  We recall criteria on the spectrum of Jacobi matrices such that the
corresponding isospectral torus consists of periodic operators. Motivated by
those known results for Jacobi matrices, we define a new class of operators
called GMP matrices. They form a certain Generalization of matrices related to
the strong Moment Problem. This class allows us to give a parametrization of
almost periodic finite gap Jacobi matrices by periodic GMP matrices. Moreover,
due to their structural similarity we can carry over numerous results from the
direct and inverse spectral theory of periodic Jacobi matrices to the class of
periodic GMP matrices. In particular, we prove an analogue of the remarkable
"magic formula" for this new class.
