# Talk:Weber–Fechner law

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## WTF is "S"

It's difficult to interpret any of these equations, since S is never defined. — Preceding unsigned comment added by 129.170.30.126 (talk) 20:16, 15 August 2017 (UTC)

## What IS the law???

I give up. I've read the page and I still couldn't tell you what the W-F law actually *is*. The article has an awful lot of discussion *about* the law, but I don't see anything that actually states what the law is. The law of gravity is that bodies are attracted to each other proportional to the product of their masses and inversely proportional to the distance between them, or F = Gm1m2/s^2. Murphy's law is that anything that can go wrong will go wrong. The Weber-Fechner law is....????? —Preceding unsigned comment added by 131.107.0.73 (talk) 19:36, 26 August 2008 (UTC)

As follows: if a stimulus varies as a geometric progression (i.e. multiplied by a fixed factor), the corresponding perception is altered in an arithmetic progression (i.e. in additive constant amounts). (See "The case of weight" section)Lestrade (talk) 20:15, 12 September 2009 (UTC)Lestrade
I agree with that but believe it should be called "Fechner's law" or "Fechner's scale": That scale does not directly derive from Weber's law, Weber objected to being part of it, and Fechner (who was a physicist by training) said there was no way of verifying it. The statement that "Fechner added to the confusion" from the Penguin dictionary is nonsense, I believe; I could not find it anywhere in Fechner's writing. When Fechner referred to "Weber's law", he indeed referred to Weber's work, i.e. meant the ratio law (delta-S/S= const.). So my answer is: There is no "W-F law". There is Weber's law and there is Fechner's law (or scale)--Strasburger (talk) 12:12, 3 April 2013 (UTC)

## 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 ${\displaystyle I}$ (in watts per square metre, ${\displaystyle W \over m^{2}}$), the (intensity) level is

${\displaystyle b=10\log \left({I \over I_{0}}\right){\mbox{dB}}}$

where ${\displaystyle I_{0}}$ is the threshold of hearing, and where ${\displaystyle \log }$ is the logarithm base 10.

For simplicity, consider just a pure tone (sine wave) of 1000 Hz; then ${\displaystyle I_{0}=10^{-12}{\frac {W}{m^{2}}}}$, and the unit dB is also called phon. According to Weber-Fechner, doubling the level ${\displaystyle b}$ should mean doubling the subjective loudness. However, experiments show that to double the subjective loudness, one should multiply the intensity ${\displaystyle I}$ by 10, or equivalently increase the level ${\displaystyle b}$ 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

${\displaystyle L=k\cdot 2^{0.1\cdot b}=k\cdot 2^{\log \left({I \over I_{0}}\right)}=k\cdot \left({I \over I_{0}}\right)^{0.301}}$

where ${\displaystyle 0.301=\log 2}$. Choosing ${\displaystyle k={\frac {1}{16}}}$, 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 ${\displaystyle I}$ level ${\displaystyle b}$ subjective loudness ${\displaystyle L}$
${\displaystyle 3\cdot 10^{-12}{\frac {W}{m^{2}}}}$ 5 dB 0.0884 sones
${\displaystyle 1\cdot 10^{-11}{\frac {W}{m^{2}}}}$ 10 dB 0.125 sones
${\displaystyle 1\cdot 10^{-10}{\frac {W}{m^{2}}}}$ 20 dB 0.25 sones
${\displaystyle 0.00000001{\frac {W}{m^{2}}}}$ 40 dB 1 sones
${\displaystyle 0.0001{\frac {W}{m^{2}}}}$ 80 dB 16 sones
${\displaystyle 10000{\frac {W}{m^{2}}}}$ 160 dB 4096 sones

But the experimental results give the following doubling sequence instead:

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

First of all, I don't see any discrepancy or contradiction in the tables that you show. Those rows which are present in both tables, match up exactly. Those which are present in a single table only, just confirm the intuition of the law.
Second, you may be misunderstanding the Weber-Fechner formulation. You say "According to Weber-Fechner, doubling the level ${\displaystyle b}$ should mean doubling the subjective loudness.". It is not. The article explains this almost directly: the perception of "next level" (intensity increment, additive increase as in an arithmetic progression) is caused by a magnitude increase (multiplying intensity by a constant factor, multiplicative increase as in the geometric progression) in the underlying physical quantity. This means A + 6 db sounds twice as loud as A db (power law); not (!) that A · 6 db sounds twice as loud.
Third, please reconsider the advice to abstain from Original Research on Wikipedia. This harms your own work (since it will not reviewed and scrutinized just as well as in "normal" science), and may be misleading or confusing for readers.
In a conclusion, I don't see any disagreement of "modern theory of sound measurement" with Weber-Fechner law that you mention.
All the best from the future, --213.227.200.124 (talk) 01:32, 30 December 2014 (UTC)
You're responding to a 9-year-old irrelevant post. Dicklyon (talk) 01:37, 30 December 2014 (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)
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)

## Keep the economics stuff

Crucial to understanding marginalism. Christofurio

## "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:
${\displaystyle \Psi =K\log(\phi +\phi _{0})}$
where ${\displaystyle \Psi }$ (Greek letter psy) is the magnitude of sensation, K is a constant that varies with sensory modality (e.g., vision vs. hearing), ${\displaystyle \phi _{0}}$ (Greek letter phi) is the magnitude of stimulation at threshold, and ${\displaystyle \phi }$ 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 ${\displaystyle \phi _{0}}$ (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)

### 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 ${\displaystyle \mathrm {P} }$ be a unit of perceptual intensity. Let ${\displaystyle P_{i}=\mathrm {P_{i}/P} }$ be the measure of the posited sensation intensity associated with a stimulus whose physical magnitude is ${\displaystyle S_{i}=\mathrm {S_{i}/S} }$. 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"

${\displaystyle P_{i}={\frac {\mathrm {P_{i}} }{\mathrm {P} }}=K\ln S_{i}}$

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)

## Conflicting Information?

This page and the page on Munsell's color notation conflict in their description of the perceived increase in luminance. This may be valid because the term used here is 'brightness' while the term used there is 'lightness,' but on the Munsell page, lightness changes with respect to the cube root of the actual intensity, rather than the logarithmic increase mentioned here. It should be noted that a cube root trend appears similar to a log, and that this is apparently used in CIELAB color space to help define the color axis. 74.74.223.195 (talk) 20:28, 11 July 2008 (UTC)

Also, the page on Visual_magnitude says that it is a misconception that the eye responds on a logarithmic scale, but this page says it does. What's with that? 122.169.103.240 (talk) —Preceding undated comment was added at 11:19, 23 October 2008 (UTC).

I suspect the eye really responds in a way that is closer to being logarithmic than linear, but that is in fact neither. Probably both articles should be modified to reflect this.--Noe (talk) 15:17, 23 October 2008 (UTC)

## Color Space and Sound Timber

Regarding Color: I'm wondering about what data anyone might have relating color space perception to Weber's law? The article on color theory might have some useful insights and I will do some research, but if any editors are able to address perception of color as it does, or does not, follow the logarithmic scale as with loudness and brightness.

Regarding sound timbre, tone and metrics: How might this be addressed in terms of Weber's law, or related concepts. Timbre perception is complex having to do with harmonic content and so on. The question of signal to noise ratio is also something I'm wondering about, that is, how to relate Weber's law and related concepts to sound quality. With regard to perception of metrical units and rate of units, otherwise known as meter and tempo in music, does Weber's law apply to our perception of these elements?

I may not even be formulating these questions all that well, but, perhaps you have some insights to share which will help my own understanding and moreover help the article expand in a useful direction. Calicocat (talk) 06:05, 20 July 2008 (UTC)

## Pitch physical quality is not "strength"

I'm not sure what this means. It seems inaccurate. It should be (re)considered, although I don't have time to parse it just this minute. 129.81.44.124 (talk) 15:59, 26 August 2010 (UTC)DGHyman

I suspect I wrote that, years ago, hoping someone else would change it to express my thought better. My point is that WF-law in most cases apply to quantities that are directly related to an energy, power, amplitude, intensity or the like. Not so with pitch; you can have a deep sound with high intensity and a high-pitched one with low intensity - or vice cersa.-- (talk) 18:25, 26 August 2010 (UTC)

## Left to right

I would presume that mental number lines have no inherent bias to be left to right, but predominantly take on that direction in cultures which write left to right. Has anyone studied this effect in populations that use the Arabic or Hebrew scripts? 4pq1injbok (talk) 02:14, 11 October 2012 (UTC)

## Relationship with the Scoville scale and perception of pungency

I'm too lazy to find reputable references which consider the two (Scoville scale and Weber-Fechner law) together, so here's some funny Finding On The Talk Page: the Scoville scale is clearly more concerned with representing magnitude rather than exact (up to the last digit) measure of capsaicin concentration (which is closely connected to spicyness, heat and, in extremes, pain sensations), and it begs for logarithmic units. — Preceding unsigned comment added by 213.227.200.124 (talk) 01:48, 30 December 2014 (UTC)

## Please someone take a moment to make math formulae easier to understand.

The symbols in the equations currently need to be looked up by a non-mathematician. Understanding would be greatly promoted and encouraged if the formulae had their symbols briefly explained in terms of which quantities each symbol represents. For example if "K" represents a constant that is specific to the different human senses, and that can vary according to which sense it represents ... then saying so right after or before the formula in which "K" appears for the first time in the article would be helpful. Having a simple equation, but for which symbols are not locally defined ... interrupts learning. Thanks. Very interesting article and concepts. — Preceding unsigned comment added by 97.125.86.191 (talk) 18:39, 13 June 2017 (UTC)

## JND really the same as Weber fraction?

From the article: "if the weight of 105 g can (only just) be distinguished from that of 100 g, the JND (or differential threshold) is 5 g. […] this minimum required fractional increase (of 5/100 of the original weight) is referred to as the "Weber fraction""

I'm no expert, but the article on JNDs says that it's the same as the difference limen. Meanwhile, the literature seems to define the Weber fraction as the relative standard deviation of the error, and in a 1975 paper by Getty, I read that "the difference limen is a constant fraction (.68) of the value of the standard deviation". So that seems to mean that JND = Weber fraction * 0.68. In our example, I would therefore expect the Weber fraction to be 5% / 0.68 = 7.35%. Could somebody clarify?

--Forlornturtle (talk) 14:53, 30 August 2018 (UTC)

## C-class?

I think that this should be C class. Anyone with me? — Preceding unsigned comment added by Just4science! (talkcontribs) 10:49, 5 March 2019 (UTC)