Beyond B A C H - the Permutation Matrix of Zoltán Göncz

Zoltán Göncz: the permutation matrix in Contrapunctus XIV
no copyright infringement intended
(http://en.wikipedia.org/wiki/File:Permutation_matrix_web.png)

I would like to come back to the question of the fourth subject of Contrapunctus XIV: the missing one. Let's recapitulate: Bach didn't finish Contrapunctus XIV (the last in the Art of Fugue), leaving it with three subjects. What differentiated Contrapunctus XIV to the rest of the cycle was that (beside it was unfinished) the main theme was missing, while present in all the other Contrapuncti: for this reason some musicologists even believed that actually it was erroneously considered as part of the Art of Fugue. On the other hand, the third subject of Contrapunctus XIV was no other musical sequence than B A C H, his signature. Naturally that incited the spirits, and people started to look for an answer to what would have been after B A C H, were the composer to finish his opus.

(Let me mention for people less familiarized with fugues, canons, polyphony and all that stuff: subject, theme, and motif are equivalent - all three define the same thing, namely a sequence of musical notes that constitute the vortex of a polyphonic piece - this sequence is played on each voice, where each voice starts this sequence a bit before the previous voice has finished it, which leads to a complex sound in which the melody is wrapping itself - also, let me mention that the fourteen fugues composing the Art of Fugue carry the title of Contrapuncti).

One of the strongest opinions about what was missing in Contrapunctus XIV was brought by Gustav Nottebohm in his article published in 1881. Nottebohm believed that the fourth subject, that was missing, should have been the main theme of the Art of Fugue: Contrapunctus XIV had three sections, and the third one was broken off shortly after the exposition of its subject (B A CH ); a fourth section would have come naturally. He came to that conclusion by studying the structure of the first three sections: it was like they claimed a fourth section to complete the whole.

Nottebohm was followed by other musicians who had more or less similar arguments. It was in the first decade of the 21th century that Zoltán Göncz had the idea of constructing a Permutation Matrix to make obvious the correctness of considering Contrapunctus XIV as potentially a quadruple fugue.

I showed the Permutation Matrix of Zoltán Göncz in the image above and I will refer briefly to each matrix within it (it is actually a set of matrices that follow a permutational logic)

The first matrix on the left shows the beginning of first section of Contrapunctus XIV: the first theme is introduced on the four voices (bass - tenor - alto - soprano). The introduction is followed by a development during which the four voices take again and again the theme and work on it (by using specific mechanisms like transposition, inversion, stretti, etc).

The second matrix shows the beginning of the second section of the Contrapunctus: the second theme is now introduced on the four voices in a different order - alto - soprano - bass - tenor. The development of the second section mixes the two themes, which is possible because each one follows its own sequence of voices: it means that never the two subjects would come on the same voice in the same time.

The third matrix shows the beginning of the third section: the third theme (the famous B A C H) is introduced on the four voices in the order tenor - alto - soprano - bass (which is different from the order followed by each of the previous two subjects).

The fourth matrix shows the continuation of the third section: a development which mixes the three themes which makes possible a fourth section where a fourth subject should be introduced in the order soprano - bass - tenor - alto (the fifth matrix). Finally, the sixth matrix shows the development of the fourth section, where all four themes are mixed.

The whole demonstration can be followed in an article by János Malina, published in 2007 in Hungarian Quarterly:

http://www.hungarianquarterly.com/no188/14.shtml

Based on this, Zoltán Göncz created a reconstruction of Contrapunctus XIV, where the third section in finished and a fourth section follows, using the main theme of the Art of Fugue.

Contrapunctus 14 - reconstruction: Part 1/2
Performed by Gyula Szilágyi (Waalse Kerk, Amsterdam)
Score published by Carus-Verlag, Stuttgart, 2006
(video by Matrix141414)

Contrapunctus 14 - reconstruction: Part 2/2
Performed by Gyula Szilágyi (Waalse Kerk, Amsterdam)
Score published by Carus-Verlag, Stuttgart, 2006
(video by Matrix141414)

(The B A C H motif)