One hundred years ago G. T. Fech-ner (1) published the fruits and find-ings of a ten-year labor-an event thatwe- celebrate as the nascence of thediscipline called psychophysics. In thecentury since the Elemente der Psycho-physik first made its stir, the simple butcontroversial logarithmic law that goesby Fechner's name has invaded almostall the textbooks that mention humanreactions to stimuli. It is fitting andproper, therefore, that we should gatherthis year in a symposium to mark theanniversary of these beginnings and toinquire how the issues stand in 1960.

Perhaps the most insistent questionon this 100th anniversary of Fechner'smonumental opus is how its authorcould have known so much and havemade such a wrong guess. (He believedthat, unlike errors in general, errors inperception are independent of the per-ceived magnitude.) Talent, erudition,originality-each of these gifts was hisin generous measure, and he applied hisskills with signal success to several dif-ferent domains. Not only did he createpsychophysics and pioneer in experi-mental esthetics, but he also laid thefoundations for what von Mises (2)later transformed into a well-knowntheory of probability based on the con-cept of a "collective."

But it was Fechner's version of thepsychophysical law that really madehim famous. With it he founded psy-chophysics and sent it off on a curioustangent-a deflection that lasted forthe better part of a century. If we re-gard a hundred years as a long time-and it certainly seems long in the fast-moving evolution of modern science-then Fechner was almost right in hisdefiant forecast of 1877 (3): "TheTower of Babel was never finished be-cause the workers could not reach an

80

understanding on how they shouldbuild it; my psychophysical edifice willstand because the workers will neveragree on how to tear it down."

These words were published 17 yearsafter the appearance of the Elemente.By that time Fechner had had full op-portunity to correct his magnum er-ror, for at least two different argu-ments had by then been made in favorof what has more recently appeared tobe the correct relation between theapparent magnitude of a sensation andthe stimulus that causes it.

1) Brentano (4) had suggested thatWeber's law may hold at both levels:stimulus 0 and sensation A'. In otherwords, A@ = kq, and A; = k'. This initself is not truth, but a simple, if illegit-imate, Fechnerian integration leadsfrom these two equations directly tothe correct general form of the psycho-physical law. It was Fechner himselfwho argued that Brentano's suggestionhad to be wrong, because it wouldentail a power-function relation be-tween stimulus and sensation.

2) Plateau (5) had suggested that,when we vary the illumination on ascene made up of different shades ofgray, it is not the subjective di/ferencesbut the subjective ratios that remainconstant. Plateau therefore conjecturedthat the psychophysical relation mightbe a power function-a conjecture thathe later renounced for a wrong reason.

Because both these suggestions led toa power law, as opposed to a logarith-mic law, Fechner was inspired to writelong and bitterly in his denunciation ofthem. It was asking too much, per-haps, to expect a professor to changehis mind after two decades of devo-tion to an ingenious theory. Moreover,by din;t of his industry and his polem-ics, Fechner succeeded in making the

HEALTH SCIENCES LIBRARY

McMASTER UNIVERSITY

logarithmic function the sole contender,so that- little or nothing was heard ofthe power function for many decades.If a change is now setting in, it is be-cause new techniques have made itplain that on some two dozen sensorycontinua the subjective magnitudegrows as a power function of thestimulus magnitude (6).

It is understandable that Fechnershould fight stubbornly throughout hislater decades to salvage his intellectualinvestment in the thesis that a measureof the uncertainty or variability in asensory discrimination can be used asa unit for the scaling of the psycho-logical continuum. He had sensed theessence of this possibility as he layabed on that famous morning of 22October 1850, and he had put theidea promptly and tenaciously to work.But why should such an unlikely notionhave persisted for so long in othercircles, and why should it have blos-somed out in such noted and provoca-tive guises as those devised by Thur-stone and his school? I have puzzled sooften about the ability of this fancy topersist and grow famous that I have ac-cumulated a list of possible reasonsfor it. I will run quickly through afew of them, not because they are thetrue causes, or are exhaustive of thepossibilities, but only because listingthem may inspire others to inquirefurther into this anomaly in the historyof scientific thought.

Excuses for Unitizing Error

These, then, are some of the excuseswe might offer for the popularity ofmethods that try to create measure-ment by unitizing the residual noise ina stimulus-response sequence of onekind or another.No competition. There are (or were)

no competing methods. Plateau, to besure, had asked eight artists each topaint a gray that appeared to lie midwaybetween black and white, but this meth-od of bisection did not catch on in away that could challenge the arsenalof procedures aimed by Fechner at thedetermination of the jnd (just notice-able difference). When a bad theoryis punctured it does not shrink quietlyaway. It retires from the field of sci-

The author is director of the PsychologicalLaboratories, Harvard University, Cambridge,Mass. This article is adapted from the text of anaddress delivered at a centennial symposiumhonoring Fechner, sponsored by the AmericanPsychological Association and the PsychometricSociety, which was held in Chicago in Septem-ber 1960.

SCIENCE, VOL. 133

-To Honor Fechner andRepeal His Law

A power function, not a log function, describes theoperating characteristic of a sensory system.

S. S. Stevens

on

Aug

ust 2

4, 2

015

ww

w.s

cien

cem

ag.o

rgD

ownl

oade

d fr

om

on

Aug

ust 2

4, 2

015

ww

w.s

cien

cem

ag.o

rgD

ownl

oade

d fr

om

on

Aug

ust 2

4, 2

015

ww

w.s

cien

cem

ag.o

rgD

ownl

oade

d fr

om

on

Aug

ust 2

4, 2

015

ww

w.s

cien

cem

ag.o

rgD

ownl

oade

d fr

om

on

Aug

ust 2

4, 2

015

ww

w.s

cien

cem

ag.o

rgD

ownl

oade

d fr

om

on

Aug

ust 2

4, 2

015

ww

w.s

cien

cem

ag.o

rgD

ownl

oade

d fr

om

on

Aug

ust 2

4, 2

015

ww

w.s

cien

cem

ag.o

rgD

ownl

oade

d fr

om

entific contest only when pushed asideby a stronger theory. Plateau shot hisperceptive shaft into the Fechnerianblimp by asserting that his bisectionresults entailed a power law rather thana logarithmic law, but he then turnedthe experimental attack over to hisfriend Delboeuf (7), who, for reasonsnot entirely clear, proceeded to obtainbisection data that approximated theFechnerian logarithmic prediction. Pla-teau thereupon reversed his view andturned to other topics, leaving Fechnerto patch up his defenses and carry theday. If Plateau, whom we rememberwell for his part in establishing theTalbot-Plateau law relating to thebrightness of intermittent light, hadfought back at this juncture instead ofcapitulating, the story might then havetaken a happier turn. Plateau's methodof bisection may not be the best of pro-cedures, but it could have carried hima long way toward the establishmentof the correct psychophysical law. Ifthat had happened, psychophysics wouldhave been spared a hundred years offutility.At this point let me interrupt the

story and try to justify my belief thata man of Plateau's great talent couldhave made an effective stand againstFechner by perfecting the method ofbisection and using it to produce asufficient body of experimental results.It is true that the method of bisection,like all methods that attempt to parti-tion the distance between a pair ofstimuli, contains systematic biases thatprevent its generating a linear segmentof the scale of sensation. Applied tosensory intensity-the so-called pro-thetic continua-these methods pro-duce "partition scales" whose curvatureis always downward when they areplotted against the ratio scale of sub-jective magnitude (8). Nevertheless, asPlateau had surmised, these proceduresare capable of verifying the power law.The estimate they give of the exponentis generally too small, but often thediscrepancy is only 10 to 15 percent.A series of brightness bisections that

R. J. Herrnstein and I obtained in1953 are shown in Fig. 1. We werewondering whether brightness bisec-tions would exhibit the hysteresis thatmanifests itself in other sense modali-ties when the order of presentation ofthe stimuli is reversed. The answer isyes (9), but more important to ourimmediate concern are three other fea-tures of the outcome. (i) The curva-ture -of the--functions in the semi-logplot of Fig. 1 demonstrates that the13 JANUARY 1961

Top-Brightnesss

1 Middle 1/ / /0'

0.

Bottom20 30 40 50 60 70 80 90 100 110

Luminance in Decibels

Fig. 1. Bisections of brightness intervals, showing the hysteresis effect produced by theorder of stimulus presentation. The abscissa shows the luminances in decibels relative to1010 lambert. The apparent midpoint is at a higher luminance when the luminances areviewed in ascending order. Each bisection was made by from 14 to 19 observers, eachof whom repeated the task three times, in both ascending and descending order. Thus,each point is based on from 45 to 57 bisections. The bisection values obtained byaveraging the ascending and descending series are consistent with a power functionhaving an exponent slightly under 0.3.

relation between sensation and stimulusis not logarithmic-that is, the bisec-tion does not occur at the geometricmean. (ii) The similarity in the curva-ture obtained over a wide range of ab-solute stimulus levels is consistent withthe power function. (iii) The bisectionvalues obtained by averaging theascending and desceiding series deter-mine that the power function has anexponent of about 0.3. This compareswith the slightly larger value, 0.33,typically found in experiments on theratio scaling of the brightness of lumi-nous targets seen in the dark.

Admittedly, modern apparatus makesthe experiment easier now, but it isdifficult to see how any ghost otherthan the Zeitgeist could have preventedthe establishment of the power lawback in the last century if Plateau hadfollowed through. Merkel (10), inci-dentally, had the necessary apparatusall set up in 1888, and his limited re-sults on bisections agree well with thosein Fig. 1. But no one paid much atten-tion to Merkel. He is the one who gaveus the method of ratio production,"stimulus doubling," as he called it.With this new procedure he was fullyequipped to rescue psychophysics, butneither he nor anyone else seemed toknow it. Oh, to have the perceptualresolving power of hindsight!

The universality of variability. Theunitizing of "noise" is attractive be-cause confusion and variability, likedeath and taxes, are always with us.Normally a nuisance to science, dis-persion among people's judgments be-comes grist for the mill when Fechner

makes dispersion into a "differencelimen" and calls it the unit of his scale;and it keeps the mill wheel whirlingwhen Thurstone enters dispersion intohis "equation of comparative judg-ment" and computes scale values forthe stimuli. While preserving the"logic," as he called it, of Fechner'sprocedure, Thurstone ventured to "ex-tend the psychophysical methods tointeresting stimuli" (11). Among otherspirited questions, he asked, "which oftwo nationalities would you prefer tomarry?" which to some people is alivelier issue than any possible questionabout lifted weights.Here I must digress to modify the

foregoing assertion regarding the uni-versal presence of confusion and varia-bility. Dispersion among data is alwayswith us, to be sure, but sometimes itis not there in sufficient bulk for thesmooth working of the Thurstoniantransformations. Mischievous as it mayseem, investigators who use thesemethods sometimes feel called uponto seek noisier data than those theyhappen to have gathered. The premiumnormally placed upon precision, repeat-ability, and the elimination of pertur-bations may turn into a liability whenthe perturbations themselves providethe alleged unit of measurement.

Wide applicability. The universalityof noise imparts an equal universalityto the Fechner-Thurstone scales. Theycan be erected with and without theknowledge of an underlying stimulusmetric. Fechner scaled subjectiveweight by nd's and determined estheticjudgment by paired comparisons. Thur-

81

stone improved the machinery for han-dling the data of paired comparisonsand made it applicable (given sufficientnoise) to such elusive matters as atti-tudes, preferences, and the goodness ofhandwriting.Wide applicability is a desirable

trait, but it does not qualify as thedecisive attribute in determining themerits of a scaling technique. For ex-

ample, the operation of empirical ad-dition, as practiced in the scaling ofsome of the "fundamental" magnitudesof physics, especially length and weight,has a very limited applicability but an

overwhelming importance. One is ledto suspect that universality may be lessimportant. than it seems at first sight.

It should be remarked that some

of the modern psychophysical tech-niques-magnitude estimation, for ex-

ample-can also be applied to stimulifor which there is no underlying metric.We have recently scaled the apparentroughness of sandpapers on both a ratioand a category scale and have demon-strated that roughness as felt by thefinger is a prothetic continuum-allthis despite our having no metric scaleof the stimulus involved. In order toerect a psychological ratio scale, theexperimenter needs only a nominalscale of his stimuli-he must be ableto keep track of which is which.The experiment on roughness was

carried out by Judith Rich and IrmaSilverman as a laboratory exercise.They presented 12 grades of sand-paper (nominally 24 to 320 grit) tothe observer, who made two sweeps

with his first and second fingers over

each paper. The papers were presentedtwice each, in an irregular order, toeach of 12 observers. For the magni-tude estimations, a paper of mediumroughness was presented first and called10. The observer then assigned num-

bers proportional to the apparentroughness of each of the sandpapers as

he felt it. For the category judgments,the smoothest paper was presented andcalled 1, and the roughest was pre-

sented and called 7. The observer thenjudged each paper twice, in irregularorder, on a 7-point scale. Again therewere 12 observers.As shown in Fig. 2, the partition

scale that results when the observerstry to divide roughness into seven

equally spaced categories is nonlinearlyrelated to the scale erected by directmagnitude estimation. This outcome isthe standard finding on prothetic con-

tinua. It is as though the observer,

82

if.s g §w sw s| ww | Ww l| || l ... 1X

6

5

0

'I)

4

0'

0 30

2

0 5 l0 Is 20 25 30

Subjective Magnitude35 40

Fig. 2. Category and ratio judgments ofthe roughness of sandpaper. The arithmet-ic means of the category judgments (ordi-nate) are plotted against the scale derivedfrom the geometric means of the magni-tude estimations (abscissa). The marksalong the abscissa show the locations ofthe sandpaper stimuli on the linear scaleof subjective roughness.

when he tries to partition a protheticcontinuum into equal intervals, findshimself biased by the fact that a givendifference at the low end of the scaleis more noticeable and impressive thanthe same difference at the high end ofthe scale. This asymmetry is not pres-ent on metathetic continua (for ex-

ample, pitch and apparent azimuth),and there the category scale is notsystematically curved.The scaling of roughness should

make it plain that the methods usedin "the new psychophysics" are notrestricted to psychological continua forwhich a stimulus metric is known atthe level of an interval or a ratio scale-the scales that we commonly callquantitative (12). If further evidenceis needed, look, for example, at Ek-man's application ,of ratio scaling pro-

cedures to the esthetic value of hand-writing (13). He cites evidence thatthe Thurstonian scale is nonlinearly re-

lated to the ratio scale in the way thatone would expect for a prothetic con-

tinuum. Ekman clings to the view that.there is still a use for the "indirect"methods that unitize confusion, becausethere may be psychological variablesthat cannot be directly observed. What-ever turn this particular argument may

take, there is no denying that the chal-lenge is out. Is there any substantiveproblem relating to the assessment of a

subjective variable whose solution can-

not be reached by direct ratio scalingprocedures?

Easy on the observer. An argumentfor the variability methods is that theyplace minimal requirements on the per-

son making the judgment. His taskis easy; all he needs to be is variable.As a more specific instance of the argu-

ment for the use of a measure of dis-persion as the unit of the subjectivescale, we find Garner (14) saying thatthese methods are "more legitimate,valid, and meaningful for the scalingof loudness than are those methodswhich make use of various types ofdirect response on the part of the ob-server." Another specific merit claimedfor these "discriminability" proceduresis the stability of the results.

If it were true (and this we may

well dispute) that jnd scales show more"stability than ratio scales of subjectivemagnitude, we could still with justiceask that rude but pointed question:So what? It is conceivable that thenoise ordinarily encountered when theobserver responds to pairs of stimuliis much the same in magnitude fromexperiment to experiment, but so alsomay be the observer's temperature.

Relevance may be more crucial thanprecision.

Model-maker's delight. Scaling bythe unitizing of variability has thefurther advantage that it poses a chal-lenge to our ingenuity and allows us toinvoke elegant formal models in one

or another aspect of the enterprise. Inthis respect it contrasts with thoseprosaic procedures of measurementthat offer only the computation of a

median or a geometric mean as theterminal ritual. As soon as it is decidedthat a measure of dispersion can beused for something more than themeasurement of dispersion, new vistasopen, and the model builders proceedto devise ingenious exercises in mattersranging from axiomatics to the pro-

gramming of computers. All this fer-ment is interesting and good. It wouldbe even better if it stood on some

firmer base than disagreement amonghuman judgments.At the risk of giving aid and com-

fort to the Thurstonians, I have else-where (15) suggested that a closer ap-proximation to the correct scale ofsubjective magnitude on a protheticcontinuum would be achieved if itwere assumed that discriminal (that is,subjective) dispersion is proportionalto the psychological magnitude. Thesubjective dispersion is usually assumedto be constant, an assumption knownas the Thurstonian "case V," which is

SCIENCE, VOL. 133

...~~~ . I .I....#..

Tactual Roughness

111: it 4 i AI. . . . . .

essentially the same as the Fechnerianassumption that each jnd unit repre-sents the same subjective differenceon the psychological continuum. Fortu-nately for a sane approach to psycho-logical measurement, the proportional-ity assumption, which I have suggestedmight be called "case VI" (16), turnsout to have two defects. First, it isnot quite true that variability in thesubjective response is always propor-tional to subjective magnitude, andsecond, even if this were a good as-

sumption, the resulting Thurstonianscale would turn out to be only a loga-rithmic interval scale. Psychologicalvalues separated by equal units of dis-persion would not stand equidistantfrom one another; they would stand ina constant ratio to one another. Butno one seems yet to have discoveredan interesting use for a logarithmicinterval scale-except to christen anddescribe it, and to point out that it re-

mains invariant under the power groupof transformations (12).

Prominent among the models con-trived for the mirroring of humanvariability is the development knownas detection theory (see 17). Herethere seems to be no thought of meas-

uring sensations or other subjectivemagnitudes; the question at issue is thenarrow problem of the rules that gov-ern the ability of a person to detect a

signal immersed in a noise. Since anal-ogous problems have been addressedby engineers and mathematicians inother domains, an elaborate model in-corporating certain aspects of statisticaldecision theory stands ready for appli-cation to the psychophysical case. Ac-tually, the model is perhaps not quiteso ready as I seem to imply, for argu-ments are in progress regarding theresemblance between the nature of thenoise assumed by the model and theproperties of the noise that mightreasonably be assumed to reside inthe observer (see 18). In any case, wehave here an instance in which thecontentions regarding the model and itsapplicability serve to generate a spark-ling interest in a problem that has littlesubstantive flesh on its bones.Pseudo differential equations. In

some quarters the popularity of themethods that try to create measure-ment out of variability has been sus-

tained by a misidentification. The stand-ard deviations and quartile points offrequency distributions of judgmentshave become identified with differen-tials. The custom is to write when13 JANUARY 1961

what we really mean is the scatter ofsome dial settings or the relative fre-quencies of some confusions. It isargued implicitly by some, explicitlyby others, that Al is determined bythe slope of the "operating character-istic" of the sensory system and there-fore that it is as useful to psychologyas differential equations are useful tophysics. Here is what we read in apaper 'by Nutting (19). "As is wellknown, the visual sensation cannot bedirectly measured, but its derivativesensibility is readily measurable, andfrom this the sensation may be readilydeduced just as the scale of an am-meter may be readily reconstructed ifthe sensibility is known for all currents."

Admittedly, that was an engineertalking, but I wish I could say thatno modern psychophysicist holds sim-ilar views. Many recent argumentshave turned on the question of whetherit is possible for the psychophysicalpower function to vary in steepness (ex-ponent) without a concomitant changein the resolving power. Some peopleclaim to enjoy a compelling intuition tothe effect that, when the psychophysicalfunction gets steeper, the resolvingpower must get better. This, I suppose,is a tacit admission that the person inquestion has not made the proper dis-tinction between a differential and astandard deviation.

Let us instruct ourselves on this is-sue by a brief sample of facts. The sen-sation of electric shock grows rapidlywith intensity (exponent = 3.5), butthe resolving power on this continuumseems to be little if at all better than inthe vibration sense, where the exponentis several times smaller (20). Themeasured resolving power for differ-ences in brightness changes by about50-fold when the area of the stimuluschanges from a smallish point to a widefield (see Hunt, 21), but no compar-able change occurs in the exponent ofthe brightness function (9). Actually,the brightness function is slightly steep-er for point sources (exponent = 0.5)than for larger targets (exponent =0.33), but the difference would be inthe opposite direction if a smaller jndwent with a steeper function. In theauditory sense modality the jnd for in-tensity can be altered merely by chang-ing the frequency of the tone. As New-man (22) pointed out, the differencein the resolving power at 80 and at1000 cycles per second is such that thesubjective size of the jnd cannot pos-sibly be constant for all pure tones.

These examples are just a few of themore dramatic instances in which varia-bility, in the guise of the jnd, has failedto behave as a proper unit of subjec-tive measurement.

Nevertheless, the number of man-hours devoted each year to the meas-urement of one or another aspect ofsensitivity, sensibility, resolving power,detectability, or just plain jnd attests apersistent belief in the utility of thesemeasures. By and large, they are re-markably tedious measurements; yetthe cumulative curves in which they aredisplayed merely gauge the noise thathappens to characterize the system un-der the circumstance chosen for the ex-periment. If the noise could somehowbe reduced, the. measured variabilitywould grow smaller, but there is noreason to expect that the subjectivemagnitude, represented by some aver-age value of the distribution, would bethereby altered. It is not required thatthe mean and the standard deviationvary together.

If so much trouble has resulted fromthe identification of A! with a measureof the slope of the magnitude function,a simple remedy suggests itself. Per-haps, if we stop writing Al when wemean a measure of dispersion on a fre-quency distribution, the confusion mayget itself cleared up in a generation ortwo. The most common measure of re-solving power in sensory psychophysicsis the median deviation-the 75-percentpoint on the cumulative "psychometricfunction" relating proportion of correctjudgments to stimulus difference. Sincethis value is sometimes called the quar-tile point, why not replace the abbrevia-tion Al by the abbreviation Q? Wecould then write the general linear formof Weber's law as Q = k(l + Jo), andin the process we would discourage theview that Q's are something that canprofitably be added up, or that can beregarded as indicative of the slope ofthe function relating sensation to stim-ulus.

The Question of the Neural Quantum

The ubiquitous variability of the hu-man response has not only provided atempting basis on which to build a de-ceptive theory of psychological meas-urement, it has also obscured the innerworkings of the discriminatory mech-anism. Under most procedures used tomeasure the jnd, some variety of noisesets bounds on the observer's resolving

83

05 H

z/

20/

0.

0

z~~~~~~~~~~~~~~~~~~~ 1000tn

~~~~~~~~~~~~~~~~~~~~Aelectric shock-60'.

C lifted weightsl l§2Ses1 | l@2@s1 2 l*|@|2|1*Dpressureon palm

E cold

0F vibration 60'~G white noise

2 H 1000-~toneI white light

10 102 1063O4 lOl opRELATIVE INTENSITY OF CRITERION STIMULUS

Fig. 3. Nine equal-sensation functions, obtained by matching force of handgrip tovarious criterion stimuli. The relative positioning of the functions along the abscissa isarbitrary. The dashed line shows a slope of 1.0 in these coordinates. Each point standsfor the median force exerted to match a criterion stimulus. Ten or more observersparticipated in each of the nine experiments.

power and thereby determines the meas-

ured size of the resulting error distribu-tion-the Q. What would happen if, bysome contrivance, we could rid the ex-

periment of noise, or at least of muchof it? No doubt, if sufficient "quiet"could be achieved, the "grain" anddiscontinuities in the action of the sen-

sory system would begin to manifestthemselves, for it is highly unlikelythat the neural processes that mediatediscrimination are devoid of some all-or-none property. As a matter of fact,direct evidence for an all-or-none stepfunction in the action of sensory dis-crimination has been observed in sev-

eral experiments that have been un-

dertaken to perfect the experimentalarrangement to the point of a sufficientsuppression of variability (for a recentreview, see 23). The success of some

of the experiments that have soughtevidence for a "neural quantum" (NQ)suggests that valuable findings may at-tend the suppression of noise and varia-bility. It may prove better to strugglefor the reduction of uncertainty than totry to enshrine it as the measure ofpsychological magnitudes.

The Psychophysical Law

All the evidence that was heaped to-gether in previous decades for the pur-

pose of refuting the Fechnerian dogmamade no great hole in the "psycho-physical edifice." Even the eloquentridicule by William James, who went

84

so far as to mix his metaphor in "strik-ing Fechner's theories hip and thighand leaving not a stick of them stand-ing," could do little to change the con-tent of the textbook discussions of"'Fechner's Law." The task of clearing

Table 1. Representative exponents of thepower functions relating psychological mag-nitude to stimulus magnitude on protheticcontinua.

Continuum Exponent Conditions

Loudness 0.6 BinauralLoudness 0.54 MonauralBrightness 0.33 50 Target, dark-

adapted eyeBrightness 0.5 Point source,

dark-adaptedeye

Lightness 1.2 Reflectance ofgray papers

Smell 0.55 Coffee odorSmell 0.6 HeptaneTaste 0.8 SaccharineTaste 1.3 SucroseTaste 1.3 SaltTemperature 1.0 Cold, on armTemperature 1.6 Warm, on armVibration 0.95 60 cy/sec, on

fingerVibration 0.6 250 cy/sec, on

fingerDuration 1.1 White-noise

stimulusRepetition rate 1.0 Light, sound,

touch, shocksFinger span 1.3 Thickness of

wood blocksPressure on 1.1 Static force onpalm skin

Heaviness 1.45 Lifted weightsForce of 1.7 Precision hand

handgrip dynamometerAutophonic 1.1 Sound pressure

response of vocalizationElectric shock 3.5 60 cy/sec

through fingers

the scientific bench top of the century-long preoccupation with the jnd, andthe consequent belief in logarithmicfunctions, demands the cleansing powerof a superior replacement. My opti-mism on this score has been recordedin other places, but I would like hereto suggest that, if I seem to feel a meas-ure of enthusiasm for the power lawrelating sensation magnitude to stimu-lus intensity, it is only because that lawseems to me to exhibit some highly de-sirable features. Not the least of thesedesiderata is its apparent generality. Onmore than a score of sensory continua,the subjective magnitude + has beenshown to grow as the stimulus magni-tude 4 raised to a power n. More spe-cifically,

+ = k(o-oo)"where 4o is the effective threshold. Asyet we have encountered no exceptionto this rule. Some of the exponents, afew of them not yet very firmly deter-mined, are listed in Table 1. If thepsychophysicists complete the sweep ofthe sensory domain and recover powerfunctions at every turn, we may an-ticipate that little room will remain forFechner's logarithmic relation. Per-haps Luce's (24) penetrating analysishas left no room for it anyhow.

William James (25) once addressedhimself to Fechner's Massformel, S =c log R, by pulling himself up to atowering indignation and letting gowith: "No human being, in any in-vestigation into which sensations en-tered, has ever used the numbers com-puted in this or any other way in orderto test a theory or reach a new result."

Whether James was precisely correctin this stricture is beside the point.What he was invoking was the prag-matic test by which all scientific prin-ciples must be judged-including myown candidate, the power law. Thatbrings us to the pay-off question: Inwhat ways has the new psychophysicallaw done better than the old one inserving a scientific purpose?

Applications and Validations

As positive answers to this question,four examples can be marshaled, eachconcerned with a different asset of thenew approach to psychophysics.

1) We have already noted that thepower function governing the growthof subjective brightness (exponent =

0.33) predicts, with only a minor sys-tematic error, the behavior of an ob-

SCIENCE, VOL. 133

server who undertakes to bisect theinterval between two levels of lumi-nance (see Fig. 1). A similar story canbe told for loudness (26). One successof the new psychophysics has been thepulling together of many loose ends ina way that discloses consistencies anduniformities where none-were apparentbefore. The nature of the biases in par-tition scales and the relation of thesescales to ratio scales are fast becomingclear. Not only that, but the relationbetween subjective magnitude and thescale that is generated by Thurstone'smethod of successive intervals, which"unitizes" the confusions among cate-gory judgments, has yielded to orderlyanalysis by Galanter and Messick (27).Since the noise and confusion in judg-ments of loudness tend to grow in di-rect proportion to the subjective magni-tude, it is not surprising that the con-fusion scale generated by discarding themean and processing the variabilityturns out to resemble a logarithmictransform of the ratio scale of loud-ness.

2) In some ways the most dramaticvalidation of the scales generated byasking observers to make numericalestimations of sensory intensity is thedemonstration that these same scalescan be generated even if no appeal ismade to "number behavior" at all. Bymeans of cross-modality comparisons,each subjective continuum can be re-lated to each other continuum, and, forthe critic who thinks he will feel betterif all reference to numerical judgmentsis avoided, the family of power func-tions governing the various sensory con-tinua can all be assigned their appro-priate exponents relative to that ofsome "base continuum," such as appar-ent length of lines. In practice, ofcourse, we have been content to goalong with results of the several proce-dures involving numerical methods, be-cause these findings have stood the testof cross-modality validation. The argu-ment runs as follows.

If, given an appropriate choice ofunits, two modalities are governed bythe equations

41= olm

4I2 = 02'

and if the subjective values Vfri and q12are equated by asking the observer tomake the one sensation seem as strongas the other at various levels, then theresulting equal-sensation function willbe given by

,km = 02n

13 JANUARY 1961

In terms of logarithni,

nlog OI = m (log 02)

In log-log coordinates, therefore, theequal sensation fultction should be astraight line with a slope equal to theratio of the two exponents.

This prediction was nicely borne outby a series of cross-modality matchesbetween all possible pairs of the threecontinua, loudness, vibration on thefinger tip, and electric shock to thefingers (20). From this encouraging be-ginning, the procedure of cross-modal-ity matching has been extended to nu-merous other pairs, with special em-phasis on what might be called scalingby squeezing.

Using a precision dynamometer, J.C. Stevens and Mack (28) worked outthe subjective scale relating the appar-ent force of handgrip to the physical

(0

0c

-c

force exerted by the subject. This re-lation turned out to be a power func-tion with an exponent of 1.7. Equippedwith this scale, we then proceeded totake the measure of other sensory con-tinua by asking observers to squeezethe dynamometer until the sensation ofstrain matched the apparent intensityof a criterion sensation in some othermodality (29). A sample of the resultsis shown in Fig. 3, where two impor-tant facts stand out. All the data ap-proximate straight lines in the log-logplot, and the slopes stand in the sameorder as the respective exponents listedin Table 1. Less obvious but even morecrucial are the exact values of the slopesin Fig. 3. If these values are multipliedby the factor 1.7, the products agreereasonably closely with the values ofthe exponents listed in Table 1 (6).

In another investigation, the cross-modality comparison of loudness and

30 40 50 60 70 80 90 100

Luminance in Decibels re 10 Lambert

Fig. 4. A generalized set of functions showing the brightness of a target disk seen in-the presence of a surround. The brightness function for the white surround is shownby the top line (slope = 0.33). Depending on the luminance of the surround, thebrightness function for the target follows one or another of the steeper straight lines.The dashed curves pass through the points (circles) representing given fixed ratios,R, between the luminances of disk and surround. These ratios are stated in decibels.Thus, the top dashed curve is for a target disk 3 decibels below the surround in bright-ness. If the surround is a white paper, a darker paper 3 decibels down would be calledlight gray, as indicated. So-called brightness constancy is a manifestation of the relativeflatness of the dashed curves. The filled circles represent data obtained by Leibowitzet al. (33), who matched a luminance seen in a black surround to a medium gray seenin a white surround under a wide range of illuminations.

85

vibration was one of the several pro-cedures used to verify the mysteriousfact that listening with two ears is dif-ferent from listening with one. Forsome strange reason, the binaural ex-ponent of the loudness function is about10 percent larger than the monauralexponent (Table 1), and this small dif-ference showed up when the two loud-ness functions were determined by com-parisons with vibration applied to thetip of the finger, just as it manifesteditself when the observers "matched"numbers to binaural and monauralloudnesses under the methods of magni-tude estimation and magnitude produc-tion (30).

These procedures established the factthat for a sound pressure level of about90 decibels the loudness of a sound asheard in two ears is precisely twice asgreat as the loudness heard when thesame sound is delivered to only oneear. This 2-to-1 relation obtains only atone level, however, because of the dif-ference in the size of the exponents forbinaural and monaural listening. Atlower levels the binaural-monaural ra-tio is smaller, whereas at levels greaterthan 90 decibels the binaural soundseems more than two times louder thanthe monaural sound.

3) In the domain of practical appli-cations, an area that is not always with-out interest to the academic mind, thesone scale of loudness, the first and themost carefully documented of the mod-ern ratio scales of sensation, has longsince proved its utility to the acousticalengineer (the sone is the subjective unitof loudness). This scale recently per-formed its bit as an essential link in thedevelopment of a method for comput-ing the total binaural loudness of acomplex sound spectrum, given an anal-ysis of the sound in terms of octave orthird-octave bands (31). The loudnessin sones of each band is determinedfrom a set of equal-loudness 'contours,and the loudness values are added upaccording to a simple weighting func-tion. To the loudness of the loudestoctave band is added 0.3 times the sumof the loudness in the remaining bands.A version of this procedure is in fairlywidespread use and is being readied asa secretariat proposal for general adop-tion by the International Standards Or-ganization (32). The relevance of allthis to our present concern is merely toshow that ratio scales of sensation havetheir utility in the world of practicaldecisions.

4) The extension of brightness scal-ing to those circumstances under whicha target is surrounded by a brighterbackground has led to some predictionsthat to me are rather startling. Thefull story is told elsewhere (9), butbriefly it is this. We know that a brightsurround depresses or inhibits the sub-jective brightness of a target in a moststriking manner. This is the same in-hibitory effect that we all experiencewhen the glare of the oncoming head-lights renders objects beside the roadeither dim or invisible. When a targetluminance is subjected to the inhibitorycontrast imposed by a brighter sur-round, the exponent governing the ap-parent brightness of the target jumpsto a larger value. At high over-all levelsthe exponent grows by a factor as greatas 10, but at lower levels the value ofthe exponent is smaller. These expo-nents are depicted by the slopes of thesteep lines in Fig. 4, where we see thatthe steepness is less at the lower levels.These functions were determined in along series of experiments aimed di-rectly at determining the slopes of thebrightness functions for targets seenunder contrast. The brightness valuesare expressed in subjective units calledbrils.

If we want to use the functions inFig. 4 to predict the appearance of agray paper of a given reflectance viewedagainst a white background under vari-ous levels of illumination, we introducethe dashed lines, each of which showsthe locus of the target luminances thatbear a fixed ratio to the luminances ofthe surround. Put more simply, eachshade of gray has its own dashed linein Fig. 4, and the line for a givengray shows how the brightness of thegray behaves when we change the il-lumination falling on the scene, includ-ing both the gray and its white sur-round.What seems startling in Fig. 4 is that

for some shades of gray the apparentbrightness is supposed to decrease whenthe illumination is increased. Turn onmore light and the target looks darker!That is the verdict of the dashed linesthat slope down toward the right. Thisprediction has been checked with sixobservers who, having spent about 10minutes in adapting to darkness, vieweda dark gray on a white background.The darkness of the target seen in dimlight grew suddenly deeper when theillumination was suddenly increased by10 or 20 decibels. The observers found

it especially interesting to watch thetarget turn gradually blacker as theillumination was gradually increased.Many other interesting deductions

can be made from the functions in Fig.4. But what we have considered isenough to show that it is indeed pos-sible to use the new psychophysical lawand the procedures by which it wasestablished in order, as James put it,"to test a theory or to reach a newresult."

References

1. G. T. Fechner, Elemente der Psychophysik(1860).

2. R. von Mises, Probability, Statistics andTruth (Macmillan, New York, 1939).

3. G. T. Fechner, In Sachen der Psychophysik(Leipzig, 1877), p. 215.

4. F. Brentano, Psychologie vom empirischenStandpunkt (Dunker and Hunblot, Leipzig,1874), vol. 1.

5. J. Plateau, Bull. Acad. roy. mdd. Belg. 33,376 (1872).

6. S. S. Stevens, Am. Scientist 48, 226 (1960).7. J. Delboeuf, Etude psychologique. Re-

cherches thioriques et experimentales sur lamesure des sensations et spJcialement dessensations de lumi0re et de fatigue (1873).

8. S. S. Stevens and E. H. Galanter, J. Exptl.Psychol. 54, 377 (1957).

9. S. S. Stevens and J. C. Stevens, "TheDynamics of Visual Brightness," Psycho-physical Project Report, Harvard University,No. PPR-246 (August 1960); J. Opt. Soc.Am. 50, 1139 (1960).

10. J. Merkel, Phil. Stud. 4, 541 (1888).11. L. L. Thurstone, The Measurement of Value

(Univ. of Chicago Press, Chicago, 1959),chap. 24.

12. S. S. Stevens, in Measurement: Definitionsand Theories, C. W. Churchman and P.Ratoosh, Eds. (Wiley, New York, 1959).

13. G. Ekman, "Some aspects of psychophysicalresearch," in Sensory Communication, W. A.Rosenblith, Ed. (Technology Press and Wi-ley, New York, in press).

14. W. R. Garner, J. Acoust. Soc. Am. 30,1005 (1958).

15. S. S. Stevens, Psychol. Rev. 64, 163 (1957).16. , Contemp. Psychol. 4, 388 (1959).17. J. C. R. Licklider, "Three auditory theories,"

in Psychology: A Study of Science, S. Koch,Ed. (McGraw-Hill, New York, 1959), vol.1.

18. M. V. Mathews, J. Acoust. Soc. Am. 32,931 (1960), abstract.

19. P. G. Nutting, Trans. Illum. Eng. Soc. N.Y.11, 939 (1916).

20. S. S. Stevens, J. Exptl. Psychol. 57, 201(1959).

21. R. W. G. Hunt, J. Phot. Sci. 1, 149 (1953).22. E. B. Newman, Trans. Kansas Acad. Sci.

36, 172 (1933).23. S. S. Stevens, "Is there a quantal threshold?"

in Sensory Communication, W. A. Rosen-blith, Ed. (Technology Press and Wiley, NewYork, in press).

24. R. D. Luce, Psychol. Rev. 66, 81 (1959).25. W. James, The Principles of Psychology

(Holt, New York, 1890), vol. 1, p. 539.26. S. S. Stevens, J. Acoust. Soc. Am. 27, 815

(1955).27. E. H. Galanter and S. Messick, "The rela-

tion between category and magnitude scalesof loudness," paper presented at the 1stannual meeting of the Psychonomic Society,Chicago, Ill., 1960.

28. J. C. Stevens and J. D. Mack, J. Exptl.Psychol. 58, 405 (1959).

29. J. C. Stevens, J. D. Mack, S. S. Stevens,ibid. 59, 60 (1960); J. C. Stevens andS. S. Stevens, Ibid. 60, 183 (1960).

30. G. S. Reynolds and S. S. Stevens. J. Acoust.Soc. Am. 32, 1337 (1960).

31. S. S. Stevens, ibid. 28, 807 (1956).32. , Noise Control 3, No. 5, 11 (1957).33. H. Leibowitz, N. A. Myers, P. Chinetti,

J. Exptl. Psychol. 50, 15 (1955).

SCIENCE, VOL. 13386