The Art of Fugue, coupled with its enigmatic unfinished fugue and anecdotes from Bach's very last years, exudes a legendary aura. Since its publication immediately after the composer's death, it has continuously captivated people's interest, receiving praise bordering on veneration. However, despite its glory, it is not frequently performed. This is largely due to the unclear instrumentation or scoring intended by Bach, but also because its legendary aura has solidified an unapproachable image. While Bach certainly imbued this collection with a rather abstract and conceptual nature, its actual performability must have been the foremost prerequisite. Therefore, to approach each piece concretely, let us first ascertain what 'techniques' appear in this work and how they succeed musically. It should be emphasized from the outset that the works contained herein were probably not intended for performance in their entirety. Treating The Art of Fugue as a set of fugal variations on a single subject seems impractical, considering the listener's concentration span.

The overall structure of the work is presented here. Furthermore, issues related to the work's genesis will be briefly discussed at the end. Hereafter, the abbreviation 'Cp.' in the text refers to Contrapunctus. Also, 'theme' refers to the melody presented at the beginning of the first piece and pervading the entire collection, while 'subject' refers to the main melody in terms of fugal form or compositional technique.

BWV 1080/ 1-5 / 6-7/ 8 / 9-10, 10a / 11 / 12-13 / 14-17 / 18 / 19 /

BWV668

Group 1: Simple Fugues on the Basic Theme (Contrapunctus 1-5)

This group uses the basic form of the theme directly. However, there are slight rhythmic alterations, such as dotted notes. Overall, it aims for an older style. This is well-represented by the Alla breve (C with a vertical line) time signature (from this perspective, the C in the published score for Cp. 5 is likely an error).

Contrapunctus 1 is the simplest fugue, with no clear counter-subject appearing, remaining almost entirely on a single subject and passing through a very narrow range of keys. The independence of voices is maintained, sounding like a strict motet. It never falls below three voices. Since there are no rests or perfect cadences in the middle of the piece, it proceeds with sustained density and weight. Furthermore, with rests where all voices stop at the end and a somewhat toccata-like coda, the whole piece becomes an old-fashioned contrapuntal work.

Contrapunctus 2: In Contrapunctus 2, the latter half of the theme takes on a dotted rhythm. Again, a simple fugue based solely on the basic form. Compared to No. 1, it feels more lively and bright, but these dotted notes should never be played sharply; rather, they must be valued as a device to create propulsion that transcends beats and bar lines. Also, voices frequently increase and decrease, and although a perfect cadence does not appear until the end, the clear entries of voices create articulation in the texture.

Contrapunctus 3 begins with an inverted form, i.e., a subject with the melodic contour of the basic form flipped (inverting intervals is called 'inversion'; the resulting subject is called an inverted subject). Furthermore, a fixed counter-subject always accompanies the subject (this contrapuntal technique is called 'invertible counterpoint' because the music remains valid regardless of which voice the counter-subject or subject appears in, i.e., even if their vertical relationship is altered). However, what gives this piece its unique power is undoubtedly the chromaticism scattered throughout the voices other than the subject. This chromaticism originates from the note C appearing in the inversion of the basic theme. In a D minor piece, this would normally be C# as the leading tone, and the key momentarily becomes ambiguous already in the subject. In the contrapuntal voices, a sense of perspective is created by combining stepwise chromatic motion with harmonic leaps.

Contrapunctus 4 also uses the inverted form, but it begins on A, the dominant of D minor, clearly establishing the key. From the clear alternation between subject exposition and episodes, and the two perfect cadences occurring in the middle (m. 53, m. 103) and the periodicity arising therefrom, the whole piece has a schematic, logical structure. Furthermore, the independence of voices is not very strict, and homophonic movements are prominent in the somewhat redundant imitative sequences in the counter-subjects and episodes. Perhaps due to being added later, it is a relatively free fugue with a more modern style of writing.

Contrapunctus 5 is an 'inversion fugue' where the original form responds to the inverted form. It is also a 'stretto fugue' where the entry of the answer begins in the middle of the subject's presentation. The stretto distance gradually shortens, and in the final presentation (from m. 86), the inverted and original forms finally appear precisely simultaneously. Then, near the final measure, it expands to six voices, leading to a majestic cadence. Fittingly concluding the group based on the basic theme, it is a piece packed with various ancient techniques.

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Group 2: Inverted Stretto Fugues (Contrapunctus 6-7)

In this pair, large-scale inverted stretto fugues in a new style appear. Cp. 5 is not included here because, as indicated by the title of Cp. 6, 'in the French style,' these pieces move away from the old style (stile antico).

Contrapunctus 6: In Contrapunctus 6, the original form of the subject with dotted rhythm is answered by a diminished inverted form, followed by successive presentations of diminished original and diminished inverted forms. 'French style' refers to the contrast and tension created by various different rhythms: the dotted rhythm of the subject, sharper ornamental rhythms formed by ties and 32nd notes in the episodes, and prominent passages of continuous 16th notes in the latter half of the piece.

Contrapunctus 7 uses the augmented form in addition to the diminished subject. However, the possibilities of presentation and inversion using various subject forms—original, inverted, augmented, and diminished—are not explored as thoroughly. Furthermore, the augmented form of the subject, appearing four times, lacks the solemnity of a cantus firmus in a chorale arrangement. This is because there is no dramatic preparation for the entry of the important subject, such as reducing voices or having multiple voices conclude together. The piece maintains almost consistently four voices, even increasing to five at the end, demonstrating robust writing; however, the subject is heard throughout various voices, resulting in a loosely structured fugue in terms of sound.

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Group 3: Double Fugues with Invertible Counterpoint (Contrapunctus 9-10)

From here, entirely new subjects are combined with the basic theme in each piece. In the autograph early draft compiled around 1742, two double fugues and two triple fugues are paired with canons in between. It is likely that the printed score disrupted this logical arrangement by placing the double fugues Cp. 9-10 between the triple fugues Cp. 8 and Cp. 11 (some scholars suggest this was intentionally done to facilitate page turns).

Contrapunctus 9 is a piece using invertible counterpoint at the 12th (or 5th). Once the opening subject, which gently alternates between descending and ascending motion, has appeared in all four voices, the augmented form of the basic theme enters, much like a cantus firmus. Invertible counterpoint is a compositional technique where two melodies can be interchanged vertically at intervals of a 5th or 12th while maintaining musical coherence; the unusually long opening subject for Bach is designed to accommodate this augmented theme. The entries of the augmented theme are well-prepared and clearly audible in all instances, and as a result, the entire piece is rich in dramatic dynamism.

Contrapunctus 10 uses invertible counterpoint at the 10th (or 3rd). Following the exposition of the new subject, the inverted form of the basic theme appears in stretto, and the two subjects are immediately combined. In the first half, the voices maintain good independence, but from the middle section onwards, the same subject begins to appear simultaneously in two voices. This when the counterpoint is, one part at the 12th, the other at the 8th invertible counterpoint. This technique creates a climax in the latter half of the piece.

Incidentally, the 14th piece in the first edition is an early draft of Cp. 10. It begins with the exposition of the basic theme, and the new subject is combined with the basic theme from its entry. In other words, Cp. 10 is the early draft with an additional 20-odd measures of exposition based on the second subject at the beginning. Comparing the two, in Cp. 10, the exposition of the basic theme was rewritten to gradually reduce the volume using stretto in the alto, preventing an abrupt reduction in voices and then gradually building up again.

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Group 4: Triple Fugues (Contrapunctus 8, 11)

The triple fugue pair might have been separated in the printed score simply because Cp. 8 is for three voices and Cp. 11 for four. However, this separation obscured the close relationship between the two pieces. In fact, these two pieces represent an attempt to apply similar subjects to three and four voices respectively, forming one of the climaxes of The Art of Fugue.

The new subject presented at the beginning of Cp. 8 corresponds to the subject appearing from m. 27 in the alto of Cp. 11, while the opening subject of Cp. 11, based on the original form of the basic theme, corresponds to a modified inverted subject of the theme appearing in m. 94 in the alto of Cp. 8. The third subject is an eighth-note melody containing repeated notes, appearing in a descending form in the alto at m. 39 in Cp. 8, and first in an ascending form (later inverted to a descending form) in the tenor at m. 90 in Cp. 11. Thus, although the order of appearance of the three subjects differs, they appear at the beginning, around m. 30, and around m. 90, indicating that the two pieces share the same structure. Furthermore, the moment when all three subjects appear simultaneously occurs at approximately three-quarters of the way through both pieces (around m. 150 out of approximately 200 measures).

These three subjects themselves possess all the typical and ideal elements for a fugue: an arch-shaped subject derived from the basic theme, a chromatic subject, and a very striking and always audible subject with repeated notes. Therefore, even with complex combinations, they avoid pedantry and maintain a beautiful sound.

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Group 5: Mirror Fugues (Contrapunctus 12-13)

The technique where the entire piece can be inverted while maintaining musical coherence is called mirror counterpoint. It is closer to a canon than a fugue. Naturally, it requires extremely strict and advanced technique. In the autograph manuscript from the 1740s, the original and inverted forms are placed one above the other, displaying a magnificent structure as if reflected in a mirror.

Contrapunctus 12 is notated in an old-fashioned 3/2 time signature and begins with a solemn subject relatively faithful to the basic theme, but the counter-subject, moving stepwise, gradually becomes dominated by eighth notes, accelerating the tempo and concluding in a breathless climax. It is astonishing that such a lively piece is actually written in mirror counterpoint.

Contrapunctus 13, however, is even more lively and full of vitality. Carried by triplets, it is truly a dance-like jig, giving no sense of the various restrictions imposed by mirror counterpoint.

Incidentally, the first edition includes an arrangement for two harpsichords as the 18th piece. When assigning the three voices to two players (four hands), Bach added one voice. This voice cannot be inverted and differs between the original and inverted forms.

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Group 6: Canons

All of them have subjects with bold variations based on the basic theme. The autograph early draft presented only a single melody in the form of an enigma canon, with the solution written on the next page in a two-stave score, but the first edition only showed the solution.

The Canon at the Augmentation and Inversion begins with the inverted form of the leading upper voice melody placed in the lower voice at double the note values. The whole consists of 104 measures + a 5-measure concluding section, with the voices interchanging from m. 53 in the lower voice. Since the upper voice in the concluding section is identical to the opening subject, it can be continued indefinitely if the lower voice is removed. Because the melody is augmented, the following voice uses only 26 measures of the leading voice. The subject contains E-flat, resulting in an intense sound with prominent chromatic harmonies throughout. It should be noted that this piece was placed last in the autograph manuscript, and the arrangement in the first edition, which places it before the other three canons at the octave, 10th, and 12th, must be considered completely confused.

The Canon at the Octave has a fugal structure. An arch-shaped subject frequently appears and develops. The section after the repeat sign completely reproduces the opening, and furthermore, the concluding section after the fermata features bold hand crossings where the notes first converge in the middle of the keyboard, then spread out again, and finally, at the last note, a voice crossing occurs in the lower register. The whole piece is carried in a lively 9/16 time signature, and it is a well-structured piece, like an invention.

The Canon at the 10th in invertible counterpoint at the 3rd consists of 78 measures + a 4-measure concluding section (including a 2-measure cadenza), with the voices interchanging at m. 40. At that point, a 4-measure free melody is inserted into the lower voice. Additionally, a free melody appears in the upper voice four measures before the concluding section. The operation of the Canon at the 10th in invertible counterpoint at the 3rd is performed as follows:

• In the first half, the following voice is at the 10th above the leading voice.
• In the second half, the original leading melody is moved up an octave to the upper voice, and the original following melody is moved down a 10th.
• As a result, it becomes an imitation at the octave.

This piece begins in a very normal range, rises to the treble clef range for both hands in the middle section, normalizes at the halfway point, but then ascends to the high register again. Accordingly, it transforms from a leisurely mood of long note values and eighth notes into an interplay of brilliant runs, and in the concluding section, the meter abruptly changes from a triple division to a duple division. The transformation of range and tempo is interesting, and it is a beautiful piece.

The Canon at the 12th in invertible counterpoint at the 5th also similar to the Canon at the 10th in terms of technique: the voices interchange at m. 34 in the middle, and the latter half becomes an imitation at the octave. This piece also has fugal subject exposition and development, similar to the Canon at the Octave, and takes a clear two-part form. Furthermore, it is a perfect perpetual canon without free melodies. To combine all these elements and still write a beautiful invention is a musically very difficult task, but Bach masterfully achieves it here.

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Group 7: Unfinished Four-Voice Fugue

The three subjects appearing here do not match the basic theme of The Art of Fugue. A slight hint of it can be felt in the perfect fifth leap of the first subject at the beginning, but it can be said to be almost unrelated. The second subject is a melody dominated by eighth notes in an organ-like idiom appearing at m. 114, and the third subject is the B-A-C-H subject appearing at m. 193. Because the theme does not appear, it has sometimes been doubted whether this piece truly belongs in The Art of Fugue, but it has been confirmed that these three subjects can be combined with the basic theme. Therefore, it is possible that this fugue is the 'final fugue containing four subjects, all four voices of which were later to be inverted without exception,' as reported in the 'Personal Obituary' and J.N. Forkel's 'Bach Biography'. However, the currently surviving portion does not sufficiently combine the three subjects and does not even reach a triple fugue. Only after the second subject is presented do the first and second subjects combine. Furthermore, when the section with the second subject begins following the first subject, a somewhat abrupt impression is undeniable. The transition to the third section is similar. From these points, it is likely that Bach connected three different works.

In what form could this work possibly be completed? No matter how one attempts to complete it, Bach's 'original plan' cannot be known, and it would likely be difficult to do it better than Bach himself. However, in performance, stopping while it is unfinished leaves a very incomplete impression. For the performer to provide some kind of conclusion here is by no means contrary to Bach's intentions.

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BWV 668: Chorale Fantasia 'Wenn wir in höchsten Nöten sein'

This work is considered Bach's last composition because approximately 25 and a half measures of this piece, with the same melody, are left written on the last page of the '17 Chorales' as the chorale 'Vor deinen Thron tret ich hiermit' (BWV 668a). However, 'Wenn wir in höchsten Nöten sein' included in The Art of Fugue is an early draft and does not completely match the revised draft found in the scorebook of the '17 Chorales'. It is likely that another, now lost, source was used for the publication of The Art of Fugue.

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The Art of Fugue constitutes one part of the conceptual collection of works conceived by Bach in his later years.

The autograph manuscript preserved in the Berlin State Library was created in 1742, suggesting that Bach began working on The Art of Fugue around 1740, prior to this. Subsequently, there were frequent interruptions—such as visiting Frederick the Great and completing The Musical Offering, joining L.C. Mizler's 'Society of Musical Sciences' and composing the Canonic Variations on 'Vom Himmel hoch' (BWV 769), revising older organ chorales to compile the so-called 'Schübler Chorales' and the '17 Chorales', and completing the Mass in B minor—and due to various changes made to the original plan of The Art of Fugue, its publication by Bach himself never materialized.

The greatest mystery is what content and arrangement Bach ultimately desired for The Art of Fugue. It is unknown who specifically arranged for the publication of the score announced in the newspaper on June 1, 1751. However, the content of this first edition likely disregarded the composer's intentions considerably. This can be inferred from, for example, the inclusion of an early draft of Cp. 10 as the 14th piece, the presence of a mere arrangement of Cp. 13 for two harpsichords as the 18th piece, the placement of a chorale arrangement at the end, the inclusion of the unfinished fugue as the 19th piece, and the significant deviation from the arrangement in the 1742 autograph manuscript. Why did Bach leave the fugue bearing his name unfinished? One might say his life simply ran out before he could finish it, but perhaps the composition did not progress precisely because he was dissatisfied with the overall outcome of this fugue. If so, including it in The Art of Fugue might be contrary to the composer's intentions. Even stranger is the placement of the chorale arrangement 'Wenn wir in höchsten Nöten sein' as the final piece. Forkel, in his 'Bach Biography', states that Bach dictated this chorale on his deathbed. The reality is that this piece was used as a supplement because the planned final fugue remained unfinished; therefore, it seems unlikely that including the chorale arrangement in The Art of Fugue was in accordance with the composer's wishes. Furthermore, there is no definitive proof that this piece was indeed his last composition. More fundamentally, according to the 'Personal Obituary' and Forkel's 'Bach Biography', there were two fugues planned but not completed. The 'Unfinished Fugue' is likely one of them (Forkel describes the 'Unfinished Fugue' as the 'second to last fugue' with 'three subjects'), but the other one is completely lost. The Art of Fugue as conceived by Bach has become an eternal mystery.

The issue of a logical arrangement has attracted as much interest from many musicians as the completion of the unfinished fugue. However, the arrangement itself is not a major issue for the performance of the work. This is because, in any case, it is not intended to be performed in its entirety.

Regarding instrumentation, it is now almost universally considered a keyboard work, specifically written for the clavier. While the long sustained notes appearing in places might indeed seem suitable for organ pedal points, the work as a whole is filled with idioms appropriate for the clavier. Furthermore, characteristics of instruments other than keyboard are rarely found. It should be noted that when performing on a modern piano, especially regarding hand crossings, the same effect as on a harpsichord or organ cannot be achieved, so ingenuity is required.

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