  First principles calculations are used to study the preferential occupation
of ternary alloying additions into the binary Ti-Al phases, namely
$\gamma$-TiAl, $\alpha_2$-Ti$_3$Al, $\beta_{\mathrm{o}}$-TiAl, and B19-TiAl.
While the early transition metals (TMs, group IVB, VB , and VIB elements)
prefer to substitute for Ti atoms in the $\gamma$-, $\alpha_2$-, and
B19-phases, they preferentially occupy Al sites in the
$\beta_{\mathrm{o}}$-TiAl. Si is in this context an anomaly, as it prefers to
sit on the Al sublattice for all four phases. B and C are shown to prefer
octahedral Ti-rich interstitial positions instead of substitutional
incorporation. The site preference energy is linked with the alloying-induced
changes of energy of formation, hence alloying-related (de)stabilisation of the
phases. We further show that the phase-stabilisation effect of early TMs on
$\beta_{\mathrm{o}}$-phase has a different origin depending on their valency.
Finally, an extensive comparison of our predictions with available theoretical
and experimental data (which is, however, limited mostly to the $\gamma$-phase)
shows a consistent picture.
