J. S. Bach’s Organ Music and Lutheran Theology

Marilyn Biery is keyboard acquisitions editor at Augsburg Fortress. She is Bridge Director of Music Ministry at Kirk in the Hills in Bloomfield Hills, Michigan. She holds bachelor and master of music degrees in organ performance from Northwestern University, Evanston, Illinois, and the Doctor of Musical Arts degree in organ performance from the University of Minnesota, Minneapolis.

Johann Sebastian Bach’s Schübler Chorales have a special place in my heart. In the winter of 1978, when I was a sophomore organ major at Northwestern University in Evanston, Illinois, I heard all six of them played (from memory) by a senior organ major during departmental organ class at Alice Millar Chapel. I was so moved by the playing that I went to the back of the chapel to greet the organist, James Biery, when he came down from the loft. It was the first time we had ever had a conversation. Forty-four years later, we are still in conversation.

Sechs Choräle von verschiedener Art, BWV 645–650, commonly referred to as the Schübler Chorales, is a beloved collection among Johann Sebastian Bach’s already treasured body of music for the organ. Toward the end of Bach’s life, he worked with his student Johann Georg Schübler (c. 1725–after 1753) to outline the collection. It is believed that Bach chose the cantata movements to be transcribed and laid out the structure, and that Schübler completed, engraved, and published the work around 1747 or 1748. Five of the six pieces are from known cantatas, the sixth is believed to be from a lost cantata. Since none of Bach’s cantatas were published during his lifetime, this collection served the purpose of getting Bach’s compositions to a wider audience and introducing more of his music to those who were not familiar with his cantatas. The Leupold Edition of the Bach Organ Works, Series 1, Volume 9, includes an excellent discussion of the Schübler Chorales by George B. Stauffer. The music and the information on the collection represent the most current scholarship and are well worth investigating.

Because of my love for these pieces and my memory of hearing Jim play them at Northwestern’s Alice Millar Chapel, at some point I started to wonder if Bach might have contemplated another volume had he lived longer. It seemed reasonable that there would be another set to be found, so it became my project to look through the cantatas for pieces that would fit into the style of the Schübler Chorales: cantata movements in trio texture with a cantus firmus. At the beginning of my search, I was looking through print copies that Jim had available in his office at Grosse Pointe Memorial Church in Grosse Pointe Farms, Michigan, but at some point I switched over to looking at the scores on the International Music Score Library Project (IMSLP, imslp.org).

The process of looking at the score while listening to or watching a recording on YouTube was good for my soul. Due to various job changes, there were stops and starts and long pauses, so that the entire process took about five years. Thanks to the support and encouragement of Augsburg Fortress and the able editing of David Sims, Cynthia Newman Edwards, and James Biery, the collection was published in the spring of 2020.

The process was daunting, but after my perusal a collection of ten movements emerged. There are quite a few movements in Bach’s cantatas that would make fine organ transcriptions, especially of trio texture, but there are a limited number of movements playable by one person at the organ that also use a cantus firmus. While I was looking for movements that qualified for this project, I was also looking for good trio movements without a cantus firmus for a future collection; I did not keep track of how many I discarded in the process. I had hoped that a Google search of “Bach cantata movements with cantus firmus in trio texture” existed but alas, it does not. I looked through the secular cantatas as well, but in the end all of these come from the sacred cantatas.

In keeping with Bach’s love of symmetry, I arranged the collection of ten transcriptions into a pleasing order of key relationships, with the cantus firmus played either by the right hand, the left hand, or the feet. Some movements presented themselves as playable only one way, some worked more than one way, and the choice was made based on playability as well as the overall structure of the collection.

Four of the eight tunes found in this collection are still sung today, most notably Lobe den Herren, Valet will ich dir geben, Jesu, meine Freude, and Christ lag in Todesbanden. The tune Wo soll ich fliehen hin is an alternate melody and not the same tune Bach used in the second Schübler chorale prelude. Three of the movements are based on the same tune, Herr Jesu Christ, du höchstes Gut, and are placed consecutively in the collection to make an effective set by themselves. Du Friedefürst, Herr Jesu Christ, and Nun lob, mein Seel, den Herren are tunes that are not found in current hymnals.

The registration suggestions were provided by James Biery as a starting point to creativity and are offered in the spirit of guidance rather than dogma. The suggestions are adapted from the specification of the large Klais organ (from Bonn, Germany) at Grosse Pointe Memorial Church. There are suggestions for each hand and pedal without specifying manuals, since specifications vary as to where stops are located. For those movements where the left hand is playing the continuo part, a 16′ is suggested, although not required. Some of the movements lend themselves well to having a solo instrument play, or a small schola sing, the cantus firmus.

The volume, published by Augsburg Fortress, Ten Organ Chorales in the Schübler Tradition, includes basic information on the cantata sources, information about the tunes, and an example of the tune as used by Bach, as well as a translation of the German verse Bach might have been working from. For those who are interested, that information is in the volume. In this article I will speak more to the editing process.

1. Du Friedefürst, Herr Jesu Christ (You Prince of Peace, Lord Jesus Christ) comes from the cantata Lobe den Herrn, meine Seele (Praise the Lord, my soul), BWV 143. It is believed to have been written when Bach was in his twenties, although the attribution to Bach is in question. The organ transcription comes from the second of the cantata’s seven movements, an aria for violins, soprano, and continuo (Example 2). The cantus firmus is played in the pedal at 4′ (Example 3). The composer of the tune is not known. It appears to be loosely based on the Heinrich Isaac (c. 1450–1517) tune Innsbruck, ich muß dich lassen. The tune appeared thus in a seventeenth-century hymnal, so it is likely the composer of BWV 143 would have used the tune in this form (Example 1).

2. Wo soll ich fliehen hin (Where Should I Fly From Here) comes from Mein Herze schwimmt im Blut (My heart is bathed in blood), BWV 199, a solo cantata. In the sixth movement Bach used an alternate cantus firmus tune (Example 4) attributed to the German poet and playwright Caspar von Stieler (1632–1707). There are two versions of this cantata, and both are included in this collection. In the 1714 version Bach called for viola obbligato, soprano, and continuo, in F major. In the 1723 Leipzig version, it is scored for violoncello piccolo solo, soprano, and continuo, in G major. In both versions the cantus firmus is in the right hand. The Leipzig version is included in the Addendum.

3. Valet will ich dir geben (I Want to Bid You Farewell) is the third movement from Christus, der ist mein Leben (Christ is my life), BWV 95, and is scored for oboe d’amores, soprano, and continuo. The preceding soprano recitative leads immediately into the chorale without any pause between movements or ritornello introduction. In the Bach-Gesellschaft example (Example 5), the first note of the tune is not shown because it is at the end of the recitative in the previous line. Bach wrote two substantial organ settings of this tune, BWV 735 in B-flat major and BWV 736 in D major. This shorter setting pairs well with BWV 735, hence the decision to change the key from D major in the cantata to B-flat for the organ chorale. The cantus firmus is in the pedal, and the key of B-flat enables the player to have a more central balance on the pedalboard (Example 6).

4, 5, 6. The next three transcriptions are all based on the same tune (Example 7): Herr Jesu Christ, du höchstes Gut (Lord Jesus Christ, O Highest Good), found in three different cantatas, BWV 113, BWV 166, and BWV 131. The tune is referred to by the name of the text and is generally believed to be of unknown authorship, traced to 1587 and 1593. Bach used different verses to inspire each movement, and none of them are based on the first verse of the text.

The first setting was transposed from its original key, F-sharp minor, to make a more pleasing order of key relationships between the three—G minor, C minor/Dorian key signature, and C minor; the original key is included in the Addendum. The first setting is my favorite of the three, and I prefer the original key. The cantus firmus is at 4′ in all three of these movements; first in the pedal, then in the left hand, then again in the pedal. The choice to do this was in honor of Bach’s love of symmetry.

The second setting can be played with the cantus firmus in the pedal instead of the left hand, as is printed, and that version would be considerably easier (and the editor will keep that in mind for a future revision). Sometimes there were differences between the two main Bach editions, and for the third setting the Bach-Gesellschaft Ausgabe has the movement in C minor, the Neue Bach-Ausgabe in D minor, and C minor was chosen for this Augsburg Fortress collection.

7. Jesu, meine Freude (Jesus, My Joy) is the last aria of Weinen, Klagen, Sorgen, Zagen (Weeping, Lamentation, Worry, Despair), BWV 12. It is scored for tenor, trumpet, and continuo (Example 8); the trumpet plays the cantus firmus. In the transcription, the right hand has the tune (Example 9).

8. Nun lob, mein Seel, den Herren (Now Praise, My Soul, the Lord) is from Gottlob! Nun geht das Jahr zu Ende (Praise God! Now the year is coming to a close), BWV 28. The second movement, scored for SATB and continuo (Example 11), was based on the melody Nun lob, mein Seel, den Herren, the composer of which is unknown (Example 10). The opening phrase bears a resemblance to the Old Hundredth hymntune. This is the only transcription in the collection that is not in trio texture, and that does not place the cantus firmus on a separate keyboard. My first draft put the melody in the pedal, but James Biery suggested I try a plenum setting with the melody in the top voice of the keyboard, which makes it more of an exact transcription from the original. Some minor editing was done when the continuo line did not precisely match the line sung by the basses. This approach reflects the style of the middle plenum section of the Fantasia in G Major, or Pièce d’Orgue, BWV 572. It has become my favorite movement (Example 12). (It has a great pedal point towards the end!)

9. Lobe den Herren, den mächtigen König der Ehren (Praise the Lord, the Mighty King of Honor), BWV 137, is based on the hymntune by Joachim Neander (1650–1680). It still appears in most current hymnals. Bach included the second movement of this cantata in the Schübler Chorales. In this fourth movement the C-major cantus firmus is presented firmly in A minor.

In this aria for tenor, trumpet, and continuo, the cantus firmus is played by the trumpet (Example 13). The composer of the tune is unknown. Due to the length of the opening ritornello, the tune does not make an appearance in these examples; it is in the pedal at 4′ (Example 14).

10. Christ lag in Todesbanden (Christ Lay in Death’s Bonds), BWV 4, is one of Bach’s earliest cantatas. It was written for Easter Sunday, on the text of the same name by Martin Luther (1483–1546). The tune was developed by Luther and Johann Walther (1496–1570) and is believed to be based on the Easter Sequence in the Catholic liturgy, Victimae paschali laudes. The third movement is scored for violins, tenor, and continuo (Example 15). The tune is in the left hand at 8′ (Example 16).

All ten of these transcriptions were played in concert by James Biery on May 22, 2022, on the Klais organ at Grosse Pointe Memorial Church, Grosse Pointe Farms, Michigan. The editor provided commentary.

Sources used:

1. Bach cantatas website: bachcantatas.com.

2. The Leupold Edition of Johann Sebastian Bach’s Complete Organ Works, Series I, Volume 9: Schübler Chorales, Canonic Variations, Chorale Partitas, edited by George B. Stauffer.

3. Cantus firmus German verses: https://www.bach-digital.de/content/index.xed.

4. Wikipedia articles.

Gavin Black, director of the Princeton Early Keyboard Center, Princeton, New Jersey, is preparing performances on Bach’s The Art of the Fugue on both harpsichord and organ for the next two concert seasons.

The Art of the Fugue, part III

This month’s installment picks up exactly where last month’s left off.

Notes on the Individual Movements

Contrapunctus 1 is a four-voice fugue on the original and simplest form of The Art of the Fugue theme (Example 1). There is no regular countersubject. The mood is forthright and powerful, but with underlying calm.

Contrapunctus 2 is a four-voice fugue on the same theme, except that the last notes are changed from eighth notes to dotted eighths and sixteenths (Example 2). This dotted rhythm pervades the piece. The sounds jump rather than flow from one beat to the next. The effect is, at least intermittently, unsettling. The dotted rhythms convey a suggestion of French musical style, though the piece is not fully worked out in that style.

Contrapunctus 3 is a four-voice fugue on the inversion of the original theme (Example 3). The piece moves slowly and gently, and the mood is quiet. The subject is accompanied (after the first entrance) by a chromatic countersubject loosely derived from the subject itself (Example 4). This countersubject also provides a short motive (Example 5) on which two episodes are based. This motive will return later in the work. In measure 23 the subject enters in a new form: with passing tones (Example 6). Although it is abandoned in the next movement this will become the most important form of the theme later on.

Contrapunctus 4 is also a four-voice fugue based on the inversion of the main theme (see Example 3). The theme is accompanied by a short chromatic motive, more explicitly dissonant than that of Contrapunctus 3 (Example 7). The movement is also characterized, however, by long episodes of almost exaggerated consonance. These are based mainly on a simple two-note descending third (which can be heard as a reference to the thirds in the original, non-inverted theme, although that theme is not explicitly present in this movement) and a four-note stepwise fourth, both ascending and descending, which is derived from the last four notes of the main theme, in both its inverted and non-inverted forms. There are several lushly beautiful internal cadences, almost hypnotic in their effect. The last measures conceal a single reference to the B-A-C-H motive.

Contrapunctus 5 opens with a statement of the inverted theme with passing tones, introduced in Contrapunctus 3. Before the end of the theme, this is joined by the non-inverted theme with passing tones in the bass (Example 8). This overlapping of themes, called “stretto” by theorists of fugue writing, is by no means characteristic of the opening of a fugue. Nor is it characteristic for the second subject entry to be an inversion rather than a transposition of the first. This is a gesture of union of the two essential forms of the main theme. They are the opposite of each other, but they can be joined together. In measure 53, an even closer union is suggested with a four-voice stretto at the interval of one quarter note. The inverted and non-inverted themes, however, are treated to this process separately (measured 53–57 and 65–69)! The mood of Contrapunctus 5 is alternately urgent and dreamy, or perhaps both at once. In the last measures of the movement the two versions of the theme are heard simultaneously, somewhat concealed by an unexpected six-voice texture.

Contrapunctus 6 is marked in Stile Francese. Thus it realizes the suggestion made by Contrapunctus 2. It is not easy to define what the “French style” is. For the purposes of this piece, however, it is enough to recognize that it involves the use of dotted rhythms, very fast short runs, and moderately fast longer runs. It also aims to create a feeling of power and grandeur in the context of rather static (non-flowing) rhythm. Native French pieces in this style would never have been fugues, so Bach was doing something new. This piece, like the last, opens with a stretto, at the interval of only one measure. The impatient quality of the second voice, created by its insistence on coming in early, is here intensified by the fact that is comes in “too fast,” that is, in diminution. The very rapid four-note runs, heard from measure 7 on, are a standard characteristic of the French style, but they can also be seen as a double diminution of the last four notes of the main theme.

Contrapunctus 7 is a fugue on the main theme with passing tones, sometimes inverted and sometimes in diminution. To this is added the theme in augmentation (Example 9) that is heard first in the bass and once in each voice. Almost every note in this contrapunctus is derived directly from the theme, by augmentation, diminution, or double diminution. The effect is one of extreme concentration and intensity, but also of rather jovial power and good humor.

Contrapunctus 8 opens with a short fugue exposition on a brand new theme, not derived in any way from what has come before (Example 10, but see under Contrapunctus 18, below). After several short episodes and a cadence in the tonic, a second exposition begins, based on the same theme and another brand new theme (Example 11), simultaneously. These two themes work themselves out with increasing intensity and exuberance, until they are joined in a somewhat hidden manner, that is, in an inner voice, by a new version of the main theme (Example 12). The intensity and exuberance, the latter quality created mainly by the almost exaggerated repetitions of the second theme (Example 11), continue to the end.

Contrapunctus 9 is a fugue on a new subject (Example 13) that is loosely derived from the main theme. Whereas Contrapunctus 8 opened with a descending whole-step (see Example 10), this movement opens with an upward leap of an octave. This is a contrast to Contrapuncti 1–7, all of which opened with the interval D–A or A–D. Contrapunctus 9 is a lively, flowing piece in which exuberance, though present, is restrained. The main theme enters in augmentation in the soprano voice in measure 35 and is heard several times thereafter.

Contrapunctus 10 opens with a fugue exposition on a subject (Example 14) whose rhythmic and melodic peculiarities create more of a sense of expectation than of fulfillment. The expectation is met with the entry of the main theme in the soprano voice in measure 23. What follows is a fugue on this subject, in which Example 14 serves as a countersubject. So many new motives are introduced, developed briefly, and then abandoned, that the movement becomes a celebration of fecundity (or perhaps of profligacy!). The countersubject (Example 15) that appears in measures 22–26, and does not appear again in this movement, will supply the subject of the second section of Contrapunctus 18 (see below).

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 motive, which looks forward to Contrapunctus 18 and also back to number 8 (see above under B-A-C-H). At measure 158, the main theme and its inversion are again, as in Contrapunctus 5, united in absolute simultaneity, this time without the concealment that marked this event in the earlier movement.

Contrapunctus 11 opens with a fugue exposition on the main theme, with passing tones and altered rhythmically as in Contrapunctus 8. The opening rhythm is the same as that of Contrapunctus 10. Following a cadence in measure 27 there is a loosely constructed fugal exposition on the two new themes: the inversion of the first theme of Contrapunctus 8 and a chromatic theme filling out the interval of a fifth. These themes are joined by a motive taken from Contrapunctus 3. This section ends with a cadence in measures 70–71. The next section is a short fugal exposition on the inversion of the opening theme, in which the motive borrowed from Contrapunctus 3 is also used. In measure 89 there begins a long, rather freely constructed fugue based primarily on the new eighth-note theme that incorporates the B-A-C-H motive. All of the thematic material that has been introduced so far in this movement participates in this section, which lasts until measure 158. At this point The Art of the Fugue theme and its inversion enter simultaneously in the soprano and alto, and then in the tenor and bass. The remaining thirty bars are concerned with a limitlessly joyous uniting of themes in all possible combinations. The main theme appears for the last time (in this movement) in the soprano in measure 180. The last note of this theme is also the last note of Contrapunctus 11 (see Example 16 a–j for all the thematic material of Contrapunctus 11).

At this point, power, grandeur, complexity, repetition, development, and intensity have gone as far as they can. The next four movements are relatively short, quiet, and simple, though not any less expressive. They are strict canons in two voices. Contrapunctus 12 is a canon at the octave based on an ornamented version of the main theme in triplet sixteenth notes (Example 17). It is a good-humored, even humorous, piece. Contrapunctus 13 is somewhat darker in mood. It is a canon at the twelfth, based on a close variant of the main theme (Example 18). Its rhythmic interest comes from an alternation between triplets and duple rhythms. Contrapunctus 14 is a canon at the tenth, based on a completely straightforward version of the inverted main theme. This is accompanied by a quiet, rather wistful triplet figure (Example 19). Contrapunctus 15 is a canon by augmentation in contrary motion. That is, the second voice enters with an inversion of the first voice, at half speed. The second voice falls progressively farther behind, and by the end it has presented less than half the musical material of the first voice. As if to be fair, Bach has arranged the work in two sections, with the upper and lower parts changing roles halfway. This canon is based on a chromatically altered version of the main theme (Example 20, p. 24). The effect is rather weird, both disquieting and humorous.

Contrapunctus 15 reintroduces a principle of inversion that was so important through number 11. Contrapuncti 16 and 17 carry this principle farther than before. Each is in two sections, and, in each case, the second section is a note-by-note inversion of the first. These are contrapuntal tours-de-force of the highest order, all the more so since in both cases the inversions sound just as natural, spontaneous, and free as the original forms. Contrapunctus 16 is in three voices, and based on a triplet version of the main theme, reminiscent in different ways both of Contrapunctus 9 and of Contrapunctus 12 (Example 21, p. 24). The style is that of a moderately lively trio sonata movement. Contrapunctus 17 is a four-voice fugue based on a version of the main theme that is identical to the original version (Example 1), except that it is in triple time (Example 22, p. 24). The texture is very thick. With many voice crossings, the contrapuntal nature of the work is somewhat obscured. The mood is dreamy and brooding. In measure 21 a new version of the subject enters (Example 23, p. 24), which is derived from a short countersubject figure (Example 24, p. 24), which was, in turn, derived from the last four notes of the subject itself. Each half of the work ends with a flourish in sixteenth notes.

Contrapunctus 18, the last movement of The Art of the Fugue, is a monumental work, surpassing even Contrapunctus 11 in scale, and equaling it in expressive power. It was left incomplete. Bach died either before he wrote the ending down or before he was even able to conceive of an appropriate ending to a work of this stature. In its present form, the movement has 239 measures. Any formally adequate continuation and ending would bring the length of the work to over four hundred measures, making it very likely the longest fugue from the Baroque era. This, however, was not to be.

The movement opens with a fugue in four voices on an apparently new theme (Example 25). The first four notes of this theme, however, are identical to the first four notes of the main theme (with passing tones). After the fourth note the theme doubles back on itself, in an allusion to the mirroring process whereby the previous two movements were created. Thus this theme is a version of the first half of The Art of the Fugue theme. This new theme can also be derived directly from the second half of the first subject of Contrapunctus 8 (see Example 10). Furthermore, the very first two notes of Contrapunctus 18 are an exact echo of the last two notes of Contrapunctus 17; and the whole of Example 25 is essentially the same as the flourish that ended the first half of that movement. The fugue built on this subject—one of extraordinary grandeur—lasts 115 measures. It overlaps with a new fugue on a new subject (Example 26). This subject is derived closely from a theme that was introduced briefly in Contrapunctus 10 (see above and Example 15). It also includes, on strong beats, all of the notes of the main theme that are missing from the first theme of Contrapunctus 18: it completes that theme. This fugue is livelier in character than the first fugue. The first theme does enter, however, five times as a kind of cantus firmus. The last two such entries are in stretto with each other. This fugue ends in measure 193. A third fugue now begins, on the B-A-C-H theme in the simplest, most straightforward possible form (Example 27). This section brings a return to the majestic quality of the first section, but it is much shorter. In measure 233 it gives way to a return of the second subject in (partial) stretto with itself. This is joined one measure later by the first theme in the bass, and, one measure later still, by the B-A-C-H theme in the tenor. The three themes are united; the main Art of the Fugue theme is present only by implication or by transfiguration. At this moment the work ends. There is no cadence, no resting place. It ends as abruptly as could be. We are challenged to accept that the meaning, quality, and completeness of a work of art, as of a human life, do not come from the timing or nature of its end, but from what it is and what it does.

Colin MacKnight is a C. V. Starr Doctoral Fellow at The Juilliard School, New York City, where he also received his bachelor’s and master’s degrees. He is in the studio of Paul Jacobs and is working on his dissertation entitled “Ex Uno Plures: A Proposed Completion of Bach’s Art of Fugue.” He currently serves as associate organist and choirmaster at the Cathedral of the Incarnation in Garden City, Long Island.

A frequent competition prizewinner, MacKnight holds the Fellow and Choirmaster certificates from the American Guild of Organists (having won the prize for top score for the latter) and is a member of The Diapason’s “20 Under 30” Class of 2019. Upcoming performance highlights include recitals in Ingelheim, Germany; Kingston, Jamaica; and at Saint Matthew’s Cathedral in Washington, DC. Colin MacKnight is represented in North America by Karen McFarlane Artists, Inc. For more information, media, and a calendar of performances, visit colinmacknight.com.

“Miss no opportunity to practice on the organ; there is no instrument that takes such immediate revenge on the impure and the careless, in composition as well as in the playing, as the organ.”¹ This description from Schumann was likely referring to the organ’s ability to execute—one might even say affinity for—complex counterpoint. It is only fitting then that his only organ work would be a set of six fugues on Johann Sebastian Bach’s surname, a homage to music’s greatest contrapuntist, one of Schumann’s principal influences, and the composer who dominates the organ repertoire to a degree that no other composer dominates any other repertoire. Schumann stated, “What art owes to Bach is to the musical world hardly less than what a religion owes to its founder.”² He also acknowledged the influence that Bach exerted on his own music; Schumann’s compositional style was unusually motivic, and he attributed his disdain for what he called “lyric simplicity” to his study of Bach and Beethoven.³

In German musical parlance, B is B-flat and H is B-natural, allowing one to turn Bach’s surname into the motive B-flat, A, C, B-natural. By composing a set of fugues based on the theme
B-A-C-H, Schumann was participating in a long tradition of composing pieces (particularly for the organ) that include the B-A-C-H motive.

This tradition began with Johann Sebastian Bach himself and continues to the present. Two notable examples of Bach encrypting his own name occur in the Toccata in F Major for organ, BWV 540i (in transposition), and Contrapuncti 8, 11, and 14 of Die Kunst der Fuga, BWV 1080. Felix Mendelssohn was the next composer of a substantial body of organ music to encrypt the B-A-C-H motive into one of his pieces. In measure 56 of the first movement of his Sonata IV in B-flat from the Six Organ Sonatas, op. 65, he prominently includes the B-A-C-H motive in the pedal, transposed down a whole-tone (Example 1).

Schumann was close friends with Mendelssohn and even wrote a glowing review of the organ sonatas in his journal, Neue Zeitschrift für Musik, so Mendelssohn’s encryption may not have been unnoticed by him. Perhaps it is not coincidence that Schumann composed his set in 1845—the year in which Mendelssohn’s organ sonatas were published.⁴ In addition to Mendelssohn’s organ sonatas, Mendelssohn’s Six Preludes and Fugues for Piano, op. 35, and Three Preludes and Fugues for Organ, op. 37, were almost certainly strong influences on Schumann’s B-A-C-H fugues.⁵

Schumann was, however, the first composer to write a large work based on this theme. This proved to be influential; other composers who would later write substantial organ works based on B-A-C-H include Franz Liszt (Prelude and Fugue on the Name B-A-C-H, S. 260, versions of which exist for piano and organ), Sigfrid Karg-Elert (Passacaglia and Fugue on B-A-C-H, op.150), Max Reger (Fantasy and Fugue on the Name B-A-C-H, op. 46), Ernst Pepping (Three Fugues on B-A-C-H), etc. Schumann’s set is the longest of these works.

The year 1845 is often called Schumann’s contrapuntal year because of his and Clara Schumann’s intense study of counterpoint, resulting in such compositions as the Six Studies in Canonic Form, op. 56, and Four Sketches, op. 58, both for pedal-piano; Six Fugues on B-A-C-H for organ, op. 60; and Four Fugues for piano, op. 72.⁶ During this year of “Fugenpassion,”⁷ to use Schumann’s own term, they studied Friedrich Wilhelm Marpurg’s Treatise on the Fugue and Luigi Cherubini’s A Course of Counterpoint and Fugue.⁸ Schumann also studied counterpoint intensely from 1831 to 1832 under Heinrich Dorn and from 1836 to 1838, a period which yielded more contrapuntally complex and rich works such as Kreisleriana.⁹

Schumann finished the first fugue on April 7 and the second on April 18.¹⁰ Soon after the second fugue was completed, a rented pedal-piano arrived at the Schumann house.¹¹ This would have been a useful tool for composing the Sketches, Canonic Studies, and Fugues, but, surprisingly, this is also around the time Schumann began to eschew the use of the piano as a compositional tool.¹² Perhaps, then, the pedal-piano was mainly used to assist in pedal-writing.

After the second fugue, progress on the set slowed for a variety of reasons including illness, work on the latter movements of the Piano Concerto in A Minor, op. 54, and organization of an orchestral concert series in Dresden.¹³ By the end of September, Schumann had drafted the third, fourth, and fifth fugues¹⁴ and in late November, Schumann completed the set.¹⁵ The influence of the B-A-C-H project can also be seen on his next opus, Symphony No. 2 in C Major; the second trio of the scherzo uses a theme beginning with the B-A-C-H motive, and the adagio contains a fugato (Example 2).¹⁶

Schumann’s set of six fugues is in many ways similar to and likely inspired by Bach’s The Art of the Fugue since both are thorough explorations of the contrapuntal potential of one musical idea. Schumann was, however, somewhat disdainful of The Art of the Fugue for being excessively cerebral.¹⁷ Part of this is probably due to what Schumann may have perceived as a lack of variety in The Art of the Fugue. Since The Art of the Fugue is based on a single subject and the Schumann B-A-C-H fugues are only based on a motive, Schumann has considerably more flexibility in his thematic material. (Interestingly, this is not unlike Bach’s use of the B-A-C-H motive in The Art of the Fugue. In Contrapunctus 8, the motive is masked with repeated notes and inversion; in Contrapunctus 11, he un-inverts it but retains the repeated notes. It is not until Contrapunctus 14 that Bach plainly reveals the motive, a technique of which Schumann surely would have been proud.)

Schumann, like Bach, derives several distinct fugue subjects from the B-A-C-H motive. The first, third, and sixth fugues, for example, all plainly feature the motive as the main substance of the subject. The fifth fugue, however, treats it just as the starting point for further elaboration. The fourth fugue also uses the motive overtly but changes its contour by leaping down a sixth from A to C, instead of up a third. By modifying and developing the theme, Schumann reveals and incorporates one of his favorite compositional genres: the character piece. John Daverio describes this important difference between fugues and character pieces by saying, “If the essence of a fugue is a fixed subject, then that of the character piece is the transformation of an eloquent motive.”¹⁸ By combining elements of these two genres, Schumann is not just taking a neo-baroque diversion but is “updating” the fugal form to include the most modern musical trends.

Each of Schumann’s fugues also has specific tempo and dynamic indications to contrast the movements. The work is framed by two large accelerando, crescendo fugues; the second fugue is a virtuosic allegro; the third, serene and lyrical; the fourth, a more austere study; and the fifth, a charming scherzo. In this way, the work is not just a compilation of fugues but also a suite of complementary movements. Schumann also provides tonal variety by including G minor and F major movements into an overarching B-flat major—something The Art of the Fugue does not do despite its much greater length. Additionally, there is greater variety of texture in Schumann’s set than The Art of the Fugue. Schumann is not averse to devolving to homophony as he does in all but the third fugue. He also frequently composed in what looks like a “lazy” five-voice texture by writing five-voice expositions but not maintaining a strict five-voice texture (until the last fugue). This is, however, more likely another attempt at textural variety than contrapuntal ineptitude on Schumann’s part; by moving the bass line between the pedal and left hand, he is varying the sound and texture even while maintaining the same number of voices.

As previously stated, the first fugue is an accelerando, crescendo fugue. The quintessential example of this, and likely inspiration for Schumann, is the first movement of Mendelssohn’s Sonata No. 3 in A Major from his Six Organ Sonatas, op. 65. Schumann’s subject comprises two bars, the first of which is the B-A-C-H motive plainly stated, and the second of which includes the B-A-C-H motive in retrograde in its first, second, fifth, and sixth pitches (Example 3). At the point at which Schumann indicates to begin crescendoing and accelerating, he combines the B-A-C-H motive (or a variant thereof) in the pedal with a two-voice stretto of the diminished form of the subject in the manuals (Example 4).

The piece builds to a very exciting homophonic climax, particularly when the performer has accelerated enough (Simon Preston starts at 92 beats per minute and comes close to doubling the tempo, reaching 168 beats per minute at the fastest), with double-pedal before a five-bar coda that returns to a polyphonic texture, although without the B-A-C-H subject. Because of the lower register and reduction in texture, it is not uncommon to decrescendo through the coda, even though this is not indicated in the score.

The second fugue is the allegro movement and the most unabashedly virtuosic. It is also the only fugue in a triple meter, a trait of which Schumann takes full advantage through the use of hemiola. Its subject begins with a quick dotted B-A-C-H before beginning a sequence that is almost certainly taken from the fugue from Bach’s Toccata and Fugue in D Minor for organ, BWV 565. Schumann would have known this work from Mendelssohn’s famous Bach recital at the Thomaskirche in Leipzig, which he enthusiastically reviewed in his Neue Zeitschrift für Musik (Examples 5 and 6).¹⁹

In measure 48, Schumann combines the subject with its augmentation in the pedal. Then in measure 74, there is a rest for performer and listener alike when Schumann quietly strettos the augmented B-A-C-H motive with occasional interruptions from the BWV 565 motive. (These fragments of the BWV 565 motive make the connection to Bach’s fugue even more obvious.) This motive gradually takes over the texture while crescendoing until the piece devolves into a virtuosic toccata, complete with double-pedal, arpeggios, octave doublings, and no hint of B-A-C-H. He then briefly alludes to the quiet B-A-C-H stretto passage again—this time with neighbor-tones on the B-flat—before a passage of triumphant homophony. There is one more fugal interruption before the chorale-style writing returns and a hemiola passage with sforzando chords every two beats. The movement ends with a coda over a B-flat pedal that continues to use the BWV 565 motive while eschewing the B-A-C-H motive, a final confirmation of this fugue’s inspiration.

The third fugue is the only one that is entirely quiet; there is a piano indication at the beginning with the description “Mit sanften Stimmen”—with gentle stops—and no further performance instructions. This is the only fugue that begins in two voices with the counter-subject present from the beginning. It is also the simplest and technically easiest fugue. The B-A-C-H theme only occurs in one form, and the emphasis is on lyricism rather than intellect or virtuosity. It is also the only fugue in a minor key, G minor, but ending in a tranquil G major.

The fourth fugue is the most austere and perhaps the least accessible of the set. This is clear from the outset when Schumann uses a jagged version of the B-A-C-H motive; instead of an ascending minor third between A and C, he writes a descending major sixth, meaning the subject outlines a pungent diminished octave. This is one of Schumann’s cleverest techniques: to utilize the B-A-C-H motive as a collection of pitch classes with no specific contour rather than a traditional theme with a set shape.   

Schumann further adds to the complexity and austerity of this movement by introducing the retrograde of B-A-C-H for the first time and immediately combining it with the subject’s normal form. The retrograde form of the subject almost always has staccato markings on the first and third notes to draw attention to itself, retrograde being among the more obscure contrapuntal techniques (Example 7).

This movement continues to use retrograde pervasively and eventually transitions into loud chordal writing with flashy pedal scales. Like the second fugue, it alternates between passages of strict counterpoint and homophony. The passagework and homophony bring what was an ascetic contrapuntal exercise—perhaps worthy of Schumann’s own questionable criticism of The Art of the Fugue—to a dramatic and exciting close. The climax in measures 96 to 99 is particularly thrilling and one of the highlights of the whole work, as if to apologize for the trials through which he has just put the listener.

As an additional conciliatory gesture, Schumann placed his charming scherzo-fugue next. The presence of a scherzo—an unusual template for a fugue—in this set is further proof that Schumann conceived this set as a suite of complementary pieces and not just miscellaneous movements based on the same theme. The fifth fugue is the only movement in F major and the shortest of the set. This movement’s subject begins with a fleeting B-A-C-H that sounds almost like a perfunctory after-thought—or pre-thought. It is a refreshing relief after the previous four movements which all employ the B-A-C-H motive so plainly.

Nevertheless, the scherzo is still a contrapuntal tour de force. It includes augmentation, inversion, retrograde, and retrograde augmentation, and combines the augmented form of the subject with the original form and the augmented retrograde form with the plain retrograde, an astonishing amount of artifice for a movement of less than three minutes. The technique, however, never hinders the charm. This fugue demonstrates well Schumann’s outlook on fugal composition: “Anyway, this will always be the best fugue the public for instance regards as a waltz by Strauss—in other words, where the artificial rootage is covered like the roots of a flower so that we can see just the flower.” For Schumann, artifice and contrapuntal ingenuity were always subservient to beauty and emotion.²⁰

The sixth and final fugue, the longest of the set, is another accelerando, crescendo fugue. It is also the only double fugue in the set, giving it more in common with the first movement of Mendelssohn’s Sonata No. 3 in A Major, op. 65, which, as previously mentioned, is also a crescendo, accelerando, double fugue. The three-bar subject begins with the B-A-C-H motive and ends with a short descending scale that is the source of the second subject. The first section also prominently features a five-note motive as a counter-subject, beginning in measure 16. This motive, intentionally or not, sounds like a quotation of the main motive in Bach’s chorale prelude on Valet will ich dir geben, BWV 736. The descending scale that closes the first subject becomes the second subject in measure 58 and begins a new section that is marked Lebhafter and più forte. This section does not use the B-A-C-H motive but instead develops the second subject, including stretto and inversion, and uses the same five-note countersubject.

In measure 95, Schumann finally combines the subjects and countersubject signaling the third and final part of this fugue, the only instance of strict five-part counterpoint in the entire set (Example 8).

The strict counterpoint continues until measure 116 when Schumann switches to grand, fortissimo, chorale writing with a few quasi-contrapuntal interruptions. The homophonic texture allows him to compose abrupt and distant modulations that are unusual in strict fugal textures: most notably, the modulation to G major in measure 139 and back to B-flat in measure 142. As in the second fugue, Schumann eventually eschews the B-A-C-H motive (after measure 144 in the sixth fugue), choosing instead to close the entire work not with either of the subjects but with the counter-subject.

As previously mentioned, Schumann “updates” the fugal form by incorporating elements of the character piece, which he does by developing and transforming the B-A-C-H motive. By examining the different transformations of B-A-C-H, as well as other motives, one can see certain relationships between movements beyond the obvious thematic unity of a monothematic work. Specifically, the movements can be organized into three related pairs: movements one and four, two and three, and five and six.

The first fugue is related to the fourth by merit of the fact that it utilizes a countersubject, itself derived from the B-A-C-H motive, which prominently features leaps of sixths, the same interval that is so characteristic of the fourth fugue’s subject (Example 9). 

While this relationship by itself may seem somewhat tenuous, Schumann retroactively confirms it with what is probably the most shocking harmony of this fugue: the false ending at measure 60, in which an F dominant-seventh chord resolves to a secondary-dominant ninth of the subdominant. What sounds like a dramatic change in register is actually the jagged contour of the fourth fugue’s subject in the soprano! The soprano and tenor lines also continue to prominently feature leaps of sixths through the coda of this movement (Example 10).

The second and third fugues are also connected by an allusion in the second fugue to the third fugue’s subject. In measure 48 of fugue two, the B-A-C-H motive appears in the pedal in augmentation with three extra notes that together with B-A-C-H will form the third fugue’s subject (Examples 11 and 12).

Just like in the first fugue, the relationship between fugues two and three may seem weak, but Schumann once again confirms it by developing the tail of the third fugue subject in the tenor and bass voices later in the second fugue (measure 135) (Example 13).

The connection between the fifth and sixth fugues is perhaps the most obvious. The fugue subject of fugue five has a five-note figure that occurs in the second (C, B-flat, A, B-flat, C) and third (G, F, E, F, G) measures of the subject (Example 14).

Schumann later inverts this theme, so it also occurs in an up-down shape. The inverted form of this motive (the aforementioned quotation of Valet will ich dir geben, BWV 736) then becomes the countersubject to the final fugue and is combined with both of the last fugue’s subjects. It is introduced in the first exposition in measure 16 and is combined with both subjects in measure 95 (see Example 8).

On the surface, nothing could have been more conservative in nineteenth-century music than a set of six fugues for organ. Further examination reveals, however, how progressive Schumann’s B-A-C-H fugues were. They were likely the first set of pieces to be based entirely on Bach’s name, they constitute a complementary “suite” of fugues—not just a collection of movements—and there are connections and developments between the movements that foreshadow Brahms’s developing variation. Schumann himself wrote, “I worked on this set for the whole of last year in order to make it somewhat worthy of the exalted name it bears; [it is] a work that will, I believe, long outlive my other works.”²¹ His prediction that his reputation would be based largely on these fugues did not prove to be accurate, but it was surely right of him to hold them in such high regard.

Notes

1. Helga Scholz-Michelitsch, “Robert Schumann and the Organ,” trans. Susanne Weber, The Franz Schmidt Organ Competition, http://orgelwettbewerb.kitz.net/Franz-Schmidt-OrganCompetition/Robert_Schumann.html. 

2. Eric Frederick Jensen, Schumann (New York: Oxford University Press, 2001), 145.

3. Ibid., 145–146. 

4. Scholz-Michelitsch, “Robert Schumann and the Organ.”

5. John Daverio, Robert Schumann: Herald of a “New Poetic Age” (New York: Oxford University Press, 1997), 308.

6. Daverio, Robert Schumann, 306.

7. Ibid., 307. 

8. Scholz-Michelitsch, “Robert Schumann and the Organ.”

9. Daverio, Robert Schumann, 306.

10. Ibid., 307.

11. Ibid.

12. Jensen, Schumann. 284.

13. Daverio, Robert Schumann, 307.

14. Ibid.

15. Ibid., 308.

16. Jensen, Schumann, 289.

17. Ibid. 144.

18. Daverio, Robert Schumann, 309.

19. Scholz-Michelitsch, “Robert Schumann and the Organ.”

20. Ibid.

21. Daverio, Robert Schumann, 308.

Bibliography

Daverio, John. Robert Schumann: Herald of a “New Poetic Age.” New York: Oxford University Press, 1997.

Jensen, Eric Frederick. Schumann. New York: Oxford University Press, 2001.

Scholz-Michelitsch, Helga. “Robert Schumann and the Organ.” Translated by Susanne Weber. The Franz Schmidt Organ Competition. http://orgelwettbewerb.kitz.net/Franz-Schmidt-OrganCompetition/Robert_Schumann.html.

Colin MacKnight, called “a stunning player of exceptional ability” by composer and conductor Bob Chilcott, is director of music at Trinity Episcopal Cathedral in Little Rock, Arkansas. At Trinity, he oversees a music program that includes among its offerings weekly choral evensong, a concert series, and chorister and choral scholar programs. Prior to Trinity, Colin was associate organist at the Cathedral of the Incarnation, Garden City, Long Island, New York; assistant organist and music theory teacher at Saint Thomas Church, Fifth Avenue, New York City; and assistant organist at Church of the Resurrection, also in New York City.

MacKnight earned his bachelor’s, master’s, and doctoral degrees from The Juilliard School, studying organ performance with Paul Jacobs. For his doctoral dissertation, “Ex Uno Plures: A Proposed Completion of Bach’s Art of Fugue,” Colin received the Richard F. French Doctoral Prize.

A frequent competition prizewinner, MacKnight’s first prizes and scholarships include the 2019 Paris Music Competition, 2017 West Chester University International Organ Competition, 2016 Albert Schweitzer Organ Competition, 2016 Arthur Poister Scholarship Competition, M. Louise Miller Scholarship from the Greater Bridgeport Chapter of the American Guild of Organists (AGO), the 2013 Rodgers North American Classical Organ Competition, and the Ruth and Paul Manz Organ Scholarship. He also won the New York City AGO Competition and advanced to the Northeast Regional Competition, when he won first place, which led to a “Rising Star” recital at the 2016 AGO national convention in Houston. He was also a laureate in the 2016 and 2019 Longwood Gardens International Organ Competitions.

In December of 2016, Colin and composer Jon Cziner were selected for an AGO Student Commissioning Project grant, resulting in Cziner’s Fantasy Chorale, which MacKnight premiered in 2017. Colin has also earned the Fellow and Choirmaster Certifications from the AGO, receiving the prize for top Choirmaster score, and he is a member of The Diapason’s 20 under 30 Class of 2019.

Colin MacKnight is represented in North America by Karen McFarlane Artists, Inc. For more information, media, and a calendar of performances, visit colinmacknight.com.

“The governing idea of the work . . . was an exploration in depth of the contrapuntal possibilities inherent in a single musical subject.”¹ So says Bach scholar Christoph Wolff of Johann Sebastian Bach’s Die Kunst der Fuge, BWV 1080. Comprising fourteen fugues and four canons, we have inherited the work in an incomplete state; soon after Bach introduces his own musical signature as Contrapunctus XIV’s third subject and combines it with two previous themes, the music abruptly stops. There is no consensus on whether the ending was lost or simply never written; what is clear is that Contrapunctus XIV is notably missing Die Kunst der Fuge theme that unifies every other movement. In the 1870s, Gustav Nottebohm discovered that Die Kunst der Fuge theme could be combined with Contrapunctus XIV’s three preexisting themes to create a quadruple fugue.² It would then seem that this was Bach’s plan.

Because of abundant structures and patterns that Bach establishes in the existing part of the work, many aspects of Bach’s plans for the conclusion of Contrapunctus XIV can be estimated with varying degrees of confidence. Indeed, it seems likely that Bach had such remarkable compositional facility that he chose to limit himself with certain external and artificial restraints, such as the use of numerology. This is especially likely in such an abstract work, which is not guided by a text or any other limitation except conservativeness of musical material. These patterns and designs mean that ascertaining Bach’s intentions concerning the conclusion of Contrapunctus XIV requires much less guesswork and subjectivity than one might think.

One of the areas that is elucidated by the study of patterns and precedent is the intended length of Contrapunctus XIV. As we will see, Bach meticulously plans the lengths, subdivisions, and proportions of the polythematic fugues in Die Kunst der Fuge. When considering whether any proportional relationships exist in the work that could indicate how much longer Contrapunctus XIV should be, it seems most logical to start by examining the two triple fugues, Contrapuncti VIII and XI. This is because the forms of the polythematic fugues are mainly governed by the introduction and combination of themes. Contrapuncti VIII and XI are especially significant because as triple fugues they have more themes and sections, and therefore more in common with Contrapunctus XIV.

First, it is necessary to establish a methodology. The lengths of sections can be quantified by beats alone as well as by measures. (When counting beats, I always count quarter notes even if the movement is in cut time.) It is important to accurately calculate the exact number of beats in a section, as opposed to rounding to the nearest whole measure. This is not to say that Bach never rounds, but merely that exactitude can yield important insights.³ The lengths of sections can often be significant in and of themselves, and furthermore, once they have been accurately calculated, there are numerous relationships that can exist between sections.

The tools that seem most likely to be revealing are those of division and subtraction: to divide the lengths of sections by each other (larger by smaller or smaller by larger), or to subtract a shorter section from a longer one. We will find that many of these relationships have numerological significance that indicate intentionality behind their proportional designs.

Numerology and gematria

According to Encyclopedia Britannica, gematria is “the substitution of numbers for letters of the Hebrew alphabet, a favourite method of exegesis used by medieval Kabbalists to derive mystical insights into sacred writings or obtain new interpretations of the texts.”⁴ This concept is naturally not limited to Hebrew, although it is more easily applied to other alphabetic languages (i.e., letters represent sounds) than those that use syllabic (i.e., characters represent syllables or moras, e.g., Japanese) or logographic (i.e., characters represent words, e.g., Chinese) writing systems. Gematria and numerology are closely related, but not synonymous. All gematria might be called a subset of numerology, but not all numerology constitutes gematria. Specifically, gematria is a compartment of numerology based on assigning numerical values to letters to encode meaning.

In Appendix 1 of her book Bach and the Riddle of the Number Alphabet, Ruth Tatlow lists 33 different number alphabets. For the purposes of Bach study, the most common number alphabet is what Tatlow calls “Latin natural-order: variant 1,” shown in Figure 1.⁵

The concept behind this number alphabet is simple, but there are two important details to note: the omission of the letters “J” and “V.” Consequently, this number alphabet goes up to 24, not 26. This is particularly significant when Bach’s gematric designs incorporate his first initial, which notably is missing from this alphabet. Up until the eighteenth century, the Roman letter “I” was used to represent both the vowel sound normally associated with that vowel as well as the sound associated with the consonant “J.” Likewise, “U” stood in for “W.”⁶

This avenue of exploration in Bach’s music came to prominence in 1947 with the publication of four volumes about Bach’s cantatas by Friedrich Smend. One of Smend’s primary observations is the prominence of the number 14, now commonly associated with Bach.⁷ This number’s significance is derived from its gematric value (Figure 2).

In her book Bach’s Numbers, Ruth Tatlow points out the astonishingly serendipitous detail that the numbers corresponding to B, A, C, and H (2, 1, 3, and 8) also correspond to Bach’s birthday, March 21, 1685. Bach would have numbered dates in the order day, month, year, so his birthday would have been numerically represented as 21-3-85, if one omits the first two digits of the year.⁸ (The concluding “5” is not a part of the numerical parallel between Bach’s name and birthday.) Another number of great importance to Bach is 41: the retrograde of 14 and the sum of the letters J, S, B, A, C, and H (Figure 3). The last gematric sum of significance in this study is 55: the sum of 14 and 41. The sum of the digits one and four is also five, so another conception is that 55 represents the sums of the constituent digits of 14 and 41, placed side by side.

These three numbers appear frequently in Bach’s oeuvre. While it is outside the scope of this disquisition to catalog such instances in Bach’s broader output,⁹ it is worth noting certain appearances of these numbers in Die Kunst der Fuge. The following is a non-comprehensive list, most of which are cited from Indra Hughes’s dissertation:

• There are 14 contrapuncti. Additionally, a preliminary version of Die Kunst der Fuge, dating from the early 1740s, had 14 total movements.

• Bach likely intended to submit the work to Lorenz Christoph Mizler’s Correspondierende Societät der musicalischen Wissenschaften, an epistolary forum for the advancement of music theory. Bach waited to be admitted into this society until he would be its 14th member and commissioned the famous Hausmann portrait to commemorate his admittance—a portrait in which Bach is holding the last of his 14 Goldberg canons.¹⁰

• The first two notes of Die Kunst der Fuge theme, D and A, gematrically correspond to the numbers 4 and 1.11 The same can also be said of the first two and last two notes of Contrapunctus XIV’s first subject.

• The gematric sum of the notes comprising the original twelve-note version of Die Kunst der Fuge theme is 55. (D + A + F + D + C + D + E + F + G + F + E + D = 4 + 1 + 6 + 4 + 3 + 4 + 5 + 6 + 7 + 6 + 5 + 4 = 55)

• Contrapunctus II has 14 subject entries.¹²

• Starting in Contrapunctus V, the regular form of Die Kunst der Fuge subject has 14 notes, due to the addition of two passing tones.¹³

• The B-A-C-H motive appears in measures 40–41 of Contrapunctus V (Example 1).¹⁴

• Contrapunctus VI has 14 inverted entries.¹⁵

• Between the two double fugues (Contrapuncti IX and X), there are 14 double combinations.

• The B-A-C-H motive occurs in the alto and soprano voices of measures 40–41 of Contrapunctus X (Example 2).

• Section 1 of Contrapunctus XIV is 114-1⁄4 measures long, a mixed number with numerous allusions to 14 and 41.

• Contrapunctus XIV’s second subject has 41 notes.¹⁶

• The B-A-C-H motive is finally plainly revealed in the 14th fugue.¹⁷

• One additional gematric detail is worth consideration, although it does not use the numbers 14, 41, or 55. Bach titled the work Die Kunst der Fuga, somewhat peculiarly eschewing the German “fuge” in favor of the Italian “fuga.” Anatoly Milka has theorized a numerological explanation for this choice: the gematric sum of the letters in “Johann Sebastian Bach” is 158, which is equal to the gematric sum of the letters in “Die Kunst der Fuga.” If, however, Bach had retained the German “fuge,” the gematric sum of the work’s title would have been 162, and would therefore not have matched the gematric sum of his full name. Additionally, the sum of the numbers 1, 5, and 8 is 14.¹⁸

There are also those who cite certain gematric elements of the final measure as evidence that Bach intentionally left the work incomplete. First, the digits of the final measure—2, 3, and 9—add up to 14. Importantly, this is also the exact number of measures in the Canonic Variations on “Vom Himmel hoch da komm’ ich her,” BWV 769, for organ, another work dating from the late 1740s that Bach submitted to Mizler’s Correspondierende Societät der musicalischen Wissenschaften. Additionally, 2 + 39 = 41 and 23 – 9 = 14.

The remaining gematric details are more easily seen in the autograph, shown in Figure 4. (Note that the top staff is in soprano clef.)

• The final complete measure (measure 238) has 14 notes.

• The final two bass notes are A and D, whose gematric equivalents are 1 and 4.¹⁹ (These notes are enclosed in a black box.)

• The final harmony consists of two Ds and an F. When these pitches are converted into numbers, D is 4 and F is 6, making the gematric sum of this chord’s notes 14 (4 + 4 + 6 = 14). (These notes are enclosed in a red box.)

• In the final measure, seven of the tenor’s eight notes are on the top staff. Only the antepenultimate note, a lone A, is transferred to the bottom staff. The result is that the only two notes in the bottom staff of the last measure are D and A, whose gematric equivalents are 4 and 1.²⁰ Bach’s decision to put the A on the lower staff seems fussy unless it was done to create a gematrically significant design. (These notes are enclosed in a blue box.)

Contrapunctus VIII

Returning to the subject of proportion, let us begin with Contrapunctus VIII—a three-voice triple fugue. In Die Kunst der Fuge’s polythematic movements (i.e., Contrapuncti VIII through XI), new sections are generally delineated through the introduction of new themes. Contrapunctus VIII, however, has one additional section because Bach delays combining all three themes. All three themes are in play in Section 3 but are not presented in triple combination until Section 4. Following that model, Section 1 begins with the introduction of Theme 1 in measure 1 (Example 3), a new theme to the work that gradually descends by one octave over the course of four measures. As previously mentioned, it is important to note that Contrapunctus VIII begins with a half rest, so Section 1 does not begin until the third quarter of measure 1.

Section 2 begins on beat three of measure 39 with the introduction of Theme 2, an inverted form of B-A-C-H utilizing a repeated-note motive, in immediate combination with Theme 1 (Example 4).

Finally, Section 3 begins on beat two of measure 94, where a new form of Die Kunst der Fuge theme, characterized by downbeat quarter rests and composed mainly of quarter notes, enters.²¹ Finally, Section 4 begins on beat three of measure 147 with the first of five triple combinations. Section 1 is 38 measures or 152 beats, Section 2 is 54.75 measures or 219 beats, Section 3 is 53.25 measures or 213 beats, and Section 4 is 41 measures or 164 beats. For ease of reference, Table 1 outlines Contrapunctus VIII’s four sections.

From the chart, one can see that there is a general arch form; the two inner sections are larger than the outer two, the inner sections are almost the same length, and the outer sections are also close in length.

Keeping in mind that the most significant numbers in Die Kunst der Fuge are 14 (the sum of the letters
B + A + C + H), 41 (the sum of the letters J + S + B + A + C + H and the retrograde of the number 14), and 55 (the sum of 14 and 41), these are the numbers that would most clearly indicate intentionality in Bach’s proportional scheme. For this reason, the first step is to look for those numbers (14, 41, and 55) in Table 1. Instantly, one can see that Section 4 is exactly 41 measures. Slightly less obvious is the fact that Section 2 is almost exactly 55 bars. Another potentially significant detail is that the first triple combination occurs in measure 147—a measure beginning with the number 14 and whose latter number is half of 14. Moreover, Section 2 is exactly 55 beats longer than Section 4, and Section 3 is 213 beats long, corresponding to the letters B - A - C.

Finally, and most importantly, sections 2 and 3 are both almost exactly 1.4 times longer than Section 1. Section 3 is slightly closer to this proportion (213 / 152 ≈ 1.4013 . . .) than Section 2 (219 / 152 ≈ 1.4407 . . .). Even more incredible, the proportion between sections 1 and 2 can be rounded to 1.441, thereby combining the numbers 14 and 41! While such a specific number may sound far-fetched, this will not be the last time we see this level of detail, or indeed this exact proportion.²²

Contrapunctus XI

We now turn to Contrapunctus XI—the most adventurous and dramatic movement of the work. Contrapunctus XI is a four-voice triple fugue with an additional chromatic countersubject. It is the sister to Contrapunctus VIII in that they use the same three subjects, but Contrapunctus XI inverts them or, depending on how one looks at it, uninverts them, since Contrapunctus VIII uses the inverted forms of Die Kunst der Fuge and B-A-C-H themes and Contrapunctus XI uses the rectus forms. In the ilk of Charles Ives’s cumulative form technique, in which Ives reversed the standard model of exposition and development by beginning with fragments and motives from a theme that culminates in a plain statement of the entire theme near the end of the piece,23 it is not until Contrapunctus XI that one hears these themes turned right side up (rectus), enabling one to more easily recognize them for what they are (particularly the B-A-C-H theme, which is still masked by the addition of repeated notes in Contrapunctus XI).²⁴

Contrapunctus XI is also, from a formal and proportional perspective, the most enigmatic movement because of the presence of multiple extra sections (i.e., there are more sections than there are themes). Section 1 begins on beat two of measure 1 and introduces Theme 1, the rectus form of the rhythmically altered Die Kunst der Fuge theme from Contrapunctus VIII. Section 2 uninverts Contrapunctus VIII’s first theme and introduces a prominent chromatic countersubject (Example 5).

Section 3 unexpectedly eschews all previously introduced material and simply exposes the inverted form of Theme 1 (i.e., Die Kunst der Fuge theme). Section 4 introduces the rectus form of the B-A-C-H theme in immediate combination with Theme 2. Finally, Section 5 uses all three themes and presents three triple combinations. Table 2 outlines these sections and their respective lengths.

From this table, one can make a number of crucial observations. First, Section 1 is exactly three fifths as long as Section 2 (175 x .6 = 105), a reference to the three themes and five sections, perhaps? Second, Section 3 is .417 times longer than Section 2.²⁵ Third, Section 5 begins in measure 146 (which begins with the number 14). Fourth, Section 4 is 1.46 times longer than Section 5, meaning sections 2 and 3 have the same proportional relationship as sections 4 and 5. Finally, Section 1 is 1.438 times as long as Section 3.

At this point, all the musical evidence indicates that Bach not only meticulously controlled the proportional scheme of Contrapunctus XI but went to extraordinary lengths to use proportions and durations of numerological significance. Yet there are even more astonishing features of this movement’s proportions. First, it is worth noting the relative lengths of the different sections—specifically, that there are two overlapping arches. Sections 1 through 3 form an arch of smaller-larger-smaller, and sections 3 through 5 also form an arch of smaller-larger-smaller.

The most notable feature of this movement’s proportions reveals itself when one adds together the two smaller sections in each arch. Section 1’s 105 beats plus Section 3’s 73 beats add up to 178 beats, which is very close to Section 2’s 175 beats. This on its own may just be coincidence, but an examination of the second arch (sections 3, 4, and 5) demonstrates that it is almost certainly not, since the second arch has the same property; Section 3’s 73 beats plus Section 5’s 155 beats add up to 228 beats, which is even closer to Section 4’s 227 beats.

The numbers are even more exact if one adds measures instead of beats: Section 1’s 26.25 measures + Section 3’s 18.25 measures = 44.5, compared to Section 2’s 43.75 measures. And for sections 3 through 5, Section 3’s 18.25 bars + Section 5’s 38.75 bars = 57—extremely close to Section 4’s 56.75 measures. (This is part of the reason for measuring durations in both beats and measures; what may seem somewhat inexact when measured in beats can seem much more precise when measured in measures, particularly when rounding.)

To put this in simpler terms, Section 1 + Section 3 = Section 2, and Section 3 + Section 5 = Section 4! It is much like the Fibonacci Sequence (1, 1, 2, 3, 5, 8, 13, . . .),
which adds together the previous two numbers in the sequence to produce the next number, except these numbers are out of order. (In this case, one adds outer numbers to produce the inner number, so the first and third numbers add up to the second, the third and fifth numbers add up to the fourth, etc.)²⁶

The enigma of Contrapunctus XI is Section 3. Why does Bach eschew all other themes and compose a section just for the inverted form of Theme I? After all, Theme I occurs in inversion only twice more, so it is not a particularly important thematic variant in this movement. Additionally, themes are typically treated cumulatively; Bach does not usually eschew a previously stated theme in favor of a different previously stated theme. (An example would be introducing Theme 2 and then reverting to using only Theme 1.) He will often temporarily abandon a previously established theme to introduce a new theme, but to do so for an old theme is regressive. (A typical example of the former is in Contrapunctus VIII, where Themes 1 and 2 drop out upon Theme 3’s entry but return later in combinations with Theme 3.) There are a couple exceptions in which Bach does backslide to a previous theme, but these are, without exception, because of permutational designs.

Furthermore, this section is rhythmically calmer than the surrounding sections; sections 2, 4, and 5 have fairly constant eighth-note motion, while Section 3 has more quarter-note motion. Bach has clearly assigned this section a special role. If Bach had a specific reason for doing so, one possibility is because Section 3 is the central section and links the two arches (sections 1 through 3 and sections 3 through 5). Additionally, he needed a way to clearly articulate five sections in this movement—two more than would be necessary in a triple fugue. Perhaps this was his way of accomplishing that.

So what do these analyses tell us about Bach’s intentions concerning Contrapunctus XIV? First, they demonstrate a high degree of care and intentionality in the proportional schemes of both triple fugues. For better or for worse, another key takeaway is that Bach’s proportional structures are inconsistent from movement to movement. Contrapunctus VIII is a three-voice triple fugue that has four sections that form an arch (shorter outer sections and longer inner sections), and Contrapunctus XI is a four-voice triple fugue that has five sections that can be organized into two arches.

The features that are consistent, and which will likely apply to Contrapunctus XIV, are a meticulously planned proportional design and the prevalence of the numbers 14, 41, and 55. Next, we will examine the extant part of Contrapunctus XIV to hopefully find the beginnings of a proportional structure that will shed light on the length of the movement and the sections it comprises.

Contrapunctus XIV

The challenge in an examination of the proportional relationships in Contrapunctus XIV is, of course, that the movement is incomplete. Sections 1 and 2 are complete, Section 3 is partially written, and Section 4 is missing completely. Since it takes three objects to establish a pattern (and we have only two objects in their entirety), any theories regarding Bach’s intended proportions cannot be positively proven; we do not have all the necessary information. The other complication is that if a study of the proportional relationships in Contrapuncti VIII through XI reveals anything, it is that Bach is not so simple-minded as to repeat proportional schemes between movements. This means that the principles we derived from previous proportional evolution can only broadly guide a completion of Contrapunctus XIV. What does seem likely to remain true, however, is that the numbers 14, 41, and 55 will guide the proportional schemes. Indeed, even the number of this contrapunctus appears to carry meaning.

Given that Contrapunctus XIV was clearly intended as a quadruple fugue (as proven by Gustav Nottebohm),²⁷ the most obvious division of sections would be one section per theme. (This is the same model that Bach used in Contrapunctus IX, albeit on a much smaller scale.) This would mean that Section 1 is dedicated to Theme 1, Section 2 uses themes 1 and 2, Section 3 uses themes 1, 2, and 3, and finally, Section 4 uses all four subjects—a straightforward, cumulative design.

The extant part of Contrapunctus XIV provides support for this hypothesis. In Contrapunctus XIV, the themes are treated cumulatively. Section 2 introduces Theme 2 while retaining Theme 1, eventually combining them. Section 3 introduces Theme 3 while retaining themes 1 and 2, eventually combining all three (Example 6). This is not so in Contrapuncti VIII or XI. In Contrapunctus VIII, Bach introduces Theme 3 in Section 3 but starts a new section (Section 4) to present triple combinations. Contrapunctus XI is even less straightforward; Theme 1 disappears in Section 2, and then Theme 2 disappears in Section 3. Section 4 finally uses all three themes, but not in triple combination until Section 5. The extant part of Contrapunctus XIV clearly shows a more direct approach to the exposition and combination of themes.

Now that we have established the likelihood that Bach intended Contrapunctus XIV to have a one-to-one ratio of subjects to sections, let us examine the durations and proportions of the surviving parts. The gigantic first section begins on beat three of measure 1 and goes to the downbeat of measure 115, giving it a duration of 457 beats or 114.25 measures. It seems highly serendipitous that the number of measures in Section 1 can be conveyed as 114-1⁄4—a number with numerous allusions to 14 and 41 (yet another reason to measure durations in both beats and measures). Section 2 is in a Stile antico idiom and is composed largely of quarter-note motion. Theme 1 comprises seven notes and has the interesting quality of being a melodic palindrome. (See Example 7.)

Importantly, Section 2 overlaps with Section 1 by four beats because Theme 2 enters before the other voices have completed their cadence marking the end of Section 1. Theme 2 is composed mainly of eighth-note motion and is exactly 41 notes. The increased rhythmic activity of Section 2 breaks the retrospective style of Section 1. Theme 2 is also the only theme in Contrapunctus XIV that is not presented in inversion in the extant sections (excluding Theme 4, which has not yet occurred in any version). Section 2 begins with the introduction of Theme 2 in the alto on beat two of measure 114 (another reference to 14) and continues through beat two of measure 193, making Section 2 317 beats or 79.25 measures. (See Example 8.)

Section 3 begins on beat three of measure 193 and trails off in measure 239.28 This gives the extant part of Section 3 a length of 186 beats or 46.5 measures, although these numbers may not be particularly significant as they represent only part of a section. Stylistically, Section 3 brings about another shift; the counterpoint becomes rhythmically less active, and the harmony becomes markedly more chromatic and adventurous.29 (This is partially by virtue of Subject 3’s chromatic profile and longer note values.) This marks the end of the extant part of Contrapunctus XIV. (See Table 3.)

One of the challenges of deducing Bach’s intentions is that with only two complete sections, there is not enough information to establish a pattern per se. Nevertheless, it seems likely that the sections should get shorter. As previously stated, Section 1 is truly massive; it has more measures than every movement of the work except the four polythematic fugues and takes about four minutes to perform on its own. The movement would likely feel bloated and disbalanced if there were another section even longer than this one, and indeed, Section 2 is about 30 measures shorter. A final, more subjective reason to believe that the sections will continue getting shorter is that Section 3 is clearly intensifying in both rhythmic and harmonic activity. In just 46.5 bars, it is already clearly approaching a climactic point, at which time Theme 4 will likely enter.

Bach scholar Dr. Gregory Butler of the University of British Columbia at one point theorized that each section of Contrapunctus XIV is intended to be approximately two thirds the length of the preceding section.30 He explains:

As it appears in the print in its incomplete state, this work occupies five pages. It seems clear that the finished version would have fit nicely on six pages. If we examine the relative proportions of the three extant sections of this fugue, we notice a consistent diminution in the lengths of successive sections. Moreover, section 2 (78 measures) is almost exactly two-thirds the length of section 1 (115 measures), and section 3, not quite complete, occupies forty-six measures and conceivably in its complete state would have occupied approximately two-thirds the length of section 2 (52 measures). Adhering to the same proportions, section 4 may well have occupied approximately two-thirds the length of section 3, that is, approximately thirty-four measures. This would leave approximately forty-six measures for the concluding sixth page which is exactly the average number of measures per page for the first five pages as they appear presently in the print.³¹

This theory has a number of issues. The reader may notice Butler’s use of language to indicate approximation. He rounds the sections to the nearest whole measures, and his proportional theory also relies on rounding. Furthermore, he makes mathematical errors, claiming that two-thirds of 52 bars is 34 bars when it is actually closer to 35. He then says that Section 3 needs six more measures and that Section 4 should be 34 measures—a clear sum of 40 measures—but states that this fits “exactly” with one additional page of manuscript, equivalent to 46 measures.

If there is one conclusion to be made from our examination of the proportions in the triple fugues, it is that Bach’s proportional schemes are anything but approximate. Another reason to doubt Butler’s theory is that it does not leave enough time in Section 3 to include even one more triple combination, if there is to be any connecting episodic material. Indra Hughes, whose dissertation, “Accident or Design? New Theories on the Unfinished Contrapunctus 14 in J. S. Bach’s The Art of Fugue, BWV 1080,” is one of the major contributions to Die Kunst der Fuge research, agrees that Butler’s theory is not sufficiently precise and proposes a much more specific and convincing theory.³²

While it seems likely that Bach intended each section to be shorter than the preceding one, this is not proof of a more detailed proportional relationship. Hughes, however, has discovered a feature in the autograph that indicates that the exact lengths of sections were of great importance to Bach. This detail arises from an edit that Bach made at the end of Section 1. Bach crossed out the original material in measures 111 and 112 and replaced them with three measures, thereby adding a bar. The edit can be seen in Bach’s hand in Figure 5; the two measures at the top right have been crossed out and replaced with three measures of tablature in the bottom margin. The two versions are presented in a more readable fashion in Examples 9 and 10.

Hughes argues that there is not a substantial musical difference between the first and second versions, so therefore it is likely that the reason for the edit was for Bach to correct a proportional error, an error that could be fixed by adding one bar to Section 1. In rejection of strong musical reasons for the replacement, Hughes writes the following:

It seems a curious and puzzling change for Bach to have made, and it does not seem easy to find a musical reason for the rejection of the original bars 111 and 112. If Bach had not made the change, but had left the score with those two bars unaltered, would scholars and analysts today point to them and identify them as a weak moment? Perhaps it might be argued that the new bar 111 strengthens the approach to the following cadence (bars 113–114) by the move to the subdominant that was not present in the original; and it might be argued that the new bar 113 is rhythmically very slightly more interesting than the old bar 112. But these are tiny points of argument and I am convinced that if the change had not been made, nobody would ever have criticized the old bars 111 and 112 as being weak.³³

Hughes further refines Butler’s theory by more accurately calculating the lengths of sections. Butler calculated only whole measure numbers and did not account for overlapping sections, giving him durations of 115 measures and 78 measures for the first two sections, respectively.³⁴ Hughes calculates sections 1 and 2 as being 114.25 and 79.25 measures, respectively. This means that Section 1 is approximately 1.44 times longer than Section 235—our first evidence that Contrapunctus XIV’s proportional scheme is deliberate and numerologically determined.

Hughes’s argument gains traction when one uses this proportion to calculate the length of Section 3. Section 2’s 79.25 measures divided by the hypothesized ratio of 1.44 gives a quotient of 55 measures for Section 3—the section that introduces Bach’s name as the third subject.³⁶ Now there is very little doubt that these relationships are deliberate. This also conveniently leaves time for one more triple combination before Section 3 concludes.³⁷

Hughes then calculates that Section 4 should be 38 measures (although this is a rounded figure), reaching the conclusion that the total number of missing measures is 47. While I agree loosely with most of what Hughes postulates, there are a few fallacies and inconsistencies in his calculations that must be addressed. First, Hughes concludes that the correct proportion between sections is 1.44 but provides no reason for the presence of an additional “4” in the hundredths decimal place. (The reasons for this specific number will be discussed in greater detail later.)

Second, Hughes does a fair amount of rounding while criticizing others for doing so. An example of this is his calculation that Section 4 should be “exactly 38 bars long.”³⁸ In reality, this number is not so exact; it should actually be 38.194. Of course, some rounding is necessary, but rounding to 38.25 measures (38 measures and one beat) would be closer than rounding to 38 measures even. Rounding to the nearest beat instead of the nearest whole measure would dictate that the movement ends with a downbeat quarter note. (This is assuming that Section 4 cannot begin on beat three of a measure, since Die Kunst der Fuge theme would almost certainly enter on a downbeat.)

Of course, one could get around this by ending the work with a quarter-note—perhaps with a fermata. It is perhaps, however, not a stretch to suggest that Bach might have rounded this to 38 measures so that the work could end with a whole note. This is not a detail that should be taken lightly though; to round Section 4 to the nearest measure effectively excises a measure of music from the section and the entire movement. If Section 4 is rounded to 38 measures, then the final measure (measure 286) will most likely be a whole note. If, however, Section 4 is rounded to 38.25 measures, then the final measure (measure 287) would be a quarter note with a fermata (likely sounding the same as a whole note), thereby making measure 286 the penultimate measure and freeing it up for an additional measure of cadential material.

Finally, Hughes is generally quite meticulous about the lengths of sections—and this allows him to make important contributions to proportional theory—but he makes one significant error when he notes that Section 3 begins on beat three of a measure but forgets this detail when determining where Section 3 ends. In his calculations, he incorrectly starts Section 3 on the downbeat of measure 193 instead of beat three of measure 193. This leads him to believe that Section 3 needs an additional nine measures, instead of 8.5. Truthfully, at 55 measures, Section 3 must end on beat two of measure 248, a measure whose digits add up to 14.

This means that Section 4 must either start on beat three or overlap with Section 3 by two beats, but as previously stated, it seems unlikely that Die Kunst der Fuge theme would start on beat three, making the overlap the more likely choice. To account for this overlap, one must subtract two beats from the final calculation of the intended duration of Contrapunctus XIV. This puts Hughes’s calculation off by one whole measure: two beats off for starting Section 3 too early and two beats off for not overlapping sections 3 and 4. The question of whether one should round Section 4 to the nearest beat may also potentially push Hughes’s solution off by an additional beat. Table 4 compares Butler’s, Hughes’s, and my own theories and calculations.

As previously mentioned, the problem that Hughes faces is why the proportion is 1.44 instead of simply 1.4. He excuses this concern by pointing out that the proportional relationship is consistent, if not exact, and that there are other works by Bach with proportional relationships that are close to 1.4 but not exact.

Before exploring this issue further, it is vital to remember that Bach’s ways of solving mathematical problems would have been very different from our own. Doing these problems by hand, as Bach would have done, yields three very interesting (and I believe previously undiscovered) observations. The first and most obvious is that when 114.25 is converted into a mixed number, it is 114-1⁄4, a number with several allusions to 14 and 41.39 The second comes from the fact that Bach likely would have made these calculations in improper fractions (or beats) rather than decimals to facilitate division. To take the two complete extant sections as examples, Section 1’s 114.25 bars become 457/4, and Section 2’s 79.25 bars become 317/4. If one subtracts 317 from 457, the remainder is exactly 140, another evidently Bachian number. Since we are working with quarter measures in common time, Section 1 is exactly 140 beats longer than Section 2.

The final observation comes from dividing 457 by 317. The benefit to doing this division by hand is that the solution will not include a long string of decimals that a calculator displays. It is not surprising that the first three digits of the solution are 1.44, as Hughes claims. If, however, one solves to three decimals instead of two, then the third decimal is 1, producing a quotient of 1.441. At first glance, this new solution works much better than Hughes’s. While Hughes struggled to explain why his ratio is 1.44 instead of simply 1.4, this new proportion is both more specific and more numerologically significant, as it combines the numbers 14 and 41. Furthermore, it has precedent in the work; in Contrapunctus VIII, Section 2 is 1.441 times longer than Section 1.

To determine more certainly whether this proportion is more accurate, one must test whether the extra one thousandth consistently yields results that are closer to the actual number of measures present. To start, let us take 114.25 measures and divide it by Hughes’s 1.44. This gives us 79.340. (I will consistently round to the nearest thousandth.) If, however, we divide it by my 1.441, we get 79.285—a number that is not insubstantially closer to Section 2’s actual 79.25 measures. (See Table 5.) The extra one thousandth makes a surprisingly significant difference. If we take Section 2’s 79.25 bars and divide it by 1.44, we get 55.035. If we divide it by 1.441, though, we get 54.997. Once again, this solution is slightly closer to the hypothesized 55 bars in Section 2. (See Table 6.)

The final section, however, creates problems with either theory (1.44 or 1.441). When one divides Section 3’s 55 bars by 1.44, the solution is 38.194. When one divides it by 1.441, the quotient is 38.168. (See Table 7.)

Neither of these answers is very close to a round number of measures or beats, but the latter solution is slightly lower and therefore closer to a whole measure. This may tip the scale toward the argument that Bach intended for Section 4 to be 38 measures even—not, as previously considered, 38.25 measures (which would have required the last chord to be a quarter note with a fermata). Bach probably recognized that, at some point, some rounding would be necessary in his numerological games. Bach’s genius is, after all, still bound by the laws of mathematics.

When mapping out this movement, Bach may have begun with one of two conditions: that Section 2 would be 114-1⁄4 measures or, alternatively, that Section 3, the section in which he introduces his musical signature, would be 55 measures. These numbers are both, after all, significant to Bach. He must have been quite tickled to discover that 55 x 1.441 x 1.441 ≈ 114-1⁄4, but nevertheless, he must have started with one or the other—55 or 114-1⁄4. For this reason, it may be enlightening to compare Hughes’s and my proportional theories starting from both of those numbers. Table 8 compares the two theories starting with 114.25 measures in Section 1 as the reference point.⁴⁰ Table 9 compares the proportions between Hughes’s theory and my own using 55 measures in Section 3 as the reference point.

These tables show that, regardless of which reference point one starts from, 1.441 is a consistently more accurate proportion than 1.44. Table 8 also demonstrates something else that indicates why 1.441 is the more likely solution. When one looks at the column for Hughes’s proportion, one can see that his ratio consistently gives an answer that is slightly too high. The solutions produced by a ratio of 1.441, however, are slightly too high in the cases of sections 2 and 4, and slightly too low for Section 3. This means that no other proportion can truly be said to be closer. If one uses a number any higher than 1.441, Section 3 will be less accurate, and if one uses a number any lower than 1.441, sections 2 and 4 will be less accurate. When all the answers are too high though, as in Hughes’s ratio, it is clear that the proportion can be refined.

The extra beat

The remaining issue concerning the length of Contrapunctus XIV is the length of Section 4 and whether it should be rounded to the nearest measure or nearest beat. There are a number of factors that should at least be considered. First is the purely musical issue of ending an hour-and-a-half-long work. It would be unusual to end such a work with a quarter note, but having a fermata on the note would alleviate that to a degree; the difference would then arguably be only visual. As previously mentioned, using the proportion 1.441 instead of 1.44 pushes the length of Section 4 down by a little over three hundredths (38.1679 versus 38.194), but this is an admittedly minor difference.

It may once again be revealing to consider how Bach would have done his calculations. If one uses long division to calculate the length of Section 4, as Bach must have done, then he would not have seen a long string of decimals. He could have merely solved for the nearest whole number, in which case he would have reached 38 as a final answer and been done with it. But even if he continued to solve for the first decimal, he would have seen only a “1” in the tenths place, which he still would have likely been content to round down to zero. It is not until one solves for the hundredths place and gets 38.16 that this issue arises at all. It seems likely that Bach would not have been particularly bothered by this discrepancy, if indeed he was even aware of it.

In summary, I conclude that Section 3 requires another 8.5 bars—plus filling in the remaining voices of measure 239—and that Section 4 should be 38 measures, keeping in mind that it will overlap with Section 3’s last half measure. This gives one enough time to include one more triple combination in Section 3’s remaining 8.5 bars, as well as multiple quadruple combinations and episodes in Section 4. It is not, however, so long that the drama that has been reached by measure 239 will become an anticlimax. According to this theory, the movement should be 285 measures. This is calculated by adding the lengths of the sections (114.25 + 79.25 + 55 + 38 = 286.5) and subtracting one measure for the overlap of sections 1 and 2 and another two beats for the overlap of sections 3 and 4 (286.5 – 1.5 = 285). The durations and overlaps of the four sections can be seen in Figure 6.

Large-scale numerology

Finally, let us consider the sum total of all the movements. If one counts the lengths of the mirror fugues and their mirrors, thereby doubling the lengths of the mirror fugues to 254 measures, then the entire length of Die Kunst der Fuge is 2,135 measures. The realization that my proportional theory put the entire work at 2,135 measures immediately gave me pause. This is, after all, three short of the numbers which gematrically correspond to Bach’s name: 2,138. This caused me to wonder whether Bach was also intentional about the length of the entire work, which in turn led me to test what would happen if I aimed for a sum total of 2,138 measures by increasing Contrapunctus XIV to 288 measures, rather than 285. Obviously, the durations of sections 1 and 2 are fixed and cannot be tailored. Bach left Section 3 incomplete, but my proportional theory is largely based on the numerological significance of Section 3 being 55 bars. This leaves Section 4, which according to my original theory was 38 bars. Amazingly though, adding three measures increases it to exactly 41 measures. Through the addition of just three bars, the length of Section 4 and the length of the entire work suddenly achieve great gematric meaning. The question, as always, is: was this Bach’s intention?

The real issue here is whether Bach would sacrifice the integrity of his proportional scheme to inject the entire design with greater personal numerological significance. To better judge this, we must determine the degree to which this change would affect the proportional scheme. Before doing so, however, it is worth remembering that Section 4 was already problematic. When one divides 55 (the conjectured duration of Section 3) by 1.441, the quotient is approximately 38.168, which we rounded down to 38 (after first considering whether to round to 38.25, or 38 measures and one beat). This is significantly less precise than the proportions of the previous sections (72.285 versus the actual 72.25 measures in Section 2, and 54.997 versus the conjectured 55 bars in Section 3).

When one takes Section 3’s hypothesized 55 bars and divides them by 41 bars for Section 4, the proportion is approximately 1.341. This is not terribly far off from the originally theorized proportion of 1.441. Additionally, it still ends with the numerologically significant digits “41.” It is actually quite intriguing that the two numbers only differ by one digit.

The question remains: if Bach had realized that adding three measures to Section 4 would give both the section and the entire work gematrically significant durations, would it have been worth the small concession of modifying a proportional scheme that, at this point, was already falling short?

I believe the answer is yes; Bach would have succumbed to such a temptation. In fact, it must have been the design all along. It would be quite a coincidence for Bach to see the original proportional scheme of 1.441 through to the end and only then realize that the work was three bars short of 2,138 measures. Far more likely, he began by assigning 288 measures to Contrapunctus XIV so that the sum of the work’s movements would be 2,138. Knowing that the sections would diminish in length, he then assigned 41 measures to Section 4 and 55 measures to Section 3. At this point, he switched to a more exact proportional scheme of 1.441, making Section 2 79-1⁄4 measures and Section 1 114-1⁄4 measures, the latter of which—he must have been pleased to learn—is a number with about as many references to 14 and 41 as a five-digit number can have. The length of each section can be seen in Table 10.

These sections add up to 289.5 measures, but because of a one-measure overlap between sections 1 and 2 and a half-measure overlap between sections 3 and 4, the total duration of Contrapunctus XIV will be 288 measures. Since Bach’s final bar is measure 239, Contrapunctus XIV requires 49 more measures: 8.5 more measures in Section 3 and 41 measures for Section 4, with these two sections overlapping by two beats. (See Figure 7.)

It is my hope that this document may have use outside of the narrow scope of determining the intended duration of Contrapunctus XIV; more broadly, I hope to shed a bit more light on Bach and his compositional tendencies. The topic of proportion in Bach’s oeuvre is ripe for further examination, and we have only begun to scratch the surface. By now, there is no question that Bach was an enthusiastic practitioner of Augenmusik and hid all sorts of elaborate structures in his music. The question is: will we seek them out?

Notes

1. Christoph Wolff, Bach: The Learned Musician (New York: W. W. Norton, 2000), 433.

2. Gustav Nottebohm, “J. S. Bach’s letzte Fuge,” Musik-Welt, nos. 20 and 21 (1881): 2, http://ks4.imslp.net/files/imglnks/usimg/f/fc/IMSLP348435-PMLP562864-No….

3. Counting beats in large sections can be more complicated than one might expect, particularly if the passage includes partial measures. The formula for doing so is as follows: (last complete measure – first complete measure + 1) x 4 beats + any beats from surrounding partial measures. For example, if one wanted to count the beats from beat 2 of measure 4 to beat 2 of measure 36, then one would start with 35 (the last full measure) and subtract five (the first full measure) for an answer of 30. Then add one for a sum of 31 and multiply that by four beats for a product of 124. Then add the extra three beats of measure 4 and the extra two beats of measure 36 for an answer of 129 (124 + 3 + 2 = 129).

4. Encyclopædia Britannica, s.v. “Gematria.”

5. Ruth Tatlow, Bach and the Riddle of the Number Alphabet (Cambridge, England: Cambridge University Press, 1991), 133.

6. Indra Hughes, “Accident or Design? New Theories on the Unfinished Contrapunctus 14 in J. S. Bach’s The Art of Fugue, BWV 1080” (D.M.A. diss., University of Auckland, 2006), 3–4.

7. Ibid., 4.

8. Ruth Tatlow, Bach’s Numbers (Cambridge, England: Cambridge University Press, 2015), 70.

9. For perhaps the preeminent study of this field, see Ruth Tatlow’s Bach’s Numbers. The trajectory of Tatlow’s opinion on this topic is intriguing. In her 1991 book Bach and the Riddle of the Number Alphabet, she argues against most numerological and gematric interpretations of Bach’s music. Now, after something of a Damascene conversion, she is its leading exponent.

10. Hughes, “Accident or Design?,” 18.

11. Ibid.

12. Ibid.

13. Ibid.

14. Ibid.

15. Ibid, 19.

16. Ibid.

17. Ibid.

18. Anatoly P. Milka, Rethinking J. S. Bach’s Art of Fugue, trans. Marina Ritzarev, ed. Esti Sheinberg (London: Routledge, 2017), 179.

19. Herbert Kellner, “Die Kunst der Fuga,” The Diapason 91, no. 5 (May 2000), 16.

20. Ibid.

21. In his article “Bach and Die Kunst der Fuge” (The Diapason, May 1998, pp. 15–17), Jan Overduin argues that downbeat quarter rests often symbolize death in Bach’s music.

22. As far as I am aware, the aforementioned observations were all previously undiscovered.

23. James Peter Burkholder, All Made of Tunes: Charles Ives and the Uses of Musical Borrowing (New Haven: Yale University Press, 1995), 138.

24. The notion of gradual revelation applies most aptly to the B-A-C-H theme, which occurs in only three movements: Contrapunctus VIII, in which its default form is inverted with repeated notes (i.e., most hidden); Contrapunctus XI, in which its default form is in rectus form with repeated notes (i.e., less hidden), and Contrapunctus XIV, in which it is finally stated plainly in Section 3.

25. I rounded to the nearest thousandth (0.417) so that one can see the numerologically significant part of the proportion (i.e., 0.41). If I had rounded to the nearest hundredth (0.42), its significance would have been obscured. This is not, however, a problem that Bach would have necessarily confronted. He would have solved these equations by hand, meaning he would see only as many decimals as he solved for, not the string of decimals that I see on a calculator. If he solved for only two decimals, he would have seen only “0.41.” After all, one only knows to round this number up to 0.42 if one also knows what the thousandth decimal is.

26. To the best of my knowledge, these proportional relationships were also previously undiscovered.

27. Gustav Nottebohm, “J. S. Bach’s letzte Fuge,” Musik-Welt, nos. 20 and 21 (1881): 2, http://ks4.imslp.net/files/imglnks/usimg/f/fc/IMSLP348435-PMLP562864-No….

28. Indra Hughes takes the final measure number, whose digits add up to 14, as evidence that Bach intentionally left the work incomplete and left a number of clues as to how it should be finished. See “Accident or Design? New Theories on the Unfinished Contrapunctus 14 in J. S. Bach’s The Art of Fugue, BWV 1080” (D.M.A. diss., University of Auckland, 2006), 20–21.

29. Indeed, the only other part of the work that could be said to be as harmonically daring is Contrapunctus XI.

30. Butler has since reversed this position, and now believes that the Fuga a 3 Soggetti was never intended to include Die Kunst der Fuge subject.

31. Gregory Butler, “Ordering Problems in J. S. Bach’s ‘Art of Fugue’ Resolved,” The Musical Quarterly 69, no. 1 (Winter 1983): 55, http://www.jstor.org/stable/741800.

32. Hughes, “Accident or Design?,” 84.

33. Hughes, “Accident or Design?,” 93–94.

34. Butler, “Ordering Problems in J. S. Bach’s ‘Art of Fugue’ Resolved,” 5.

35. Hughes, “Accident or Design?,” 86.

36. Ibid., 87.

37. This is because any combination will take as long as its longest theme. At six bars long, Theme 2 is the longest subject, meaning another triple combination would take at least six measures.

38. Hughes, “Accident or Design?,” 88.

39. A skeptical reader who double-checks my calculations may notice that Section 1’s 114.25 measures include the opening two beats of rest, even though I have not counted beginning rests in other movements. The original reason for this was that the calculations are much more exact if one counts 114.25 measures instead of 113.75, but this reason alone struck me as intellectually indolent. The realization though that 114.25 converts to 1141⁄4, and that counting those two beats makes Section 1 exactly 140 beats longer than Section 2 gave me some measure of peace with this decision.

40. In these calculations, it is important to start with the actual number of measures. For example, if one is determining the length of Section 3 from Section 2, then one should start from 55, not 55.035 or 54.997.

Sources

Burkholder, James Peter. All Made of Tunes: Charles Ives and the Uses of Musical Borrowing. New Haven: Yale University Press, 1974.

Butler, Gregory. “Ordering Problems in J. S. Bach’s ‘Art of Fugue’ Resolved.” The Musical Quarterly 69, no. 1 (Winter 1983): 55. http://www.jstor.org/stable/741800.

Hughes, Indra. “Accident or Design? New Theories on the Unfinished Contrapunctus 14 in J. S. Bach’s The Art of Fugue, BWV 1080.” D.M.A. diss., University of Auckland, 2006.

Kellner, Herbert Anton. “Die Kunst der Fuga.” The Diapason 91, no. 5 (May 2000): 15–17.

Milka, Anatoly P. Rethinking J. S. Bach’s Art of Fugue. Translated by Marina Ritzarev. Edited by Esti Sheinberg. London: Routledge, 2017.

Nottebohm, Gustav. “J. S. Bach’s letzte Fuge.” Musik-Welt, nos. 20 and 21 (1881): 2. http://ks4.imslp.net/files/imglnks/usimg/f/fc/IMSLP348435-PMLP562864-No….

Overduin, Jan. “Bach and Die Kunst der Fuge.” The Diapason (May 1998), pp. 15–17.

Wolff, Christoph. Johann Sebastian Bach: The Learned Musician. New York: W. W. Norton & Company, 2001.

Michael Gailit graduated from the University of Music and Performing Arts in Vienna with both performance and pedagogy diplomas in organ as well as in piano. Teaching piano at this institute since 1980, he has also conducted the organ studio at the Musik und Kunst Universität in Vienna since 1995. As church organist he served at St. Augustine’s Church, 1979–2008; in 2011 he was appointed organist at the Jesuit Church (Old University Church).

Both in his performance and teaching repertoire, Gailit includes all style areas on the base of their individual performance practices. He toured with solo recitals on both instruments in Europe as well as in North America and appeared with leading orchestras and renowned conductors. Recordings, masterclasses, invitations to juries, musicological publications, editing sheet music, compositions, arrangements, supporting the piano-organ duo repertoire, commissioned works, first performances, and finally occasional trips into the theatre and silent movie repertoire should be noted.

Particular attention was received in 1989 for the first performance of the complete piano and organ works of Julius Reubke, the performance of the complete organ works of Franz Schmidt the same year, as well as in September 2005 a series of six recitals with the trio sonatas of Johann Sebastian Bach, the organ sonatas of Felix Mendelssohn-Bartholdy, and the organ symphonies of Louis Vierne. Currently Gailit is working on a book The Enigma BWV 565, a study elucidating new answers and new questions.

Editor’s note: Part 1 of this series appeared in the June 2021 issue of The Diapason, pages 18–19.

The analysis of the first twelve measures revealed a compositional technique that can most aptly be described as metamorphosis. The list of the motives that appear in BWV 565, given in detail in the first part of this article, shows how one motive becomes the other (Example 15). In the last consequence, the mordent motive creates all others. This evolutionary style of composition appears on all levels. The mordent creates what we called a nucleus idea, which in turn creates the next two phrases. A part of the nucleus idea creates the ascending triplets, which in turn develop into the descending triplets. The closing section varies measures 1 and 2 and ends with the mordent motive.

From three to four

The second section, measures 12 through 30, continues this concept, only that the focus changes from three-note motives to four-note ones. In order to achieve a smooth transformation, the section starts with an upbeat to a trichord. The following tetrachords are rhythmically shifted in the same way, so that they appear as trichords with upbeats. The passage also prepares the fugue theme that shows a very similar structure of four consecutive tetrachords. Pitch notation of the opening phrase of the first part had assisted to discover the 3 x 3 form of the nucleus idea. Pitch notation of the current phrase now reveals a 4 x 4 structure (Examples 16 and 17). Both lines add another 4 x 4 notes as inserted repetitions of the dominant tone.

Simultaneous notation of the immanent two voices in measures 13 through 15 helps to understand the process. Edition Peters gives the original text of the earliest manuscript. Bärenreiter and Breitkopf & Härtel in their Urtext editions align measures 14 and 15 by substituting measure 14 with the octave transposition of measure 15. As we observe now, this contradicts the overall concept of continuous transformation. Measure 14 quotes the nucleus idea; measure 15 prepares the next section by changing the ending with the surprise of an eighth-note rest on beat one of measure 16 (Example 18).

In 1845–1846, the complete organ works by Johann Sebastian Bach appeared in print for the first time. This release by Edition Peters is considered a milestone, due in part to the editorial policy of Friedrich Conrad Griepenkerl (1782–1849), which abstained from any amendments. The eighth-note rest at the beginning of measure 16, however, obviously came as too great a surprise to the editor. He replaced it with a manual D31 without any reference to a manuscript source. In view of the preceding three-fold thundering D2 on the pedals, the eighth-note rest is only one of the many skillfully calculated effects in BWV 565.

The motivic substance of measures 16 through 20 consists of the tetrachord and the turn motive. Both textures, the figurations as well as the syncopated chords, follow a harmonic progression commonly known as the Phrygian cadence.2 This harmonic pattern is frequently inserted after a slow movement in order to prepare the listener for a finale. The transition from Adagio to Allegro in a voluntary by John Stanley (1712–1786)—with quite similar motives as BWV 565—demonstrates this function (Example 19).

The Phrygian cadence in BWV 565 comprises four chords consisting of four motives, two tetrachords and two turn motives (Example 20). The middle voice can count for two as the doubling of these notes in the thirty-second-note figurations suggests.

The syncopated chords appear four times, with a different pedal phrase each time (Example 21). The evolutionary concept of motive metamorphosis in BWV 565 requires abstention from the editorial practice of unification, in this case to replace phrase 2 by phrase 1. In measure 17, the bass and tenor notes are exchanged. The tenor varies the tetrachord E–D–C–B-flat in a new way by placing the first note at the end and an octave lower.

Whereas the Phrygian cadence aims to the last note in measures 16 through 18, its third and fourth appearances in measures 19 and 20 shifts the accented beat to the second note. This gives room for another chord on the last eighth-note beat in measure 20. On this beat, the new note of a G-sharp surprises; it gives the base for another diminished seventh chord. The G-sharp descends to the dissonance G-natural of the dominant seventh chord that spans over five more measures to be resolved only in measure 27.

Pitch notation shows that the run in measure 18 is a variation of measure 11, with some added notes (Example 22). It consists of a collection of the prime suspects—three tetrachords, a mordent, and a turn motive. Also, the arpeggio appears, this time following the tonic chord.

The passage ends in measure 19 with a scale comprising the complete tone material including the B-natural. This run is repeated in measure 21 without the passing notes (Example 23).

Four descending tetrachords run into a halt on the diminished seventh chord, before a prestissimo tremolo breaks loose. The toccata returns here to triplets and the mordent, whose intervals are stretched to thirds, with alternating upper and lower neighbor notes. The pedal surprisingly takes on a melodic role, worthy of the designation Recitativo. With the exception of the final cadence, the pedal lines up ten seconds, the last one repeating the first (Example 24).

Probably just a coincidence: we discover the first six notes of the pedal line in the Advent hymn, Nun komm, der Heiden Heiland (Example 25). More striking is that the pedal line cites the nucleus idea in the form of the second phrase in measure 1 (Example 26). The manual parts, note repetitions, and voice doubling omitted, consist only of seconds, which in turn are condensed tetrachords (Example 27).

The purpose

On the second page of Ringk’s manuscript, the last four staves are left blank. The fugue begins on a new third page. Ringk wrote, “Verte. Fuga.” (in Latin: Turn! Fugue.) below the final cadence on page 2. So before we turn the page and continue with the fugue, let us look at what has happened so far. BWV 565 is a work that undoubtedly attracts attention. So what was the purpose of the composition; why was it created? We need to leave the objective path of note-by-note analysis here. The meaning of a text cannot be grasped comparatively by counting the letters or discussing grammar and spelling.

Measures 16–17 and 19–20 were the first to open a new perspective. As mentioned earlier, we often find the harmonic pattern of the Phrygian cadence by default as a bridge between a slow movement and the following finale, as a musical announcement, and to increase the tension that a finale or something similar is to be expected. In measures 16 and 17, the Phrygian cadence is repeated four times, but it remains without consequence. After an extended run, the cadence is repeated again, a full three times, and it again remains without consequence. Is this a joke? A prank?

The impression is reinforced by the inverted echos of the section. It is in the nature of an echo that the return sounds weaker. However, in the Phrygian cadential measures in BWV 565, the reverse is true. The harmonic pattern is introduced with a thin texture of broken chords, only to be answered with heavy chords above a pedal line, plus hammering syncopations. Is this a provocation?

Flawless part-writing includes avoidance of parallel octaves and fifths. Let us imagine a student, required to submit a clean four-part piece, presenting the strict teacher something hitherto unheard like BWV 565. Seated at the organ console, the teacher begins to play. On the first page parallel octaves as far as one can see. The first note isolated, nailed with a mordent and a fermata, immediately followed by a rest. Short phrases, constantly interrupted with more rests. Laughter in the background, when the teacher has to take note that his organ is missing the bottom C-sharp. Broken chords are based on parallel fifths. In the pedal, there are only three roaring Ds, above them long-held quadruple dissonances. Then one finds a run pointing to a D, but silence instead. The inverted echoes startle. The Phrygian cadence remains without consequence, sounding seven times in vain.

There is no evidence whatsoever that BWV 565 was intended as a joke or prank, just as a joke does not have the caption, “This is a joke.” The nature of a text can be discerned from its content and setting alone. The amazing thing about this theory is, however, that it can answer all hitherto raised questions, among them:

• Only one copy of the autograph survived, some of the notation is only sketched: BWV 565 was intended only for private use and not for publication.

• All compositional features described as problematic: deliberate violations of the rules to show a new style.

• Bach named as author on the title page, the work possibly not by Bach: the most plausible explanation for this contradiction.

• The wrong key signature on the title page “ex . d . # .:” one cannot seriously claim that someone copied a multi-page piece in D minor and the same person mistakenly writes D major on the title page. Furthermore, Ringk notated minor keys adding a flat to the key letter, major keys adding a natural, not a sharp. The sharp could be a hint pointing to the unusual content, which is so-to-speak not natural, but sharp. In German, the word for sharp is Kreuz, which conveys the saying, ein Kreuz haben, having problems with something or someone.

• The simple composition style: simplified overdrawing as a device of parody.

• The low C-sharp in measure 2 being a rare key on organs of the time: deliberately pointing to the deficiency of a missing key.

• The late Peter Williams’s theory of a lost piece for violin: an allusion to J. S. Bach’s skills as a string player.

Following this theory, somewhere at an early stage someone took the piece as serious. The consequences are outrageous and at the same time incredibly humorous. It is singular that BWV 565 beats any other organ work in popularity. Then there are also all the discussions about the deliberately composed “misbehaving” features, if they can be credited to Bach or not. It is a work brilliant on several levels: as a provocation, fun, or a prank with a highly skillful compositional technique not common in Baroque music, but pointing to the future.

This theory may or may not be the case, but the journey into the unknown of BWV 565 is far from over. We now turn the second page of the manuscript and explore not only the fugue, but also lift the anchors on some findings that are considered certain. Stay tuned.

To be continued.

Notes

^1. Note designations in scientific orthography: C2-C3-C4-C5-C6 (= traditionally C-c°-c′-c′′-c′′′).

^2. The Phrygian church mode has the half-tone step between the first and second and the fifth and sixth scale degrees. Church modes, however, are melodic, not harmonic entities.