The Hamiltonian describing interacting two-dimensional electrons in a high magnetic field is diagonalized numerically for a small number of particles to obtain the low-lying excitation spectra. The results include estimates of energy gaps for values of $\ensuremath{\nu}$ (the lowest-Landau-level filling factor) equal to certain multiples of $\frac{1}{5}$, $\frac{1}{7}$, $\frac{1}{9}$, and $\frac{1}{11}$. These $\ensuremath{\nu}'\mathrm{s}$ are characterized by the existence of periodic rigid parent states which generate maximum phase space. The even-denominator cases are markedly different.