Anatomy of a CanonThe musical examples in this portion of the study are designed to be played from compact disks specified at the top of each web page. This method for obtaining sound was intended for my students who have access to these disks, but you may listen in if you wish. If you have obtained the FREE helper application from Voyager (and configured your computer to receive these files), you should be able to insert the specified disk in the CD drive of your computer and hear the examples.

Origin and Definition of the word "Canon"

"Canon" comes from the Greek word for rule or law. Musically, it designates the strictest form of counterpoint in which one voice is bound to imitate the rhythm, and interval content of another voice.

Requirements of a Canon

To qualify as a canon three conditions must be met: 1. The 2nd voice must be an exact repetition or a contrapuntal derivation of the 1st. 2. The 2nd voice must enter later than the 1st (cancrizans and proportional canon

excepted) 3. The 2nd voice may not deviate from the 1st voice or its contrapuntal variations.

Thus, the 2nd voice is thought to be strictly generated by the 1st. The two voices of a canon have been called dux/comes, antecedens/consequens, or proposta/ risposta; but this study uses the terms "leader" and "follower."

If all of the above conditions are met, the canon is said to be "strict." If liberties are taken with one or more of the above conditions, the canon is said to be "free." Canons of the 18th and 20th centuries tend to be strict, while canons of the 19th century may be free.

Canons are based, in theory, upon the principle of contrapuntal inversion...two melodic lines that can be performed simultaneously with either line functioning as the bass.

Categories of Canonic Imitation

The second voice of a canon may imitate the first voice exactly, at a different pitch level, in contrary motion, with change of rhythmic proportions, backward, or any combination thereof.

Canon at the Unison or Octave

In a unison canon the follower performs precisely the same melody as the leader. As the name implies, canon at the octave involves repetition of the leader an octave higher or lower. Var. 3 and Var. 24 of the Goldberg Variations are at the unison and octave respectively. If the end of the canon returns smoothly to the beginning it might

be called a round, circular canon, or perpetual canon like Canon 7 of the Musical Offering and Bach's Canon a 2 Perpetuus (BWV 1075) .

Canon at Intervals Other than the Octave or Unison

Many canons are contrived so that the follower begins on a pitch other than the starting pitch of the leader. The canons of the Goldberg Variations, for example, are ordered systematically so that each successive canon employs a larger interval between leader and follower. The follower may be a tonal imitation of the leader, that is, it may alter the interval qualities somewhat so as to stay in the same key as the leader, or it may be an exact transposition to a new key. Var. 18 of the Goldberg Variations is a canon at the sixth, but the interval may be a major or minor sixth depending upon the scale degrees that are involved. By contrast, the follower of the Fuga Canonica in Epidiapente from the Musical Offering is a strict transposition of the leader up a perfect fifth (each note of the follower reposing a perfect fifth above its counterpart in the leader). A third type of interval canon is exemplified in the second of the Canonic Variations on Vom Himmel hoch , where Bach inflects the pitches of the follower quite freely in order that the canon might conform to the tonality of the cantus firmus which it accompanies.

Retrograde Canon (Cancrizans, or "crab" canon)

One of the more exotic forms, retrograde canon involves the playing of a melody forward and backward at the same time. It is the custom, with canons of this sort, for each player to read the music once from left to right (forward) and then to return from right to left (backward). Thus, retrograde canons are sometimes called "crab" or cancrizans (after the sideways manner of that creature). Because both parts begin simultaneously, the terms "leader" and "follower" hardly apply to crab canons, examples of which include: the Cancrizans from the Musical Offering, and the First and second canons from the Fourteen on the Goldberg Ground.

Canon in Contrary Motion

When the canon leader and follower progress by the same melodic intervals, but move in the opposite direction, the canon is said to be in contrary motion. In the context of canons, the term "inverted canon" is synonymous with "canon in contrary motion." Canons in contrary motion exemplify the technique of "melodic inversion," and should not be confused with contrapuntal inversion (also known as "double counterpoint") in which two contrapuntal lines exchange registers...the low becoming the high and visa versa. The fourth canon of Die Kunst employs both techniques: its follower is in contrary motion to its leader, and its second half involves an exchange of registers (double counterpoint) between leader and follower. Of the Bach's canons extant, many involve contrary motion, including the following: the Trias Harmonica, Canon Concordia Discors, most of the Fourteen Canons on the Goldberg Ground, Goldberg #12, Goldberg #15, Vom Himmel hoch #3 , and Musical Offering No. 3.

Notice that canons in contrary motion are normally constructed so that if the leader begins on the tonic pitch the follower will begin on the dominant, and visa versa.

Mirror Canon

Ordinarily, canons in contrary motion freely inflect interval qualities in order to stay within the key. Composers with exceptional skill have constructed a rigorous sub-category, called "mirror" canon, in which followers mimic the precise quality of intervals stated by leaders (albeit in the opposite direction). As the technique is difficult, mirror canons are quite rare. The rule of qualitative correspondence between intervals implies that mirror canons invoke more than the usual number of chromatic pitches as No. 6, No. 8, and No. 11 from the Fourteen on the Goldberg Ground, Canon perpetuus and Canon a 2 Quaerendo invenietis from the Musical Offering demonstrate.

Proportional Canon

More commonly termed "canon in augmentation or diminution," proportional canons re-articulate the rhythm of the leader at a ratio other than one to one. Thus, the follower might progress at half, twice, or three times, the speed of the leader. Voices in proportional canon may start at the same time or at different times. Bach's proportional canons include: the fourth canon of the Musical Offering the final canon of the Variations on Vom Himmel hoch , the fourth canon from the Art of Fugue, and, Bach's tour de forcein this genre, the final canon of the 14 on the Goldberg Ground.

Spiral Canon

Whereas most canons are repeatable, when a spiral canon repeats it does so at some other pitch. If the new pitch is the same scale degree (in a new key), the canon is a modulating spiral, like Bach's canon a 2 per tonus of the Musical Offering . If the new pitch is a different scale degree (in the same key), the canon is a modal spiral. This study contains no examples of modal spiral, but the four-voice canon Bach composed for Walther puts each of the voices into a different mode with an overall effect of Dorian.

Accompanied Canon

Music that contains canonic voices to which have been added one or more voices in free counterpoint is said to be "accompanied." In most of Bach's accompanied canons this added voice is the bass. Obviously, when a bass part is added the requirement that the upper canonic voices be able to function as bass no longer applies. This liberates the composer to involve the canonic voices in counterpoint that might not otherwise have been possible. With the exception of the last (Var. 27), all of the canons of the Goldberg Variations are accompanied. The added voice may represent a pre-existing melody, such as the "royal theme" in the second canon of the Musical Offering, or the canonic voices themselves may be cantus firmi. All of the canonic

preludes of the Orgelbuchlein, Clavier-Ubung III, most of the Canonic Variations on Vom Himmel hoch, and five of the canons from the Fourteen on the Goldberg Ground fall into this latter category.

Double and Triple Canon

A canon that has two leaders and two followers is a double canon...see the fifth canon of the Fourteen on the Goldberg Ground. Double canons are sometimes referred to as "canon four in two." In the thirteenth canon of that cycle Bach managed even to construct a triplex canon, or "canon six in three."

Combining More than One Technique

After listening to the canons of this study the casual listener might come to the conclusion that they are not difficult to compose. Nothing could be further from the truth! Even the simplest types present challenges beyond the abilities of most musicians. Yet Bach imposed upon himself not only the strictures of contrary motion, augmentation, and retrograde motion, but in many instances the simultaneous adherence to more than one canonic rule! Thus, the third canon of the Musical Offering is an accompanied canon in contrary motion as are the sixth, seventh, and eighth canons from the Fourteen on the Goldberg Ground. The eighth canon of the Musical Offering is an accompanied mirror, while the eleventh canon of the Fourteen on the Goldberg ground is an accompanied double mirror. Perhaps the most difficult combinations of Bach's repertory are his three canons in augmentation and contrary motion: No. 14 from the cycle on the Goldberg ground, No. 4 from the Art of Fugue, and No. 4 from the Musical Offering.

Cryptic Notation

Before 1600 polyphonic music was normally written in parts, not score. This meant that a musician could see but one line of music and not the accompanying voices. As a consequence it was customary in the writing of canons from this era to notate only the canon leader, with some rule whereby the follower would be generated from it: a second starting point, another interval or a time proportion. Many of Josquin's chansons, for example, contain a vocal line intended to be sung as two--in canon. Today we call this type of notation "cryptic," meaning that it is concise not that the composer was wanting to be secretive.

There does exist, however, a genre of canons where the composer engages in deliberate obfuscation. Many of Bach's canons are of this type. This study, for example, contains instances where he hints at the canon by means of a monogram, symbol, or other cryptic device. When the solution is not obvious the work is said to be a "riddle" or "enigmatic" canon. J. S. Bach encrypted the cancrizans from the Musical Offering for example, by placing a backward clef at its conclusion. He used the same technique in the 1st and 2nd canons on the Goldberg Ground.

Bach encrypted canons in contrary motion by inverting clefs (see canons four, and nine of the Musical Offering). Because inverting a C-clef effects no apparent change, contrary motion is signified by the inversion of key signatures or by the placement of accidentals on "wrong" lines and spaces. The third and fourth canons from the Fourteen on the Goldberg Ground, for example, contain C-clefs with sharps in the key signature that appear "incorrectly" on the pitch G. Only after the clefs have been inverted do the sharps appear correctly on F!

Canons in which the follower begins at a pitch other than that of the leader (e.g. numbers five and six from the Musical Offering) are indicated by the imposition of two or more clefs upon the staff. Bach's canon for Walther contains four such clefs, while his canon for Hudemann contains no fewer than eight (four inverted with different key signatures).

Finally, if the musical symbolism is not enough, the composer might write clues in prose. The fourth and fifth canons of the Musical Offering are accompanied by Latin riddles indicating the nature of the canonic technique, while the Canon Fa Mi et Mi Fa contains a dedicatory acrostic spelling the composer's name.

Canons of the Goldberg Variations

The musical examples on this page are designed to be played from the Trevor Pinnock recording of the Goldberg Variations (Archiv 415-130-2). This method for obtaining sound was intended for my students who have access to this particular disk, but you may listen in if you wish. If you have obtained the FREE helper application from Voyager (and configured your computer to receive these files), you should be able to insert the Archiv disc in the CD drive of your computer and hear the examples.

How the Variations Came to Be

The eminent harpsichordist, Ralph Kirkpatrick, writes: "However much it is an act of impudence thus to discuss something which is far too profound and complex to be grasped in words, it seems necessary in order to explain all that has been said before, to confess some of the feelings which inevitably come with the playing of this music." The music about which Kirkpatrick writes is the set of variations commissioned of Johann Sebastian Bach by Dresden's Count Von Keyserlingk for his court harpsichordist, Johann Gottlieb Goldberg.

It seems that the Count, an insomniac, had asked Bach to compose something that might occupy his restless nights. For his trouble, Bach was rewarded with a golden goblet filled with 100 louis-d'or. Keyserlingk was so pleased that he thereafter referred to Bach's music as "my variations," but history has remembered "Keyserlingk's Variations" by the name of his harpsichordist: Goldberg.

The Goldberg Variations begin with an Aria Bach had composed in 1725, possibly for his wife This Aria appears in Anna Magdalena's Notebook, where it is written in her own hand. The Sarabande-like Aria becomes theme for a set of 30 variations to follow.

Analysis of Each Canon

Canon 1: canone all' Unisuono

Structurally, the Goldberg Variations demonstrate the Baroque ideal of balance and internal coherence. Every third variation is a canon, of which this is the first. Canone all' Unisuono means "canon at the unison," implying that the canon leader and follower begin on the same pitch. This title suggests that subsequent canons might use other intervals as indeed they do.

Canon 2: canone all' Seconda

In the second canon, the follower chases the leader but seems never to catch up. This is because the follower commences each phrase a step higher than the leader--canone all' Seconda. Kirkpatrick describes this canon as having "an almost nostalgic tenderness."

Canon 3: canone alla Terza

Canone alla Terza means "canon at the third." Here the leader begins on the pitch "B" while the follower begins a third lower, on "G." Each variation divisible by three (except No. 30) is a canon. The quotient becomes the interval between leader and follower. Such a mathematical conception makes the Goldberg Variations unique in music literature.

Canon 4: canone alla Quarta

It is difficult to recognize the canone alla Quarta as a canon at all. Not only does the follower begin a perfect 4th lower than the leader, but it moves each of its intervals in the opposite direction: a moto contrario. In the "B" section, the leader and follower exchange positions; instead of the high voice leading (with the follower down a 4th), the middle voice leads (with the follower UP a 4th). This balancing of opposing registers and melodic contours represents a Bachian fondness for what I call "back 'n forth." The fundamental bass is clearly heard in the triple articulations of the lowest voice.

Canon 5: canone alla Quinta

Like the canon at the fourth, this Canone alla Quinta is in contrary motion. This time, however, the interval separating leader and follower is a fifth. That Bach could have used a compositional procedure so abstruse to produce something so musical is an evidence of his genius. Minor mode, for the first time in the Goldberg Variations, lends a dark and tragic aura. Each of the variations is centered in G, but three of them are in minor rather than major. Of the two other minor mode variations, No. 21 is a canon, and No. 25 is not. The constant G major/minor tonality does not tire the ear because each variation also modulates to related keys (typically D-major and e-minor), providing a variety of tonal centers.

Canon 6: alla marcia

Variation 18, "in a marching style," has the distinction of being the canon in which the follower comes closest to "catching" the leader. The time interval separating the two is one beat: the pitch interval, a sixth.

In theme and variation one expects the soprano to become the object of variation. This does not happen in the Goldbergs. Instead, Bach follows an Italian model known as ruggiero which varies the bass instead. Bach's ruggiero consists of eight phrases with every other phrase ended by the same ground bass but in different keys. Thus Bach's ruggiero is very nearly a chaconne (ostinato form in which the repeated element is a chord progression rather than a melody). The bass does not repeat verbatim, but its implied harmonies are always intact. If the bass had repeated exactly, the cycle might have been called a passacaglia--the favorite Baroque cousin of theme and variation. This similarity to the ground bass forms of chaconne and passacaglia led Albert Schweitzer to call the Goldberg Variations "a passacaglia worked out in chiaroscuro." There is nothing quite like it in all of music literature.

Canon 7: canone alla Settima

Filling in the ruggiero by five descending semitones (lamento bass), the seventh canon weaves its tapestry of counterpoint in what may be the most evocative melody of the Goldberg Variations. This Canone alla Settima is the second of three variations to be set in minor mode (another Trinitarian symbol...perhaps). The follower imitates the leader at the interval of a seventh.

Canon 8: canone alla Ottava

After the rollicking exuberance of the variation that precedes it (No. 23), the canon at the octave rocks the listener in a gentle three beats per measure, each beat divided into three parts.

Canon 9: canone alla Nona

The cycle closes with a sprightly variation in which leader and follower imitate each other at the ninth. The figure "spun out" in this canon is characterized by running sixteenth notes punctuated by expressive leaps of a sixth (an interval of which Bach seems to have been especially fond). For the first time the canon is abandoned by its ruggiero (having heard the bass theme many times before, our minds eagerly fill it in). Whereas the leader in part "A" is the low voice, the leader in part "B" is the high. Notice how the leader in part "B" is nearly the melodic inversion of part "A" (another example of "back 'n forth").

Fourteen Canons on the First Eight Notesof the Goldberg Ground (BWV 1087)

The musical examples on this page are designed to be played from Don Dorsey's "Bachbusters" (Telarc CD-80123) recording of the "Diverse Kanons" BWV 1087. This method for obtaining sound was intended for my students who have access to this particular disk, but you may listen in if you wish. If you have obtained the FREE helper application from Voyager (and configured your computer to receive these files), you

should be able to insert the Telarc disc in the CD drive of your computer and hear the examples.

Suggested Reading: Christoph Wolff. "The Handexemplar of the Goldberg Variations" Bach: Essays on His Life and Music (Cambridge: Harvard University Press, 1991) pp. 162-177.

Historical Background and Significance

Discovery of BWV 1087 In 1974 a published copy of the Goldberg Variations, first owned by Bach himself, was discovered in private possession in France. Accompanying the manuscript, in Bach's hand, there was attached a single page with fourteen canons on the first eight notes of the Goldberg ground. The discovery of the hitherto unknown manuscript was immediately hailed as the most important addition of a Bach source in recent decades. Of the fourteen canons, only numbers 11 and 13 had been known before 1974.

Significance of the Cycle The fourteen canons are important for four reasons. Not only are they delightful to hear, but they represent a germinal stage of the mature variations that is highly instructive as to compositional processes that Bach may have used. Thirdly, the canon cycle bridges the gap between the canons of the Goldberg and the more esoteric canons of the Musical Offering of 1747. Finally, the enigmatic notations of the fourteen canons represent Bach's affinity for musical riddles and cryptographic symbols.

Signature Number? It is a coincidence, perhaps, that there are fourteen canons in the Goldberg addendum. But to those who are cognizant of Bach's fascination with the number 14 as the sum of the ordinal values of the letters of his name (B+A+C+H), the number of canons in this cycle is more than coincidental. Bach's last major work, Art of the Fugue contains fourteen Contrapuncti, the last of which is the unfinished quadruple fugue in which the third subject is the BACH motive. If the number of canons can be understood to represent the composer's signature number, we might infer that Bach wished for the cycle of canons to represent, just as the Art of the Fugue represents, the last word on the subject.

Analysis of Each Canon The first four canons of the cycle use the Goldberg ground as leader. While not

the most interesting things to hear, from a theoretical perspective these canons are necessary inasmuch as Bach demonstrates in them the inherent contrapuntal possibilities of his soggetto, or "subject," by combining it with itself in four ways:

Canon 1 - soggetto with its retrograde

Play Soggetto Stop Animate Canon 1

Canon 2 - inverted soggetto with its retrograde

Play Inverted Soggetto Stop Animate Canon 2

Canon 3 - soggetto with its inversion

Play Soggetto Stop Animate Canon 3

Canon 4 - inverted soggetto with its inversion

Play Inverted Soggetto Stop Animate Canon 4

Canon 5 - Canon Duplex a 4 From the fifth canon onward, the Goldberg ground is employed either by itself or canonically as ostinato beneath other canonic voices. Now that the composer has established the contrapuntal viability of his soggetto, he is ready to add more complex counterpoint above it. In canon 5 Bach repeats the soggetto as it was stated canonically in 3, above which he adds a second canon in contrary motion.

Canon 6 - Canon simplex uber besagtes Fundament a 3 In this "mirror canon," the intervals of the follower are exactly the same as those of the leader, but moving in the opposite direction.

Canon 7 - Idem a 3 Like the sixth canon, No. 7 is in contrary motion above a ground bass.

Canon 8 - Canon simplex a 3 il soggetto in Alto In the eighth canon, the ground is put into the middle voice (il soggetto in Alto). Bach applies accidentals to make the follower the mirror image of its leader. This makes the canon sounds more like it is in d-minor than the G-major to which we have become accustomed.

Canon 9 - Canon in unisono post semifusam a 3 Bach titled this one "Canon in unison followed at the sixteenth," drawing attention to the unusually short time interval between leader and follower.

Canon 10 - Alio modo per syncopationes et per ligaturas a 2 Bach stipulates that this counterpoint above the ground is to be a piece for two voices. Thus it was not intended to be a canon (else the follower would have added an unspecified third voice). Instead Bach inverts both parts and has them played as a second counterpoint which he call the Evolutio. This technique is the precursor to two of the fugues in Die Kunst which are likewise contrived so that they can be played with intervals moving in the opposite direction.

Play Counterpoint Stop Animate Evolutio

Canon 11 - Canon duplex ubers Fundament a 5 Suggested Reading: That Crown of Thorns by Tim Smith

When the Goldberg canon cycle was discovered in 1974, only two of its fourteen canons had been known before that date; number eleven was one of

them. On October 15, 1747, Bach scribbled this canon (right) on the flyleaf of a notebook owned by Johann Gottlieb Fulda (1718-1796), theology student and some-time player in Leipzig's orchestras. In Fulda's notebook the canon is accompanied by two cryptic inscriptions: Symbolum Christus Coronabit Crucigeros "Symbol: Christ will crown the Cross-bearers" and Domino Possessori hisce notulis commendare se volebat J. S. Bach "J. S. Bach wanted to commend himself to the lord possessor by means of these notes." 1 The meaning of the first inscription is found in the five descending semitones of the top voice representing the five wounds of Christ (stigmata ). No. 11 is the third mirror canon in this series of fourteen. The 1747 (Fulda) version is somewhat different from that which is found in the Goldberg cycle.

Canon 12 - Canon duplex uber Fundamental - Noten a 5

Canon 13 - Triplex canon

This is the second canon from BWV 1087 that was known before the discovery of the cycle in 1974. In 1746 Elias Haussmann painted what we now know as the most authentic portrait of Bach (right). In his right hand the composer holds the triplex canon (before 1972 known as BWV 1076) which he would present the following year, with his variations on Vom Himmel hoch (BWV 769) for membership in Mizler's Society of Musical Sciences. Because the canon's soggetto is similar to the cantus firmus of the Vom Himmel hoch variations it was thought that the canon was composed in or about 1746. With the discovery of the fourteen canons of the Goldberg cycle it was learned that the erstwhile Vom

himmel hoch canon was actually written in conjunction with the Goldberg Variations.

Canon 14 - Canon a 4 per Augmentationem et Diminutionem The last canon is for four voices in rhythmic proportions (mensuration canon). Bach labeled it "canon for 4 voices in augmentation and diminution," but for the Goldberg ground to appear the canon must also move in contrary motion.

Notes

1. It is likely that the "lord possessor" of the inscription has a double meaning. It obviously refers to the owner of the book (Fulda), but may also refer to the Lord God as well.

Canons from the Art of Fugue

The musical examples on this page are designed to be played from the Archiv (431 704-2) recording of Die Kunst der Fuge: Musica Antiqua Koln, Reinhard Goebel conducting. This method for obtaining sound was intended for my students who have access to this particular disk, but you may listen in if you wish. If you have obtained the FREE helper application from Voyager (and configured your computer to receive these files), you

should be able to insert the Archiv disc in the CD drive of your computer and hear the examples.

Background:

Die Kunst der Fuge contains four canons which are among Bach's longest and formally complex. True to the cyclical conception of which they are part, each of the four develops aspects of fugal writing that are difficult to maintain in canon.

While from head to tail a strict canon at the octave, the Canon alla Ottava, for example, contains fugal expositions of the subject at the dominant, again at the dominant in contrary motion, and at the tonic in contrary motion. Invertible counterpoint at the 10th and 12th characterize the canons alla Decima and Duodecima, respectively, while the former contains a meandering diminution of the subject in its coda. The Canon per Augmentationem in contrario Motu continues its invertible counterpoint (at the 8va) while developing the subject, as its title implies, by means of augmentation and contrary motion.

It is possible that the inspiration for these four canons came from Bach's own Fuga Canonica in Epidiapente which he had earlier composed for the Musical Offering, and to which they bear a marked resemblance.

Canon 1: The Canon alla Ottava explores fugal technique in its reiteration of the head motive at the level of the dominant and in contrary motion. This canonic "subject" is the melodic inversion of the main theme of the Art of Fugue.

Show Complete Score

Formally, the Canon alla Ottava is the least complicated in that its high voice consistently represents the leader, and its low voice, the follower. The colored sections in the following picture represent analogous units. If you click a section, the CD will begin playing there:

Stop

Canon 2: The subject of this Canon alla Decima consists of the unadorned Die Kunst theme in contrary motion.

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The second, third, and fourth canons of the Art of Fugue each divide neatly into two sections. In section one of this canon (mm. 1-33), the lower voice leads while the upper voice follows a 10th higher (canon at the 10th). After a short bridge passage (mm. 34-40), there is an exchange of registers in which the follower, this time in the lower voice, imitates not a 10th lower, but at the octave. Thus sections one and two exemplify double counterpoint at the 10th. The colored sections in the following picture link analogous units. If you click a section, the CD will begin playing there:

Stop

Canon 3: The subject of this Canon alla Duodecima consists of Die Kunst's main theme with complete upper and lower neighbor diminutions of each factor in an ascending triad outlining the rising P5 of the theme's head motive.

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The formal organization of this canon is identical to that of its predecessor. Other than the substantial change of affect in head motives, the primary innovation of Canon No. 3 is that its follower imitates at the interval of the 12th (8va+5th higher) rather than the 10th. The low voice takes the lead in section one whereafter the high voice follows a 12th higher, exemplifying the fugal principle of "real" answer. Midway through the canon these registers are exchanged but with the follower echoing the leader, not a 12th lower, but at the octave (double counterpoint at the 12th). The colored sections in the following picture link analogous units. If you click a section, the CD will play it:

Stop

Canon 4: The subject of this Canon per Augmentationem in contrario Motu is a disjunct variation of the Die Kunst theme.

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This canon ranks among the most complicated in Bach's oeuvre. While simple in terms of form--two sections in double counterpoint--the canon's follower is in contrary motion AND rhythmic augmentation to its leader! Bach manages to sustain this display of contrapuntal fireworks for fifty-two measures whereafter he causes the voices to exchange registers and do it again in double counterpoint at the 8va (mm. 53-109)! This procedure is quickly recognized, aurally, in the two places where the leader executes ascending chromatics in eighths (which are answered, of course, by descending chromatics in quarters). The colored sections in the following picture represent analogous units. If you click a section, the CD will begin playing there:

Anatomy of a Fugue

The musical examples on this page are designed to be played from the Musica Antiqua Koln (Archiv 431 704-2) recording of the "Art of Fugue" BWV 1080. This method for obtaining sound was intended for my students who have access to this particular disk, but you may listen in if you wish. If you have obtained the FREE helper application from Voyager (and configured your computer to receive these files), you should be able to insert the Archiv disc in the CD drive of your computer and hear the examples.

I. Definition of a Fugue

Polyphonic procedure involving a specified number of voices in which a motive (subject) is exposed, in each voice, in an initial tonic/dominant relationship, then developed by contrapuntal means.

II. Character of a Fugue

A fugue generally consists of a series of expositions and developments with no fixed number of either. At its simplest, a fugue might consist of one exposition followed by optional development. A more complex fugue might follow the exposition with a series of developments, or another exposition followed by one or more developments. Fugues that are tonally centered will expose the subject without venturing out of an initial tonic/dominant constellation.

III. Parts of a Fugue

A. Main Idea of the Fugue and How It Is Stated 1. Subject: Melody that comprises the primary melodic/rhythmic material of the fugue. Subjects typically have two parts: the "head" is calculated to attract attention either by unusual rhythmic or intervallic emphasis, while the "tail" is typically more conjunct, rhythmically uniform, and sometimes modulatory. The head and/or tail itself may employ variation of one or two smaller motives or figures...each comprised of a characteristic rhythm and/or interval. 2. Answer: Subject imitation which immediately follows the first statement of the subject: in a different voice and usually fifth higher. Answers are a subclass of subjects which bear certain interval characteristics in relationship to the subject as it was originally stated. o Tonal Answer: An answer that typically (though not always) stays in the same key as the subject. To do this it is necessary for the intervals of the subject to change somewhat. In a tonal answer "do" and "sol" switch places: The position occupied by "do," in the subject, becomes "sol" in the answer and vice versa. Analytical technique: Subjects having many skips (disjunct) that focus upon the tonic and dominant scale degrees lend themselves to a tonal answer. o Real Answer: An answer that is a transposition of the subject to another key, usually the dominant. Analytical technique: Subjects having mostly steps (conjunct) that don't focus upon "do" and "sol" lend themselves to a real answer. 3. Countersubject: Substantive figure that sometimes recurs immediately following the subject or answer (in the same voice). Countersubjects serve as counterpoint to subjects (or answers) sounding simultaneously in a different voice. Not every fugue will have a countersubject. Some fugues may have more than one countersubject. 4. False Subject: Some people use the term "false subject" to describe an entry of the subject (or answer) that begins but never finishes. This term should be reserved for instances where the subject appears to enter, breaks off, then

follows immediately with a complete statement. Most other instances of incomplete subjects are developmental and should be termed "imitation."

B. Main Sections of the Fugue 1. Exposition: Portion(s) of the fugue consisting of subject(s) with at least one answer, and possibly countersubject(s). To qualify as an exposition, the subject (or answer) must appear in all voices and answers must be in the proper relationship (tonal or real) to subjects. The exposition normally concludes immediately after the subject (or answer) appears in the last voice. Expositions may defer the cadence until after a codetta. Differentiation between exposition subtypes is based upon the order in which voices enter (as compared to the first exposition) and whether or not the subject has changed. o Re-Exposition: An exposition, following the initial exposition, in which the voices enter in the same order as the first exposition. o Counterexposition: An exposition following the initial exposition in which the voices enter in a different order than they did in the first exposition, or the subject of the new exposition is a contrapuntal variation of the original. o Double Exposition: Exposition utilizing a brand new subject (i.e. not contrapuntally derived from the first). If the new subject is unique, then the fugue is a double fugue (or, in the case of three subjects, triple fugue).

2. Developmental Episode: Section in which motives from the exposition are treated in sequence, modulation, contrary motion, double counterpoint, stretto, augmentation/diminution, pedal, etc. Episodes are generally terminated by a cadence and may follow one after the other. Developmental episodes characteristically begin by departing from the subject, to fragment or vary it in some way, but gradually building up to a restatement of the subject in at least one voice. These statements of the subject are typically not in the tonic/dominant relationship of the exposition and are called "middle entries" (or in German Durchführung). Episodes typically do not enunciate the subject in all voices.

3. Coda or Codetta: Concluding segment of a section (codetta) or of the entire fugue (coda). Codas and codettas often sound as if they are something added after the structural end of the section or work. The function of codettas is often modulatory (to return the tonality to the key of the subject after an answer at the dominant). Not all fugues have these.

IV. Compositional Techniques of the Fugue

A. Tonal Variation 1. Modulation: Repetition of a motive in another key. Bach typically

arranges his fugues around closely related keys (major and minor keys immediately adjacent to each other on the circle of fifths).

2. Mutation (also called "change of mode"): Statement of the subject or answer (or any other primary material) in the opposing mode. A subject first stated in minor and later stated in major is said to have "mutated."

B. Contrapuntal Variation 1. Stretto: Entry of a motive in a second voice before the first voice has finished its statement. Motive can mean subject, answer, countersubject, or any other substantive melodic/rhythmic entity in imitation. 2. Augmentation/Diminution: Statement of a motive in rhythmic durations that are proportionately doubled or halved. 3. Pedal Point: Suspension of one pitch, often the bass, in such a manner that it is alternately consonant then dissonant with the chord progression. Fugues often conclude with episodes of pedal point. 4. Retrograde: (rare) Statement of the motive's pitches in reverse order. 5. Melodic Inversion: (Contrary Motion) Statement of a motive where interval directions have been made to move in the opposite direction of the original motive. If the quality of the intervals is preserved the motion is said to be the "mirror inversion." 6. Sequence: Repetition of a motive at another pitch level, usually up or down a step. Each repetition is called a "leg." Sequences in which each leg itself contains a sequential pattern are said to be nested. Bach's sequences tend to be of this latter variety, with the overall sequence comprised of two or three legs, each leg comprised of two subsidiary units. For example: study the sequences in the mirror fugues of Art of Fugue . Sequential episodes seldom appear in fugal expositions but are frequent accouterments to developments. 7. Contrapuntal Inversion: (Double/Triple Counterpoint) Reappearance of a pair of voices (double ctpt.) or trio of voices (triple ctpt.) in which registers have been reassigned in such a way that the voices have crossed and the interval relationship between voices is fundamentally altered.

a. Types of Contrapuntal Inversions:

o At the Octave: Fourths become fifths, unisons become octaves, etc. While parallel 4ths sound fine, they do not invert contrapuntally, and double ctpt. at the octave avoids them. See the Canon per Augmentationem in

contrario Motu from the Art of Fugue for an example of double counterpoint at the octave. o At the Tenth (8va+3rd): Parallel motion tends to be avoided altogether. This is because intervals that parallel acceptably in one texture (e.g. 3rds & 6ths) become unacceptable when inverted (8vas & 5ths). Study the Canon alla Decima of the Art of Fugue. o At the Twelfth (8va+5th): With the exception of 3rds (which remain 3rds), acceptable parallels become unacceptable when inverted at the 12th. Thus, in the Canon alla Duodecima of the Art of Fugue (which features this type of double ctpt.) the composer uses many parallel thirds.

b. How to Calculate Type of Contrapuntal Inversion:

1. Determine interval that the lower voice has been moved UP 2. Determine interval that the higher voice has been moved DOWN. Note: if the voices have not exchanged registers, the higher voice becoming the lower and vice versa, then contrapuntal inversion has not occurred. 3. If steps 1 and 2 are each octaves, then the double counterpoint is at the octave. Otherwise, add the results of steps 1 and 2, then subtract 1.

c. How to Calculate What Intervals Become After Inversion:

4. Double counterpoint @8va: Subtract the interval (before inversion) from 9 to get the interval after inversion. For example: a 4th before inversion will become a 5th after inversion.

5. Double counterpoint @10th: Subtract the interval (before inversion) from 11 to get the interval after inversion. For example: a 4th before inversion will become a 7th after inversion.

6. Double counterpoint @12th: Subtract the interval (before inversion) from 13 to get the interval after inversion. For example: a 4th before inversion will become a 9th after inversion.

For Practice: Recognizing Contrapuntal Inversions

The following examples of contrapuntal inversion are designed to be played from the Musica Antiqua Koln (Archiv 431 704-2) recording of the "Art of Fugue" BWV 1080. The first system, in each example, is paired with the analogous measures, in double counterpoint, several bars later. Answer the following questions:

1. In what direction and how far is each part moved in analogous sections? 2. What happens to vertical intervals between analogous sections?

3. What is the basic interval of contrapuntal inversion? 4. What types of motion (parallel, contrary, oblique) characterize each

example?

Exchange of Registers (contrapuntal inversion) in Canon per Augmentationem in contrario Motu

Play mm. 5-8 Play Double Ctpt. @ 8va (mm. 57-60)

Exchange of Registers (contrapuntal inversion) in Canon alla Decima

Play mm. 5-7 Play Double Ctpt. @ 10th (mm. 44-46)

Exchange of Registers (contrapuntal inversion) in Canon alla Duodecima

Play mm. 9-12 Play Double Ctpt. @ 12th (mm. 42-45)

How to Analyze a Fugue

Before creating a time-line you will need to analyze your fugue. The following questions and techniques are intended to facilitate such an analysis. A thorough and accurate analysis will save time and effort when you come to the time-line phase of the CD Counterpoint Companion. This document contains many links to the Anatomy of a Fugue that you might find helpful to have read first.

How many sections are there and in which measures do these sections begin and end? Analytical Technique: look for cadences.

1. Not every cadence represents the end of a section, but every section ends in a cadence.

2. Sections typically cadence in keys that are closely related to the home key. Closely-related keys differ by no more than one sharp or flat.

3. In the context of the 18th century fugue, authentic cadences (V-I) predominate, half (?-V) and deceptive (V-vi) cadences appear less often, and plagal cadences (IV-I) appear seldom.

4. Fugal cadences are difficult to recognize because the composer does not pause on the cadence chord; contrapuntal and harmonic motion normally continues directly into the next section.

Cadences are often elided (the cadence chord serves dual function of concluding one section and beginning the next).

What is the function of each section? Analytical Technique: determine if the section exposes, develops, or concludes material.

1. Exposition : Analytical Technique: mark all instances where the main idea is stated or answered. Because the exposition "exposes" new material, all fugues begin with an exposition. In the sense that it consists of the subject stated and answered in all voices, the exposition is the most predictable and form-defining section of the fugue. The subject (or answer) may appear in any order, but in Bach's fugues the bass voice will often take the last entry. When the subject is answered in a second voice, the first voice may continue with a countersubject. Note all instances of countersubject. Fugues may have more than one exposition. To qualify as an exposition the subject must appear in all voices and in the prescribed subject/answer relationship (tonal or real). In a re-exposition, the prior subject is voiced in the same order; in a counterexposition, the prior subject is voiced in a different order. A "double exposition" consists of the exposition of a second subject (triple-exposition = third subject, etc.)

2. Developmental Episode : Analytical Technique: note all instances of contrapuntal elaboration. Sections that elaborate upon the subject by contrapuntal means are called "developmental episodes." While the primary function of a development is to elaborate upon the subject, developments often contain statements of the fugue's subject outright, and these subjects are often answered. Possible contrapuntal elaborations include: augmentation/diminution, melodic inversion (contrary motion), contrapuntal inversion (double counterpoint), pedal point, modulation, sequence, stretto, canon, simple imitation, rhythmic and melodic permutation, fragmentation (separation of the subject's head from its tail, etc), truncation of the subject, anticipatory statements of the subject's head (false subject). 3. Concluding Sections : Analytical Technique: listen for sections that sound as if they are bringing things to a close. A section that concludes an interior exposition or development is called a "codetta." Codettas sound as if they are appended after the structural close of a main section. Codettas seldom last more than two or three bars. More common is the "coda," which designates the conclusion of the entire fugue. Codas often modulate to the key of the subdominant. Codas often

contain stretti, often visit the subject one last time (usually in the bass voice), and often employ pedal point.