  We consider the probability for a colour-singlet qqbar pair to emit a gluon,
in strongly and smoothly ordered antenna showers. We expand to second order in
alphaS and compare to the second-order QCD matrix elements for Z -> 3 jets,
neglecting terms suppressed by 1/NC^2. We give a prescription that corrects the
shower to the matrix-element result at this order for both soft and hard
emissions, thereby explicitly reducing its dependence on evolution- and
renormalization-scale choices. We confirm that the choice of pT for both of
these scales absorbs all logarithms through order alphaS^2, and contrast this
with various alternatives. We include these corrections in the VINCIA shower
generator and study the impact on LEP event-shape and fragmentation
observables. An uncertainty estimate is provided for each event, in the form of
a vector of alternative weights.
