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 rationalised in terms of the average valence-electron count $bar{N}$ and the composition-dependent relative volume-difference $overline{Delta V/V}$. 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, $chi$, $mu$, and $sigma$ 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 stabilisation of C14/C15/C36/$mu$ by $overline{Delta V/V}$ at $Delta N=0$ in V-Ta. In the systems V/Nb-Re, we focus on the well-known trend of A15$- sigma - chi$ stability with increasing $bar{N}$ and show that the influence of $overline{Delta V/V}$ is too weak to stabilise C14/C15/C36/$mu$ in Nb-Re. As an example for a significant influence of both $bar{N}$ and $overline{Delta V/V}$, we also consider the systems Cr/Mo-Co. Here the sequence A15$- sigma - chi$ is observed in both systems but in Mo-Co the large size-mismatch stabilises C14/C15/C36/$mu$. We also include V/Nb-Co that cover the entire valence range of TCP stability and also show the stabilisation of C14/C15/C36/$mu$. Moreover, the combination of a large volume difference with a large mismatch in valence-electron count reduces the stability of the A15/$sigma$/$chi$ 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.
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