  We present a detailed description of the extraction of the highly excited
isovector meson spectrum on dynamical anisotropic lattices using a new
quark-field construction algorithm and a large variational basis of operators.
With careful operator construction, the combination of these techniques is used
to identify the continuum spin of extracted states reliably, overcoming the
reduced rotational symmetry of the cubic lattice. Excited states, states with
exotic quantum numbers (0+-, 1-+ and 2+-) and states of high spin are resolved,
including, for the first time in a lattice QCD calculation, spin-four states.
The determinations of the spectrum of isovector mesons and kaons are performed
on dynamical lattices with two volumes and with pion masses down to ~ 400 MeV,
with statistical precision typically at or below 1% even for highly excited
states.
