We present numerical simulations of forced rotating stratified turbulence dominated by vortical motion (i.e. with potential vorticity). Strong stratification and various rotation rates are considered, corresponding to a small Froude number and a wide range of Rossby numbers $\hbox{\it Ro}$ spanning the regimes of stratified turbulence ($\hbox{\it Ro}\,{=}\,\infty$) to quasi-geostrophic turbulence ($\hbox{\it Ro}\,{\ll}\,1$). We examine how the energy spectra and characteristic vertical scale of the turbulence vary with Rossby number between these two regimes. The separate dependence on $N/f$, where $N$ is the Brunt–Väisälä frequency and $f$ is the Coriolis parameter, is found to be of secondary importance. As the macroscale $\hbox{\it Ro}$ decreases below 0.4 and the microscale $\hbox{\it Ro}$ (at our resolution) decreases below 3, the horizontal wavenumber energy spectrum steepens and the flat range in the vertical wavenumber spectrum increases in amplitude and decreases in length. At large Rossby numbers, the vertical scale $H$ is proportional to the stratified turbulence value $U/N$, where $U$ is the root mean square velocity. At small $\hbox{\it Ro}$, $H$ takes the quasi-geostrophic form $(f/N)L$, where $L$ is the horizontal scale of the flow. Implications of these findings for numerical atmosphere and ocean modelling are discussed.