THE JOURNAL OF THE ACOUSTICAL SOCIETY OF AMERICA Volume27Number5 SEPTEMBER. 1955 The Measurement of Loudness* S.S. STEVENS Psycho-AcousticLaboratory, Harvard University,Cambrfigge, Massachusetts (ReceivedJune 17, 1955) This paperreviews the availableevidence (published and unpublished)on the relationbetween loudness and stimulusintensity. The e.vidence suggests that for the typical listenerthe loudnessL of a 1000-cycle tone can be approximatedby a powerfunction of intensity I, of which the exponentis log•02.The equation is: L--kI ø.a.Intensity here is assumedto be proportionalto the squareof the soundpressure. In terms of sones,where 1 soneis the loudnessproduced by a tone at 40 db above the standardreference level, the equationfor loudnessL as a function of the number of decibelsN becomes'logL=0.03N--1.2. Otherwisesaid, a loudnessratio of 2:1 is producedby a pair of stimuli that differ by 10 db, and this re- lation appearsto hold over the entire range of audibleintensities. At low levelsof intensity,the loudnessof white noisegrows more rapidly than the loudnessof a 1000-cycle tone, but abovethe level of approximately50 db the two loudnessesremain more nearly proportional.The suggestionis madethat for all levelsgreater than 50 db the loudnessof continuousnoises may be calculated from the equation:logL=O.O3N-t--S, where $ is a spectrumparameter to be determinedempirically. HE purposeof this reviewis to examinethe the nature of the central issue. What we want to know availabledata on the measurementof subjective is how loud variousstimuli soundto people.By this we loudness.It is hoped that by assemblingthe relevant mean, perforce,what do people say when they try to informationin one place we may be able to reach a describeloudness in quantitative terms?In askingthis reasonableconclusion concerning the relation between questionwe are not at the outset trying to solve the loudnessand intensity.The variousresults obtained by problem of how the ear works or how the nervous workersin this fieldmake it plain that the scalerelating systemperforms its integrations.We are not trying to loudnessto intensity is not somethingthat can be countnerve impulses nor provea theory.We are merely determinedwith high precision,but these efforts also looking for the empirical answer to a very empirical make it plain that peopleare able to make quantitative question'How do peopledescribe sounds when we ask estimates of loudness and that it is not unreasonable to them to usea numericallanguage instead of adjectives? try to determinea loudnessscale that will be representa- Our interestin askingthis questionmay be academic tive of the typical listener. or it may be practical.The academicside of the issue Despite its many pitfalls, I think we can probably hasa long historyfull of side-takingand polemics.• The make senseof this problemprovided we are sufficiently practicalside of the problemhad its originin acoustical modestin our demands.Not only mustwe renouncethe engineering.Not long after they had developedthe hopefor highprecision, but alsowe mustkeep in mind conventionaldecibel scale for measuringsound inten- sity, the engineersnoted that equalsteps on the decibel * This work was carried out under Contract N5ori-76 between Harvard Universityand the Officeof Naval Research,U.S. Navy scaledo not soundlike equalsteps, and that a level of (ProjectNR142-201, Report PNR-168). Reproductionfor any purposeof the U.S. Governmentis permitted. • E.G. Boring,Am. J. Psychol.32, 449-471 (1921). 815 Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 165.123.230.136 On: Tue, 25 Aug 2015 06:45:44 816 S. S. STEVENS 50 db does not sound like half of 100 db. Since the periment.Thus bisectionmight providea test of the engineeroften faces the problemof communicatingwith generalityof a ratio scale,but by itselfbisection cannot a customer, it was soon realized that there was a need generate a ratio scale. for a scale whose numbers would make more sense to In the tabulationsthat follow only the resultsob- the customer than do the numbers on the decibel scale. tainedby magnitudeestimation and ratio determination The generationof a loudnessscale is in principle will be considered. quite simple.All we need to do is producean array of Magnitudeestimation means the following:A stand- soundsand ask a groupof listenersto assignnumbers to ard tone is presentedand an arbitrary number is as- thesesounds in sucha way that the numbersreflect the signedto its loudness,e.g., 1 or 100.Then a comparison perceivedloudness of the sounds.In practice,of course, tone is presented,and the subjectdecides what number it turns out that many alternative techniquesare pos- he thinks shouldbe assignedto the loudnessof the com- sible, and that subtle differencesin experimentalpro- parison stimulus.Under another versionof the method cedure sometimesinfluence what the listener says or of magnitude estimation, the standard is omitted does. When the listener tries to tell us about the relative entirely.The subjecthears a seriesof intensitiespre- loudnessof two sounds,he is subjectto a host of poten- sented in random order, and to these intensities he tially biasing factors. Some of these factors are built assignsnumbers proportioned to theirapparent loudness. into the listener;someare suppliedby the experimenter. Ratio determinationmeans those procedures that aim Someare easyto discover;others are as elusiveas foxes to discoverwhat intensityproduces a loudnessbearing in a forest. The parametersthat affect experimentson a prescribedratio to a givenloudness. The ratio may be loudnessare numerousand sometimesso conflicting that a fraction or a multiple of the standard. it may be impossibleunder any one procedureto opti- We can further divide the method of ratio determina- mize all of them. Consequently,the definitive experi- tion into two classes'(1) The subjectis allowedto ment in this field, if we may hopefor such,will probably adjust the intensity to produce the desired ratio needto involvea multipleattack of the sort that might (methodof adjustment).(2) The experimentersets the balance out the various sources of distortion and bias. intensityand asksthe subjectto say whetherit is too Nevertheless, under a wide variety of conditions highor too low (methodof constantstimuli). Of course, rather similar results continue to be obtained. Provided from a seriesof magnitudeestimations a ratio deter- we are willing to be content with modestprecision, we minationcan be made by a processof interpolation. can predict pretty well what the averagelistener will Consequently,it is possibleto combinethe resultsof say about loudnessin most ordinary situations.It is experimentsthat usethese two differentprocedures. my own beliefthat this predictabilityis sufficientlyhigh Each of these methodshas its advantagesand its to justify the standardizationof a loudnessscale to disadvantages.And, of course,each has many sub- representfor the "standardobserver" the relation be- varieties, someof which are better or worsethan others. tween loudnessand intensity. Merits of the Methods METHODS Let me try to list someof the assetsand liabilities of Three principal methodshave been used to obtain these methods as they seem to be disclosedin the direct estimates of the relation between loudness and studiesof other experimentersand in our work at the intensity:bisection (or equisection),ratio determination Psycho-AcousticLaboratory. (includingfractionation), and direct magnitudeestima- tion. Methods that involve the comparisonof one-rs- Methodof MagnitudeEstimation (ME) two tones and one-rs-two ears are not included here This is the mostdirect and, in someways, the most becausethey rest on assumptionsthat can be tested efficientmethod. Each presentationof the stimulusis only by the direct comparisontechniques that are re- rated numerically,and no information need be thrown quiredto establisha loudnessscale in the first place. The methodof bisection,involving the determination away. Like all procedures,the method of magnitude estimation is susceptibleto various biases, some of of the loudnessthat lies midway between two given whichcan probably be avoidedor counterbalanced.One loudnesses,has its limitations, owing principally to the biasarises from the fact that the subject'sestimates are fact that it leadsonly to an interval scale2 (analogous to the temperaturescales, Fahrenheit and Celsius).Its influencedby the order in which the stimuli are pre- sented. Since the subject usually tries to be self-con- resultsdo not allowus to setup a ratio scale(analogous sistent,what he saysabout a givencomparison stimulus to scalesof weight and length). Ratio scalescan, in dependsto someextent on what he has said about the principle,be determinedby the methodsof fractionation precedingones. This particular bias doesnot affect the and magnitudeestimation, and suchratio scalescan be first judgment he makes, however, and it is therefore usedto predictwhat oughtto happenin a bisectionex- instructiveto comparethese first judgmentswith the • S.S. Stevens, editor, Handbookof ExperimentalPsychology later ones.Actually, thesefirst judgmentsare usually (John Wiley and Sons,Inc., New York, 1951),Chap. 1. consistent with the later ones. Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 165.123.230.136 On: Tue, 25 Aug 2015 06:45:44 MEASUREMENT OF LOUDNESS 817 Another bias arisesfrom the fact that, although the potentiometerso that the