  I construct predictive models of neutrino mass and mixing that have fewer
parameters, both in the lepton sector and overall, than the default seesaw
model. The predictions are theta13=0 and one massless neutrino, with the models
having a Z4 or Z2 symmetry and just one extra degree of freedom: one real
singlet Higgs field. It has been shown that models with an unbroken family
symmetry, and with no Higgs fields other than the Standard Model Higgs doublet
produce masses and mixing matrices that have been ruled out by experiment.
Therefore, this article investigates the predictions of models with Abelian
family symmetries that involve Higgs singlets, doublets and triplets, in the
hope that they may produce the maximal and minimal mixing angles seen in the
best fit neutrino mixing matrix. I demonstrate that these models can only
produce mixing angles that are zero, maximal or unconfined by the symmetry. The
maximal mixing angles do not correspond to physical mixing, so an Abelian
symmetry can, at best, ensure that theta13=0, while leaving the solar and
atmospheric mixing angles as free parameters. To generate more features of the
best-fit mixing matrix a model with a non-Abelian symmetry and a complicated
Higgs sector would have to be used.
