This study aims to identify the multidimensional variables and sub-variables and study their relative weight in music recommender systems when maximizing the rating function R. To undertake the task, a optimization formula and variables for a research model were derived from the review of prior works on recommender systems, which were then used to establish the research model for an empirical test. With the research model and the actual log data of real customers obtained from an on line music provider in Korea, multiple regression analysis was conducted to induce the optimal correlation of variables in the multidimensional model. The results showed that the correlation value against the rating function R for Items was highest, followed by Social Relations, Users and Contexts. Among sub-variables, popular music from Social Relations, genre, latest music and favourite artist from Items were high in the correlation with the rating function R. Meantime, the derived multidimensional recommender systems revealed that in a comparative analysis, it outperformed two dimensions(Users, Items) and three dimensions(Users, Items and Contexts, or Users, items and Social Relations) based recommender systems in terms of adjusted $R^2$ and the correlation of all variables against the values of the rating function R.