Cross sections for e⁺e⁻ → n-jets, as functions of the jet resolution parameter y_cut, are computed according to a new clustering algorithm. The jet multiplicity n is defined in such a way that jets i and j with energies E_i and E_j at relative angle angle θ_ij are resolved if y _ij = 2(1− cos θ _ij) min (E _i², E _j²) s>y _cut , where s is the centre-of-mass energy squared. Using this algorithm, large higher-order corrections at small values of y_cut can easily be evaluated. Our calculations include resummation of leading and next-to-leading logarithms of y_cut to all orders in QCD perturbation theory. This enables us to predict the jet cross sections at small y_cut for arbitrary n. Simple analytical results for n⩽5 are presented.