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
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
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
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
(i) Weber's law holds very approximately for a limited range of
(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.