Abstract <p>Masses of the ground and orbitally excited states of triply heavy tetraquarks with one open and another hidden heavy flavors are calculated within the relativistic quark model, based on the quasipotential approach and QCD, within the diquark–antidiquark picture of tetraquarks. The relativistic quasipotential equation takes into account all spin-dependent and spin-independent relativistic contributions and the internal structure of the diquarks. The calculated masses of such tetraquarks are compared with the strong fall–apart decay thresholds into a heavy and heavy–light mesons. The states that could be observed as narrow resonances in other decay modes are identified.</p>