psychoacoustics.erb_scale

The ERB_N scale: auditory-filter bandwidth and the Cam frequency scale.

The cochlea behaves as a bank of overlapping band-pass auditory filters. The width of the filter centred on a given frequency is summarised by its equivalent rectangular bandwidth ERB_N: the bandwidth of the rectangular filter that passes the same power and has the same peak response. Glasberg and Moore (1990), fitting notched-noise data for young listeners at moderate levels, give it as a straight line in frequency (Moore, An Introduction to the Psychology of Hearing 6th ed., p. 76):

$${\text{ERB}}_N=24.7(4.37F+1)\text{Hz},F\text{in kHz}$$

Integrating $df/{\text{ERB}}_N(f)$ turns that into a frequency scale whose unit is one auditory-filter width, the ERB_N number, whose unit is called the Cam (after Cambridge, following a suggestion by Hartmann):

$${\text{ERB}}_N\text{number}=21.4{\log }_{10}(4.37F+1)\text{Cam}$$

The Cam scale plays the same role as the Bark scale of Zwicker and Terhardt or the mel scale of pitch: it is the axis on which masking patterns, excitation patterns and specific loudness are naturally expressed. It differs from Bark numerically, mostly below 500 Hz where the “old” critical-band function flattens out but direct measurements show ERB_N continuing to shrink.

This module states the constants to the precision used by the ISO 532-2 implementation of the Moore-Glasberg loudness model (phonometry.psychoacoustics.loudness.moore_glasberg), which shares them:

$${\text{ERB}}_N=24.673(0.004368f+1)\text{Hz},f\text{in Hz}$$ $${\text{ERB}}_N\text{number}=21.366{\log }_{10}(0.004368f+1)\text{Cam}$$

They are the same fit written with one more digit: the two forms agree to better than 0.2 % over the whole audible range. The extra digits matter for the round trip through frequency_from_cam, and they are what reproduce the check value Moore prints on p. 77, that 1000 Hz corresponds to 15.59 Cam (the two-significant-digit constants give 15.62).

Reference: Glasberg, B. R. and Moore, B. C. J. (1990), “Derivation of auditory filter shapes from notched-noise data”, Hearing Research 47, 103-138.

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CAM_C

Constant (float).

CAM_C = 21.366

cam_from_frequency

cam_from_frequency(frequency: ArrayLike) -> np.ndarray | float

ERB_N number (Cam) of a frequency.

$i=21.366{\log }_{10}(0.004368f+1)$ (Glasberg and Moore 1990; Moore 6th ed., p. 76, printed there as $21.4{\log }_{10}(4.37F+1)$ for F in kHz). One Cam is one auditory-filter width.

Parameters

Name	Description
frequency	Frequency f, Hz (scalar or array, non-negative).

Returns: The ERB_N number, Cam; a float for a scalar input.

erb_bandwidth

erb_bandwidth(frequency: ArrayLike) -> np.ndarray | float

Equivalent rectangular bandwidth of the auditory filter, Hz.

${\text{ERB}}_N=24.673(0.004368f+1)$ (Glasberg and Moore 1990; Moore 6th ed., p. 76, printed there with the rounded constants $24.7(4.37F+1)$ for F in kHz).

Parameters

Name	Description
frequency	Filter centre frequency f, Hz (scalar or array, non-negative).

Returns: The bandwidth ERB_N, Hz; a float for a scalar input.

ERB_C1

Constant (float).

ERB_C1 = 24.673

ERB_C2

Constant (float).

ERB_C2 = 0.004368

frequency_from_cam

frequency_from_cam(cam: ArrayLike) -> np.ndarray | float

Frequency of an ERB_N number, the inverse of cam_from_frequency.

$f=(10^{i/21.366}-1)/0.004368$.

Parameters

Name	Description
cam	ERB_N number i, Cam (scalar or array, non-negative).

Returns: The frequency f, Hz; a float for a scalar input.