  We measure the metallicities of 374 red giant branch (RGB) stars in the
isolated, quenched dwarf galaxy Tucana using Hubble Space Telescope (HST)
narrow-band (F395N) Calcium H & K (CaHK) imaging. Our sample is a factor of
$\sim7$ larger than what is published. Our main findings are: (i) A global
metallicity distribution function (MDF) with $\langle \mbox{[Fe/H]} \rangle =
-1.55 \pm 0.04$ and $\sigma_{\mbox{[Fe/H]}}=0.54\pm0.03$; (ii) A metallicity
gradient of $-0.54 \pm 0.07$ dex $R_e^{-1}$ ($-2.1 \pm 0.3$ dex kpc$^{-1}$)
over the extent of our imaging ($\sim 2.5 R_e$), which is steeper than
literature measurements. Our finding is consistent with predicted gradients
from the publicly-available FIRE-2 simulations, in which bursty star formation
creates stellar population gradients and dark matter cores; (iii) Tucana's
bifurcated RGB has distinct metallicities: a blue RGB with $\langle
\mbox{[Fe/H]} \rangle = -1.78 \pm 0.06$ and
$\sigma_{\mbox{[Fe/H]}}=0.44^{+0.07}_{-0.06}$, and a red RGB with $\langle
\mbox{[Fe/H]} \rangle = -1.08 \pm 0.07$ and $\sigma_{\mbox{[Fe/H]}}=0.42 \pm
0.06$; (iv) At fixed stellar mass, Tucana is more MR than MW satellites by
$\sim 0.4$ dex, but its blue RGB is chemically comparable to the satellites.
Tucana's MDF appears consistent with star-forming isolated dwarfs, though MDFs
of the latter are not as well-populated; (v) $\sim2$% of Tucana's stars have
$\mbox{[Fe/H]} < -3$ and 20% $\mbox{[Fe/H]} > -1$. We provide a catalog for
community spectroscopic follow-up.
