Crystals of varying compensation mechanism (extra oxygen bridges or ${\mathrm{Co}}^{2+}$ substitution) and phase ($\ensuremath{\beta}$ or ${\ensuremath{\beta}}^{\ensuremath{'}\ensuremath{'}}$) have been grown by adding cobalt oxide to a melt containing NaC${\mathrm{O}}_{3}$, ${\mathrm{Al}}_{2}$${\mathrm{O}}_{3}$, and ${\mathrm{Bi}}_{2}$${\mathrm{O}}_{3}$ in the proportion 2.9:4.4:100. In $(1+x){\mathrm{Na}}_{2}\mathrm{O}\ifmmode\cdot\else\textperiodcentered\fi{}11{\mathrm{Al}}_{2}{\mathrm{O}}_{3}$ the value of $x$ is \ensuremath{\sim}0.16(3) for crystals grown from the melt and \ensuremath{\sim}0.38(3) for crystals grown from the flux. Diffuse x-ray scattering measurements confirm the formation of Al Frenkel defects which lock the extra oxygen ions in the diffusing plane. Rods of diffuse scattering are observed in $\ensuremath{\beta}$-alumina grown from the flux with Ag, K, Rb, and Eu substituted for Na in the diffusing plane. The rods are at ($h+\frac{1}{3},k+\frac{1}{3}$) and ($h+\frac{2}{3},k+\frac{2}{3}$) but not at integral ($\mathrm{hk}$). A two-dimensional short-range-order model consisting of three types of cells: (i) containing one diffusing ion, (ii) two ions, and (iii) 2 ions plus an extra bridge [cells (ii) and (iii) have no diffusing ions in the Eu case] agrees semiquantitatively with the observed data. The conditional probability for doubly occupied cells or defect cells being nearest neighbors is ${P}_{22}(1)={P}_{33}(1)\ensuremath{\sim}0.05$ (random value is 1/6) and the correlations are assumed to fall off as $\frac{{e}^{\frac{\ensuremath{-}R}{\ensuremath{\xi}}}}{{R}^{\frac{1}{2}}}$ with the signs of the coefficients given by a superlattice which isolates cell (ii) and (iii). The correlation lengths in the plane and the activation energy for ionic conductivity both vary linearly with the concentration of extra bridges for K $\ensuremath{\beta}$-alumina. The correlation lengths in units of ${a}_{0}$ are 0.9(1), 1.2(1), 2.5(1), and 6.1(5) for Na, Ag, K, and Rb from the flux and 1.0(1), 1.2(1), 1.3(1) for Ag, K, and Rb from melt-grown crystals.