Why do tetrahedral complexes have approximately 4/9 the field split of octahedral complexes? I am trying to calculate the relationship between the octahedral field splitting parameter ($\Delta_\mathrm{o}$) and the square planar field splitting parameter ($\Delta_\mathrm{sp}$) and thought a good way to start would be to understand why $\Delta_\mathrm{t}=4/9\Delta_\mathrm{o}$ for tetrahedral fields but couldn't find any information on it.

How does one arrive at this particular ratio mathematically?
The ratio is derived in The angular overlap model. How to use it and why J. Chem. Educ., vol. 51, page 633-640.

First, relative expressions (Shaffer Angular Overlap Factors) are derived for overlap integrals between metal and ligand orbitals as a function of angles. 

For ligand sigma orbitals, the expressions are a functions of 2 angles, specifying the direction of the ligands in the coordinate system of the metal.  

For pi-x and pi-y ligand orbits, there is a third angle for orientation about the ligand-metal axis.    

Then, assuming the ligand-metal distances are the same in both the tetrahedral and octahedral cases, and considering the coordinate geometry of the ligands in the two cases, the relationship:

$$\Delta_{T} = -4/9 \Delta_{O}$$

is obtained.  

For another discussion of this ratio in the framework of the angular overlap model, as well as extension to other geometries such as square planar, see A New Look at Structure and Bonding in Transition Metal Complexes, Advances in Inorganic Chemistry and Radiochemistry, Volume 21, pages 113-146.