  Both the CO(2-1) and CO(1-0) lines are used to trace the mass of molecular
gas in galaxies. Translating the molecular gas mass estimates between studies
using different lines requires a good understanding of the behaviour of the
CO(2-1)-to-CO(1-0) ratio, $R_{21}$. We compare new, high quality CO(1-0) data
from the IRAM 30-m EMPIRE survey to the latest available CO(2-1) maps from
HERACLES, PHANGS-ALMA, and a new IRAM 30-m M51 Large Program. This allows us to
measure $R_{21}$ across the full star-forming disc of nine nearby, massive,
star-forming spiral galaxies at 27" (${\sim} 1{-}2$ kpc) resolution. We find an
average $R_{21} = 0.64\pm0.09$ when we take the luminosity-weighted mean of all
individual galaxies. This result is consistent with the mean ratio for disc
galaxies that we derive from single-pointing measurements in the literature,
$R_{\rm 21, lit}~=~0.59^{+0.18}_{-0.09}$. The ratio shows weak radial
variations compared to the point-to-point scatter in the data. In six out of
nine targets the central enhancement in $R_{21}$ with respect to the
galaxy-wide mean is of order $\sim 10{-}20\%$. We estimate an azimuthal scatter
of $\sim$20% in $R_{21}$ at fixed galactocentric radius but this measurement is
limited by our comparatively coarse resolution of 1.5 kpc. We find mild
correlations between $R_{21}$ and CO brightness temperature, IR intensity,
70-to-160$ \mu$m ratio, and IR-to-CO ratio. All correlations indicate that
$R_{21}$ increases with gas surface density, star formation rate surface
density, and the interstellar radiation field.
