The golden mean falls on the

tonic in the third beat of measure 3.

Utilizing the most basic of formations Bach crafted a subject that could smoothly

link harmonic and motivic movement. The subject is divided into one triad and two

tetrachords, which by shifting one note or tetrachord stepwise, fluidly shifts the subject’s

harmonic direction, all the while maintaining its characteristic shape. Throughout the

course of the work this basic subject is brought into contact with counter-subjects, and

‘counter-voices’ which, though not strictly used as a counter-subject, rhetorically fulfills

the counter-subject role. Through the counter-voice’s contact and influence on the

subject, a developmental progression or transformation of the subject takes place. As we

will see, the contour of the subject is strong enough to allow us to trace its form through

its twists and turns as it gradually accumulates rhetoric weight. We will return to the

gradual motivic transformation of the subject, but first let us turn our attention to seeds of

the entire super-structure, which can be seen in the structure of Contrapunctus I, the first

of the ‘simple’ fugues. As stated above, the subject itself is in the shape of a tiny 12 note,

17 beat fugal form. Its implications are immediately expanded in the exposition of the

first fugue (Example 3).

Observing the entries of the subject and its counter-subject(s) in Contrapunctus I,

the qualities found in each of the four simple fugues can be seen in its latent seed form.

The first statement in the alto is alone – corresponding to the first simple fugue – in its

last note forms an elision with the entry of the soprano falling on the down beat of

measure 5. _ After the formal structure of the subject as stated alone, this elision is

perhaps the first rhetorical element that Bach sets forth in the whole Art of Fugue.

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As alto continues from its opening entry, it states the answer-form counter-subject

against the tonal answer in the soprano. And here it begins to move in suspension figures

– reminiscent of the displaced tonic return falling on the weaker beat 3 in the second

measure of the subject. The first of the suspension ties from the quarter note on beat 4 in

measure 6, over the barline and again in measure 7 on beat 2, resolving up on the second

eighth on beat 3 is characteristic of a dotted rhythm and will shift into this momentarily.

As the soprano continues after its exposition entry to the counter-subject in

statement-form against the bass statement in measure 9 it moves in syncopation with the

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alto voice. Completing the counter-subject in measure 10, the alto continues the tied

figure but shifts to the strong beats changing the notation to the implied dotted rhythm.

The statement-form counter-subject in the soprano, begun in measure 9, tonicizes D

repeating the neighbor-tone D-C#-D figure from the subject to measure 10. Quickly, the

C# is cancelled out by a C (natural) after which the soprano picks up the quarter-eighth

tie pattern as in the answer-form counter-subject, suspending over beat 3 and resolves

(interrupted by a leap-step, down a 3rd, up a step) to Bb which begins a B-A-C-H

statement (momentarily interrupted by a leap down to E and back) as early as measure

10.

The final answer of the exposition enters in measure 13 (tenor), with the bass

picking up the answer-form counter-subject after a short rest in measure 14. The upper

voices continue their tied/dotted suspension figure in syncopation through the first

measure of the answer (measure 13), and then shift in measure 14 to stepwise figures

filling in thirds – corresponding to the final descending tetrachord found in the primary

subject form. The tenor opens the first episode in measure 17, stating the ‘B-A-C-H-C#-

D’ theme, as dubbed by H. Eggebrecht (interrupted by first two notes then one note), and

continues repetition of the ‘step-leap’ motive, in canonic imitation along with the bass

and soprano. This ‘step-leap’ motive originates from the subject’s F-D-C# figure, the

leap interval now expanded by a step. Referring back to Example 1, you can see the

beginning of the structural telescoping gradually taking form.

The role of the counter-voices as observed in Contrapunctus I, is to reflect and

elaborate on the inner motions implied in a given subject. Through the counter-voices,

the potential of a subject is fragmented into archetypal actions. The potential elements of

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the subject are depicted through the counter-subject, expanding through the counterpoint

against the first four basic statements of the subject. These expanding motives become

the four basic types of motion that push an entire work forward. In this case the types of

movement are from the principal form: triadic, syncopation and stepwise motion; dotted

rhythms; chromatic stepwise motion; and movement in thirds. Motivically, just as the

structure of the subject forms the characteristics of the four voice exposition in this first

fugue, the character of the counter-voices in Contrapunctus I parallel the character in the

first four simple fugues which form the Art of Fugue’s large-scale exposition. The

progression of the first four fugues lays out the rhetorical presentation of the subject. As

the work continues to unfold the pattern repeats itself in the progression of larger

groupings of works, ever increasing in contrapuntal complexity as the inner motivations

of the subject are brought to light and examined. As I put forward in the introduction,

this exponential growth of the subject telescopes from seed to the whole, and becomes

the image of the fugue itself. To illustrate this, let us return to Contrapunctus I, to look at

this over-all shape of the first ‘simplest’ fugue (Example 4).

In the 78 measures of Contrapunctus I, there are 11 entries, just as there are 11

notes before the elision between statement and answer. The elision of the last note of the

subject and the first note of the answer is a semantic point for Bach, and he reinforces this

through the enlarged elision (or minor stretto) in measure 32. This is where a second

exposition would have ended if the fugue had continued to repeat another 16-measure

period. However Bach breaks the 4 bar grid with the 6 measure episode, which pushes

the entrance of the first post-exposition entry to measure 23 (also forming a relationship

to the falling of the second D in the subject on the weak beat – see Example 4). The

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following entry is back to the 4 bar grid of the exposition beginning in measure 29

(soprano), only to be undermined by the entry of the bass in measure 32 which forms an

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elision of one measure. This type of rhythmic phrase displacement increases the fluidity

of motion and flexibility of harmonic motion – seeming to extend and wind, avoiding the

sense of conclusion or cadence. Following the bass entry in 32 by 8 measures, the tenor

voice enters at exactly the mid-point of the piece, measure 40. The original 4 bar grid is

once again established by an entry of the soprano in measure 49 – fittingly the highest

entry in the piece, as well as the highest collected register of all voices. Measure 49 also

marks the golden mean point of the 78 measures. The mid-point of the piece, measure

39, is treated as the center of symmetry. Counting measure 39 as 1 in both directions, we

see that at a distance of 5 measures forward and backward from the mid-point,

corresponds to the 4th measure of the theme (see bass in 35 and tenor in 43). Counting 11

measures in either direction (again including measure 39 as 1) points to the 1st measure of

the subject (the soprano entrances in measures 29 and 49). Also apparent is the balance

between the 16 measures of the exposition and the dominant pedal in the bass 16

measures from the end (measure 63). These symmetries can be seen to reflect the mirror

shape of the subject.

There is one more proportional relationship in this first fugue, which I would like

to point out between the elision in the exposition and the overlapping of one measure

between the entries in measures 29 and 32. The first elision occurs on beat 1 of the

soprano’s answer, (measure 5) which is the first beat of the second quarter of the

exposition. Similarly, if you subtract the exposition (16 measures) from the total piece

(78 measures) you are left with 62. Divided by 4, we find that one quarter of the

remaining fugue following the exposition is equal to 15.5 measures, which counted from

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the end of the exposition places the elision of one measure at the corresponding 1/4 mark

to the rest of the fugue, mirroring the proportion of the first elision to the exposition.

Turning attention now to the episodes and the treatment counter-voice material in

relation to the subject entries, we will be able to see a larger-scale model emerge, and of

the rhetorical structure of the fugue. The general process of a rhetorical argument are as

follows: introduction and statement of the case, followed by a related story which leads

back to the restatement of the idea; this is followed by a period of debate and rebuttal

from the contrary parties; and eventually the culmination of the case is reiterated

forcefully as conclusion, and truth.

In terms of the first fugue, we can follow the rhetorical process as progressing

from the statement of the case to the introduction of related ideas, to be parallel to the

statements of the exposition leading to the first episode. In Contrapunctus I, the episode

material refers to the step-leap motive of the subject – here moving in the soprano and

tenor, holding half-notes moving up in thirds. The bass compresses the step-leap motif to

a neighbor-tone figure, ascending chromatically (measure 17 – 20) up to D, as in the

rhetoric model, leading back to the statement.

The shorter second episode (measures 36 – 39), continues the step-leap figure

with half-notes moving up in thirds, now in the alto and bass with the soprano moving in

dotted rhythms elaborating another version of the step-leap figure – this all leading up to

the entry in the tenor at the half-way mark. Immediately following the mid-point entry is

episode 3 leading to the golden-mean entrance in the soprano. Here at the end of the

third episode (32 measures from the end, measure 46), a fragment of the counter-subject

emerges in the bass and introduces a figure related to an inverted version of the 4th part of

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subject, moving stepwise in eighths. According to our rhetorical fugue scheme, this

introduction of a new figure is a dissenting voice, commenting on the until-now, un-

questioned motion of the counter-voice step-leap material. Continuing into the last third

of the piece (measure 53 – 78), and above the return of the bass in its entry in measure

56, this stepwise motion dominates the texture – making a final argument against the

step-leap motive.

The final stage in the rules of rhetoric is to culminate with a forceful conclusive

statement of the case, usually by weighing the options and breaking them into smaller

pieces, stacked against each other in an accumulation of meaning. In Contrapunctus I we

can observe two key elements in Bach’s technique of culminating the fugue’s rhetorical

drama. One successful method of convincing an audience through rhetoric is to take a

subject and portray it from two opposing vantage points, contrapuntally this is the

equivalence of a motive in two contrary directions, otherwise known as contrary motion

or inversion if moving exactly. Once rhetorically viewing a subject from contrasting

vantage points, the speaker then frequently summarizes the points stacking them in short

succession one after another to create an overwhelming sense of the logic which ties

together the threads of the argument. In the fugue, this is typically achieved through the

fragmenting of materials and treated in close canonic imitation or through stretto. In

Contrapunctus I we can see that Bach does just this. In measures 60 – 62 (Example 5),

leading to the bass pedal on the dominant in 63, the tied half and stepwise eighth-note

figure, brought into the discussion in the last 32 measures (originating from the final

gesture of the subject), is placed in contrary motion and to the furthest voices from each

other (bass and soprano) which is then treated in canonic-imitation. Reinforcing the

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contrary motion, in measure 60 the alto and tenor voices move from above and below by

semi-tone cadencing on the downbeat of measure 61 (forming a first inversion

14

subdominant harmony above the Bb in the bass). Shortly after the dominant pedal

appears in the bass in measure 63, the triadic motion and step-leap figure from the

counter-subject reappear, gaining prevalence once more in canonic imitation, and now in

a four voice texture for the first time. The soprano joins the texture on a high A, the

highest note in the piece. In 66, as the bass joins the rest of the group in the step-leap

figure, a critical mass it met, with the soprano now returning to a combination of both

stepwise eighth-note and step-leap figures. Continuing as an obbligato until the

rhetorically potent Bb-C# leap, where the texture suddenly breaks off dramatically. A

lone tonic second inversion chord is stated between a frame of three beats of rest, then

momentarily cadencing on the dominant and winding to the final statement of the subject

over a tonic pedal. A partially interrupted B-A-C-H statement in the alto (another is

scrambled in the soprano), brings the final cadence, ending once more in the soprano with

the leap from Bb-C# and cadencing on D (The Bb-C# figure are the resulting two

chromatics formed when combining the rectus and inversus forms of the subject).

Having described the model form for our introductory fugue, Contrapunctus I, let

us now look to see how this design might be echoed in the macro or super-form of Die

Kunst der Fuge. In a similar manner to the subject’s motion from the opening fifth, the

triad and then increasingly segmented and complex development – the works that form

the Art of Fugue start with clear relationships to the fugal unfoldment of the subject, and

become more fluid towards the central dialog of the work.

Separating the four basic categories of contrapuntal characters that Bach lays out

in the exposition of Contrapunctus I, we can then observe that these counter-

subject/motion characteristic may be applied to the opening four ‘exposition’ fugues as

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follows: I – suspended, II – dotted, III – chromatic, on the inverted subject, and IV –

movement by thirds, on the inverted. The inverted subjects in III and IV additionally

mirror the first two fugues – either as dominant to tonic, or perhaps as a mirror to the

mid-point symmetrical structure. The characteristic textures can again be telescoped

from the ‘exposition’ fugues to the expansion of the remaining movements (Example 6).

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Exposition Fugues:

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= original

= inverted

= diminution

= augmentation

GENERALIZED

CONNECTION

TO EXPOSITION

I suspentions

II dotted notes

III chromatics

IV thirds

Example 6: Relationship of Exposition Fugues to Total Form

XIV quadruple fugue

open triadic theme

running theme

b-a-c-h-c#-d theme

ground theme

V dotted

VI dotted

VII dotted

Final Fugue:

canon at the octave

canon at the twelth

canon at the tenth

canon in augmentation and contrary motion

Stretto Fugues:

XII a,b subject in complex meter

XIII a,b subject in triplets

Canons:

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Mirror Fugues:

( )�

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VIII double fugue + counter-subject

(or third theme)

step-leap

rest-theme

IX double fugue

octave-leap

original subject

X double fugue

mirror-rest

dotted

XI double fugue + 2 counter-subjects

(or third theme and 1 counter subject)

rest-theme

step-leap

GENERALIZED RELATIONSHIP TO SUBJECT

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Double Fugues:

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The progression of contrapuntal treatment and the subject and counter-subject materials

can be thought of as an intertwining formal and motivic dramatic arch, following the

rhetorical fugue design as described above as a forum for motivically continuing the

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discussion between the subject and counter-subject, gradually transforming through their

rhetorical conflict.

As the exposition of the macro-structure, the first four ‘exposition’ fugues, project

the image of the subject and counter-subject themes onto the remaining cycle. After the

first two fugues, placed in balance by fugues III and IV, as though forming statement and

answer, begins the following three counter-fugues, or ‘stretto’ fugues in dotted rhythm.

The level of contrapuntal intensity increases through the ‘stretto’ fugues – V uses normal

and inverted forms (with a mirror statement at the final cadence), VI – uses normal &

inverted – primary & diminution forms, and VII uses normal & inverted – diminution &

augmentation forms.

Note that the placement of the Stretto Fugues is closely related to the elision we

looked at in Contrapunctus I. Contrapunctus II also has the same single measure elision

as in Contrapunctus I. This is very likely in reference to the second elision that occurs in

measure 9 of the first three fugues – the one measure elision in II occurs at 9 measures

after the conclusion of the exposition. In Contrapunctus III the one measure elision again

appears, this time flipped inverted towards the end of the piece, reflecting the inverted

version of the subject. This occurs 15 measures from the end, symmetrical with the final

entry of the exposition in III at measure 15 (the delayed final statement in the exposition,

expands the exposition to 18 measures – reflecting the number of pieces in the Art of

Fugue). Contrapunctus IV continues the elision and 4 beat grid dialog by the insertion of

a complete four voice counter-exposition beginning in measure 27, grouping all of its

entries by one beat elisions. The central entries are again placed symmetrically around

the mid-point (+/- 8 measures), before rhetorically building up through a 26 measure

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episode to entries of the subject in one beat syncopation (measures 107-115) and

concluding the ‘exposition’ fugue section with two elision entries, with an un-interrupted

B-A-C-H statement in the tenor (measure 135-136). The first ‘stretto’ fugue,

Contrapunctus V, begins immediately with a one-measure elision (measure 4), and

continues in this way, continuing from the close syncopated entries from IV and

gradually moving the voices further apart. The one solo entry in V is place

symmetrically balanced by -12 bars from the mid-point with the first stretto spaced one

measure apart (an inversion of the bar elision) +12 from the mid-point, and the highest

entry. After a pseudo-four-voice-stretto in measure 65, and two more 4 beat spaced

strettos, the movement culminates in the simultaneous two voice, inverted mirror

statement the close. VI continues much this way, the details I will leave for the reader to

explore. Of note, in VI the final entry is by elision of one note - the first to occur in the

‘stretto’ fugues. There are no more elisions until VII 3 measures after the mid-point,

where there are three elision entries in a single voice (alto, measures 35-47).

The rhetorical function served by the episode in the ‘simple’ fugue, is transferred

to the central section of fugal movements in the Art of Fugue. Movements VIII – XI are

the ‘double’ (or ‘triple’) fugues, which now in turn balance the ‘stretto’ fugues by

shifting the focus from the subject to the counter-subject materials. I note the ‘triple’ as

parenthetical due to what is sometimes called the second subject being developed out of

the counter-subject and consistently grouped with the first subject of VIII, and the fact

that it never occurring alone here (in VIII), as pointed out by H. Eggebrecht. The same

‘counter-subject’ or ‘subject’ does occur in XI alone however so it is certainly close to

being a subject. But since its origin can be clearly traced to the ascending chromatic bass

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in Contrapunctus I (measure 17-20 – see Example 3), and as in the more reduced form of

the counter-subject in III – I view this ‘subject’ more as a solidifying of the counter-

subject, or the ‘counter-subject-theme’. Referring back to the rhetorical progression, the

solidifying of the counter-subject is also balanced by the increasing fluctuation of the

original subject. Moving towards the central axis of the Art of Fugue, the subject itself

internally undergoes the type of rhetorical debate previously reserved for interplay

between contrapuntal voices. In the center four fugues or ‘episode’ fugues, the subject is

subjected to transformation, or modulation of its expressive content through its contact

with the influence of the counter-subject(s). As seen in the canonic imitation of the

counter-voice materials in Contrapunctus I, the counter-subjects, and literally the

contrapuntal treatments of the subject itself, which have occurred thus far (i.e. inversion,

diminution and augmentation), now work their way into the subject itself as the large

scale fugal rhetoric examines its potential implications.

Incremental alteration of the subject occurs throughout the ‘episodic’

double/triple fugue group (VIII – XI). This appears first by the initial solidifying of the

chromatic counter-subject material of Contrapunctus III (and episode 1, in Contrapunctus

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I) into the first subject of Contrapunctus VIII (Example 7). The falling chromatic line

stated D-C-H-B-A in measures 5-6 in Contrapunctus III, is now merged with the subject

form into the first subject of VII. The step-leap motive from the counter-subject of I (and

from within the subject itself) stretches its contour becoming a stepwise descending scale

smoothing over fragmented implications of the retrograde version of the ground theme. I

will refer to this subject as the ‘chromatic-step-leap’ subject. The counter-subject of VII,

which begins on the second beat of measure 5 continuing to measure 10 (Example 8),

carries on the residue of the previous two fugue’s contrapuntal treatment, now transferred

to the counter-subject, as visible by the quarter-note motion in the last 3 beats of 6 to bar

7 (label ‘a’ in Example 8), which are then re-stated in eighth-note diminution at the end

of measure 7. The ‘falling-quarter’ note figure in measures 5-6, on beats 2-4 is an

intervallically augmented version of the final gesture of the subject, as first seen in

Contrapunctus I, measure 15 (alto – see Example 3). This figure is then continued

through the counter-voices until it solidifies into a fully formed counter-subject or

‘counter-subject-theme’ (as referred to above) in the last beat of measure 39. At the mid-

point of VIII (measure 94), the subject returns stating a second exposition as the fugue’s

second subject. Through its contact with the ‘falling-quarters’ counter-subject, the

ground theme has taken the rhythm of the falling notes of measure 5-6, replacing the

downbeats with quarter rests and states the subject in its inverted form which I will refer

to as the ‘rest-theme’.

Notably, accompanying the ‘rest-theme’ at its alto entrance in measure 94, in the

bass is the diminution second part of the counter-subject to the 'chromatic-step-leap’

theme. At the golden mean (measure 114), a pedal tonic note is sounded in the bass, as

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an incomplete entry of the ‘chromatic-step-leap’ and ‘counter-subject’ themes trail off

into another diminution of the counter-subject from measure 7, now sixteenths. Three

full statements of the two, coupled themes, and then four statements of all three themes in

triple-invertible counterpoint then follow.

Contrapunctus IX and X surround the central axis of the Art of Fugue’s 18

movements. Just as Contrapunctus III and IV, reflected the subject through its inversion,

IX and X reflect the rhetorical progress by combining the subject’s normal and inverted

forms internally with the counter-subject dialog as detailed in VIII. The first subject of

IX, begins with a leap of an octave (an augmentation of the original 5th), and runs in

eighths down stepwise, back up and down again, globally in the familiar shape of the

ground theme. All 12 notes of both the primary-form theme, and inverted version are

21

buried within the running eighth-note texture – and are modulated by the integration of

the ‘counter-subject-theme’ from the previous movement (Example 9). A variation of the

‘falling-quarters’ counter-material appears in the high alto register, C#-D-A, in measure

36-37 accompanying the arrival of the ground theme in its original form (entering in

measure 35). This occurs 31 measures before the mid-point and is the first pure

appearance since Contrapunctus II, which leads us to the place that H. H. Eggebrecht

refers to as ‘the silent axis’ of the work as a whole – the mid-point between the two

sections of 9 movements.

The ‘mirror-rest’ theme, to use Eggebrecht’s terminology, represents the most

compressed version of the subject/counter-subject combination. The first note of the

theme is C#, the center note of the subject from which the entirety of the harmonic

22

language of the piece unfolds from. In the inverted form of the subject, the C# becomes

Bb, symmetrically surrounding the D minor triad that is the Art of Fugue.

Looking at the first two measures of this subject (Example 10), the first measure

states a form of the ‘falling-quarter’ counter-subject in a step-leap variation, as first seen

in the high alto of the previous fugue (IX, measure 36-37). The figure is immediately

inverted in the second measure as if echoing the symmetrical mid-point which occurred

in the space between Contrapunctus IX and X. Reinforcing the mirror imagery, the

second half of the theme is another approximate mirror, in stepwise motion up to C and

down again – also the shape of the second half of the subject. The pitches themselves are

multilayered in their organization, creating another type of mirroring; these reflections

are interwoven layers that thread together the ground theme and its inversion in at least

five ways (Example 10). Numerically mirroring the total work, there are 18 notes in the

theme – but the answer enters forming a one and a half measure elision on the 12th pitch

of the subject. Therefore 11 notes played alone, thus mirroring the subject again.

Looking back to the other side of the sectional mid-point between movements into

Contrapunctus IX, the chromatic notes of the ‘mirror-rest’ theme can be found in vertical

orientation in the alto and soprano at the final cadence (in reverse order – the A in the

tenor and bass completes the pitch collection). Counting 11 measures in both direction

from the final D major measure of Contrapunctus IX, reveals that not only does the last

pure statement of the ground theme occur 11 measures prior, but that counting forward,

the exposition of Contrapunctus X is also 11 measures long. This also parallels the +/- 11

measure symmetry found in Contrapunctus I. Surely this is not a coincidence!

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After a second exposition re-introducing the ‘dotted-step’ version of the ground

theme that permeated the stretto fugues, the ‘mirror-rest’ theme is brought back in its

inversion and placed in invertible counterpoint with the ‘dotted-step’ version of the

subject. At the golden mean of Contrapunctus X (measure 75) a triple entry with the

‘dotted-step’ subject in parallel 6ths with the ‘mirror-rest’ theme. This is followed 11

measures later by another triple entry, this time the ‘mirror-rest’ theme appears in parallel

3rds, with the ‘dotted-theme’ in the bass (measure 85). 18 measures later, and again 18

measures after that, 18 measures from the end, there are two more statements of the

‘mirror-rest’ theme in thirds with the ‘dotted-theme’ accompanying bring Contrapunctus

X to its end.

Contrapunctus XI serves as a summary of the episodal-rhetoric development of

the counter-subjects with the ground theme. As the merging of the counter-subject and

subject solidifies, a more formally complex structure reiterates the ‘new’ vantage points

on the ground theme as portrayed in the ‘episode’ fugue group. The inverted version of

the ‘rest-theme’ is the first subject of XI. Notably, since the downbeats of the theme are

quarter rests, no elision forms from the direct movement from statement to answer. The

first half of XI is broken up into three partial fugues, the first spans from measure 1 to 26

where the texture thins to one voice. At measure 27, the inverted version of the

‘chromatic-step-leap’ theme is presented in a second exposition, accompanied by the

extended chromatic stepwise motion from the counter-subject of III. It should be noted

that this second subject exposition is broken by short episodic counterpoint, which points

to the rhetorical relationship between the ‘chromatic-step-leap’ theme and its original

counter-subject origins. Measure 89 re-introduces the ‘counter-subject-subject’ placed in

24

stretto against the ‘chromatic-step’ theme. The second entrance of the characteristically

chromatic ‘counter-subject-subject’ in measure 90 (alto), states B-A-C-H and continues

in stretto through measure 107. 27 measures before the end (measure 158), and again 21

measure from the end (measure 164), two simultaneous mirror statements enter first in

the soprano and alto, followed by the tenor and bass, foreshadowing the mirror canons

which are to follow.

The two mirror fugues, Contrapunctus XII and XIII represent the ultimate vertical

balancing of the two contrapuntal elements of the subject and counter-subject. Just as the

counter-fugues exalt or re-examine the subject through stretto, diminution and

augmentation, and the ‘episodic’ fugues merge the subject and counter-subject into new

poly-structural themes taking on characteristics of both, the ‘mirror’ fugues place the

subject/counter-subject dynamic into perfect vertical counterpoint – frozen on the page,

to be played in the mind of the reader (the limitation being the impossibility of a single

entry should both versions be played simultaneously – among other limitations

presumably). Rhetorically, XII and XIII reiterate the combination process of the subject

and counter-subject. The subject in XII is an original version of the basic theme, placed

in complex meter (and inverted in the mirror version); the counter-subject recalls those

found in I and IV. In XIII the merged subject/counter-subject is re-introduced as a

relative of the ‘octave-leap’ theme, now in triplets. The ‘dotted-step’ version of the

subject is in inversion and buried within the triplet motion, with the kind of interweaving

of subject that we saw in X. Both original and inverted versions of the ‘triplet-mirror’

theme are used (in both rectus and inversus versions of XIII).

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According to the ascertains of G. Butler’s paper on the evidential marks found on

the printing plates for the Art of Fugue, the famous incomplete quadruple fugue,

Contrapunctus XIV was designed to follow the mirror fugues. Rhetorically this follows

the form of we have been tracing through our examination. As Butler illustrates in his

article, this ordering creates two sets of alternating types of contrapuntal technique or

rhetorical motion (Example 11). One group is made up of those fugues featuring the

introduction of counter-subjects, and thus the rhetorical interrelation between motivic

concepts. These are the active and flexible aspects of the subject and counter-subject.

The second group examines the static elements, attempting to culminate the materials

developed in the first group through rhetorical reiteration into a convincing truth.

Fugal Progression Rhetorical Progression

I. Simple

II. "III. "IV. "

V. Stretto Fugue

VI. "VII. "

VIII. Double/TripleIX. Double

X. "XI. Double/Triple

XII. Mirror

XIII. "

XIV. Quadruple

1. Canon (octave)

2. " (12th)3. " (10th)

4. " (aug./contrary)

} Development of

Thematic Writing

Exposition

Episode

Introduction of Subject

Digression

Statement of Case} Exaltation of the Subject Canonic Statement

}

}

}

}

Exaltation of the Influenceof the Counter-Subject

(Merging of Subject and Inversionwith Counter Subject)

Fragmented versions of the Subject

Treated Episodically

Canon as Stretto Reiteration;Rhetorical Climax Perfect Horizontal Mirroring

(and in final canon Horizontal & Vertical) of Subject/Counter-Subject

Perfect Vertical Mirroring ofSubject/Counter-subject Dynamic

Example 11: Rhetorical Outline of Fugal Form

Episodic Inclusionof Counter-Subject

'Simulataneous-Canon'

in Contrary Motion

Climax of EpisodicFragmentation and

Restatement

Canonic Culmination

Weighing of Opposites

Opposing

Counter-Statements

Summary And Verdict

Final Arguments

Bach’s unfinished Contrapunctus XIV was planned as the climactic point of

episodic fragmentation of the subject. Introduced as two basic elements, the first and

second halves of the subject appear separately in XIV. The first, ‘open’ theme,

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expresses the opening of the subject (in its ‘dotted-theme’ version with passing tones),

stating the tonic, dominant and then down to the subdominant. This ‘open’ theme of the

quadruple fugue for H. Eggebrecht was a symbol of the divine, while he saw the second

‘running’ theme, created out of the second half of the ground theme, as symbolizing

man’s imperfection. This very well may be what Bach intended, but for however much

Bach was a devout Lutheran, he was also a devout composer, and these two figures

which make up the incredibly flexible and characteristic subject of the Art of Fugue, are

here as opposing musical elements. The logic allowing Bach so much flexibility with

these elements is due to their immense simplicity, and through their simplicity they form

the archetypes of rhetorical imagery. The form of fugue XIV is structured around a

reducing telescope, each of the four themes’ exposition become increasingly shorter, and

posthumously the fourth subject was to be the re-entry of the original subject. This

restatement makes sense in terms of rhetorical structuring. After the mirroring of the

subject/counter-subject relationship in XII and XIII, we find XIV, the final fugue of the

‘episodic’ or ‘subject/counter-subject’ fugue group, with the composer’s own name

carved out of the chromatics resulting from the combination of the original and inverted

forms of the subject, the chromatic counter-subject from III, and the ‘chromatic-step-

leap’ theme. These streams combine creating the third subject, B-A-C-H-C#-D (which

Eggebrecht interprets as Bach’s striving to unite with the tonic). The intertwining of

subject and counter-subject themes now in their most separated forms (aside from the

original presentation in the ‘exposition’ fugues), brings the rhetorical interplay between

the subject and counter-subject to a climactic apex, but not a rhetorical culmination. The

final fragmentation and reconstruction of the subjects may have been achieved, and

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presented in mirror order leading back to the opening subject from its constitutive parts,

but as stated above, to conclude, Bach reserves the forceful presentation through

reiteration – canonic repetition.

To form a convincing rhetorical statement of verdict, the canons are placed at the

culminating place as a final, complete merging of subject/counter-subject dynamic. Thus

the canon may be thought of as forming a mirror to the four ‘exposition’ fugues. Each of

the four canons approximate a fugal form, increasing in complexity from one to the next,

from the canon at the octave, twelfth, tenth, and finally to augmentation and contrary

motion. Bringing the cycle to cadence, each canon ends at an open octave on tonic. Just

as in the opening ‘exposition’ fugues, the subjects in the canons reflect each other, here

by alternating between simpler variations, and ‘merged’ versions. Canons 1 and 3 mirror

the simple subject by inversion, while 2 and 4 incorporate the elaborations of the counter-

subjects. Canon 4 in augmentation and contrary motion simultaneously mirrors 1 and 2

through using both rectus and inversus forms of the subject, and the passing tone figures

of canon 2. Conclusively, the incorporation of the counter-subject dialog into the contour

of canon 4 is then treated as an extended stretto, which is of course the nature of a canon

treated fugally. It is a rhetorical summation, a stacked and contrapuntally forceful

statement of the unfoldment of the subject and counter-subject as put forth by the opening

fugues.

Looking back to exposition in the first ‘simple’ fugues, we are reminded of the

pseudo-canonic gesture that is the fugal exposition. The fugal introduction, where each

voice enters as though in canon, but then breaks off, sometimes after an elision, into the

counter-subject. The counter-voices formed by gradually fragmenting the motivic fabric

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into discrete ideas and re-assembled, rhetorically search for the convincing solution.

Surrounding the mid-point of the form by 11 measures, entries are obscured rhythmically

from the implied 4 beat grid of the exposition. At the soprano’s high register entry at

golden mean, the counter-subject begins to form from a new type of material pulled from

the subject contour which is pushed forward and brought to a climax, in which the

counter-subjects are placed in contrary motion leading to a dominant pedal in the bass.

12 measures before the end, the rhetorical gears lock in place, reiterating the earlier

counter-subject material coupled with the second counter-subject, the bass and tenor

voices canonically imitate each other sequentially. After the dramatic pause, the B-A-C-

H theme appears in the counter-voice (alto). The final tenor entry of the subject is stated

over a tonic pedal in the bass, and within the intertwining counter-motives in the alto and

soprano. Here the subject and counter-subject dynamic is just beginning to expose what

will in the rest of the Art of Fugue become the central theme of development and

transformation. Stated together at the end of Contrapunctus I, Bach foreshadows the

characteristic elements which will be contrasted, fragmented and recombined – forming

an accumulative vertical and horizontal unfoldment of the rhetoric. The dual premise of

the subject as ‘developed’ through episodic, and contrapuntal polarities, will be

ultimately reiterated into a fixed, static, truth.

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BIBLIOGRAPHY:

Butler, Gregory. “Ordering Problems in J. S. Bach's ‘Art of Fugue’ Resolved”, TheMusical Quarterly, Vol. 69, No. 1. (Winter, 1983), pp. 44-61.

Eggebrecht, Hans Heinrich. J. S. Bach’s Art of Fugue: the work and its interpretation;English translation by Jeffery L. Prater. Iowa State University Press: Ames. 1993.

Rivera, Benito V. “Bach's Use of Hitherto Unrecognized Types of Countersubjects in the‘Art of Fugue’ ”, Journal of the American Musicological Society, Vol. 31, No. 2.(Summer, 1978), pp. 344-362.

Sheldon, David A. “The Stretto Principle: Some Thoughts on Fugue as Form”,The Journal of Musicology, Vol. 8, No. 4. (Autumn, 1990), pp. 553-568.

Tovey, Donald Francis. A Companion to ‘The Art of Fugue (Die Kunst Der Fuge), J. S.Bach, Oxford University Press, London: Humphrey Milford, 1931.