On Teaching: The Art of the Fugue, part 3

Colin MacKnight

Colin MacKnight is a C. V. Starr Doctoral Fellow at The Juilliard School, New York City, where he also received his bachelor’s and master’s degrees. He is in the studio of Paul Jacobs and is working on his dissertation entitled “Ex Uno Plures: A Proposed Completion of Bach’s Art of Fugue.” He currently serves as associate organist and choirmaster at the Cathedral of the Incarnation in Garden City, Long Island.

A frequent competition prizewinner, MacKnight holds the Fellow and Choirmaster certificates from the American Guild of Organists (having won the prize for top score for the latter) and is a member of The Diapason’s “20 Under 30” Class of 2019. Upcoming performance highlights include recitals in Ingelheim, Germany; Kingston, Jamaica; and at Saint Matthew’s Cathedral in Washington, DC. Colin MacKnight is represented in North America by Karen McFarlane Artists, Inc. For more information, media, and a calendar of performances, visit colinmacknight.com.

“Miss no opportunity to practice on the organ; there is no instrument that takes such immediate revenge on the impure and the careless, in composition as well as in the playing, as the organ.”¹ This description from Schumann was likely referring to the organ’s ability to execute—one might even say affinity for—complex counterpoint. It is only fitting then that his only organ work would be a set of six fugues on Johann Sebastian Bach’s surname, a homage to music’s greatest contrapuntist, one of Schumann’s principal influences, and the composer who dominates the organ repertoire to a degree that no other composer dominates any other repertoire. Schumann stated, “What art owes to Bach is to the musical world hardly less than what a religion owes to its founder.”² He also acknowledged the influence that Bach exerted on his own music; Schumann’s compositional style was unusually motivic, and he attributed his disdain for what he called “lyric simplicity” to his study of Bach and Beethoven.³

In German musical parlance, B is B-flat and H is B-natural, allowing one to turn Bach’s surname into the motive B-flat, A, C, B-natural. By composing a set of fugues based on the theme
B-A-C-H, Schumann was participating in a long tradition of composing pieces (particularly for the organ) that include the B-A-C-H motive.

This tradition began with Johann Sebastian Bach himself and continues to the present. Two notable examples of Bach encrypting his own name occur in the Toccata in F Major for organ, BWV 540i (in transposition), and Contrapuncti 8, 11, and 14 of Die Kunst der Fuga, BWV 1080. Felix Mendelssohn was the next composer of a substantial body of organ music to encrypt the B-A-C-H motive into one of his pieces. In measure 56 of the first movement of his Sonata IV in B-flat from the Six Organ Sonatas, op. 65, he prominently includes the B-A-C-H motive in the pedal, transposed down a whole-tone (Example 1).

Schumann was close friends with Mendelssohn and even wrote a glowing review of the organ sonatas in his journal, Neue Zeitschrift für Musik, so Mendelssohn’s encryption may not have been unnoticed by him. Perhaps it is not coincidence that Schumann composed his set in 1845—the year in which Mendelssohn’s organ sonatas were published.⁴ In addition to Mendelssohn’s organ sonatas, Mendelssohn’s Six Preludes and Fugues for Piano, op. 35, and Three Preludes and Fugues for Organ, op. 37, were almost certainly strong influences on Schumann’s B-A-C-H fugues.⁵

Schumann was, however, the first composer to write a large work based on this theme. This proved to be influential; other composers who would later write substantial organ works based on B-A-C-H include Franz Liszt (Prelude and Fugue on the Name B-A-C-H, S. 260, versions of which exist for piano and organ), Sigfrid Karg-Elert (Passacaglia and Fugue on B-A-C-H, op.150), Max Reger (Fantasy and Fugue on the Name B-A-C-H, op. 46), Ernst Pepping (Three Fugues on B-A-C-H), etc. Schumann’s set is the longest of these works.

The year 1845 is often called Schumann’s contrapuntal year because of his and Clara Schumann’s intense study of counterpoint, resulting in such compositions as the Six Studies in Canonic Form, op. 56, and Four Sketches, op. 58, both for pedal-piano; Six Fugues on B-A-C-H for organ, op. 60; and Four Fugues for piano, op. 72.⁶ During this year of “Fugenpassion,”⁷ to use Schumann’s own term, they studied Friedrich Wilhelm Marpurg’s Treatise on the Fugue and Luigi Cherubini’s A Course of Counterpoint and Fugue.⁸ Schumann also studied counterpoint intensely from 1831 to 1832 under Heinrich Dorn and from 1836 to 1838, a period which yielded more contrapuntally complex and rich works such as Kreisleriana.⁹

Schumann finished the first fugue on April 7 and the second on April 18.¹⁰ Soon after the second fugue was completed, a rented pedal-piano arrived at the Schumann house.¹¹ This would have been a useful tool for composing the Sketches, Canonic Studies, and Fugues, but, surprisingly, this is also around the time Schumann began to eschew the use of the piano as a compositional tool.¹² Perhaps, then, the pedal-piano was mainly used to assist in pedal-writing.

After the second fugue, progress on the set slowed for a variety of reasons including illness, work on the latter movements of the Piano Concerto in A Minor, op. 54, and organization of an orchestral concert series in Dresden.¹³ By the end of September, Schumann had drafted the third, fourth, and fifth fugues¹⁴ and in late November, Schumann completed the set.¹⁵ The influence of the B-A-C-H project can also be seen on his next opus, Symphony No. 2 in C Major; the second trio of the scherzo uses a theme beginning with the B-A-C-H motive, and the adagio contains a fugato (Example 2).¹⁶

Schumann’s set of six fugues is in many ways similar to and likely inspired by Bach’s The Art of the Fugue since both are thorough explorations of the contrapuntal potential of one musical idea. Schumann was, however, somewhat disdainful of The Art of the Fugue for being excessively cerebral.¹⁷ Part of this is probably due to what Schumann may have perceived as a lack of variety in The Art of the Fugue. Since The Art of the Fugue is based on a single subject and the Schumann B-A-C-H fugues are only based on a motive, Schumann has considerably more flexibility in his thematic material. (Interestingly, this is not unlike Bach’s use of the B-A-C-H motive in The Art of the Fugue. In Contrapunctus 8, the motive is masked with repeated notes and inversion; in Contrapunctus 11, he un-inverts it but retains the repeated notes. It is not until Contrapunctus 14 that Bach plainly reveals the motive, a technique of which Schumann surely would have been proud.)

Schumann, like Bach, derives several distinct fugue subjects from the B-A-C-H motive. The first, third, and sixth fugues, for example, all plainly feature the motive as the main substance of the subject. The fifth fugue, however, treats it just as the starting point for further elaboration. The fourth fugue also uses the motive overtly but changes its contour by leaping down a sixth from A to C, instead of up a third. By modifying and developing the theme, Schumann reveals and incorporates one of his favorite compositional genres: the character piece. John Daverio describes this important difference between fugues and character pieces by saying, “If the essence of a fugue is a fixed subject, then that of the character piece is the transformation of an eloquent motive.”¹⁸ By combining elements of these two genres, Schumann is not just taking a neo-baroque diversion but is “updating” the fugal form to include the most modern musical trends.

Each of Schumann’s fugues also has specific tempo and dynamic indications to contrast the movements. The work is framed by two large accelerando, crescendo fugues; the second fugue is a virtuosic allegro; the third, serene and lyrical; the fourth, a more austere study; and the fifth, a charming scherzo. In this way, the work is not just a compilation of fugues but also a suite of complementary movements. Schumann also provides tonal variety by including G minor and F major movements into an overarching B-flat major—something The Art of the Fugue does not do despite its much greater length. Additionally, there is greater variety of texture in Schumann’s set than The Art of the Fugue. Schumann is not averse to devolving to homophony as he does in all but the third fugue. He also frequently composed in what looks like a “lazy” five-voice texture by writing five-voice expositions but not maintaining a strict five-voice texture (until the last fugue). This is, however, more likely another attempt at textural variety than contrapuntal ineptitude on Schumann’s part; by moving the bass line between the pedal and left hand, he is varying the sound and texture even while maintaining the same number of voices.

As previously stated, the first fugue is an accelerando, crescendo fugue. The quintessential example of this, and likely inspiration for Schumann, is the first movement of Mendelssohn’s Sonata No. 3 in A Major from his Six Organ Sonatas, op. 65. Schumann’s subject comprises two bars, the first of which is the B-A-C-H motive plainly stated, and the second of which includes the B-A-C-H motive in retrograde in its first, second, fifth, and sixth pitches (Example 3). At the point at which Schumann indicates to begin crescendoing and accelerating, he combines the B-A-C-H motive (or a variant thereof) in the pedal with a two-voice stretto of the diminished form of the subject in the manuals (Example 4).

The piece builds to a very exciting homophonic climax, particularly when the performer has accelerated enough (Simon Preston starts at 92 beats per minute and comes close to doubling the tempo, reaching 168 beats per minute at the fastest), with double-pedal before a five-bar coda that returns to a polyphonic texture, although without the B-A-C-H subject. Because of the lower register and reduction in texture, it is not uncommon to decrescendo through the coda, even though this is not indicated in the score.

The second fugue is the allegro movement and the most unabashedly virtuosic. It is also the only fugue in a triple meter, a trait of which Schumann takes full advantage through the use of hemiola. Its subject begins with a quick dotted B-A-C-H before beginning a sequence that is almost certainly taken from the fugue from Bach’s Toccata and Fugue in D Minor for organ, BWV 565. Schumann would have known this work from Mendelssohn’s famous Bach recital at the Thomaskirche in Leipzig, which he enthusiastically reviewed in his Neue Zeitschrift für Musik (Examples 5 and 6).¹⁹

In measure 48, Schumann combines the subject with its augmentation in the pedal. Then in measure 74, there is a rest for performer and listener alike when Schumann quietly strettos the augmented B-A-C-H motive with occasional interruptions from the BWV 565 motive. (These fragments of the BWV 565 motive make the connection to Bach’s fugue even more obvious.) This motive gradually takes over the texture while crescendoing until the piece devolves into a virtuosic toccata, complete with double-pedal, arpeggios, octave doublings, and no hint of B-A-C-H. He then briefly alludes to the quiet B-A-C-H stretto passage again—this time with neighbor-tones on the B-flat—before a passage of triumphant homophony. There is one more fugal interruption before the chorale-style writing returns and a hemiola passage with sforzando chords every two beats. The movement ends with a coda over a B-flat pedal that continues to use the BWV 565 motive while eschewing the B-A-C-H motive, a final confirmation of this fugue’s inspiration.

The third fugue is the only one that is entirely quiet; there is a piano indication at the beginning with the description “Mit sanften Stimmen”—with gentle stops—and no further performance instructions. This is the only fugue that begins in two voices with the counter-subject present from the beginning. It is also the simplest and technically easiest fugue. The B-A-C-H theme only occurs in one form, and the emphasis is on lyricism rather than intellect or virtuosity. It is also the only fugue in a minor key, G minor, but ending in a tranquil G major.

The fourth fugue is the most austere and perhaps the least accessible of the set. This is clear from the outset when Schumann uses a jagged version of the B-A-C-H motive; instead of an ascending minor third between A and C, he writes a descending major sixth, meaning the subject outlines a pungent diminished octave. This is one of Schumann’s cleverest techniques: to utilize the B-A-C-H motive as a collection of pitch classes with no specific contour rather than a traditional theme with a set shape.   

Schumann further adds to the complexity and austerity of this movement by introducing the retrograde of B-A-C-H for the first time and immediately combining it with the subject’s normal form. The retrograde form of the subject almost always has staccato markings on the first and third notes to draw attention to itself, retrograde being among the more obscure contrapuntal techniques (Example 7).

This movement continues to use retrograde pervasively and eventually transitions into loud chordal writing with flashy pedal scales. Like the second fugue, it alternates between passages of strict counterpoint and homophony. The passagework and homophony bring what was an ascetic contrapuntal exercise—perhaps worthy of Schumann’s own questionable criticism of The Art of the Fugue—to a dramatic and exciting close. The climax in measures 96 to 99 is particularly thrilling and one of the highlights of the whole work, as if to apologize for the trials through which he has just put the listener.

As an additional conciliatory gesture, Schumann placed his charming scherzo-fugue next. The presence of a scherzo—an unusual template for a fugue—in this set is further proof that Schumann conceived this set as a suite of complementary pieces and not just miscellaneous movements based on the same theme. The fifth fugue is the only movement in F major and the shortest of the set. This movement’s subject begins with a fleeting B-A-C-H that sounds almost like a perfunctory after-thought—or pre-thought. It is a refreshing relief after the previous four movements which all employ the B-A-C-H motive so plainly.

Nevertheless, the scherzo is still a contrapuntal tour de force. It includes augmentation, inversion, retrograde, and retrograde augmentation, and combines the augmented form of the subject with the original form and the augmented retrograde form with the plain retrograde, an astonishing amount of artifice for a movement of less than three minutes. The technique, however, never hinders the charm. This fugue demonstrates well Schumann’s outlook on fugal composition: “Anyway, this will always be the best fugue the public for instance regards as a waltz by Strauss—in other words, where the artificial rootage is covered like the roots of a flower so that we can see just the flower.” For Schumann, artifice and contrapuntal ingenuity were always subservient to beauty and emotion.²⁰

The sixth and final fugue, the longest of the set, is another accelerando, crescendo fugue. It is also the only double fugue in the set, giving it more in common with the first movement of Mendelssohn’s Sonata No. 3 in A Major, op. 65, which, as previously mentioned, is also a crescendo, accelerando, double fugue. The three-bar subject begins with the B-A-C-H motive and ends with a short descending scale that is the source of the second subject. The first section also prominently features a five-note motive as a counter-subject, beginning in measure 16. This motive, intentionally or not, sounds like a quotation of the main motive in Bach’s chorale prelude on Valet will ich dir geben, BWV 736. The descending scale that closes the first subject becomes the second subject in measure 58 and begins a new section that is marked Lebhafter and più forte. This section does not use the B-A-C-H motive but instead develops the second subject, including stretto and inversion, and uses the same five-note countersubject.

In measure 95, Schumann finally combines the subjects and countersubject signaling the third and final part of this fugue, the only instance of strict five-part counterpoint in the entire set (Example 8).

The strict counterpoint continues until measure 116 when Schumann switches to grand, fortissimo, chorale writing with a few quasi-contrapuntal interruptions. The homophonic texture allows him to compose abrupt and distant modulations that are unusual in strict fugal textures: most notably, the modulation to G major in measure 139 and back to B-flat in measure 142. As in the second fugue, Schumann eventually eschews the B-A-C-H motive (after measure 144 in the sixth fugue), choosing instead to close the entire work not with either of the subjects but with the counter-subject.

As previously mentioned, Schumann “updates” the fugal form by incorporating elements of the character piece, which he does by developing and transforming the B-A-C-H motive. By examining the different transformations of B-A-C-H, as well as other motives, one can see certain relationships between movements beyond the obvious thematic unity of a monothematic work. Specifically, the movements can be organized into three related pairs: movements one and four, two and three, and five and six.

The first fugue is related to the fourth by merit of the fact that it utilizes a countersubject, itself derived from the B-A-C-H motive, which prominently features leaps of sixths, the same interval that is so characteristic of the fourth fugue’s subject (Example 9). 

While this relationship by itself may seem somewhat tenuous, Schumann retroactively confirms it with what is probably the most shocking harmony of this fugue: the false ending at measure 60, in which an F dominant-seventh chord resolves to a secondary-dominant ninth of the subdominant. What sounds like a dramatic change in register is actually the jagged contour of the fourth fugue’s subject in the soprano! The soprano and tenor lines also continue to prominently feature leaps of sixths through the coda of this movement (Example 10).

The second and third fugues are also connected by an allusion in the second fugue to the third fugue’s subject. In measure 48 of fugue two, the B-A-C-H motive appears in the pedal in augmentation with three extra notes that together with B-A-C-H will form the third fugue’s subject (Examples 11 and 12).

Just like in the first fugue, the relationship between fugues two and three may seem weak, but Schumann once again confirms it by developing the tail of the third fugue subject in the tenor and bass voices later in the second fugue (measure 135) (Example 13).

The connection between the fifth and sixth fugues is perhaps the most obvious. The fugue subject of fugue five has a five-note figure that occurs in the second (C, B-flat, A, B-flat, C) and third (G, F, E, F, G) measures of the subject (Example 14).

Schumann later inverts this theme, so it also occurs in an up-down shape. The inverted form of this motive (the aforementioned quotation of Valet will ich dir geben, BWV 736) then becomes the countersubject to the final fugue and is combined with both of the last fugue’s subjects. It is introduced in the first exposition in measure 16 and is combined with both subjects in measure 95 (see Example 8).

On the surface, nothing could have been more conservative in nineteenth-century music than a set of six fugues for organ. Further examination reveals, however, how progressive Schumann’s B-A-C-H fugues were. They were likely the first set of pieces to be based entirely on Bach’s name, they constitute a complementary “suite” of fugues—not just a collection of movements—and there are connections and developments between the movements that foreshadow Brahms’s developing variation. Schumann himself wrote, “I worked on this set for the whole of last year in order to make it somewhat worthy of the exalted name it bears; [it is] a work that will, I believe, long outlive my other works.”²¹ His prediction that his reputation would be based largely on these fugues did not prove to be accurate, but it was surely right of him to hold them in such high regard.

Notes

1. Helga Scholz-Michelitsch, “Robert Schumann and the Organ,” trans. Susanne Weber, The Franz Schmidt Organ Competition, http://orgelwettbewerb.kitz.net/Franz-Schmidt-OrganCompetition/Robert_Schumann.html. 

2. Eric Frederick Jensen, Schumann (New York: Oxford University Press, 2001), 145.

3. Ibid., 145–146. 

4. Scholz-Michelitsch, “Robert Schumann and the Organ.”

5. John Daverio, Robert Schumann: Herald of a “New Poetic Age” (New York: Oxford University Press, 1997), 308.

6. Daverio, Robert Schumann, 306.

7. Ibid., 307. 

8. Scholz-Michelitsch, “Robert Schumann and the Organ.”

9. Daverio, Robert Schumann, 306.

10. Ibid., 307.

11. Ibid.

12. Jensen, Schumann. 284.

13. Daverio, Robert Schumann, 307.

14. Ibid.

15. Ibid., 308.

16. Jensen, Schumann, 289.

17. Ibid. 144.

18. Daverio, Robert Schumann, 309.

19. Scholz-Michelitsch, “Robert Schumann and the Organ.”

20. Ibid.

21. Daverio, Robert Schumann, 308.

Bibliography

Daverio, John. Robert Schumann: Herald of a “New Poetic Age.” New York: Oxford University Press, 1997.

Jensen, Eric Frederick. Schumann. New York: Oxford University Press, 2001.

Scholz-Michelitsch, Helga. “Robert Schumann and the Organ.” Translated by Susanne Weber. The Franz Schmidt Organ Competition. http://orgelwettbewerb.kitz.net/Franz-Schmidt-OrganCompetition/Robert_Schumann.html.

Gavin Black

The Art of the Fugue, part 7

I begin this column with an account of something that happened in a recent lesson, something surprisingly germane, by complete coincidence, to what I had already been planning to write about this month. I notice recently that between when I finish one column and when I am due to start writing the next, something often crosses my path, completely by chance, that is relevant to what is coming next in the column. Sometimes it provides an example, other times an interesting sidelight. Often it essentially ratifies and strengthens what I had been planning to write; yet sometimes it suggests a bit of a change.

Before I get to my story, it occurs to me that the fact that I encounter these coincidences as often as I do is itself a commentary on the philosophy and purpose of the column. My hypothesis is that everything is relevant to performing music and to teaching others about music and performance. I believe everything we encounter in life might be potentially fruitful for our music and our teaching.

This series of columns on The Art of the Fugue is meant to be an account of my working process. Part of that working process is to be aware of anything that might lead to an interesting thought about the piece or the performance. If we notice things and assume that they will be interesting and relevant, we cannot lose. If they are relevant, the gain from having noticed them is clear and describable; if they are not, the thought process sparked by noticing them is still rewarding.

The teacher-student experience

A week or so ago a harpsichord student of mine brought up The Goldberg Variations, partly as a step in thinking about whether to work on the piece, partly to ask a few questions. What is the overall structure of the piece? Did the composer write the movements in the order in which we see them? Does the order matter in performance? If so, why? Is it about individual transitions or something else? We talked about all of this a bit, mentioning “official” sources of structure and continuity. These start with the basic fact that each variation shares some version of the same underlying harmonic pattern and includes the phenomenon of every third variation’s being a canon. We talked about the formal layout of the canons, in which each one is a canon at a one-greater interval—Variation 3 a canon at the unison, Variation 6 at the second, Variation 9 at the third, and so on, up to Variation 27, which is a canon at the interval of a ninth. We discussed, but did not resolve the question of whether this formal layout is palpably meaningful to a listener, or is just meaningful as a formality.

Then we tried a live-action version of something similar to an experiment I described in a recent column, about electronic listening. I played a movement, or the last few lines of one, and my student turned some pages at random. I then started the movement on the new page, randomly chosen, after the one that I had just completed. We did not go through the whole piece in this fashion—not enough time, and other things to do. But we got a good sample of what it felt like to move from one variation to another when that transition is neither what the composer had in mind nor what we are used to.

The result was that each transition sounded fine (“worked”, nothing jarring) but also sounded odd. The oddness came from the confounding of fixed expectations, fixed by years (or for me, decades) of listening. It is hard or impossible to sort out what each of those transitions would have sounded like to someone who had never heard the piece before. I strongly suspect that they would each have been just fine, but I cannot be sure of that. This was a reminder that structure comes in part from expectation. The kind of expectation that comes simply from having heard the work before can be very powerful. But I assume (or perhaps I hope) that expectations created intrinsically by the music are even more powerful.

Sources of overall structure in The Art of the Fugue

It is my working hypothesis that in The Art of the Fugue there are two main sources of overall structure. The first is the nature of the theme itself and the way that it sets up other themes and musical gestures to be meaningful. The second is the recurrence of specific themes. I mentioned both of these in my 1985 program notes that were reprinted here in two recent columns. But I want to delve into both of them in more detail in this column and in next month’s. Of these two ideas, the recurrence of specific themes is, perhaps, the more clear-cut. It is not remotely unusual to include as a source of continuity and of structure. This includes all of the uses of the main theme and its variants: the top level of what we get by contrapuntal analysis. But there is also more below the surface.

My thoughts about the nature of The Art of the Fugue theme are perhaps more speculative. That theme starts on a note, goes up a perfect fifth, goes down a major third, then down a minor third, then down a semitone. After that it goes up by that same semitone, and up and down by step until it ends. Every musical theme, of course, has some pattern of intervals that I or anyone could describe in words. What is striking to me about this theme is how comprehensively, systematically, almost encyclopedically, it lays out all of the intervals that define tonal music. They occur in what could plausibly be considered their order of importance, there is essentially no redundancy: almost no interval is repeated until each interval has been heard (the exception is the semitone), and the inversions of intervals are assumed rather than stated.

My own experience as a listener has been that throughout the piece I hear any interval as a reference to this theme. This starts right away, as soon as there is anything to hear other then the theme itself. The scale notes that make up the counterpoint to the second entry of the theme at measure 5 fill in the opening interval of the theme itself (Example 1).

Does a listener hear it this way? Does a listener spontaneously think, “that is that same theme” or “that is a reference to that same theme?” I believe I do, and that I began to do so after repeated listening years ago, before I thought that I might have any idea why.

The chromatic countersubject in Contrapunctus III seems to arise out of that original semitone that is the interval from the fourth note to the fifth note of the opening theme. The leaping thirds that are one of the characteristic gestures of Contrapunctus IV seem to answer the thirds that make up measure 2 of the theme. Contrapunctus V plays around further with the idea of the third. First of all, when the theme comes in the second time, it is a third away from the note that accompanies it. (In Contrapuncti I–IV, this has always been the interval of a fifth.) Second, this movement contains a pervasive gesture that is the interval of a third filled in by step, and there are also a lot of parallel thirds!

The thirty-second-note flourish that pervades Contrapunctus VI, and that is a principal justification of the heading of this movement as being “in French Style,” sounds like a reference to the last four notes of the opening theme. The mordent-like figure that enters Contrapunctus VII in measure 3 is related to the semitone-based gesture in the opening theme going (in Contrapunctus I) from the end of measure 2 into measure 3. For me, the nature of the opening theme itself turns the whole piece into a tapestry of familiar, known, referential themes and gestures.

A complement to this is the gradual introduction of the octave into the picture. The opening subject, while displaying all of the discrete intervals of its tonal language, almost pointedly fails to encompass an octave. However, the first thing that happens after the filled-in reference to the opening interval that I pointed out in Example 1 is a drop of an octave: the first octave in the piece. When the second voice to enter (the soprano voice) finishes its statement of the fugue subject, it immediately makes an octave leap: one not in any way required by the counterpoint or harmony (Example 2).

Then, later on, whereas each of the first seven contrapuncti begins with the interval of either a fifth or a fourth, Contrapunctus VIII beings with a step, a very striking change. While the compass of the theme of each of the first seven movements has been never more than a sixth, the compass of the opening subject of Contrapunctus VII is a tenth, but one that could also be seen as outlining an octave. (The first and last notes of the subject are an octave apart, and the note that creates the tenth is off the beat and somewhat ornamental.) Then Contrapunctus IX begins with the brand-new gesture of a leap of an octave. The compass of this subject is a ninth, and that of the next movement a tenth, so that we are in a region of expanded compass of themes.

All of this is enough to make me feel that it makes sense to say that the subject or fundamental building block of The Art of the Fugue is not “the AOTF fugue subject” but the very concept of the melodic interval. This in itself does not create structure in the sense of linear shape. But it establishes the conditions for the creation of that structure.

The repetition, recurrence, and referencing of identifiable individual themes is the foundation of counterpoint, and probably the major defined source of contrapuntal structure. At the level of “this theme is the inversion of that theme” or “this theme is the same as that theme, but with altered rhythm,” this is fundamental and definitional. And there is an abundance of that sort of correspondence in The Art of the Fugue. What interests me beyond that is the more fleeting or hidden thematic connections. There are quite a few of these in the piece. The phenomenon that I have tried to describe above (the tendency of this piece to permit any interval, even in an isolated occurrence, to seem meaningfully thematic) is a background against which it becomes clear, I think, that small, individual thematic connections are meaningful to a listener and almost certainly intentional on the part of the composer. Some of the ones that stand out to me are as follows:

This seems to be an isolated event in the bass line in measure 35 of Contrapunctus IV. The notes seem like a filler in a kind of quasi-cadence (Example 3).

All of the eighth-note motion in the movement prior to this has been by step, as is almost all of it after. This is then picked up in Contrapunctus X (Example 4). The figure that enters in the tenor voice at the beginning of this example is passed back and forth among all of the voices and then culminates in an exact statement (at a different pitch level) of the theme from Contrapunctus IV cited above.

The beginning of the main theme of Contrapunctus X (Example 5) is referenced in passing in the bass voice near the end of Contrapunctus XI (Example 6) and again in the middle of the final movement (Example 7).

That latter one is an “answer” rather than a literal quoting of the motif. Are these parallels valid? It could certainly be argued that the fragments of themes that I am pointing to here are just routine cadential figures or other tropes that are too non-specific to be meaningful. I do not hear them that way. Again, I think that the structure of The Art of the Fugue subject itself predisposes any motivic entity to be significant, and the ways in which some of these fleeting motifs are deployed seems too systematic to be non-intentional.

As I said above, I think that there is more to be gained by assuming that correspondences are real than by resisting hearing them that way. Next month I will bring forward several more examples—perhaps even more important to the actual overall structure of the work as a whole—and talk more expressly about that structure.

To be continued.

Gavin Black

The Art of the Fugue, II

For discussion in this the next two columns, I offer the program notes I wrote for my first performance of The Art of the Fugue in May 1985. This performance, on the Fisk organ at Westminster Choir College, was one of my two graduate recitals. I prepared these notes over more or less an entire semester and had some input and help from my teacher Eugene Roan and from William Hays, who was the advisor for degree recital program notes. I have been pleased with this essay, and I have used it as partial program notes for subsequent performances. It has an integrity to its overall structure—thanks in significant part to Dr. Hays’s assistance—such that I have not changed it or excerpted it. Despite that, if I were to write these notes today, there are a number of things I would phrase differently.

It could be fruitful to use some of those theoretical revisions to frame future columns about the learning process, the evolution of my relationship with this work, and the relationship between my own work on this piece and teaching. Some of what I wrote about the order of the movements was too cut-and-dried, rather too simple, failing to reflect some of the complexities of what we do and do not know about the piece. In later columns, I will discuss that, including some new ideas.

History and form

J. S. Bach wrote The Art of the Fugue during the last years of his life, probably beginning work on what turned out to be his longest and most complex instrumental composition in 1743, leaving the opus incomplete at his death in July 1750. It was published in 1751 in Leipzig in a poorly engraved edition, the preparation of only part of which had been supervised by Bach himself. The publication was not a commercial success, and the project was soon abandoned by Bach’s heirs.

Copies of The Art of the Fugue circulated among musicians, however, from that time on. In 1799 a scholar referred in print to the work as “celebrated,” and both Mozart and Beethoven owned copies. The Art of the Fugue was studied extensively by musicians throughout the nineteenth century, and nearly twenty editions or arrangements were published during those years. The first known public performance of the whole work took place in 1927 in Leipzig under the direction of Karl Straube, one of Bach’s successors as Kantor of Saint Thomas School in that city.

The Art of the Fugue is a work of well over an hour in length, consisting of eighteen movements all based in one way or another on the same musical theme. This theme occurs in something like one hundred different forms throughout the piece. The first and simplest form of the theme is shown in Example 1.

The theme is closely based on the tonic triad of the key of D minor, or, looking at it another way, on the interval of a fifth, and on the idea of filling that interval in. The first gesture creates a perfect fifth; the next gesture fills in that fifth, in the simplest possible way. The rest of the theme provides the remaining notes needed to fill in the perfect fifth, D–A, by step, and outlines a diminished fifth, C-sharp–G. In the tonal world of Bach the perfect fifth is the source of security and repose, while the diminished fifth is a source of tension, unrest, and striving. The two are antithetical to one another. This antithesis, with the one side represented not only by the perfect fifth as such but also by all diatonicism, and the other side mainly represented by the chromaticism implicit in the diminished fifth, is a major source of direction, growth, and meaning throughout The Art of the Fugue.

The opening theme also contains, in significant contexts, all the intervals from the semitone to the perfect fifth. This is in spite of the brevity, compactness, and apparent simplicity of the theme. The use of such a theme creates a situation in which any interval, either open or filled in by step, can be used by the composer as a motive significantly related to the main theme of the work. This possibility for motivic interrelation is an important source of unity and coherence in The Art of the Fugue in spite of considerable variety and diversity.

Most of the movements of The Art of the Fugue are fugues or are largely constructed through fugal procedures. Four movements are strict two-voice canons. Bach did not designate any of the movements as fugues, but rather as contrapuncti. (He may well also not have been responsible for the title under which the work is known, since the title page was engraved after his death.) He seems to have been concerned in his use of nomenclature to suggest that the movements were not autonomous fugues such as the organ fugues or the fugues of the Well-Tempered Clavier (all of which are paired with non-fugal preludes), but rather stages in the working out of a musical idea, or a set of musical ideas, through a variety of contrapuntal techniques. Several of the movements, even apart from the canons, would probably not have satisfied Bach’s own definition of a fugue as such, because of serious irregularities in the construction of their opening sections. These irregularities, however, make perfect sense as stages in the contrapuntal development of the work as a whole. They serve invariably as responses to what has come before and as preparations for what will follow. These relationships are described in detail below in the comments on the individual contrapuncti.

The four two-voice canons (numbers 12–15) are lighter in texture and mood than any of the other movements and are simpler in construction. Coming after the most complex of all the contrapuncti, and before the movements in which contrapuntal ingenuity is carried to its farthest extremes, they provide for performer and listeners a moment of repose. This makes possible a renewal of energy and of momentum towards the climax of the final movement. Many individual Bach organ fugues contain within their structure a similar “relaxed” passage, which serves a similar function of providing a breathing space before the final climactic musical gesture. (Measures 121–139 of the Fugue in C minor, BWV 546ii, and measures 141–155 of the Fugue in E minor, BWV 548ii, are particularly good examples of this.) This suggests that The Art of the Fugue should be thought of not as a collection of fugues, but as one structure analogous to a single giant fugue. Further facts bear this analogy out (assuming it is not pressed into too detailed a form). The first movements of the work introduce the main musical ideas in a straightforward way, as does the exposition of a fugue.

The middle movements of The Art of the Fugue develop those musical ideas and others, with increasing complexity, contrapuntal and harmonic, and with increasing variety of texture. This is similar to the middle section (sometimes called “development”) of many fugues, especially, longer ones. The four canons fulfill the purpose described above. In the final three movements harmonic complexity is reduced, and anything even approaching the almost impenetrable density of Contrapunctus 11 is abandoned. In Contrapunctus 17, the original theme is reintroduced in a form closer to the opening of Contrapunctus 1 than anything that has been heard since Contrapunctus 4. This is analogous to the return of the initial subject that characterizes the final section of many fugues. The extraordinary contrapuntal ingenuity of Contrapuncti 16 and 17 (see below) is analogous to the increase in contrapuntal complexity that is found at the end of many Bach fugues, usually in the form of stretto.

Neither the first edition of The Art of the Fugue nor any of the eighteenth-century manuscript copies say on what instrument or instruments the work was meant to be performed. Over the years many different performing forces have been used, including piano, chamber ensembles of various composition, symphony orchestra, jazz combo, harpsichord, and organ. Many scholars believe that Bach actually meant the work for organ, some that he meant it for harpsichord, even though the posthumous title page says neither. The first edition was published in open score, that is, with a separate line for each voice. This was an old Italian and German way of presenting keyboard music used, for example, by Samuel Scheidt in his Tabulatura Nova (1624). It was certainly not the standard keyboard notation in 1750, but Bach had used it shortly before, in his Canonic Variations, BWV 769. The contrapuncti all fit very well under two hands and two feet, and with some difficulty under two hands alone. The pedal parts work as pedal parts: that is, they can be learned using the kinds of pedal technique known to Bach and his students, and when so learned they are comfortable (though occasionally challenging) to play. This would not be true of the bass lines of Bach chamber works or harpsichord works, by and large. The editors of the first edition chose to include a short additional piece by Bach, to compensate the purchaser for the incomplete state of the last movement. The piece they selected was an organ chorale, which they also presented in open score. It is thus likely that they assumed that the users of the work would be organists, even though they did not say so on the title page. It is also quite possible that Bach himself wanted musicians to use their own judgment as to how the piece can be realized in sound.

B-A-C-H

The third subject of the last movement of The Art of the Fugue is made up of notes that, in the standard German musical nomenclature, spell the name “Bach” (Example 2). In the German system, B-flat is called B, and B-natural is called H. Bach was aware throughout his life that the letters of his name made a plausible musical theme—it was certainly known to his musical ancestors as well—but he used it sparingly in his music. The only extensive use he made of it was in The Art of the Fugue. The final appearance of the B-A-C-H theme as the subject of a powerfully climactic fugue in Contrapunctus 18 is prepared by a chain of musical developments running through the whole work. This chain is best followed retrospectively. Before Contrapunctus 18, the B-A-C-H theme appears in Contrapunctus 11. Here, the four relevant notes form part of a lively and insistent eighth-note motive (Example 3). They do not stand on their own, but they are clearly present. This eighth-note motive, however, is an inversion of one of the main themes of Contrapunctus 8. That movement is thus revealed to have contained the B-A-C-H theme in a highly disguised form. The motive also occurs in once in Contrapunctus 8, casually, without repetition or development, in the bass voice at measure 143, transposed up a whole step. The first appearance of the B-A-C-H theme in the work occurs at the end of Contrapunctus 4, where the four notes form part of an otherwise meandering free chromatic countersubject to the main theme. This serves to underline the essential chromaticism of the B-A-C-H theme, and to tie that theme to the other chromaticism in The Art of the Fugue. The seeds of the chromaticism in the work, and thus the seeds of the B-A-C-H motive itself, are found, as explained above, in the initial statement of the main theme. The four contrapuncti in which the B-A-C-H theme is found (4, 8, 11, and 18) are by a considerable margin the four longest movements in the work, and each of the four is longer than the last.

To be continued.

The Clavier-Übung Third Part

Michael Radulescu

Michael Radulescu, born in Bucharest, Romania, studied organ and conducting in Vienna at the Academy (now University) of Music and Performing Arts where he taught as professor of organ from 1968 to 2008. His career encompasses work as a composer, organist, and conductor. Since his debut in 1959 he has presented concerts throughout Europe, North America, Australia, South Korea, and Japan. He regularly gives guest lectures and masterclasses in Europe and overseas, focusing mainly on the interpretation and elucidation of Bach’s organ and major choral works.

As a composer, Radulescu has written sacred music, works for organ, voice and organ, choral and chamber music, and orchestral works. He is also in demand as a jury member in international organ and composition competitions and as an editor of early and ancient organ music. Radulescu conducts international vocal and instrumental ensembles in performances of major vocal works. As an organist, he has recorded among other things Bach’s complete works for organ, without any technical manipulation.

For his musical and pedagogical contributions Radulescu was awarded the Goldene Verdienstzeichen des Landes Wien in 2005. In 2007 he received Würdigungspreis für Musik from the Austrian Ministry of Education and Art. In December 2013 Michael Radulescu’s book on J. S. Bach’s spiritual musical language, Bey einer andächtig Musiq . . .,
focusing on the two Passions and the B Minor Mass was published.

When approaching Baroque music in general and spiritual music in particular, it is of greatest importance to take into consideration the fundamental difference between the function and the aims of music in the Roman Catholic rite and the Lutheran conception of music. While Roman Catholic music mainly embellishes and adorns the liturgy, Lutheran music wants to preach, to impress, to move, to convince every single listener. Whereas the mystery of the Canon is at the center of the Roman Catholic Mass, the announcing and the elucidation of the Word of God, spoken by the minister and sung or performed by the church musician, stand at the core of the Lutheran Divine Service.

From this dichotomy results the overwhelming importance of rhetoric, of the musical speech (Klangrede) in Lutheran music. Both the ancient rules of rhetoric and the use of the rhetorical-musical figures determine respectively the overall formal concept of a work as well as the invention of characteristic “speaking motifs.”

In the case of J. S. Bach’s music, however, there also seems to be a more subtle, profound, and hidden means of communicating a message, an interpretation of a text. This happens through the ample use of symbols such as allegories, certain characteristic motifs and specific numerical ratios between different sections of the overall formal concept of a piece, and also, most controversial of all, as numerological entities. The latter aspect has been both heartily emphasized and strongly questioned and even ruled out by scholars and practical performers in recent decades. Nevertheless, a surprising hint at the possibility of Bach’s interest in the use of the “numeric alphabet” seems to be, among others, the theoretical work called Cabbalologia by Johannes Henningius (Johann Henning), published in Leipzig in 1683. This publication is said to have been found also in the famous private library of Bach’s neighbor and colleague Johann Heinrich Ernesti, former rector of Saint Thomas Church in Leipzig.

I

Bach published the Third Part of his Clavier-Übung for the feast of Saint Michael at the end of September 1739 on the occasion of the bicentenary of the Lutheran Reformation in Leipzig. This collection of keyboard compositions is generally known under the titles “The Organ Mass” or “The Dogma Chorales,” neither of which can suggest the complex meaning and the message of the entire opus.

It should be remembered that when Luther introduced his Reformation in Leipzig in 1539 he preached on Pentecost Monday in the Leipzig Pleissenburg Castle on two most crucial themes: the Mystery of the Trinity in the Lutheran Mass and the Lutheran Catechism. Most significantly, Bach takes both these theological categories into consideration and, obviously referring to Luther’s sermon of 1539, treats them consistently in his Third Part of the Clavier-Übung. Of the total of twenty-one chorale settings in the collection, the first nine deal with the Lutheran Missa brevis (which includes only the “Kyrie-Christe-Kyrie” and the “Gloria”), while the remaining twelve chorales follow exactly, chapter by chapter, Luther’s Catechism of 1529.

Seen as a whole, the entire Clavier-Übung III seems to suggest a most striking resemblance to Bach’s own organ improvisations as described by his first biographer, J. N. Forkel, in 1802:

a) a great prelude and fugue in Organo Pleno as an opening;

b) a long series of different kinds of chorale settings with a varying number of parts;

c) a great fugue in Organo Pleno at the end.

In Bach’s Clavier-Übung III, these correspond to the following sections:

a) the E-flat Preludium in Organo Pleno also containing the two fugal sections;

b) the 21 chorale settings in 3, 4, 5, or 6 parts, as well as four duettos;

c) the E-flat Fugue in Organo Pleno.

Two further allusions to the Trinity are most interesting in the overall plan of the entire collection. These are manifest already in the title, “Third Part of the Clavier-Übung,” and also in the use of the majestic key of E-flat major, with its three flats in the signature, for both the opening Prelude and the closing Fugue. Also striking is the fact that both the Prelude and the Fugue appear to be determined by the number 3 (three main musical ideas in the prelude and three themes in the triple fugue).

Another obvious hint at the Trinity is the fact that the first 9 chorales dealing with the Lutheran Mass are organized in 3 groups of 3 each: 3 “great” settings for Kyrie-Christe-Kyrie, 3 “small” alio modo settings for the same cantus firmi Kyrie-Christe-Kyrie, and 3 settings for the German Gloria, “Allein Gott in der Höh’ sei Ehr.”

The remaining 12 chorales, which follow Luther’s Catechism, are arranged by 3 + 3 groups of 2 each, the first group dealing with the 3 main chapters of the Catechism (The Law of the Lord = The Ten Commandments, The Creed, and The Prayer of the Lord = The “Our Father”), and the second with the 3 chapters concerning the Sacraments and the Penitence respectively (Baptism, Penitence as continual renewal of Baptism, and the Communion). Each of these cantus firmi is treated twice, in a “great” version with pedal and in a “small” version without pedal, mostly in another key.

It has often been suggested that these two contrasting versions may allude to Luther’s “Great Catechism” versus its reduced form, the “Small Catechism” for younger and “more modest people.” This double treatment of the “catechism settings,” however, seems also to allude to the double form of liturgy: as the great, official one “in churches,” versus its “small,” intimate, personal form “at home,” within each Christian family. Interestingly enough, this dualism appears also in the original subtitle of the Clavier-Übung III dedicated to both amateurs (Liebhaber) and connoisseurs (Kenner).

II

The opening Praeludium pro Organo pleno, Bach’s largest organ prelude, suggests, in spite of the original slurring of the dotted rhythms of its beginning, the pattern of a French overture:

a) majestic homophonic section with dotted rhythm, measures 1 to 70;

b) Fugato section, measures 71 to 97;

c) majestic homophonic section with dotted rhythm, measures 98 to 129;

d) Fugato section, measures 130 to 173;

e) majestic homophonic section with dotted rhythm, measures 174 to the end.

The three different musical ideas used by Bach seem to illustrate in a marvelous way the three Persons of the Trinity:

1. majestic five-part homophonic section for God the Father (Example 1);

2. transition passage with staccato notes suggesting drops of tears (as in the Passions and in several cantatas) and a plaintive theme in the right hand, full of suspensions and chromaticisms and going to the “extreme” keys B-flat minor and E-flat minor, respectively (musical-rhetorical figure of parrhesia), suggesting the human sufferings, the Passion and Death of God, the Son (Examples 2 and 3);

3. The fugal sections using the most spiritual writing, the fugue, and a theme which by its shape (musical-rhetorical figure of hypotyposis) suggests the movement and the shape of the flames, the fire of God, the Holy Spirit (Example 4).

III

Considering the 9 chorale settings of the Missa brevis, the great “Kyrie-Christe-Kyrie,” the small “Kyrie-Christe-Kyrie,” and the 3 “Allein Gott” settings, one notes the following characteristics:

• The first three settings of the great “Kyrie-Christe-Kyrie” are written in the ancient vocal, a cappella style, the stylus gravis, using the so called white notation (breves, whole notes, half notes, quarter and, more rarely, eighth notes as note values). According to Bach’s cousin
J. G. Walther the stylus gravis is “majestic, serious . . . and best appropriate to elevate the human soul to God.”

• The respective cantus firmus descends within this first triad from the soprano in Kyrie, Gott Vater in Ewigkeit (highest part ~ God Father as the Highest) into the tenor in Christe aller Welt Trost (middle part ~ God the Son as the Mediator) and finally into the pedal-bass in Kyrie, Gott, heiliger Geist (bass part ~ God, the Holy Spirit as the universal Basis). This katabasis, i.e., “descending movement,” suggests the descending of God’s mercy upon us and depicts the “eleison” (“have mercy”).

• The tenor cantus firmus in Christe aller Welt Trost stresses the idea of Christ as the Mediator between God and Man, as strongly emphasized by Luther.

• The bass cantus firmus in Kyrie, Gott, heiliger Geist, on the other hand, represents the fundamental Lutheran idea of Justification through the power of Faith; the text of the chorale also prays for “the reinforcement of our Faith.” The final section of this setting, “eleison,” is excruciatingly dissonant, once again stressing human misery awaiting God’s mercy.

• The total number of measures of all three large chorale-settings is a primary, indivisible number:

Kyrie (42 measures) + Christe (61 measures) + Kyrie (60 measures) = 163 ~ indivisibility of the Holy Trinity!

• The three small settings of “Kyrie-Christe-Kyrie” strongly contrast with the preceding three works. The cantus firmus is only hinted at by quotation of its first phrase. Their writing is manualiter, without pedal, and in a soft “cantabile clavier style.” This might suggest love and the soft breath of the Holy Spirit by its “cantability.”

• All three small settings end modally on an E-major chord.

• The time signatures of all these 3 chorales also allude to the Trinity, being “progressions” of the number 3: 3/4; 6/8; 9/8 (= 1 x 3/4; 2 x 3/8; 3 x 3/8).

• The three Allein Gott in der Höh’ sei Ehr settings fulfill a wonderful anabasis (ascending movement) by the sequence of their keys: following the small “Kyrie-Christe-Kyrie” settings ending all on E major, they rise up to F major, G major, and finally to A major, thus obviously alluding to Gloria in excelsis (Allein Gott in der Höh’/“Glory to the Lord in the Highest”).

• All three settings are trios and written in an “instrumental keyboard style,” the first and the last in a brilliant, light style, the second à 2 Claviers et Pedale imitating violins or flutes accompanied by a basso continuo in the pedal.

• The G-major trio on “Allein Gott” seems to stress Jesus’s role as Lamb of God, alluding to the third stanza of the chorale, “Lamb of God, holy Lord and God, accept the prayer of our misery,” by citing these two verses in canon, a most simple symbol for “one part following another part:” first between the right hand and pedal in measures 78 to 83, and in measures 87 to 92 between the left hand and pedal, and thus alluding to the Gospel of John, 1:29–30: “The next day John seeth Jesus coming unto him, and saith, Behold the Lamb of God, which taketh away the sin of the world. / This is he of whom I said, After me cometh a man which is preferred before me: for he was before me” (Example 5).

IV

The density and complexity of Bach’s dealing with the theological message through music is most impressively revealed in the settings of chorales treating the main chapters of Luther’s Catechism: the Ten Commandments, the Creed, and the Lord’s Prayer.

The large setting of Dies sind die heil’gen zehn Gebot is written in five parts distributed on two manuals and pedal. The cantus firmus is carried out as a canon between the two tenor parts played by the left hand while the right hand plays the two free upper parts. Surprisingly, these free parts never imitate or cite the cantus firmus.

Most interesting is the fact that in Bach’s treatments of this cantus firmus (Orgelbüchlein, the cantata BWV 77 Du sollst Gott, deinen Herrn, lieben, and the two settings in the Clavier-Übung), he uses the same key, Mixolydian on G, the “pure” key without accidentals in its signature. Never does this cantus firmus appear transposed: this obviously suggests the “immutability of the Divine Law.” Most consistently, the treatment of the cantus firmus as a canon also evokes the “severity of God’s Law.”

A further symbolic meaning of the musical texture is the setting of the canonic cantus firmus in the two middle parts, which clearly refers to Luther’s commentary in his Catechism, regarding the way to keep the Divine Law through “Christ’s the Mediator’s Intercession.”

The beginning of the chorale is most serene, diatonic, and calm, and takes place over an organ point in the pedal. After four measures of “complete harmony” the character changes in the fifth measure: the alto plays a “harsh” descending chromatic figure (the figure of parrhaesia) while the soprano plays three times a “sighing figure” consisting of a sixteenth rest followed by three sixteenth notes, and followed by two groups of stepwise descending eighth notes (Example 6).

This seems to be a strong allusion to the Book of Genesis describing the Garden of Eden (= full harmony~4 measures) and Adam’s Fall in the fifth measure (Adam in Hebrew meaning man and being symbolized, according to Andreas Werckmeister, by the number 5 for man’s 5 senses, 5 fingers and toes, and also hinting at Jesus’s 5 wounds on the Cross).

Interestingly enough, this “sighing” figura suspirans is played by the two upper parts during the whole piece exactly 33 times, reminding of the 33 years of Jesus’s earthly life.

From measure 6 on this figure appears also “transformed” into another figure called kyklosis or circulatio and suggesting a “turning around,” an “insecurity” or, as in our case, a great joy.

This “transformation” of suffering (“sighing figure”) into joy (“turning around in joy”) perfectly matches Luther’s commentary about the Commandments, stressing that those who keep the Law apparently suffer in this earthly world, but that through Christ they shall live in joy.

Luther also considers the First Commandment as being the most important of the Decalogue. It is this very commandment that is cited in the second stanza of the cantus firmus, the stanza to which the great chorale setting seems to allude the most: “I alone am your God and Lord. Thou shalt not have other gods; thou shalt love me from the bottom of your heart. Kyrieleis.”

It is when the cantus firmus expounds the phrase “Thou shalt not have other gods” that the pedal plays a “huge” and “exaggerated” interval of two octaves,
C – c′ (the figure of hyperbole = exaggeration) and obviously referring to God’s immensity (Example 7).

Astonishing is the fact that the motif of measures 47 and 48 appears altered in measures 51 and 52, transformed insofar as it is now divided between the two upper parts: one part continuing the other, and thus suggesting the idea of “two parts becoming one” (the figure called heterolepsis = meaning this continuity, the unification of two parts, i.e., love, as described by J. G. Walther). It is striking to note how often Bach makes use of this figure when alluding to love, to unification in and through love. Not surprisingly, this figure appears in our chorale setting only two times, exactly where each of the two canonic cantus firmus parts play the notes for lieben mich (love me); as one can easily see in the “transformed” version, the motive is played by two “unified” parts according to the text line “Thou shall love me” (Example 8).

If we take a look at the pedal part we note that it is divided into several sections either by rests or by the recurring long organ point on A in measure 29. A most intriguing and striking speculation presents itself in this context when considering the number of notes of each of these sections:

a) measure 1 to 10 = 37 notes

b) measure 10 to 20 = 60 notes

c) measure 21 to 28 = 47 notes

d) measure 29 to 55 = 147 notes

e) measure 56 = 5 notes

f) measure 57 = 5 notes

g) measure 58 to 60 = 14 notes

 

a) Could 37 represent the monogram JCHR for Jesus Christ? (the number alphabet with the correspondence between the letters of the alphabet and the natural numerical order: A = 1, B = 2, C = 3, . . ., Z = 24, with I = J and U = V as in old Latin: J (9) + C (3) + H (8) + R (17) = 37);

b) Could 60 allude to the Old Testament, to the 6 Days of God’s Creation, and also to the 10 Commandments = 60?

NB! Bach occasionally uses the number 6 as allegory for the Creation, for the Entire World (also Orgelbüchlein: Christum wir sollen loben schon, measure 6, where the whole range of the organ is encompassed by the lowest C in the pedal and the highest C in the treble part).

NB! Luther always sees and treats the Old Testament considering the New Testament and vice versa.

c) Could 47 recall the 47th Psalm, mentioned by Luther in his Great Catechism: “O, clap your hands, all ye people; shout unto God with the voice of triumph. / For the Lord most high is terrible; He is a great King over all the earth”?

NB! This third section of the pedal starts in measure 21 where the cantus firmus plays the phrase “Thou shalt not have other gods.” Also, it is here where the pedal plays the enormous, exaggerated interval of the double octave, which also perfectly matches the second verse of Psalm 47.

d) Could 147 recall the 11th verse of the 147th Psalm: “The Lord taketh pleasure in them that fear Him, in those that hope in His mercy”?

NB! Luther himself quotes Ps. 147, 11 in his Catechism, in the chapter dedicated to the Ten Commandments. This could make the assumption mentioned above quite plausible!

e) & f) Could the number 5 possibly allude in this context to mankind (five senses; the five wounds on Jesus’s crucified body) as “the Old” vs. “the New Man”?

g) 14 might well suggest Bach’s own name (B [2] + A [1] + C [3] + H [8] = 14) as his personal commitment as a believer, as the pro me (= “for me”), a central point in Luther’s theology.

Another interesting symbolic connotation is suggested by the general form of the chorale setting. The total of 60 measures is clearly divided into two unequal sections considering the sort of “recapitulation” of the beginning, in measure 29:

28 measures (= 7 x 4) + 32 measures (= 8 x 4) = 60 measures, or 28 : 32 = 7 : 8.

Could 7 allude to the seven days of the week, of the 6 + 1 days of the Creation of the earthly world and 8 to the eighth day (the day of Messiah)? Could this overall form and its “articulation” transmit the message of Redemption?

The “small,” manualiter version of Dies sind die heiligen zehn Gebot seems to have a more obvious, more straightforward approach to the text. It is a fughetta using the theme in both normal and inverted position. The gigue-like theme is characterized by the strongly repeated notes at its beginning and by strong leaps followed by stepwise passages. It is most interesting to note some aspects of this piece:

1) the title in the original print from 1739 is Dies sind die heiligen zehen Gebot consisting of exactly 10 syllables (Ten Commandments?)

2) the repeated G in the theme appears 14 times (BACH’s commitment? See above).

3) the theme appears 4 times in normal, 4 times in inverted, and again 2 times in its normal forms, i.e., 4 + 4 + 2 = 10 times (see above).

4) there is quite a long interlude without the theme between measures 18 and 31, lasting 14 measures (see above).

V

The large chorale setting dealing with the Creed, Wir gläuben all’an einen Gott (Schöpfer) is striking because of its dynamism, abundant syncopations, “modern” 2/4 time signature, constant movement in sixteenth notes, and lack of organ points in the pedal, by the six times of the pedal ostinato, and the flamboyant movement of the manual parts. The theme treated in the manual is rooted in the first phrase of the cantus firmus, and it is this very phrase that appears literally quoted in the tenor in the last 12 measures of the piece. The overall flamboyant, dynamic character of this setting might be surprising, but it seems in perfect coherence with Luther’s idea of a willful, powerful, and passionate personal commitment of each believer aiming to attain personal justification.

Some characteristics of this composition might elucidate its possible further message:

a) the total of exactly 100 measures of the piece might suggest the idea of the totality of the Creation (Gott Schöpfer = God, the Creator);

b) the 6-fold appearance of the pedal ostinato might hint at the 6 “working” days of God’s Creation (see above);

c) the quotation of the first cantus firmus phrase in the tenor, starting in measure 89 might allude to Christ as the Mediator;

d) the last pedal entry is longer than its other entries and has exactly 43 notes; this may well mean: (C [3] + R [17] + E [5] + D [4] + O [14] = 43: CREDO) “I believe.”

NB! Interestingly enough, the score of the first Credo chorus in the B Minor Mass shows the word “Credo” written 43 times and heard 41 times, i.e., J-S-B-A-C-H’s creed.

The small version of the same chorale is written as a short manualiter fughetta in the style of a brilliant French overture. This surprising setting can be seen as an introduction to the large version of The Lord’s Prayer, Vater unser im Himmelreich, written in the same key of E Dorian. More likely, however, it also seems to have the function of dividing the whole set of 21 chorales into 12 + 9. One should remember that, on the other hand, the 21 chorales are also divided into 9, dealing with the Lutheran Mass, and 12, treating Luther’s Catechism and the Sacraments. A very beautiful parallel, indeed!

VI

The large version of Vater unser im Himmelreich is possibly Bach’s most difficult and intricate organ work. It is written in 5 parts distributed once again among the two manuals and the pedal, with the cantus firmus in canon. Unlike the Ten Commandments however, each hand here plays a free voice and a canonic cantus firmus part.

Some characteristics may help understand and elucidate the enormous complexity of this composition:

a) the slow, majestic tempo in the 3/4 time signature suggests the austere character of a slow sarabande;

b) the pedal is treated as a basso continuo without quoting the cantus firmus;

c) the cantus firmus is treated in canon suggesting our intimately repeating the prayer spoken by Jesus according to Saint Mark and Saint Matthew;

d) the alternating order of the canonic parts at each new entry seems to suggest a still dialogue between the believer and Jesus;

e) the free manual parts are based on a theme quoting the richly embellished first phrase of the cantus firmus (Example 9);

f) each hand expounds this theme 3 times, alluding probably once more to the Trinity;

g) the two free manual parts display an enormous rhythmical richness with frequent use of the “plaintive” Lombard rhythms and the staccato triplets (Example 10);

h) this “plaintive” Lombardian rhythm and the overall rhythmical complexity seem to depict Luther’s comment on The Lord’s Prayer expressing the “multitude of human miseries;”

i) the staccato triplets obviously describe Saint Matthew 7:7: “Ask and it shall be given to you; seek and ye shall find; knock and it shall be opened unto you.” As a matter of fact, this very verse appears quoted in practically all older Lutheran hymn books on the page where the chorale Vater unser im Himmelreich is printed. The staccato triplets may also allude to drops of tears;

j) there is only one spot where the pedal quotes the “plaintive” Lombardian rhythm and this happens in measure 41 (J [9] + S [18] + B [2] + A [1] + C [3] + H [8] = J. S. BACH), alluding to the composer’s personal commitment.

After this extraordinary piece, the alio modo manualiter version of the same cantus firmus is a simple, quiet meditation on the Prayer, devoid of all further speculative symbols.

VII

Following Luther’s Large Catechism exactly, Bach now treats the Sacraments of Baptism in Christ, unser Herr zum Jordan kam, Penitence in Aus tiefer Not schrei ich zu dir, the latter considered by Luther as the continuation and constant renewal of baptism, and finally the Sacrament of Communion in Jesus Christus, unser Heiland.

The large version of Christ unser Herr zum Jordan kam treats Jesus’s baptism as described in Saint John, Chapter 1. The piece is set for two manuals and pedal with the cantus firmus in the latter, the bass in the left hand and the two upper parts in the right hand. This setting is quite full of important symbolic meanings:

a) the tenor cantus firmus in the pedal suggests, as the middle part of the setting, Christ’s role as Mediator between God Father and mankind;

b) the almost constant movement in sixteenth notes in the left hand bass part seems to allude to the flow of the waters of the Jordan River;

c) the two upper parts of the right hand can be seen as a symbol for the Holy Spirit floating above the scene of Christ’s Baptism by Saint John the Baptist. The beginning four notes in each of the two upper parts seem to depict, as a hypotyposis, a cross motif. Also, the most intricate imitations between the small motives of the two upper parts can be seen as a hint to the Holy Spirit proceeding from the consubstantiality of God Father and God Son, as mentioned in the Nicene Creed (Example 11).

d) NB: the final note of the fifth chorale phrase in the pedal d° seems to generate a “wrong” 6/4-chord d° - g° - b′: This is to be seen as a hint to avoid the wrong harmony by the use of a 4′ reed in the pedal if the left hand were based on 8′, or a 16′ basis for the left hand, should the pedal be played only on an 8′ basis!

e) The total number of measures, 81, equals 3 x 3 x 3 x 3 as a most impressive symbol for the Trinity.

The small manualiter version of the same chorale is quite a short fughetta based on the first phrase of the chorale, combined with an “obbligato” counter-subject, both treated in normal and inverted position. Could the theme itself represent Christ and its inverted form Christ’s descent on Earth? Could the countersubject stand for Saint John the Baptist? Interesting enough is the fact that this fughetta consists of 27 measures (3 x 3 x 3) with exactly 81 quarter notes (see above).

The large version of Aus tiefer Not schrei ich zu dir, the German version of Psalm 130, “De profundis,” is an exceptional work, as it is written in the old, solemn, majestic vocal stylus gravis or motet style, which, according to Johann Gottfried Walther’s Musicalisches Lexicon of 1732, is able to “elevate the soul to God.” This setting marks a pinnacle in Bach’s entire organ music insofar as it is written in six parts, four in the manual and two in the pedal, with the augmented cantus firmus of Luther’s chorale melody in the right foot’s part. This obviously seems to be an allusion to the significance of the upper bass part as the voice of the Old Testament psalmist. Most impressive is also the fact that at the beginning of the last verse of the chorale Wer kann, Herr, vor dir bleiben? (Who can, Lord, stand before Thee?) in the seventh to last measure, the upper bass part playing the cantus firmus is the highest part in the whole texture (Example 12).

• The registration should be the Organo pleno, i.e., an 8′ based Plenum  in the (coupled) manual(s) and 16′ Plenum in the pedal, without mixtures but with reeds 16′, 8′, and 4′.

• This setting is obviously inspired by the great pleno settings in five parts, with double pedal, in Matthias Weckmann’s great chorale settings with the cantus firmus in the upper pedal part.

NB! In one of the Lüneburg tablatures containing Weckmann’s majestic hymn on O lux, beata Trinitas the opening first movement in five parts with double pedal and the cantus firmus in the upper bass bears the indication that the cantus firmus of the upper bass could be played in the pedal by the right foot, or on the manual by the left hand, or also by both the pedal and the left hand together. This comment seems to confirm the registration mentioned above, with the result that the left foot bass is playing in the reeds-pleno, the manual parts in the mixture-pleno and the cantus firmus in both the reeds- and the mixture-pleno, and thus strengthening the cantus firmus.

The following alio modo manualiter version of the same chorale is written in four parts. Learned contrapuntal imitations in the three lower parts—in normal and inverted form—of each phrase of the chorale, anticipate each phrase of the augmented cantus firmus expounded each time by the treble part.

• Each section of the piece begins with five contrapuntal measures in intricate counterpoint between the three lower parts, followed by eight bars expounding the respective phrase of the chorale in the treble and one supplementary bar concluding each section.

• The overall organization of the piece is quite extraordinary:

Sections a), b), c) & d): 5 + 8 + 1 bars; section e): 5 + 8 + 5

• But 5 + 8 + 1 = 14  [= B-A-C-H = 2 + 1 + 3 + 8] and 8 : 5 stands for the golden ratio.

The large version of Jesus Christus, unser Heiland, der von uns den Zorn Gottes wandt is a trio for the two manuals and pedal with the cantus firmus in the latter. It seems quite interesting that the pedal oscillates between playing the tenor and bass parts. Could this hint at Jesus’s double nature, as God and Man?

• The two manual parts seem to actually symbolize the “Wrath of God” by their extremely virtuosic, agitated, and aggressive movements in sixteenth notes and eighth notes.

• The main theme in the manuals starting with big and then diminishing intervals (tenth-octave-sixth) could possibly hint at Man’s approach to God, whereas, on the other hand, these leaps sometimes occur also in the opposite direction, from smaller to larger (sixth-octave-tenth). The message of these patterns seems to be the “struggle” between God and sinful mankind expecting redemption through communion, Luther’s second sacrament.

The following alio modo version of the same chorale is a very complex fugue in F minor, using as a main theme the first phrase of the cantus firmus. The extremely rich counterpoint and the surprisingly daring new motives seem to recall the big, learned fugues of the Well-Tempered Clavier, Book II.

• The augmented entry of the main-theme in the tenor part in measure 57 might be another symbol for the praise of Christ the Lord, as the mediator between God and Man.

• NB! In order to emphasize this augmented theme in the tenor it should be helpful to use a registration of foundations (principals) 8′ and 4′ and a trumpet 8′.

VIII

Most intriguing and surprising part of the work are the following four duettos preceding the final Fugue in E-flat Major. Some speculations might help justify their presence:

a) Luther adds a “Short Admonition of Confession” after the chapter about Communion. In this short appendix he quotes the various ways of confessing: 1. to the priest/pastor; 2. as an open and common confession in front of the congregation; 3. to the neighbor; and 4. to God;

b) in the first part of his Large Catechism Luther quotes the four elements of the world: 1. Fire; 2. Air; 3. Water; 4. Earth;

c) in his Neu vermehrtes Hamburgisches Gesangbuch (New Hymn Book) from 1739, Vopelius inserts after the Catechism Hymns other hymns for: 1. the morning; 2. the evening; 3. before meals; 4. after meals;

d) taking into consideration the Baroque Theory of Affects one can easily imagine a certain parallel with the four temperaments: 1. choleric; 2. sanguine; 3. phlegmatic; and 4. melancholic temperament;

e) the duettos form a tight unity: their tonal progression ascending from E to F, to G, and finally to A corresponds strikingly to the sequence of keys in the “Trinity chorales” 4 to 9, and thus leading to the first note, B-flat, starting the following fugue;

f) two of the duettos are in a major (II and III) and two in a minor key (I and IV);

g) two are in a ternary (I: 3/8 and III: 12/8) and two in a binary (II: 2/4 and IV: 2/2) time signature.

h) two start with the right hand (I and II) and two with the left hand (II and IV).

It also seems quite remarkable how well the duettos match—by their astonishing variety and by their individual character—both the conception of the four elements (mentioned by Luther in his Great Catechism) and that of the four temperaments and even maybe of the four archangels (Michael, Gabriel, Raphael, and Uriel) as well as the four Evangelists (?).

Duetto I: E-minor key; 3/8 time signature; right hand starts, left hand follows; 73 measures; perfectly symmetrical form based upon the golden ratio (28 measures + 17 measures + 28 measures = 73 measures // 28:17 = ~ 1.64; 45 (= 28 + 17) : 28 = ~ 1.7; 73 (= 28 + 17 + 28) : 45 = ~ 1.62); flamboyant themes and countersubjects suggesting flames of fire; Archangel Michael (with attributes: fire, sword, perfect balance); choleric temperament (?); element Fire (?).

Duetto II: F-major key; 2/4 time signature; right hand starts, left hand follows; 149 measures; perfectly symmetrical form of: 37 + 31 + 13 + 31 + 37 measures. NB! 37 could stand for Christ’s monogram in the Greek alphabet [ChRistos]: X ~ CH (= 20) + P ~ R (= 17); 31 may stand for the Latin “In Nomine Jesu” (In Jesus’s Name): [I (= 9) + N (= 13) + I (= 9) = 31]; 13 could allude to Jesus and his Twelve Apostles at the Last Supper. NB! This section of 13 measures from measures 69 to 78 is the center, the middle of the whole piece in which the measures 74 to 78 are the exact “inversion” of measures 69 to 73; could that maybe hint to Jesus’s death?; element Air (?) (Example 13).

The overall form of the piece is quite complex, insofar as the first section and its da capo recapitulation (both 37 measures) are in major and in a serene, joyous mood, whereas the second and penultimate sections (both 31 measures) are in minor and written as canons; might this “discrepancy” remind one of the sanguine temperament (?); Air; could the three references to Jesus Christ (see above) suggest a link to the Archangel Gabriel, Jesus’s messenger (with the attributes: lily and fish); could the perfect formal symmetry represent the symmetrical beauty of a lily?; could the inversion, the crossing of the parts in measures 69–78 hint at a symbol for Christ’s Cross and Death?

Duetto III: G-major key; 12/8 time signature; left hand starts, right hand follows; 39 measures:

15 + 8 + 15 + 1 = 39 measures; 15 (= 3 x 5) + 24 (= 3 x 8) = 39 (= 3 x 13) = golden ratio (cf. Fibonacci); melancholic temperament (?); could the very serene character of the piece remind of the Archangel Raphael (with attribute: fish)? element Water?

Duetto IV: A-minor key; 2/2 (Alla breve) time signature; left hand starts, right hand follows; two themes are used (a and b); 108 measures arranged as 8 (a) + 8 (a) + 16 (b) + 8 (a) + 8 (a) + 8 (b) + 13 (b) + 8 (a) + 8 (b) + 10 (b) + 13(a); NB! The grouping of measures and themes reveals the scheme of: 9 x 8 (= 72 measures) + 2 x 13 (= 26 measures) + 2 x 5 (= 10 measures), an order once more based upon the progression 5, 8, and 13 as quantities of the Fibonacci progression hinting at the “golden ratio;” the quite robust character of the music seems to allude to the strong phlegmatic temperament, while the very intricate formal scheme of the piece might possibly be a hint to the archangel Uriel (with attribute: fire); element Earth?

IX

The concluding Fuga à 5 Pro Organo pleno in E-flat major perfectly continues the ascending keys movement of the duettos (E-F-G-A) by its starting with a B-flat in the tenor.

The main theme suggests by its shape the form of a cross: connecting on paper the first note with the fourth and the second with the third, respectively the second with the fifth and the third with the fourth, respectively the third with the sixth and the fourth with the fifth, respectively the fourth with the seventh and the fifth with the sixth, one obtains three times (Trinity again!) the Greek letter X = Chi used as a symbol of the Cross, for crossing: cf. also Bach’s original title Da Jesus an dem X stund’ and the English No X-ing or Merry X-mas (Example 14).

This majestic theme dominates the whole first section of the fugue written in the ancient stylus gravis (see above, chorales 1 to 3). The second section of the fugue is in 6/4 meter and based on a strongly contrasting theme characterized by its constant movement representing a lengthy kyklosis (“turning around-figure”), with the main notes E-flat—F—G and thus quoting the first phrase of the first large chorale Kyrie, Gott Vater in Ewigkeit (Example 15).

Exactly in the middle of this second section, the majestic first theme reappears, rhythmically strengthened by its syncopations, and dividing the whole fugue into two equal parts of 36 + 22½ : 22½ + 36 (Example 16).

Finally, the third and last section of the fugue written in 12/8 time signature, expounds a third theme that will later be combined with the first and with a varied form of the second theme. This third theme seems to use a bass cadence formula of C–F, and B-flat–E-flat (Example 17).

Most impressive is the perfect formal symmetry of the whole fugue organized in: (20 + 16 =) 36 measures + (22½ + 22½ =) 45 measures + (16 + 20 =) 36 measures.

Considering the fugue as a whole and the most natural tempo relationship of its three time signatures (half note = dotted half note = dotted quarter note), one can conclude the following:

a) the first and the third sections of the fugue are equal in length lasting 36 measures each, divided into 20 + 16, respectively, into 16 + 20;

b) applying the tempo relationship “half note = dotted half note = dotted quarter note” and taking as a common unity of measurement the smaller quantity, i.e., the measure length of the second fugue (which has only two beats per measure vs. the four beats of the first and the third sections respectively), one obtains the following measurements for the three sections:

72 (= 36 x 2) half-measures; 45 measures and again 72 (= 36 x 2) half-measures

c) all these numbers being multiples of 9, these ratios can be reduced to:

72 (= 8 x 9); 45 (= 5 x 9); 72 (= 8 x 9), or just 8 + 5 (= 13) + 8 = 21

d) this series of numbers 8, 5, 13, 21 belongs to the famous “Fibonacci progression” starting by 1:1:2:3:5:8:13:21 and reaching the golden ratio or divine proportion (= “proportio divina”) in the infinite.

e) NB! according to the Italian Renaissance mathematician Luca Pacioli the golden ratio might symbolize the Holy Trinity:

A (the greater quantity/God Father) : B (the smaller quantity/God the consubstantial Son) = (A + B) : A, or, theologically speaking:

A (God Father) engenders B (the consubstantial Son) and, out of these two, proceeds A + B (the Holy Spirit);

f) could this majestic, astonishingly built fugue thus represent once more the ultimate Symbol of the Holy Trinity?

g) its perfectly symmetrical construction is most impressive:

First section (40 half measures—cadence—32 half measures),

Second section (22½—22½ measures)

Third section (32 half measures—cadence—40 half measures), or, more simply:

40 – 32 – 22½ – 22½ – 32 – 40 measure lengths of the second section.

X

Taking a more attentive, new look at the Third Part of Bach’s Clavier-Übung, one discovers some interesting facts concerning the overall compositional plan, a plan corresponding also to Bach’s work, the B Minor Mass:

a) both cycles contain a total of 27 movements each.

b) these 27 movements are divided into two groups of: 6 “free” works without a cantus firmus (prelude in E-flat, the four Duettos, and the final fugue) and the 21 chorales; NB! the “Missa” and the “Symbolum Nicenum” in the
B Minor Mass have together 12 + 9 = 21 movements and the last section of the B Minor Mass (“Sanctus,” “Osanna,” “Benedictus,” “Osanna,” “Agnus Dei,” and “Dona nobis pacem”) also contains 6 movements.

c) the 21 chorales in the Clavier-Übung are divided twice into: 9 for the Lutheran Mass (“Kyrie-Christe-Kyrie – Gloria:” Trinity) and 12 dealing with Luther’s Catechism plus Sacraments.

d) The 21 chorales are also divided (“musically”) into 12 and 9 chorales by the 13th chorale written as a French overture and thus opening the rest of 9 chorales.

[NB! All these numbers are multiples of 3 (Trinity again!).]

e) could the total number of 27 pieces possibly recall in both the Clavier-Übung and the B Minor Mass the 27 books of the New Testament?

f) could the number of 21 pieces allude to the “Teaching Books” of the New Testament, the 21 Epistles, and the 6 “free works” to the four Gospels, the Acts of the Apostles, and the prophetic Apocalypse of John?

g) could one not consider the overall architecture of Bach’s most impressive cycles, Clavier-Übung III and the B Minor Mass, as huge symbols for the New Testament and thereby also for Martin Luther’s Theology?

Colin MacKnight

Colin MacKnight, called “a stunning player of exceptional ability” by composer and conductor Bob Chilcott, is director of music at Trinity Episcopal Cathedral in Little Rock, Arkansas. At Trinity, he oversees a music program that includes among its offerings weekly choral evensong, a concert series, and chorister and choral scholar programs. Prior to Trinity, Colin was associate organist at the Cathedral of the Incarnation, Garden City, Long Island, New York; assistant organist and music theory teacher at Saint Thomas Church, Fifth Avenue, New York City; and assistant organist at Church of the Resurrection, also in New York City.

MacKnight earned his bachelor’s, master’s, and doctoral degrees from The Juilliard School, studying organ performance with Paul Jacobs. For his doctoral dissertation, “Ex Uno Plures: A Proposed Completion of Bach’s Art of Fugue,” Colin received the Richard F. French Doctoral Prize.

A frequent competition prizewinner, MacKnight’s first prizes and scholarships include the 2019 Paris Music Competition, 2017 West Chester University International Organ Competition, 2016 Albert Schweitzer Organ Competition, 2016 Arthur Poister Scholarship Competition, M. Louise Miller Scholarship from the Greater Bridgeport Chapter of the American Guild of Organists (AGO), the 2013 Rodgers North American Classical Organ Competition, and the Ruth and Paul Manz Organ Scholarship. He also won the New York City AGO Competition and advanced to the Northeast Regional Competition, when he won first place, which led to a “Rising Star” recital at the 2016 AGO national convention in Houston. He was also a laureate in the 2016 and 2019 Longwood Gardens International Organ Competitions.

In December of 2016, Colin and composer Jon Cziner were selected for an AGO Student Commissioning Project grant, resulting in Cziner’s Fantasy Chorale, which MacKnight premiered in 2017. Colin has also earned the Fellow and Choirmaster Certifications from the AGO, receiving the prize for top Choirmaster score, and he is a member of The Diapason’s 20 under 30 Class of 2019.

Colin MacKnight is represented in North America by Karen McFarlane Artists, Inc. For more information, media, and a calendar of performances, visit colinmacknight.com.

“The governing idea of the work . . . was an exploration in depth of the contrapuntal possibilities inherent in a single musical subject.”¹ So says Bach scholar Christoph Wolff of Johann Sebastian Bach’s Die Kunst der Fuge, BWV 1080. Comprising fourteen fugues and four canons, we have inherited the work in an incomplete state; soon after Bach introduces his own musical signature as Contrapunctus XIV’s third subject and combines it with two previous themes, the music abruptly stops. There is no consensus on whether the ending was lost or simply never written; what is clear is that Contrapunctus XIV is notably missing Die Kunst der Fuge theme that unifies every other movement. In the 1870s, Gustav Nottebohm discovered that Die Kunst der Fuge theme could be combined with Contrapunctus XIV’s three preexisting themes to create a quadruple fugue.² It would then seem that this was Bach’s plan.

Because of abundant structures and patterns that Bach establishes in the existing part of the work, many aspects of Bach’s plans for the conclusion of Contrapunctus XIV can be estimated with varying degrees of confidence. Indeed, it seems likely that Bach had such remarkable compositional facility that he chose to limit himself with certain external and artificial restraints, such as the use of numerology. This is especially likely in such an abstract work, which is not guided by a text or any other limitation except conservativeness of musical material. These patterns and designs mean that ascertaining Bach’s intentions concerning the conclusion of Contrapunctus XIV requires much less guesswork and subjectivity than one might think.

One of the areas that is elucidated by the study of patterns and precedent is the intended length of Contrapunctus XIV. As we will see, Bach meticulously plans the lengths, subdivisions, and proportions of the polythematic fugues in Die Kunst der Fuge. When considering whether any proportional relationships exist in the work that could indicate how much longer Contrapunctus XIV should be, it seems most logical to start by examining the two triple fugues, Contrapuncti VIII and XI. This is because the forms of the polythematic fugues are mainly governed by the introduction and combination of themes. Contrapuncti VIII and XI are especially significant because as triple fugues they have more themes and sections, and therefore more in common with Contrapunctus XIV.

First, it is necessary to establish a methodology. The lengths of sections can be quantified by beats alone as well as by measures. (When counting beats, I always count quarter notes even if the movement is in cut time.) It is important to accurately calculate the exact number of beats in a section, as opposed to rounding to the nearest whole measure. This is not to say that Bach never rounds, but merely that exactitude can yield important insights.³ The lengths of sections can often be significant in and of themselves, and furthermore, once they have been accurately calculated, there are numerous relationships that can exist between sections.

The tools that seem most likely to be revealing are those of division and subtraction: to divide the lengths of sections by each other (larger by smaller or smaller by larger), or to subtract a shorter section from a longer one. We will find that many of these relationships have numerological significance that indicate intentionality behind their proportional designs.

Numerology and gematria

According to Encyclopedia Britannica, gematria is “the substitution of numbers for letters of the Hebrew alphabet, a favourite method of exegesis used by medieval Kabbalists to derive mystical insights into sacred writings or obtain new interpretations of the texts.”⁴ This concept is naturally not limited to Hebrew, although it is more easily applied to other alphabetic languages (i.e., letters represent sounds) than those that use syllabic (i.e., characters represent syllables or moras, e.g., Japanese) or logographic (i.e., characters represent words, e.g., Chinese) writing systems. Gematria and numerology are closely related, but not synonymous. All gematria might be called a subset of numerology, but not all numerology constitutes gematria. Specifically, gematria is a compartment of numerology based on assigning numerical values to letters to encode meaning.

In Appendix 1 of her book Bach and the Riddle of the Number Alphabet, Ruth Tatlow lists 33 different number alphabets. For the purposes of Bach study, the most common number alphabet is what Tatlow calls “Latin natural-order: variant 1,” shown in Figure 1.⁵

The concept behind this number alphabet is simple, but there are two important details to note: the omission of the letters “J” and “V.” Consequently, this number alphabet goes up to 24, not 26. This is particularly significant when Bach’s gematric designs incorporate his first initial, which notably is missing from this alphabet. Up until the eighteenth century, the Roman letter “I” was used to represent both the vowel sound normally associated with that vowel as well as the sound associated with the consonant “J.” Likewise, “U” stood in for “W.”⁶

This avenue of exploration in Bach’s music came to prominence in 1947 with the publication of four volumes about Bach’s cantatas by Friedrich Smend. One of Smend’s primary observations is the prominence of the number 14, now commonly associated with Bach.⁷ This number’s significance is derived from its gematric value (Figure 2).

In her book Bach’s Numbers, Ruth Tatlow points out the astonishingly serendipitous detail that the numbers corresponding to B, A, C, and H (2, 1, 3, and 8) also correspond to Bach’s birthday, March 21, 1685. Bach would have numbered dates in the order day, month, year, so his birthday would have been numerically represented as 21-3-85, if one omits the first two digits of the year.⁸ (The concluding “5” is not a part of the numerical parallel between Bach’s name and birthday.) Another number of great importance to Bach is 41: the retrograde of 14 and the sum of the letters J, S, B, A, C, and H (Figure 3). The last gematric sum of significance in this study is 55: the sum of 14 and 41. The sum of the digits one and four is also five, so another conception is that 55 represents the sums of the constituent digits of 14 and 41, placed side by side.

These three numbers appear frequently in Bach’s oeuvre. While it is outside the scope of this disquisition to catalog such instances in Bach’s broader output,⁹ it is worth noting certain appearances of these numbers in Die Kunst der Fuge. The following is a non-comprehensive list, most of which are cited from Indra Hughes’s dissertation:

• There are 14 contrapuncti. Additionally, a preliminary version of Die Kunst der Fuge, dating from the early 1740s, had 14 total movements.

• Bach likely intended to submit the work to Lorenz Christoph Mizler’s Correspondierende Societät der musicalischen Wissenschaften, an epistolary forum for the advancement of music theory. Bach waited to be admitted into this society until he would be its 14th member and commissioned the famous Hausmann portrait to commemorate his admittance—a portrait in which Bach is holding the last of his 14 Goldberg canons.¹⁰

• The first two notes of Die Kunst der Fuge theme, D and A, gematrically correspond to the numbers 4 and 1.11 The same can also be said of the first two and last two notes of Contrapunctus XIV’s first subject.

• The gematric sum of the notes comprising the original twelve-note version of Die Kunst der Fuge theme is 55. (D + A + F + D + C + D + E + F + G + F + E + D = 4 + 1 + 6 + 4 + 3 + 4 + 5 + 6 + 7 + 6 + 5 + 4 = 55)

• Contrapunctus II has 14 subject entries.¹²

• Starting in Contrapunctus V, the regular form of Die Kunst der Fuge subject has 14 notes, due to the addition of two passing tones.¹³

• The B-A-C-H motive appears in measures 40–41 of Contrapunctus V (Example 1).¹⁴

• Contrapunctus VI has 14 inverted entries.¹⁵

• Between the two double fugues (Contrapuncti IX and X), there are 14 double combinations.

• The B-A-C-H motive occurs in the alto and soprano voices of measures 40–41 of Contrapunctus X (Example 2).

• Section 1 of Contrapunctus XIV is 114-1⁄4 measures long, a mixed number with numerous allusions to 14 and 41.

• Contrapunctus XIV’s second subject has 41 notes.¹⁶

• The B-A-C-H motive is finally plainly revealed in the 14th fugue.¹⁷

• One additional gematric detail is worth consideration, although it does not use the numbers 14, 41, or 55. Bach titled the work Die Kunst der Fuga, somewhat peculiarly eschewing the German “fuge” in favor of the Italian “fuga.” Anatoly Milka has theorized a numerological explanation for this choice: the gematric sum of the letters in “Johann Sebastian Bach” is 158, which is equal to the gematric sum of the letters in “Die Kunst der Fuga.” If, however, Bach had retained the German “fuge,” the gematric sum of the work’s title would have been 162, and would therefore not have matched the gematric sum of his full name. Additionally, the sum of the numbers 1, 5, and 8 is 14.¹⁸

There are also those who cite certain gematric elements of the final measure as evidence that Bach intentionally left the work incomplete. First, the digits of the final measure—2, 3, and 9—add up to 14. Importantly, this is also the exact number of measures in the Canonic Variations on “Vom Himmel hoch da komm’ ich her,” BWV 769, for organ, another work dating from the late 1740s that Bach submitted to Mizler’s Correspondierende Societät der musicalischen Wissenschaften. Additionally, 2 + 39 = 41 and 23 – 9 = 14.

The remaining gematric details are more easily seen in the autograph, shown in Figure 4. (Note that the top staff is in soprano clef.)

• The final complete measure (measure 238) has 14 notes.

• The final two bass notes are A and D, whose gematric equivalents are 1 and 4.¹⁹ (These notes are enclosed in a black box.)

• The final harmony consists of two Ds and an F. When these pitches are converted into numbers, D is 4 and F is 6, making the gematric sum of this chord’s notes 14 (4 + 4 + 6 = 14). (These notes are enclosed in a red box.)

• In the final measure, seven of the tenor’s eight notes are on the top staff. Only the antepenultimate note, a lone A, is transferred to the bottom staff. The result is that the only two notes in the bottom staff of the last measure are D and A, whose gematric equivalents are 4 and 1.²⁰ Bach’s decision to put the A on the lower staff seems fussy unless it was done to create a gematrically significant design. (These notes are enclosed in a blue box.)

Contrapunctus VIII

Returning to the subject of proportion, let us begin with Contrapunctus VIII—a three-voice triple fugue. In Die Kunst der Fuge’s polythematic movements (i.e., Contrapuncti VIII through XI), new sections are generally delineated through the introduction of new themes. Contrapunctus VIII, however, has one additional section because Bach delays combining all three themes. All three themes are in play in Section 3 but are not presented in triple combination until Section 4. Following that model, Section 1 begins with the introduction of Theme 1 in measure 1 (Example 3), a new theme to the work that gradually descends by one octave over the course of four measures. As previously mentioned, it is important to note that Contrapunctus VIII begins with a half rest, so Section 1 does not begin until the third quarter of measure 1.

Section 2 begins on beat three of measure 39 with the introduction of Theme 2, an inverted form of B-A-C-H utilizing a repeated-note motive, in immediate combination with Theme 1 (Example 4).

Finally, Section 3 begins on beat two of measure 94, where a new form of Die Kunst der Fuge theme, characterized by downbeat quarter rests and composed mainly of quarter notes, enters.²¹ Finally, Section 4 begins on beat three of measure 147 with the first of five triple combinations. Section 1 is 38 measures or 152 beats, Section 2 is 54.75 measures or 219 beats, Section 3 is 53.25 measures or 213 beats, and Section 4 is 41 measures or 164 beats. For ease of reference, Table 1 outlines Contrapunctus VIII’s four sections.

From the chart, one can see that there is a general arch form; the two inner sections are larger than the outer two, the inner sections are almost the same length, and the outer sections are also close in length.

Keeping in mind that the most significant numbers in Die Kunst der Fuge are 14 (the sum of the letters
B + A + C + H), 41 (the sum of the letters J + S + B + A + C + H and the retrograde of the number 14), and 55 (the sum of 14 and 41), these are the numbers that would most clearly indicate intentionality in Bach’s proportional scheme. For this reason, the first step is to look for those numbers (14, 41, and 55) in Table 1. Instantly, one can see that Section 4 is exactly 41 measures. Slightly less obvious is the fact that Section 2 is almost exactly 55 bars. Another potentially significant detail is that the first triple combination occurs in measure 147—a measure beginning with the number 14 and whose latter number is half of 14. Moreover, Section 2 is exactly 55 beats longer than Section 4, and Section 3 is 213 beats long, corresponding to the letters B - A - C.

Finally, and most importantly, sections 2 and 3 are both almost exactly 1.4 times longer than Section 1. Section 3 is slightly closer to this proportion (213 / 152 ≈ 1.4013 . . .) than Section 2 (219 / 152 ≈ 1.4407 . . .). Even more incredible, the proportion between sections 1 and 2 can be rounded to 1.441, thereby combining the numbers 14 and 41! While such a specific number may sound far-fetched, this will not be the last time we see this level of detail, or indeed this exact proportion.²²

Contrapunctus XI

We now turn to Contrapunctus XI—the most adventurous and dramatic movement of the work. Contrapunctus XI is a four-voice triple fugue with an additional chromatic countersubject. It is the sister to Contrapunctus VIII in that they use the same three subjects, but Contrapunctus XI inverts them or, depending on how one looks at it, uninverts them, since Contrapunctus VIII uses the inverted forms of Die Kunst der Fuge and B-A-C-H themes and Contrapunctus XI uses the rectus forms. In the ilk of Charles Ives’s cumulative form technique, in which Ives reversed the standard model of exposition and development by beginning with fragments and motives from a theme that culminates in a plain statement of the entire theme near the end of the piece,23 it is not until Contrapunctus XI that one hears these themes turned right side up (rectus), enabling one to more easily recognize them for what they are (particularly the B-A-C-H theme, which is still masked by the addition of repeated notes in Contrapunctus XI).²⁴

Contrapunctus XI is also, from a formal and proportional perspective, the most enigmatic movement because of the presence of multiple extra sections (i.e., there are more sections than there are themes). Section 1 begins on beat two of measure 1 and introduces Theme 1, the rectus form of the rhythmically altered Die Kunst der Fuge theme from Contrapunctus VIII. Section 2 uninverts Contrapunctus VIII’s first theme and introduces a prominent chromatic countersubject (Example 5).

Section 3 unexpectedly eschews all previously introduced material and simply exposes the inverted form of Theme 1 (i.e., Die Kunst der Fuge theme). Section 4 introduces the rectus form of the B-A-C-H theme in immediate combination with Theme 2. Finally, Section 5 uses all three themes and presents three triple combinations. Table 2 outlines these sections and their respective lengths.

From this table, one can make a number of crucial observations. First, Section 1 is exactly three fifths as long as Section 2 (175 x .6 = 105), a reference to the three themes and five sections, perhaps? Second, Section 3 is .417 times longer than Section 2.²⁵ Third, Section 5 begins in measure 146 (which begins with the number 14). Fourth, Section 4 is 1.46 times longer than Section 5, meaning sections 2 and 3 have the same proportional relationship as sections 4 and 5. Finally, Section 1 is 1.438 times as long as Section 3.

At this point, all the musical evidence indicates that Bach not only meticulously controlled the proportional scheme of Contrapunctus XI but went to extraordinary lengths to use proportions and durations of numerological significance. Yet there are even more astonishing features of this movement’s proportions. First, it is worth noting the relative lengths of the different sections—specifically, that there are two overlapping arches. Sections 1 through 3 form an arch of smaller-larger-smaller, and sections 3 through 5 also form an arch of smaller-larger-smaller.

The most notable feature of this movement’s proportions reveals itself when one adds together the two smaller sections in each arch. Section 1’s 105 beats plus Section 3’s 73 beats add up to 178 beats, which is very close to Section 2’s 175 beats. This on its own may just be coincidence, but an examination of the second arch (sections 3, 4, and 5) demonstrates that it is almost certainly not, since the second arch has the same property; Section 3’s 73 beats plus Section 5’s 155 beats add up to 228 beats, which is even closer to Section 4’s 227 beats.

The numbers are even more exact if one adds measures instead of beats: Section 1’s 26.25 measures + Section 3’s 18.25 measures = 44.5, compared to Section 2’s 43.75 measures. And for sections 3 through 5, Section 3’s 18.25 bars + Section 5’s 38.75 bars = 57—extremely close to Section 4’s 56.75 measures. (This is part of the reason for measuring durations in both beats and measures; what may seem somewhat inexact when measured in beats can seem much more precise when measured in measures, particularly when rounding.)

To put this in simpler terms, Section 1 + Section 3 = Section 2, and Section 3 + Section 5 = Section 4! It is much like the Fibonacci Sequence (1, 1, 2, 3, 5, 8, 13, . . .),
which adds together the previous two numbers in the sequence to produce the next number, except these numbers are out of order. (In this case, one adds outer numbers to produce the inner number, so the first and third numbers add up to the second, the third and fifth numbers add up to the fourth, etc.)²⁶

The enigma of Contrapunctus XI is Section 3. Why does Bach eschew all other themes and compose a section just for the inverted form of Theme I? After all, Theme I occurs in inversion only twice more, so it is not a particularly important thematic variant in this movement. Additionally, themes are typically treated cumulatively; Bach does not usually eschew a previously stated theme in favor of a different previously stated theme. (An example would be introducing Theme 2 and then reverting to using only Theme 1.) He will often temporarily abandon a previously established theme to introduce a new theme, but to do so for an old theme is regressive. (A typical example of the former is in Contrapunctus VIII, where Themes 1 and 2 drop out upon Theme 3’s entry but return later in combinations with Theme 3.) There are a couple exceptions in which Bach does backslide to a previous theme, but these are, without exception, because of permutational designs.

Furthermore, this section is rhythmically calmer than the surrounding sections; sections 2, 4, and 5 have fairly constant eighth-note motion, while Section 3 has more quarter-note motion. Bach has clearly assigned this section a special role. If Bach had a specific reason for doing so, one possibility is because Section 3 is the central section and links the two arches (sections 1 through 3 and sections 3 through 5). Additionally, he needed a way to clearly articulate five sections in this movement—two more than would be necessary in a triple fugue. Perhaps this was his way of accomplishing that.

So what do these analyses tell us about Bach’s intentions concerning Contrapunctus XIV? First, they demonstrate a high degree of care and intentionality in the proportional schemes of both triple fugues. For better or for worse, another key takeaway is that Bach’s proportional structures are inconsistent from movement to movement. Contrapunctus VIII is a three-voice triple fugue that has four sections that form an arch (shorter outer sections and longer inner sections), and Contrapunctus XI is a four-voice triple fugue that has five sections that can be organized into two arches.

The features that are consistent, and which will likely apply to Contrapunctus XIV, are a meticulously planned proportional design and the prevalence of the numbers 14, 41, and 55. Next, we will examine the extant part of Contrapunctus XIV to hopefully find the beginnings of a proportional structure that will shed light on the length of the movement and the sections it comprises.

Contrapunctus XIV

The challenge in an examination of the proportional relationships in Contrapunctus XIV is, of course, that the movement is incomplete. Sections 1 and 2 are complete, Section 3 is partially written, and Section 4 is missing completely. Since it takes three objects to establish a pattern (and we have only two objects in their entirety), any theories regarding Bach’s intended proportions cannot be positively proven; we do not have all the necessary information. The other complication is that if a study of the proportional relationships in Contrapuncti VIII through XI reveals anything, it is that Bach is not so simple-minded as to repeat proportional schemes between movements. This means that the principles we derived from previous proportional evolution can only broadly guide a completion of Contrapunctus XIV. What does seem likely to remain true, however, is that the numbers 14, 41, and 55 will guide the proportional schemes. Indeed, even the number of this contrapunctus appears to carry meaning.

Given that Contrapunctus XIV was clearly intended as a quadruple fugue (as proven by Gustav Nottebohm),²⁷ the most obvious division of sections would be one section per theme. (This is the same model that Bach used in Contrapunctus IX, albeit on a much smaller scale.) This would mean that Section 1 is dedicated to Theme 1, Section 2 uses themes 1 and 2, Section 3 uses themes 1, 2, and 3, and finally, Section 4 uses all four subjects—a straightforward, cumulative design.

The extant part of Contrapunctus XIV provides support for this hypothesis. In Contrapunctus XIV, the themes are treated cumulatively. Section 2 introduces Theme 2 while retaining Theme 1, eventually combining them. Section 3 introduces Theme 3 while retaining themes 1 and 2, eventually combining all three (Example 6). This is not so in Contrapuncti VIII or XI. In Contrapunctus VIII, Bach introduces Theme 3 in Section 3 but starts a new section (Section 4) to present triple combinations. Contrapunctus XI is even less straightforward; Theme 1 disappears in Section 2, and then Theme 2 disappears in Section 3. Section 4 finally uses all three themes, but not in triple combination until Section 5. The extant part of Contrapunctus XIV clearly shows a more direct approach to the exposition and combination of themes.

Now that we have established the likelihood that Bach intended Contrapunctus XIV to have a one-to-one ratio of subjects to sections, let us examine the durations and proportions of the surviving parts. The gigantic first section begins on beat three of measure 1 and goes to the downbeat of measure 115, giving it a duration of 457 beats or 114.25 measures. It seems highly serendipitous that the number of measures in Section 1 can be conveyed as 114-1⁄4—a number with numerous allusions to 14 and 41 (yet another reason to measure durations in both beats and measures). Section 2 is in a Stile antico idiom and is composed largely of quarter-note motion. Theme 1 comprises seven notes and has the interesting quality of being a melodic palindrome. (See Example 7.)

Importantly, Section 2 overlaps with Section 1 by four beats because Theme 2 enters before the other voices have completed their cadence marking the end of Section 1. Theme 2 is composed mainly of eighth-note motion and is exactly 41 notes. The increased rhythmic activity of Section 2 breaks the retrospective style of Section 1. Theme 2 is also the only theme in Contrapunctus XIV that is not presented in inversion in the extant sections (excluding Theme 4, which has not yet occurred in any version). Section 2 begins with the introduction of Theme 2 in the alto on beat two of measure 114 (another reference to 14) and continues through beat two of measure 193, making Section 2 317 beats or 79.25 measures. (See Example 8.)

Section 3 begins on beat three of measure 193 and trails off in measure 239.28 This gives the extant part of Section 3 a length of 186 beats or 46.5 measures, although these numbers may not be particularly significant as they represent only part of a section. Stylistically, Section 3 brings about another shift; the counterpoint becomes rhythmically less active, and the harmony becomes markedly more chromatic and adventurous.29 (This is partially by virtue of Subject 3’s chromatic profile and longer note values.) This marks the end of the extant part of Contrapunctus XIV. (See Table 3.)

One of the challenges of deducing Bach’s intentions is that with only two complete sections, there is not enough information to establish a pattern per se. Nevertheless, it seems likely that the sections should get shorter. As previously stated, Section 1 is truly massive; it has more measures than every movement of the work except the four polythematic fugues and takes about four minutes to perform on its own. The movement would likely feel bloated and disbalanced if there were another section even longer than this one, and indeed, Section 2 is about 30 measures shorter. A final, more subjective reason to believe that the sections will continue getting shorter is that Section 3 is clearly intensifying in both rhythmic and harmonic activity. In just 46.5 bars, it is already clearly approaching a climactic point, at which time Theme 4 will likely enter.

Bach scholar Dr. Gregory Butler of the University of British Columbia at one point theorized that each section of Contrapunctus XIV is intended to be approximately two thirds the length of the preceding section.30 He explains:

As it appears in the print in its incomplete state, this work occupies five pages. It seems clear that the finished version would have fit nicely on six pages. If we examine the relative proportions of the three extant sections of this fugue, we notice a consistent diminution in the lengths of successive sections. Moreover, section 2 (78 measures) is almost exactly two-thirds the length of section 1 (115 measures), and section 3, not quite complete, occupies forty-six measures and conceivably in its complete state would have occupied approximately two-thirds the length of section 2 (52 measures). Adhering to the same proportions, section 4 may well have occupied approximately two-thirds the length of section 3, that is, approximately thirty-four measures. This would leave approximately forty-six measures for the concluding sixth page which is exactly the average number of measures per page for the first five pages as they appear presently in the print.³¹

This theory has a number of issues. The reader may notice Butler’s use of language to indicate approximation. He rounds the sections to the nearest whole measures, and his proportional theory also relies on rounding. Furthermore, he makes mathematical errors, claiming that two-thirds of 52 bars is 34 bars when it is actually closer to 35. He then says that Section 3 needs six more measures and that Section 4 should be 34 measures—a clear sum of 40 measures—but states that this fits “exactly” with one additional page of manuscript, equivalent to 46 measures.

If there is one conclusion to be made from our examination of the proportions in the triple fugues, it is that Bach’s proportional schemes are anything but approximate. Another reason to doubt Butler’s theory is that it does not leave enough time in Section 3 to include even one more triple combination, if there is to be any connecting episodic material. Indra Hughes, whose dissertation, “Accident or Design? New Theories on the Unfinished Contrapunctus 14 in J. S. Bach’s The Art of Fugue, BWV 1080,” is one of the major contributions to Die Kunst der Fuge research, agrees that Butler’s theory is not sufficiently precise and proposes a much more specific and convincing theory.³²

While it seems likely that Bach intended each section to be shorter than the preceding one, this is not proof of a more detailed proportional relationship. Hughes, however, has discovered a feature in the autograph that indicates that the exact lengths of sections were of great importance to Bach. This detail arises from an edit that Bach made at the end of Section 1. Bach crossed out the original material in measures 111 and 112 and replaced them with three measures, thereby adding a bar. The edit can be seen in Bach’s hand in Figure 5; the two measures at the top right have been crossed out and replaced with three measures of tablature in the bottom margin. The two versions are presented in a more readable fashion in Examples 9 and 10.

Hughes argues that there is not a substantial musical difference between the first and second versions, so therefore it is likely that the reason for the edit was for Bach to correct a proportional error, an error that could be fixed by adding one bar to Section 1. In rejection of strong musical reasons for the replacement, Hughes writes the following:

It seems a curious and puzzling change for Bach to have made, and it does not seem easy to find a musical reason for the rejection of the original bars 111 and 112. If Bach had not made the change, but had left the score with those two bars unaltered, would scholars and analysts today point to them and identify them as a weak moment? Perhaps it might be argued that the new bar 111 strengthens the approach to the following cadence (bars 113–114) by the move to the subdominant that was not present in the original; and it might be argued that the new bar 113 is rhythmically very slightly more interesting than the old bar 112. But these are tiny points of argument and I am convinced that if the change had not been made, nobody would ever have criticized the old bars 111 and 112 as being weak.³³

Hughes further refines Butler’s theory by more accurately calculating the lengths of sections. Butler calculated only whole measure numbers and did not account for overlapping sections, giving him durations of 115 measures and 78 measures for the first two sections, respectively.³⁴ Hughes calculates sections 1 and 2 as being 114.25 and 79.25 measures, respectively. This means that Section 1 is approximately 1.44 times longer than Section 235—our first evidence that Contrapunctus XIV’s proportional scheme is deliberate and numerologically determined.

Hughes’s argument gains traction when one uses this proportion to calculate the length of Section 3. Section 2’s 79.25 measures divided by the hypothesized ratio of 1.44 gives a quotient of 55 measures for Section 3—the section that introduces Bach’s name as the third subject.³⁶ Now there is very little doubt that these relationships are deliberate. This also conveniently leaves time for one more triple combination before Section 3 concludes.³⁷

Hughes then calculates that Section 4 should be 38 measures (although this is a rounded figure), reaching the conclusion that the total number of missing measures is 47. While I agree loosely with most of what Hughes postulates, there are a few fallacies and inconsistencies in his calculations that must be addressed. First, Hughes concludes that the correct proportion between sections is 1.44 but provides no reason for the presence of an additional “4” in the hundredths decimal place. (The reasons for this specific number will be discussed in greater detail later.)

Second, Hughes does a fair amount of rounding while criticizing others for doing so. An example of this is his calculation that Section 4 should be “exactly 38 bars long.”³⁸ In reality, this number is not so exact; it should actually be 38.194. Of course, some rounding is necessary, but rounding to 38.25 measures (38 measures and one beat) would be closer than rounding to 38 measures even. Rounding to the nearest beat instead of the nearest whole measure would dictate that the movement ends with a downbeat quarter note. (This is assuming that Section 4 cannot begin on beat three of a measure, since Die Kunst der Fuge theme would almost certainly enter on a downbeat.)

Of course, one could get around this by ending the work with a quarter-note—perhaps with a fermata. It is perhaps, however, not a stretch to suggest that Bach might have rounded this to 38 measures so that the work could end with a whole note. This is not a detail that should be taken lightly though; to round Section 4 to the nearest measure effectively excises a measure of music from the section and the entire movement. If Section 4 is rounded to 38 measures, then the final measure (measure 286) will most likely be a whole note. If, however, Section 4 is rounded to 38.25 measures, then the final measure (measure 287) would be a quarter note with a fermata (likely sounding the same as a whole note), thereby making measure 286 the penultimate measure and freeing it up for an additional measure of cadential material.

Finally, Hughes is generally quite meticulous about the lengths of sections—and this allows him to make important contributions to proportional theory—but he makes one significant error when he notes that Section 3 begins on beat three of a measure but forgets this detail when determining where Section 3 ends. In his calculations, he incorrectly starts Section 3 on the downbeat of measure 193 instead of beat three of measure 193. This leads him to believe that Section 3 needs an additional nine measures, instead of 8.5. Truthfully, at 55 measures, Section 3 must end on beat two of measure 248, a measure whose digits add up to 14.

This means that Section 4 must either start on beat three or overlap with Section 3 by two beats, but as previously stated, it seems unlikely that Die Kunst der Fuge theme would start on beat three, making the overlap the more likely choice. To account for this overlap, one must subtract two beats from the final calculation of the intended duration of Contrapunctus XIV. This puts Hughes’s calculation off by one whole measure: two beats off for starting Section 3 too early and two beats off for not overlapping sections 3 and 4. The question of whether one should round Section 4 to the nearest beat may also potentially push Hughes’s solution off by an additional beat. Table 4 compares Butler’s, Hughes’s, and my own theories and calculations.

As previously mentioned, the problem that Hughes faces is why the proportion is 1.44 instead of simply 1.4. He excuses this concern by pointing out that the proportional relationship is consistent, if not exact, and that there are other works by Bach with proportional relationships that are close to 1.4 but not exact.

Before exploring this issue further, it is vital to remember that Bach’s ways of solving mathematical problems would have been very different from our own. Doing these problems by hand, as Bach would have done, yields three very interesting (and I believe previously undiscovered) observations. The first and most obvious is that when 114.25 is converted into a mixed number, it is 114-1⁄4, a number with several allusions to 14 and 41.39 The second comes from the fact that Bach likely would have made these calculations in improper fractions (or beats) rather than decimals to facilitate division. To take the two complete extant sections as examples, Section 1’s 114.25 bars become 457/4, and Section 2’s 79.25 bars become 317/4. If one subtracts 317 from 457, the remainder is exactly 140, another evidently Bachian number. Since we are working with quarter measures in common time, Section 1 is exactly 140 beats longer than Section 2.

The final observation comes from dividing 457 by 317. The benefit to doing this division by hand is that the solution will not include a long string of decimals that a calculator displays. It is not surprising that the first three digits of the solution are 1.44, as Hughes claims. If, however, one solves to three decimals instead of two, then the third decimal is 1, producing a quotient of 1.441. At first glance, this new solution works much better than Hughes’s. While Hughes struggled to explain why his ratio is 1.44 instead of simply 1.4, this new proportion is both more specific and more numerologically significant, as it combines the numbers 14 and 41. Furthermore, it has precedent in the work; in Contrapunctus VIII, Section 2 is 1.441 times longer than Section 1.

To determine more certainly whether this proportion is more accurate, one must test whether the extra one thousandth consistently yields results that are closer to the actual number of measures present. To start, let us take 114.25 measures and divide it by Hughes’s 1.44. This gives us 79.340. (I will consistently round to the nearest thousandth.) If, however, we divide it by my 1.441, we get 79.285—a number that is not insubstantially closer to Section 2’s actual 79.25 measures. (See Table 5.) The extra one thousandth makes a surprisingly significant difference. If we take Section 2’s 79.25 bars and divide it by 1.44, we get 55.035. If we divide it by 1.441, though, we get 54.997. Once again, this solution is slightly closer to the hypothesized 55 bars in Section 2. (See Table 6.)

The final section, however, creates problems with either theory (1.44 or 1.441). When one divides Section 3’s 55 bars by 1.44, the solution is 38.194. When one divides it by 1.441, the quotient is 38.168. (See Table 7.)

Neither of these answers is very close to a round number of measures or beats, but the latter solution is slightly lower and therefore closer to a whole measure. This may tip the scale toward the argument that Bach intended for Section 4 to be 38 measures even—not, as previously considered, 38.25 measures (which would have required the last chord to be a quarter note with a fermata). Bach probably recognized that, at some point, some rounding would be necessary in his numerological games. Bach’s genius is, after all, still bound by the laws of mathematics.

When mapping out this movement, Bach may have begun with one of two conditions: that Section 2 would be 114-1⁄4 measures or, alternatively, that Section 3, the section in which he introduces his musical signature, would be 55 measures. These numbers are both, after all, significant to Bach. He must have been quite tickled to discover that 55 x 1.441 x 1.441 ≈ 114-1⁄4, but nevertheless, he must have started with one or the other—55 or 114-1⁄4. For this reason, it may be enlightening to compare Hughes’s and my proportional theories starting from both of those numbers. Table 8 compares the two theories starting with 114.25 measures in Section 1 as the reference point.⁴⁰ Table 9 compares the proportions between Hughes’s theory and my own using 55 measures in Section 3 as the reference point.

These tables show that, regardless of which reference point one starts from, 1.441 is a consistently more accurate proportion than 1.44. Table 8 also demonstrates something else that indicates why 1.441 is the more likely solution. When one looks at the column for Hughes’s proportion, one can see that his ratio consistently gives an answer that is slightly too high. The solutions produced by a ratio of 1.441, however, are slightly too high in the cases of sections 2 and 4, and slightly too low for Section 3. This means that no other proportion can truly be said to be closer. If one uses a number any higher than 1.441, Section 3 will be less accurate, and if one uses a number any lower than 1.441, sections 2 and 4 will be less accurate. When all the answers are too high though, as in Hughes’s ratio, it is clear that the proportion can be refined.

The extra beat

The remaining issue concerning the length of Contrapunctus XIV is the length of Section 4 and whether it should be rounded to the nearest measure or nearest beat. There are a number of factors that should at least be considered. First is the purely musical issue of ending an hour-and-a-half-long work. It would be unusual to end such a work with a quarter note, but having a fermata on the note would alleviate that to a degree; the difference would then arguably be only visual. As previously mentioned, using the proportion 1.441 instead of 1.44 pushes the length of Section 4 down by a little over three hundredths (38.1679 versus 38.194), but this is an admittedly minor difference.

It may once again be revealing to consider how Bach would have done his calculations. If one uses long division to calculate the length of Section 4, as Bach must have done, then he would not have seen a long string of decimals. He could have merely solved for the nearest whole number, in which case he would have reached 38 as a final answer and been done with it. But even if he continued to solve for the first decimal, he would have seen only a “1” in the tenths place, which he still would have likely been content to round down to zero. It is not until one solves for the hundredths place and gets 38.16 that this issue arises at all. It seems likely that Bach would not have been particularly bothered by this discrepancy, if indeed he was even aware of it.

In summary, I conclude that Section 3 requires another 8.5 bars—plus filling in the remaining voices of measure 239—and that Section 4 should be 38 measures, keeping in mind that it will overlap with Section 3’s last half measure. This gives one enough time to include one more triple combination in Section 3’s remaining 8.5 bars, as well as multiple quadruple combinations and episodes in Section 4. It is not, however, so long that the drama that has been reached by measure 239 will become an anticlimax. According to this theory, the movement should be 285 measures. This is calculated by adding the lengths of the sections (114.25 + 79.25 + 55 + 38 = 286.5) and subtracting one measure for the overlap of sections 1 and 2 and another two beats for the overlap of sections 3 and 4 (286.5 – 1.5 = 285). The durations and overlaps of the four sections can be seen in Figure 6.

Large-scale numerology

Finally, let us consider the sum total of all the movements. If one counts the lengths of the mirror fugues and their mirrors, thereby doubling the lengths of the mirror fugues to 254 measures, then the entire length of Die Kunst der Fuge is 2,135 measures. The realization that my proportional theory put the entire work at 2,135 measures immediately gave me pause. This is, after all, three short of the numbers which gematrically correspond to Bach’s name: 2,138. This caused me to wonder whether Bach was also intentional about the length of the entire work, which in turn led me to test what would happen if I aimed for a sum total of 2,138 measures by increasing Contrapunctus XIV to 288 measures, rather than 285. Obviously, the durations of sections 1 and 2 are fixed and cannot be tailored. Bach left Section 3 incomplete, but my proportional theory is largely based on the numerological significance of Section 3 being 55 bars. This leaves Section 4, which according to my original theory was 38 bars. Amazingly though, adding three measures increases it to exactly 41 measures. Through the addition of just three bars, the length of Section 4 and the length of the entire work suddenly achieve great gematric meaning. The question, as always, is: was this Bach’s intention?

The real issue here is whether Bach would sacrifice the integrity of his proportional scheme to inject the entire design with greater personal numerological significance. To better judge this, we must determine the degree to which this change would affect the proportional scheme. Before doing so, however, it is worth remembering that Section 4 was already problematic. When one divides 55 (the conjectured duration of Section 3) by 1.441, the quotient is approximately 38.168, which we rounded down to 38 (after first considering whether to round to 38.25, or 38 measures and one beat). This is significantly less precise than the proportions of the previous sections (72.285 versus the actual 72.25 measures in Section 2, and 54.997 versus the conjectured 55 bars in Section 3).

When one takes Section 3’s hypothesized 55 bars and divides them by 41 bars for Section 4, the proportion is approximately 1.341. This is not terribly far off from the originally theorized proportion of 1.441. Additionally, it still ends with the numerologically significant digits “41.” It is actually quite intriguing that the two numbers only differ by one digit.

The question remains: if Bach had realized that adding three measures to Section 4 would give both the section and the entire work gematrically significant durations, would it have been worth the small concession of modifying a proportional scheme that, at this point, was already falling short?

I believe the answer is yes; Bach would have succumbed to such a temptation. In fact, it must have been the design all along. It would be quite a coincidence for Bach to see the original proportional scheme of 1.441 through to the end and only then realize that the work was three bars short of 2,138 measures. Far more likely, he began by assigning 288 measures to Contrapunctus XIV so that the sum of the work’s movements would be 2,138. Knowing that the sections would diminish in length, he then assigned 41 measures to Section 4 and 55 measures to Section 3. At this point, he switched to a more exact proportional scheme of 1.441, making Section 2 79-1⁄4 measures and Section 1 114-1⁄4 measures, the latter of which—he must have been pleased to learn—is a number with about as many references to 14 and 41 as a five-digit number can have. The length of each section can be seen in Table 10.

These sections add up to 289.5 measures, but because of a one-measure overlap between sections 1 and 2 and a half-measure overlap between sections 3 and 4, the total duration of Contrapunctus XIV will be 288 measures. Since Bach’s final bar is measure 239, Contrapunctus XIV requires 49 more measures: 8.5 more measures in Section 3 and 41 measures for Section 4, with these two sections overlapping by two beats. (See Figure 7.)

It is my hope that this document may have use outside of the narrow scope of determining the intended duration of Contrapunctus XIV; more broadly, I hope to shed a bit more light on Bach and his compositional tendencies. The topic of proportion in Bach’s oeuvre is ripe for further examination, and we have only begun to scratch the surface. By now, there is no question that Bach was an enthusiastic practitioner of Augenmusik and hid all sorts of elaborate structures in his music. The question is: will we seek them out?

 

Notes

1. Christoph Wolff, Bach: The Learned Musician (New York: W. W. Norton, 2000), 433.

2. Gustav Nottebohm, “J. S. Bach’s letzte Fuge,” Musik-Welt, nos. 20 and 21 (1881): 2, http://ks4.imslp.net/files/imglnks/usimg/f/fc/IMSLP348435-PMLP562864-No….

3. Counting beats in large sections can be more complicated than one might expect, particularly if the passage includes partial measures. The formula for doing so is as follows: (last complete measure – first complete measure + 1) x 4 beats + any beats from surrounding partial measures. For example, if one wanted to count the beats from beat 2 of measure 4 to beat 2 of measure 36, then one would start with 35 (the last full measure) and subtract five (the first full measure) for an answer of 30. Then add one for a sum of 31 and multiply that by four beats for a product of 124. Then add the extra three beats of measure 4 and the extra two beats of measure 36 for an answer of 129 (124 + 3 + 2 = 129).

4. Encyclopædia Britannica, s.v. “Gematria.”

5. Ruth Tatlow, Bach and the Riddle of the Number Alphabet (Cambridge, England: Cambridge University Press, 1991), 133.

6. Indra Hughes, “Accident or Design? New Theories on the Unfinished Contrapunctus 14 in J. S. Bach’s The Art of Fugue, BWV 1080” (D.M.A. diss., University of Auckland, 2006), 3–4.

7. Ibid., 4.

8. Ruth Tatlow, Bach’s Numbers (Cambridge, England: Cambridge University Press, 2015), 70.

9. For perhaps the preeminent study of this field, see Ruth Tatlow’s Bach’s Numbers. The trajectory of Tatlow’s opinion on this topic is intriguing. In her 1991 book Bach and the Riddle of the Number Alphabet, she argues against most numerological and gematric interpretations of Bach’s music. Now, after something of a Damascene conversion, she is its leading exponent.

10. Hughes, “Accident or Design?,” 18.

11. Ibid.

12. Ibid.

13. Ibid.

14. Ibid.

15. Ibid, 19.

16. Ibid.

17. Ibid.

18. Anatoly P. Milka, Rethinking J. S. Bach’s Art of Fugue, trans. Marina Ritzarev, ed. Esti Sheinberg (London: Routledge, 2017), 179.

19. Herbert Kellner, “Die Kunst der Fuga,” The Diapason 91, no. 5 (May 2000), 16.

20. Ibid.

21. In his article “Bach and Die Kunst der Fuge” (The Diapason, May 1998, pp. 15–17), Jan Overduin argues that downbeat quarter rests often symbolize death in Bach’s music.

22. As far as I am aware, the aforementioned observations were all previously undiscovered.

23. James Peter Burkholder, All Made of Tunes: Charles Ives and the Uses of Musical Borrowing (New Haven: Yale University Press, 1995), 138.

24. The notion of gradual revelation applies most aptly to the B-A-C-H theme, which occurs in only three movements: Contrapunctus VIII, in which its default form is inverted with repeated notes (i.e., most hidden); Contrapunctus XI, in which its default form is in rectus form with repeated notes (i.e., less hidden), and Contrapunctus XIV, in which it is finally stated plainly in Section 3.

25. I rounded to the nearest thousandth (0.417) so that one can see the numerologically significant part of the proportion (i.e., 0.41). If I had rounded to the nearest hundredth (0.42), its significance would have been obscured. This is not, however, a problem that Bach would have necessarily confronted. He would have solved these equations by hand, meaning he would see only as many decimals as he solved for, not the string of decimals that I see on a calculator. If he solved for only two decimals, he would have seen only “0.41.” After all, one only knows to round this number up to 0.42 if one also knows what the thousandth decimal is.

26. To the best of my knowledge, these proportional relationships were also previously undiscovered.

27. Gustav Nottebohm, “J. S. Bach’s letzte Fuge,” Musik-Welt, nos. 20 and 21 (1881): 2, http://ks4.imslp.net/files/imglnks/usimg/f/fc/IMSLP348435-PMLP562864-No….

28. Indra Hughes takes the final measure number, whose digits add up to 14, as evidence that Bach intentionally left the work incomplete and left a number of clues as to how it should be finished. See “Accident or Design? New Theories on the Unfinished Contrapunctus 14 in J. S. Bach’s The Art of Fugue, BWV 1080” (D.M.A. diss., University of Auckland, 2006), 20–21.

29. Indeed, the only other part of the work that could be said to be as harmonically daring is Contrapunctus XI.

30. Butler has since reversed this position, and now believes that the Fuga a 3 Soggetti was never intended to include Die Kunst der Fuge subject.

31. Gregory Butler, “Ordering Problems in J. S. Bach’s ‘Art of Fugue’ Resolved,” The Musical Quarterly 69, no. 1 (Winter 1983): 55, http://www.jstor.org/stable/741800.

32. Hughes, “Accident or Design?,” 84.

33. Hughes, “Accident or Design?,” 93–94.

34. Butler, “Ordering Problems in J. S. Bach’s ‘Art of Fugue’ Resolved,” 5.

35. Hughes, “Accident or Design?,” 86.

36. Ibid., 87.

37. This is because any combination will take as long as its longest theme. At six bars long, Theme 2 is the longest subject, meaning another triple combination would take at least six measures.

38. Hughes, “Accident or Design?,” 88.

39. A skeptical reader who double-checks my calculations may notice that Section 1’s 114.25 measures include the opening two beats of rest, even though I have not counted beginning rests in other movements. The original reason for this was that the calculations are much more exact if one counts 114.25 measures instead of 113.75, but this reason alone struck me as intellectually indolent. The realization though that 114.25 converts to 1141⁄4, and that counting those two beats makes Section 1 exactly 140 beats longer than Section 2 gave me some measure of peace with this decision.

40. In these calculations, it is important to start with the actual number of measures. For example, if one is determining the length of Section 3 from Section 2, then one should start from 55, not 55.035 or 54.997.

Sources

Burkholder, James Peter. All Made of Tunes: Charles Ives and the Uses of Musical Borrowing. New Haven: Yale University Press, 1974.

Butler, Gregory. “Ordering Problems in J. S. Bach’s ‘Art of Fugue’ Resolved.” The Musical Quarterly 69, no. 1 (Winter 1983): 55. http://www.jstor.org/stable/741800.

Hughes, Indra. “Accident or Design? New Theories on the Unfinished Contrapunctus 14 in J. S. Bach’s The Art of Fugue, BWV 1080.” D.M.A. diss., University of Auckland, 2006.

Kellner, Herbert Anton. “Die Kunst der Fuga.” The Diapason 91, no. 5 (May 2000): 15–17.

Milka, Anatoly P. Rethinking J. S. Bach’s Art of Fugue. Translated by Marina Ritzarev. Edited by Esti Sheinberg. London: Routledge, 2017.

Nottebohm, Gustav. “J. S. Bach’s letzte Fuge.” Musik-Welt, nos. 20 and 21 (1881): 2. http://ks4.imslp.net/files/imglnks/usimg/f/fc/IMSLP348435-PMLP562864-No….

Overduin, Jan. “Bach and Die Kunst der Fuge.” The Diapason (May 1998), pp. 15–17.

Wolff, Christoph. Johann Sebastian Bach: The Learned Musician. New York: W. W. Norton & Company, 2001.