  We consider quantum oscillation experiments in
$\mathrm{YBa_{2}Cu_{3}O_{6+\delta}}$ from the perspective of an incommensurate
Fermi surface reconstruction using an exact transfer matrix method and the
Pichard-Landauer formula for the conductivity. The specific density wave order
considered is a period-8 $d$-density wave in which the current density is
unidirectionally modulated. The current modulation is also naturally
accompanied by a period-4 site charge modulation in the same direction, which
is consistent with recent magnetic resonance measurements. In principle Landau
theory also allows for a period-4 bond charge modulation, which is not
discussed, but should be simple to incorporate in the future. This scenario
leads to a natural, but not a unique, explanation of why only oscillations from
a single electron pocket is observed, and a hole pocket of roughly twice the
frequency as dictated by two-fold commensurate order, and the corresponding
Luttinger sum rule, is not observed. However, it is possible that even higher
magnetic fields will reveal a hole pocket of half the frequency of the electron
pocket or smaller. This may be at the borderline of achievable high field
measurements because at least a few complete oscillations have to be clearly
resolved.
