Talk:Weber-Fechner Law
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Sound intensity
The Weber-Fechner law of logarithmic sensitivity may be valid for some of our senses, but modern theory of sound measurement is in disagreement with it. If the intensity is <math>I<math> (in watts per square metre, <math>W \over m^2<math>), the (intensity) level is
- <math>
b = 10 \log \left( {I \over I_0} \right) \mbox{dB} <math>
where <math>I_0<math> is the threshold of hearing, and where <math>\log<math> is the logarithm base 10.
For simplicity, consider just a pure tone (sine wave) of 1000 Hz; then <math>I_0 = 10^{-12} \frac{W}{m^2}<math>, and the unit "dB" is also called "phones". According to Weber-Fechner, doubling the level <math>b<math> should mean doubling the subjective loudness. However, experiments show that to double the subjective loudness, one should multiply the intensity <math>I<math> by 10, or equivalently increase the level <math>b<math> by 10 dB. It takes 10 violins to sound twice as loud as one violin! (Some sources give a value smaller than 10. The article sone mentions the value 3.16; this discrepancy is due to a misunderstanding - on my part, or on the part of the author of that article; I am not sure.) Therefore, the subjective loudness is better represented by
- <math>
L = k \cdot 2^{0.1 \cdot b} = k \cdot 2^{ \log \left( {I \over I_0} \right) } = k \cdot \left( {I \over I_0} \right) ^{0.301} <math>
where <math>0.301 = \log 2<math>. Choosing <math>k = \frac{1}{16}<math>, the unit of this measure is called "sones".
Accoring to Weber-Fechner, the following should be a doubling sequence in terms of subjective loudness:
| intensity <math>I<math> | level <math>b<math> | subjective loudness <math>L<math> |
|---|---|---|
| <math> 3 \cdot 10^{-12} \frac{W}{m^2} <math> | 5 dB | 0.0884 sones |
| <math> 1 \cdot 10^{-11} \frac{W}{m^2} <math> | 10 dB | 0.125 sones |
| <math> 1 \cdot 10^{-10} \frac{W}{m^2} <math> | 20 dB | 0.25 sones |
| <math> 0.00000001 \frac{W}{m^2} <math> | 40 dB | 1 sones |
| <math> 0.0001 \frac{W}{m^2} <math> | 80 dB | 16 sones |
| <math> 10000 \frac{W}{m^2} <math> | 160 dB | 4096 sones |
But the experimental results give the following doubling sequence instead:
| intensity <math>I<math> | level <math>b<math> | subjective loudness <math>L<math> | examples |
|---|---|---|---|
| <math> 1 \cdot 10^{-12} \frac{W}{m^2} <math> | 0 dB | 0.0625 sones | limit of hearing |
| <math> 1 \cdot 10^{-11} \frac{W}{m^2} <math> | 10 dB | 0.125 sones | |
| <math> 1 \cdot 10^{-10} \frac{W}{m^2} <math> | 20 dB | 0.25 sones | |
| <math> 1 \cdot 10^{-9} \frac{W}{m^2} <math> | 30 dB | 0.5 sones | |
| <math> 0.00000001 \frac{W}{m^2} <math> | 40 dB | 1 sones | ppp |
| <math> 0.0000001 \frac{W}{m^2} <math> | 50 dB | 2 sones | pp |
| <math> 0.000001 \frac{W}{m^2} <math> | 60 dB | 4 sones | p |
| <math> 0.00001 \frac{W}{m^2} <math> | 70 dB | 8 sones | |
| <math> 0.0001 \frac{W}{m^2} <math> | 80 dB | 16 sones | f |
| <math> 0.001 \frac{W}{m^2} <math> | 90 dB | 32 sones | ff |
| <math> 0.01 \frac{W}{m^2} <math> | 100 dB | 64 sones | fff |
| <math> 0.1 \frac{W}{m^2} <math> | 110 dB | 128 sones | |
| <math> 1 \frac{W}{m^2} <math> | 120 dB | 256 sones | limit of pain |
The notations ppp = piano pianissimo, etc., are used in musical scores. Their correspondence to sound levels are approximate only.
--Niels Ø 13:53, Mar 20, 2005 (UTC)
Pythagoras and 12-tone
... in this article there was something about "Pythagoras finding out that every (n+1) tone is the "twelveth root of 2" * (n)tone. that is weird because the 12-tone-western music (that is what this root-thingy revers to i guess) was introduced around 2000 years after Pythagoras died...
- You are right, that is wrong! Greek and other classical theories of music as well as of artistic proportion only involve commensurable quantities, i.e. rational ratios, i.e. quantities where one is a multiple of a fraction of the other. The 12th root of 2 is irrational. Its introduction into music is often attributed to Bach. nø
- I fixed this part, but I don't think I made it very clear. someone else tweak it. - Omegatron 18:53, Jun 29, 2004 (UTC)