A novel synthesis method, "arc-melting-self-flux", has been developed and a series of five Tsai-type 1/1 approximant crystals in the Tb-Au-Si system have been synthesized. The synthesis method, by employing a temperature program which oscillates near the melting and nucleation points of the approximants, has provided high-quality and large single crystals in comparison to those obtained from the standard arc-melting-annealing and self-flux methods. The atomic structures of the approximants have been determined from single-crystal X-ray diffraction data and described using concentric atomic clusters with icosahedral symmetry. The compounds are nearly isostructural with subtle variations; two types of atomic clusters which mainly vary at their cluster centers are observed. One type contains a Tb site at the center, and the other contains a disordered tetrahedron decorated with Au/Si mixed sites. Both cluster types can be found coexisting in the approximants. The compounds have different average weighted ratios of central Tb to disordered tetrahedron in the bulk material. Furthermore, a strategy for chemically tuning magnetic behavior is presented. Magnetic property measurements on the approximants revealed that the magnetic transition temperature (T-c) decreases as the occupancy of the central Tb site increases. T-c decreased from 11.5 K for 0% occupancy of the central Tb to 8 K for 100% occupancy. Enhanced magneto crystalline anisotropy is observed for the approximants with higher central Tb occupancy in comparison to their low central Tb occupancy counterparts. Hence, the previously reported "ferrimagnetic-like" magnetic structure model remains valid.