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J.S.Bach : The Art of Fugue
A portrait that may depict Bach in
175
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
culminates 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 to increase in 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]
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Sources
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.
● Structure
Each of the canons and fugues use some
variation of the principal subject in D minor:
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":
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).
● Instrumentation
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]
- 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.
However, opponents of Leonhardt's theory such as Reinhard Goebel argue
that:
- 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.
- 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
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
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
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]
