Canon (music)

Not to be confused with Canon (hymnography) or Qanun (instrument).

Contrived example of a canon in three voices at the unison, two beats apart.

Example of a canon in three voices at the unison sung with a text of a German poem, four beats apart.

In music, a canon is a contrapuntal (counterpoint-based) compositional technique that employs a melody with one or more imitations of the melody played after a given duration (e.g., quarter rest, one measure, etc.). The initial melody is called the leader (or dux), while the imitative melody, which is played in a different voice, is called the follower (or comes). The follower must imitate the leader, either as an exact replication of its rhythms and intervals or some transformation thereof. Repeating canons in which all voices are musically identical are called rounds—familiar singalong versions of "Row, Row, Row Your Boat" and "Frère Jacques" that call for each successive group of voices to begin the same song a bar or two after the previous group began are popular examples.

An accompanied canon is a canon accompanied by one or more additional independent parts that do not imitate the melody.

History

Medieval and Renaissance

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". Only in the 16th century did the word "canon" begin to be used to describe the strict, imitative texture created by such a procedure. The word is derived from the Greek "κανών", Latinised as canon, which means "law" or "norm". 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. 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; the best known is "Sumer is icumen in" (composed around 1250), called a rota ("wheel") in the manuscript source. The term "round" only first came to be used in English sources in the 16th century.

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 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 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, "This virelai has two canonic voices over a free and textless tenor."

"La harpe de melodie"

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 describes as a "florid top line":

Dufay, "Resvelons nous"

Dufay, "Resvelons nous amoureux"

Baroque

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 Chaconne HWV 442, variation 62

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

Classical

An example of a classical strict canon is the Minuet of Haydn's String Quartet in D Minor, Op. 76, No. 2. "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":

Haydn, Minuet from Quartet in D minor, Op. 76

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

Beethoven

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

Beethoven Symphony No. 4, canonic passage from the 1st movement

Beethoven Symphony No. 4, first movement, canonic passage

Antony Hopkins 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 canon from piano sonata in A, 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", accompanied by orchestration of the utmost delicacy and refinement. "Each of the four participants delivers his or her quatrain", "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". "Everyone sings the same music to very different words, sinking their private thoughts into musical or at least linear anonymity". "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."

Romantic era

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" from Waldszenen

Schumann, "Vogel als Prophet"

According to Nicholas Cook, "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 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:

Brahms Intermezzo Op. 118, no. 4

From Brahms Intermezzo Op. 118, no. 4

Michael Musgrave 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."

20th century to present

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.) Anton Webern employed canonic textures in his work; his Op. 16 work is a collection of five canons for soprano, clarinet, and bass clarinet.

Types

Considering the many types of canon "in the tonal repertoire", it may be ironic that "canon—the strictest type of imitation—has such a wide variety of possibilities". The most rigid and ingenious forms of canon are not strictly concerned with pattern but also with content. Canons are classified by various traits including the number of voices, the interval at which each successive voice is transposed in relation to the preceding voice, whether voices are inverse, retrograde, or retrograde-inverse; the temporal distance between each voice, whether the intervals of the second voice are exactly those of the original or if they are adjusted to fit the diatonic scale, and the tempo of successive voices. However, canons may use more than one of the above methods.

Contour Canon

A Contour Canon can be recognized in the traditional sense, similar to a strict canon or to a canon by inversion, where an original theme or design is presented, and is then followed by a response of the same theme, as well as in an untraditional fashion, where Subcontouric Cells are positioned in such a way that they assemble a canon. In this untraditional fashion, a contour’s cells are presented and altered in a rotational motion, until the entire image or contour can be seen in its Prime form. Each cell in a pairing of Subcontouric Cells cycles through their rotational variations, until they have established themselves in their intended contour position, or Prime Form, such as (1-1)(1-2), referred to as a contour’s Cell Cycle.

Terminology

Although, for clarity, this article uses leader and follower(s) to denote the leading voice in a canon and those that imitate it, musicological literature also uses the traditional Latin terms dux and comes for "leader" and "follower", respectively.

Number of voices

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.

Simple

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.

Interval

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.

Contrapuntal derivations

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

Canon by inversion

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.

Retrograde or crab canon

Main article: Crab canon

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.

Mensuration and tempo canons

Main article: prolation canon

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. This self-similarity of sloth canons makes it "fractal like".

Other types

The most familiar of the canons is the perpetual/infinite canon (in Latin: canon perpetuus) or round. As each voice of the canon arrives at its end it can begin again, in a perpetuum mobile fashion; e.g., "Three Blind Mice". Such a canon is also called a round or, in medieval Latin terminology, a rota. Sumer is icumen in is one example of a piece designated rota.

Additional types include the spiral canon, accompanied canon, and double or triple canon. A double canon is a canon with two simultaneous themes; a triple canon has three.

Double canon

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."

Bach, passage from duet aria "Herr, du siehst statt guter Werke" in Cantata BWV 9

Bach, passage from duet aria "Herr, du siehst statt guter Werke" in Cantata BWV 9

Mirror canon

Main article: Mirror canon

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 minor, K. 388/384a), Anton Webern, and other composers.

Table canon

Main article: Table canon

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.

Rhythmic canon

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. Peter Maxwell Davies was another post-tonal composer who favoured rhythmic canons, where the pitch materials are not obliged to correspond.

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 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,...

— Andranik Tangian

Computational methods for finding rhythmic canons, both infinite and finite, with arbitrary generative rhythmic patterns were developed in the 2000s with further generalization to so-called "rhythmic fugues" with a few generative rhythmic patterns.

Puzzle canon

Three voice canon by Ernst Friedrich Richter

Same canon, presented by the composer as a puzzle, with multiple clefs provided as clues

"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. "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." Clues hinting at the solution may be provided by the composer, in which case the term "riddle canon" can be used. J. S. Bach presented many of his canons in this form, for example in The Musical Offering. Mozart, after solving Father Martini's puzzles, 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").

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. 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.

Thomas Morley complained that sometimes a solution, "which being founde (it might bee) was scant worth the hearing", 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".

Elaborate use of canon technique

• Josquin des Prez, Missa L'homme armé super voces musicales, Agnus Dei 2: One voice with the words 'ex una voce tres' (three voice parts out of one), a mensuration canon in three voices.
• Josquin des Prez, Missa L'homme armé sexti toni, Agnus Dei 2: two simultaneous canons in the four upper voices, and at the same time a crab canon in the two lower voices.
• Johann Sebastian Bach's Goldberg Variations contains nine canons of increasing interval size, ranging from unison to ninth. Each canon additionally obeys the overall structure and harmonic sequence common to all variations in the composition.

Contemporary canons

A famous piano piece, "River Flows in You" in the key of F# minor by South Korean Pianist Lee Ru-ma or Yiruma, features a repetitive canon using the same key progression (F#, D, A, E x2). Since its recognition online, there have been multiple covers of the song, including a mashup of it with Johann Pachelbel's Canon and Gigue in D Major.

In his early work, such as Piano Phase (1967) and Clapping Music (1972), Steve Reich used a process he calls phasing which is a "continually adjusting" canon with variable distance between the voices, in which melodic and harmonic elements are not important, but rely simply on the time intervals of imitation.

See also

• Canon, a 1964 animated representation of a musical canon
• Pachelbel's Canon

References

Sources

• Albrechtsberger, Johann Georg. 1855. J. G. Albrechtsberger's Collected Writings on Thorough-bass, Harmony, and Composition, for Self Instruction, edited by Sabilla Novello, translated by Ignaz von Seyfried, revised by Vincent Novello. London: Novello, Ewer, & Company. [ISBN unspecified].
• Amiot, Emmanuel (2011). "Structures, algorithms, and algebraic tools for rhythmic canons" (PDF). Perspectives of New Music. 49 (2): 93–142. doi:10.1353/pnm.2011.0018. Retrieved 16 January 2021.
• Barrett, Margaret S. (ed.). 2014. Collaborative Creative Thought and Practice in Music. Farnham: Ashgate Publishing, Ltd. ISBN 9781472415868.
• Benjamin, Thomas. 2003. The Craft of Tonal Counterpoint New York: Routledge. ISBN 0-415-94391-4 (accessed 14 April 2011)
• Bridge, J. Frederick. [1881]. Double Counterpoint and Canon. London: Novello & Co., Ltd.; New York: The H. W. Gray Co., Inc.
• Carvalho, Mário Vieira de. 1999. "Towards Dialectic Listening: Quotation and Montage in the Work of Luigi Nono". Contemporary Music Review 18, no 2: 37–85.
• Chafe, Eric Thomas. 2000. Analyzing Bach Cantatas. Oxford and New York: Oxford University Press. ISBN 978-0-19-512099-8 (cloth); ISBN 978-0-19-516182-3 (pbk).
• Ciconia, Johannes. 1993. Nova musica and De Proportionibus, edited and translated by Oliver B. Ellsworth. Greek and Latin Music Theory 9. Lincoln: University of Nebraska Press. ISBN 9780803214651.
• Cook, Nicolas. 1990. Music, Imagination and Culture. Oxford: Clarendon Press; New York: Oxford University Press.
• Davidian, Teresa. 2015. Tonal Counterpoint for the 21st-Century Musician: An Introduction. Lanham and London: Rowman and Littlefield. ISBN 978-1-4422-3458-1 (cloth); ISBN 978-1-4422-3459-8 (pbk); ISBN 978-1-4422-3460-4 (ebook).
• Davies, Peter Maxwell. 1971. "Canon: In Mem. IS". Tempo 97 (June): [Supplement]: In Memoriam: Igor Fedorovich Stravinsky. Canons and Epitaphs, Set I: one unnumbered page.
• Davies, Peter Maxwell. 1972. [Supplement]: "Canons and Epitaphs in Memoriam Igor Stravinsky. A Solution by Peter Maxwell Davies of the Puzzle-Canon He Contributed to Set I, Published in Tempo 97". Tempo, no. 100: three unnumbered pages.
• Fallows, David. 1982. Dufay. London: J. M. Dent. ISBN 9780460031806.
• Griffiths, Paul. 2001. "Messiaen, Olivier (Eugène Prosper Charles)". The New Grove Dictionary of Music and Musicians, second edition, edited by Stanley Sadie and John Tyrrell. London: Macmillan.
• Hartmann, Günter. 1989. "Ein Albumblatt für Eliza Wesley: Fragen zu Mendelssohns Englandauenthalt 1837 und eine spekulative Antwort". Neue Zeitschrift für Musik 150, no. 1:10–14.
• Hewett, Helen. 1957. "The Two Puzzle Canons in Busnois's Maintes femmes". Journal of the American Musicological Society 10, no. 2 (Summer): 104–110.
• Hopkins, Antony. 1981. The Nine Symphonies of Beethoven. London: Heinemann.
• Hoppin, Richard. 1978. Medieval Music. New York: W. W. Norton.
• Hughes, R. 1966. Haydn String Quartets. London: BBC.
• Johnson, David. 2001. "Round". The New Grove Dictionary of Music and Musicians, second edition, edited by Stanley Sadie and John Tyrrell. London: Macmillan.
• Johnson, Stephen. 1994. Beethoven. New York: Simon and Schuster. ISBN 9780671887896.
• Jones, David Wyn. 2009. The Life of Haydn. Cambridge and New York: Cambridge University Press. ISBN 978-0-521-89574-3 (cloth); ISBN 978-1-107-61081-1 (pbk).
• Karhausen, Lucien. 2011. The Bleeding of Mozart. Foreword by H. C. Robbins Landon. Xlibris. ISBN 9781456850760.
• Kennedy, Michael (ed.). 1994. "Canon". The Oxford Dictionary of Music, associate editor, Joyce Bourne. Oxford and New York: Oxford University Press. ISBN 0-19-869162-9.
• Kerman, Joseph. 1996. "Augenblicke in Fidelio". In Ludwig van Beethoven: Fidelio, edited by Paul Robinson, 132–144. Cambridge and New York: Cambridge University Press. ISBN 0-521-45221-X.
• Latham, Peter. 1948. Brahms. London: Dent.
• Leven, Louise W. 1948. "An Unpublished Mendelssohn Manuscript". The Musical Times 89, no. 1270 (December): 361–363.
• Litterick, Louise. 2000. "Chansons for Three and Four Voices". In The Josquin Companion, edited by Richard Sherr, 335–392. Oxford and New York: Oxford University Press. ISBN 0-19-816335-5.
• Mann, Alfred, J. Kenneth Wilson, and Peter Urquhart. n.d. "Canon (i)". Grove Music Online. Oxford Music Online (Accessed 2 January 2011) (subscription required).
• Matthews, Denis. 1978. Brahms Piano Music. London: BBC.
• Mellers, Wilfrid. 1983. Beethoven and the Voice of God. London, Faber & Faber. ISBN 057111718X
• Merriam-Webster. n.d. "Puzzle Canon"". Merriam-Webster Dictionary online edition (subscription required).
• Messiaen, Olivier (1944). La technique de mon langage musical (in French). Paris: Alphonse Leduc. ISBN 9782856890332.
• Morley, Thomas. 1597. A Plaine and Easie Introduction to Practicall Musicke, Set Downe in Forme of a Dialogue. London: Peter Short. [ISBN unspecified]
• Mortensen, Jørgen. n.d. "The 'Open Hierarchies' of the Infinity Series". In Per Nørgård: En introduktion til komponisten og hans musik (Danish and English), edited by Jørgen Mortensen. www.pernoergaard.dk (Accessed 20 January 2013).
• Musgrave, Michael. 1985. The Music of Brahms. London: Routledge.
• Perkins, Leeman L. 2001. "Ockeghem [Okeghem, Hocquegam, Okegus etc.], Jean de [Johannes, Jehan]". The New Grove Dictionary of Music and Musicians, second edition, edited by Stanley Sadie and John Tyrrell. London: Macmillan.
• Richter, Ernst Friedrich. 1888. A Treatise on Canon and Fugue: Including the Study of Imitation. Boston: Oliver Ditson. Translated from third German edition by Foote, Arthur W. [ISBN unspecified].
• Robinson, Paul (ed.). 1996. Ludwig van Beethoven: Fidelio. Cambridge and New York: Cambridge University Press. ISBN 0-521-45221-X.
• Sanders, Ernest H. 2001a. "Rota". The New Grove Dictionary of Music and Musicians, second edition, edited by Stanley Sadie and John Tyrrell. London: Macmillan.
• Sanders, Ernest H. 2001b. "Sumer is icumen in". The New Grove Dictionary of Music and Musicians, second edition, edited by Stanley Sadie and John Tyrrell. London: Macmillan.
• Scholes, Percy, Judith Nagley, and Arnold Whittall. "Canon". The Oxford Companion to Music, edited by Alison Latham. Oxford Music Online. Oxford University Press (accessed 13 December 2014) (subscription required).
• Tangian, Andranik (2002–2003). "Eine kleine Mathmusik I and II". IRCAM, Seminaire MaMuX, 9 February 2002, Mosaïques et pavages dans la musique. Retrieved January 16, 2021.
• Tangian, Andranik (2003). "Constructing rhythmic canons" (PDF). Perspectives of New Music. 41 (2): 64–92. Retrieved January 16, 2021.
• Tangian, Andranik (2010). "Constructing rhythmic fugues". IRCAM, Seminaire MaMuX, 9 February 2002, Mosaïques et pavages dans la musique (PDF) ((unpublished addendum to Constructing rhythmic canons)). Retrieved January 16, 2021.
• Taruskin, Richard. 2010. Music from the Earliest Notations to the Sixteenth Century: The Oxford History of Western Music, Volume 1. Oxford and New York: Oxford University Press. ISBN 978-0-19-538481-9.
• Tatlow, Ruth. 1991. Bach and the Riddle of the Number Alphabet. Cambridge and New York: Cambridge University Press. ISBN 978-0-521-36191-0.
• Todd, R. Larry. 2003. Mendelssohn. Oxford and New York: Oxford University Press. ISBN 978-0-19-511043-2.
• Tusa, Michael C. 1996. "Music as Drama: Structure, Style and Process in Fidelio". In Ludwig van Beethoven: Fidelio, edited by Paul Robinson, 101–131. Cambridge and New York: Cambridge University Press. ISBN 0-521-45221-X.
• van Rij, Inge. 2006. Brahms's Song Collections. Cambridge and New York: Cambridge University Press. ISBN 978-0-521-83558-9.
• Vuza, Dan Tudor (1991a). "Supplementary sets and regular complementary unending canons. Part 1". Perspectives of New Music. 29 (2): 22–49. doi:10.2307/833429. JSTOR 833429.
• — (1991b). "Supplementary sets and regular complementary unending canons. Part 2". Perspectives of New Music. 30 (1): 184–207. doi:10.2307/833290. JSTOR 833290.
• — (1991c). "Supplementary sets and regular complementary unending canons. Part 3". Perspectives of New Music. 30 (2): 102–125. doi:10.2307/3090628. JSTOR 3090628.
• — (1991d). "Supplementary sets and regular complementary unending canons. Part 4". Perspectives of New Music. 31 (1): 270–305. doi:10.2307/833054. JSTOR 833054.
• — (1995). "Supplementary sets: theory and algorithms". Muzika. 1: 75–99.
• Walker, Paul Mark. 2000. Theories of Fugue from the Age of Josquin to the Age of Bach. Rochester, NY: University of Rochester Press. ISBN 9781580461504.
• Watkins, Glenn. 1986. "Canon and Stravinsky's Late Style". In Confronting Stravinsky: Man, Musician, and Modernist, edited by Jann Pasler, 217–246. Berkeley and Los Angeles: University of California Press. ISBN 0-520-05403-2.
• White, John David. 1976. The Analysis of Music. Englewood Cliffs, N.J.: Prentice-Hall. ISBN 0-13-033233-X.
• Zaslaw, Neal, and William Cowdery (eds.) 1990. The Compleat Mozart: A Guide to the Musical Works of Wolfgang Amadeus Mozart. New York: W. W. Norton & Company. ISBN 9780393028867.

Further reading

• Agon, Carlos, and Moreno Andreatta. 2011. "Modeling and Implementing Tiling Rhythmic Canons in the OpenMusic Visual Programming Language". Perspectives of New Music 49, no. 2 (Summer): 66–91.
• Andreatta, Moreno. 2011. "Constructing and Formalizing Tiling Rhythmic Canons: A Historical Survey of a 'Mathematical' Problem". Perspectives of New Music 49, no. 2 (Summer): 33–64.
• Blackburn, Bonnie J. 2012. "The Corruption of One Is the Generation of the Other: Interpreting Canonic Riddles". Journal of the Alamire Foundation 4, no. 2 (October):182–203.
• Cooper, Martin. 1970. Beethoven: The Last Decade. London and New York: Oxford University Press.
• Davalan, Jean-Paul. 2011. "Perfect Rhythmic Tiling". Perspectives of New Music 49, no. 2 (Summer): 144–197.
• Johnson, Tom. 2011. "Tiling in My Music". Perspectives of New Music 49, no. 2 (Summer): 9–21.
• Lamla, Michael. 2003 Kanonkünste im barocken Italien, insbesondere in Rom. 3 vols. Berlin: Dissertation.de—Verlag im Internet. ISBN 3-89825-556-5.
• Lévy, Fabien. 2011. "Three Uses of Vuza Canons". Perspectives of New Music 49, no. 2 (Summer): 23–31.
• Messiaen, Olivier. Traité de rythme, de couleur, et d'ornithologie (1949–1992). I-II, edited by Yvonne Loriod, preface by Pierre Boulez. Paris: Leduc, 1994.
• Schiltz, Katelijne, and Bonnie J. Blackburne (eds.). 2007. Canons and Canonic Techniques, 14th–16th Centuries: Theory, Practice, and Reception History. Proceedings of the International Conference Leuven, 4–5 October 2005. Analysis in Context: Leuven Studies in Musicology 1. Leuven and Dudley, Massachusetts: Peeters. ISBN 978-90-429-1681-4.
• Ziehn, Bernhard. Canonic Studies: A New Technique in Composition, edited and introduced by Ronald Stevenson. New York: Crescendo Pub., 1977. ISBN 0-87597-106-7.

External links

• Anatomy of a Canon Archived 2012-06-18 at the Wayback Machine
• The Musical Offering – A Musical Pedagogical Workshop by J. S. Bach, or, The Musical Geometry of Bach's Puzzle Canons, schillerinstitut.dk (in English)
• Visualization of J. S. Bach's crab canon (requires Adobe Flash)
• Software SonneLematine to produce canons
• Electro-Acoustic Music Dartmouth.edu: Larry Polansky's Four Voice Canons
• Video canon on YouTube, on "My Favorite Things"