Die Kunst der Fuge contains four canons which are among Bach's longest and formally complex. True to the cyclical conception of which they are part, each of the four develops aspects of fugal writing that are difficult to maintain in canon.
It is possible that the inspiration for these four canons came from Bach's own Fuga Canonica in Epidiapente which he had earlier composed for the Musical Offering, and to which they bear a marked resemblance.
Canon 1: The Canon alla Ottava explores fugal technique in its reiteration of the head motive at the level of the dominant and in contrary motion. This canonic "subject" is the melodic inversion of the main theme of the Art of Fugue.
Formally, the Canon alla Ottava is the least complicated in that its high voice consistently represents the leader, and its low voice, the follower. The colored sections in the following picture represent analogous units.
Canon 2: The subject of this Canon alla Decima consists of the unadorned Die Kunst theme in contrary motion.
The second, third, and fourth canons of the Art of Fugue each divide neatly into two sections. In section one of this canon (mm. 1-33), the lower voice leads while the upper voice follows a 10th higher (canon at the 10th). After a short bridge passage (mm. 34-40), there is an exchange of registers in which the follower, this time in the lower voice, imitates not a 10th lower, but at the octave. Thus sections one and two exemplify double counterpoint at the 10th. The colored sections in the following picture link analogous units.
Canon 3: The subject of this Canon alla Duodecima consists of Die Kunst's main theme with complete upper and lower neighbor diminutions of each factor in an ascending triad outlining the rising P5 of the theme's head motive.
The formal organization of this canon is identical to that of its predecessor. Other than the substantial change of affect in head motives, the primary innovation of Canon No. 3 is that its follower imitates at the interval of the 12th (8va+5th higher) rather than the 10th. The low voice takes the lead in section one whereafter the high voice follows a 12th higher, exemplifying the fugal principle of "real" answer. Midway through the canon these registers are exchanged but with the follower echoing the leader, not a 12th lower, but at the octave (double counterpoint at the 12th). The colored sections in the following picture link analogous units.
Canon 4: The subject of this Canon per Augmentationem in contrario Motu is a disjunct variation of the Die Kunst theme.
This canon ranks among the most complicated in Bach's oeuvre. While simple in terms of form--two sections in double counterpoint--the canon's follower is in contrary motion AND rhythmic augmentation to its leader! Bach manages to sustain this display of contrapuntal fireworks for fifty-two measures whereafter he causes the voices to exchange registers and do it again in double counterpoint at the 8va (mm. 53-109)! This procedure is quickly recognized, aurally, in the two places where the leader executes ascending chromatics in eighths (which are answered, of course, by descending chromatics in quarters). The colored sections in the following picture represent analogous units.
