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Topologically close-packed (TCP) phases in single crystal Ni-based superalloys have a detrimental effect on the mechanical properties. In order to gain a microscopic understanding of the factors that control TCP phase stability, we carry out atomistic calculations based on the electronic structure. In particular, we use a hierarchy of methods that treat the electronic structure at different levels of coarse-graining, i.e. at different levels of computational cost and accuracy. The applied levels of approximation range from density functional theory (DFT) to tight-binding (TB) to bond-order potentials (BOPs). This hierarchy of electronic structure methods allows us to interpret the findings of a recently derived structure map of experimentally observed TCP stability. The TB and BOP calculations are compared to extensive high-throughput DFT calculations for the TCP phases A15, C14, C15, C36, mu, sigma, and chi of transition-metal elements. These findings are extended to binary systems based on DFT heat-of-formations for TCP phases in the systems V/Nb-Ta, Nb/Mo-Ru, V/Cr/Nb/Mo-Re, V/Cr/Nb/Mo-Co. By pairwise comparisons of selected systems, we illustrate the interplay of the difference in average valence electron concentration N and the composition-dependent relative volume difference dV/V . Such an approach could be useful to predict the change of expected TCP phase stability due to changes of the composition for a given multi-component alloy.