Polyrhythm

For the Perfume song, seePolyrhythm (song).

Polyrhythm:Tripletsoverdupletsin all fourbeats

2:3 polyrhythm (cross rhythm) as bounce inside oval

Polyrhythm(/ˈpɒlirɪðəm/) is the simultaneous use of two or morerhythmsthat are not readily perceived as deriving from one another, or as simple manifestations of the same meter.The rhythmic layers may be the basis of an entire piece of music (cross-rhythm), or a momentary section. Polyrhythms can be distinguished fromirrational rhythms,which can occur within the context of a singlepart;polyrhythms require at least two rhythms to be played concurrently, one of which is typically an irrational rhythm. Concurrently in this context means within the same rhythmic cycle. The underlying pulse, whether explicit or implicit can be considered one of the concurrent rhythms. For example, theson claveis poly-rhythmic because its 3 section suggests a different meter from the pulse of the entire pattern.

In western art music

In someEuropean art music,polyrhythm periodically contradicts the prevailing meter. For example, inMozart's operaDon Giovanni,two orchestras are heard playing together in different metres (³
₄and²
₄):

MozartDon Giovanni2 dances together

Mozart,Don Giovannidances from act 1, scene 5

They are later joined by a third band, playing in³
₈time.

Polyrhythm is heard near the opening ofBeethoven'sSymphony No. 3.(See alsosyncopation.)

Chopinoften explored the rhythmic possibilities inherent in the independence of a pianist's two hands. A spectacular example may be found in hisÉtude, Op, 10 No. 10.Alan Walkercomments that while this piece is straightforward for listeners, "From the player’s point of view, however, nothing is straightforward. Chopin has placed him inside a veritable hornets’ nest of cross-rhythms and syncopations. The melody first emerges from a background of triplets, then of duplets. Accents are changed without warning, shifting the balance of the phrase sideways, so to speak, together with the place of each note within it."

Polyrhythm is a particularly common feature of the music ofBrahms.Writing about theViolin Sonata in G major, Op. 78,Jan Swafford(1997, p. 456) says "In the first movement Brahms plays elaborate games with the phrasing, switching the stresses of the⁶
₄meter back and forth between3+3and2+2+2,or superimposing both in violin and piano. These ideas gather at the climax at measure 235, with the layering of phrases making an effect that perhaps during the 19th century only Brahms could have conceived. "

Brahms Violin Sonata in G, 1, bars 235ff

Brahms Violin Sonata in G, 1, bars 235ff

In "The Snow Is Dancing" from hisChildren's Cornersuite,Debussyintroduces a melody "on a static, repeated B-flat, cast in triplet-division cross rhythms which offset this stratum independently of the sixteenth notes comprising the two dancing-snowflake lines below it.""In this section great attention to the exactitude of rhythms is demanded by the polyrhythmic superposition of pedals,ostinato,and melody. "

Debussy, "The Snow Is Dancing", bars 34–38

Debussy, "The Snow Is Dancing", bars 34–38

Hemiola

Concerning the use of a two-over-three (2:3)hemiolain Beethoven'sString Quartet No. 6,Ernest Walker states, "The vigorously effectiveScherzois in³
₄time, but with a curiously persistentcross-rhythmthat does its best to persuade us that it is really in⁶
₈."

Beethoven Scherzo from String Quartet, Op. 18, No. 6

Beethoven Scherzo from Op. 18, No. 6, violin and cello only

Polyrhythm, not polymeter

The illusion of simultaneous³
₄and⁶
₈,suggestspolymeter:triple metercombined withcompound duple meter.

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 arecross-beatswithin a triple beat scheme.

Composite hemiola

The four-noteostinatopattern ofMykola Leontovych's "Carol of the Bells"(the first measure below) is the composite of the two-against-threehemiola(the second measure).

Another example of polyrhythm can be found in measures 64 and 65 of the first movement ofMozart'sPiano Sonata No. 12.Three evenly-spaced sets of three attack-points span two measures.

Cross-rhythm

Cross-rhythmrefers to systemic polyrhythm. TheConcise Oxford Dictionary of Musicdefines it as “The Regular shift of some beats in a metric pattern to points ahead of or behind their normal positions.”The finale ofBrahmsSymphony No. 2features a powerful passage where the prevailing metre of four beats to the bar becomes disrupted. Here is the passage as notated in the score:

Brahms Symphony No. 2, finale, bars 126-142

Here is the same passage re-barred to clarify how the ear may actually experience the changing metres:

Brahms Symphony No. 2, finale, bars 126-142

Brahms Symphony No. 2, finale, bars 126-142, re-barred

“Polyrhythms run through Brahms’s music like an obsessive-compulsive streak...For Brahms, subdividing a measure of time into different units and layering different patterns on top of one another seemed to be almost a compulsion — as well as a compositional device and an engine of expression.”

Another straightforward example of a cross-rhythm is 3 evenly spaced notes against 2 (3:2), also known as ahemiola.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 of⁶
₈time, quarter note triplets over 2 quarter notes within one bar of²
₄time. Other cross-rhythms are 4:3 (with 4 dotted eighth notes over 3 quarter notes within a bar of³
₄time as an example in standard western musical notation), 5:2, 5:3, 5:4, etc.

Representation of 4 beats parallel to 5 beats

In auditory processing, rhythms are perceived as pitches once they have been sufficiently sped up. Furthermore, intervals of rhythms are perceived as intervals of pitch once sufficiently sped up. As such, there is a parallel between cross-rhythms andmusical intervals:in an audible frequency range, the 2:3 ratio produces the musical interval of aperfect fifth,the 3:4 ratio produces aperfect fourth,and the 4:5 ratio produces amajor third.All theseinterval ratiosare found in theharmonic series.These are called harmonic polyrhythms.

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 non-SaharanAfrican 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 non-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 non-Saharan languages do not have a word forrhythm,or evenmusic.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 ormeter... 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 thegenerativeortheoretic formof non-Saharan rhythmic principles.Victor Kofi Agawusuccinctly 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. The example below shows the African 3:2 cross-rhythm within its proper metric structure.

The music of Africanxylophones,such as thebalafonandgyil,is often based on cross-rhythm. In the following example, a Ghanaian gyil sounds a 3:2-basedostinatomelody. 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 thekushaurapart of the traditionalmbirapiece "Nhema Mussasa". The mbira is alamellophone.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 linearbasstotreblestructure that is so common to many western instruments such as thepiano,harp,ormarimba.

Hugh Tracey TrebleKalimba

Lamellophonesincludingmbira,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. Thekalimbais a modern version of these instruments originated by the pioneer ethnomusicologistHugh Traceyin the early 20th century which has over the years gained worldwide popularity.

Signature SeriesGravikord

Chordophones,such as the West Africankora,and doussn'gouni, part of theharp-lutefamily of instruments, also have this African separated double tonal array structure.Another instrument, theMarovanyfrom Madagascar is a double sided box zither which also employs this divided tonal structure.Trough zithersalso have the ability to play polyrhythms.

TheGravikordis 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 ofsyncopation.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 beatpattern 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.

Jazz

3:2 cross-rhythm

Polyrhythm is a staple of modernjazz.Although not as common, use of systemiccross-rhythmis also found in jazz. In 1959,Mongo Santamariarecorded "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 of¹²
₈(6:4). The following example shows the originalostinato"Afro Blue" bass line. The cross noteheads indicate the mainbeats.

2:3 cross-rhythm

The famous jazz drummerElvin Jonestook the opposite approach, superimposing two cross-beats over every measure of a³
₄jazz waltz (2:3). This swung³
₄is perhaps the most common example of overt cross-rhythm in jazz.In 1963John Coltranerecorded "Afro Blue" with Elvin Jones on drums.Coltrane reversed the metric hierarchy of Santamaria's composition, performing it instead in³
₄swing (2:3).

In popular music

Nigerian percussion masterBabatunde Olatunjiarrived on the American music scene in 1959 with his albumDrums of Passion,which was a collection of traditionalNigerianmusic 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 theYorubasakarastyle 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, includingAirto Moreira,Carlos SantanaandMickey Hartof theGrateful Dead.Olatunji reached his greatest popularity during the height of theBlack Arts Movementof the 1960s and 1970s.

Afro-Cuban musicmakes extensive use of polyrhythms.Cuban Rumbauses 3-based and 2-based rhythms at the same time. For example, the lead drummer (playing thequinto) might play in⁶
₈,while the rest of the ensemble keeps playing²
₂.Afro-Cubanconguero,orcongaplayer,Mongo Santamaríawas another percussionist whose polyrhythmic virtuosity helped transform both jazz and popular music. Santamaria fused Afro-Latin rhythms withR&Band jazz as a bandleader in the 1950s, and was featured in the 1994 albumBuena Vista Social Club,which was the inspiration forthe like-titled documentaryreleased five years later.

Another form of polyrhythmic music is south Indian classicalCarnatic music.A kind of rhythmicsolfegecalledkonnakolis 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 beforeTony Williamsused it when playing withMiles Davis); and finally³
₄time against⁴
₄,which along with 2:3 was used famously byElvin JonesandMcCoy Tynerplaying withJohn Coltrane.

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 highly avant garde album produced by Frank Zappa,Trout Mask Replicaby Captain Beefheart and his Magic Band found extensive use of polyrhythm and cross-rhythm. ThemetalbandsMudvayne,Nothingface,Threat Signal,Lamb of God,also use polyrhythms in their music.Contemporaryprogressive metalbands such asMeshuggah,Gojira,Periphery,Textures,TesseracT,Tool,Animals as Leaders,Between the Buried and MeandDream Theateralso incorporate polyrhythms in their music, and polyrhythms have also been increasingly heard in technical metal bands such asIon Dissonance,The Dillinger Escape Plan,Necrophagist,Candiria,The ContortionistandTextures.Muchminimalistandtotalistmusic makes extensive use of polyrhythms.Henry CowellandConlon Nancarrowcreated music with yet more complex polytempo and using irrational numbers likeπ:e.

Peter Magadini's albumPolyrhythm,with musiciansPeter Magadini,George Duke,David Young, andDon Menza,features different polyrhythmic themes on each of the six songs.

1. Doin' Time and a Half: Has the polyrhythmic theme of 6 over 4.
2. Five For Barbara: Has the polyrhythmic theme of 5 over 4.
3. The Modulator: The beginning tempo modulates to two times faster and then modulates back to two times slower.
4. Seventy Fourth Ave: Has the polyrhythmic theme of 7 over 4.
5. Samba de Rollins: Includes a drum solo based on 3 over 4.
6. Midnight Bolero: In³
   ₄,has a continuous interlude of 2 over 3 and then 4 over 3.

King Crimsonused polyrhythms extensively in their 1981 albumDiscipline.Above allBill Brufordused polyrhythmic drumming throughout his career.

The bandQueenused polyrhythm in their 1974 song "The March of the Black Queen"with⁸
₈and¹²
₈time signatures.

Talking Heads'Remain in Lightused dense polyrhythms throughout the album, most notably on the song "The Great Curve".

Megadethfrequently tends to use polyrhythm in its drumming, notably from songs such as "Sleepwalker" or the ending of "My Last Words", which are both played in 2:3.

Carbon Based Lifeformshave a song named "Polyrytmi",Finnishfor "polyrhythm", on their albumInterloper.This song indeed does use polyrhythms in its melody.

Aphex Twinmakes extensive use of polyrhythms in his electronic compositions.

Japanese girl groupPerfumemade use of the technique in their single, appropriately titled "Polyrhythm",included on their second albumGame.The bridge of the song incorporates⁵
₈,⁶
₈in the vocals, common time (⁴
₄) and³
₂in the drums.

TheBritney Spearssingle "Till the World Ends"(released March 2011) uses a 4:3 cross-rhythm in its hook.

The outro of the song "Animals" from the albumThe 2nd Lawby the bandMuseuses⁵
₄and⁴
₄time signatures for the guitar and drums respectively.

TheAaliyahsong "Quit Hatin" uses⁹
₈against⁴
₄in the chorus.

TheJapanese idolgroup3776makes use of polyrhythm in a number of their songs, most notably on their 2014 mini-album "Love Letter",which features five songs that all include several rhythmic references to the number 3776. A secret track on the album has the group's leader, Ide Chiyono, explain some of the uses of polyrhythm to the listener.

The Nationalsong "Fake Empire"uses a 4 over 3 polyrhythm.

The Cars' song "Touch and Go"has a⁵
₄rhythm in the drum and bass and a⁴
₄rhythm in the keys and vocals.

Harpist and pop folk musicianJoanna Newsomis known for the use of polyrhythms on her albumsThe Milk-Eyed MenderandYs.

King Gizzard & the Lizard Wizardused polyrhythms extensively throughout their discography, most notably on the albumPolygondwanaland.

InVietnam,bolerosongs are composed with³
₄against⁴
₄.

TheBritney Spearssong "...Baby One More Time"contains a 16:9 polyrhythm in the beginning of the music video (the boot tapping 16 times every 9 times the clock ticks - a full cycle can be heard from 0:05 to 0:14).

The Sky Signals song "Amaterasu" is constructed around a 5:4 polyrhythm.

Examples

The following is an example of a 3 against 2 polyrhythm, given intime 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.

3 against 2 polyrhythm

3-beat rhythm

X

X

X

X

X

X

X

X

X

X

X

X

2-beat rhythm

X

X

X

X

X

X

X

X

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", and "ring, Christmas bells".

Similar phrases for the 4 against 3 polyrhythm are "pass the golden butter"or "pass the goddamn butter"and "what atrocious weather" (or "what a load of rubbish" in British English); the 4 against 3 polyrhythm is shown below.

4 against 3 polyrhythm

4-beat rhythm

X

X

X

X

X

X

X

X

3-beat rhythm

X

X

X

X

X

X

Polyrhythm4 4with3 4simultaneously (cross rhythm) as bounce inside oval

As can be seen from above, the counting for polyrhythms is determined by thelowest 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 (123456and123456). 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 asclassical Indian musicians,can intuitively play high polyrhythms such as 7 against 8.

Polyrhythms are quite common in lateRomantic Musicand20th-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 inGrieg'sConcerto in A Minor;the left hand playsarpeggiosof 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 inRachmaninoff'sPiano 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 'quasiglissando', fits 52 notes into the space of onemeasure,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.

List of basic polyrhythms

First rhythm	Second rhythm	Least common multiple
2	3	6
2	5	10
3	4	12
2	7	14
3	5	15
2	9	18
4	5	20
3	7	21
3	8	24
4	6	12
4	7	28
5	6	30
5	7	35
4	9	36
5	8	40
6	7	42
5	9	45
6	8	24
6	9	18
7	8	56
7	9	63
7	11	77
8	9	72

See also

• 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.
• Euclidean rhythm
• Ewe music
• Xenochrony

References

Sources

• Peñalosa, David (2009).The Clave Matrix; Afro-Cuban Rhythm: Its Principles and African Origins.Redway, California: Bembe.ISBN9781886502802.

Further reading

• Peter Magadini(1993).Polyrhythms for the Drumset.ISBN978-0-89724-821-1.[Study in polymetric independence for drummers.]
• Peter Magadini(2001).The Musician's Guide to Polyrhythms.ISBN978-0-63403-283-7.[Polyrhythm reference book.]

External links

• Superimposed Subdivisions (Polyrhythm Hell)– An article by Heikki Malmberg
• C.K. Ladzekpo'sFoundation 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
• polyPulseAn athenaCL netTool for on-line, web-based MIDI polyrhythm generation
• polygnomeA polyrhythmic metronome application for Linux
• Polyrhythm experiments using Improvisor and AudioCubes
• polymathAnother 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 PolyrhythmsAudio clips of many polyrhythms
• How to Play Polyrhythms – VideosVideo clips of many polyrhythms like the ones used to illustrate this article
• 4 different time signatures polymeter,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 Magadinivia YouTube
• Drum Solo with Metric Modulations – Peter Magadini (2006) from the Hal Leonard DVDJazz Drums
• The 26 Official Polyrhythm Rudiments (2012)Modern Drummer Magazine– Pete Magadini