Irrational rhythm
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In music, an irrational rhythm is any rhythm in which an odd number of beats is superimposed on an even number in the predominating tempo, or vice versa. The name is therefore, from a mathematical perspective, quite wrong: rhythms of this sort are precisely defined as the ratio of beats played to beats in the underlying tempo.
The most familiar example is the triplet, in which three beats are played in the space of two. In compound time, the triplet can form the basic rhythmic unit (one triplet is 38, two triplets is 68, and so on), and so a common irrational rhythm in compound time is the duplet. Claude Debussy's famous composition Au Clair du Lune is written mostly in 98 but makes characteristic use of duplets and their derivatives, including 6:9 (which is really just three successive duplets).
Irrational rhythms are hence to be distinguished from polyrhythms, which are two separate rhythms played against one another, whereas an irrational rhythm can occur in the context of a single part. When irrational rhythms in one part are played against the underlying rhythm in another part, however, the outcome is a polyrhythm.
Historical development
Until the nineteenth century triplets were the only irrational rhythms that were commonly seen in written music; the Romantic composers then introduced the quintuplet, in which five beats are played in the space of four, creating a hurried, rushing effect. Such groupings are often written with figures of the form "5:4" above the notes; here the colon can be read off as "in the space of".
In many forms of modern classical music irrational rhythms have been greatly extended, with groupings such as 7:8 and even 11:8 or 11:16 appearing fairly commonly. This reflects a general tendency away from regular beat-based rhythms.
Outside classical music, rhythms that may be best expressed notationally using irrational groups are found all over the world.
Practical considerations
Irrational rhythms can be challenging for performers, particularly when they stretch over several beats -- a quaver (eighth-note) triplet in 4/4, which occupies one beat, is considerably more intuitive for most musicians than a semibreve (half-note) triplet that occupies an entire bar.
One solution is to take the number of superimposed beats (in this case, 3) and mentally subdivide each beat in the bar into that number. Then tie together n notes at a time, where n is the ratio of the note you are counting to the note you you need to play. So to play a half-note triplet accurately in a bar of 4/4, count eighth-note triplets and tie them together in groups of four. With a stress on each target note, you would count:
- 1-2-3 / 1-2-3 / 1-2-3 / 1-2-3
This gives a semibreve triplet in 4/4 meter, and can be done quite slowly and accurately. The same principal can be applied to quintuplets, septuplets and so on.
To some degree, the time unit box system of notation formalises this approach.