- Weber–Fechner law
The

**Weber–Fechner law**attempts to describe the relationship between the physical magnitudes of stimuli and the perceived intensity of the stimuli.Ernst Heinrich Weber (1795–1878) was one of the first people to approach the study of the human response to aphysical stimulus in aquantitative fashion. Gustav Theodor Fechner (1801–1887) later offered an elaborate theoretical interpretation of Weber's findings, which he called simply**Weber's law**, though his admirerswho made the law's name a hyphenate.fact|date=July 2008**Background**Stevens' power law is sometimeswhen considered more accurate and general, although both make assumptions about the measurement of perceived intensity. The Weber–Fechner law assumes thatjust noticeable difference s are additive.L. L. Thurstone uses this assumption for the concept of discriminal dispersion in theLaw of comparative judgment .Fechner believed that Weber had discovered the fundamental principle of mind-body interaction, a mathematical analog of the function

René Descartes once assigned to thepineal gland .**The case of weight**In one of his experiments, Weber gradually increased the weight that a blindfolded man was holding and asked him to respond when he first felt the increase. Weber found that the smallest noticeable difference in weight (the least difference that the test person can still perceive as a difference), was proportional to the starting value of the weight. That is to say, if the weight is 1 kg, an increase of a few grams will not be noticed. Rather, when the mass is increased by a certain factor, an increase in weight is perceived. If the mass is doubled, the threshold called smallest noticeable difference also doubles. This kind of relationship can be described by a differential equation as,

:$dp\; =\; kfrac\{dS\}\{S\}\; ,\; ,!$

where "dp" is the differential change in perception, "dS" is the differential increase in the stimulus and "S" is the stimulus at the instant. A constant factor "k" is to be determined experimentally.

Integrating the above equation gives

:$p\; =\; k\; ln\{S\}\; +\; C,\; ,!$

where $C$ is the

constant of integration , ln is thenatural logarithm .To determine $C$, put $p\; =\; 0$, i.e. no perception; then

:$C\; =\; -kln\{S\_0\},\; ,!$

where $S\_0$ is that threshold of stimulus below which it is not perceived at all.

Therefore, our equation becomes

:$p\; =\; k\; ln\{frac\{S\}\{S\_0.\; ,!$

The relationship between stimulus and perception is logarithmic. This logarithmic relationship means that if a stimulus varies as a

geometric progression (i.e. multiplied by a fixed factor), the corresponding perception is altered in anarithmetic progression (i.e. in additive constant amounts). For example, if a stimulus is tripled in strength (i.e, 3 x 1), the corresponding perception may be two times as strong as its original value (i.e., 1 + 1). If the stimulus is again tripled in strength (i.e., 3 x 3 x 1), the corresponding perception will be three times as strong as its original value (i.e., 1 + 1 + 1). Hence, for multiplications in stimulus strength, the strength of perception only adds.This logarithmic relationship is valid, not just for the sensation of weight, but for other stimuli and our sensory perceptions as well.

In addition, the mathematical derivations of the torques on a simple beam balance produce a description that is strictly compatible with Weber's law (see [

*http://cogprints.org/4094/ link1*] or [*http://www.bio-balance.com/Ref.htm link2*] ).**The case of vision**The eye senses

brightness logarithmically. Hencestellar magnitude is measured on a logarithmic scale. This magnitude scale was invented by the ancient Greek astronomerHipparchus in about 150 B.C. He ranked the stars he could see in terms of their brightness, with 1 representing the brightest down to 6 representing the faintest, though now the scale has been extended beyond these limits. An increase in 5 magnitudes corresponds to a decrease in brightness by a factor of 100.**The case of sound**Still another logarithmic scale is the

decibel scale ofsound intensity. And yet another is pitch, which, however, differs from the other cases in that the physical quantity involved is not a "strength".In the case of perception of pitch, humans hear pitch in a logarithmic or geometric ratio-based fashion: For notes spaced equally apart to the human ear, the frequencies are related by a multiplicative factor. For instance, the frequency of corresponding notes of adjacent octaves differ by a factor of 2. Similarly, the perceived difference in pitch between 100 Hz and 150 Hz is the same as between 1000 Hz and 1500 Hz. Musical scales are always based on geometric relationships for this reason. Notation and theory about music often refers to pitch intervals in an additive way, which makes sense if one considers the logarithms of the frequencies, as $log(a\; imes\; b)=log\; a+log\; b$.

Loudness : Weber's law does not quite hold for loudness. It is a good approximation for higher amplitudes, but not for lower amplitudes. This is usually referred to as "near miss" to Weber's law.Fact|date=June 2008**See also***

Stevens' power law

*Sone

*nervous system

*human nature **References***Jianhong (Jackie) Shen and Yoon-Mo Jung (2006), Appl. Math. Optim., 53(3):331-358, [

*http://www.springerlink.com/content/172467181w543245/?p=cab484b467b64e729a4666703e273f95&pi=3 "Weberized Mumford-Shah model with Bose-Einstein photon noise"*] (Integrating Weber's Law*quantitatively*into modern computational vision and image processing).

*Jianhong (Jackie) Shen (2003), Physica D: Nonlinear Phenomena, 175(3/4):241-251, [*http://dx.doi.org/10.1016/S0167-2789(02)00734-0 "On the foundations of vision modeling I. Weber's law and Weberized TV (total variation) restoration".*]

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**Weber-Fechner law**— Vėberio ir Fechnerio dėsnis statusas T sritis fizika atitikmenys: angl. Weber Fechner law vok. Weber Fechnersches Gesetz, n rus. закон Вебера Фехнера, m pranc. loi de Weber et Fechner, f … Fizikos terminų žodynas**Weber-Fechner law**— We·ber Fech·ner law web ər fek nər , vā bər fek nər n an approximately accurate generalization in psychology: the intensity of a sensation is proportional to the logarithm of the intensity of the stimulus causing it called also Fechner s law… … Medical dictionary**Weber-Fechner law**— noun (psychophysics) the concept that the magnitude of a subjective sensation increases proportional to the logarithm of the stimulus intensity; based on early work by E. H. Weber • Syn: ↑Fechner s law • Topics: ↑psychophysics • Hypernyms: ↑law,… … Useful english dictionary**Weber's law**— or Weber Fechner law In psychophysics, a historically important law quantifying the perception of change in a given stimulus. Originated by the German physiologist Ernst Heinrich Weber (1795–1878) in 1834 and elaborated by his student Gustav… … Universalium**Weber's law**— We·ber s law web ərz , vā bərz n an approximately accurate generalization in psychology: the smallest change in the intensity of a stimulus capable of being perceived is proportional to the intensity of the original stimulus E. H. Weber see WEBER … Medical dictionary**law of weber-fechner**— Usage: usually capitalized W&F : weber fechner law … Useful english dictionary**ley de Weber-Fechner**— Eng. Weber Fechner law Ver ley de Fechner … Diccionario de oftalmología**Law**— /law/, n. 1. Andrew Bonar /bon euhr/, 1858 1923, English statesman, born in Canada: prime minister 1922 23. 2. John, 1671 1729, Scottish financier. 3. William, 1686 1761, English clergyman and devotional writer. * * * I Discipline and profession… … Universalium**Weber (disambiguation)**— Weber is a surname.Weber may also refer to:; Places * Weber City, Virginia * South Weber, Utah * Weber County, Utah * Weber Street, (pronounced weeber ) is a main city street in Waterloo, Ontario * Weber, New Zealand, in the region of Hawke Bay;… … Wikipedia**Law of comparative judgment**— The law of comparative judgment was conceived by L. L. Thurstone. In modern day terminology, it is more aptly described as a model that is used to obtain measurements from any process of pairwise comparison. Examples of such processes are the… … Wikipedia