Talk:Weber–Fechner law

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[edit] 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 I (in watts per square metre, W \over m^2), the (intensity) level is

b = 10 \log \left( {I \over I_0} \right) \mbox{dB}

where I0 is the threshold of hearing, and where log is the logarithm base 10.

For simplicity, consider just a pure tone (sine wave) of 1000 Hz; then I_0 = 10^{-12} \frac{W}{m^2}, and the unit dB is also called phon. According to Weber-Fechner, doubling the level b should mean doubling the subjective loudness. However, experiments show that to double the subjective loudness, one should multiply the intensity I by 10, or equivalently increase the level b 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

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}

where 0.301 = log2. Choosing k = \frac{1}{16}, the unit of this measure is called "sones".

Accoring to Weber-Fechner, the following should be a doubling sequence in terms of subjective loudness:

Weber-Fechner doubling sequence
intensity I level b subjective loudness L
3 \cdot 10^{-12} \frac{W}{m^2} 5 dB 0.0884 sones
1 \cdot 10^{-11} \frac{W}{m^2} 10 dB 0.125 sones
1 \cdot 10^{-10} \frac{W}{m^2} 20 dB 0.25 sones
0.00000001 \frac{W}{m^2} 40 dB 1 sones
0.0001 \frac{W}{m^2} 80 dB 16 sones
10000 \frac{W}{m^2} 160 dB 4096 sones

But the experimental results give the following doubling sequence instead:

Experimental doubling sequence
intensity I level b subjective loudness L examples
1 \cdot 10^{-12} \frac{W}{m^2} 0 dB 0.0625 sones limit of hearing
1 \cdot 10^{-11} \frac{W}{m^2} 10 dB 0.125 sones  
1 \cdot 10^{-10} \frac{W}{m^2} 20 dB 0.25 sones  
1 \cdot 10^{-9} \frac{W}{m^2} 30 dB 0.5 sones  
0.00000001 \frac{W}{m^2} 40 dB 1 sones ppp
0.0000001 \frac{W}{m^2} 50 dB 2 sones pp
0.000001 \frac{W}{m^2} 60 dB 4 sones p
0.00001 \frac{W}{m^2} 70 dB 8 sones  
0.0001 \frac{W}{m^2} 80 dB 16 sones f
0.001 \frac{W}{m^2} 90 dB 32 sones ff
0.01 \frac{W}{m^2} 100 dB 64 sones fff
0.1 \frac{W}{m^2} 110 dB 128 sones  
1 \frac{W}{m^2} 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)

[edit] 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)
I didn't think such specific knowledge about musical scale construction was relevant, so I replaced it with a more general relationship of the law to music practice. Rainwarrior 23:41, 28 January 2006 (UTC)

[edit] Keep the economics stuff

Crucial to understanding marginalism. Christofurio

[edit] "Intensity"

I'm trying to figure out the origin of the bel unit, now most commonly used as a decibel. I've discovered that it was originally derived from Fechner's law. This article describes it as so:

Fechner’s law can be stated as follows:
Ψ = Klog(φ + φ0)
where Ψ (Greek letter psy) is the magnitude of sensation, K is a constant that varies with sensory modality (e.g., vision vs. hearing), φ0 (Greek letter phi) is the magnitude of stimulation at threshold, and φ is the magnitude of stimulation above threshold. Note that sensation requires psychological measurement and stimulation requires physical measurement.
The bel modified Fechner’s law in an important way: the summation within the parentheses was replaced by a division, making the expression a ratio. This allowed use of φ0 (or something very much like it) as a reference quantity for level. Originally, the constant K was given a value of 1. Because the result (in bels) was too small for practical applications, K was later changed to 10. Hence, the unit of level was changed from the bel to the decibel.

Physicists are sticklers for the dB being only used for 10·log intensity/power ratios (and not 20·log amplitude/voltage/pressure ratios). Since Fechner's law is about "perceived intensity", does it really refer to a ratio of a specific type of unit? — Omegatron 17:29, 6 January 2006 (UTC)

[edit] Good question Omegatron

Fechner's interpretation depends on having some unit of sensation intensity. In my view, this gets lost by proceeding directly to an expression of the so-called law in terms of ratios. The implied unit is more difficult to interpret than a unit of a physical quantity. Let P be a unit of perceptual intensity. Let Pi = Pi / P be the measure of the posited sensation intensity associated with a stimulus whose physical magnitude is Si = Si / S. I'm using non-italicized symbols to represent quantities rather than ratios of quantities (i.e. rather than measurements in some unit). I'm using italicized symbols only for ratios of quantities; i.e. for measurements in which some unit is implicit.

Making the perceptual unit explicit, according to Fechner's "law"

P_i={\frac{\mathrm{P_i}}{\mathrm{P}}} = K \ln S_i

I think your question is whether it is possible to obtain measurements of sensation intensity relative to a fixed unit such as P? If so, I think the answer is far from clear as things currently stand.

There is only stochastic information about sensation intensity, such as the proportion of occasions on which differences between stimuli of different magnitudes are noticed under specified conditions. Thurstone argued that Weber's law and Fechner's law are only equivalent if the so-called discriminal dispersions are constant, which gives a unit (see law of comparative judgment). What this amounts to is that the sensation intensity associated with the so-called JND is the unit (or equivalently some multiple of this intensity is the unit). For one thing, why should we be able to obtain sensory units without controlling conditions when we cannot obtain physical units without using instruments deliberately designed to measure in a particular unit under controlled conditions? In my view it is still very much an open question. smhhms 06:40, 7 January 2006 (UTC)

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