Modulating subject

In music, a fugue () is a contrapuntal compositional technique in two or more voices, built on a subject (a musical theme) that is introduced at the beginning in imitation (repetition at different pitches) and which recurs frequently in the course of the composition. It is not to be confused with a ''fuguing tune'', which is a style of song popularized by and mostly limited to early American (i.e. shape note or "Sacred Harp") music and West Gallery music. A fugue usually has three main sections: an exposition, a development and a final entry that contains the return of the subject in the fugue's tonic key. Some 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 completed the subject, 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."Fugue", ''The Concise Oxford Dictionary of Music'', fourth edition, ed. Michael Kennedy (Oxford and New York: Oxford University Press, 1996). 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 Jan Pieterszoon Sweelinck (1562-1621), 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 such as Dmitri Shostakovich (1906–1975).

Etymology

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).

Musical outline

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.

::S = subject; A = answer; CS = countersubject; T = tonic; D = dominant

Exposition

A fugue begins with the exposition of its subject in one of the voices alone in the tonic key.
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). After the statement of the subject, a second voice enters and states the subject with the subject transposed to another key (usually the dominant or subdominant), 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 new key, with identical intervals to the first statement, it is classified as a ''real answer''; if the intervals are altered to maintain the key 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''.

The countersubject is written in invertible counterpoint at the octave or fifteenth. 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 (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 J.S. Bach's '' Well-Tempered Clavier, Book 1''. 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 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).

Episode

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 the primary 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.

Development

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 major is marked by a quasi perfect cadence across the bar line, 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 the second countersubject to accommodate the change of mode.

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.

Counter-exposition

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.

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 ' choruses from J.S. 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

Simple 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, a triple fugue has three subjects. There are two kinds of double (triple) fugue: (a) a fugue in which the second (third) subject is (are) presented simultaneously with the subject in the exposition (e.g. as in Kyrie Eleison of Mozart's Requiem in D minor or the fugue of Bach's Passacaglia and Fugue in C minor, BWV 582), and (b) a fugue in which all subjects have their own expositions at some point, and they are not combined until later (see for example, the three-subject Fugue No. 14 in F minor from Bach's ''Well-Tempered Clavier'' Book 2, or more famously, Bach's "St. Anne" Fugue in E major, BWV 552, a triple fugue for organ.)

Counter-fugue

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''.

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. (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 number 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 eighth and final chorus of J.S. Bach's cantata, ''Himmelskönig, sei willkommen'', BWV 182.

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. Invertible counterpoint is essential to permutation fugues but is not found in simple fugues.

Fughetta

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.

History

Middle Ages and Renaissance

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.

"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 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. 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.

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 Johann Joseph Fux's ''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.

Bach's most famous fugues are those for the harpsichord in ''The Well-Tempered Clavier'', which many composers and theorists look at as the greatest model of fugue. ''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 is also known for his organ fugues, which are usually preceded by a 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.

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

Joseph 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 String 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 to 1763. 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; and the late string quartets, Opus 71 no. 3 and (especially) Opus 76 no. 6.

Mozart

The young Wolfgang Amadeus Mozart studied counterpoint with Padre Martini in Bologna. Under the employment of Archbishop Colloredo, and the musical influence of his predecessors and colleagues such as Johann Ernst Eberlin, Anton Cajetan Adlgasser, Michael Haydn, and his own father, Leopold Mozart at the Salzburg Cathedral, the young Mozart composed ambitious fugues and contrapuntal passages in Catholic choral works such as Mass in C minor, K. 139 "Waisenhaus" (1768), Mass in C major, K. 66 "Dominicus" (1769),