Art of Fugue – Contrapunctus XI

Here it is, the most stupendous entry in The Art of Fugue. Its three themes are inversions of the ones in Contrapunctus VIII, meaning that they have the same rhythms and intervals but are upside down. Bach also added lots of subthemes and counterthemes. I used Ableton Live to visualize Angela Hewitt’s recording, drawing extensively on Dennis Collins’ analysis in Music Theory Online. I didn’t color-code every little thing that happens, just enough to show the major events in the piece.

Here are the three subjects, with a kick drum underneath for rhythmic clarity:

Some details:

• Subject 1 (green in the video) is the inverted version of subject 3 from Contrapunctus VIII. The little riff in the last two bars also appears in various places in the fugue as an independent entity.
• Subject 2 (blue in the video) is the inverted version of subject 1 from Contrapunctus VIII. The rising chromatic notes are the basis for some delightfully weird harmony.
• Subject 3 (purple in the video) is the inverted version of subject 2 from Contrapunctus VIII. It’s even more chromatic, and the harmonies it produces are even weirder. The little scale motif at the end shows up all over the place throughout the fugue, so many times in all its variants that I couldn’t even mark them all. In one of its transpositions, subject 3 starts off with the famous “BACH” riff: B-flat, A, C, B natural. In German convention, B-flat is called B, and B natural is called H. This riff is the subject of the last section of the unfinished last piece in The Art of Fugue.

Subject 3 comes with a countersubject, the ascending or descending chromatic scale. A descending chromatic scale is easy to fit into functional harmony, because it naturally follows the voice leading of V7-I cadences moving counterclockwise around the circle of fifths. (Fun fact, the circle of fifths and the chromatic scale are involutes of each other.) However, the ascending chromatic scale is much harder to fit into functional harmony. Measures 122-124 have a particularly long rising chromatic climb, with a series of dominant seventh chords that don’t even resolve. It sounds more like the backcycling harmony in a rock song.

Some other points of particular interest:

• At the beginning of the piece, the alto voice enters on subject 1 in D minor. Then the soprano voice enters on subject 1 in A minor. Then the bass voice voice enters on subject 1 in D minor. Finally, the tenor voice enters on subject 1 in A minor. Symmetry!
• Measure 27 is an obvious section change because the texture thins out and the harmony changes. Otherwise, though, the structure of this piece was opaque to me until I marked up the MIDI following Denis Collins’ analysis. I would have thought that measure 89 would be a section change, because the texture and harmony change, and because it’s the entrance of subject 3. But I defer to Denis Collins on this.
• In measure 46, a D augmented triad “resolves” to a D diminished chord. That is cool! These are probably not meant to be “real” chords, just sounds that emerge out of the voice leading, but the vertical relationships exist, and they are fascinating.
• At measure 57, subject 2 enters inverted so it’s the same as subject 1 from Contrapunctus VIII.
• In measures 64-66, Bach modulates from C minor all the way out to A major remarkably quickly. Each step makes its own logical sense, but that is a lot of key changes in a short span of time.
• There are places where the chromatic voice leading defies harmonic logic completely. The first half a beat of measure 69 includes the pitches E, D-sharp, G-sharp, and C-sharp. That’s coming in the middle of a long stretch of A minor. I hear these as spelling out either an Emaj7 chord or maybe an Amaj7(#11) with E in the bass. Bach wouldn’t have been thinking of either of those chords; once again, this is just a weird harmony that emerges out of the voice-leading. The best explanation I can think of is that it’s a temporary visit to parallel A major, and the D-sharp is just a quick apoggiatura. It’s still pretty wild, especially for that time and place.
• For the first two thirds of the piece, the harmonic rhythm aligns with the barlines. The rate at which the chords and keys change is uneven and unpredictable, but the changes always fall on even subdivisions of the bar. However, in measures 130-133, the chord changes are “misaligned” so they’re half a beat later than you’re expecting. It happens again in measures 169-171. Hip!

This analysis by Gavin Black says that the entire Art of Fugue is organized like one big fugue, with Contrapunctus XI as moment of peak contrapuntal intensity .

The first movements of the work introduce the main musical ideas in a straightforward way, as does the exposition of a fugue. The middle movements of The Art of the Fugue develop those musical ideas and others, with increasing complexity, contrapuntal and harmonic, and with increasing variety of texture. This is similar to the middle section (sometimes called “development”) of many fugues, especially, longer ones. The four canons fulfill the purpose described above. In the final three movements harmonic complexity is reduced, and anything even approaching the almost impenetrable density of Contrapunctus 11 is abandoned. In Contrapunctus 17, the original theme is reintroduced in a form closer to the opening of Contrapunctus 1 than anything that has been heard since Contrapunctus 4. This is analogous to the return of the initial subject that characterizes the final section of many fugues. The extraordinary contrapuntal ingenuity of Contrapuncti 16 and 17 (see below) is analogous to the increase in contrapuntal complexity that is found at the end of many Bach fugues, usually in the form of stretto.

Contrapunctus 11 is the centerpiece of The Art of the Fugue. It is probably the most complex contrapuntal creation of J. S. Bach, as well as one of the most deeply and powerfully expressive. It is a contrapuntal fantasy making significant use of at least ten different themes, and including five fully worked out fugue expositions. It is related to what has come before and what will come after in more ways than it is possible to describe. All of the thematic material is taken from earlier contrapuncti, mostly from Contrapunctus 8, but also at least from 3 and 10. The richness of thematic material is reminiscent of 10, while the compactness with which it is used is reminiscent of 7. The sectional construction is reminiscent of 5 and 8. The compellingly powerful cadences suggest number 4. The most powerful of these cadences is followed by the introduction of the B-A-C-H theme, which looks forward to Contrapunctus 18 and also back to number 8.

In his book Rethinking J. S. Bach’s the Art of Fugue, Anatoly Milka says that no one knows exactly why Bach wrote all these fugues. One theory is that he wrote them to fulfill the membership requirement for the Society of Musical Sciences. Another theory says that he wrote them as a kind of practical treatise on, well, the art of fugue. It’s perfectly possible that both theories are true; Bach recycled his own work for multiple purposes many times in his life. Milka says:

Although officially becoming a member as late as June 1747, Bach followed closely the Socie­ty’s activities from its very beginning, 10 years earlier. He kept contact with its members and was regularly updated about its ongoing issues, difficulties and creative aspirations.

This constant flow of information and interchange of ideas expanded the composer’s views on music creativity. Indeed, during his last decade, Bach was focused on traditional strict coun­terpoint and its compositional techniques. He was fascinated by the mathematical aspects of music, which were the subject of animated debates among the Society’s members.

Another indication of Bach’s active interest in the state­-of-­the-­art research of music theory, as well as of its practice, is the translation of Fux’s Gradus ad Parnassum from Latin to German that Mizler published in 1742. It is known that Bach owned both Latin and German versions of this treatise. A work that inspects contrapuntal techniques is, therefore, in full accord with the trends and tastes that characterised the Society during all the time of its existence.

The bulk of musical examples in the German edition looks like a collection of counterpoints, in which one section of short fugues stands apart. Written in score layout and in vocal clefs, almost all these fugues are based on just five themes, three of which are variants of each other, thus suggesting a tendency toward monothematicism. The first fugue of this section is based on the Gradus’s most prevalent theme. All the fugues on this theme are written in the first church mode (Dorian on D) and ordered in ascending degrees of complexity: first, the number of parts is increased from two to three and then four; then, the themes themselves are elaborated with added sophistication in the relations among the parts. Such organisation resembles that of The Art of Fugue in its early versions.

Both Bach’s and Fux’s fugue themes share their metre and order of voice introduction—alto, soprano, bass and tenor—which are more characteristic of strict choral style than of baroque instrumental music. Even the themes themselves could be related, Bach’s theme seemingly an elaboration of Fux’s. Bach’s countersubjects, with their gradual progression into quarter notes and syncopations after a leap, are, too, conversant with the style of Gradus. It is thus hard to avoid detecting an intentional attempt to signal that no earlier scholarly treatise, not even the best—and undoubtedly Gradus ad Parnassum was a work of that rank—had ever manifested such mastership in displaying the richest potentials of fugal writing. The Art of Fugue would establish Bach’s unique offering of the truly artistic illustration of fugue.

I put beats under the piece to help me hear what’s going on:

Bruce Berr uses the term “prosaic rhythm” to describe the “mere timing of events decoded from the printed page.” This is as opposed to the “poetic” rhythm, which is the timing as actually performed by someone like Angela Hewitt. I wanted to hear the prosaic rhythms, because this piece is confusing enough without having to listen around Hewitt’s performance choices. To be clear, her choices are excellent and well-motivated! But I want to “hear” the notes on the page, and I don’t audiate nearly well enough to do that from the score. This is why it’s so nice to be able to quantize things in Ableton. Putting the beats underneath helps me distinguish between the front and back of each bar, the front and back of each phrase, and each hyperphrase. It also just sounds cool! We are very far removed from Afrodiasporic dance music here, but Afrodiasporic grooves make everything sound good.