China (1987), where the sole use of irrational signatures would quickly produce massive numerators and denominators. Historically, this device has been prefigured wherever composers wrote tuplets. For example, a bar of 3 triplet quarter notes could be written as a bar of . Henry Cowell's piano piece Fabric (1920) employs separate divisions of the bar (1 to 9) for the three contrapuntal parts, using a scheme of shaped noteheads to visually clarify the differences, but the pioneering of these signatures is largely due to Brian Ferneyhough, who says that he finds 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 used them extensively—for example in Traced Overhead (1996), the second movement of which contains, among more conventional meters, bars in such signatures as , and . A gradual process of diffusion into less rarefied musical circles seems underway. For example, John Pickard's Eden, commissioned for the 2005 finals of the National Brass Band Championships of Great Britain, contains bars of and . 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 is written as a normal quarter note, four quarter notes complete the bar, but the whole bar lasts only of a reference whole note, and a beat of one (or 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. Some video samples are shown below. These video samples show two time signatures combined to make a polymeter, since , say, in isolation, is identical to . Variants Some composers have used fractional beats: for example, the time signature appears in Carlos Chávez's Piano Sonata No. 3 (1928) IV, m. 1. Both and appear in the fifth movement of Percy Grainger's Lincolnshire Posy. Music educator Carl Orff proposed replacing the lower number of the time signature with an actual note image, as shown at right. This system eliminates the need for compound time signatures, which are confusing to beginners. While this notation has not been adopted by music publishers generally (except in Orff's own compositions), it is used extensively in music education textbooks. Similarly, American composers George Crumb and Joseph Schwantner, among others, have