The crystal structures of potassium tetratellurite and potassium ditellurite are reported, completing the series of known phases of alkali tellurites including lithium, sodium, potassium and cesium. Potassium tetratellurite (K-2-Te4O9, a = 7.572(1) Angstrom, b = 17.821(3) Angstrom, c = 7.829(1) Angstrom, beta = 108.62 degrees, monoclinic, P2(1)/c, Z = 4) contains linked 12-membered tellurite rings, coordinated by potassium. The ditellurite (K2Te2O5, a = 5.454(1) Angstrom, b = 15.142(1) Angstrom, c = 7.731(1) Angstrom, beta = 93.71(1)degrees, Z = 4), monoclinic, P2(1)/a, Z = 4) consists of infinite chains separated by rough sheets of cations. Comparison of the different alkali tellurite structures confirms the qualitative picture that as alkali oxide is added, the tellurite network is increasingly cleaved. We show, however, that the extent of network cleavage depends significantly on the nature of the modifying ion, with lithium inducing the fewest nonbridging oxygens per equivalent and potassium and cesium inducing as many as two nonbridging oxygens per equivalent of added cation. This behavior is due to the stability in tellurites of delocalized electrons across several nonbridging oxygens and is contrasted with the much simpler and well-known case of silicate modification.