In music, a fugue ( ) is a contrapuntal compositional technique in two or more voices, built on a subject (theme) that is introduced at the beginning in imitation (repetition at different pitches) and recurs frequently in the course of the composition.
The English term fugue originated in the 16th century and is derived from the French word fugue or the Italian fuga. This in turn comes from Latin, also fuga, which is itself related to both fugere ("to flee") and fugare ("to chase"). The adjectival form is fugal. Variants include fughetta (literally, "a small fugue") and fugato (a passage in fugal style within another work that is not a fugue).
A fugue usually has three sections: an exposition, a development, and finally, a recapitulation that contains the return of the subject in the fugue's tonic key, though not all fugues have a recapitulation. In the Middle Ages, the term was widely used to denote any works in canonic style; by the Renaissance, it had come to denote specifically imitative works. Since the 17th century, the term fugue has described what is commonly regarded as the most fully developed procedure of imitative counterpoint.
Most fugues open with a short main theme, the subject, which then sounds successively in each voice (after the first voice is finished stating the subject, a second voice repeats the subject at a different pitch, and other voices repeat in the same way); when each voice has entered, the exposition is complete. This is often followed by a connecting passage, or episode, developed from previously heard material; further "entries" of the subject then are heard in related keys. Episodes (if applicable) and entries are usually alternated until the "final entry" of the subject, by which point the music has returned to the opening key, or tonic, which is often followed by closing material, the coda. In this sense, a fugue is a style of composition, rather than a fixed structure.
The form evolved during the 18th century from several earlier types of contrapuntal compositions, such as imitative ricercars, capriccios, canzonas, and fantasias. The famous fugue composer Johann Sebastian Bach (1685–1750) shaped his own works after those of Johann Jakob Froberger (1616–1667), Johann Pachelbel (1653–1706), Girolamo Frescobaldi (1583–1643), Dieterich Buxtehude (c. 1637–1707) and others. With the decline of sophisticated styles at the end of the baroque period, the fugue's central role waned, eventually giving way as sonata form and the symphony orchestra rose to a dominant position. Nevertheless, composers continued to write and study fugues for various purposes; they appear in the works of Wolfgang Amadeus Mozart (1756–1791) and Ludwig van Beethoven (1770–1827), as well as modern composers like Dmitri Shostakovich (1906–1975).
Musical outline 1
- The exposition 1.1
- The episode 1.2
- The development 1.3
- Example and analysis 1.4
- False entries 1.5
- Counter-exposition 1.6
- Stretto 1.7
- Final entries and coda 1.8
Types of fugue 2
- Simple fugue 2.1
- Double (triple, quadruple) fugue 2.2
- Counter-fugue 2.3
- Permutation fugue 2.4
- Fughetta 2.5
- Middle Ages and Renaissance Period 3.1
- Baroque era 3.2
- Classical era 3.3
- Romantic era 3.4
- 20th century 3.5
- Musical form or texture? 4
- Perceptions and aesthetics 5
- References 6
- Sources 7
- Further reading 8
- External links 9
A fugue begins with the exposition and is written according to certain predefined rules; in later portions the composer has more freedom, though a logical key structure is usually followed. Further entries of the subject will occur throughout the fugue, repeating the accompanying material at the same time. The various entries may or may not be separated by episodes.
What follows is a chart displaying a fairly typical fugal outline, and an explanation of the processes involved in creating this structure.
|Final Entries in Tonic|
|Tonic||Dom.||T||(D-redundant entry)||Relative Maj/Min||Dom. of Rel.||Subdom.||T||T|
S = subject; A = answer; CS = countersubject
A fugue begins with the exposition of its subject in one of the voices alone in the tonic key. After the statement of the subject, a second voice enters and states the subject with the subject transposed to another (often closely related) key, which is known as the answer. To make the music run smoothly, it may also have to be altered slightly. When the answer is an exact copy of the subject to the dominant, it is classified as a real answer; if it has to be altered in any way it is a tonal answer.
A tonal answer is usually called for when the subject begins with a prominent dominant note, or where there is a prominent dominant note very close to the beginning of the subject. To prevent an undermining of the music's sense of key, this note is transposed up a fourth to the tonic rather than up a fifth to the supertonic. Answers in the subdominant are also employed for the same reason.While the answer is being stated, the voice in which the subject was previously heard continues with new material. If this new material is reused in later statements of the subject, it is called a countersubject; if this accompanying material is only heard once, it is simply referred to as free counterpoint. invertible or double counterpoint. In tonal music invertible contrapuntal lines must be written according to certain rules because several intervallic combinations, while acceptable in one particular orientation, are no longer permissible when inverted. For example, when the note "G" sounds in one voice above the note "C" in lower voice, the interval of a fifth is formed, which is considered consonant and entirely acceptable. When this interval is inverted ("C" in the upper voice above "G" in the lower), it forms a fourth, considered a dissonance in tonal contrapuntal practice, and requires special treatment, or preparation and resolution, if it is to be used. The countersubject, if sounding at the same time as the answer, is transposed to the pitch of the answer. Each voice then responds with its own subject or answer, and further countersubjects or free counterpoint may be heard.
When a tonal answer is used, it is customary for the exposition to alternate subjects (S) with answers (A), however, in some fugues this order is occasionally varied: e.g., see the SAAS arrangement of Fugue No. 1 in C Major, BWV 846, from the Well-Tempered Clavier, Book 1 by J. S. Bach. A brief codetta is often heard connecting the various statements of the subject and answer. This allows the music to run smoothly. The codetta, just as the other parts of the exposition, can be used throughout the rest of the fugue.
The first answer must occur as soon after the initial statement of the subject as possible; therefore the first codetta is often extremely short, or not needed. In the above example this is the case: the subject finishes on the quarter note (or crotchet) B-flat of the third beat of the second bar which harmonizes the opening G of the answer. The later codettas may be considerably longer, and often serve to (a) develop the material heard so far in the subject/answer and countersubject and possibly introduce ideas heard in the second countersubject or free counterpoint that follows (b) delay, and therefore heighten the impact of the reentry of the subject in another voice as well as modulating back to the tonic.
The exposition usually concludes when all voices have given a statement of the subject or answer. In some fugues, the exposition will end with a redundant entry, or an extra presentation of the theme. Furthermore, in some fugues the entry of one of the voices may be reserved until later, for example in the pedals of an organ fugue (see J. S. Bach's Fugue in C major for Organ, BWV 547).
Further entries of the subject follow this initial exposition, either immediately (as for example in Fugue No. 1 in C major, BWV 846 of the Well-Tempered Clavier), or separated by episodes. Episodic material is always modulatory and is usually based upon some element heard in the exposition. Each episode has primarily the function of transitioning for the next entry of the subject in a new key, and may also provide release from the strictness of form employed in the exposition, and middle-entries. André Gedalge states that the episode of the fugue is generally based on a series of imitations of the subject that have been fragmented.
Further entries of the subject, or middle entries, occur throughout the fugue. They must state the subject or answer at least once in its entirety, and may also be heard in combination with the countersubject(s) from the exposition, new countersubjects, free counterpoint, or any of these in combination. It is uncommon for the subject to enter alone in a single voice in the middle-entries as in the exposition; rather, it is usually heard with at least one of the countersubjects and/or other free contrapuntal accompaniments. Middle-entries tend to occur at pitches other than the initial. As shown in the typical structure above, these are often closely related keys such as the relative dominant and subdominant, although the key structure of fugues varies greatly. In the fugues of J. S. Bach, the first middle-entry occurs most often in the relative major or minor of the work's overall key, and is followed by an entry in the dominant of the relative major or minor when the fugue's subject requires a tonal answer. In the fugues of earlier composers (notably Buxtehude and Pachelbel), middle entries in keys other than the tonic and dominant tend to be the exception, and non-modulation the norm. One of the famous examples of such non-modulating fugue occurs in Buxtehude's Praeludium (Fugue and Chaconne) in C, BuxWV 137.
When there is no entrance of the subject and answer material, the composer can develop the subject by altering the subject . This is called an episode, often by inversion, although the term is sometimes used synonymously with middle-entry and may also describe the exposition of completely new subjects, as in a double fugue for example (see below). In any of the entries within a fugue the subject may be altered, by inversion, retrograde (a less common form where the entire subject is heard back-to-front) and diminution (the reduction of the subject's rhythmic values by a certain factor), augmentation (the increase of the subject's rhythmic values by a certain factor) or any combination of them.
Example and analysis
The excerpt below, bars 7–12 of J. S. Bach's Fugue no. 2 in C minor, BWV 847, from the Well-Tempered Clavier, Book 1 illustrates the application of most of the characteristics described above. The fugue is for keyboard and in three voices, with regular countersubjects. This excerpt opens at last entry of the exposition: the subject is sounding in the bass, the first countersubject in the treble, while the middle-voice is stating a second version of the second countersubject, which concludes with the characteristic rhythm of the subject, and is always used together with the first version of the second countersubject. Following this an episode modulates from the tonic to the relative major by means of sequence, in the form of an accompanied canon at the fourth. Arrival in E-flat major is marked by a quasi perfect cadence across the barline, from the last quarter note beat of the first bar to the first beat of the second bar in the second system, and the first middle entry. Here Bach has altered countersubject 2 to accommodate the change of mode.
At any point in the fugue there may be false entries of the subject, which include the start of the subject but are not completed. False entries are often abbreviated to the head of the subject, and anticipate the "true" entry of the subject, heightening the impact of the subject proper.
The counter-exposition is a second exposition. However, there are only two entries, and the entries occur in reverse order. The counter-exposition in a fugue is separated from the exposition by an episode, and is in the same key as the original exposition.
StrettoSometimes counter-expositions or the middle entries take place in stretto, whereby one voice responds with the subject/answer before the first voice has completed its entry of the subject/answer, usually increasing the intensity of the music. Dona nobis pacem choruses from Bach's Mass in B Minor).
Final entries and coda
The closing section of a fugue often includes one or two counter-expositions, and possibly a stretto, in the tonic; sometimes over a tonic or dominant pedal note. Any material that follows the final entry of the subject is considered to be the final coda and is normally cadential.
Types of fugue
A simple fugue has only one subject, and does not utilize invertible counterpoint.
Double (triple, quadruple) fugue
A double fugue has two subjects that are often developed simultaneously; similarly, it follows that a triple fugue has three subjects. There are two kinds of double fugue: (a) a fugue in which the second subject is presented simultaneously with the subject in the exposition (e.g. as in prelude or toccata. The Art of Fugue, BWV 1080, is a collection of fugues (and four canons) on a single theme that is gradually transformed as the cycle progresses. Bach also wrote smaller single fugues, and put fugal sections or movements into many of his more general works.
J. S. Bach's influence extended forward through his son C.P.E. Bach and through the theorist Friedrich Wilhelm Marpurg (1718–1795) whose Abhandlung von der Fuge ("Treatise on the fugue", 1753) was largely based on J. S. Bach's work.
During the Classical era, the fugue was no longer a central or even fully natural mode of musical composition. Nevertheless, both Haydn and Mozart had periods of their careers in which they in some sense "rediscovered" fugal writing and used it frequently in their work.
Haydn was the leader of fugal composition and technique in the Classical era. Haydn's most famous fugues can be found in his Sun quartets (op. 20, 1772), of which three have fugal finales. This was a practice that Haydn repeated only once later in his quartet-writing career, with the finale of his quartet op. 50 no. 4 (1787). Some of the earliest examples of Haydn's use of counterpoint, however, are in three symphonies (No. 3, No. 13, and No. 40) that date from 1762–63. The earliest fugues, in both the symphonies and in the baryton trios, exhibit the influence of Joseph Fux's treatise on counterpoint, Gradus ad Parnassum (1725), which Haydn studied carefully. Haydn's second fugal period occurred after he heard, and was greatly inspired by, the oratorios of Handel during his visits to London (1791–1793, 1794–1795). Haydn then studied Handel's techniques and incorporated Handelian fugal writing into the choruses of his mature oratorios The Creation and The Seasons, as well as several of his later symphonies, including No. 88, No. 95, and No. 101.
Wolfgang Amadeus Mozart studied counterpoint when young with Padre Martini in Bologna. However, the major impetus to fugal writing for Mozart was the influence of Baron Gottfried van Swieten in Vienna around 1782. Van Swieten, during diplomatic service in Berlin, had taken the opportunity to collect as many manuscripts by Bach and Handel as he could, and he invited Mozart to study his collection and also encouraged him to transcribe various works for other combinations of instruments. Mozart was evidently fascinated by these works, and wrote a set of transcriptions for string trio of fugues from Bach's Well-Tempered Clavier, introducing them with preludes of his own. Mozart then set to writing fugues on his own, mimicking the Baroque style. These included the fugues for string quartet, K. 405 (1782) and a fugue in C Minor K. 426 for two pianos (1783). Later, Mozart incorporated fugal writing into the finale of his Symphony No. 41 and his opera Die Zauberflöte. The parts of the Requiem he completed also contain several fugues (most notably the Kyrie, and the three fugues in the Domine Jesu; he also left behind a sketch for an Amen fugue which, some believe, would have come at the end of the Sequentia).
Ludwig van Beethoven was familiar with fugal writing from childhood, as an important part of his training was playing from The Well-Tempered Clavier. During his early career in Vienna, Beethoven attracted notice for his performance of these fugues. There are fugal sections in Beethoven's early piano sonatas, and fugal writing is to be found in the second and fourth movements of the Eroica Symphony (1805). Beethoven incorporated fugues in his sonatas, and reshaped the episode's purpose and compositional technique for later generations of composers. Nevertheless, fugues did not take on a truly central role in Beethoven's work until his "late period." A massive, dissonant fugue forms the finale of his String Quartet, Op. 130 (1825); the latter was later published separately as Op. 133, the Große Fuge ("Great Fugue"). The finale of Beethoven's Hammerklavier Sonata contains a fugue, which was practically unperformed until the late 19th century, due to its tremendous technical difficulty and length. Furthermore, the last movement of his Cello Sonata, Op. 102 No. 2 is a fugue, and the whole work contains fugal passages. Fugal passages are also found in the Missa Solemnis and all movements of the Ninth Symphony (except the third one).
By the beginning of the Romantic era, fugue writing had become specifically attached to the norms and styles of the Baroque. Verdi included a whimsical example at the end of his opera Falstaff. Bruckner and Mahler also included them in their respective symphonies. Felix Mendelssohn wrote many fugues inspired by his study of the music of Johann Sebastian Bach.
Twentieth-century composers brought fugue back to its position of prominence, realizing its uses in full instrumental works, its importance in development and introductory sections, and the developmental capabilities of fugal composition.
Igor Stravinsky also incorporated fugues into his works, including the Symphony of Psalms and the Dumbarton Oaks concerto. Stravinsky recognized the compositional techniques of Bach, and in the second movement of his Symphony of Psalms, he lays out a fugue that is much like that of the Baroque era. It employs a double fugue with two distinct subjects, the first beginning in C and the second in E♭. Techniques such as stretto, sequencing, and the use of subject incipits are frequently heard in the movement.Olivier Messiaen, writing about his Vingt Regards sur l'Enfant Jésus, wrote of the sixth piece of that collection, "Par Lui tout a été fait" ("By Him were all things made"):
It expresses the Creation of All Things: space, time, stars, planets - and the Countenance (or rather, the Thought) of God behind the flames and the seething - impossible even to speak of it, I have not attempted to describe it ... Instead, I have sheltered behind the form of the Fugue. Bach's Art of Fugue and the fugue from Beethoven's Opus 106 (the Hammerklavier sonata) have nothing to do with the academic fugue. Like those great models, this one is an anti-scholastic fugue.
György Ligeti wrote a five-part double fugue for his "Requiem's" (1966) second movement, the "Kyrie", in which each part (S,M,A,T,B) is subdivided in four-voice "bundless" that make a canon. The melodic material in this fugue is totally chromatic, with melismatic (running) parts overlaid onto skipping intervals, and use of polyrhythm (multiple simultaneous subdivisions of the measure), blurring everything both harmonically and rhythmically so as to create an aural aggregate, thus highlighting the theoretical/aesthetic question of the next section as to whether fugue is a form or a texture.
Canadian pianist and musical thinker Glenn Gould composed So You Want to Write a Fugue?, a full-scale fugue set to a text that cleverly explicates its own musical form.
Musical form or texture?
A widespread view of the fugue is that it is not a musical form (in the sense that, say, sonata form is) but rather a technique of composition.
The Austrian musicologist
- Downloadable PDFs of J.S. Bach's Well-Tempered Clavier on Mutopia
- Fugues of the Well-Tempered Clavier (viewable in Adobe Flash or Shockwave)
- Theory on fugues
- Fugues and Fugue Sets
- with accompanying recordingsWell-Tempered ClavierAnalyses of J. S. Bach's
- Video Visualization of Bach's "Little" Fugue in G minor, Organ
- "Fugue, n." The Concise Oxford English Dictionary, eleventh edition, revised, ed. Catherine Soanes and Angus Stevenson (Oxford and New York: Oxford University Press, 2006).
- "Fugal, adj." The Concise Oxford English Dictionary, eleventh edition, revised, ed. Catherine Soanes and Angus Stevenson (Oxford and New York: Oxford University Press, 2006).
- "Fugue", The Concise Oxford Dictionary of Music, fourth edition, ed. Michael Kennedy (Oxford and New York: Oxford University Press, 1996). ISBN 0-19-280037-X
- "Fugue [Fr. fugue; Ger. Fuge; Lat., It., Sp., fuga]." The Harvard Dictionary of Music (Cambridge: Harvard University Press, 2003),
- for discussion of the changing use of the term throughout Western music history.
- Ratner 1980, p. 263
- Gedalge 1964, p. 7
- (subscription required)
- (subscription required)
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- Gedalge 1964, p. 70
- G. M. Tucker and Andrew V. Jones, "Fugue", in The Oxford Companion to Music, ed. Alison Latham (Oxford and New York: Oxford University Press, 2002). ISBN 0-19-866212-2
- Gedalge 1964, p. 12
- Verrall 1966, p. 12
- Gedalge 1964, p. 59
- "Invertible Counterpoint" The Oxford Companion to Music, ed. Alison Latham (Oxford and New York: Oxford University Press, 2002)
- (subscription required)
- Gedalge 1964, p. 61
- Gedalge 1964, pp. 71–72
- Paul Walker, "Fugue, §1: A Classic Fugue Analysed"
- Verrall 1966, p. 33
- Gedalge 1964
- (subscription required)
- Gedalge 1964, p. 108
- (subscription required)
- Verrall 1966, p. 77
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- "double fugue" The Oxford Companion to Music, Ed. Alison Latham, Oxford University Press, 2002,
- "Double Fugue", The Concise Oxford Dictionary of Music, fourth edition, ed. Michael Kennedy (Oxford and New York: Oxford University Press, 1996)
- (subscription required)
- (subscription required)
- Walker 1992, p. 56
- Walker 2000, p. 7
- Walker 2000, pp. 9–10
- Mann 1960, p. 9
- Walker 2000, p. 165
- Walker 2000, p. 316
- Walker 2000, p. 317
- Mann 1960, p. 53
- Walker 2000, p. 2
- Graves 1962, p. 64
- Ratner 1980, p. 266
- Graves 1962, p. 65
- Graves 1962, p. 67
- Notes to Vingt Regards sur l'Enfant Jésus. Translator not indicated. Erato Disques S.A. 4509-91705-2, 1993. Compact Disc.
- Eric Drott, "Lines, Masses, Micropolyphony: Ligeti's Kyrie and the 'Crisis of the Figure'". Perspectives of New Music 49, no. 1 (Winter 2011): 4–46. Citation on 10.
- Ratz 1951, Chapter 3
- Ratz 1951, p. 259
- Kivy 1990, p. 206
- Kivy 1990, p. 210
- Mann 1965, p. 16
the composer, having completed the initial imitative entrances, take the passage which has served as accompaniment to the theme and make it the basis for new imitative treatment, so that "he will always have material with which to compose without having to stop and reflect". This formulation of the basic rule for fugal improvisation anticipates later sixteenth-century discussions which deal with the improvisational technique at the keyboard more extensively.
This is related to the idea that restrictions create freedom for the composer, by directing their efforts. He also points out that fugue writing has its roots in improvisation, and was, during the Renaissance, practiced as an improvisatory art. Writing in 1555, Nicola Vicentino, for example, suggests that:
In presenting Bach's fugues as among the greatest of contrapuntal works, Peter Kivy points out that "counterpoint itself, since time out of mind, has been associated in the thinking of musicians with the profound and the serious" and argues that "there seems to be some rational justification for their doing so."
Fugue is the most complex of contrapuntal forms. In Ratz's words, "fugal technique significantly burdens the shaping of musical ideas, and it was given only to the greatest geniuses, such as Bach and Beethoven, to breathe life into such an unwieldy form and make it the bearer of the highest thoughts."
Perceptions and aesthetics
works), use panchromatic or even denser harmonic spectra. Witold Lutosławski, Requiem's György Ligeti in Kyrie/Christe. This is unlike later forms such as the sonata, which clearly prescribes which keys are explored (typically the tonic and dominant in an ABA form). Then, many modern fugues dispense with traditional tonal harmonic scaffolding altogether, and either use serial (pitch-oriented) rules, or (as the subdominant and the supertonic, the relative minor explores the Well Tempered Clavier major from book one of the ♭ are more commonly explored in fugal development, the overall structure of a fugue does not limit its harmonic structure. For example, a fugue may not even explore the dominant, one of the most closely related keys to the tonic. Bach's Fugue in Brelated keys, whereas the episodes usually explore more distant tonalities. Ratz stressed, however, that this is the core, underlying form ("Urform") of the fugue, from which individual fugues may deviate. Thus it is to be noted that while certain tonic key In particular, the exposition and coda tend to emphasize the  Bach's most famous fugues are those for the harpsichord in
The Baroque period also saw a rise in the importance of music theory. Some fugues during the Baroque period were pieces designed to teach contrapuntal technique to students. The most influential text was published by Johann Joseph Fux (1660–1741), his Gradus Ad Parnassum ("Steps to Parnassus"), which appeared in 1725. This work laid out the terms of "species" of counterpoint, and offered a series of exercises to learn fugue writing. Fux's work was largely based on the practice of Palestrina's modal fugues. Mozart studied from this book, and it remained influential into the nineteenth century. Haydn, for example, taught counterpoint from his own summary of Fux, and thought of it as the basis for formal structure.
It was in the oratorios. Keyboard suites from this time often conclude with a fugal gigue. Domenico Scarlatti has only a few fugues among his corpus of over 500 harpsichord sonatas. The French overture featured a quick fugal section after a slow introduction. The second movement of a sonata da chiesa, as written by Arcangelo Corelli and others, was usually fugal.
"Fugue" as a theoretical term first occurred in 1330 when Jacobus of Liege wrote about the fuga in his Speculum musicae. The fugue arose from the technique of "imitation", where the same musical material was repeated starting on a different note. Gioseffo Zarlino, a composer, author, and theorist in the Renaissance, was one of the first to distinguish between the two types of imitative counterpoint: fugues and canons (which he called imitations). Originally this was to aid improvisation, but by the 1550s, it was considered a technique of composition. The Renaissance composer Giovanni Pierluigi da Palestrina (1525?-1594) wrote masses using modal counterpoint and imitation, and fugal writing became the basis for writing motets as well. Palestrina's imitative motets differed from fugues in that each phrase of the text had a different subject which was introduced and worked out separately, whereas a fugue continued working with the same subject or subjects throughout the entire length of the piece.
The term fuga was used as far back as the Middle Ages, but was initially used to refer to any kind of imitative counterpoint, including canons, which are now thought of as distinct from fugues. Prior to the 16th century, fugue was originally a genre. It was not until the 16th century that fugal technique as it is understood today began to be seen in pieces, both instrumental and vocal. Fugal writing is found in works such as fantasias, ricercares and canzonas.
Middle Ages and Renaissance Period
A fughetta is a short fugue that has the same characteristics as a fugue. Often the contrapuntal writing is not strict, and the setting less formal. See for example, variation 24 of Beethoven's "Diabelli Variations" Op. 120.
Invertible counterpoint is essential to permutation fugues but is not found in simple fugues.
Permutation fugues differ from conventional fugue in that there are no connecting episodes, nor statement of the themes in related keys. So for example, the fugue of Bach's Passacaglia and Fugue in C minor, BWV 582 is not purely a permutation fugue, as it does have episodes between permutation expositions.
Permutation fugue describes a type of composition (or technique of composition) in which elements of fugue and strict canon are combined. Each voice enters in succession with the subject, each entry alternating between tonic and dominant, and each voice, having stated the initial subject, continues by stating two or more themes (or countersubjects), which must be conceived in correct invertible counterpoint. (In other words, the subject and countersubjects must be capable of being played both above and below all the other themes without creating any unacceptable dissonances.) Each voice takes this pattern and states all the subjects/themes in the same order (and repeats the material when all the themes have been stated, sometimes after a rest). There is usually very little non-structural/thematic material. During the course of a permutation fugue, it is quite uncommon, actually, for every single possible voice-combination (or "permutation") of the themes to be heard. This limitation exists in consequence of sheer proportionality: the more voices in a fugue, the greater the amount of possible permutations. In consequence, composers exercise editorial judgment as to the most musical of permutations and processes leading thereto. One example of permutation fugue can be seen in the opening chorus of Bach's cantata, Himmelskönig, sei willkommen, BWV182.
A counter-fugue is a fugue in which the first answer is presented as the subject in inversion (upside down), and the inverted subject continues to feature prominently throughout the fugue. Examples include Contrapunctus V through Contrapunctus VII, from Bach's The Art of Fugue.