Goal Orientation in Music Composition and Other Social Behaviors Leading to the Common Quantitative Law

The power-law distribution is ubiquitous and common in many natural, engineering, and social systems. In this article, we found that goal orientation is the driving reason on the power-law distribution in typical social behaviors. A unified model is constructed to formulate the goal orientation mechanism in many social systems. Three mathematical characteristics of tonal music melody are found based on music composition theory. The constrained entropy maximization model (CEMM) is constructed, and the power-law distribution of melodic intervals is the result of solving the problem formulated in this model. Three other typical power-law models in economics, linguistics/informatics, and design of complex engineering systems are reviewed, and their goal orientation behaviors are analyzed. It is found that the models quantifying different social behaviors with goal orientation are consistent with the unified model. The power law is actually the result of the well-known social distribution mechanisms with limited resources, such as the maximal wealth with limited economic capability, the minimal cost under constrained resources, and the maximal entropy with restrictions. If without limit or constraint, these social systems would not observe the power law. This finding could inspire the study on other common laws in computational social systems.