Definitions

triple fugue

Fugue

[fyoog]

In music, a fugue is a type of contrapuntal composition or technique of composition for a fixed number of parts, normally referred to as "voices". 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. A fugue opens with one main theme, the subject, which then sounds successively in each voice in imitation; when each voice has entered, the exposition is complete; this is occasionally 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, fugue is a style of composition, rather than fixed structure. Though there are certain established practices, in writing the exposition for example, composers approach the style with varying degrees of freedom and individuality.

The form evolved during the 18th century from several earlier types of contrapuntal compositions, such as imitative ricercars, capriccios, canzonas, and fantasias. Middle and late Baroque composers such as Dietrich Buxtehude (1637–1707) and Johann Pachelbel (1653–1706) contributed greatly to the development of the fugue, and the form reached ultimate maturity in the works of Johann Sebastian Bach (1685–1750). With the decline of sophisticated contrapuntal styles at the end of the baroque period, the fugue's popularity as a compositional style waned, eventually giving way to sonata form. Nevertheless, composers from the 1750s to the present day continue to write and study fugue for various purposes; they appear in the works of Mozart (e.g. Kyrie Eleison of the Requiem in D minor) and Beethoven (e.g. end of the Credo of the Missa Solemnis), and many composers such as Anton Reicha (1770–1836) and Dmitri Shostakovich (1906–1975) wrote cycles of fugues.

The English term fugue originates in the 16th century and is derived from either the French or Italian fuga, which 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).

Musical outline

A fugue begins with what is known as the exposition and is characteristically written according to certain predefined rules; in later portions the composer has somewhat more freedom, though a logical key structure is usually followed, and 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 a detailed explanation of the processes involved in creating this structure, with examples.

Example of Key/Entry Structure, in a Three-Voice Baroque Fugue
Exposition 1st Middle-Entry 2nd Middle-Entry Final Entries in Tonic
Tonic Dom. T (D-redundant entry) Relative Maj/Min Dom. of Rel. Subdom. T T
Sop. Subj. CS1 C
O
D
E
T
T
A
CS² A E
P
I
S
O
D
E

CS1 CS² E
P
I
S
O
D
E
S E
P
I
S
O
D
E
CS1 Free Counterpoint C
O
D
A
Alto Ans. CS1 CS² S CS1 CS² S CS1
Bass S CS1 CS² A CS1 CS² S

The Exposition

A fugue begins with the exposition of its subject sounding in one of the voices alone in the tonic key. After the statement of the subject, a second voice enters with the subject transposed to the dominant, which is known as the answer. Sometimes the answer is the tonic or subdominant (see J.S.Bach's Toccata and Fugue in D minor, BWV 565, and the opening fugato of the Partita No 2 in C minor, BWV 826); to avoid disturbing the sense of key, it may also have to be altered slightly. When the answer is an exact transposition 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.

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The first note of the subject, D, (in red) is a prominent dominant note, demanding that the first note of the answer (in blue) sounds as the tonic, G, rather than A.]]

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 and tend to occur in specific circumstances: a) when the subject begins with the following scale tones: 5-4-5 or 5-4-3; and b) when subjects themselves modulate to the dominant, in which case, the answer begins in the subdominant, and subsequently modulates to the tonic.

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. Each voice then responds with its own subject or answer in turn, 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 either a) return to the tonic, following an answer in the dominant, or b) to modulate to the dominant to enable a statement of the answer.

The first answer must occur as soon after the initial statement of the subject as possible, therefore the first codetta is often extremely short, and in many cases is not even 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 second and 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 many fugues, however, there is one more entry of the subject, called a redundant entry, in which all of the voices sound simultaneously. 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). Whilst the voices of a fugue may enter in any order desired, Bach frequently reserves the bass entry until last, for dramatic effect. Opening with the subject stated in the bass, whilst unusual, is to be preferred when other actions would present illegal inversions in the exposition.

The Countersubject(s)

The distinction is made between the use of free counterpoint and regular countersubjects accompanying the fugue subject/answer, because in order for a countersubject to be heard accompanying the subject in more than one instance, it must be capable of sounding correctly above or below the subject, and must be conceived, therefore, in invertible or double counterpoint. In tonal music invertible contrapuntal lines must be written according to certain rules because several intervallic combinations, whilst 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 is to be used. Likewise, the use of a 4-3 suspension is somewhat ineffectual in invertible counterpoint; in inversion it becomes 5-6, and since both the fifth and sixth are considered consonant intervals, is not a suspension at all, whereas the 7-6 suspension may be used successfully as it inverts to 2-3.

When writing invertible countersubjects in a fugue, composers therefore usually restrict themselves to the intervals of the third, which inverts to a sixth, and octaves, which invert to the unison, in addition the careful use of the 7-6 suspension.

The Episode

Further entries of the subject follow this initial exposition, either immediately (as for example in Fugue no.1 in C major, BWV 846, WTC Bk 1), 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 modulating 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. Whilst not all of Bach's fugues contain episodes, most have at least one serving as a modulatory bridge passage between the middle-entries, on which the greater emphasis is usually laid.

An episode is an interlude between enteries of the subject for modulatry purposes.

The Middle-Entries

Further entries of the subject, or middle entries, occur at various intervals throughout the fugue, 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 just as it is 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 in keys other than the tonic - as shown in the typical structure above, these are often closely related keys such as the relative dominant and subdominant, but key structure of fugues varies greatly. In the fugues of J.S. Bach (see for example Fugue no. 2 in C Minor, BWV 847, from J.S. Bach's Well-Tempered Clavier Book 2, quoted below) the first middle-entry very commonly occurs in the relative major or minor of the work's overall key, and is followed by an entry the dominant of the relative major or minor when the fugue's subject requires a tonal answer.

Some fugues may also have one or more counter-expositions, similar to the opening exposition, but in which the subject is altered in some way, 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, the second countersubject in the middle-voice. 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 4th. 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.

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False entries

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.

Stretto

Sometimes 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. Only one entry of the subject must be heard in its completion in a stretto. However, a stretto in which the subject/answer is heard in completion in all voices is known as stretto maestrale or grand stretto. Strettos may also occur by inversion, augmentation and diminution. A fugue in which the opening exposition takes place in stretto form is known as a close fugue or stretto fugue (see for example, the Gratias agimus tibi and Dona nobis pacem choruses from Bach's Mass in B Minor). In general, fugues that are densely strettoed will not contain countersubjects, and vice versa. One notable exception is the E Major fugue from Bach's Well-Tempered Clavier, Book II, which initially exposes the subject accompanied by its countersubject, followed by counterexposition of the two ideas, separated in time, and each in stretto with itself.

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

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. It is not widely agreed whether a double fugue can be defined as a) a fugue in which the second subject is presented simultaneously with the subject in the exposition (e.g. as in Kyrie Eleison of Mozart's Requiem in D minor), or b) a fugue in which the second subject has its own exposition at some later point, and the two subjects are not combined until later (see for example, fugue no. 14 in f-sharp minor from Bach's Well-tempered Clavier Book 2).

Counter-Fugue

A counter-fugue is a fugue in which the first answer is presented as the subject in inversion, and the inverted subject continues to feature prominently throughout the fugue, as for example in Bach's The Art of Fugue, Contrapunctus IV.

Permutation Fugue

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. 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. It is common during the course of a permutation fugue for every single possible voice-combination (or 'permutation') of the themes to be heard at some point. One example of permutation fugue can be seen in the opening chorus of Bach’s cantata, Himmelskönig, sei willkommen BWV182.

Permutation fugues differ from conventional fugue in that there are no connecting episodes, nor statement of the themes in related keys.

History

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. 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.

The fugue arose from the technique of "imitation", where the same musical material was repeated starting on a different note. 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.

Baroque era

It was in the Baroque period that the writing of fugues became central to composition, in part as a demonstration of compositional expertise. Fugues were incorporated into a variety of musical forms. Jan Pieterszoon Sweelinck, Girolamo Frescobaldi, Johann Jakob Froberger and Dieterich Buxtehude all wrote fugues, and George Frideric Handel included them in many of his oratorios. Keyboard suites from this time often conclude with a fugal gigue. 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.

The Baroque period also saw a rise in the importance of music theory. 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. 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.

Johann Sebastian Bach often entered into contests where he would be given a subject with which to spontaneously improvise a fugue on the organ or harpsichord. This musical form was also apparent in chamber music he would later compose for Weimar; the famous Concerto for Two Violins in D Minor (BWV 1043) (although not contrapuntal in its entirety) has a fugal opening section to its first movement.

Bach's most famous fugues are those for the harpsichord in The Well-Tempered Clavier and the Art of Fugue, and his organ fugues, which are usually preceded by a prelude or toccata. The Art of Fugue is a collection of fugues (and four canons) on a single theme that is gradually transformed as the cycle progresses. The Well-Tempered Clavier comprises two volumes written in different times of Bach's life, each comprising 24 prelude and fugue pairs, one for each major and minor key. Bach also wrote smaller single fugues, and put fugues into many of his works that were not fugues per se.

Although J. S. Bach was not well known as a composer in his lifetime, his 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.

Classical era

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'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.

Mozart studied counterpoint when young with Padre Martini in Rome. 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 would have come at the end of the Sequentia).

A common characteristic of the Classical composers is that they usually wrote fugues not as isolated works but as part of a larger work, often as a sonata-form development section or as a finale. It was also characteristic to abandon fugal texture just before the end of a work, providing a purely homophonic resolution. This is found, for instance, in the final fugue of the chorus "The Heavens are Telling" in Haydn's The Creation (1798).

Romantic era

By the beginning of the Romantic era, fugue writing had become specifically attached to the norms and styles of the Baroque. 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). Nevertheless, fugues did not take on a truly central role in Beethoven's work until his "late period." A fugue forms the development section of the last movement of his piano sonata op. 101 (1816), and massive, dissonant fugues form the finales of his Hammerklavier piano sonata (1818) and string quartet op. 130 (1825); the latter was later published separately as op. 133, the Grosse Fuge ("Great Fugue"). Also his piano sonata op. 110 and his cello sonata op. 102,2 have fugue movements. Beethoven's last piano sonata, op. 111 (1822) integrates fugal texture throughout the first movement, written in sonata form. Fugues are also found in the Missa Solemnis and in the finale of the Ninth Symphony.

One manual explicitly stated that the hallmark of contrapuntal style was the style of J. S. Bach. The 19th century's taste for academicism - setting of forms and norms by explicit rules - found Marpurg, and the fugue, to be a congenial topic. The writing of fugues also remained an important part of musical education throughout the 19th century, particularly with the publication of the complete works of Bach and Handel, and the revival of interest in Bach's music.

Examples of fugal writing in the Romantic era are found in the last movement of Berlioz's Symphonie Fantastique, and Wagner's Meistersinger, in particular the triple fugue at the conclusion of the second act. The finale of Giuseppe Verdi's opera Falstaff is a ten-voice fugue. Felix Mendelssohn was obsessed with fugal writing, as it can be found prominently in the Scottish Symphony, Italian Symphony, and the Hebrides Overture. In the last movement of his Fifth Symphony Anton Bruckner wrote the development section in form of a big double fugue. The unfinished Finale of his Ninth Symphony has a fugue section, too. Another composer of this time, whose work is strongly influenced by fugal textures, was Felix Draeseke. Especially in his church music highly artificial fugues could be found.

Robert Schumann, and Johannes Brahms also included fugues in many of their works. The final part of Schumann's Piano Quintet is a double fugue, and his opus numbers 126, 72 and 60 are all sets of fugues for the piano (opus 60 based on the BACH motif). The recapitulation of Liszt's B minor sonata is cast in the form of a 3-part fugato. The Quasi-Faust movement of Charles-Valentin Alkan's Grande Sonate contains a bizarre but musically convincing fugue in 9 parts. Brahms' Variations and Fugue on a Theme by Handel ends with a fugue, as does his Cello Sonata No. 1. Towards the end of the Romantic era, Richard Strauss included a fugue in his tone poem, Also sprach Zarathustra, to represent the high intelligence of science. Sergei Rachmaninoff, despite writing in a lush post-romantic idiom, was highly skilled in counterpoint (as is highly evident in his Vespers); a well known fugue occurs in his Symphony No. 2. Alexander Glazunov wrote a very difficult Prelude and Fugue in D minor, his Op. 62, for the piano.

20th century

The late Romantic composer Max Reger had the closest association with the fugue among his contemporaries. Many of his organ works contain, or are themselves fugues. Two of Reger's most-played orchestral works, the Hiller variations and the Mozart variations, end with a large-scale orchestral fugue.

A number of other twentieth century composers made extensive use of the fugue. Béla Bartók opened his Music for Strings, Percussion and Celesta with a fugue in which the tritone, rather than the fifth, is the main structural interval. He also included fugal sections in the final movements of his String Quartet No. 1, String Quartet No. 5, Concerto for Orchestra, and Piano Concerto No. 3. The second movement of his Sonata for Solo Violin is also a fugue. The Czech composer Jaromir Weinberger studied fugue with Max Reger, and had an uncommonly facile skill in fugal writing. The fugue of the "Polka and Fugue" from his opera "Schwanda the Bagpiper" is an example.

Igor Stravinsky also incorporated fugues into his works, including the Symphony of Psalms and the Dumbarton Oaks concerto. The last movement of Samuel Barber's famous Sonata for Piano is a sort of "modernized" fugue, which, instead of obeying the constraint of a fixed number of voices, develops the fugue subject and its head-motif in various contrapuntal situations. Elliott Carter uses a fugue for the second movement of his Piano Sonata (1945–1946). In a different direction, the tonal fugue movement of Charles Ives' fourth symphony evokes a nostalgia for an older, halcyon time. The practice of writing fugue cycles in the manner of Bach's Well-Tempered Clavier was perpetuated by Paul Hindemith in his Ludus Tonalis, Kaikhosru Sorabji in a number of his works including the Opus clavicembalisticum, and Dmitri Shostakovich in his Preludes and Fugues, opus 87 (which, like the Well-Tempered Clavier, contains a prelude and fugue in each key, although the order of Shostakovich's pieces follows the cycle of fifths, whereas Bach's progressed chromatically). Several Bachianas Brasileiras of Heitor Villa-Lobos feature a fugue as one of the movements. Ástor Piazzolla also wrote a number of fugues in his Nuevo tango style. György Ligeti wrote a Fugue for his "Requiem" (1966), which consists of a 5 part fugue in which each part (S,M,A,T,B) is subsequently divided in four voices that make a canon.

Benjamin Britten composed a fugue for orchestra in his The Young Person's Guide to the Orchestra, consisting of subject entries by each instrument once. Leonard Bernstein wrote a "Cool Fugue" as part of his musical West Side Story. Stephen Schwartz wrote a song from his 1974 Broadway hit The Magic Show called "The Goldfarb Variations" which uses fugue-like vocal counterpoint. Jazz musician Alec Templeton even wrote a fugue (recorded subsequently by Benny Goodman): "Bach Goes to Town." Canadian pianist 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. John Cage wrote a fugue for unpitched percussion instruments which substitutes complex rhythmic ratios for key areas.

20th-century fugue writing explored many of the directions implied by Beethoven's Grosse Fuge, and what came to be termed "free counterpoint" as well as "dissonant counterpoint." Fugal technique as described by Marpurg became part of the theoretical basis for Schoenberg's twelve-tone technique. Indeed, the aforementioned Glenn Gould, wrote a 12-tone Fugue in his Bassoon Sonata (later arranged for Flute and Piano) in 1950. In this Fugue, the Subject in the Bassoon later becomes the Answer in the Piano at the Tritone. It is important to note that Gould wrote traditional tonal-style canons with his rows as opposed to adopting Webern-like canonic principles where the tone row dictates the phrase entry.

Is the fugue a 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. For instance, Donald Francis Tovey wrote that "Fugue is not so much a musical form as a musical texture," that can be introduced anywhere as a distinctive and recognizable technique, often to produce intensification in musical development.

On the other hand, composers almost never write music in a purely cumulative fashion, and usually a work will have some kind of overall formal organization—hence the rough outline given above, involving the exposition, the sequence of episodes, and the concluding coda. When scholars say that the fugue is not a musical form, what is usually meant is that there is no single formal outline into which all fugues reliably can be fitted.

The Austrian musicologist Erwin Ratz argues that the formal organization of a fugue involves not only the arrangement of its theme and episodes, but also its harmonic structure. In particular, the exposition and coda tend to emphasize the tonic key, whereas the episodes usually explore more distant tonalities. However, it is to be noted that while certain related keys are more commonly explored in fugal development, the overall structure of a fugue does not limit its harmonic structure as much as Ratz would have us believe. For example, a fugue may not even explore the dominant, one of the most closely related keys to the tonic. Bach's Fugue in Bb from the Well Tempered Clavier explores the relative minor, the supertonic and the subdominant. 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).

Fugues are also not limited in the way the exposition is structured, the number of expositions in related keys, or the number of episodes (if any). So, the fugue may be considered a compositional practice rather than a compositional form, similar to the invention. The fugue, like the invention and sinfonia, employs a basic melodic subject and spins out additional melodic material from it to develop an entire piece. Fugal technique is really just a way to develop pieces of a particular contrapuntal style.

Perceptions and aesthetics

Fugue is the most complex of contrapuntal forms and as a result, gifted composers have used it to express the profound. The philosopher Theodor Adorno, a skilled pianist and interpreter of Beethoven's music, expressed a sense of the arduousness and also the inauthenticity of modern fugue composition, or any composing of fugue in a contemporary context, i.e., as an anachronism . Adorno's conservative and historically bound view of Bach is not found among most modern fugue composers, such as David Diamond, Paul Hindemith or Dmitri Shostakovich. The most classicist fugues that have appeared after Beethoven are those of Felix Mendelssohn, who as a child impressed Goethe and others with his mastery of counterpoint while improvising at the piano. In the words of Ratz, "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.

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. Because of the way fugue is often taught, the form can be seen as dry and filled with laborious technical exercises. The term "school fugue" is used for a very strict form of the fugue that was created to facilitate teaching.

Others, such as Alfred Mann, argued that fugue writing, by focusing the compositional process actually improves or disciplines the composer towards musical ideas. 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 baroque, practiced as an improvisatory art.

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