One of the most important functions of the peripheral auditory
system is its function as a frequency analyzer. The crucial
role in this respect is played by the basilar membrane, on which
every location is characterized by a best or
characteristic frequency.
Physiologically the auditory filter can, e.g., be measured by:
 neural tuning curves
Slide: Tuning curves
(Durrant & Lovrinic, 3rd ed., p. 217)
 neural impulse responses
Slide: Impulse response basilar membrane
(Moore, p. 27)
 Slide: Neural impulse responses
(Yost, p. 126)
 Slide: Simulated impulse responses
(Moore, p. 193)
Psychoacoustically, this has lead to the concept of critical
band,
which represents the frequency resolution of the auditory system. Psychoacoustically the critical bandwidth can, e.g., be measured
by:
 masking
Auditory demonstrations. Tracks 26. "Critical
bands by masking"
 loudness comparison
Auditory demonstrations. Track 7. "Critical
bands by loudness comparison"
The Bark scale
This has lead to the definition of the Bark scale, a frequency scale
on which equal distances correspond with perceptually equal distances.
Above about 500 Hz this scale is more or less equal to a logarithmic
frequency axis. Below 500 Hz the Bark scale becomes more and more
linear.
The ERBrate scale
Measuring the critical bands below 500 Hz appeared to be quite
difficult, due to the fact that, at low frequencies, the
sensitivity and the efficiency of the auditory system rapidly diminishes,
while headphone technology caused leakage of acoustic energy at low
frequencies. More accurate measurements of the auditoryfilter
bandwidth have now lead to the ERBrate scale. These measurements
have used notchednoise maskers to measure the auditory filter
bandwidth. In general, on this ERBrate scale the auditoryfilter
bandwidth, expressed in equivalent rectangular bandwidth (ERB), is
smaller than on the Bark scale, a difference which becomes larger for
lower frequencies. (From the towns of the research groups where they
were developed, Hartmann, 1997, has doped the ERBrate scale the
Cambridge scale, while he doped the Bark scale, the Munich scale.)
If the unit on the ERBrate scale is E, the number of ERBs, and f is
frequency, the conversion formula's are:
 f = 229 · (10^{E/21.4}  1),
 E = 21.4 · log_{10}(0.00437·f+1).
All this has lead to the notion of an auditory filter
bank. In this view, the peripheral hearing system consists of a large
number of bandpass filters. The frequency characteristics of
these filters are constant on an ERBrate scale. Mathematically
they can be expressed as socalled rounded exponentials. When we define
g = (f  f_{c} )/
f_{c},
 H(f) ^{2} = (1+pg) · e ^{ pg}.
In the time domain, the impulse response is modelled as a socalled
gamma tone:
 h(t) = c · t ^{(g
1)}
· e^{t /t} ·
cos(2pf_{c}t +
j).
The notion of critical bandwidth or auditoryfilter bandwidth arises
in many different contexts in hearing research.
How to construct perceptual frequency scales
The ERBrate scale z
We now want to construct a frequency scale z in such a way that
equal intervals on that scale represent the same number of auditory
filter bandwidths (for further details see Hartmann, 1997). Let
E(f) be the ERB at f. Point of departure is that the
change in z, in number of ERBs, is such that
dz = (Dz/
Df) df.
In the equation 1/(Dz/
Df) =
Df / Dz
represents the change in f per unit z which, at that f,
is exactly one ERB. Or, dz = (1/E(f)) df. Integration then yields:
which gives us the equation for calculating the ERBrate scale from
measured ERBs. The unit for this scale E is number of ERBs.
The Bark scale
The Bark scale is defined as a table (Zwicker, 1961). The italics are the lower
and higher bound of the critical bands with integer values.
Bark 

frequency 

Bark 

frequency 

Bark 

frequency 
0 

20 

8 

920 

16 

3150 


50 



1000 



3400 
1 

100 

9 

1080 

17 

3700 


150 



1170 



4000 
2 

200 

10 

1270 

18 

4400 


250 



1370 



4800 
3 

300 

11 

1480 

19 

5300 


350 



1600 



5800 
4 

400 

12 

1720 

20 

6400 


450 



1850 



7000 
5 

510 

13 

2000 

21 

7700 


570 



2150 



8500 
6 

630 

14 

2320 

22 

9500 


700 



2500 



10500 
7 

770 

15 

2700 

23 

12000 


840 



2900 



13500 








24 

15500 
Both Bark en ERBrate scale are presented in the figure below. The
formula used for the Bark scale is from Traunmüller (1990), JASA 88,
971000.
Literature
Hartmann W.M.
Auditory filters.
In: Signals, sound, and sensation, Chapter 10.
Woodbury, NY: American Institute of Physics, 1997.
Moore B.C.J.
An introduction to the psychology of hearing, 3rd edition.
San Diego, CA: Academic Press, 1997.
