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We calculate the energies of three-quark states with definite permutationsymmetry (i.e. of SU(6) multiplets) in the N=0,1,2 shells, confined by theY-string three-quark potential. The exact Y-string potential consists of one,so-called three-string term, and three angle-dependent two-string terms. Due tothis technical complication we treat the problem at three increasingly accuratelevels of approximation: 1) the (approximate) three-string potential expandedto first order in trigonometric functions of hyper-spherical angles; 2) the(approximate) three-string potential to all orders in the power expansion inhyper-spherical harmonics, but without taking into account the transition(s) totwo-string potentials; 3) the exact minimal-length string potential to allorders in power expansion in hyper-spherical harmonics, and taking into accountthe transition(s) to two-string potentials. We show the general trend ofimprovement %convergence of these approximations: The exact non-perturbativecorrections to the total energy are of the order of one per cent, as comparedwith approximation 2), yet the exact energy differences between the$[20,1^{+}], [70,2^{+}], [56,2^{+}], [70,0^{+}]$-plets are shifted to 2:2:0.9,from the Bowler and Tynemouth separation rule 2:2:1, which is obeyed byapproximation 2) at the one per cent level. The precise value of the energyseparation of the first radial excitation ("Roper") $[56^{\prime},0^{+}]$-pletfrom the $[70,1^{-}]$-plet depends on the approximation, but does not becomenegative, i.e. the "Roper" remains heavier than the odd-parity$[70,1^{-}]$-plet in all of our approximations.

Low-lying spectrum of the Y-string three-quark potential using hyper-spherical coordinates