Although the statement is debatable, it is often said that the fugue is the most complex and highly developed type of composition in Western music. The term fugue, derived from fuga, the Latin word for "flight," was first used about 1330 by Jacques de Liége, the author of Speculum musicae, an important medieval treatise. At that time it referred to a technique of musical writing based on strict imitation. Later, after its emergence as an independent musical form in the 17th century, the fugue became a composition in counterpoint based on a generating theme, in which different pans, or voices, enter successively in imitation, as if in pursuit of each other. The heir of all the compositional techniques that had developed earlier, it differsfrom its ancestors (the motet, the ricercar, the canzona) in having a more specifically tonal character, unity of form, and a greater economy.

Description of the fugue

The fugue is written in counterpoint, two or more lines that sound simultaneously. Counterpoint's laws and techniques have developed from the 10th century to the Renaissance, a period during which Western music was essentially polyphonic. One of the main problems was the harmonic aspect of the meeting of the voices, and the rules of counterpoint are always precise regarding the use of consonance and dissonance. Counterpoint deals also with movement between the parts. It includes various techniques of development, among which imitation is probably the most remarkable feature of polyphonic music. There arc many kinds of imitations. The strictest is the canon, in which the melody slated by the first voice is later reproduced by the second voice. A good example is the song "Frère Jacques." Other common types of imitation include inversion of all the intervals, augmentation (in which the rhythmic values are doubled), diminution (in which they are reduced), or even retrograde imitation, in which the last note of one voice becomes the first note of the next. All these techniques are used in fugal composition, which is characterized more by its "language" than by its form.

Elements. The fugue is written for a certain number of voices, or instrumental parts. The most frequent are fugues for three or four voices, but there arc also fugues for two, five, or more voices. Although the fugal form varies from composer to composer, there arc certain commonelements.

Subject. The subject is the theme of the fugue. It is stated "alone by the first voice before being taken up by the others. In the course of the fugue, it will bestated in different keys, and it will be sometimes slightly modified or inverted. Some of its elements may be developed separately.

Answer. 1 he second voice brings in the answer, generally stated in the key of the dominant (the fifth degree of the major or minor scale). If it reproduces the subject exactly, it is called a real answer. But in most cases, in order topreserve the tonal unity of the fugue, the answer has to undergo a "mutation' that alters some of its melodic intervals and makes the modulation to the dominant key smoother. This is called the tonal answer.

Countersubject. The countersubject accompanies the answer. If it is maintained throughout the fugue, it is called sustained or obligate countersubject and will follow the subject like its shadow for each new statement. Subject and countersubject arc the two principal "actors" of the fugue, and theoretically all the musical substance must he derived from them.

Exposition. The first part of the fugue, which includes the successive entrance of the voices, in subject-answer alternation, is called exposition. This progressive enrichment of the polyphonic web is one of the most striking traits of the fugue. In some fugues, after the exposition, the composer brings in the answer followed by the subject. This is called counterexposition.

Episode. An episode is any passage, developed or not. that links two statements of the subject. It is characteristically written in imitative style. Generally it uses a motive from the subject or the countersubject, but sometimes a new element is introduced. There is a great variety of episodes.

Stretto. When the subject overlaps the answer (or the answer the subject), it is called a stretto. This device, whereby the entries are drawn more closely together, is often used at the end of the fugue, where it achieves spectacular effects.

Plan of the fugue: Once past the exposition, the plan ofthe fugue depends on the will of the composer and the resources of its thematic elements. For instance, the Danish composer Dietrich Buxtchude (died 1707) often presents the subject and the answer only in the principal and in the dominant keys, in short expositions linked by small episodes. With J.S. Bach, the tonal plan becomes more elaborate and includes a journey to the principal neighbouring keys. Since the composer adapts his plan to the character and to the potential of his themes, the itinerary is always different. That is why in his hands the fugue becomes the most versatile of musical forms; each fugue of Bach brings a new solution to the problem of the relation between form and content.

The "school fugue." Theorists created an ideal plan of the fugue and gradually perfected it, in the 19th century, as fewer and fewer real fugues were being written. They devised a tripartite form consisting of the exposition, the development, and the stretto.

Varieties of fugue

Simple fugue This is monothematic, without a maintained countersubject (such as the Fugue in D Major of Bach's work The Well-Tempered Clavier, Book I, No. S). More elaborate fugues use one or more countersubjects. In a counterfugue, the answer imitates the subject by inversion. There are beautiful examples of this technique in Bach's work The Art of the Fugue, numbers 5 to 7.

Double fugue. There are two ways of writing a double fugue; either the two subjects may be presented simultaneously, in which case the fugue is not very different from a fugue with countersubject (Bach, Fugue in B Minor on a Theme of Corelli, BWV 579), or else the second subject has a special exposition. The latter yields, in general, a tripartite scheme: exposition and development of the first subject, exposition and development of the second subject, and finally combination of the two elements, which are devised so that they can be superimposed. A splendid example is given by Bach's Toccata end Fugue in F Major for organ, BWV 540. The same principles apply toa triple or quadruple fugue.

Fugitetta. This is a miniature fugue but strictly written, whereas the fugato starts like a fugue but gives up its discipline once past the exposition.

Choral fugues. The fugue plays a considerable role in works for chorus and orchestra. Generally the chorus sings in strict counterpoint, while the instruments play an expressive or decorative accompaniment. The composition techniques of the fugue can also be used in forms as universal as the prelude, the aria, the chorus, the overture, the concerto, and others.

Literature of the fugue

Since vocal polyphony was based on a text that had to be sung by each of the parts (either simultaneously or, mure frequently, in imitation), it was not very much concerned with the problem of form. That is why theorists and musicians concentrated on questions of texture.

At first the term fuga applied to strict imitations (which would now be called canons); the conceptevolved in a more general sense when it was realized that a freer use of all kinds of imitations offered much more stimulating opportunities. Besides the purely vocal polyphony, the first independent forms of keyboard music (the ricercar, the anzona the Capriccio, the fantasia) testify to the remarkable development of what can be called the fugal style of the 16th and 17th centuries. In this evolution, the role of the Italian composer Andrea Gabrieliand of the English virginalists was preponderant. The two great precursors of the fugue proper were the Dutch composer J.P. Sweelinck (died 1621) and the Italian Girolamo Frescobaldi (died 1643). Both greatly influenced the keyboard music of the 17th century, in particular through their students Samuel Scheidt and Johann Jakob Froberger. In Germany, a generation of musicians dominated by Buxtehude gave the fugue its modern form by putting it in a tonal perspective and abandoning the fragmentary style of their predecessors. Almost all the composers of the 17th century contributed to the history of the fugue.

J.S. Bach. The genius of Bach found particular expression in the fugue, perhaps because it allies the strictest economy of language to a relative freedom of form. Each of his fugues amazes by the freshness of its inspiration, the wonders of its writing, or by its gigantic proportions, all marvellously represented in the two volumes of The Well-TemperedClavier (1722-44), two sets of 24 preludes and fugues going through the cycle of the 24 major and minor Keys. Some of his organ fugues lend toward development, some toward symmetry, and some toward virtuosity; others take the form Of double fugues. His last work, The Art ofthe Fugue, is a collection of 14 fugues and four canons, all based on a theme in D minor and its inversion. All the resources and procedures of the fugue are demonstrated in what constitutes the most inspiring treatise on fugue, a treatise without-words, in which music speaks alone. After being long slighted as a purely theoretical work. The Art of the Fugue has won a high place in the hearts of music lovers, who see it as Bach's musical testament. Bach's cantatas, passions, and oratorios abound in admirable fugues.

Handel and after. Handel's fugues are less erudite than those of Bach and sometimes employ looser counterpoint, but they touch the listener by their vitality and their harmonious proportions. The great fugal sections of his oratorios are more important than his keyboard fugues.

After Bach the fugue lost much of its importance. With the appearance of the sonata, the musical taste changed, and composers tended to consider counterpoint as an archaic discipline. Nonetheless, the fugue retained a place in choral works, and fugal methods were kept in the sonata form, particularly in the development section. A great passion for Bach led Mozart (died 1791) to a more contrapuntal style. This influence is obvious in the Fugue in C Minor, for two pianos, K. 426, which, though pure Mozart, is nonetheless a. homage to Bach. In ingenuity and mastery, Mozart rivalled the greatest contrapuntists in, for instance, the great choruses of the Mass in C Minor, K. 427, or in the final development of the Jupiter Symphony.

Beethoven. Beethoven resorted more and more to fugal technique in his last works. He confessed that he wrote his fugues with the greatest difficulty, and it is true that his counterpoint gives an impression of effort. Mostof the fugal passages integrated into his last sonatas and quartets create a dramatic tension. Far from being a scholastic technique, the fugue was for Beethoven a means to reach the expressive limits of an idea. He used this language in particular circumstances. The strange and desolate atmosphere of the opening fugue of the String Quartet in C Sharp Minor, Opus 131, brings to mind the 20th-century fugue in the first section of the Music for Strings, Percussion and Celesta of Béla Bartók. The Promethean side of Beethoven asserts itself particularly in the Great Fugue from the String Quartet in B Flat Major, Opus 130, and the noble fugal section of the Missa Solemnis. Unlike Mozart's classical fugues, Beethoven's arc rather irrational in form but are justified by their creative power alone.

The Romantic era. Compared with the fugues of Beethoven. those of Felix Mendelssohn defer to traditional rules. Critics sense in them a nostalgia for Bach, sometimes weakened by a touch of sentimentalism. A more authentic romantic breath animates the fugues of Koben Schumann on B.A.C.H. (the German letters for the notes B-A-C-B). but the fugue is not his natural language. The genre is more suited to César Franck, as may be seen in his Prélude, fugue et variation, or in bis . Prelude, chorale et fugue.His harmonic sensitivity enriches a contrapuntal technique while not breaking with tradition. The fugues of Franz Liszt arc entirely different: once past the exposition, he cannot renounce symphonic developments. There is an original use of the fugato before the re-exposition in his Piano Sonata in B Minor. The fugal style was used to varying degree by the major composers of the 19th century, including Brahms in his German Requiem, Richard Wagner in his opera Die Meistersinger,and Verdi in his Requiem and at the end of his last opera, Falstafj.

Among the post-Romantics, who cultivated their own exaggerated form of counterpoint, mention must be made of Max Reger, whose admirers took, him as the heir of J.S. Bach. Though that appears to be going too far, his counterpoint does possess vitality in works such as the Variations and Fugue on a Theme by Mozart, for orchestra, or the fugue on the chorale melody "Wachet auf, ruftuns die Stimme," for organ.

The 20th century. As an autonomous form, the fugue played only a modest role in the first half of the 20th century. The most beautiful example is the already mentioned fugue of the Music for Strings, Percussion and Celesta of Béla Bartók, a born contrapuntist. This is a true model of fugal treatment in a post-tonal style. The methods of the fugue are also found in his Sonata for Two Pianos and Percussion and in his admirable quartets.

Stravinsky, though influenced by the composers of the I7th and 18th centuries, showed no particular interest in the fugue. Although the second part of his Symphony of Psalms can be considered a double fugue, it docs not strike the listener as such.

In his collection of interludes and fugues, called Ludus Tonalis, Paul Hindemith seems to have drawn his inspiration from Bach's Well-Tempered Clavier. Although an interesting work, it did not herald a rebirth of the fugue A new conception of counterpoint appeared in the works of Arnold Schoenberg and Anton von Webern. The serial techniques of composers like Pierre Boulez can. to a certain extent, claim some kind of kinship with fugal language, and a work such as the Passion According so St. Luke of Krzysztof Penderecki testifies to the permanence of a musical form the history of which is probably unfinished.

Canons and Fugues

The idea of a canon is that one single theme is played against itself. This is done by having "copies" of the theme played by the various participating voices. But there are many ways todo this. The most straight forward of all canons is the round, such as "Three Blind Mice", "Row, Row, Row Your Boar", or "Frére Jacques", Here, the theme enters in the first voice and, after a fixed lime-delay, a "copy" of it enters, in precisely the same key. Alter the same fixed time-delay in the second voice, the third voice enters carrying the theme, and so on. Most themes will not harmonize with themselves in this way. In order for a theme to work as a canon theme, each of its notes must he able to serve in a dual (or triple, or quadruple) role: it must firstly he part of a melody, and secondly it must be part of a harmonization of the same melody. When there are three canonical voices, for instance, each note of the theme must act in two distinct harmonic ways, as well as melodically. Thus, each note in a canon has more than one musical meaning; the listener's ear and brain automatically figure out the appropriate meaning, by referring to context.

There are more complicated sorts of canons, of course. The first escalation in complexity conies when the "copies" of the theme are staggered not only in time, but also in pitch; thus, the first voice might sing the theme starting on C, and the second voice, overlapping with the first, voice, might sing the identical theme starting five notes higher, on G. A third voicestarting on the D yet five notes higher, might overlap with the first two, and so on. The next escalation in complexity comes when the speeds of the different voices are not equal; thus, the second voice might sing twice as quickly, or twice as slowly35 the first voice. The former is called diminution, the latter augmentation (since the theme seems to shrink or to expand).

We are not yet done! The next stage of complexity in canon construction is to invert the theme, which means to make a melody which jumps down wherever the original theme jumps up, and by exactly thesame number of semitones. This is a rather weird melodic transformation, but when one has heard many themes inverted, it begins to stem quite natural. Bach was especially fond of inversions, and they show up often in his work-and the Musical Offering is no exception. (For a simple example of inversion, try the tune "Good King Wenceslas". When the original and its inversion are sung together, starting an octave apart and staggered with à time-delay of two beats, a pleasing canon results.) Finally, the most esoteric of "copies" is the retrograde copy-where the theme is played backwards in time. A canon which uses this trick is affectionately known as a crab canon, because of the peculiarities of crab locomotion. Bach included a crab canon in the Musical Offering, needless to say. Notice that every type of "copy" preserves all the information in the original theme, in the sense that the theme is fully recoverable from any of the copies. Such an informationpreserving transformation is often called an isomorphism, and we will have much traffic with isomorphisms in this book.

Sometimes it is desirable to relax the tightness of the canon form. One way is to allow slight departures from perfect copying, for the sake of more fluid harmony. Also, some canons have "free" voices-voices winch do not employ the canon's theme, but which simply harmonize agreeably with the voices that are in canon with each other,

Each of the canons in the Musical Offering has for its theme a different variant of the King's Theme, and all the devices described above for making canons intricate are exploited to the hilt: in fact, they are occasionally combined. Thus, one three-voice canon is labeled "Canon per Augmentationem, contrario Motu"; its middle voice is free (in fact, it sings the Royal Theme), while the other two dance canonically above and below it. using the devices of augmentation and inversion. Another bears simply the cryptic label "Quaerendo invenietis" ("By seeking, you will discover"). All of the canon puzzles have been solved. The canonical solutions were given by one of Bach's pupils, Johann Philipp Kirnberger. But one might still wonder whether there are more solutions to seek!

I should also explain briefly what a fugue is. A fugue is like a canon, in that it is usually based on one theme which gets played in different voices and different keys, and occasionally at different speeds or upside down or backwards. However, the notion of fugue is much less rigid than that of canon, and consequently it allows for more emotional and artistic expression. The telltale sign of a fugue is the way it begins: with a single voice singing its theme. When it is done, then a second voice enters, either five scale-notes up. or four down. Meanwhile the first voice goes on. singing the "countersubject": a secondary theme, chosen to provide rhythmic, harmonic, and melodic contrasts to the subject. Bach of the voices enters in turn, singing the theme, often to the accompaniment of the countersubject in some other voice, with the remaining voices doing whatever fanciful things entered the composer's mind. When all the voices have "arrived", then there are no rules. There are, to be sure, standard kinds of things to do-but not so standard that one can merely compose a fugue by formula. The two fugues in the Musical Offering are outstanding examples of fugues that could never have been "composed by formula". There is something much deeper in than than mere fugality.

All in all, the Musical Offering represents one of Bach's supreme accomplishments in counterpoint. It is itself one large intellectual fugue, in which many ideas and forms have been woventogether, and in which playful double meanings and subtle allusions are commonplace. And it is a very beautiful creation of the human intellect which we can appreciate forever. (The entire work is wonderfully described in the book J.. S. Bach's Musical Offering, by H.T. David.)

An Endlessly Rising Canon

There is one canon in the Musical Offering which is particularly unusual. Labeled simply "Canon per Tonos", it has three voices. The uppermost voice sings a variant of the Royal Theme, while underneath it, two voices provide a canonic harmonization based on a second theme. The lower of this pair sings its theme in C minor (which is the key of the canon as a whole), and the upper of the pair sings the same theme displaced upwards in pitch by an interval of a fifth. What makes this canon different from any other, however, is that when it concludes-or, rather, seems to conclude-it is no longer in the key of C minor, but now is in D minor. Somehow Bach has contrived to modulate (change keys) right under the listener's nose. And it is so constructed that this "ending" ties smoothly onto the beginning again; thus one can repeat the process and return in the key of E, only to join again to the beginning. These successive modulations lead the ear to increasingly remote provinces of tonality, so that after several of them, one would expect to be hopelessly far away from the starting key. And yet magically, after exactly six such modulations, the original key of C minor has been restored! All the voices are exactly one octave higher than they were at the beginning, and here the piece may be broken off in a musically agreeable way. Such, one imagines, was Bach's intention; but Bach indubitably also relished the implication that this process could go on ad infinitum, which is perhaps why he wrote in the margin "As the modulation rises, so may the King's Glory." To emphasize its potentially infinite aspect, I like to call this the "Endlessly Rising Canon".

In this canon, Bach has given us our first example of the notion of Strange Loops. The "Strange Loop" phenomenon occurs whenever, by moving upwards (or downwards) through the levels of some hierarchical system, we unexpectedly find ourselves right back where we started. (Here, the system is that of musical keys.) Sometimes I use the term Tangled Hierarchy to describe a system in which a Strange Loop occurs. As we go on, the theme of Strange Loops will recur again and again. Sometimes it will be hidden, other times it will be out in the open; sometimes it will be right side up, other times it will be upside down, or backwards. "Quaerendo invenietis" is my advice to the reader.