Ti-Ta alloys show a high martensitic transformation temperature and good workability which make them promising candidates for high-temperature shape memory alloys (HTSMAs). Insight in the composition dependence of the martensitic start temperature (Ms) and other properties is, however, still missing. In our study we use density functional theory (DFT) calculations to predict the free energies of the martensite and austenite phase as a function of composition and obtain an estimate of composition dependence of Ms. The predicted change in Ms with composition is in good agreement with available experimental results. We also investigate the composition dependence of the elastic properties of the involved phases, which not only are important for Ms but also indicate phase stability and anisotropic properties of the martensite. From our calculations we identify a one dimensional descriptor that can be used to obtain estimates of the transformation temperature in the high-throughput screening of ternary or multicomponent alloys for computationally guided development of novel HTSMAs.