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In steels and single-crystal superalloys the control of the formation of topologically close-packed (TCP) phases is critical for the performance of the material. The structural stability of TCP phases in multi-component transition-metal alloys may be rationalized in terms of the average valence-electron count N and the composition-dependent relative volume-difference dV/V. We elucidate the interplay of these factors by comparing density-functional theory calculations to an empirical structure map based on experimental data. In particular, we calculate the heat of formation for the TCP phases A15, C14, C15, C36, χ, μ and σ for all possible binary occupations of the Wyckoff positions. We discuss the isovalent systems V/Nb–Ta to highlight the role of atomic-size difference and observe the expected stabilization of C14/C15/C36/μ by dV/V at N=0 in V–Ta. In the systems V/Nb–Re, we focus on the well-known trend of A15→σ→χ stability with increasing N and show that the influence of dV/V is too weak to stabilize C14/C15/C36/μ in Nb–Re. As an example for a significant influence of both N and dV/V, we also consider the systems Cr/Mo–Co. Here the sequence A15→σ→χ is observed in both systems but in Mo–Co the large size-mismatch stabilizes C14/C15/C36/μ. We also include V/Nb–Co that cover the entire valence range of TCP stability and also show the stabilization of C14/C15/C36/μ. Moreover, the combination of a large volume-difference with a large mismatch in valence-electron count reduces the stability of the A15/σ/χ phases in Nb–Co as compared to V–Co. By comparison to non-magnetic calculations we also find that magnetism is of minor importance for the structural stability of TCP phases in Cr/Mo–Co and in V/Nb–Co.