Canon (music)

During the Middle Ages, Renaissance, and Baroque—that is, through the early 18th century—any kind of imitative musical counterpoints were called fugues, with the strict imitation now known as canon qualified as fuga ligata, meaning "fettered fugue" (Bridge 1881, 76; Mann, Wilson, and Urquhart n.d.; Walker 2000, 1). Only in the 16th century did the word "canon" begin to be used to describe the strict, imitative texture created by such a procedure (Mann, Wilson, and Urquhart n.d.). The word is derived from the Greek "κανών", Latinised as canon, which means "law" or "norm", and may be related to 8th century Byzantine hymns, or canons, like the Great Canon by St. Andrew of Crete. In contrapuntal usage, the word refers to the "rule" explaining the number of parts, places of entry, transposition, and so on, according to which one or more additional parts may be derived from a single written melodic line. This rule was usually given verbally, but could also be supplemented by special signs in the score, sometimes themselves called canoni (Bridge 1881, 76). The earliest known non-religious canons are English rounds, a form first given the name rondellus by Walter Odington at the beginning of the 14th century (Mann, Wilson, and Urquhart n.d.); the best known is Sumer Is Icumen In (composed around 1250), called a rota ("wheel") in the manuscript source (Sanders 2001a; Sanders 2001b). The term "round" only first came to be used in English sources in the 16th century (Johnson 2001).

Canons featured in the music of the Italian Trecento and the 14th-century ars nova in France. An Italian example is "Tosto che l'alba" by Gherardello da Firenze. In both France and Italy, canons were often featured in hunting songs. The medieval and modern Italian word for hunting is "caccia", while the medieval French word is spelled "chace" (modern spelling: "chasse"). A well-known French chace is the anonymous Se je chant mains. Richard Taruskin (2010, 331) describes Se je chant mains as evoking the atmosphere of a falcon hunt: "The middle section is truly a tour de force, but of a wholly new and off-beat type: a riot of hockets set to 'words' mixing French, bird-language, and hound-language in an onomatopoetical mélange." Guillaume de Machaut also used the 3-voice "chace" form in movements from his masterpiece Le Lai de la Fontaine (1361). Referring to the setting of the fourth stanza of this work, Taruskin (2010, 334–35) says "a well-wrought chace can be far more than the sum of its parts; and this particular chace is possibly Machaut's greatest feat of subtilitas."

An example of late 14th century canon which featured some of the rhythmic complexity of the late 14th century ars subtilior school of composers is La harpe de melodie by Jacob de Senleches. According to Richard Hoppin (1978, 468), "This virelai has two canonic voices over a free and textless tenor."

Jacob de Senleches, La harpe de melodie

In many pieces in three contrapuntal parts, only two of the voices are in canon, while the remaining voice is a free melodic line. In Dufay's song "Resvelons nous, amoureux", the lower two voices are in canon, but the upper part is what David Fallows (1982, 89) describes as a "florid top line":

Dufay, "Resvelons nous amoureux"

Both J. S. Bach and Handel featured canons in their works. The final variation of Handel's keyboard Chaconne in G major (HWV 442) is a canon in which the player's right hand is imitated at the distance of one beat, creating rhythmic ambiguity within the prevailing triple time:

Handel, final Variation (no. 62) from Chaconne in G major, HWV 442

An example of a classical strict canon is the Minuet of Haydn's String Quartet in D Minor, Op. 76, No. 2 (White 1976, 66). "Throughout its sinewy length, between upper and lower strings. Here is the superbly logical fulfilment of the two-part octave doubling of Haydn's earliest divertimento minuets" (Hughes 1966, p. 49):

Minuet from Haydn, String Quartet in D minor, Op. 76, No. 2

Beethoven's works feature a number of passages in canon. The following comes from his Symphony No. 4:

Beethoven Symphony No. 4, first movement, canonic passage

Antony Hopkins (1981, 108) describes the above as "a delightfully naïve canon". More sophisticated and varied in its treatment of intervals and harmonic implications is the canonic passage from the second movement of his Piano Sonata 28 in A major, Op. 101:

Beethoven, canonic passage from the second movement of Piano Sonata Op. 101

Beethoven's most spectacular and dramatically effective use of canon occurs in the first act of his opera Fidelio. Here, four of the characters sing a quartet in canon, "a sublime musical wonder" (Mellers 1993, 441), accompanied by orchestration of the utmost delicacy and refinement. "Each of the four participants delivers his or her quatrain" (Robinson 1996, 10), "The use of canon to embody the differing perspectives of the participants a first glance seems odd, but the rigid form allows for some character differentiation and does in fact make a dramatic point" (Tusa 1996, 108). "Everyone sings the same music to very different words, sinking their private thoughts into musical or at least linear anonymity" (Kerman 1996, 140). "The softly padding gait, the dove-tailed perfection of the counterpoint, induce a trance that, carrying the protagonists outside Time, hints that there are realms of truth beyond the masks they pathetically or comically present to the world" (Mellers 1993, 441).

In the Romantic era, the use of devices such as canon was even more often subtly hidden, as for example in Schumann's piano piece "Vogel als Prophet" (1851).

Schumann, "Vogel als Prophet"

According to Nicholas Cook (1990, 164), "the canon is, as it were, absorbed into the texture of the music—it is there, but one doesn't easily hear it." Peter Latham (1948, 117) describes Brahms' Intermezzo in F minor, Op. 118 No. 4 as a piece "rich in canons". In the following passage, the left hand shadows the right at the time distance of one beat and at the pitch interval of an octave lower:

From Brahms Intermezzo Op. 118, no. 4

Michael Musgrave (1985, 262) writes that as a result of the strict canon at the octave, the piece is "of an anxious, suppressed nature, [...] in the central section this tension is temporarily eased through a very contained passage which employs the canon in chordal terms between the hands." According to Denis Matthews, "[what] looks on paper like another purely intellectual exercise... in practice it produces a warmly melodic effect" (Matthews 1978, 68).

Stravinsky composed canons, including a "Canon on a Russian Popular Tune" and the Double Canon. Conlon Nancarrow composed a number of canons for player piano. (See Mensuration and tempo canons below.)

A canon of two voices may be called a canon in two, similarly a canon of x voices would be called a canon in x. This terminology may be used in combination with a similar terminology for the interval between each voice, different from the terminology in the following paragraph.

Another standard designation is "Canon: Two in One", which means two voices in one canon. "Canon: Four in Two" means four voices with two simultaneous canons. While "Canon: Six in Three" means six voices with three simultaneous canons, and so on.

A simple canon (also known as a round) imitates the leader perfectly at the octave or unison. Well-known canons of this type include the famous children's songs Row, Row, Row Your Boat and Frère Jacques.

Beginning of psalm motet De profundis by Josquin des Prez, featuring a canon at the fourth between the two upper voices in the first six bars.

If the follower imitates the precise interval quality of the leader, then it is called a strict canon; if the follower imitates the interval number (but not the quality—e.g., a major third may become a minor third), it is called a free canon (Kennedy 1994).

The follower is by definition a contrapuntal derivation of the leader.

An inversion canon (also called an al rovescio canon) has the follower moving in contrary motion to the leader. Where the leader would go down by a particular interval, the follower goes up by that same interval (Kennedy 1994).

In a retrograde canon, also known as a canon cancrizans (Latin for crab canon, derived from the Latin cancer = crab), the follower accompanies the leader backward (in retrograde). Alternative names for this type are canon per recte et retro or canon per rectus et inversus (Kennedy 1994).

In a mensuration canon (also known as a prolation canon, or a proportional canon), the follower imitates the leader by some rhythmic proportion. The follower may double the rhythmic values of the leader (augmentation or sloth canon) or it may cut the rhythmic proportions in half (diminution canon). Phasing involves the application of modulating rhythmic proportions according to a sliding scale. The cancrizans, and often the mensuration canon, take exception to the rule that the follower must start later than the leader; that is, in a typical canon, a follower cannot come before the leader (for then the labels 'leader' and 'follower' should be reversed) or at the same time as the leader (for then two lines together would constantly be in unison, or parallel thirds, etc., and there would be no counterpoint), whereas in a crab canon or mensuration canon the two lines can start at the same time and still respect good counterpoint.

Many such canons were composed during the Renaissance, particularly in the late fifteenth and early sixteenth centuries; Johannes Ockeghem wrote an entire mass (the Missa prolationum) in which each section is a mensuration canon, and all at different speeds and entry intervals. In the 20th century, Conlon Nancarrow composed complex tempo or mensural canons, mostly for the player piano as they are extremely difficult to play. Larry Polansky has an album of mensuration canons, Four-Voice Canons. Arvo Pärt has written several mensuration canons, including Cantus in Memoriam Benjamin Britten, Arbos and Festina Lente. Per Nørgård's infinity series has a sloth canon structure (Mortensen n.d.). This self-similarity of sloth canons makes it "fractal like".

A double canon is a composition that unfolds two different canons simultaneously. A duet aria, "Herr, du siehst statt guter Werke" from J. S. Bach's Cantata BWV 9, Es ist das Heil uns kommen her features a double canon "between flute and oboe on the one hand and the soprano and alto voices on the other. But what is most interesting in this movement is that the very attractive melodic surface of the canon belies its dogmatic message by offering a moving simplicity of tone to indicate the comfort that particular doctrine provides for the believer. Canonic devices often bear the association of strictness and the law in Bach's work" (Chafe 2000, 155).

Bach, passage from Duet aria "Herr, du siehst statt guter Werke" in Cantata BWV9

In a mirror canon (or canon by contrary motion), the subsequent voice imitates the initial voice in inversion. They are not very common, though examples of mirror canons can be found in the works of Bach, Mozart (e.g., the trio from Serenade for Wind Octet in C, K. 388), Webern, and other composers.

A Table canon is a retrograde and inverse canon meant to be placed on a table in between two musicians, who both read the same line of music in opposite directions. As both parts are included in each single line, a second line is not needed. Bach wrote a few table canons (Benjamin 2003, 120).

Olivier Messiaen employed a technique which he called "rhythmic canon", a polyphony of independent strands in which the pitch material differs. An example is found in the piano part of the first of the Trois petites liturgies de la présence divine, where the left hand (doubled by strings and maracas), and the right hand (doubled by vibraphone) play the same rhythmic sequence in a 3:2 ratio, but the right hand adapts a sequence of 13 chords in the sixth mode (B-C-D-E-F-F-G-A-B) onto the 18 duration values, while the left hand twice states nine chords in the third mode (Griffiths 2001). Peter Maxwell Davies was another post-tonal composer who favoured rhythmic canons, where the pitch materials are not obliged to correspond (Scholes, Nagley, and Whittall n.d.).

The notion of rhythmic canon transfers Messiaen's idea of mode of limited transposition from the domain of pitch to the domain of time (Messiaen 1944):

Messiaen considered a set of disjoint pitch classes with the same interval content which covers the twelve-tone tempered scale. For instance, four pitch classes {C, E♭, F# , A} and two transpositions, by one and by two semitones, cover the twelve-tone scale and, consequently, meet this requirement. This is similar to what is called in mathematics tiling, that is, covering an area, e.g., a square, by disjoint equal figures.

...By analogy with covering the scale by a few pitch classes and their transpositions, the pulse train was covered by a certain rhythmic pattern with different delays. The disjointedness of pitch classes implied no common beats in different instances of the rhythmic pattern.

...A rhythmic canon is one whose tone onsets result in a regular pulse train with no simultaneous tone onsets at a time. In that sense, a rhythmic canon tiles time, covering a regular pulse train by disjoint equal rhythms from different voices. Note that the established term "rhythmic canon" is somewhat misleading, and "disjoint rhythm canon" might be more exact.

...It turned out, however, that solutions to the time-tiling problem are mainly trivial and musically not interesting. A typical solution is a metronome rhythm entering with equal delays, e.g., a sequence of every fourth beat, entering at the first, at the second, and at the third beat, which is a rhythm analogy of the transpositions of pitch class classes {C, E♭, F♯, A}. Non-trivial solutions have been found by Dan Tudor Vuza for a circular time with periods 72, 108, 120,... (Vuza 1991a, Vuza 1991b, Vuza 1991c, Vuza 1991d, Vuza 1995)

— Tangian 2003

Computational methods for finding rhythmic canons, both infinite and finite, with arbitrary generative rhythmic patterns were developed in the 2000s (Tangian 2003) with further generalization to so-called "rhythmic fugues" with a few generative rhythmic patterns (Tangian 2010); for scores and recordings see Tangian 2002–2003. For a later survey see Amiot 2011.

Three voice canon by Ernst Friedrich Richter (1888, 29)

Same canon, presented by the composer as a puzzle, with multiple clefs provided as clues (Richter 1888, 38)

"Wann?", canon for soprano and alto by Brahms

A puzzle canon, riddle canon, or enigma canon is a canon in which only one voice is notated and the rules for determining the remaining parts and the time intervals of their entrances must be guessed (Merriam-Webster n.d.). "The enigmatical character of a [puzzle] canon does not consist of any special way of composing it, but only of the method of writing it down, of which a solution is required" (Richter 1888, 38). Clues hinting at the solution may be provided by the composer, in which case the term "riddle canon" can be used (Scholes, Nagley, and Whittall n.d.). J. S. Bach presented many of his canons in this form, for example in The Musical Offering. Mozart, after solving Father Martini's puzzles (Zaslaw and Cowdery 1990, 98), composed his own riddles, K. 73r, using Latin epigrams such as Sit trium series una and Ter ternis canite vocibus ("Let there be one series of three parts" and "sing three times with three voices") (Karhausen 2011, 151).

Other notable contributors to the genre include Ciconia, Ockeghem, Byrd, Beethoven, Brumel, Busnois, Haydn, Josquin des Prez, Mendelssohn, Pierre de la Rue, Brahms, Schoenberg, Nono and Maxwell Davies (Carvalho 1999, 38–39; Davidian 2015, 136; Davies 1971; Davies 1972; Hartmann 1989, passim; Hewett 1957, passim; Johnson 1994, 162–63; Jones 2009, 152; Leven 1948, 361 Litterick 2000, 388; Mann, Wilson, and Urquhart n.d.; Morley 1597, 173, 176; Perkins 2001; Tatlow 1991, 15, 126; Todd 2003, 165; van Rij 2006, 215; Watkins 1986, 239). According to Oliver B. Ellsworth, the earliest known enigma canon appears to be an anonymous ballade, En la maison Dedalus, found at the end of a collection of five theory treatises from the third quarter of the fourteenth century collected in the Berkeley Manuscript (Ciconia 1993, 411n12).

Thomas Morley complained that sometimes a solution, "which being founde (it might bee) was scant worth the hearing" (Morley 1597, 104, cited, inter al., by Barrett 2014, 123). J. G. Albrechtsberger admits that, "when we have traced the secret, we have gained but little; as the proverb says, 'Parturiunt montes, etc.'" but adds that, "these speculative passages...serve to sharpen acumen" (Albrechtsberger 1855, 234).