Last year, Carus-Verlag, a Stuttgart publisher specialising in early vocal church music, issued a slender volume containing Zoltán Göncz's reconstruction of the last movement of Bach's The Art of Fugue, a late work considered to be one of his crowning achievements. The 19th movement, Contrapunctus 14, is arguably the most famous unfinished composition next to Mozart's Requiem and Schubert's "Unfinished" Symphony.
Göncz's reconstruction is based on his research in print both in Hungary and abroad as early as the 1990s. It was a decade and a half before the final result was published.
What is so special about Contrapunctus 14 that impels someone to have a go at completing it every five years or so? Putting the professional aspects of the challenge aside, the answer is surely that this fugue is clearly Bach's great summation. Bach himself no doubt intended this way. Even in its fragmentary form the movement is one of his most breathtaking of pure polyphonic essays. Completing it is like solving a puzzle, a test of the interpreter's mastery of Bach's legacy. To pass it is to take symbolic possession of that legacy.
The Art of Fugue is a summing up, a précis not only of various technical devices but of a compositional approach which, by the mid-18th century, had been honed over a good three centuries. Polyphony— the dynamic interplay of melodic voices— provided a technical means towards a greater end. Namely, a piece of music should be organic; it should achieve unity through a dense web of internal connections. Contrapuntal possibities should be explored to the full, as being a musician means to labour incessantly at perfecting ourselves and the work. The Art of Fugue, several movements of which Bach revised and expanded over the years, presents an ageless model.
We now know that many important Bach compositions have been lost. Many others survived by happy accident, and The Art of Fugue is perhaps the happiest of chance survivors. There are two main sources: a 1742 manuscript containing twelve fugues and two canons, and the first edition, published not long after Bach's death. The latter contains fourteen fugues, four canons and—intriguingly—a chorale arrangement. Some of the earlier pieces appear in new versions. Bach called the fugues contrapuncti, an archaic term he never used elsewhere, as if to emphasise his own role in transmitting an ancient tradition. Contrapunctus 14 is one of the movements that are absent from the 1742 manuscript. What is remarkable in the first edition is that it is separated from the other thirteen contrapuncti by four intervening canons. It is labelled Fuga a 3 soggetti (fugue with three themes), and, being incomplete, is "rounded off" by the chorale "Wenn wir in höchsten Nöthen sein."
The editors of the first edition were not exactly at the top of their game and were unsure of the order of the movements, which significantly differs from that of the manuscript.

He wanted to walk an even tighter rope and to push the work of reconstruction in the ideal direction that best suited Bach's intent. Thus he made a long-term, in-depth study of Bach's counterpoint and the The Art of Fugue in particular. His goal was to ask questions, draw conclusions and learn as much as possible about Bach's compositional methods and way of thinking. He strove to introduce some new points of view that, although relevant to the topic, had been previously neglected. While his conclusions, like those of his predecessors, cannot always be accepted with absolute certainty, his methods are not as speculative as theirs. Göncz avoided all preconceptions, attractive as they might seem, regarding the formal symmetries in Contrapunctus 14. He also resisted estimating the length of the missing part by extrapolation. He was both stubborn and humble: by dint of sheer analytical observation, he, in a sense, questioned Bach about his way of thinking and his intentions until the dilemmas were solved by the Master himself.
Göncz's meticulous and labour-intensive method involved making a comprehensive survey of all possible contrapuntal combinations and then sifting these combinations from the standpoint of texture and the fluidity of voice-leading. Moreover, while assembling various fugal sections and stretti, Göncz devotes particular attention to the solutions employed in other fugues within the cycle. A model is also provided by noticing the way Bach changed keys in the other fugues and existing sections of Contrapunctus 14. Göncz explores every major possible inherent combination too, striving for the greatest possible economy to follow another Bachian requirement.
On the one hand, this method yields a stylistically correct reconstruction that satisfies all expectations; on the other, it reduces the possibilities of arbitrary decisions at almost every turn. Göncz always finds a clear reason which is textural, tonal or other to distinguish between a good solution and a less good one. In other words, the solutions are suggested by the musical material itself. This method is made possible in the first place by the specific nature of Bach's thematic material and the conceptual framework on which the quadruple fugue apparently rested.
As already noted, Bach's manuscript breaks off just after the first simultaneous presentation of the first three themes. Göncz completes the third section by artfully distributing stretto entries among the voices and taking into account the direction of key changes. In this way, he manages to introduce all three themes in all four voices.
It is Göncz's realisation of the fourth and last section of the four-subject fugue where his most important contribution lies. This accomplishment owes itself to a "key" he found in the course of his analysis and made possible by the interesting structural rules he had discovered in the course of analysing the expositions in the extant sections of the fugue. A simple diagram will help clarify his thought process.

Some movements were even switched during the printing process! Small wonder then that the debate concerning Contrapunctus 14, its place in the cycle and its state of completion (a more or less complete version might have later been lost) has continued unabated ever since. Even his contemporaries recognised the extraordinary significance of this movement. Bach's Obituary—published four years after his death—not only mentions it, but goes on to discuss it in considerable technical detail.
For a long time it was thought that the fugue was supposed to be based only on the three themes introduced in the fragment. This opinion is reflected in the movement's title in the first edition. Since the actual main theme of the The Art of Fugue, used in several different versions throughout the cycle, is not among the three themes of Contrapunctus 14, it seemed that this movement did not originally belong to the cycle at all. One of the three themes is a distant, simplified variant of the absent main theme, the second is a longer, sinuous melodic idea, while the third is none other than the B-AC- H theme, made up of the letters of Bach's name (B being German for B flat and H standing for B natural), with a short cadence added for closure. Being a personal statement, it was understandable that the B-A-C-H theme came to be seen as the movement's culmination, excluding the possibility of a fourth theme.

An 1881 article by music scholar Gustav Nottebohm changed all that. Nottebohm discovered that the three themes could be combined with the principal theme of the cycle to produce flawless four-part counterpoint, opening the door to a re-evaluation of the fragment and raising the theoretical possibility of completing it as a fugue with four subjects—i.e., a quadruple fugue.
At this point we must briefly pause to explain how the extant portion of Contrapunctus 14 unfolds and how fugues work in general.
It begins with the exposition of the first theme. In a four-part fugue, this means that the theme is heard in each voice (soprano, tenor, alto, bass). It alternates between two slightly different forms known as dux "leader" and comes "companion" (the reason for the alteration of a note here and there is to avoid incongruent harmonies). The theme then undergoes several transformations. For instance, it can be inverted and transposed. Sometimes successive entrances are brought closer together, a device which is called stretto. Both the original and the inversion appear at least once in each voice.
The second theme is introduced in the second section, appearing in each of the four voices, just as in the first theme. Both themes are combined and each theme appears again in each of the four voices. This time, however, there are no inversions.
The third section, which brings in the B-A-C-H theme, is presented in inversion and then in stretto. Finally, all three themes are played simultaneously in stretto. That is the point where the manuscript breaks off.
Even though the third section is incomplete, the overall structure of all three sections is clear:
1. Exposition – first theme – elaboration
2. Exposition – first and second theme – elaboration
3. Exposition – 1+2+3 – elaboration
After Nottebohm's discovery, it is safe to assume that Bach intended Contrapunctus 14 as a quadruple fugue to crown the whole of the The Art of Fugue, and the main theme of the cycle was saved for last. Therefore, the task was to complete section 3 by a few additional stretto elaborations of themes 1–3—making sure that each of the three themes appears in each of the four voices— and write the fourth and last section of the fugue following scheme 4: Exposition – 1+2+3+4 – elaboration. Many musicians since the late 19th century have tackled it. Some, notably Donald Francis Tovey, Helmut Walcha and Erich Bergel, achieved solutions of a high standard.

It was around 1990 that Zoltán Göncz first took up the challenge. Göncz, a musical editor at Hungarian Radio, is also a composer. His interest in music theory and history, though not informed by formal academic work, is evident from his not too extensive but exciting compositional output: he uses archaic forms and complicated structures in his works. Earlier attempts at reconstructing Bach's fugue left him dissatisfied. This "system of equations" had too many unknowns and therefore allowed too many arbitrary choices. Many widely divergent solutions had seemed all too equally acceptable. Göncz, in contrast, wanted to discover something inherent in the music that narrowed the range of solutions.

In the exposition of the first fugal section, the theme appears in the bass first and moves up from there to the tenor, the alto and the soprano. By following the diagram, you can see which voice leads each subject entry. Incidentally, the empty spaces simply denote each voice's "filler" material unrelated to the themes. In the second fugal exposition, the theme is first introduced by the alto. From there it moves up to the soprano; then, since it cannot ascend any further, it goes to the bass and finally one voice up, to the tenor. The third exposition starts in the tenor and, like the first two expositions, keeps moving upward to the next higher voice.
At this point, Göncz takes a step that has far-reaching and surprising consequences. He superimposed the first three matrices on top of one another. The four empty spaces outline the fourth exposition. Now Göncz projected the matrix of the fourth exposition onto the combined matrix of the first three.
Göncz noticed that this pattern belonged to a well-known type of Bach fugue, the "permutation fugue" found in several of his vocal works. This is perhaps the strictest of all fugue types, almost like a canon in its structural rigour. It has as many themes as it has voices, a different theme in each voice in any of the fugal sections; yet each theme makes the rounds and is presented in each voice as in a canon. Of course, the order of the entries may change from case to case. However, there are no a-thematic interludes between the themes; in other words, the permutation fugue is a hundred per cent thematically saturated.
Göncz found that this matrix, which he dubbed "permutation matrix", functioned as a concrete operative command. In other words, the thematic voices of the first three fugal expositions, if superimposed one on top of the other, produce a flawless contrapuntal texture, and the exposition of the fourth theme fits into that texture perfectly, filling in the "empty" spots in the voices. This can be no accident. It rests on the assumption that Bach planned the musical material of the permutation matrix before composing the movement and derived the first three expositions from that material. Thus the permutation section forms the essence of the entire movement and is a precondition of its very existence.
This is how Göncz achieved his objective. The exposition of the fourth theme coincides with the permutation section that brings the four themes together— perhaps because the theme and its close variants have already served as fugue themes throughout the cycle. Furthermore, musical logic demands the continuation of the "intensification" process that started towards the end of the third section; the extremely concentrated permutation fugue follows naturally from there.
This, however, does not mean that the permutation fugue, which rotates four themes in four voices in the most concise musical process imaginable, is all there is to this section. In fact, Göncz has two other surprises in store. First, he discovered that the fourth theme fits the others not only in its original form but in inversion as well. This is another possibility that Bach "preprogrammed" into the material and it therefore has to be used. So, after the permutation fugue has unfolded, Göncz immediately begins combining the first three themes with the inversion of the fourth, the latter appearing first in the bass and then in the soprano. When the remaining two voices take over the theme, it would seem that all the contrapuntal possibilities have been exploited. Yet, at this point, Göncz serves up his final surprise: simultaneously with the inversion (now in the alto and the tenor), the fourth theme appears in its original form as well, producing a five-part contrapuntal texture in the last eighteen bars of the fugue. All the themes appear together, including the Urtheme, the seed from which the entire cycle grows, together with its own mirror image. (As a sixth voice, a pedal point on A, later D, definitively confirming the D-minor tonality, is here added to the texture.)
In the 1990s, György Ligeti called Göncz's essay, outlining the reconstruction, "excellent and convincing". To György Kurtág, reading the essay was an "overwhelming and illuminating experience". Göncz's reasoning is all the more convincing because he has probed the material with unflagging devotion and sagacity, without trying to formulate any facile personal hypotheses. He brings to the project the total freedom of an independent scholar; his approach is refreshingly un-dogmatic.
Göncz's work on Contrapunctus 14 will surely make its mark on the musical consciousness of the world, and, before long, reference books on the subject will count it among the most seminal findings of recent times. The new Carus edition is the guarantor of Göncz's important legacy.