Simple closed geodesics on regular tetrahedra in spherical space

On a regular tetrahedron in spherical space there exist the finite number of simple closed geodesics. For any pair coprime integers $(p,q)$ it was found numbers $\alpha_1$ and $\alpha_2$ depending on $p$, $q$ satisfying inequalities $\pi/3< \alpha_1 < \alpha_2 2\pi/3$ such that with faces angle $\alpha \in \left( \pi/3, \right)$ exists unique, up to rigid motion tetrahedron, geodesic type $(p,q)$, \alpha_2, 2\pi/3 is no $(p,q)$.