We show that an independent four-body momentum scale μ₍₄₎ drives the tetramer binding energy for fixed trimer energy (or three-body scale μ₍₃₎) and large scattering length (a). The three- and four-body forces from the one-channel reduction of the atomic interaction near a Feshbach resonance disentangle μ₍₄₎ and μ₍₃₎. The four-body independent scale is also manifested through a family of Tjon lines, with slope given by μ₍₄₎/μ₍₃₎ for a⁻¹ = 0. There is the possibility of a new renormalization group limit cycle due to the new scale.