A simple tight binding picture shows how allowed states in a solid can be quantized with allowed wavenumber $$k_i = i \frac{2 \pi}{N a}$$ where $N$ is the number of atoms in a circular chain. For large system size, the spacing $k_{i+1}-k_i$ tend to zero and the spectrum is nearly continuous: the electronic bands.
An alternative explanation for the formation of (nearly) continuous energy levels (i.e., the bands) goes as follows: "... if a large number N of identical atoms come together to form a solid, such as a crystal lattice, the atoms' atomic orbitals overlap. Since the Pauli exclusion principle dictates that no two electrons in the solid have the same quantum numbers, each atomic orbital splits into N discrete molecular orbitals, each with a different energy. Since the number of atoms in a macroscopic piece of solid is a very large number ($N\approx10^{22}$) the number of orbitals is very large and thus they are very closely spaced in energy (of the order of $10^{−22}$ eV). The energy of adjacent levels is so close together that they can be considered as a continuum, an energy band." [copied from Wikipedia] ].
How to reconcile the two concepts?
