  We present a new lattice QCD analysis of heavy-quark
pseudoscalar-pseudoscalar correlators, using gluon configurations from the MILC
collaboration that include vacuum polarization from $u$, $d$, $s$ and
$c$~quarks($n_f=4$). We extract new values for the QCD coupling and for the $c$
quark's $\overline{\mathrm{MS}}$ mass:
$\alpha_{\overline{\mathrm{MS}}}(M_Z,n_f=5) = 0.11822(74)$ and
$m_c(3\mathrm{GeV}, n_f=4) = 0.9851(63)$GeV. These agree well with our earlier
simulations using $n_f=3$ sea quarks, vindicating the perturbative treatment of
$c$ quarks in that analysis. We also obtain a new nonperturbative result for
the ratio of $c$~and $s$~quark masses: $m_c/m_s=11.652(65)$. This ratio implies
$m_s(2\,\mathrm{GeV}, n_f=3)=93.6(8)$MeV when it is combined with our
new~$c$~mass. Combining $m_c/m_s$ with our earlier $m_b/m_c$ gives
$m_b/m_s=52.55(55)$, which is several standard deviations (but only 4%) away
from the Georgi-Jarlskop prediction from certain GUTs. Finally we obtain an
$n_f=4$ estimate for $m_b/m_c=4.528(54)$ which agrees well with our earlier
$n_f=3$ result. The new ratio implies~$m_b(m_b,n_f=5)=4.162(48)$GeV.
