  We give a simple expression for the integral of the canonical holomorphic
volume form in degenerating families of varieties constructed from wall
structures and with central fiber a union of toric varieties. The cycles to
integrate over are constructed from tropical 1-cycles in the intersection
complex of the central fiber. One application is a proof that the mirror map
for the canonical formal families of Calabi-Yau varieties constructed by Gross
and the second author is trivial. We also show that these families are the
completion of an analytic family, without reparametrization, and that they are
formally versal as deformations of logarithmic schemes. Other applications
include canonical one-parameter type III degenerations of K3 surfaces with
prescribed Picard groups. As a technical result of independent interest we
develop a theory of period integrals with logarithmic poles on finite order
deformations of normal crossing analytic spaces.
