In recent decades, a great variety of dependence models for data analysis have been elaborated. Among them, those based on copulas have demonstrated a great ability to capture the possible dependence between quantitative measures. The three-dimensional case has received particular interest in several important applications in the last few years. The most commonly used three-dimensional copulas have one or two parameters and are exchangeable; they are often members of the well-known Archimedean family. In this paper, we go beyond this standard framework by proposing a brand-new three-dimensional copula that depends on three parameters, is mainly non-exchangeable, and is constructed from ratio and power functions. It is thus of the ratio-power type. Our findings are purely theoretical. In particular, wide ranges of valid parameter values are determined, the main related functions (density, survival, etc.) are exhibited, and various correlation measures (medial, Spearman rho, etc.) are examined. Subsequently, a unique two-dimensional copula, derived from the three-dimensional one, is discussed and studied. When comparing it to the famous two-dimensional Farlie-Gumbel-Morgenstern copula, some noteworthy advantages are emphasized. As a result, the restrictions imposed by the exchangeable property, which are typical of traditional three-dimensional copulas in the literature, are removed, creating a promising future for cutting-edge methods of dependent modeling.