Traditional music notation privileges duple divisions of a steady beat or prevailing time unit. A whole note divides into two half notes, a half note into two quarters, etc. Up to any given tolerance, by tying together sufficiently many notes, purely duple notation can express any time point or duration. In mathematical terms, taking any particular bar line as the time origin, duple notation affords a notation for expressing directly any time point or duration with denominator a power of 2. That these fractions form a dense subset of the real numbers explains the approximate universality of the duple system. The same principle underlies binary radix expansions.
An irrational rhythm occurs when a musical score indicates an exact time point or duration that lies outside the scope of the duple system. The notation customarily achieves this through the specification of, in effect, a highly temporary tempo change over a span of duration which itself belongs to the default duple system.
Rather than specifying the new tempo by means of a metronome marking, the prevailing notation indicates the proportional increase or decrease relative to the prevailing tempo. For example, a bracket labelled 5:4 (read five in the space of four) might group together durations (occurring as a sequence of notes and rests) that total to the equivalent of, say, five sixteenth notes. A tempo 5/4 faster than usual then compresses these events into the space of four sixteenth notes. While, in principle, one can increase the pace of any sequence of rhythmic events by 20%, the completion of an irrational rhythm will usually return the count to the duple system. For this to occur with a 5:4 bracket, the total bracketed duration must have a 5 in its numerator, 5/16 in the example. Note that one obtains the actual duration of the bracketed events by dividing two fractions, the notated duration and the indicated tempo increase, (5/16)/(5/4) = 1/4, in this example.
Several uncommon variants occur. One involves nesting the temporary tempo changes. One might see, say, six eighth notes, the first three grouped by a 3:2 bracket, and the whole sequence grouped by a 5:4 bracket; the whole sequence will thus unfold within the span of a normal half note. Note that the first three notes will undergo a combined (3/2)*(5/4)=15/8-fold tempo increase, almost a twofold compression. As a result the first three eighth notes will actually sound as very close to sixteenth notes.
Another variant involves a tempo increase that does not return to the original duple rhythm framework. For example, one might have merely three sixteenth notes grouped by a bracket marked 3 of 5:4. Such an occurrence generally entails a nontraditional time signature. The indication 3 of constitutes a mere courtesy to the player (who might otherwise expect five time units), but does not affect the tempo change itself.
A third variant has the notes of a particular irrational rhythm occur non-sequentially, dovetailed with various duple notes or notes from separate and distinct irrational rhythms.
The best notation for a given rhythmic passage can depend on a number of factors and involve subtle trade-offs, particularly between precision, simplicity and clarity. Tolerating at least negligible approximation multiplies the admissible solutions, but even an exact representation admits a range of equivalent forms (for example, see the discussion of duplets, below). Concerning simplicity, a notation that, say, keeps to a minimum the number of ties, may nevertheless pose more serious difficulties for human performers than a visually cluttered notation that avoids, say, nesting or dovetailing. But nesting or dovetailing will often project with clarity a compositional idea potentially obscured by a more playable notation. Composers will sometimes include an ossia to get the combined benefits of multiple notations for the same passage.
In metrical music, an irrational rhythm will usually create rhythmic variety by borrowing from another tempo, in much that way that some chromatic chords create harmonic variety by borrowing from another key. In this case one can properly speak of an irrational rhythm as a musical effect and not merely a form of notation. Irrational rhythms also arise in non-metrical music, as a means to render into musical notation durations arising out of some alien logic (real world event streams, random processes, mathematical algorithms). In this case, while they will contribute to the overall non-metrical effect, they will not project a distinctive character of their own. Composer will sometimes employ irrational rhythms less for the sake of a precise temporal effect, but rather to affect the players' attention to and perception of, and thus interpretation of a passage.
In mathematics, an irrational number simply has no expression at all as a ratio of integers, while the durations and time-points achieved by an irrational rhythm in music, as described above, definitely do. From this viewpoint one must regard irrational rhythm as a misnomer, and indeed, confusion sometimes arises. Nevertheless, (real) irrational numbers admit approximation by integer fractions, with larger denominators necessary for more precise results. This suggests viewing ratios that require large denominators as less rational than ratios expressible as fractions of small integers. Rational, of course, also carries a psychological meaning, and simple ratios do indeed make themselves more vividly evident to human reason than complex ratios.
The simplest irrational rhythm, 3:2, has a long history in Western music and a special name, the triplet.
Some metered music moves in multiples of three beats, as in the case of the time signatures 38 and 68. In such context, the irrational rhythm 2:3, called a duplet can arise naturally. Claude Debussy's famous Clair de Lune carries the 98 time signature, but makes characteristic use of duplets and their derivatives, including 6:9 (six notes in the space of nine, really just three successive duplets). Actually, as one can achieve the duplet effect of two eighth notes in the space of three by simply writing two dotted eighth notes, one should not consider the duplet a true irrational rhythm. Some experts on music notation frown on dotted eighth notes without an explicit sixteenth note to fill out the beat. But notating a dotted eighth followed by a sixteenth tied to an eighth would entail clutter and misleading visual asymmetry, making duplets a more intuitive notation.
Polyrhythms arise when a musical work moves simultaneously in distinct tempos. While irrational rhythms constitute a means for effecting polyrhythms, one must consider the two notions distinct. Accentual patterns in purely duple music can create polyrhythms. Irrational rhythms in music with just a single part, or coordinated irrational rhythms in all parts will have no polyrhythmic implications.
Until the nineteenth century triplets were the only irrational rhythms commonly seen in written music; 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.
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 minim (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 need to play. So to play a half-note (minim) 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:
The same principle can be applied to quintuplets, septuplets and so on.
To some degree, the time unit box system of notation formalises this approach.