Polyrhythm is the simultaneous sounding of two or more independent rhythms. 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.
A simple example of a polyrhythm is 3 evenly-spaced notes against 2, with the 3-beat pattern being faster than the 2-beat pattern, so that they both take the same amount of time. Other simple polyrhythms are 5-2, 5-4, etc. Where one of the parts involves an irrational rhythm, the resulting rhythm could be said to be an "irrational polyrhythm".
Another form of polyrhythm, which might also be termed polymeter, would be phrasing to suggest a different meter than the one being played by the rest of the ensemble. A common example of this in jazz would be phrasing quarter notes in groupings of 3 to suggest 3/4 time while the ensemble plays in 4/4. Compare with hemiola (not a polyrhythm).
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 Santamaria 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.
Probably the most sophisticated polyrhythms in the world are found in 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-eight 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.
Frank Zappa, especially towards the end of his career, experimented with complex polyrhythms, such as 11:17, and even nested polyrhythms. The metal bands Meshuggah and Mudvayne also use polyrhythms in their music. Contemporary progressive metal bands such as Tool and Dream Theater also incorporate polyrhythms in their music, and polyrhythms have also been increasingly heard in techmetal bands such as Ion Dissonance and The Dillinger Escape Plan. Maybe the best example of this style of rhythmic sound is produced/composed/performed by the fusion metal band Candiria. 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" off of his album, The Fragile. The piano holds a 3/4 riff while the drums and bass back it with a standard 4/4 signature.
Polyrhythm is also called "measure preserving polymeter," because there exists more than one meter, but the measure stays constant. "Tactus preserving polymeter" is used to describe what is most commonly referred to as polymeter. These terms are found in the writings of Keith Waters and Steve Larson. Waters' 1996 article "Blurring the Barline: Metric Displacement in the Piano Solos of Herbie Hancock" from the Annual Review of Jazz Studies" and Larson's 2006 "Rhythmic Displacement in the Music of Bill Evans" are two examples.
In Dr. Kagliavanna and the Optical Machine by Cyeniic Apparatus, the opening/closing sections entail a 4/4 melody in the bass and piano (Ethan Brauer and Wil Bywaters, respectively) played over a 5/8 hi-hat rhythm (David Belcher).
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, although this is irrelevant since the pattern is symmetric. 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 below. Another phrase with the same rhythm is "cold cup of tea". This polyrhythm has a beat akin to the song Carol of the Bells. So another phrase is "Bells Caroling":
Similar phrases for the 4 against 3 polyrhythm are "pass the gold-en but-ter" and "what atrocious weather"; The 4 against 3 polyrhythm is shown below.
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.