Emotion is a core fundamental attribute of humans. Using music to induce emotional responses from subjects to better facilitate human behavior shaping have been effective across domains of health, education, and retail. Computationally model the musically-induced emotion provides necessary content-based analytics for large-scale and wide-applicability of such human-centered applications. In this work, we propose a relationship neural network architecture to learn to regress the induced emotion attributes with an auxiliary task of genre classification. Our proposed Genre-Affect Relationship Network with homoscedastic uncertainty weighting embeds the relationship between affect and genre as tensor normal prior within task-specific layers; the architecture is optimized further by incorporating task-specific uncertainty. The proposed architecture achieves a state-of-art 0.564 average Pearson correlation computed over nine induced emotion ratings in the Emotify database. Furthermore, we provide an analysis to understand the relationship between the induced emotions of these musical pieces and their associated genres.