Additive Meter - Irrational Meters

Irrational Meters

These are time signatures, used for so-called irrational measure lengths, which have a denominator which is not a power of two (1, 2, 4, 8, 16, 32, etc.). These are "based on beats expressed in terms of fractions of full beats in the prevailing tempo," for example 3
10 or 5
24. For example, where 4
4 implies a bar construction of four quarter-parts of a whole note (i.e., four quarter notes), 4
3 implies a bar construction of four third-parts of it. These signatures are only of utility when juxtaposed with other signatures with varying denominators; a piece written entirely in 4
3, say, could be more legibly written out in 4
4.

Metric modulation is "a somewhat distant analogy". It is arguable whether the use of these signatures makes metric relationships clearer or more obscure to the musician; it is always possible to write a passage using non-"irrational" signatures by specifying a relationship between some note length in the previous bar and some other in the succeeding one. Sometimes, successive metric relationships between bars are so convoluted that the pure use of irrational signatures would quickly render the notation extremely hard to penetrate. Good examples, written entirely in conventional signatures with the aid of between-bar specified metric relationships, occur a number of times in John Adams' opera Nixon in China (1987), where the sole use of "irrational" signatures would quickly produce massive numerators and denominators.

Historically, this device has been prefigured wherever composers have written tuplets; for example, a 2
4 bar consisting of 3 triplet crotchets could arguably more sensibly be written as a bar of 3
6. Henry Cowell's piano piece "Fabric" (1920) throughout employs separate divisions of the bar (anything from 1 to 9) for the three contrapuntal parts, using a scheme of shaped noteheads to make the differences visually clear, but the pioneering of these signatures is largely due to Brian Ferneyhough, who says that he "find that such 'irrational' measures serve as a useful buffer between local changes of event density and actual changes of base tempo". Thomas Adès has also made extensive use of them, for example in his piano work "Traced Overhead" (1996), the second movement of which contains, among more conventional meters, bars in such signatures as 2
6, 9
14 and 5
24. His "Piano Quintet" (2000) makes such extensive use of these, including different lines juxtaposed with varying meters, that an alternate form of notation is not immediately obvious, or arguably desirable. A gradual process of diffusion into less rarefied musical circles seems to be underway, hence for example, John Pickard's work "Eden", commissioned for the 2005 finals of the National Brass Band Championships of Great Britain, which contains bars of 3
10.

Notationally, rather than using Cowell's elaborate series of notehead shapes, the same convention has been invoked as when normal tuplets are written; for example, one beat in 4
5 is written as a normal quarter note, four quarter notes complete the bar, but the whole bar lasts only 4/₅ of a reference whole note, and a beat 1/₅ of one (or 4/₅ of a normal quarter note). This is notated in exactly the same way that one would write if one were writing the first four quarter notes of five quintuplet quarter notes.

The term "irrational" is not being used here in its mathematical sense: an irrational number is one that cannot be written as a ratio of whole numbers, which all these signatures obviously are. Nevertheless, the term appears to be established now, although at least one such piece with a truly irrational signature already exists: one of Conlon Nancarrow's "Studies for Player Piano" contains a canon where one part is augmented in the ratio √42:1 (approximately 6.48:1).