Weber-Fechner Law

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The Weber - Fechner law describes the human perception of various physical stimuli. Ernst Heinrich Weber (1795-1878), was one of the first people to quantitatively study the human response to a physical stimulus. In one of his classic 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 response was proportional to a relative increase in 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 in increased by a certain factor, an increase in weight is perceived. If the mass is doubled, the threshold is also doubled. 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 equstion

with C is the constant of integration, ln is the natural logarithm.

To determine C, put p = 0, i.e. no perception; then

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

Therefore, our equation becomes

The relationship between stimulus and perception is logarithmic. This logarithmic relationship means that if the perception is altered in an arithmetic progression (.ie. add constant amounts) the corresponding stimulus varies as a geometric progression (i.e. multiply by a fixed factor).

The point is that this logarithmic relationship is valid, not just for the sensation of weight, but for other stimulii as well. Take the case of vision. The eye senses brightness logarithmically. Hence stellar magnitude is measured in a logarithmics scale. This magnitude scale was invented by the ancient Greek astronomer Hipparchus 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 thought now it has been extended beyond these limits. An increase in 5 magnitudes corresponds to a decrease in brightness by a factor 100. Still another logarithmic scale is the decibel scale of sound intensity. And so is pitch. In the case of perception of pitch, the frequency of the sound gets multiplied by a factor. As it turns out this factor is (twelfth root of 2). This relationship was discovered by Pythagoras. So the frequency of the A# note is the frequency of A times 12th root of 2. The frequency of corresponding notes of adjacent octaves differ by a factor of 2. Weber's findings were later popularized by Gustav Theodor Fechner (1801-1887) and hence the name.