Being a musician and composer himself and teacher of music in his young adulthood years, though with no formal training in composition, Rousseau left several marks of influence in opera. Some influence was negative, like the uproar caused in the French music Académie des Sciences against his new system of musical notation, which he thought would bring him fame and wealth. Some influence led to develops in musical genres, like the (1) development of the French opéra comique spurred by the success of his Italianate intermède "petit opera"--itself influenced by Italian intermezzos--Le devin du village (1752, Fontainebleau) and like the (2) growth of the new genre "melodrama" spurred by his Pygmalion (1770, Lyons), which was spoken with musical interludes and not sung. While his early operas were failures, these works did leave a lasting influence as did his philosophy of music. He stated that music must express and inspire deep feeling, not great ideas nor moral principles, and that this feeling might only be expressed through the words of language and not through music alone; hence the combination of words and music in Pygmalion.
Gale Encyclopedia of Biography: Jean Jacques Rousseau
Oxford Grove Music Encyclopedia: Jean-Jacques Rousseau
Oxford Grove Art: Jacques Rousseau
Rosseau was a strong advocate of humanism. He believed that people should experience all of the arts. This would lead to unfettered experiences with music. He would also say that there would be a connection between different types of art, an interdisciplinary approach.
As I understand it, Rousseau helped to influence the Romantic movement in music. Rousseau is famous in philosophical circles for having proposed that human beings are free in the state of nature but that they chain themselves in societies and reduce their level of freedom. He believed that music should therefore not try to be too bound by rules and conventions. He wanted a more expressive kind of music that showed emotions in a natural way. Rousseau felt that this sort of music was more beneficial to people as it helped to free them from the constraints of artificial societal rules.