Origin and Definition of the word "Canon"

Requirements of a Canon

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.

Categories of Canonic Imitation

• 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.