Fano resonances between plasmons and diffracted light offer tunable energies and locales, but attribution of Fano resonance features to geometry and physicochemistry of metal nanostructures and adjacent dielectrics has been confounded by complexity and computational expense. This work shows predictable modal shifts of Fano resonance in square lattices of plasmonic nanostructures can be attributed directly to changes in medium wavenumber, particle size, and lattice constant that alter plasmon polarizability and diffractive interference. For 45 to 80 nm radius particles, a window of lattice constants that support Fano resonances is identified in a range from 500 to 900 nm. Lattice constants that support high intensity resonances are determined by individual particle polarizability and medium wavenumber. Fano resonance wavelengths redshift from diffracted photon energies as local refractive index (RI) changes due to coupling with particle polarizability in the window. Redshift sensitivities for quadrupole, dipole, and Fano resonances are 150, 348, and 541 nm, respectively, per RI unit. Fano resonance intensity may be enhanced more than tenfold by selecting nanoparticle sizes and lattice constants. The quantitative effects of such parametric changes are rapidly and intuitively distinguished using a semi-analytic approach, consisting of an exact expression for particle polarizability, a trigonometric description of diffraction, and a semianalytical coupled dipole approximation.