I [1.1] As an ex-mathematician, I do not use the word “impossible” lightly. When a mathematician says that something is impossible, it means the thing involves some inherent logical contradiction. Adding two even numbers and getting an odd number, for example, is impossible, no matter how much one practices one’s addition. This paper, accordingly, is not about rhythms that are merely very difficult, as might be encountered in the music of Elliott Carter or Brian Ferneyhough. Rather, it is about rhythms with the curious property that, no matter how faultless a performer’s mental metronome and digital dexterity may be, it is logically, mathematically, absolutely impossible to play exactly what the composer wrote. (1) [1.2] Example 1 is full of impossible rhythms. The trouble starts in measure 3, where two voices are notated on the lower staff. One voice, written with stems up, moves in sixteenth notes, duple subdivisions of the beat in the moderately slow Volume 17, Number 4, December 2011 Copyright © 2011 Society for Music Theory How to Perform Impossible Rhythms Julian Hook NOTE: The examples for the (text-only) PDF version of this item are available online at: http://www.mtosmt.org/issues/mto.11.17.4/mto.11.17.4.hook.php KEYWORDS: rhythm, meter, metric conflict, triplets, piano, 19th century, Brahms, Scriabin ABSTRACT: This paper investigates a fairly common but seldom-studied rhythmic notation in the nineteenth-century piano literature, in which duplets in one voice occur against triplets in another, and the second duplet shares its notehead with the third triplet—a logical impossibility, as the former note should theoretically fall halfway through the beat, the latter two-thirds of the way. Examples are given from the works of several composers, especially Brahms, who employed such notations throughout his career. Several alternative realizations are discussed and demonstrated in audio examples; the most appropriate performance strategy is seen to vary from one example to another. Impossibilities of type 1 ⁄ = 2 ⁄ , as described above, are the most common, but many other types occur. Connections between such rhythmic impossibilities and the controversy surrounding assimilation of dotted rhythms and triplets are considered; the two phenomena are related, but typically arise in different repertoires. A few other types of impossible notations are shown, concluding with an example from Scriabin’s Prelude in C Major, op. 11, no. 1, in which triplets and quintuplets occur in complex superposition. The notation implies several features of alignment that cannot all be realized at once; recorded examples illustrate that a variety of realizations are viable in performance. Received April 2011 2 3 1 of 14