A dynamo driven by motions unaffected by viscous forces is termed ‘magnetostrophic’, but cannot be found through today’s numerical simulations, which require substantial viscosity to stabilize solutions of the full magnetohydrodynamic (MHD) dynamo equations. By using an alternative numerical technique, proposed by Taylor (Proc. R. Soc. Lond. A, vol. 274, 1963, pp. 274–283), we recently obtained the first magnetostrophic dynamo solutions ever derived (Wu & Roberts, Geophys. Astrophys. Fluid Dyn., vol. 109, 2014, pp. 84–110). These were axisymmetric and of mean-field type. In an earlier paper (Roberts & Wu, Geophys. Astrophys. Fluid Dyn., vol. 108, 2014, pp. 696–715), we proposed an extension of Taylor’s method. Here we explore its numerical implications, comparing them to the consequences of Taylor’s original proposal. One of the differences between the two approaches is that our modification retains torsional waves but Taylor’s theory does not. A more important difference is that our extension of Taylor’s method is, for reasons presented here, the most general possible that does not suffer from the limitations imposed by viscosity on today’s numerical simulations.