The Art of Fugue

Title page of the first edition, 1751
The Art of Fugue, part 1

The Art of Fugue, part 2
Performed by David Ezra Okonşar on organ and harpsichord

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The Art of Fugue (or The Art of the Fugue; German: Die Kunst der Fuge), BWV 1080, is an incomplete work 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.

This work consists of 14 fugues and 4 canons in D minor, each using some variation of a single principal subject, and generally ordered to increase in complexity. "The governing idea of the work", as put by Bach specialist Christoph Wolff, "was an exploration in depth of the contrapuntal possibilities inherent in a single musical subject."[1] The word "contrapunctus" is often used for each fugue.


The earliest extant source of the work is an autograph manuscript[2] of the early 1740s, containing 12 fugues and 2 canons. This autograph is typically referred to by its call number of P200 in the Berlin State Library. Three manuscripts for pieces that appear in the revised edition were bundled with P200 at some point before its acquisition by the library.

The revised version was published in May 1751, slightly less than a year after Bach's death. In addition to changes in the order, notation, and material of pieces which appeared in the autograph; it contained 2 new fugues, 2 new canons, and 3 pieces of ostensibly spurious inclusion. A second edition was published in 1752, but differed only in its addition of a preface by Friedrich Wilhelm Marpurg.

In spite of its revisions, the printed edition of 1751 contained a number of glaring editorial errors. The majority of these may be attributed to Bach's relatively sudden death in the midst of publication. Three pieces were included that do not to appear to have been part of Bach's intended order: an unrevised (and thus redundant) version of the second double fugue, Contrapunctus X; a two-keyboard arrangement[3] of the first mirror fugue, Contrapunctus XIII; and a chorale harmonization "Vor deinen Thron tret ich hiermit" ("Herewith I come before Thy Throne"), derived from BWV 668a, and noted in the introduction to the edition as a recompense for the work's incompleteness, having purportedly been dictated by Bach on his deathbed.

The anomalous character of the published order and the Unfinished Fugue have engendered a wide variety of theories which attempt to restore the work to that state originally intended by Bach.


The Art of Fugue is based on a single subject ( play on piano  or  play on organ ):

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

which each canon and fugue employs in some variation.

The work divides into seven groups, according to each piece's prevailing contrapuntal device; in both editions, these groups and their respective components are generally ordered to increase in complexity. In the order in which they occur in the printed edition of 1751 (without the aforementioned works of spurious inclusion), the groups, and their components are as follows.

Simple fugues:

1. Contrapunctus I: 4-voice fugue on principal subject,  play principal subject  or  play whole contrapunctus 
2. Contrapunctus II: 4-voice fugue on principal subject, accompanied by a 'French' style dotted rhythm motif,  play principal subject 
3. Contrapunctus III: 4-voice fugue on principal subject in inversion, employing intense chromaticism,  play principal subject 
4. Contrapunctus IV: 4-voice fugue on principal subject in inversion, employing counter-subjects,  play principal subject 

Counter-fugues, in which the subject is used simultaneously in regular, inverted, augmented, and diminished forms:

5. Contrapunctus V: Has many stretto entries, as do Contrapuncti VI and VII,  play principal subject 
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]  play principal subject 
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.

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

8. Contrapunctus VIII, a 3: Triple fugue, with three subjects, having independent expositions,  play principal subject 
9. Contrapunctus IX, a 4 alla Duodecima: Double fugue, with two subjects occurring dependently, and in invertible counterpoint at the 12th,  play principal subject 
10. Contrapunctus X, a 4 alla Decima: Double fugue, with two subjects occurring dependently, and in invertible counterpoint at the 10th,  play principal subject 
11. Contrapunctus XI, a 4: Triple fugue, employing the three subjects of Contrapunctus VIII in inversion,  play subject 

Mirror fugues, in which a piece is notated once and then with voices and counterpoint completely inverted, without violating contrapuntal rules or musicality:

12. Contrapunctus XII, a 4
13. Contrapunctus XIII, a 3

Canons, labelled by interval and technique:

14. Canon per Augmentationem in Contrario Motu: Canon in which the following voice is both inverted and augmented.
15. Canon alla Ottava: Canon in imitation at the octave
16. Canon alla Decima in Contrapunto alla Terza: Canon in imitation at the tenth
17. Canon alla Duodecima in Contrapunto alla Quinta: Canon in imitation at the twelfth

The Unfinished Fugue:

18. Fuga a 3 Soggetti (Contrapunctus XIV): 4-voice triple fugue (not completed, but likely to have become a quadruple fugue: see below), the third subject of which is based on the BACH motif, B – A – C – B ('H' in German letter notation).  play motif .


Both editions of the Art of Fugue are written in open score, where each voice is written on its own staff. This has led some to conclude[6] that the Art of Fugue was intended as an intellectual exercise, meant to be studied more than heard. The renowned keyboardist and musicologist Gustav Leonhardt,[7] argued 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. (Although Leonhardt notes that there are similarly 'unplayable' passages in The Well-Tempered Clavier.)
  2. The absence of the basso continuo is only logical since a fugue for string quartet wouldn't have one by default.

There is also the possibility that The Art of Fugue was not intended for one single (type of) instrument, but instead for whatever instruments were at hand. This hypothesis is given weight by the modern recording and concert history of the work: it is variously performed by string quartets, wind quartets, solo keyboardists, and orchestras.

The Unfinished Fugue

The final page of Contrapunctus XIV

A handwritten manuscript of the piece known as the Unfinished Fugue is among the three bundled with the autograph manuscript P200. It breaks off abruptly in the middle of its third section, with an only partially written measure 239. This autograph carries a note in the handwriting of Carl Philipp Emanuel Bach, stating "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.") This account is disputed by modern scholars, as the manuscript is clearly written in Bach's own hand, and thus dates to a time before his deteriorating health and vision would have prevented his ability to write, 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", on which the composer attempted to work out the counterpoint between the four subjects. The 2016 completion by pianist Kimiko Douglass-Ishizaka rejects the fourth subject theory, opting instead to develop the extant materials in the fugue to completion.[10]

A number of musicians and musicologists have composed conjectural completions of Contrapunctus XIV, notably Boëly, 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 2001 Luciano Berio completed the contrapunctus for orchestra. Conductor Rudolf Barshai completed the last fugue and recorded the entire Art of Fugue with the Moscow Chamber Orchestra.[11]

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.[12][13]

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. According to Gödel, 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". 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.

Sylvestre and Costa[14] reported a mathematical architecture of The Art of Fugue, based on bar counts, which shows that the whole work was conceived on the basis of the Fibonacci series and the golden ratio. The significance of the mathematical architecture can probably be explained by considering the role of the work as a membership contribution to the Correspondierende Societät der musicalischen Wissenschaften, and to the "scientific" meaning that Bach attributed to counterpoint.

Notable recordings




String quartet



See also

Notes and references

  1. Johann Sebastian Bach, the Learned Musician by Christoph Wolff, p. 433, ISBN 0-393-04825-X.
  2. The autograph manuscript bears the title Die Kunst der Fuga, written in the hand of Bach's son-in-law Johann Christoph Altnickol. This implies that the title was conceived at some point before the printed edition, which is titled Die Kunst der Fuge, but after the completion of the autograph.
  3. The printed indication of "a 2 Clav." and the counterpoint of the added voices do not appear to follow Bach's practice, evidencing that the parts were likely included by the editors of the printed edition to bolster the work.
  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. 1 2 Leonhardt, Gustav (July 1953). "The Art of Fugue – Bach's Last Harpsichord Work: An Argument". The Musical Times. 39 (3): 463–466. JSTOR 740009.
  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. "Perfectly right for this performer at this time - Kimiko Ishizaka performs The Art of Fugue with her own completion".
  11. "Rudolf Barshai obituary".
  12. University of Auckland News, Volume 37, Issue 9 (May 25, 2007)
  13. The thesis is available online:
  14. Loïc, Sylvestre; Costa, Marco (2011). THE_ART_OF_FUGUE "The Mathematical Architecture of Bach's The Art of Fugue". Il Saggiatore Musicale. 17: 175–196.
  15. 1 2 3 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.
  16. 1 2 Partial performances on organ (Contrapuncti I–IX) and piano (I, II, IV, IX, XI, XIII inversus, and XIV).
  17. 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.
  18. Recordings of Musical Offering and Art of Fugue
  19. Published by Accentus Music: CD – J. S. Bach Kunst der Fuge – Zhu Xiao-Mei, Piano, No. ACC 30308
  20. Paolo Borciani and Elisa Pegreffi with Tommaso Poggi and Luca Simoncini, as Quartetto Italiano, CD Nuova Era 7342, recording 1985.See
  21. Except the canons, which are played by harpsichordist Kenneth Gilbert on the recording.
  22. Jack Stratton: Contrapunctus IX (talkbox) on YouTube
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