  We investigate the effect that density-wave states have on the Hofstadter
Butterfly. We first review the problem of the $d$-density wave on a square
lattice and then numerically solve the $d$-density wave problem when an
external magnetic field is introduced. As the $d$-density wave condensation
strength is tuned the spectrum evolves through three topologically distinct
butterflies, and an unusual quantum Hall effect is observed. The chiral
$p+ip$-density wave state demonstrates drastically different Hofstadter
physics--inducing a destruction of the gaps in the butterfly which causes
electrons' cyclotron orbits to not obey any type of Landau quantization, and
the creation of a large gap in the spectrum with Hall conductance
$\sigma_{xy}$=0. To investigate the quantum phases in the system we perform a
multifractal analysis of the single particle wavefunctions. We find that tuning
the $d$-density wave strength at a generic value of magnetic flux controls a
metal-metal transition at charge neutrality where the wavefunction
multifractality occurs near band touching events. In the $p+ip$ case we observe
another metal-metal transition near a band touching event which is seperated by
a quasi-insulating island state occuring at charge neutrality near strip
dimerization of the lattice.
