  In this paper, we suggest a new Heterogeneous Multiscale Method (HMM) for the
Helmholtz equation with high contrast. The method is constructed for a setting
as in Bouchitt\'e and Felbacq (C.R. Math. Acad. Sci. Paris 339(5):377--382,
2004), where the high contrast in the parameter leads to unusual effective
parameters in the homogenized equation. We revisit existing homogenization
approaches for this special setting and analyze the stability of the two-scale
solution with respect to the wavenumber and the data. This includes a new
stability result for solutions to the Helmholtz equation with discontinuous
diffusion matrix. The HMM is defined as direct discretization of the two-scale
limit equation.
  With this approach we are able to show quasi-optimality and an a priori error
estimate under a resolution condition that inherits its dependence on the
wavenumber from the stability constant for the analytical problem. Numerical
experiments confirm our theoretical convergence results and examine the
resolution condition. Moreover, the numerical simulation gives a good insight
and explanation of the physical phenomenon of frequency band gaps.
