# Bark scale

The Bark scale (after Heinrich Barkhausen ) is a psychoacoustic scale for the perceived pitch ( tonality ). A doubling of the Bark value means that the corresponding tone is perceived as twice as high. The Bark scale is defined from 0.2 to 25 Bark.

The Bark scale is linked to the pitch in Mel according to Eberhard Zwicker :

1 bar = 100 mel

Both the Bark and Mel scales are normalized to the musical note  c (131 Hz):

1.31 Bark = 131 Mel = 131 Hz

## Relation to the frequency

Relationship between tonality in Mel (100 Mel = 1 Bark) and frequency

The further relationships between frequency and Bark value result from psychoacoustic experiments (a rule of thumb in brackets):

• at frequencies below 500 Hz there is an almost linear relationship: a tone with twice as high a frequency (one octave ) is perceived as twice as high (a difference of 1 bark corresponds to an increase in frequency of 100 Hz in this frequency range).
• at frequencies above 500 Hz there is more of a logarithmic relationship: in order to be felt twice as high, e.g. E.g. at 1000 Hz a second tone already has 4 times the frequency, at 1600 Hz even 10 times the frequency (a difference of 1 bark in this frequency range corresponds to an increase in frequency by a minor third , i.e. by a factor of 1 , 19).

Diagrams that use the Bark scale instead of the linearly plotted frequency therefore correspond better to the auditory impression.

A frequency  f can be converted into the associated Bark value  z using the following formula:
(Note: The formula is not entirely exact: 131 Hz results in a little more than 1.31 Bark)

${\ displaystyle z = 13 \ cdot \ arctan \ left (0 {,} 00076 \ cdot f \ right) +3 {,} 5 \ cdot \ arctan \ left (\ left ({\ frac {f} {7500}} \ right) ^ {2} \ right)}$

### Bark frequency scale

 z / Bark f / Hz z / Bark f / Hz z / Bark f / Hz 1 100 9 1080 17th 3700 2 200 10 1270 18th 4400 3 300 11 1480 19th 5300 4th 400 12 1720 20th 6400 5 510 13 2000 21st 7700 6th 630 14th 2320 22nd 9500 7th 770 15th 2700 23 12000 8th 920 16 3150 24 15500

### Critical bandwidth

According to Traunmüller (1990) the critical bandwidth rate  z in Bark is:

${\ displaystyle z '= {\ dfrac {26 {,} 81} {1 + 1960 / f}} - 0 {,} 53}$
 if ${\ displaystyle z '<2 \}$ then ${\ displaystyle z = z '+ 0 {,} 15 \ cdot \ left (2-z' \ right) \}$ if ${\ displaystyle z '> 20 {,} 1 \}$ then ${\ displaystyle z = z '+ 0 {,} 22 \ cdot \ left (z'-20 {,} 1 \ right) \}$ otherwise is ${\ displaystyle z = z '\}$

The critical bandwidth  f in Hz can be calculated from  z in Bark:

${\ displaystyle f = {\ dfrac {52548} {z ^ {2} -52 {,} 56z + 690 {,} 39}}}$

## Excitation of nerve cells in the inner ear

Relationship between Bark value or frequency group , location on the basilar membrane , perceived height  Z and frequency of a tone

Sound stimulates the basilar membrane in the cochlea of the inner ear to vibrate , which in turn excites nerve cells along the membrane . The Bark value of the perceived pitch depends approximately linearly on how far the excited nerve cells are from the end of the basilar membrane (see adjacent figure):

1 Bark = 1.3 mm basilar membrane length

## Signal processing of the human hearing

In order to determine the volume , the sound or the direction of sound, the human ear divides the audible frequencies into defined ranges, the frequency groups . The actual information evaluation takes place within a frequency group, and finally the information from different frequency groups is compiled to form an overall impression.

With listening tests, 24 frequency groups can be determined, each with position and width. This suggests that the human ear divides the basilar membrane of the inner ear into approximately 24 sections of equal length, for which the nerve impulses generated are evaluated together. This corresponds exactly to the definition of the Bark scale (see above):

The width of a frequency group corresponds exactly to one bark.

The Bark scale is therefore also used to designate the different frequency groups (and thus the signal analysis areas of the hearing).

Another scale for frequency groups is the ERB scale .