  We use neutron scattering, to study magnetic excitations in crystals near the
ideal superconducting composition of FeTe$_{0.5}$Se$_{0.5}$. Two types of
excitations are found, a resonance at (0.5, 0.5, 0) and incommensurate
fluctuations on either side of this position. We show that the two sets of
magnetic excitations behave differently with doping, with the resonance being
fixed in position while the incommensurate excitations move as the doping is
changed. These unusual results show that a common behavior of the low energy
magnetic excitations is not necessary for pairing in these materials.
