  In this paper, we construct a family of Berezin-Toeplitz type quantizations
of a compact symplectic manifold. For this, we choose a Riemannian metric on
the manifold such that the associated Bochner Laplacian has the same local
model at each point (this is slightly more general than in almost-K\"ahler
quantization). Then the spectrum of the Bochner Laplacian on high tensor powers
$L^p$ of the prequantum line bundle $L$ asymptotically splits into clusters of
size ${\mathcal O}(p^{3/4})$ around the points $p\Lambda$, where $\Lambda$ is
an eigenvalue of the model operator (which can be naturally called a Landau
level). We develop the Toeplitz operator calculus with the quantum space, which
is the eigenspace of the Bochner Laplacian corresponding to the eigebvalues
frrom the cluster. We show that it provides a Berezin-Toeplitz quantization. If
the cluster corresponds to a Landau level of multiplicity one, we obtain an
algebra of Toeplitz operators and a formal star-product. For the lowest Landau
level, it recovers the almost K\"ahler quantization.
