# Bark scale

A440  . 440 Hz = 4.21 or 4.39

The Bark scale is a psychoacoustical scale proposed by Eberhard Zwicker in 1961. It is named after Heinrich Barkhausen who proposed the first subjective measurements of loudness.[1] One definition of the term is "...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."[2]

The scale ranges from 1 to 24 and corresponds to the first 24 critical bands of hearing.[3]

It is related to, but somewhat less popular than[citation needed], the mel scale, a perceptual scale of pitches judged by listeners to be equal in distance from one another.

## Bark scale critical bands

Chart of the critical bands of the Bark scale
Number Center Frequency (Hz) Cut-off Frequency (Hz) Bandwidth (Hz)
20
1 60 100 80
2 150 200 100
3 250 300 100
4 350 400 100
5 450 510 110
6 570 630 120
7 700 770 140
8 840 920 150
9 1000 1080 160
10 1170 1270 190
11 1370 1480 210
12 1600 1720 240
13 1850 2000 280
14 2150 2320 320
15 2500 2700 380
16 2900 3150 450
17 3400 3700 550
18 4000 4400 700
19 4800 5300 900
20 5800 6400 1100
21 7000 7700 1300
22 8500 9500 1800
23 10500 12000 2500
24 13500 15500 3500

Since the direct measurements of the critical bands are subject to error, the values in this table have been generously rounded.[1]

In his letter "Subdivision of the Audible Frequency Range into Critical Bands", Zwicker states:

"These bands have been directly measured in experiments on the threshold for complex sounds, on masking, on the perception of phase, and most often on the loudness of complex sounds. In all these phenomena, the critical band seems to play an important role. It must be pointed out that the measurements taken so far indicate that the critical bands have a certain width, but that their position on the frequency scale is not fixed; rather, the position can be changed continuously, perhaps by the ear itself."

Thus the important attribute of the Bark scale is the width of the critical band at any given frequency, not the exact values of the edges or centers of any band.

## Conversions

To convert a frequency f (Hz) into Bark use:

${\displaystyle {\text{Bark}}=13\arctan(0.00076f)+3.5\arctan((f/7500)^{2})\,}$

or (Traunmüller, 1990)[4]

${\displaystyle {\text{Bark}}=[(26.81f)/(1960+f)]-0.53\,}$

or (Wang, Sekey & Gersho, 1992)[5]

${\displaystyle {\text{Bark}}=6\sinh ^{-1}(f/600)}$

## References

1. ^ a b Zwicker, E. (1961), "Subdivision of the audible frequency range into critical bands," The Journal of the Acoustical Society of America, Volume 33, Issue 2, pp. 248-248 (1961)
2. ^ Hermes, Dik J. "Sound Perception: The Science of Sound Design". home.ieis.tue.nl. Retrieved 17 September 2015.
3. ^ Julius O. Smith III and Jonathan S. Abel. "The Bark Frequency Scale", CCRMA.Stanford.edu.
4. ^ Traunmüller, H. (1990). "Analytical expressions for the tonotopic sensory scale". The Journal of the Acoustical Society of America. 88: 97. doi:10.1121/1.399849.
5. ^ "Sonification seminar - 10/9/03", CCRMA.Stanford.edu.