  We derive two-dimensional spatial power spectra of four distinct interstellar
medium tracers, HI, $^{12}$CO($J$=1--0), $^{13}$CO($J$=1--0), and dust, in the
Perseus molecular cloud, covering linear scales ranging from $\sim$0.1 pc to
$\sim$90 pc. Among the four tracers, we find the steepest slopes of
$-3.23\pm0.05$ and $-3.22\pm0.05$ for the uncorrected and opacity-corrected HI
column density images. This result suggests that the HI in and around Perseus
traces a non-gravitating, transonic medium on average, with a negligible effect
from opacity. On the other hand, we measure the shallowest slope of
$-2.72\pm0.12$ for the 2MASS dust extinction data and interpret this as the
signature of a self-gravitating, supersonic medium. Possible variations in the
dust-to-gas ratio likely do not change our conclusion. Finally, we derive
slopes of $-3.08\pm0.08$ and $-2.88\pm0.07$ for the ${}^{12}$CO(1--0) and
${}^{13}$CO(1--0) integrated intensity images. Based on theoretical predictions
for an optically thick medium, we interpret these slopes of roughly $-3$ as
implying that both CO lines are susceptible to the opacity effect. While simple
tests for the impact of CO formation and depletion indicate that the measured
slopes of ${}^{12}$CO(1--0) and ${}^{13}$CO(1--0) are not likely affected by
these chemical effects, our results generally suggest that chemically more
complex and/or fully optically thick media may not be a reliable observational
tracer for characterizing turbulence.
