10. The most direct and most easily utilizable results derived from these methods of comparison are given by the second method, or the method of minimal differences as it is called.
The difference between the Physical stimuli which corresponds to the just noticeable difference between psychical quantities is called the difference-threshold of the stimulus. The stimulus from which the resulting psychical process, for example, a sensation, can be just apperceived, is called the stimulus-threshold. Observation shows that the difference-threshold of the stimulus increases in proportion to the distance from the stimulus-threshold, in such a way that the relation between the difference-threshold and the absolute quantity of the stimulus, or the relative differ threshold, remain constant. If, for example, a certain sound whose intensity is 1 must be increased 1/3 in order that the sensation may, be just noticeably greater, one whose intensity is 2 must be increased 2/3, one 3, 3/3, etc., to reach the difference-threshold. This law is called Weber's law, after its discoverer E. H. Weber. It is easily understood when we look upon it as a law of apperceptive comparison. From this point of view it mull obviously be interpreted to mean that psychical quantities are compared according to their relative values. [p. 255]
This view that Weber's law is an expression of the general law of the relativity of psychical quantities, assumes that the psychical quantities that are compared, themselves increase in proportion to their stimuli within the limits of the validity of the law. It has not yet been possible to demonstrate the truth of this assumption on its physiological side, on account of the difficulties of measuring exactly the stimulation of nerves and sense-organs. Still, we have evidence in favor of it in the psychological experience that in certain special cases, where the conditions of observation lead very naturally to a comparison of absolute differences in quantity, the absolute difference threshold, instead of the relative threshold, is found to be constant.
We have such a case, for example, in the comparison, within wide limits, of minimal differences in pitch. Then, too, in many cases where large differences in sensations are compared according to the third method described above (p. 254), equal absolute stimuli-differences, not relative differences, are perceived as equal. This shows that apperceptive comparison follows two different principles under different conditions: a principle of relative comparison that finds its expression in Weber's law and must be regarded as the more general, and a principle of absolute comparison of differences which takes the place of the first under special conditions which favor such a form of apperception.
10 a. Weber's law has been shown to hold, first of all, for the intensity of sensations and then, within certain limits, for the comparison of extensive compounds, especially temporal ideas, also, to some extent, for spacial ideas of sight and for motor ideas. On the other hand, it does not hold for the spacial ideas of external touch, obviously on account of the complexity of the local signs (p. 105); and it can not be verified for sensational qualities. In fact, for the comparison of pitches the absolute, not the relative difference-threshold is constant within wide limits. Still, the scale of tonal intervals is relative, for every interval corresponds to a [p. 256] certain ratio between the number of vibrations (for example, an octave 1 : 2, a fifth 2 : 3, etc.). This is probably due to the relationship between clangs which is due to the relation of the fundamental tone to its overtones (comp. p.
95 sq.). Even where an absolute comparison takes place instead of a comparison according to Weber's law of relativity, we must not, of course, confuse this with the establishment of an absolute measure. That would presuppose an absolute unit, that is, the possibility of finding a constant standard, which, as noted above (p. 253), is in the psychical world impossible. Absolute comparison must take the form of a recognition of the equality of equal absolute difference. This is possible in the various single cases without a constant unit. Thus, for example, we compare two sensational lines AB and BC according to their relative values, when we think in both cases of the relation of the upper to the lower extreme sensation. In such a case we judge AB and BC to be equal when B/A = C/B (Weber's law). On the other hand, we compare A B and B C according to their absolute values when the difference between C and B in the single sensational dimension in question appears equal to that between B and A, that is, when C - B = B - A (law of proportionality). Weber's law has sometimes been regarded as the expression of the functional relation between sensation and stimulus, and it has been assumed that the law holds for infinitely small changes on both sides. On this basis there has been given to it the mathematical form of the logarithmic function: sensation increases in proportion to the logarithm of the stimulus (Fechner's psycho-physical law).
