A review of work done on Weber's law as applied to the intensity of sound

Abstract

The following summary recapitulates the main points put forward in my review of Weber's law applied to the intensity of sound and the accompanying discussion. The topics dealt with are numbered to correspond with the division into sections. Certain general conclusions, not previously made, are also included here.

I. Weber's law, or the Weber-Fechner law, if the name is to survive at all, must be interpreted on a psychological basis. That is to say, it must deal in some way with events of consciousness, since Weber's original statement dealt with the act of comparison, and Fechner's interest was in the relation between objective and subjective phenomena. Johnson's objection (152) that Fechner's formulation involved an undefined variable (sensation) is, on this view, invalid, since common experiences recognizes the real occurrence of sensation: as Boring (77) puts it, 'Everybody known what γ ( Fechner's S) is, independently of ß ( Fechner's R)', whereas what (ß itself is cannot be said with certainty. It is therefore more logical to define stimulus in terms of sensation than sensation in terms of stimulus. A psycho - physical interpretation must further be ruled out, since practically no -one now believes (though Kümmel (164) is a comparatively recent exception) that every smallest alteration of stimulus produces a sensation change.

The possibility of a logarithmic relation between S and R, in whatever sense they may be interpreted, stands or falls not so much by the equality of just noticeable differences, but by the measurability of sensation. Although sensation is certainly not measurable in any sort of objective units, the general opinion seems now to be that in some sense or other sensation is measurable. This seems to have been the conclusion drawn at the Leicester meeting of the British Association (1932). In the symposium already quoted, Houston (149) stated that he personally was certain that sensation could be measured. The view that sensory discrimination is based on the occurrence or absence of awareness of sameness or difference, and that judgements of sameness or difference may be treated by the Law of Error was represented at that symposium by Bartlett (60). A synthesis of this view with a recognition of the effects of absolute 'bigness' and 'littleness' characterizing sensory events, would seem to afford a pragmatic justification of the measurement of sensation. The results of such measurement, if subjected to mathematical analysis, may or may not be found to take the form of a logarithmic equation.

II. In the investigation of sensibility to intensity differences of sound many difficulties due to the physical properties of sound waves, and to peculiarities of the hearing mechanism, are encountered. Divergent results may be due to a variety of factors very hard to control; Sivian and White (229) call attention in particular to the possibility of effects of wave motion in the meatus, and of diffraction caused by the listener's head.

The best results are probably to be obtained by using some apparatus which converts acoustical energy into equivalent units of electric current. In any case it, is advisable to use the decibel scale, since the response of the ear is at all events approximately logarithmic, so that simple units of pressure or energy flow soon reach unmanageable proportions.

III. Certain other psychological properties of sound must also be considered, as well as certain phenomena which may be said to be of an intermediate nature. In dealing with any sound other than a pure tone, masking phenomena must be recognized, and compensated for if absolute measurement by summation is required. These phenomena, as Wegel and Lane (272) point out, are largely responsible for the non-occurrence of a linear relation at high intensities between sound pressure and response of the ear.

In addition, sensory discrimination is affected by a large number of purely subjective factors, such as those listed by Fernberger (114) as governing equality judgements, e.g., attention, instructions, comprehension, temperament, etc. Some of the phenomena of hearing seem to have withstood all attempts at explanation in physical terms; on the other hand, one cannot agree with Watt(269) , who stated that all sensory experiences can be accounted for in systematic terms without recourse to the discoveries of physics or physiology.

IV. A summary of work on auditory intensive thresholds has already been given (pp. 152-4). At best, Δ R/R for sound intensity is constant only for a limited range.

V. Direct estimation or judgement of loudness in absolute units provides rather inconclusive results. Constancy is usually obtained within the bounds of an individual investigation, even among the different subjects participating in that investigation. On the other hand, agreement between investigators is rather exceptional, except in very general terms. Analysis of results, however, seems now to favour an exponential equation relating loudness to intensity of sound measured in db. The decibel scale, though useful in its own way is very misleading if interpreted as giving numerical indications of loudness.

Measurement may also be accomplished by a method of balancing loudness, preferably against a definite reference tone. This is most difficult at the extremes of the frequency range, where the number of distinguishable intensities is small, and consequently, as Tucker (257) points out, it may not be possible to match with the reference tone.

VI. Many writers have reduced the status of Weber's law to that of a special case of a generalized relativity law. Various attempts have also been made to determine a more accurate mathematical formulation representing sensory discrimination in the different fields.

On the whole it is best to take Weber's law from the phenomenological point of view - i.e., as an approximation best describing the general form of a number of observed facts. Thus, we may, for example follow Lloyd Morgan (181), who proposed the modification: 'For constant increments of sensation, the concomitant increments of stimulus are in geometrical progression', introducing the qualification 'approximately'. Better still, it is possible to eliminate intensity from the problem altogether, and consider Weber's law as expressing a relation between different modes of consciousness (cf. Gatti (127), Cobb (98) ).

The problem has also been approached from a physiological angle; In this cases sound intensity discrimination is bound up with the theory of hearing. Since it is not possible to work with living human ears, experimental work has been confined to work with animals, and with models of an 'artificial ear', (e.g. by Békésy (63) ), and Langenheck (167) who showed the limitations imposed upon this method by the use of imperfect materials.), The results of recent work seem to indicate that some theory based on the all-or-none principle must hold good, but work on the different sense-departments still awaits synthesis. The recent discoveries alluded to have rendered many standard and semi-popular books (e.g. that of Ogden (196)) out-of-date, and a complete survey of recent work on hearing would no doubt prove valuable.

VII. The practical implications of the difference threshold are numerous, and in many cases too obvious to attract attention. A case in point is comprehension of speech, discussed by Marx (30).

VIII. The original experiments here described lead to the following conclusions: (i) Weber's law holds very approximately for a limited range of sound intensities. (ii) The deviations are of a continuous nature. This bears out the findings of Kenneth and Thouless (23), and Riesz (36). Upper deviations seem to have been conclusively demonstrated. (iii) Individual differences are very noticeable, and often even surprising (cf. Weiss (274) ) , and day-to-day variations for the same observer may also occur. (iv) All the subjects tended, to quote a phrase used by Banister (56), to objectify their experience to a high degree. Thus in the watch-tick experiment, the criterion was often intuited nearness rather than apparent loudness. In the phonometer experiment, again, one subject (B) found it helpful to think of the sounds as produced by hammer -blows of varying force; another subject (C) equated the tuning -fork tones with the energy he would have required to sing them.

(v) Different statistical procedures yield widely different thresholds. It is doubtful whether the labour of the Constant method ever justifies results. Interpolation, as advocated by Newhall (191, 192) is probably sufficient for most purposes. Thresholds based on Spearman's (295) or Wirth's (see 258)formulae seem to yield too low values.