The "Unfinished" Fugue
Obviously the components of AOF proceed from the simple to the complex, both in the arrangement of the contrapuncti, as well as in the ordering and placement (at least as I prefer to do it) of the canons. However, in terms of sheer scope, none of the preceding parts comes close to the mighty proportions of the concluding fugue, which was left "unfinished" and which is one of the fascinating stories in all of Western music.
I noted in the introduction how C. P. E. labeled the final page of Bach's manuscript with the message that Bach died while introducing B-A-C-H as a countersubject in the third part of the fugue. Ignoring for the moment that B-A-C-H is here a subject, not a countersubject, the whole story is apocryphal. The piece was further labeled Fuga a 3 Soggetti ("Fugue with Three Subjects") in the 1751 published edition, despite that fact that it was patently Bach's intention that it be a quadruple (four subject) fugue. How can we be sure of that? Let's begin by backing up to square one.
To encourage you to visit the individual parts of AOF, I've put most discussion about fugues (episodes, strettos, augmentations, diminutions, inversions, etc.) in the narratives for each piece. But to clarify the present discussion, we'll cover subjects here.
Camille Saint-Saëns, in one of his more cynical moments, called a fugue "a piece in which the voices come in – and the listeners go out – one by one." Well, the first part is right. The first voice in a fugue appears by introducing the subject, the principal melody upon which the fugue is based. In AOF, the main subject, and the opening notes of the first fugue, Contrapunctus I, is this.
These notes are the foundation of the entire 80-minute work, and even if you don't read music, you can see some essential things here about a fugue subject. It looks to the eye as if it more or less has two distinct parts that divide it roughly in half (a common characteristic of fugue subjects), and also that the note values tend to get shorter and a little more active toward the end of the subject (again, a fairly common tactic). Once the initial voice has stated the subject, a second voice will state the subject, either a fifth above or a fourth below, while the original voice plays new material in counterpoint. Then the third voice plays the subject, and finally the fourth voice, and all the while, the previously entered voices accompany it in an increasingly complex texture.
Now, if a fugue has a subject, then a double fugue must have...right, two subjects! (An example of a splendid double fugue in a familiar work is the Kyrie from Mozart's Requiem.) Contrapunctus IX and Contrapunctus X in AOF are also double fugues. Not surprisingly, double fugues have latitude for substantial complexity above and beyond one-subject fugues, since the composer now not only has to deal with each subject and all its supporting material, but to achieve a true double fugue, must combine the two subjects simultaneously and satisfactorily. And a triple fugue (a gold star for anyone who can tell how many subjects it has!) is more complex still. Listen to Contrapunctus VIII or Contrapunctus XI for AOF's massive examples of triple fugues.
The Quadruple Fugue
Obviously, a quadruple fugue must be one with a staggering four subjects, all of which must be combined into an aesthetic whole. By the time the manuscript of Contrapunctus XIV breaks off after 239 measures, Bach has introduced three subjects, both separately and in combination. Significantly, though, the main AOF subject is not one of them. Here they are, in the order they are introduced.
Subject 1 is an austere and somewhat distant variant of the main AOF subject. Subject 2 has a rotating character that gives it a propulsive feeling, a perfect foil to the long notes of the first subject. Subject 3 stirs the pot even further with its two chromatic intervals.
(By the way, look at the first four notes of Subject 3: B flat - A - C - B natural, which in the German key spelling is B-A-C-H! Bach has coded himself into his ultimate creation, and this is what C. P. E. was referring to in his inscription on the manuscript. The BACH theme has been used in fond tribute by many composers since; for one example, see my realization of the thrilling Prelude and Fugue on the Theme BACH by Franz Liszt.)
Early AOF commentators suggested that the Fuga a 3 soggetti did not belong with AOF since as written it fails to exploit the main AOF subject. But as has been amply demonstrated since the late 19th century, the AOF subject melds so perfectly with the other three that is it inconceivable Bach did not intend it to appear as the apotheosis of the piece, making it a true quadruple fugue. In 1991, Hungarian composer Zoltán Göncz pointed out that Contrapunctus XIV is structured using a permutation formula, in which the voice order of the subjects' appearance follows a strictly controlled pattern, as shown below.
As the matrix shows, the first subject appears in rising order through the voices: bass - tenor - alto - aoprano (BTAS). The second subject appears in ASBT order, and the third in TASB order. That takes care of the part that Bach wrote, and when these are put together into a combined matrix, only one permutation remains for the fourth subject, one that has the main AOF theme appearing most satisfyingly as a triumphant statement in the soprano. That this is not the only fugue in which Bach employed a permutation formula adds to the plausibility of this interpretation.
It is a pleasure for me to serve Dr. Göncz's completion of Contrapunctus XIV with this realization of AOF. I thank him for supplying the graphic above and for permission to make his completion available on this site.
On "Finishing" Bach
It might seem the ultimate in chutzpah to presume to finish a Bach work, especially the final bars of his crowning achievement. To acknowledge the many scholars besides Göncz who have studied the matter, I should note there are numerous completions of Contrapunctus XIV in the literature, dating all the way back to no less than Donald Tovey, and that there is ongoing disagreement about the nature and length of the ideal completion of this work. In that fracas, Göncz's completion appears regarded as one of the substantial and influential final words in the discussion, but not necessarily the final word. Regardless, I find it to be an enjoyable and respectful conclusion to a masterpiece, allowing us to give it its proper name: Contrapunctus XIV, rather than the misleading Fuga a 3 Soggetti.
These comments beg the question, though, of why we would have the nerve to write music for Bach in the first place. Interesting arguments suggest it is a plausible thing to do -- in this case. There appear to be two schools of thought regarding the incompleteness of Contrapunctus XIV. First, there is the possibility that Bach did finish it on a separate manuscript, out of the necessity of making sure the four subjects could be combined before the full fugue was composed. This missing manuscript has been called Fragment X by Bach scholar Christoph Wolff, and if it did exist, either C. P. E. or the publishers lost it sometime after Bach's death. (And if that is true, it must go down as one of the great boneheaded moves in the history of art.) Second, it has been posited that Bach deliberately left Contrapunctus XIV unfinished, as a puzzle for future generations of composers. That is not as farfetched as it might sound; writing canons in quirky "puzzle" form was a common practice for Bach and his contemporaries; see my comments on the Fourteen Canons on the Goldberg Ground for an example.
These two possibilities can be argued at far greater length than the tolerable span of a Web page. Suffice to say that in either case, Bach intended this magnum opus to have a resolution, and that he has coded so many clues into its structure, and with such impeccable logic, that it is possible for musicians to construct a reasonably faithful completion. Surely such efforts will not be note for note what Bach would have written; but equally surely they can capture the spirit of what Bach had in mind for the conclusion of one of the great works of the repertoire.
So, if you're not familiar with Contrapunctus XIV, see if you can tell where Bach ends and Göncz begins. And whether you can or not, I hope you'll find it as miraculous as I when the four subjects appear in their full glory, pervading the final moments of this 2400 bar, 80 minute odyssey in a dizzying array of superpositions and inversions. This is music with an eye to infinity, as much as any of the images accompanying the narratives, a work of profound depth to be appreciated for a lifetime...and beyond.