A Mathematical and Computational Framework for Modeling Melodic Structures in North Indian Classical Music Using Raga-Restricted Operations

This paper presents a unified mathematical and computational framework for analyzing melodic structures in North Indian classical music through a novel concept termed the raga-restricted operation. This operation provides an algebraic mechanism to model permissible note transitions while enforcing the grammatical constraints of a raga. The proposed framework integrates algebraic structures, graph-theoretic representations, and stochastic modeling using Markov chains to achieve a coherent and computationally tractable description of melodic behavior. The methodology is applied to two representative ragas, Raga Yaman and Raga Bhupali, where melodic sequences are generated under raga constraints and analyzed using directed graphs, transition probability matrices, and stationary distributions. The results demonstrate that the framework preserves the intrinsic structure of ragas while enabling quantitative analysis of transition dynamics, probabilistic dependencies, and long-term behavior. From a computational perspective, the model supports systematic sequence generation, prediction of note transitions, and structural analysis of melodic patterns. Thus, the proposed approach establishes a rigorous connection between music theory and mathematical modeling, contributing to the advancement of computational musicology and probabilistic analysis of raga-based systems.