The Art of the Fugue closes in spectacular fashion, with two mirror fugues. As one musicologist explained:
A mirror fugue is a pair of fugues in which each voice (or line) in the second fugue is a mirror image of the first – where the first goes up, the other goes down. In the previous movements only the theme was inverted; in the mirror fugues the entire piece is inverted. This requires Bach to play even more difficult games with his themes, since everything must be designed with its inversion in mind. Bach actually manages to achieve six different types of mirrors in these two pairs of fugues – a particularly stunning feat in that no matter how much Bach is bound up by the mirror fugue’s strict techniques, he still manages to make the music dance.
While some consider these fugues to be too mathematical to be truly musical, they are marvels of composition—among the most difficult ever produced. In the Twelfth, you get the main theme and then its mirror; in the Thirteenth, you get the main theme, which returns inverted, and then in its mirror. The following graphical videos help to make sense of what is going on in the music.
J.S. Bach, The Art of the Fugue
Contrapunctus XII (piano with graphic):
Contrapunctus XII (two harpsichords—use the links in the comments to skip to 1:09:21):
Contrapunctus XIII (piano with graphic):
Contrapunctus XIII (two harpsichords—use the links in the comments to skip to 1:14:44):
Bach left the final Fourteenth fugue unfinished—four themes, both upright and inverted, and then, presumably in mirror form. Only three themes were completed before Bach died. I like to think that he left this unfinished on purpose—as a challenge for every composer who followed him.
In the Art of the Fugue, a central mathematics is clearly at work. Figure that out and you can compose mirror fugues for as many subjects as you choose. Many composers have tried to finish it; all have failed, choosing to simplify rather than truly engage. Bach’s core sequence remains a mystery: It is the great unsolved puzzle in music history. At least that’s how I like to think of it (please don’t tell me I’m wrong and some stupid computer figured it out).
A Gertus History of Music 