Aspects of the Tetrahedral Neutrino Mass Matrix

The four-parameter tetrahedral neutrino mass matrix introduced earlier by the author is studied in two specific limits, both having only two parameters and resulting in theta_13=0, theta_23=pi/4, and tan^2 theta_12=1/2. One limit corresponds to a recent proposal which predicts a normal ordering of neutrino masses; the other is new and allows both inverted and normal ordering.