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where f is in Hz and ERB(f) is also in Hz. The approximation is applicable at moderate sound levels and for values of f between 100 and 10000Hz.cite-ref-glasbergmoore-2-2[2]

The ERB-rate scale, or ERB-number scale, can be defined as a function ERBS(f) which returns the number of equivalent rectangular bandwidths below the given frequency f. The units of the ERB-number scale are known ERBs, or as Cams, following a suggestion by Hartmann.cite-ref-3[3] The scale can be constructed by solving the following differential system of equations:

{ E R B S ( 0 ) = 0 d f d E R B S ( f ) = E R B ( f ) {\displaystyle {\begin{cases}\mathrm {ERBS} (0)=0\\{\frac {df}{d\mathrm {ERBS} (f)}}=\mathrm {ERB} (f)\\\end{cases}}}

The solution for ERBS(f) is the integral of the reciprocal of ERB(f) with the constant of integration set in such a way that ERBS(0) = 0.cite-ref-mooreglasberg-1-2[1]

Using the second order polynomial approximation (Eq.1) for ERB(f) yields:

E R B S ( f ) = 11.17 ⋅ ⋅ ln ⁡ ⁡ ( f + 0.312 f + 14.675 ) + 43.0 {\displaystyle \mathrm {ERBS} (f)=11.17\cdot \ln \left({\frac {f+0.312}{f+14.675}}\right)+43.0} cite-ref-mooreglasberg-1-3[1]

where f is in kHz. The VOICEBOX speech processing toolbox for MATLAB implements the conversion and its inverse as:

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