  QCD lattice simulations with 2+1 flavours (when two quark flavours are mass
degenerate) typically start at rather large up-down and strange quark masses
and extrapolate first the strange quark mass and then the up-down quark mass to
its respective physical value. Here we discuss an alternative method of tuning
the quark masses, in which the singlet quark mass is kept fixed. Using group
theory the possible quark mass polynomials for a Taylor expansion about the
flavour symmetric line are found, first for the general 1+1+1 flavour case and
then for the 2+1 flavour case. This ensures that the kaon always has mass less
than the physical kaon mass. This method of tuning quark masses then enables
highly constrained polynomial fits to be used in the extrapolation of hadron
masses to their physical values. Numerical results for the 2+1 flavour case
confirm the usefulness of this expansion and an extrapolation to the physical
pion mass gives hadron mass values to within a few percent of their
experimental values. Singlet quantities remain constant which allows the
lattice spacing to be determined from hadron masses (without necessarily being
at the physical point). Furthermore an extension of this programme to include
partially quenched results is given.
