  It is demonstrated that the TaN molecule is the best candidate to search for
T,P-violating nuclear magnetic quadrupole moment (MQM), it also looks promising
to search for other T,P-odd effects. We report results of coupled-cluster
calculations of T,P-odd effects in TaN produced by the Ta nucleus MQM, electron
electric dipole moment (EDM), scalar$-$pseudoscalar nucleus$-$electron
interactions, also of the molecule-axis hyperfine structure constant and dipole
moment. Nuclear calculations of $^{181}$Ta MQM are performed to express the
T,P-odd effect in terms of the strength constants of T,P-odd nuclear forces,
proton and neutron EDM, QCD parameter $\theta$ and quark chromo-EDM.
