There is great controversary over the last section of J.S. Bach’s great work, “**Die Kunst der Fuge**” (The Art of the Fugue) as to why it was left unfinished or even if the last section was even meant to be a part of the work since it does not even contain the subject. Those who have studied the Schillinger System are well aware of his rhymic engineering devices, one of which is Permutations (General & Circular) – so I thought it interesting when I found this article in Wikipedia , which if true, shows how J.S. Bach permuted the entrance of his subjects:

### The permutation matrix

In 1991 a theory was published by Zoltán Göncz answering the question of how Bach planned the appearance of the fourth subject, the main subject of the cycle:

In the course of the exposition of the first three subjects (first subject: mm. 1–21, second subject: mm. 114–141, third subject: mm. 193–207), Bach applied a serial sequence of voice entries decided in advance, by which he determined the space and time parameters of the subject entries. The superimposition of the three exposition matrices foreshadows, and develops as a negative, the sequence of the voice entries of the fourth subject. The copying of the four subjects onto each other displays a characteristic construction of Bach’s oeuvre occurring mainly in the vocal fugues: that of the permutation fugue.

However paradoxical, it follows from the logic of composing a quadruple fugue that the combinations joining all four subjects (i.e. those combinations which appear *last* when performing the work) were already completed in the *very first stage* of composition, because the possibility of overlapping the four subjects (1+2+3+4) is the *sine qua non* of writing a quadruple fugue. The process of composition does not proceed in a linear way from the beginning, but with all four parts in view.

One of the striking features of Contrapunctus XIV is that in this movement Bach applied the stretto of *whole expositions*, layering the first two expositions atop each other prior to introducing the third subject. In the exposition of the first three subjects he “programmed” the later permutation stretti, then applied the expositions as “programs”, “algorithms”. The *permutation matrix*, apart from originating authentically with Bach, can be proved to have been ready at the time of the genesis of the work (that is, earlier than the surviving section).

The discovery of the *permutation matrix* was one of the most essential requirements for achieving a reconstruction of Contrapunctus XIV which might approach *the original form planned by Bach*. (Göncz, Z.: Reconstruction of the Final Contrapunctus of The Art of Fugue, in: *International Journal of Musicology* Vol. 5, pp. 25–93. 1997 ISBN 3-631-49809-8; Vol. 6, pp. 103–119. 1998 ISBN 3-631-33413-3)