  We calculate the energies of three-quark states with definite permutation
symmetry (i.e. of SU(6) multiplets) in the N=0,1,2 shells, confined by the
Y-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 to
this technical complication we treat the problem at three increasingly accurate
levels of approximation: 1) the (approximate) three-string potential expanded
to first order in trigonometric functions of hyper-spherical angles; 2) the
(approximate) three-string potential to all orders in the power expansion in
hyper-spherical harmonics, but without taking into account the transition(s) to
two-string potentials; 3) the exact minimal-length string potential to all
orders in power expansion in hyper-spherical harmonics, and taking into account
the transition(s) to two-string potentials. We show the general trend of
improvement %convergence of these approximations: The exact non-perturbative
corrections to the total energy are of the order of one per cent, as compared
with 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 by
approximation 2) at the one per cent level. The precise value of the energy
separation of the first radial excitation ("Roper") $[56^{\prime},0^{+}]$-plet
from the $[70,1^{-}]$-plet depends on the approximation, but does not become
negative, i.e. the "Roper" remains heavier than the odd-parity
$[70,1^{-}]$-plet in all of our approximations.
