Fractal Dimensions and Entropies of Meragi Songs

Abstract

Melodies can be treated as time series systems with the pitches (or frequencies of the notes) representing the values in subsequent intervals. The pattern of a melody can be revealed in a scattering diagram where pitches represent vertices, and the directed pathways which connect the former pitches to the next ones signify the relations established during the performance. The pathways form a pattern which is called animal diagram (or lattice animal) in the vocabulary of graph theory. The slopes of pathways can be used to characterize an animal diagram and thus to characterize a melody; and the scattering diagram can be used to find out the fractal dimension. In addition, the entropy, the maximum entropy, and the negentropy (or the order) of melodies can be determined. The analysis of Meragi songs in terms of fractal dimension and entropy was carried out in this work. It was found out that there is not a correlation between the fractal dimension and the entropy; therefore, the fractal dimension and the entropy each characterizes different aspects of Meragi songs.