The mathematical analysis of the structure of some songs

The mathematical structures of six songs were studied by treating them as complex systems. The studies were based on (i) scattering diagram, (ii) spiral structure, (iii) graph theory and animal diagrams, and (iv) entropy and organization. In (i), the fractal dimension and the radius of gyration was determined for each melody. In (ii), it was shown that the total number of notes used in a song display a spiral, which can be expressed by a mathematical equation. In (iii), the animal diagram was obtained by using the pathways between the consecutive notes of the song. A ratio defined as ‘the number of edges to the total number of unit edge’ for the animal diagram was used as a parameter to characterize a melody. In (iv), the entropy, the maximum entropy, and the negentropy or the organization of the songs were determined. In addition, the rate of change of organization of the songs was also determined.