Mini–Canons and Symmetries in Bach’s Chorale #29,
"Mein Gott, öffne mir die Pforten"

by Luke Dahn

When considering the great contrapuntal masterpieces of Bach, those intricately designed works that cause one to marvel at Bach’s genius for fusing the linear and the vertical, works such as the Art of the Fugue or the Saint Matthew Passion or perhaps the Well-Tempered Clavier fugues come to mind. Rarely, though, would any of his four-part chorales be considered (though, perhaps, his chorale preludes may). The chorales are often considered, rather, for their didactic potential, serving as models for proper part-writing and voice-leading (though models in which "rules" are often shirked). However, in working on my Bach Chorale pedagogical index, I came across Chorale Number 29, which is an absolute gem. I began noticing symmetries and motivic connections at every phrase and involving every voice. With its tightly-woven musical fabric, this chorale stands alongside every other masterpiece attesting to the genius of the Great Contrapuntalist.

Chorale 29 (Riemenscheider edition numbering) is one of four Bach settings of the Louis Bourgeois tune, "Freu’ dich sehr, o meine Seele." This particular setting comes from the cantata, Liebster Jesu, mein Verlangen, BWV32. (The English translation in the example below is by Z. Philip Ambrose.



An initial read through of this chorale may not yield much by way of intricate contrapuntal devices, but a closer look reveals that symmetries and canonic devices abound. There are six phrases in total, the first two of which are repeated as a phrase period. These first two phrases serve as a kind of exposition, presenting us with the motives and materials that will be intricately woven into the fabric of the entire chorale. The single most important motive of the chorale is given at the tune’s very opening: G-A-B-A-G. (Example 1) This will henceforth be referred to as “motive X” or simply “X.” The palindromic structure of motive X informs other parts of the chorale, as palindromic structures appear frequently.

The melody of phrase 2 (Example 2) is an elaboration of phrase 1. It is simply an expanded version of motive X, encompassing four notes instead of three: G-A-B-C-B-A-G. Naturally, this version of X is also palindromic. This expanded form of X is then transferred to the bass in measures 5-6 (Example 3), where it appears in inversion. The bass here is the exact mirror of the soprano’s phrase in measures 3-4, with an pitch axis, and common starting point, of G. Bach then continues his method of expansion in measures 7-8 in the bass part, presenting a further expansion of motive X, now encompassing five notes. Example 3 also shows three other manifestations of X in measures 1-8: a quick appearance via passing tones in the tenor in measure 3, a third consecutive appearance in the hymn tune in measures 5-6 (made possible by a slight alteration of Bourgeois’s tune), and a rhythmically askew appearance in the alto leading to the half cadence in measure 8.

Almost every note in every part of phrase 5 (mm.9-11) figures in a presentation of motive X (Example 4). The melody in the soprano has three overlapping presentations of motive X, with the two outer occurrences being partial. The first instance is missing its second note (C), while the third instance is missing its fourth note (A), thus revealing another palindromic arrangement. A similar thing happens in the tenor part, with an inverted form of motive X overlapping an uninverted form. (Notice the palindromic rhythmic nature of the two forms when taken as a unit.) The alto and bass lines are also related to each other. The alto part has two inverted presentations of X, with the final note of the first serving as a link starting the second. (One could also say than an uninverted form of X binds these two linked presentations together: E-F#-G-F#-E.) The result is an undulating wave. The bass likewise has an undulating wave, but a wave created by overlapping two expanded four-note versions of X.

In the final phrase of the chorale, motive X is found mirroring itself in the soprano and bass parts in measures 11-12. The bass presents an inverted form of the soprano retaining the exact intervallic structure (m2, M2, M2, m2), while its rhythm is retrograded. The result, of course, is that the bass is a retrograde inversion of the soprano. Finally, the alto and tenor parts end the chorale with closing instances of motive X: the expanded four-note version in the alto (with a missing penultimate note), and the three-note version in the tenor. The rearticulation of the motive’s peak note in the tenor (D) increases the satisfaction of its downward resolution.


Measures 1-4
Unsurprisingly perhaps, canonic devices frequently appear as phrases and phrase fragments are transferred from part to part. Such transference has already been seen in Example 3. The fragmentary nature of some of these canonic moments may cause them to seem coincidental, but the frequency with which such devices are used throughout the chorale does not allow for coincidence. Example 6 shows the first two phrases and two very simple mini-canons occurring between the tenor and soprano parts. Again, taken alone, one might certainly suggest the connection between these 3-note groupings is made arbitrarily. When taken along with all of the other canonic devices discussed in this essay, and when considering the fact that each of these canonic connections is perfectly audible, one is left only to conclude that canon and symmetry are a guiding principle.

Measures 3-6
Example 3 above demonstrated a relationship between the soprano line in measures 3-4 and the bass line in measures 5-6. The bass is an exact mirror of the soprano’s expanded version of motive X, with an axis on the pitch G. (The soprano moves upward away from G; the bass moves downward away from G.) A closer look at the passage reveals a similar relationship between the alto and tenor parts. The alto’s counterpoint to the soprano’s melody in measures 3-4 (itself nearly palindromic), is transferred to the tenor in measures 5-6. The tenor gives with a mirrored version of the alto’s counterpoint. The mirror is not exact, for the tenor should end on E rather than D, but this slight alteration does in no way diminish the audible connection between the two. The tenor materializes out of the alto in the same way the bass materializes out of the soprano. The slight alteration of the tenor does two things here: it provides an emphasis on the pitch D (m.6), the dominant of the home key of G, and it creates a palindromic structure in the tenor (D-E-A-E-D). This transference of two grouped counterpointing voices in mirroring fashion from the soprano and alto to the bass and tenor (which then carry the responsibility of harmonically supporting a new melodic phrase) is an example of the stroke of genius we come to expect from Bach, though usually only in grander contrapuntal genres.

The way in which the tenor and bass "develop" ideas presented previously in the soprano and alto might cause one to envision (anachronistically, to be sure) a kind of proto-sonata form. Phrases 1 and 2 serve "expositional" purposes; phrases 3 and 4 and perhaps 5 (though 5 is melodically connected with 6) demonstrate imaginative and thorough "developmental" devices; and phrase 6 provides closure.

A few palindromic structures in Chorale 29 have already been demonstrated: in the various forms of motive X itself, in the structure of the soprano’s tune in phrase 5 (Example 4), and in the tenor’s rhythm in the same phrase. With the liberal use of various palindromic forms of the chorale’s primary motive X, palindromic structures naturally emerge. However, it is measures 5-6 that truly confirm the palindrome as a guiding principle, for not only does Bach alter the original tune to produce a palindromic phrase (substituting the original second note (B) for a C), and not only does he alter the tenor’s mirrored version of the alto’s previous line (Example 7), he also gives every part a palindromic phrase, producing a progression that clearly and quite audibly doubles back on itself (Example 8). The palindromic generative motive X is now clothed with its own palindromic harmonic attire. This is hardly coincidence.

Measures 7-8
Phrase 4 in measures 7-8 has already been mentioned in dealing with the various appearances of X. Another look reveals other carefully constructed symmetries (Example 9). The bass line of measure 7 is a retrograde inversion of the six notes of the phrase’s melody (the two Ds that start the phrase counting as one). This motive is labeled Y. Motive Y also appears in the tenor in inversion and with rhythmic diminution, starting with the A on beat three of measure 7 (A-G-A-B-C#-D). Notice also the further linkage between bass and tenor. The tenor is presenting a retrograde form of the bass’s first eight notes in this phrase, with the rhythm more or less also remaining in tact. Finally, the alto’s role is to preserve the prominence of motive X, presenting a rhythmically (and harmonically) deviant version.

Measures 9-11
The next several examples demonstrate that no phrase in this chorale is more tightly and intricately knit as the fifth phrase (mm.9-11). Example 4 showed the extent to which motive X pervades the phrase’s fabric. No fewer than eight instances of X emerge. The next four examples show other symmetries and canonic devices. Example 10 presents the longest canonic passage of the entire chorale, extending eight notes. The soprano is the dux, and the bass, the comes. Interestingly, Bach has varied the bass’s rhythm, mostly with diminution, to the extent that the canonic connection is rather obscured. (Notice the soprano’s final G being greatly stressed in its metric placement, while the corresponding G in the bass part occupying perhaps the weakest metric placement of all.)

Example 11 shows a symmetry of an entirely different kind. Bach uses twice in this chorale a particular device in the lower parts, once in measure 8 to close phrase four, and once in measure 12 of the final phrase. The device consists of three ascending parallel thirds between the bass and tenor, followed by three more ascending thirds between the tenor and alto. The tenor, thus, serves as a link between the two sets of parallel thirds. Turning now to the phrase under discussion (mm.9-11), Bach flips the device upside down and places it in the middle of the phrase. The first part of Example 11 shows the inverted device. Three parallel thirds between the soprano and alto are now descending, and are followed by three more descending thirds between the tenor and bass. Bach here involves all four parts, with the alto voice linking up with the tenor at the device’s midway point (on E).

Looking again at the same passage (mm.9-10) shows yet other canonic and palindromic structures. Notice the connection between the soprano and the two lower parts demonstrated in Example 12A. First, the tenor eighth notes that begin the phrase (D-C-B) anticipate the same notes in the soprano on beats 2-4, the soprano’s version being rhythmically augmented. The following measure reveals a reversal of roles. The soprano takes on the anticipatory role with its G-A-B quarter notes being followed by the same notes in the bass, though now in rhythmic diminution. The pitch B serves as an axis here, with the measure 9 phrase descending to B, and measure 10 phrase ascending to B (reminiscent of the soprano/bass relationship in Example 7). Readily apparent is the palindromic nature of the composite rhythm.

Example 12B illustrates other cross-staff connections. The descending parallel thirds have already been discussed and shown in Example 11. What was not shown there are the ascending thirds (or tenths) in that very same passage. While the soprano and alto parts have descending thirds in eighths, the tenor and bass have ascending tenths in quarters. Measure 10, the moment of transference of the descending thirds, is also the moment of transference of the ascending thirds. The two sets of parallel thirds simply trade staves, so to speak. Furthermore, the ascending quarter-note thirds can be viewed as an augmented version of the device illustrated in Example 11.

Example 13 (rather confusingly) shows all of the canonic connections in this wonderful, intricately-woven passage.

Measures 11-13
Bach concludes the chorale with a canonic treatment of motive X. The soprano begins motive X on B, scale-degree 3, with two eighth-notes followed by three quarters. This is then mimicked in the tenor part starting at the moment the soprano reaches its final note of X. Bach then reiterates the D in the tenor voice before resolving back down to B, in effect heightening the satisfaction of the final resolution. Furthermore, the B-C-D beginning of the canon with its eighth-eighth-quarter rhythm is likewise mimicked in the bass part, with the same rhythm. The result is a kind of echo of the gesture, descending to three distinct registral realms.

It is perhaps of little surprise that Bach uses voice-exchange to a great extent in this chorale, given both Bach’s propensity for using mirroring devices and the construction of motive X itself. We hear the first instance of voice-exchange at the very outset of the chorale: the soprano’s G-A-B is mirrored by the tenor’s B-A-G (Example 15). In a sense, the tenor’s three notes combine with those of the soprano to create a compressed form of motive X. Other straightforward examples of voice-exchange occur in measure 3 (soprano/tenor), measures 7-8 (soprano/bass), and two instances in measure 9 (both involving soprano/tenor). The three other instances are not quite as straightforward. The soprano and bass undergo a 5-note voice-exchange extending from the final beat of measure 1 to the final beat of measure 2. Bach slightly adjusts the bass line here, accelerating its rhythm so that it may harmonically support the oncoming cadence. The soprano/bass voice-exchange in measures 3-4 also receives slight rhythmic alteration to accommodate cadential harmonic implications. Finally, measure 10 possesses a slightly altered soprano/bass exchange.

Bach’s Chorale 29 is a marvelous demonstration of the composer’s unmatched facility with contrapuntal devices. It is also a demonstration that the 371(!) chorales should not be neglected as analytically insignificant or marginalized as didactic models. The chorales are beautiful expressions that move both heart and thrill the mind.