The Art of Fugue

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A portrait that may depict Bach in 1750

The Art of Fugue (or The Art of the Fugue, original German: Die Kunst der Fuge), BWV 1080, is an incomplete work[1] of unspecified instrumentation by Johann Sebastian Bach (1685–1750). Written in the last decade of his life, The Art of Fugue is the culmination of Bach's experimentation with monothematic instrumental works. It consists of 14 fugues and 4 canons, each using some variation of a single principal subject, and generally ordered by increasing complexity.

"The governing idea of the work", as Bach specialist Christoph Wolff put it, is "an exploration in depth of the contrapuntal possibilities inherent in a single musical subject."[2]


The earliest extant source for the work is an autograph manuscript[3] of the early 1740s, containing 12 fugues and 2 canons. The revised version was published in 1751, containing 14 fugues and 4 canons.

The order of the work's component pieces has been debated, especially as there are differences between the manuscript and the printed editions appearing immediately after Bach's death. Also musical reasons have been invoked to propose different orders for later publications and/or the execution of the work, e.g. by Wolfgang Graeser in 1927, who also published his own "completion" of the final Contrapunctus XIV.

The 1751 printed edition contained—apart from a high number of errors and other flaws—a four-part version of Contrapunctus XIII, arranged to be played on two keyboards (rectus BWV 1080/18,1 and inversus BWV 1080/18,2). It is however doubtful whether the printed indication "a 2 Clav.", and the fourth added voice, that is not mirrored according to Bach's usual practice, derive from him, or from his son(s) that supervised this first edition.

The engraving of the copper plates for the printed edition would however have started shortly before the composer's death, according to contemporary sources, but it is unlikely that Bach had any real supervision in that preparation of the printed edition, due to his illness at the time.

The first printed edition also includes an unrelated work as a kind of "encore", the chorale prelude Vor deinen Thron tret Ich hiermit (Herewith I come before Thy Throne), BWV 668a, which Bach is said to have dictated on his deathbed.

A 1742 fair copy manuscript contains Contrapuncti I–III, V–IX, and XI–XIII, plus the octave and augmented canons and an earlier version of Contrapunctus X.


Each of the canons and fugues use some variation of the principal subject in D minor:

  \relative c'' {                 \clef treble                 \key d \minor                 \time 4/4                  d,2 a' |                 f d |                 cis d4 e |                 f2~ f8 g f e |                 d4         }

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In the 1751 printed edition, the various movements are roughly arranged by increasing order of sophistication of the contrapuntal devices used. The Arabic number in the title indicates the number of voices in the fugue, with the exception of the last one, where a 3 Soggetti means "with 3 subjects":

Simple fugues:

1. Contrapunctus I, and
2. Contrapunctus II: Simple monothematic 4-voice fugues on main theme, accompanied by a 'French' style dotted rhythm motif.
3. Contrapunctus III, and
4. Contrapunctus IV: Simple monothematic 4-voice fugues on inversion of main theme, i.e. the theme is "turned upside down":

 \relative a' { \key d \minor         \time 4/4          a2 d, |         f a |         bes a4 g |         f2~ f8 e f g |         a4         }

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Counter-fugues, in which a variation of the main subject is used in both regular and inverted form:

5. Contrapunctus V: Has many stretto entries, as do Contrapuncti VI and VII.
6. Contrapunctus VI, a 4 in Stylo Francese: This adds both forms of the theme in diminution[4] (halving note lengths), with little rising and descending clusters of semiquavers in one voice answered or punctuated by similar groups in demisemiquavers in another, against sustained notes in the accompanying voices. The dotted rhythm, enhanced by these little rising and descending groups, suggests what is called "French style" in Bach's day, hence the name Stylo Francese.[5]
7. Contrapunctus VII, a 4 per Augmentationem et Diminutionem: Uses augmented (doubling all note lengths) and diminished versions of the main subject and its inversion.
The two subjects of Contrapunctus IX. Excepting the first four entries of the eighth note subject the two always enter together, sometimes an octave apart as shown here, sometimes a twelfth (an octave plus a fifth) apart.

Double and triple fugues, with two and three subjects respectively:

8. Contrapunctus VIII, a 3: Triple fugue.
9. Contrapunctus IX, a 4 alla Duodecima: Double fugue
10. Contrapunctus X, a 4 alla Decima: Double fugue.
11. Contrapunctus XI, a 4: Triple fugue.

Mirror fugues, in which the complete score can be inverted without loss of musicality:

12. Contrapunctus XII, a 4: The rectus (normal) and inversus (upside-down) versions are generally played back to back.
13. Contrapunctus XIII, a 3: The second mirror fugue in 3 voices, also a counter-fugue.

Canons, labeled by interval and technique:

14. Canon per Augmentationem in Contrario Motu: Augmented canon in inverted motion.
15. Canon alla Ottava: Canon at the Octave. The two imitating voices are separated by an octave.
16. Canon alla Decima in Contrapunto alla Terza: Canon at the tenth, counterpoint at the third.
17. Canon alla Duodecima in Contrapunto alla Quinta: Canon at the twelfth, counterpoint at the fifth.

An arrangement of Contrapunctus XIII, see below.

18. Fuga a 2 (rectus), and Alio modo Fuga a 2 (inversus)
19. Fuga a 4 Soggetti (Contrapunctus XIV): 4-voice triple, possibly quadruple, fugue, the third subject of which is based on the BACH motif, B – A – C – B ('H' in German letter notation).


Manuscript copies of the Art of Fugue, as well as the first printed edition, use open scoring, where each voice is written on its own staff. This has led to the assumption[6] that the Art of Fugue was an intellectual exercise, meant to be studied and not heard. Some musicologists today, such as Gustav Leonhardt,[7] argue that the Art of Fugue was probably intended to be played on a keyboard instrument (and specifically the harpsichord).[8] Leonhardt's arguments included the following:[7]

  1. It was common practice in the 17th and early 18th centuries to publish keyboard pieces in open score, especially those that are contrapuntally complex. Examples include Frescobaldi's Fiori musicali (1635), Samuel Scheidt's Tabulatura Nova (1624), works by Johann Jakob Froberger (1616–1667), Franz Anton Maichelbeck (1702–1750), and others.
  2. The range of none of the ensemble or orchestral instruments of the period corresponds to any of the ranges of the voices in The Art of Fugue. Furthermore, none of the melodic shapes that characterize Bach's ensemble writing are found in the work, and there is no basso continuo.
  3. The fugue types used are reminiscent of the types in The Well-Tempered Clavier, rather than Bach's ensemble fugues; Leonhardt also shows an "optical" resemblance between the fugues of the two collections, and points out other stylistic similarities between them.
  4. Finally, since the bass voice in The Art of Fugue occasionally rises above the tenor, and the tenor becomes the "real" bass, Leonhardt deduces that the bass part was not meant to be doubled at 16-foot pitch, thus eliminating the pipe organ as the intended instrument, leaving the harpsichord as the most logical choice.

However, opponents of Leonhardt's theory such as Reinhard Goebel argue that:

  1. The Art of Fugue is not completely playable on a keyboard. Contrapunctus XII and XIII, for instance, cannot be played on a single keyboard without making awkward jumps or neglecting the main theme, especially on the keyboard instruments of Bach's day, such as the harpsichord or the early pianoforte, both of which lacked a sustain pedal. This is something Bach would never have allowed to happen.
  2. The absence of the basso continuo is only logical since a fugue for string quartet wouldn't have one by default.

This leaves only two options, being either two keyboard instruments or a classical string quartet. Fact is that a lot of the Baroque chamber music was not intended for one single (type of) instrument and the performance depended on which instruments were ready at hand. The open score probably means that Bach didn't suggest any preference and that therefore the Art of Fugue can be performed by various ensembles, to personal taste.

The unfinished fugue[edit]

The final page of Contrapunctus XIV

Contrapunctus XIV breaks off abruptly in the middle of the third section at bar 239. The autograph carries a note in the handwriting of Bach's son Carl Philipp Emanuel Bach saying "Über dieser Fuge, wo der Name B A C H im Contrasubject angebracht worden, ist der Verfasser gestorben." ("At the point where the composer introduces the name BACH [for which the English notation would be B-A-C-B] in the countersubject to this fugue, the composer died.") However, modern scholarship disputes this version, in particular because the musical notes are indisputably in Bach's own hand, written in a time before his deteriorating vision led to erratic handwriting, probably 1748–1749.[9]

Many scholars, including Gustav Nottebohm (1881), Wolff and Davitt Moroney, have argued that the piece was intended to be a quadruple fugue, with the opening theme of Contrapunctus I to be introduced as the fourth subject. The title Fuga a 3 soggetti, in Italian rather than Latin, was not given by the composer but by CPE Bach, and Bach's Obituary actually makes mention of "a draft for a fugue that was to contain four themes in four voices". The combination of all four themes would bring the entire work to a fitting climax. Wolff also suspected that Bach might have finished the fugue on a lost page, called "fragment X" by him, on which the composer attempted to work out the counterpoint between the four subjects.

A number of musicians and musicologists have conjectured completions of Contrapunctus XIV, notably music theoretician Hugo Riemann, musicologists Donald Tovey and Zoltán Göncz, organists Helmut Walcha, David Goode and Lionel Rogg, and Davitt Moroney. Ferruccio Busoni's Fantasia Contrappuntistica is based on Contrapunctus XIV, but is more a work by Busoni than by Bach.

In 2007, New Zealand organist and conductor Indra Hughes completed a doctoral thesis about the unfinished ending of Contrapunctus XIV, proposing that the work was left unfinished not because Bach died, but as a deliberate choice by Bach to encourage independent efforts at a completion.[10][11]

Douglas Hofstadter's book Gödel, Escher, Bach discusses the unfinished fugue and Bach's supposed death during composition as a tongue-in-cheek illustration of Austrian logician Kurt Gödel's first incompleteness theorem. To be more specific, the idea in that theorem is that the very power of a "sufficiently powerful" formal mathematical system can be exploited to "undermine" the system, by leading to statements that assert such things as "I cannot be proven in this system". Because of this twisty kind of self-reference, such assertions are true but unprovable. In Hofstadter's discussion, Bach's great compositional talent is used as a metaphor for a "sufficiently powerful" formal system; however, Bach's insertion of his own name "in code" into the fugue is not, even metaphorically, a case of Gödelian self-reference; and Bach's failure to finish his self-referential fugue serves as a metaphor for the unprovability of the Gödelian assertion, and thus for the incompleteness of the formal system.

A book entitled "Bach: Essays on His Life and Music" includes an article about the unfinished fugue, stating that Bach never intended to write the rest of the fugue on the last sheet of music paper used for the fugue because of the unalignment of the bottom staves. It also says that because of the above-mentioned reason, Bach wrote the rest of the fugue on another sheet of music paper, called "fragment x" that would have completed, or almost completed, the fugue. However, even if there is a fragment x, it has been lost.

The permutation matrix[edit]

In 1991 a theory was published by Zoltán Göncz answering the question of how Bach planned the appearance of the fourth subject, the main subject of the cycle:

In the course of the exposition of the first three subjects (first subject: mm. 1–21, second subject: mm. 114–141, third subject: mm. 193–207), Bach applied a serial sequence of voice entries decided in advance, by which he determined the space and time parameters of the subject entries. The superimposition of the three exposition matrices foreshadows, and develops as a negative, the sequence of the voice entries of the fourth subject. The copying of the four subjects onto each other displays a characteristic construction of Bach's oeuvre occurring mainly in the vocal fugues: that of the permutation fugue.


However paradoxical, it follows from the logic of composing a quadruple fugue that the combinations joining all four subjects (i.e. those combinations which appear last when performing the work) were already completed in the very first stage of composition, because the possibility of overlapping the four subjects (1 + 2 + 3 + 4) is the sine qua non of writing a quadruple fugue. The process of composition does not proceed in a linear way from the beginning, but with all four parts in view.[12]

One of the striking features of Contrapunctus XIV is that in this movement Bach applied the stretto of whole expositions, layering the first two expositions atop each other prior to introducing the third subject. In the exposition of the first three subjects he "programmed" the later permutation stretti, then applied the expositions as "programs", "algorithms". The permutation matrix, apart from originating authentically with Bach, can be proved to have been ready at the time of the genesis of the work (that is, earlier than the surviving section).

The discovery of the permutation matrix was one of the most essential requirements for achieving a reconstruction of Contrapunctus XIV which might approach the original form planned by Bach.[13][14]

A Pythagorean enigma[edit]

The theory is advanced[15] by the cellist Hans-Eberhard Dentler (a pupil of Pierre Fournier's, and Fellow of the World Academy of Art and Science) that the Art of the Fugue was written to display Pythagorean philosophical principles. The arguments revolve upon Bach's friendship with Johann Matthias Gesner, whom he had known in Weimar and who in 1730 moved to the Thomasschule at Leipzig (where Bach was Cantor) as rector. There Gesner taught Greek philosophy with an emphasis on Pythagorean thought.

Among Gesner's students was Lorenz Christoph Mizler, who became a pupil and friend of Bach's. Bach was one of four distinguished dedicatees of Mizler's 1734 doctoral dissertation on Music as part of a Philosophical Education. Mizler founded the Korrespondierenden Sozietät der Musikalischen Wissenschaften (Corresponding Society of Musical Sciences) in 1738, which Bach joined in June 1747, and of which Handel and Telemann were also members. The society was concerned with the union of music, philosophy, mathematics and science in Pythagorean theory, and required each member to contribute a practical work in demonstration of this approach, for which Bach produced his Canonic Variations on "Vom Himmel hoch da komm' ich her" for organ, BWV 769, and the Canon triplex a 6 voci. The Society's work commenced with the publication of a Bibliography (in its Musikalische Bibliotek) referencing works of Marcus Meibom, John Wallis, Leibniz, Kepler and Robert Fludd.[16]

The points of this analysis are that the work constitutes an enigma in the classical sense of a puzzle contained within its structure. This subsists in the numerical and philosophical relations of Unity (one key signature throughout and the thematic synthesis); Tetraktys (the relation of 1, 2, 3 and 4 as arranged to form the perfect triangle), the mirror or speculum principle, Contrapunctus as derived from Aristotelian terminology referring to balancing opposites, the Music of the Spheres is possibly reflected in Fugues 1–7, and in the term Fugue, meaning 'flight', which refers to the flight of the musical phrases.[17]

Against the theory is Bach's apparent indifference to the Society in its early years, and his hesitancy in joining it. The Society had in fact attempted to establish principles for the writing of cantatas which were not in line with his own approach.[18] Since any musical structure was susceptible to Mizler's Pythagorean analysis, the case for any specific precedent influence on The Art of Fugue remains conjectural.

It has also been argued that the hidden theme in Edward Elgar's Enigma Variations may derive from The Art of Fugue.[19]

Films about the Art of Fugue[edit]

The documentary film Desert Fugue is a 90 minute documentary about the history of the Art of Fugue and its suitability for performance on the organ. The film features interviews with scholar Christoph Wolff, George Ritchie (organist) and organ builders Ralph Richards and Bruce Fowkes.

Notable recordings[edit]

See and for more complete lists.

String quartet


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See also[edit]

Notes and references[edit]

  1. ^ Some consider it a work which was completed, but is incompletely preserved today, either because its publication by engraving was not completed, or because the last pages of the manuscript were misplaced by Bach's son. See notes below.
  2. ^ Johann Sebastian Bach, the Learned Musician by Christoph Wolff, page 433, ISBN 0-393-04825-X.
  3. ^ This manuscript bears a slightly different title, added afterwards by his son-in-law Johann Christoph Altnickol: Die Kunst der Fuga.
  4. ^ Helmut Walcha, 'Zu meiner Wiedergabe', in Die Kunst Der Fuge BWV 1080, St Laurenskerk Alkmaar 1956 (Archiv Production, Polydor International 1957), Insert pp 5–11, at p.7.
  5. ^ "The Art of the Fugue". American Public Media. Retrieved 25 November 2010. 
  6. ^ The Art of the Fugue
  7. ^ a b
  8. ^ D. Schulenberg. "Expression and Authenticity in the Harpsichord Music of J.S. Bach". The Journal of Musicology, Vol. 8, No. 4 (Autumn, 1990), pp. 449–476
  9. ^ See e.g. the discussion in Johann Sebastian Bach, the Learned Musician by Christoph Wolff, ISBN 0-393-04825-X.
  10. ^ University of Auckland News, Volume 37, Issue 9 (May 25, 2007)
  11. ^ The thesis is available online:
  12. ^ Hence Schweitzer remarks, 'It is an error to say he did not complete The Art of the Fugue. He died before the engraving was completed; hence the work has come down to us in a seemingly incomplete form.' (A. Schweitzer, J.S. Bach, trans. E. Newman, 1911 (1938 reissue, A & C Black, London, I, 423.)
  13. ^ Göncz, Z.: Reconstruction of the Final Contrapunctus of The Art of Fugue, in: International Journal of Musicology Vol. 5, pp. 25–93. 1997 ISBN 3-631-49809-8; Göncz, Z.: Bach's Testament
  14. ^ Score published by Carus-Verlag [CV 18.018]. [1]
  15. ^ H.-E. Dentler, L'Arte della fuga di Johann Sebastian Bach: un'opera pitagorica e la sua realizzazione (Skira, Milano 2000). Presented at the Accademia nazionale Santa Cecilia, Rome. An elaboration in a series of lectures was offered by Dentler at the Scuola Communale de Musica de Grosseto, 27–29 January 2001.
  16. ^ F. David Peat, 'J.S. Bach's The Art of the Fugue: An Enigma Resolved', see external site [2]
  17. ^ The theory is developed in the German edition of Dentler's work, Johann Sebastien Bachs "Kunst der Fuge": Ein Pythagoreisches Werk Und Seine Verwirklichung (Schott Music, Mainz 2004), ISBN 3-7957-0490-1, and in his more recent work Johann Sebastien Bachs "Musikalisches Opfer": Music Als Abbild der Sphärenharmonie (Schott Music, Mainz 2008), ISBN 3-7957-0181-3.
  18. ^ Schweitzer, J.S. Bach (Black, 1923), Chapter XI.
  19. ^ The Answer to Elgar's Enigma Marshall A. Portnoy, Musical Quarterly 1985 LXXI: 205–210; doi:10.1093/mq/LXXI.2.205
  20. ^ a b c The recordings by Walcha (1970) and Moroney include both their completion of Contrapunctus XIV and the unfinished original, while Bergel's includes only his attempt.
  21. ^ a b Partial performances on organ (Contrapuncti I–IX) and piano (I, II, IV, IX, XI, XIII inversus, and XIV).
  22. ^ The recording, which includes both the unfinished original and Rogg's completion, in the year of its release won the Grand Prix du Disque from the Charles Cros Academy.
  23. ^ Source:
  24. ^ Paolo Borciani and Elisa Pegreffi with Tommaso Poggi and Luca Simoncini, as Quartetto Italiano, CD Nuova Era 7342, recording 1985.See [3]
  25. ^ Except the canons, which are played by harpsichordist Kenneth Gilbert on the recording.

External links[edit]