The Art of Fugue

{{Listen|type=music The Art of Fugue is based on a single subject, which each canon and fugue employs in some variation: \relative c'' { \set Staff.midiInstrument = #"church organ" \clef treble \key d \minor \time 4/4 d,2 a' | f d | cis d4 e | f2~ f8 g f e | d4 } 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: • Contrapunctus 1: four-voice fugue on principal subject • Contrapunctus 2: four-voice fugue on principal subject, accompanied by a 'French' style dotted rhythm • Contrapunctus 3: four-voice fugue on principal subject in inversion, employing intense chromaticism • Contrapunctus 4: four-voice fugue on principal subject in inversion, employing counter-subjects Stretto-fugues (counter-fugues), in which the subject is used simultaneously in regular, inverted, augmented, and diminished forms: • Contrapunctus 5: has many stretto entries, as do Contrapuncti 6 and 7 • Contrapunctus 6, a 4 in Stylo Francese: adds both forms of the theme in diminution, (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. • Contrapunctus 7, a 4 per Augment[ationem] et Diminut[ionem]: 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: • Contrapunctus 8, a 3: triple fugue with three subjects, having independent expositions • Contrapunctus 9, a 4, alla Duodecima: double fugue, with two subjects occurring dependently and in invertible counterpoint at the twelfth • Contrapunctus 10, a 4, alla Decima: double fugue, with two subjects occurring dependently and in invertible counterpoint at the tenth • Contrapunctus 11, a 4: triple fugue, employing the three subjects of Contrapunctus 8 in inversion Mirror fugues, in which a piece is notated once and then with voices and counterpoint completely inverted, without violating contrapuntal rules or musicality: • Contrapunctus inversus 12 a 4 [forma inversa and recta] • Contrapunctus inversus 13 a 3 [forma recta and inversa] Canons, labeled by interval and technique: • Canon per Augmentationem in Contrario Motu: Canon in which the following voice is both inverted and augmented. The following voice, running at half-speed, eventually lags the first voice by 20 bars, making the canon effect hard to hear. Three versions have appeared in the autograph Mus. ms. autogr. P 200: Canon in Hypodiatesseron, al roversio e per augmentationem, perpetuus, Canon al roverscio et per augmentationem, and Canon p. Augmentationem contrario Motu, the third of which appears on the second supplemental Beilage. • Canon alla Ottava: canon in imitation at the octave; titled Canon in Hypodiapason in Mus. ms. autogr. P 200. • Canon alla Decima [in] Contrapunto alla Terza: canon in imitation at the tenth • Canon alla Duodecima in Contrapunto alla Quinta: canon in imitation at the twelfth Alternate variants and arrangements: • Contra[punctus] a 4: alternate version of the last 22 bars of Contrapunctus 10. • Fuga a 2 Clav: and Alio modo. Fuga a 2 Clav.: two-keyboard arrangements of Contrapunctus inversus a 3, the forma inversa and recta, respectively. Incomplete fugue: • [Contrapunctus 14] Fuga a 3 Soggetti: four-voice triple fugue (not completed by Bach, but likely to have become a quadruple fugue: see below), the third subject of which begins with the BACH motif, B–A–C–B ('H' in German letter notation). == Instrumentation ==

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 that the Art of Fugue was intended as an intellectual exercise, meant to be studied more than heard. The renowned keyboardist Gustav Leonhardt argued that the Art of Fugue was intended to be played on a keyboard instrument, and specifically the harpsichord. Leonhardt's arguments included the following: • 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. • 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. • 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. • 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. It is now generally accepted by scholars that the work was envisioned for keyboard. == Fuga a 3 Soggetti ==

that the composer died at this point. Fuga a 3 Soggetti ("fugue in three subjects"), also called the "Unfinished Fugue" and Contrapunctus 14, was contained in a handwritten manuscript bundled with the autograph manuscript Mus. ms. autogr. P 200. 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." ("While working on this fugue, which introduces the name BACH [for which the English notation would be B–A–C–B] in the countersubject, the composer died.") This account is disputed by modern scholars, as the manuscript was written in Bach's own handwriting, and thus dates to a time before his deteriorating health and vision prevented him from writing, in their view probably 1748–1749. Attempts at completion Several musicologists and musicians have composed conjectural completions of Contrapunctus XIV which include the fourth subject, including: • Musicologists Donald Tovey (1931); Zoltán Göncz (1992, which makes use of what he refers to as a permutational matrix of the 4 subjects); Yngve Jan Trede (1995); and Thomas Daniel (2010). • Organists Helmut Walcha, David Goode, Lionel Rogg, and Davitt Moroney (1989). • Conductor Rudolf Barshai (2010). • Composer Kalevi Aho (2011), including versions for string quartet (first performed by saxophone quartet) and for string orchestra. • Pianist Daniil Trifonov (2021). Ferruccio Busoni's Fantasia contrappuntistica is based on Contrapunctus 14, but it develops Bach's ideas to Busoni's own purposes in Busoni's musical style, rather than working out Bach's thoughts as Bach himself might have done. In total, there have been over 80 conjectural attempts to complete Contrapunctus 14. In his 2007 doctoral thesis about the unfinished ending of Contrapunctus 14, the New Zealand organist and conductor Indra Hughes proposed that the work was left unfinished not because Bach died, but as a deliberate choice by Bach to encourage independent efforts at a completion. 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 the 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. == Significance ==

Principles of construction Loïc Sylvestre and Marco Costa reported a mathematical architecture of The Art of Fugue, based on bar counts, which shows that the whole work could have been conceived on the basis of the Fibonacci series and the golden ratio. Dominic Florence proposes that a concept he calls "opposition" governs all the methods that Bach uses in Contrapuncti 1, 2, 3, and 5 to create variety. These include changes in "melody (contrary motion), polyphony (contrapuntal inversion), harmony (dissonance), [rhythmic] density (texture), rhythm (syncopation), and tonality (modulation}". Religious interpretations In 1984, the German musicologist Hans Heinrich Eggebrecht suggested a possible religious interpretation of The Art of Fugue, which he agreed was impossible to prove: that the work illustrated in musical terms the Christian doctrine of redemption by God's grace alone, sola gratia, rather than by any action an individual can take. Eggebrecht noted the presence in fugue 3 of the presence of the composer's surname Bach in its theme, the sequence of notes B-A-C-H-C#-D. In Eggebrecht's view, this could mean that the composer is not just signing the work, but is placing himself by his grave, accepting sola gratia as he reaches towards the tonic note, which marks the end of the fugue and symbolically the end of his life. Further, the six-note fragment is chromatic, denoting sinful humanity, whereas the work as a whole is diatonic, symbolising God's perfection. == Recordings ==