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Timothy A. Smith

D. M. A. Professor, Music Theory, NAU



Anatomy of a Canon


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.






Why Did Bach Write Canons?


  • More than a Game
    Bach may have composed canons for the same reason that we solve crossword puzzles; they were entertainment...a game. Perhaps he composed canons because he found in them a challenge of the first order. Or Bach may have written canons in order to stimulate his muse; composers often generate new ideas by employing canonic techniques. There is reason to suspect, however, that the Baroque taste for canon was more than a game, challenge, or method for generating ideas.
  • Window into the laws of music
    Bach and his contemporaries were of the notion that music was a science...sounding mathematics in Mizler's words...therefore reducible to theorem and law. Bach's concurrence with the notion was demonstrated by his membership in Mizler's Society for Musical Sciences...joined in 1747 while composing the Musical Offering...for which he not only submitted the thirteenth canon of the Fourteen on the Goldberg Ground but also composed the Canonic Variations on Vom Himmel hoch. If Bach believed that music was a science, he may have conceived of canon as a window through which it might be possible to glimpse its laws. Given as we are to understand the eighteenth-century compositional ideal of elaboratio...the development of ideas from a single theme (inventio)...Bach's fascination with canon was more than entertainment.
  • Mystery of Musical Creation
    There remains a third possible explanation why Bach and his contemporaries practiced the art of writing canons. The technique may have stood for them as a symbol of all that was NOT understood...that which was transcendent, therefore symbolic of themselves as creators and the processes of musical creation. Thus, while Bach may have composed canons in an effort to understand these processes, he could just as well have composed them as an expression of the very mystery of musical creation itself. We know by their enigmatical notations that Baroque composers viewed canon as something to be figured out, if not mediated by that select few (namely, composers) who understood it.
  • Genitum non factum
    The theological implications of the canon, while speculative to be sure, invite contemplation no less. Just as a well-composed leader had the potential to animate itself in multifarious followers, so, too, according to Christian cosmogony, the Creator animates all things, including music. And just as the canon follower was not created, but begotten (so to speak), it is conceivable that the process of canonic generation represented, to the Lutheran way of thinking, the second person of the Trinity: genitum non factum...per quem omnia facta sunt (begotten, not whom all things were created).

It is revealing that when Bach set the foregoing words to music in the Creed of his Mass in B Minor, he utilized canonic processes. While not technically a canon, the movement is very canon-like, but with pitch and time intervals continuously mutating. These mutations bring to mind the Trinitarian doctrine...three in one and one in much the same manner as the third canon of the Variations on Vom Himmel hoch portrays heaven and earth by its mutating consonant and dissonant intervals separating leader and follower. Underlying both movements is the Christian doctrine of the dual nature of Christ incarnate, one with God, yet fully human. As sign of this duality, Bach does not allow the canonic followers to engage in unthinking mimicry but gives them independence. Yet, in its independence of time and pitch interval, the follower (Christ) is audibly generated by the leader (the Father), and always doing the Father's will.

  • About the Nature of Meaning
    In Tonal Allegory in the Vocal Music of J. S. Bach, Chafe suggests that Bach's use of the term Symbolum in association with at least one of his canons suggests a metaphoric dimension permitting us to take the canons as statements about the nature of meaning and the relationship between art and theology. He writes: Artists of the baroque period had means of expressing basic relationships between art and transcendent meaning. One means was the poetic epigram wherein the author used antithesis, conceit and paradox to instruct and familiarize the reader with basic concepts of faith in highly condensed form (p. 14). Also called paradoxa, a typical epigram included juxtapositions of passages such as God wills all men should be saved with Few are chosen (I Timothy 2:4 and Matthew 22:14).

Chafe likens the allegorical canons of Bach's oeuvre with Lutheran paradoxa noting that they, too, are rooted in antithesis (p. 15). Bach's use of inversion, contrary motion, retrograde, major/minor and sharp/flat contrasts represented a microcosm of musical devices. But Bach's canons are more than compressed tonal materials. With enigmatical notations such as mi contra fa, concordia discors, cross/crown, and beginning/ending, Bach associates his canons with a peculiarly Lutheran dialectic in which antithesis (what Augustine called antinomy) is a symbol for the cross of Christ. Thus Bach's canons may have stood for the affirmation of Lutheran precept as much, or more, than commentary on Baroque art.

  • In the Final Analysis
    We may never know why Bach wrote canons, nor is this necessary. Beauty in any form has reason enough to exist. But to continue to wonder at their creation can only add mystery to beauty, making the canons of J. S. Bach the most beautiful of all!



Published on Kunst der Fuge with the kind permission by the Author © 1996, Timothy A. Smith, D. M. A. Professor, Music Theory, NAU.

The Treatise on canons and fugues

Anatomy of a Canon
Why Bach writes canons?
Anatomy of a Fugue
How to analyze a Fugue?
Fugue Timeline
The Art of the Fugue

Timothy A. Smith is a professor of music theory at Northern Arizona University in Flagstaff, Arizona, USA. His web site, Canons & Fugues of J. S. Bach, is referenced, on average, over 8000 times a month from individuals in 70 countries.


The above texts or parts of them may not be published without prior permission of the Author.

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