The optimal relative bandwidth-mapping error (RBME) for the ERB case is plotted in Fig.E.15 for a kHz sampling rate. The peak error has grown from close to 20% for the Bark-scale case to more than 60% for the ERB case. Thus, frequency intervals are mapped to the ERB scale with up to three times as much relative error (60%) as when mapping to the Bark scale (20%). The continued narrowing of the auditory filter bandwidth as frequency decreases on the ERB scale results in the conformal map not being able to supply sufficient stretching of the low-frequency axis. The Bark scale case, on the other hand, is much better provided at low frequencies by the first-order conformal map.

Figure E.16 shows the rms and peak ERB RBME as a function of sampling rate. Near a 10 kHz sampling rate, for example, the Chebyshev ERB RBME is increased from 12% in the Bark-scale case to around 37%, again a tripling of the peak error. We can also see in Fig.E.16 that the arctangent formula gives a very good approximation to the optimal Chebyshev solution at all sampling rates. The optimal least-squares and weighted equation-error solutions are quite different, with the weighted equation-error solution moving from being close to the least-squares solution at low sampling rates, to being close to the Chebyshev solution at the higher sampling rates. The rms error is very similar in all four cases, as it was in the Bark-scale case, although the Chebyshev and arctangent formula solutions show noticeable increase in the rms error at low sampling rates where they also show a reduction in peak error by 5% or so.

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Center for Computer Research in Music and Acoustics (CCRMA), Stanford University