Structure of The Art of Fugue

A work with as many parts as AOF offers the composer numerous avenues to introduce structure and form on both small and large scales. Here I will discuss a few of these.

Divisions of Four

Immediately obvious is what we might call the “4-symmetry” of AOF. Generally, each “section” (indicated by the black headers in the program in the table of contents) contains four fugues, with all the fugues in each section being of related types (simple, stretto, and so forth). There are four such sections with collections of fugues; and AOF is capped off by a a quadruple fugue that has four sections. The work includes four canons. With only two exceptions, the fugues are in four voices.

Given this embedded 4-symmetry, it seems most natural to me to place the canons between each of the major sections, providing a change of texture at logical places for those listening to the work from start to finish, and emphasizing the “4-ness” of AOF.

The Bach Number

AOF contains 14 Contrapuncti, and 14 was a number of great significance to Bach. The letters of his name add up to 14 (B+A+C+H = 2+1+3+8 = 14), and he codes the number 14 into many of his works. As a particularly relevant case in point, the Fourteen Canons on the Goldberg Ground (BWV 1087) were, like AOF, written in the last decade of Bach’s life, a time when he was devoting himself increasingly to large, cyclic works demonstrating his transcendental mastery of his craft. The canons of BWV 1087 are intimately linked to the Goldberg Variations, which are as deeply imbued with “3-ness” as AOF is with 4. But it seems there are some structural links between AOF and BWV 1087 as well, one example being the methodical development of the simplest forms of the thematic germ in the first four Contrapuncti of AOF (two rectus and two inversus) that parallels the idea, if executed on a much smaller scale, in the first four canons of BWV 1087.

For this reason, I view the 4-symmetry of AOF as its medium scale structure. Arching above this is the large scale structure of the Bach Number, which links the various parts of AOF in ways beyond the simple divisions of 4. In the narratives accompanying each part, we will see clear motivic links, for example, in Contrapuncti III and XI, in V, X, and XIV, and so forth. No doubt I have missed many more such links than I have noticed; small wonder scholars have found AOF worthy of a lifetime of study!

Incidentally, Bach’s predilection for 14s provides a natural explanation for why he “shorted” the stretto fugues by one; by so doing, he contrived to have the final Contrapunctus be XIV rather than XV. This explanation requires two comments. First, I’m counting the mirror fugues of Contrapuncti XII and XIII as one piece; despite the fact that you’ll hear two fugues with each one; the rectus and inversus performances are precisely isomorphic and thus quite reasonably count as only one piece of music. Second, the explanation for the missing stretto fugue is Tim Smith’s, not mine, and I thank him for the idea.

“Missing” Pieces in This Realization

The principal parts of AOF that you may find in other performances, but not here, are the keyboard reductions of Contrapunctus XIII and the concluding chorale. In the case of the former, as marvelous as the music is, it has always seemed a bit redundant, and even detracting from the impact of the music, to play the same work twice. In the case of the chorale, BWV 668a, it simply has no relationship to the rest of AOF. It is “traditional” for it to be be performed after the end of Contrapunctus XIV, but in a performance that includes a completion of the final fugue, I find 668a to be a dreadful anticlimax — lovely as it is when taken on its own merits. It is therefore not included here.

Gematrics in AOF

As with BWV 1087, we should be careful regarding Bach numerology. I believe there is some “4-ness” embedded in AOF’s large scale structure, and the numbers 3, 4, 11, and 14 figure prominently in the narratives on this site. However, it is all too easy to find accidental symbolism. For example, I noted above in the discussion of 4-symmetry that all but two of the fugues have four voices (and Contrapuncti VIII and XIII have three…there’s that 3 again!). Skeptics will correctly note that most fugues have four voices; and if not four, three is the next choice. One voice would be a bit boring, two…well, see the canons. Five and six voice fugues exist, but are monstrously difficult to bring off and are correspondingly less common. So while Bach must have been aware that most of the AOF fugues had the same number of voices as the fourfold nature of its structure (not to mention the same as the number of letters in his name), he likely saw that as an amusing necessity rather than a symbolic gesture.

The same caution applies to any ramblings about numeric symbolism, and you should take all such commentary in my narratives with a grain of salt. Undoubtedly some of it is symbolic, but much of it must be driven purely by Bach’s desire to write sensible and effective music, and by the constraints on the structure of musical phrases that this desire imparts.

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