# Bravais lattice with sublattices : why multiple bands?

I have a very naive question : given a tight-binding model (with nearest-neighbor hoping) on a lattice defined by a Bravais lattice with a number of sublattices (for instance the honeycomb lattice is a triangular lattice with two sublattices), why is there a band associated with each sublattice ? For instance when a lattice is defined by a Bravais lattice with a 2 sublattice basis, the tight-binding model will have 2 bands.. But why is that ? There must be something very simple that I am not getting here... is it just because that to define a band structure, you need some translation invariance ?

Another way to look at it would be the Hamiltonmatrix, of which the eigenenvalues are the energies we look for: When you have one atom in the unit cell and 4 orbitals (e.g. s,p$_x$,p$_y$,p$_z$) you have a 4x4 Matrix. A 4x4 matrix has 4 eigenvalues, which means 4 bands. So if you had 2 atoms in the unit cell with the same amount of orbitals you'd have an 8x8 matrix, and therefore 8 eigenvalues (= 8 bands).
The number of bands you get in a tight binding model is given by: $$n_{\text{bands}} = n_{\text{orbitals}}*n_{\text{atoms in unit cell}}$$