  A periodic tridiagonal matrix is a tridiagonal matrix with additional two
entries at the corners. We study the space $X_{n,\lambda}$ of Hermitian
periodic tridiagonal $n\times n$-matrices with a fixed simple spectrum
$\lambda$. Using the discretized Shr\"{o}dinger operator we describe all
spectra $\lambda$ for which $X_{n,\lambda}$ is a topological manifold. The
space $X_{n,\lambda}$ carries a natural effective action of a compact
$(n-1)$-torus. We describe the topology of its orbit space and, in particular,
show that whenever the isospectral space is a manifold, its orbit space is
homeomorphic to $S^4\times T^{n-3}$. There is a classical dynamical system: the
flow of the periodic Toda lattice, acting on $X_{n,\lambda}$. Except for the
degenerate locus $X_{n,\lambda}^0$, the Toda lattice exhibits Liouville--Arnold
behavior, so that the space $X_{n,\lambda}\setminus X_{n,\lambda}^0$ is fibered
into tori. The degenerate locus of the Toda system is described in terms of
combinatorial geometry: its structure is encoded in the special cell
subdivision of a torus, which is obtained from the regular tiling of the
euclidean space by permutohedra. We apply methods of commutative algebra and
toric topology to describe the cohomology and equivariant cohomology modules of
$X_{n,\lambda}$.
