  We use new ALMA observations to investigate the connection between dense gas
fraction, star formation rate, and local environment across the inner region of
four local galaxies showing a wide range of molecular gas depletion times. We
map HCN (1-0), HCO$^+$ (1-0), CS (2-1), $^{13}$CO (1-0), and C$^{18}$O (1-0)
across the inner few kpc of each target. We combine these data with short
spacing information from the IRAM large program EMPIRE, archival CO maps,
tracers of stellar structure and recent star formation, and recent HCN surveys
by Bigiel et al. and Usero et al. We test the degree to which changes in the
dense gas fraction drive changes in the SFR. $I_{HCN}/I_{CO}$ (tracing the
dense gas fraction) correlates strongly with $I_{CO}$ (tracing molecular gas
surface density), stellar surface density, and dynamical equilibrium pressure,
$P_{DE}$. Therefore, $I_{HCN}/I_{CO}$ becomes very low and HCN becomes very
faint at large galactocentric radii, where ratios as low as $I_{HCN}/I_{CO}
\sim 0.01$ become common. The apparent ability of dense gas to form stars,
$\Sigma_{SFR}/\Sigma_{dense}$ (where $\Sigma_{dense}$ is traced by the HCN
intensity and the star formation rate is traced by a combination of H$\alpha$
and 24$\mu$m emission), also depends on environment.
$\Sigma_{SFR}/\Sigma_{dense}$ decreases in regions of high gas surface density,
high stellar surface density, and high $P_{DE}$. Statistically, these
correlations between environment and both $\Sigma_{SFR}/\Sigma_{dense}$ and
$I_{HCN}/I_{CO}$ are stronger than that between apparent dense gas fraction
($I_{HCN}/I_{CO}$) and the apparent molecular gas star formation efficiency
$\Sigma_{SFR}/\Sigma_{mol}$. We show that these results are not specific to
HCN.
