Why does silicon have 6 phonon modes instead of 24? As is known, silicon has a structure made up of two intertwined face-centered cubic (FCC) lattices, with bases $[000]$ and $[\frac 1 4,\frac 1 4,\frac 1 4]$.
This equates to $\frac{1}{8}\cdot 8 + \frac{1}{2}\cdot 6 + 4 = 8$ atoms total in the unit cell. This would result in $24$ phonon modes ($3\cdot 8$). However, according to the literature, silicon has $6$ phonon modes. Why is this? 
 A: This confusion of unit cells is not uncommon. So, lets establish a few things: 
There are 14 Bravais lattices covering 7 point groups. These Bravais lattices are a periodic array in which repeated units of a crystal can be arranged. The Bravais lattice is a description of the underlying periodic structure of the crystal, regardless of what 'stuff' you put at each Bravais lattice point. 
A primitive unit cell is a volume that, when translated through all the vectors of a Bravais lattice, just fills space (no voids, no overlap). When a primitive unit cell contains a lattice point, it only contains one such lattice point. (Note that the volume of a primitive cell could have multiple lattice points on an edge or face.) The primitive unit cell is not unique. A Wigner-Seitz primitive unit cell is, however, a unique way to generate a primitive cell that fully reflects the symmetry of the Bravais lattice.
A conventional unit cell is one or more primitive unit cells, drawn in a way that is pleasing to a human eye. For bcc or fcc the usual conventional unit cell is based on a cube with lattice points on the corners as well as either the cube center (bcc) or the face centers (fcc). In reality, a bcc conventional cubic unit cell is twice the volume of the bcc primitive unit cell, and the fcc conventional cubic unit cell is four times the volume of the fcc primitive unit cell.
Diamond cubic is a fcc Bravais lattice with a 2-atom basis on the lattice point. The cubic conventional unit cell is still 4 times bigger in volume than the primitive unit cell. 
(I'd also add that the fcc conventional unit cell is horrible in clearly showing the packing symmetry along the <111> axis, but that another story.) 
