where, dp is the differential change in perception, dS is the differential increase in the stimulus and S is the stimulus at the instant. k is a constant factor that is to be determined experimentally.
Integrating the above eqn.,
C is the constant of integration, ln is the natural logarithm
To determine C, put p = 0, ie. there is no perception, then
where is that threshold of stimulus below which it is not perceived at all.Therefore, our eqn. 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 (ie. multiply by a fixed factor). The point is that this logarithmic relationship is valid for not just the sensation of weight alone, 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.