Polyrhythm is the simultaneous sounding of two or more independent rhythms.
Polyrhythm in general is a nonspecific term for the simultaneous occurrence of two or more conflicting rhythms, of which cross-rhythm is a specific and definable subset.—Novotney (1998: 265)
Below is a simple polyrhythm found in sub-Saharan African musics.
This is the duple-pulse (duple or quadruple subdivision) version of the same polyrhythm.
These polyrhythms are not cross-rhythms because they do not involve cross-beats.
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.
The terms polyrhythm and cross-rhythm are often used interchangeably.
Cross-rhythm. 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.—New Harvard Dictionary of Music (1986: 216).
A simple example of a cross-rhythm is 3 evenly-spaced notes against 2 (3:2), with the 3-beat pattern being faster than the 2-beat pattern, so that they both take the same amount of time. Other cross-rhythms are 5:2, 5:4, etc.
Another form of cross-rhythm 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.
European (Western) art music
In some European art music cross-rhythm periodically contradicts the prevailing meter. For example, cross-rhythm 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) cross-rhythm in Beethoven ‘s sixth quartet in B flat, Ernest Walker says:
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