The *slope* of the frequency versus warped-frequency curve can be
interpreted as being proportional to critical bandwidth, since a unit
interval (one Bark) on the warped-frequency axis is magnified by the slope
to restore the band to its original size (one critical bandwidth). It is
therefore interesting to look at the *relative slope error*, i.e., the
error in the slope of the frequency mapping divided by the ideal Bark-map
slope. We interpret this error measure as the *relative
bandwidth-mapping error* (RBME). The RBME is plotted in Fig.7 for
a kHz sampling rate. The worst case is 21% for the Chebyshev case
and 20% for both least-squares cases. When the mapping coefficient is
explicitly optimized to minimize RBME, the results of Fig.8 are
obtained: the Chebyshev peak error drops from 21% down to 18%, while the
least-squares cases remain unchanged at 20% maximum RBME. A 3% change in
RBME is comparable to the 0.03 Bark peak-error reduction seen in
Fig.6 when using the Chebyshev norm instead of the norm;
again, such a small difference is not likely to be significant in most
applications.

Similar observations are obtained at other sampling rates, as shown in Fig.9. Near a 10 kHz sampling rate, the Chebyshev RBME is reduced from 17% when minimizing absolute error in Barks (not shown in any figure) to around 12% by explicitly minimizing the RBME, and this is the sampling-rate range of maximum benefit. At 15.2, 19, 41, and 54 kHz sampling rates, the difference is on the order of only 1%. Other cases generally lie between these extremes. The arctangent formula generally falls between the Chebyshev and optimal least-squares cases, except at the highest (extrapolated) sampling rate 54kHz. The rms error is very similar in all four cases, although the Chebyshev case has a little larger rms error near a 10 kHz sampling rate, and the arctangent case gives a noticeably larger rms error at 54 kHz.

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