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Polyrhythm is the simultaneous use of two or more conflicting rhythms, that are not readily perceived as deriving from one another, or as simple manifestations of the same meter. The rhythmic conflict may be the basis of an entire piece of music (cross-rhythm), or a momentary disruption. Polyrhythms can be distinguished from irrational rhythms, which can occur within the context of a single part; polyrhythms require at least two rhythms to be played concurrently, one of which is typically an irrational rhythm.
In western art music 
In some European art music, polyrhythm periodically contradicts the prevailing meter. For example, polyrhythm is heard in the first few minutes of Beethoven's Third Symphony and in the first movement of Brahms's Violin Concerto.
Concerning the use of a two-over-three (2:3) hemiola in Beethoven's Sixth String Quartet, Ernest Walker states, "The vigorously effective Scherzo is in 3/4 time, but with a curiously persistent cross-rhythm that does its best to persuade us that it is really in 6/8."
Polyrhythm, not polymeter 
However, the two beat schemes interact within a metric hierarchy (a single meter). The triple beats are primary and the duple beats are secondary; the duple beats are cross-beats within a triple beat scheme.
Composite hemiola 
Another example of polyrhythm can be found in measures 64 and 65 of the first movement of Wolfgang Amadeus Mozart's Twelfth Piano Sonata. Three evenly-spaced sets of three attack-points span two measures.
Cross-rhythm refers to systemic polyrhythm. The New Harvard Dictionary of Music states that cross-rhythm is: "A rhythm in which the regular pattern of accents of the prevailing meter is contradicted by a conflicting pattern and not merely a momentary displacement that leaves the prevailing meter fundamentally unchallenged" (1986: 216). The physical basis of cross-rhythms can be described in terms of interference of different periodicities.
A simple example of a cross-rhythm is 3 evenly-spaced notes against 2 (3:2), also known as a hemiola. Two simple and common ways to express this pattern in standard western musical notation would be 3 quarter notes over 2 dotted quarter notes within one bar of 6/8 time, quarter note triplets over 2 quarter notes within one bar of 2/4 time. Other cross-rhythms are 4:3 (with 4 dotted eight notes over 3 quarter notes within a bar of 3/4 time as an example in standard western musical notation), 5:2, 5:3, 5:4, etc.
There is a parallel between cross rhythms and musical intervals: in an audible frequency range, the 2:3 ratio produces the musical interval of a perfect fifth, the 3:4 ratio produces a perfect fourth, and the 4:5 ratio produces a major third. All these interval ratios are found in the harmonic series.
Sub-Saharan African music traditions 
Comparing European and Sub-Saharan African meter 
In traditional European ("Western") rhythms, the most fundamental parts typically emphasize the primary beats. By contrast, in rhythms of sub-Saharan African origin, the most fundamental parts typically emphasize the secondary beats. This often causes the uninitiated ear to misinterpret the secondary beats as the primary beats, and to hear the true primary beats as cross-beats. In other words, the musical "background" and "foreground" may mistakenly be heard and felt in reverse—Peñalosa (2009: 21).
The generating principle 
In Sub-Saharan African music traditions, cross-rhythm is the generating principle; the meter is in a permanent state of contradiction. Cross-rhythm was first explained as the basis of sub-Saharan rhythm in lectures by C.K. Ladzekpo and the writings of David Locke.
From the philosophical perspective of the African musician, cross-beats can symbolize the challenging moments or emotional stress we all encounter. Playing cross-beats while fully grounded in the main beats, prepares one for maintaining a life-purpose while dealing with life's challenges. Many sub-Saharan languages do not have a word for rhythm, or even music. From the African viewpoint, the rhythms represent the very fabric of life itself; they are an embodiment of the people, symbolizing interdependence in human relationships—Peñalosa (2009: 21).
At the center of a core of rhythmic traditions within which the composer conveys his ideas is the technique of cross-rhythm. The technique of cross-rhythm is a simultaneous use of contrasting rhythmic patterns within the same scheme of accents or meter... By the very nature of the desired resultant rhythm, the main beat scheme cannot be separated from the secondary beat scheme. It is the interplay of the two elements that produces the cross-rhythmic texture—Ladzekpo (1995).
Eugene Novotney observes: "The 3:2 relationship (and [its] permutations) is the foundation of most typical polyrhythmic textures found in West African musics." 3:2 is the generative or theoretic form of sub-Saharan rhythmic principles. Victor Kofi Agawu succinctly states, "[The] resultant [3:2] rhythm holds the key to understanding... there is no independence here, because 2 and 3 belong to a single Gestalt."
The two beat schemes interact within the hierarchy of a single meter. The duple beats are primary and the triple beats are secondary. [Watch: Stepping to the primary beats within 3:2 cross-rhythm. Afro-Cuban Obatalá dance (Marta Ruiz).] The example below shows the African 3:2 cross-rhythm within its proper metric structure.
The music of African Xylophones such as the balafon and gyil is often based on cross-rhythm. In the following example, a Ghanaian gyil sounds a 3:2-based ostinato melody. The left hand (lower notes) sounds the two main beats, while the right hand (upper notes) sounds the three cross-beats. The cross-beats are written as quarter-notes for visual emphasis.
The following notated example is from the kushuara part of the traditional mbira piece "Nhema Mussasa". The mbira is a lamellophone. The left hand plays the ostinato bass line while the right hand plays the upper melody. The composite melody is an embellishment of the 3:2 cross-rhythm.
Adaptive instruments 
Sub-Saharan instruments are constructed in a variety of ways to generate polyrhythmic melodies. Some instruments organize the pitches in a uniquely divided alternate array – not in the straight linear bass to treble structure that is so common to many western instruments such as the piano, harp, or marimba.
Lamellophones including mbira, mbila, mbira huru, mbira njari, mbira nyunga, marimba, karimba, kalimba, likembe, and okeme. This family of instruments are found in several forms indigenous to different regions of Africa and most often have equal tonal ranges for right and left hands. The kalimba is a modern version of these instruments originated by the pioneer ethnomusicologist Hugh Tracey in the early 20th century which has over the years gained world-wide popularity.
The Gravikord is a new American instrument closely related to both the African kora and the kalimba was created in the latter 20th century to also exploit this adaptive principle in a modern electro-acoustic instrument.
On these instruments, one hand of the musician is not primarily in the bass nor the other primarily in the treble, but both hands can play freely across the entire tonal range of the instrument. Also, the fingers of each hand can play separate independent rhythmic patterns, and these can easily cross over each other from treble to bass and back, either smoothly or with varying amounts of syncopation. This can all be done within the same tight tonal range, without the left and right hand fingers ever physically encountering each other. These simple rhythms will interact musically to produce complex cross rhythms including repeating on beat/off beat pattern shifts that would be very difficult to create by any other means. This characteristically African structure allows often simple playing techniques to combine with each other to produce polyrhythmic music.
3:2 cross-rhythm 
Polyrhythm is a staple of modern jazz. Although not as common, use of systemic cross-rhythm is also found in jazz. In 1959, Mongo Santamaria recorded "Afro Blue," the first jazz standard built upon a typical African 6:4 cross-rhythm (two cycles of 3:2). The song begins with the bass repeatedly playing 6 cross-beats per each measure of 12/8 (6:4). The following example shows the original ostinato "Afro Blue" bass line. The slashed noteheads indicate the main beats, where you would normally tap your foot to "keep time."
2:3 cross-rhythm 
The famous jazz drummer Elvin Jones took the opposite approach, superimposing two cross-beats over every measure of a 3/4 jazz waltz (2:3). This swung 3/4 is perhaps the most common example of overt cross-rhythm in jazz. In 1963 John Coltrane recorded "Afro Blue" with Elvin Jones on drums. Coltrane reversed the metric hierarchy of Santamaria's composition, performing it instead in 3/4 swing (2:3).
In popular music 
Nigerian percussion master Babatunde Olatunji arrived on the American music scene in 1959 with his album Drums of Passion, which was a collection of traditional Nigerian music for percussion and chanting. The album stayed on the charts for two years and had a profound impact on jazz and American popular music. Trained in the Yoruba sakara style of drumming, Olatunji would have a major impact on Western popular music. He went on to teach, collaborate and record with numerous jazz and rock artists, including Airto Moreira, Carlos Santana and Mickey Hart of the Grateful Dead. Olatunji reached his greatest popularity during the height of the Black Arts Movement of the 1960s and '70s.
Afro-Cuban music makes extensive use of polyrhythms. Cuban Rumba uses 3-based and 2-based rhythms at the same time, for example, the lead drummer (playing the quinto) might play in 6/8, while the rest of the ensemble keeps playing 2/2. Afro-Cuban conguero, or conga player, Mongo Santamaría was another percussionist whose polyrhythmic virtuosity helped transform both jazz and popular music. Santamaria fused Afro-Latin rhythms with R&B and jazz as a bandleader in the 1950s, and was featured in the 1994 album Buena Vista Social Club, which was the inspiration for the like-titled documentary released five years later.
Among the most sophisticated polyrhythmic music in the world is south Indian classical Carnatic music. A kind of rhythmic solfege called konnakol is used as a tool to construct highly complex polyrhythms and to divide each beat of a pulse into various subdivisions, with the emphasised beat shifting from beat cycle to beat cycle.
Common polyrhythms found in jazz are 3:2, which manifests as the quarter-note triplet; 2:3, usually in the form of dotted-quarter notes against quarter notes; 4:3, played as dotted-eighth notes against quarter notes (this one demands some technical proficiency to perform accurately, and was not at all common in jazz before Tony Williams used it when playing with Miles Davis); and finally 3/4 time against 4/4, which along with 2:3 was used famously by Elvin Jones and McCoy Tyner playing with John Coltrane.
The Beatles used polyrhythm in their 1968 song "Happiness Is a Warm Gun"(from the White Album). The song also changes time signature frequently. The Beatles use polyrhythm again on Abbey Road's "Mean Mr. Mustard".
Jimi Hendrix had the distinct ability to play polyrhythmic melodies on his guitar during live concerts and jam sessions. This ability was facilitated by the impressive length and size of his hands, and his unorthodox fretting method, in which he would maintain rhythm and lead melodies while using his thumb to fret underlying basslines. Examples are live concerts from 1968 to 1970, in particular a performance of "Killing Floor" live at Winterland 1968, an Improvisation during Woodstock 1969, a solo guitar jam for his song titled "Valleys of Neptune", among several other recordings.
Frank Zappa, especially towards the end of his career, experimented with complex polyrhythms, such as 11:17, and even nested polyrhythms (see "The Black Page" for an example). The metal bands Meshuggah, Nothingface, Periphery, Threat Signal, Lamb of God, Textures and TesseracT also use polyrhythms in their music. Contemporary progressive metal bands such as Tool, Animals as Leaders, Between the Buried and Me and Dream Theater also incorporate polyrhythms in their music, and polyrhythms have also been increasingly heard in technical metal bands such as Ion Dissonance, The Dillinger Escape Plan, Necrophagist, Candiria, The Contortionist and Textures. Much minimalist and totalist music makes extensive use of polyrhythms. Henry Cowell and Conlon Nancarrow created music with yet more complex polytempo and using irrational numbers like pi:e.
Nine Inch Nails front man Trent Reznor uses polyrhythm frequently. One notable appearance is in the song "La Mer" from the album The Fragile. The piano holds a 3/4 riff while the drums and bass back it with a standard 4/4 signature. Talking Heads' Remain in Light used dense polyrhythms throughout the album, most notably on the song "The Great Curve".
Japanese girl group Perfume made use of the technique in their single, appropriately titled as Polyrhythm, from their second album Game. The bridge of the song incorporates 5/8, 6/8 in the vocals, common time (4/4) and 3/2 in the drums.
The Aaliyah song "Quit Hatin" uses 9/8 against 4/4 in the chorus
The following is an example of a 3 against 2 polyrhythm, given in time unit box system (TUBS) notation; each box represents a fixed unit of time; time progresses from the left of the diagram to the right. It is in bad form to teach a student to play 3/2 polyrhythms as simply quarter note, eighth note, eighth note, quarter note. The proper way is to establish sound bases for both the quarter-notes, and the triplet-quarters, and then to layer them upon each other, forming multiple rhythms. Beats are indicated with an X; rests are indicated with a blank.
A common memory aid to help with the 3 against 2 polyrhythm is that it has the same rhythm as the phrase "not difficult"; the simultaneous beats occur on the word "not"; the second and third of the triple beat land on "dif" and "cult", respectively. The second 2-beat lands on the "fi" in "difficult." Try saying "not difficult" over and over in time with the sound file above. This will emphasize the "3 side" of the 3 against 2 feel. Now try saying the phrase "not a problem", stressing the syllables "not" and "prob-". This will emphasize the "2 side" of the 3 against 2 feel. More phrases with the same rhythm are "cold cup of tea", "four funny frogs", "come, if you please".
As can be seen from above, the counting for polyrhythms is determined by the lowest common multiple, so if one wishes to count 2 against 3, one needs to count a total of 6 beats, as lcm(2,3) = 6 (123456 and 123456). However this is only useful for very simple polyrhythms, or for getting a feel for more complex ones, as the total number of beats rises quickly. To count 4 against 5, for example, requires a total of 20 beats, and counting thus slows the tempo considerably. However some players, such as classical Indian musicians, can intuitively play high polyrhythms such as 7 against 8.
Polyrhythms are quite common in late Romantic Music and 20th century classical music. Works for keyboard often set odd rhythms against one another in separate hands. A good example is in the soloist's cadenza in Grieg's Concerto in A Minor; the left hand plays arpeggios of seven notes to a beat; the right hand plays an ostinato of eight notes per beat while also playing the melody in octaves, which uses whole notes, dotted eighth notes, and triplets. Other instances occur often in Rachmaninoff's Piano Concerto No. 2. The piano arpeggios that constitute much of the soloist's material in the first movement often have anywhere from four to eleven notes per beat. In the last movement, the piano's opening run, marked 'quasi glissando', fits 52 notes into the space of one measure, making for a glissando-like effect while keeping the mood of the music. Other instances in this movement include a scale that juxtaposes ten notes in the right hand against four in the left, and one of the main themes in the piano, which imposes an eighth-note melody on a triplet harmony. Another example is the fluid 7:3 polyrhythm at the beginning of Charles Griffes' The White Peacock.
In fine art 
Although most often discussed in music (in time) an analogous form of the aesthetic of polyrhythms can also be an organizing principle in the fine arts (in space). Polyrhythmic design aesthetics are a deep and rich vein in two dimensional pattern creation especially in African and Asian art. It is a prominent design motif and can be found in many Japanese kimonos and African fabric prints and also especially in the fabric batiks of India, and in many other eastern cultures. In western art, polyrhythmic design elements can also prominently be seen in the latter works of the 20th century European painters Matisse, Joan Miró, and Mondrian among others. There is also a strong polyrhythmic approach apparent in many current original American quilt makers designs, and also in some of the quilts that exist from earlier periods of this popular ongoing folk art.
See also 
- Ewe music
- Beat (acoustics) - another example of the same effect (mathematically), but with two continuous waves rather than a hit of the instrument only at every peak and trough of either wave.
Further reading 
- Peter Magadini (2001). Polyrhythms: The Musicians Guide. ISBN 0-634-03283-6. Polyrhythm reference book.
- Peter Magadini (1993). Polyrhythms For The Drumset. ISBN 0-89724-821-X. Study in polymetric independence for drummers.
- Slenczynska (1976). Music At Your Fingertips: Advice For The Artist And Amateur On Playing The Piano, p.43. ISBN 0-306-80034-9.
- New Harvard Dictionary of Music (1986: 646). Cambridge, MA: Harvard University Press.
- Novotney, Eugene D. (1998: 265). The Three Against Two Relationship as the Foundation of Timelines in West African Musics Urbana, IL: University of Illinois. UnlockingClave.com. http://www.unlockingclave.com/free-download-32-thesis.html
- Walker, Ernest (1905: 79) The Music of the Masters; Beethoven. New York: Brentano's Union Square.
- New Harvard Dictionary of Music (1986: 216). Cambridge, MA: Harvard University Press.
- Joseph Schillinger.1941. Theory of Rhythm. In: Joseph Schillinger. 1941. The Schillinger System of Musical Composition, p. 4-12
- Peñalosa, David (2009: 21).
- Peñalosa, David (2009: 21) The Clave Matrix; Afro-Cuban Rhythm: Its Principles and African Origins. Redway, CA: Bembe Inc. ISBN 1-886502-80-3.
- Ladzekpo, C.K. (1995: webpage). "The Myth of Cross-Rhythm", Foundation Course in African Dance-Drumming.
- Novotney, Eugene D. (1998). The Three Against Two Relationship as the Foundation of Timelines in West African Musics Urbana, IL: University of Illinois. UnlockingClave.com.
- Agawu, Kofi (2003: 92). Representing African Music: Postcolonial Notes, Queries, Positions New York: Routledge. ISBN 0-415-94390-6.
- Peñalosa (2010). The Clave Matrix p. 22.
- Peñalosa (2010). The Clave Matrix p. 35.
- The Gravikord web site : http://www.gravikord.com/instrument.html#gravikord
- "Afro Blue," Afro Roots (Mongo Santamaria) Prestige CD 24018-2 (1959).
- Conor Guilfoyle demonstrates 3/4 swing. http://www.youtube.com/watch?v=OEAyWsTLrYY&feature=related
- "Afro Blue," Impressions (John Coltrane) Pablo CD (1963).
- John Coltrane performs "Afro Blue" with Elvin Jones on drums. http://www.youtube.com/watch?v=olOYynQ-_Hw
- The songwriting secrets of the Beatles By Dominic Pedler.[page needed]
- Markus Heuger's Beabliography (Abstract).[page needed]
- Pellerin, Adrien (2011). Britney Spears is using Tuplets? http://adrienpellerin.tumblr.com/post/6274133096/britney-spears-is-using-tuplets.
- Sep 2007. Guitar World, p.102. Vol. 28, No. 9. ISSN 1045-6295.
- Superimposed Subdivisions (Polyrhythm Hell) - An article by Heikki Malmberg
- C.K. Ladzekpo's Foundation Course in African Dance-Drumming
- Novotney, Eugene D. (1998) The Three Against Two Relationship as the Foundation of Timelines in West African Musics, PhD thesis. Urbana, IL: University of Illinois.
- African Music Encyclopedia: Babatunde Olatunji
- "Africano, the mother of groove," an article on polyrhythms and the African drumming tradition
- Tempo Mental- Article by Steve Vai on Polyrhythms
- Drums Database- More Polyrhythms
- polyPulse An athenaCL netTool for on-line, web-based MIDI polyrhythm generation
- polygnome A polyrhythmic metronome application for GNU/Linux
- Polyrhythm experiments using Improvisor and AudioCubes
- polymath Another on-line, web-based MIDI polyrhythm generation tool which uses symbolic input
- Polyrhythm Lessons- Information on applying polyrhythms on the guitar
- How many tuplets- Post on triplets and duplets in West African music
- Metronome for Rhythms and Multi-Beat Polyrhythms Audio clips of many polyrhythms
- How to Play Polyrhythms - Videos Video clips of many polyrhythms like the ones used to illustrate this article
- 4 different time signatures Polymeter An example of 4 layer Time Signatures (2:4, 3:8, 5:16 & 7:8) played by Denny AJD simultaneously on a drumset.
- Polyrhythms ...an Introduction Peter Magadini - as seen on " YouTube "