Fractals and music. A reconnaissance

Among the many definitions of the fractal employed by mathematicians, one of the most suggestive holds that ‘the fractal is a self-similar figure displaying an invariability in respect to the transformations of scaling’. This article is an effort to present the overview of fractals in mathematics and nature and then to describe the current state of research on fractal nature of music. It is shown that self-similarity and scaling are properties of many canonic works of Western music (e.g. Johann Sebastian Bach, Ludwig van Beethoven), appearing in various forms in all historical periods. It is found in binary and ternary divisions of form and in melodic structures. It is also noted that a frequent point of reference in fractal studies of the properties of music is twentieth-century repertoire (e.g. Per Norgárd, Conlon Nancarrow, Gyórgy Ligeti, Charles Wuorinen). The case of l/f noise in which frequency (pitch) scaling naturally occurs is also discussed. Such ‘scaling noise’ is typical of many natural phenomena; it is observed, for example, in the variable tension of nerve cells and in heartbeats. It was also discovered in music. The article summarizes the results of the research made by Voss and Clarke (1975, 1978), Hsü and Hsü (1990,1991), Henze and Cooper (1997) who analyzed stylistically diverse works - classical, jazz, blues, rock and non-European music - and found in them l/f relationships referring to Fourier spectra, notes or intervals. The article reports also the psychological experiments raising the statements about a close relationship between fractal structure and the human sense of beauty. It is stressed that the fractal orientation of modern mathematics provides interesting cognitive tools allowing us to discover hitherto unexplored links between nature and art, both in the area of listeners’ aesthetic preferences and also in the fascinating realm of artistic creation