  We extend an effective field theory developed to describe rotational bands in
even-even nuclei to the odd-mass case. This organizes Bohr and Mottelson's
treatment of a particle coupled to a rotor as a model-independent expansion in
powers of the angular velocity of the overall system. We carry out this
expansion up to fourth order in the angular velocity and present results for
$^{99}$Tc, ${}^{159}$Dy, ${}^{167, 169}$Er, ${}^{167, 169}$Tm, ${}^{183}$W,
${}^{235}$U and ${}^{239}$Pu. In each case, the accuracy and breakdown scale of
the effective field theory can be understood based on the single-particle and
vibrational energy scales in that nucleus.
