Figure 1 illustrates the surprisingly good match between the allpass transformation and a Bark frequency warping when the map parameter is properly chosen. In the following, a simple direct-form expression is developed for the map parameter giving the best least-squares fit to a Bark scale for a chosen sampling rate. As Fig.1 shows, the error is so small that the solution is also very close to the optimal Chebyshev fit. In fact, the L2 optimal warping is within 0.04 Bark of the optimal warping. Since the experimental uncertainty when measuring critical bands is on the order of a tenth of a Bark or more [20,22,31,38], we consider the optimal Chebyshev and least-squares maps to be equivalent psychoacoustically.