  If our aesthetic preferences are affected by fractal geometry of nature,
scaling regularities would be expected to appear in all art forms, including
music. While a variety of statistical tools have been proposed to analyze time
series in sound, no consensus has as yet emerged regarding the most meaningful
measure of complexity in music, or how to discern fractal patterns in
compositions in the first place. Here we offer a new approach based on
self-similarity of the melodic lines recurring at various temporal scales. In
contrast to the statistical analyses advanced in recent literature, the
proposed method does not depend on averaging within time-windows and is
distinctively local. The corresponding definition of the fractal dimension is
based on the temporal scaling hierarchy and depends on the tonal contours of
the musical motifs. The new concepts are tested on musical 'renditions' of the
Cantor Set and Koch Curve, and then applied to a number of carefully selected
masterful compositions spanning five centuries of music making.
