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The martensitic start temperature M_s is one of the key characteristics of shape memory materials. High-temperature shape memory alloys are a special class of materials where transformation temperatures between the martensite and austenite phase above 373 K are desirable. For the design of new high-temperature shape memory alloys it is therefore important to understand and predict the dependence of M_s on the composition of the material. Using density functional theory in combination with the quasiharmonic Debye model, we evaluate the different contributions to the free energy to determine the transition temperature T₀ over a wide range of compositions in Ti-Ta alloys. Our approach provides physical insight into the various contributions that explain the strong composition dependence of M_s that is observed experimentally. Based on our calculations, we identify the relative phase stability at T=0 K and the vibrational entropy difference between the involved phases as critical parameters to predict changes in T₀. We propose a simple, one-dimensional descriptor to estimate the transition temperature that can be used in the identification of new alloys suitable for high-temperature shape memory applications.