Large Language Models (LLMs) have demonstrated remarkable proficiency in diverse tasks. This success raises a fundamental question in machine composition: Can symbolic music be considered a special form of language that can be jointly modeled with natural language for composition tasks? Recent studies validate that symbolic music can be modeled as a human language, yet composing structured music from partial symbolic inputs through natural language interaction remains underexplored. Even LLMs struggle to generate structurally coherent compositions in such hybrid input-output scenarios, highlighting a fundamental gap that calls for a domain-specific learning paradigm. To this end, we propose Inspiration-to-Structure (IoS), a cognitively inspired framework that enables LLMs to generate structured musical sections from melodic ideas. IoS employs a three-phase process—semantic, structural, and collaborative cognition—and is supported by two key components: (1) a new dataset and construction protocol called Structured Triplet Data (STD), and (2) a training method, Dual-Instance Structural Contrastive Optimization (DiSCO), designed to enhance structural awareness. Experiments show that IoS improves structural coherence by 47.8% and artistic creativity by 21.8% compared to conventional language modeling paradigm, supervised fine-tuning, and even enables smaller LLMs to surpass larger LLMs. These results suggest that symbolic music, while language-like, demands specialized modeling beyond standard language modeling paradigms. IoS enables LLMs to transform music theory knowledge into structured composition, empowering users to compose music interactively via language and advancing toward general creative AI.