The

**Weber–Fechner law** is a confusing term, because it combines two different laws. Some authors use the term to mean

**Weber's law**, and others

**Fechner's law**. Fechner himself added confusion to the literature by calling his own law Weber's law.

Ernst Heinrich WeberErnst Heinrich Weber was a German physician who is considered one of the founders of experimental psychology.Weber studied medicine at Wittenberg University...

(1795–1878) was one of the first people to approach the study of the human response to a

physical stimulusIn physiology, a stimulus is a detectable change in the internal or external environment. The ability of an organism or organ to respond to external stimuli is called sensitivity....

in a

quantitativeIn the social sciences, quantitative research refers to the systematic empirical investigation of social phenomena via statistical, mathematical or computational techniques. The objective of quantitative research is to develop and employ mathematical models, theories and/or hypotheses pertaining to...

fashion. His law states that the just-noticeable difference between two stimuli is proportional to the magnitude of the stimuli.

Gustav Theodor FechnerGustav Theodor Fechner , was a German experimental psychologist. An early pioneer in experimental psychology and founder of psychophysics, he inspired many 20th century scientists and philosophers...

(1801–1887) later offered an elaborate theoretical interpretation of Weber's findings, in which he attempted to describe the relationship between the physical magnitudes of

stimuliIn physiology, a stimulus is a detectable change in the internal or external environment. The ability of an organism or organ to respond to external stimuli is called sensitivity....

and the perceived intensity of the stimuli. Fechner's law states that subjective sensation is proportional to the logarithm of the stimulus intensity.

## Derivation of Fechner's law for weight perception

Weber found that the just noticeable difference (jnd) between two weights was approximately proportional to the mass of the weights. Thus, if 105 g can (only just) be distinguished from 100 g, the jnd (or differential threshold) is 5 g. If the mass is doubled, the differential threshold also doubles to 10 g, so that 210 g can be distinguished from 200 g. This kind of relationship can be described by a differential equation as,

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

where

is the constant of integration, ln is the

natural logarithmThe natural logarithm is the logarithm to the base e, where e is an irrational and transcendental constant approximately equal to 2.718281828...

.

To determine

, put

, i.e. no perception; then subtract

from both sides and rearrange:

where

is that threshold of stimulus below which it is not perceived at all.

Substituting this value in for

above and rearranging, our equation becomes:

The relationship between stimulus and perception is

logarithmicA logarithmic scale is a scale of measurement using the logarithm of a physical quantity instead of the quantity itself.A simple example is a chart whose vertical axis increments are labeled 1, 10, 100, 1000, instead of 1, 2, 3, 4...

. This logarithmic relationship means that if a stimulus varies as a

geometric progressionIn mathematics, a geometric progression, also known as a geometric sequence, is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed non-zero number called the common ratio. For example, the sequence 2, 6, 18, 54, ... is a geometric progression...

(i.e. multiplied by a fixed factor), the corresponding perception is altered in an

arithmetic progressionIn mathematics, an arithmetic progression or arithmetic sequence is a sequence of numbers such that the difference between the consecutive terms is constant...

(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. The mathematical derivations of the torques on a simple beam balance produce a description that is strictly compatible with Weber's law (see

link1 or

link2).

Fechner did not conduct any experiments on how perceived heaviness increased with the mass of the stimulus. Instead, he assumed that all jnds are subjectively equal, and argued mathematically that this would produce a logarithmic relation between the stimulus intensity and the sensation. These assumptions have both been questioned. Most researchers nowadays accept that a power law is a more realistic relationship, or that a logarithmic function is just one of a family of possible functions.

Other sense modalities provide only mixed support for either Weber's law or Fechner's law.

## The case of vision

The eye senses

brightnessBrightness is an attribute of visual perception in which a source appears to be radiating or reflecting light. In other words, brightness is the perception elicited by the luminance of a visual target...

approximately logarithmically over a moderate range (but more like a

power lawA power law is a special kind of mathematical relationship between two quantities. When the frequency of an event varies as a power of some attribute of that event , the frequency is said to follow a power law. For instance, the number of cities having a certain population size is found to vary...

over a wider range), and stellar magnitude is measured on a logarithmic scale.

This magnitude scale was invented by the ancient Greek astronomer

HipparchusHipparchus, the common Latinization of the Greek Hipparkhos, can mean:* Hipparchus, the ancient Greek astronomer** Hipparchic cycle, an astronomical cycle he created** Hipparchus , a lunar crater named in his honour...

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.

Modern researchers have attempted to incorporate such perceptual effects into mathematical models of vision.

## The case of sound

Weber's law does not quite hold for

loudnessLoudness is the quality of a sound that is primarily a psychological correlate of physical strength . More formally, it is defined as "that attribute of auditory sensation in terms of which sounds can be ordered on a scale extending from quiet to loud."Loudness, a subjective measure, is often...

. It is a fair approximation for higher intensities, but not for lower amplitudes. Similarly, the law applies loosely to pitch perception. The

mel scaleThe mel scale, named by Stevens, Volkman and Newman in 1937is a perceptual scale of pitches judged by listeners to be equal in distance from one another. The reference point between this scale and normal frequency measurement is defined by assigning a perceptual pitch of 1000 mels to a 1000 Hz...

is a proposed perceptual scale of pitch that maps sound frequency change to perceived pitch change. Below 500 Hz, this mapping is near linear, but above 500Hz, the relationship is approximately logarithmic.

The Weber–Fechner law holds approximately for musical

tempoIn musical terminology, tempo is the speed or pace of a given piece. Tempo is a crucial element of any musical composition, as it can affect the mood and difficulty of a piece.-Measuring tempo:...

(the difference between a tempo of 60 beats per minute and 61 bpm is perceived as a much larger difference than between 200bpm and 201bpm).

### "Near miss" of Weber's law in the auditory system

Weber's law does not hold at perception of higher intensities. Intensity discrimination improves at higher intensities. The first demonstration of the phenomena were presented by Riesz in 1928, in the Journal of Acoustical Society of America. This deviation of the Weber's law is known as the "near miss" of the Weber's law. This term was coined by McGill and Goldberg in their paper of 1968 in the Journal of Acoustical Society of America (JASA). Their study consisted of intensity discrimination in pure tones. Further studies have shown that the near miss is observed in noise stimuli as well. Jesteadt et al. in their 1976 paper demonstrated that the near miss holds across all the frequencies, and that the intensity discrimination is not a function of frequency, and that the change in discrimination with level can be represented by a single function across all frequencies.

## The case of numerical cognition

Psychological studies show that numbers are thought of as existing along a mental number line. Larger entries are on the right and smaller entries on the left. It becomes increasingly difficult to discriminate among two places on a number line as the distance between the two places decreases—known as the

*distance effect*. This is important in areas of magnitude estimation, such as dealing with large scales and estimating distances.

See this article

http://johnhawks.net/weblog/reviews/brain/culture/math/dehaene-2008-number-logarithmic-space.html on logarithmic number representation.

## See also

- Stevens' power law
Stevens' power law is a proposed relationship between the magnitude of a physical stimulus and its perceived intensity or strength. It is often considered to supersede the Weber–Fechner law on the basis that it describes a wider range of sensations, although critics argue that the validity of the...

- Sone
The sone was proposed as a unit of perceived loudness by Stanley Smith Stevens in 1936. In acoustics, loudness is the subjective perception of sound intensity...

- Nervous System
The nervous system is an organ system containing a network of specialized cells called neurons that coordinate the actions of an animal and transmit signals between different parts of its body. In most animals the nervous system consists of two parts, central and peripheral. The central nervous...

- Human nature
Human nature refers to the distinguishing characteristics, including ways of thinking, feeling and acting, that humans tend to have naturally....

- Ricco's Law
Several laws describe a human's ability to visually detect targets on a uniform background. One such law is Riccò's law. This law explains the visual relationship between a target area and target contrast required for detection when that target is unresolved...