Fractal and Complex Patterns Existing in Music: Application to the Composition DIAPHONIES of Michael Paouris

This paper reports fractal patterns identified in the complex musical composition DIAPHONIES by Michael Paouris via power-law fractal analysis with sliding-windows of size 1024. From 7,647,232 analysed musical segments of DIAPHONIES, 3,222,832 (42.4%) are fractional Brownian motion (fBm) fractal segments and 4,424,400 (57.6 %) are fractional Gaussian noise (fGn) stochastic ones. From the fBm fractal segments 295,294 (9.1 %) exhibit strong persistency-P with power-law segments in the range of $2.3\le b\le 3$. These are the very strong fractal areas in DIAPHONIES. Numerous segments with strong antipersistency $1.7\le b<2$ are reported together with segments with AP changes ($1.7\le b<2.3$). In DIAPHONIES continuous fractal fBm areas are dipped in non-fractal fGn areas of deterministic music. The results are within the fBm fractal areas reported in existing papers. Very importantly, the simple composition called Nocturnal-Angel by Michael Paouris, exhibited limited fBm areas of average $\overline{b}=1.98$ ($\sigma =0.3$), namely of pure statistical, deterministic music, while simultaneously, the fractal analysis profile was completely different from the profiles of DIAPHONIES, hence reinforcing, the fractal findings of DIAPHONIES in relation to trivial music harmony.