  We present new 2 cm and 6 cm maps of H2CO, radio recombination lines, and the
radio continuum in the W51 star forming complex acquired with Arecibo and the
Green Bank Telescope at ~50" resolution. We use H2CO absorption to determine
the relative line-of-sight positions of molecular and ionized gas. We measure
gas densities using the H2CO densitometer, including continuous measurements of
the dense gas mass fraction (DGMF) over the range $10^4$ cm$^{-3}$ < n(H$_2$) <
$10^6$ cm$^{-3}$- this is the first time a dense gas mass fraction has been
measured over a range of densities with a single data set. The DGMF in W51A is
high,f >~70% above $n>10^4$ cm$^{-3}$, while it is low, f<20%, in W51 B. We did
not detect any H2CO emission throughout the W51 GMC; all gas dense enough to
emit under normal conditions is in front of bright continuum sources and
therefore is seen in absorption instead. The data set has been made public at
http://dx.doi.org/10.7910/DVN/26818. Conclusions. (1) The dense gas fraction in
the W51 A and B clouds shows that W51 A will continue to form stars vigorously,
while star formation has mostly ended in W51 B. The lack of dense, star-forming
gas around W51 C indicates that collect-and-collapse is not acting or is
inefficient in W51. (2) Ongoing high-mass star formation is correlated with n ~
1x10$^5$ cm$^{-3}$ gas. Gas with n > 10$^4$ cm$^{-3}$ is weakly correlated with
low and moderate mass star formation, but does not strongly correlate with
high-mass star formation. (3) The nondetection of H$_2$CO emission implies that
the emission detected in other galaxies, e.g. Arp 220, comes from high-density
gas that is not directly affiliated with already-formed massive stars. Either
the non-star-forming ISM of these galaxies is very dense, implying the star
formation density threshold is higher, or H II regions have their emission
suppressed.
