Self-Similar Melodies by Tom Johnson (review)

SELF-SIMBLAR MELODIES by TomJohnson. Two-Eighteen Press, New York, NY, U.S.A., 1996. 292 pp. ISBN: 2-907-20001-1. Reviewed by David Feldman, Department of Mathematics, University ofNew Hampshire, Durham, NH 03824, U.S.A. E-mail: . John Cage insinuates Nature, understood broadly, into music. Unlike late nineteenth-century composers, for example , Cage never merely depicts Nature with his music, but rather unfolds music in many ways from nature. Cage's Nature extends far beyond the wild, the savage or the pristine, beyond any mere antithesis to the urban or civilized; it embraces the whole of contingent experience , the world as what happens. The multiple radios of Imaginary Landspaces make audible the complexity of our radio-wave environment, including the various human productions transmitted thereby; Child of Tree uses sounds made by amplified plant materials . In Cheap Imitation, Empty Words and elsewhere, Cage makes new works by processing the works of other artists much as he processes star charts to make the notes of Etudes Australes. Cage's famous anechoic chamber anecdote emphasizes the impossibility of experiencing true sonic neutrality because of the natural sounds our living bodies constantly make. Cage's silent piece 4'33"presents contingent experience minimally mitigated. Nevertheless, Cage's Nature suffers a profound limitation. By dint of his interest in meditation and contemplation , Cage privileges the actual and so neglects the potential. The laws of mathematics constrain Nature by logically delimiting the possible; confronting those laws brings us to Nature unbridled by the chains of history. Beyond the challenge of exploring the physical space about us lies the difficulty of charting the vast conceptual space that comprises Nature's mathematical objects which proliferate exponentially. TomJohnson, among other composers, makes music by unfolding Nature in this larger aspect. The popular imagination reflexively links Mathematics and Music even as it habitually dissociates mathematics and music. On the one hand, the mythic marriage of Mathematics and Music— made in Platonic heaven—posits, perhaps , their primordial unity in the distinct past, or else their potential convergence in a distant future. On the other hand, popular wisdom reviles as mechanical, inhuman and unexpressive the direct application of mathematical methods employed in the quotidian business of actually composing music; indeed, such an application is regarded as tantamount to blasphemy against the mydi. Popular wisdom permits the myth to manifest itself only by the occasional but telling coincidence of these multiple talents in certain individuals. Of course, mathematical materials have entered the works of various composers over the course of this century—notably Arnold Schoenberg, Conlon Nancarrow, Milton Babbitt and Iannis Xenakis—but the public celebrates these composers for the rich range of other resonances offered by their music; even devotees rarely claim to hear the mathematical aspect unfold in real time. Let us leave die myth aside and examine with an open mind the possible fundamental affinities between mathematics and music. Suppose for a moment that we do not know the meaning of either word—"mathematics" or "music ." Suppose we have a knowledgeable informant, someone sensitized to the nuances raised by recent developments from whom we ask for definitions. Our informant's first approximations— mathematics as the science of number and shape, music as the art of sound— offer us few clues in search of any resonance between these two spheres of activity . When pressed, our informant confesses that neither definition suffices , that both include and exclude too much. Ultimately, our informant might take refuge in circularity, defining mathematics as what mathematicians do and music as what musicians do. But this seemingly empty circularity actually has unexpected content. Both mathematics and music emerge from historical processes: mathematicians and musicians of one generation spin variations on ideas and concepts of their counterparts in previous generations . Moreover, both music and mathematics make claims on the protean and the universal, as well as the representation and even the ultimate synthesis —each in its own way—of virtually every aspect of human experience. Picture the body of knowledge and practice that constitutes either mathematics or music as a neuron, a cell with a central nucleus graced with long dendritic extensions that trail off into invisibility and fill—however tenuously—a space reaching...