The rapid degradation of the functional properties of many Ti-based alloys is due to the precipitation of the omega phase. In the conventional high-temperature shape memory alloy Ti-Ta, the formation of this phase compromises completely the shape memory effect, and high (>100 degrees C) transformation temperatures cannot be maintained during cycling. A solution to this problem is the addition of other elements to form Ti-Ta-X alloys, which often modifies the transformation temperatures; due to the largely unexplored space of possible compositions, very few elements are known to stabilize the shape memory effect without decreasing the transformation temperatures below 100 degrees C. In this study, we use transparent descriptors derived from first-principles calculations to search for new ternary Ti-Ta-X alloys that combine stability and high temperatures. We suggest four alloys with these properties, namely Ti-Ta-Sb, Ti-Ta-Bi, Ti-Ta-In, and Ti-Ta-Sc. Our predictions for the most promising of these alloys, Ti-Ta-Sc, are subsequently fully validated by experimental investigations, the alloy Ti-Ta-Sc showing no traces of omega phase after cycling. Our computational strategy is transferable to other materials and may contribute to suppress omega phase formation in a large class of alloys.