Empirically, we find that parametric plots of mechanical loss angle versus complex shear modulus may depend neither on temperature [Van Gurp and Palmen, Rheol. Bull, 67, 5 (1998)], nor on average molecular weight [Hatzikiriakos (2000), Polymer Engineering and Science, 40, 2279 (2000)]. Moreover, Hatzikiriakos (2000) discovered that these Van Gurp-Palmen curves descend with long-chain branching content. In this paper, we explain these descents from general rigid bead-rod theory [Hassager, J Chem Phys, 60, 4001 (1974)]. We explore the effects of branching, number of branches, branch length, branch position and branch distribution along a straight chain in small-amplitude oscillatory shear flow