THE
JOURNAL
OF
ACOUSTICAL
THE
OF
SOCIETY
AMERICA
Volume27
Number5
SEPTEMBER.
1955
The Measurement
S.S.
of Loudness*
STEVENS
Psycho-Acoustic
Laboratory,Harvard University,Cambrfigge,
Massachusetts
(ReceivedJune 17, 1955)
This paperreviewsthe availableevidence(publishedand unpublished)on the relationbetweenloudness
and stimulusintensity.The e.videncesuggests
that for the typical listenerthe loudnessL of a 1000-cycle
tone can be approximatedby a powerfunctionof 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 loudnessproducedby a tone at 40 db abovethe standardreference
level, the equationfor loudnessL as a function of the number of decibelsN becomes'logL=0.03N--1.2.
Otherwisesaid, a loudness
ratio of 2:1 is producedby a pair of stimulithat differby 10 db, and this relation appearsto hold over the entire rangeof audibleintensities.
At low levelsof intensity,the loudnessof white noisegrowsmorerapidly than the loudnessof a 1000-cycle
tone, but abovethe level of approximately50 db the two loudnesses
remainmorenearly proportional.The
suggestion
is madethat for all levelsgreaterthan 50 db the loudnessof continuousnoisesmay be calculated
from the equation:logL=O.O3N-t--S,
where$ is a spectrumparameterto be determinedempirically.
HE purposeof this reviewis to examinethe
availabledata on the measurement
of subjective
loudness.It is hopedthat by assemblingthe relevant
informationin one place we may be able to reach a
reasonableconclusionconcerningthe relation between
loudness
and intensity.The variousresultsobtainedby
workersin thisfieldmakeit plain that the scalerelating
loudnessto intensity is not somethingthat can be
determinedwith high precision,but theseefforts also
makeit plain that peopleare able to makequantitative
estimates of loudness and that it is not unreasonable to
try to determinea loudness
scalethat will be representative of the typical listener.
Despite its many pitfalls, I think we can probably
makesenseof this problemprovidedwe are sufficiently
modestin our demands.Not only mustwe renouncethe
hopefor highprecision,but alsowe mustkeepin mind
* This work was carried out under Contract N5ori-76 between
Harvard Universityand the Officeof Naval Research,U.S. Navy
the nature of the central issue. What we want to know
is howloud variousstimulisoundto people.By this we
mean, perforce,what do peoplesay when they try to
describeloudnessin quantitative terms?In askingthis
questionwe are not at the outset trying to solve the
problem of how the ear works or how the nervous
systemperformsits integrations.We are not trying to
countnerveimpulsesnor provea theory.We are merely
looking for the empiricalanswerto a very empirical
question'How do peopledescribesoundswhen we ask
them to usea numericallanguageinsteadof adjectives?
Our interestin askingthis questionmay be academic
or it may be practical.The academicsideof the issue
hasa longhistoryfull of side-takingand polemics.
• The
practicalsideof the problemhad its originin acoustical
engineering.Not long after they had developedthe
conventionaldecibel scalefor measuringsound intensity, the engineers
notedthat equalstepson the decibel
scaledo not soundlike equalsteps,and that a level of
(ProjectNR142-201, Report PNR-168). Reproduction
for any
purposeof the U.S. Governmentis permitted.
• E.G. Boring,Am. J. Psychol.32, 449-471 (1921).
815
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06:45:44
816
S.
S.
STEVENS
50 db does not sound like half of 100 db. Since the
engineeroftenfacesthe problemof communicating
with
periment.Thus bisectionmight providea test of the
generalityof a ratio scale,but by itselfbisectioncannot
a customer,it was soonrealized that there was a need
generate a ratio scale.
for a scale whose numbers would make more sense to
the customer than do the numbers on the decibel scale.
In the tabulationsthat follow only the resultsobtainedby magnitudeestimationand ratio determination
The generationof a loudnessscale is in principle
quite simple.All we need to do is producean array of
soundsand ask a groupof listenersto assignnumbersto
thesesoundsin sucha way that the numbersreflect the
perceivedloudnessof the sounds.In practice,of course,
it turns out that many alternative techniquesare possible, and that subtle differencesin experimentalprocedure sometimesinfluencewhat the listener says or
will be considered.
parisonstimulus.Under another versionof the method
of magnitude estimation, the standard is omitted
does. When the listener tries to tell us about the relative
entirely.The subjecthearsa seriesof intensitiespre-
loudnessof two sounds,he is subjectto a hostof potentially biasing factors. Some of these factors are built
into the listener;someare suppliedby the experimenter.
Someare easyto discover;othersare as elusiveas foxes
in a forest. The parametersthat affect experimentson
loudnessare numerousand sometimessoconflictingthat
it may be impossibleunder any oneprocedureto optimize all of them. Consequently,the definitive experiment in this field, if we may hopefor such,will probably
needto involvea multipleattack of the sortthat might
balance out the various sources of distortion
and bias.
Nevertheless, under a wide variety of conditions
rather similar results continue to be obtained. Provided
we are willing to be contentwith modestprecision,we
can predict pretty well what the averagelistenerwill
say about loudnessin most ordinary situations.It is
my ownbeliefthat thispredictabilityis sufficientlyhigh
to justify the standardizationof a loudnessscale to
representfor the "standardobserver"the relationbetween loudnessand intensity.
Magnitudeestimationmeansthe following:A standard tone is presentedand an arbitrary numberis assignedto its loudness,
e.g.,1 or 100.Thena comparison
toneis presented,and the subjectdecideswhat number
he thinksshouldbe assignedto the loudnessof the com-
sented in random order, and to these intensities he
assigns
numbers
proportioned
to theirapparentloudness.
Ratio determinationmeansthoseprocedures
that aim
to discoverwhat intensityproducesa loudness
bearing
a prescribed
ratio to a givenloudness.
The ratio may be
a fractionor a multipleof the standard.
We can further divide the method of ratio determina-
tion into two classes'(1) The subjectis allowedto
adjust the intensity to produce the desired ratio
(methodof adjustment).(2) The experimenter
setsthe
intensityand asksthe subjectto say whetherit is too
highor toolow (methodof constantstimuli).Of course,
from a seriesof magnitudeestimationsa ratio determinationcan be made by a processof interpolation.
Consequently,it is possibleto combinethe resultsof
experiments
that usethesetwo differentprocedures.
Each of thesemethodshas its advantagesand its
disadvantages.
And, of course,each has many subvarieties, someof which are better or worsethan others.
Merits
of the Methods
METHODS
Let me try to list someof the assetsand liabilities of
Three principalmethodshave 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),
ratiodetermination Psycho-Acoustic
Laboratory.
(includingfractionation),and directmagnitudeestimation. Methods that involve the comparisonof one-rsMethodof 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 comparisontechniquesthat are rerated numerically,and no information need be thrown
quiredto establisha loudnessscalein the first place.
away.
Like all procedures,the methodof magnitude
The methodof bisection,involvingthe determination
estimation is susceptibleto various biases,some of
of the loudnessthat lies midway between two given
whichcanprobablybe avoidedor counterbalanced.
One
loudnesses,
has its limitations, owingprincipally to the
biasarisesfrom the fact that the subject'sestimatesare
fact that it leadsonly to an interval scale
2 (analogous
influencedby the order in which the stimuli are preto the temperaturescales,Fahrenheitand Celsius).Its
sented. Since the subject usually tries to be self-conresultsdo not allowus to setup a ratio scale(analogous
sistent,what he saysabouta 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 scalescanbe first judgment he makes,however, and it is therefore
usedto predictwhat oughtto happenin a bisection
ex- instructiveto comparethesefirst judgmentswith the
• S.S. Stevens, editor, Handbookof ExperimentalPsychology later ones.Actually, thesefirst judgmentsare usually
consistent with the later ones.
(JohnWiley and Sons,Inc., New York, 1951),Chap. 1.
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06:45:44
MEASUREMENT
OF
LOUDNESS
817
Another bias arisesfrom the fact that, although the
subjecttries to judge the comparisonrelative to the
standard,he may be slightlyinfluencedby the absolute
level of the comparisontone.He may overestimatethe
relative loudnessof high intensitiesand underestimate
potentiometersothat the positionof its knobis approximatelyproportionalto loudness.
We find that theresults
obtainedwith sucha "sonepotentiometer"are different
from, and presumablysuperiorto, the resultsobtained
the relative loudness of low intensities. Since in estimat-
The designof a sonepotentiometercan be achieved,
of course,only after we have somenotion of the form
of the loudnessfunction.In principlewe might proceed
by successive
approximations.Beginningwith an arbitrary potentiometer(e.g.,logarithmic)we woulddetermine a first approximationto the loudnessfunction,
which we would then use to modify the characteristic
of the potentiometer.With this modifiedpotentiometer
we would redeterminethe loudnessfunction, and again
modify the potentiometer,and soon. In practice,however, the bias due to a decibelattenuator can probably
be effectivelyeliminatedby substitutinga potentiometer whoseangular turn is roughly proportionalto the
ing subjectivemagnitudesthe subjectis completelyunconstrainedin the choiceof the numbershe assigns,it
turnsout that an occasional
subjectmay giveestimates
that are far out of line with thoseof the group.When
this happens,the distributionof the estimatesis considerablyskewed.And, sincethe arithmeticmeanis not
a good measureto use with skeweddistributions,it is
usually advisable to compute medians rather than
means.
Methodof ConstantStimuli (CS)
with decibel attenuators?
This method differs from magnitude estimation in loudness function first determined with the decibel
that the subject merely indicateswhether the com- attenuator.
parisonstimulusis greaterthan, lessthan, or equal to,
The intensityof the comparisontonewhenthe subject
some criterion, such as "half as loud." Some experifirst hears it at the beginningof his adjustment also
mentershave changedthe comparisonin the direction
seemsto make a difference.This biascanpresumablybe
indicatedby the subject'sresponseand recordedonly
the value the subjectsaid was equal to the criterion. avoided if the experimenter sets the comparisonat
random about the criterion point.
Othershavelimited the subject'sresponses
to "greater"
Whatever methodis used,it probably makesa lot of
and "less"only, and presentedcomparisons
at random
difference
how the subjectsare treated. Their attention
until sufficientjudgmentshave beenmadeto determine
flags
quickly,
and whatever biasingforcesare at work
a "psychometricfunction" from which the "equal"
point couldbe found. In this more rigorousform, the can producemoreformidabledistortionswhen the subject's attention has wandered.Judgingloudnessis at
methodof constantstimuliis relativelytime-consuming
best a delicate,difficult business.
and laborious.
Another sourceof bias arisesfrom the commonpracThe principalsourceof biasin the methodof constant
tice
of combiningresults by averagingdecibels.This
stimuliseemsto stemfrom the fact that the range,level,
would be a proper averageif loudnesswere linearly
order, and spacing of the comparisonstimuli may
relatedto decibels,whichit is not. Sinceit is loudnesswe
influencethe subject'sjudgments.
want to average,it might be better, beforewe average,
There is also an order effect called the "time error"
that seemsto operate when stimuli are presentedin
succession.
This effect is evidencedby the fact that
when two successive
stimuli are physicallyequalthe
secondtends to be judged louder than the first. The
magnitudeof the effectis usuallynot large,and its role
in loudnessratio judgmentsis not easy to determine.
It couldmeanthat whena comparison
tonefollowinga
standardis set to half loudnessthe settingis slightly
biasedin the directionof beingtoo low.
to turn
all the decibel values
into
loudness values
(sones)by meansof a reasonablyapproximateloudness
scale.Then the averagedsonesmay be convertedback
to decibels via the same loudness scale. 4 When
we are
concernedwith the decibelreductionneededto produce
half loudness,the differencebetweenaveragingdecibels
and averagingsonesmay amountto a decibelor more,
dependingon the originalvariability. Evidenceof this
fact is seen in Table I where the median decibel reduc-
tionsreportedby Geigerand Firestone5are smallerthan
the decibel averages.The median is a less efficient
Methodof Adjustment(adj)
statisticthan the mean,but in experimentsonfractionaThis method puts the task of finding the criterion tion the median of the decibel values usually comes
levelunderthe:control
ofthesubject.
He turnsa dialto closer to what we would get if we averaged sones
adjust the comparisontone.A potentialsourceof bias instead of decibels.
in the methodof adjustmentderivesfrom the relation
As a matter of fact, experimentersseem generally
betweenloudnessand the angularturning of the dial.
to have paid too little attention to the problem of
Most experimentershave usedlogarithmicattenuators
on whichthe scale(decibels)
is very nonlinearlyrelated
a S.S. Stevensand E. C. Poulton, "The estimation of loudness
to loudness.
The useof thisnonlinearcontrolapparently by unpractisedobservers,"J. Exptl. Psychol.(to be published).
Stevens,Science121, 113-116 (1955).
leadsthe subjectto set the dial toolow whenadjusting 4* S.S.
P. I-I. Geigerand F. A. Firestone,J. Acoust.Soc.Am. 5, 25-30
for half loudness.It is possible,of course,to arrangea (1933).
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06:45:44
818
S.
S.
STEVENS
averagingtheir data. It is a difficultproblemand there stimulusjudgedrelative to the standard,but also to a
is no universalsolutionto it. There are many potential slight extent it appearsto be judged relative to the
sourcesof skewness
and no oneaveragingprocedurecan "comfortablelisteninglevel." It is as thoughthe effecundo all of them.
tive standardstimulus,from the point of view of the
For experimentson loudnessa defensiblerule might subject, were systematicallyshifted toward the combe this' When a conversion of the decibel values to sone
fortable level. The resulting constanterrors can probvalues eliminates the skewnessin the data, it is advis- ably be ascertainedand eliminatedby a balancedproable to averagein sones.When this conversionto sones cedure that makes each stimulus serve once as the
does not eliminate the skewnessin the data, medians standard and once as the variable. In ratio determinashouldbe computed.Medians must, of course,be used tion, this balancingprocedurewouldcall for both fracwith caution when the number of cases is small.
tional and multipleloudnessjudgments,and an averagAnother generalsourceof distortionin loudnessjudg- ing of the two results.
ments, sharedmore or lessby all the methods,arises
Many of the foregoingproblemsrelatingto loudness
from the preferencethe subjectsshowfor listeningto judgmentsare further discussed
and illustratedin the
moderate levels of loudness.They seem to avoid Appendix.
listening to soundsthat are too loud or too weak by
DATA
adjustingthe comparisonstimulusa little toohighwhen
Tables I to IV record the decibel values that corresit is in a low range and a little too low when it is in a
high range. Consistentwith this they underrate the pondto a loudness
ratio of 2' 1. Other ratioshave not
loudnessof a faint soundrelative to a faint standard, been recordedin thesetables,but wheneveran experiand they overestimatethe loudnessof an intensesound ment has determined a ratio of 4' 1 it has been used to
relative to an intensestandard.The centerof the range, estimatean additional entry of 2' 1 by taking half the
between60 and 80 db above threshold,seemsto be the decibelratio corresponding
to 4:1. These entriesare
preferredlevel for listeningø and it is as thoughdepar- italicized.Ratios greater than 4:1 have not been used
tures from this regionwere "resisted"by the subjects. for thispurpose,
despitethefact that higherratiosoften
The consequences
of this tendencyshowup in almost give resultsquite consistentwith thoseobtainedfor
every experimentthat requiresa subjectto judge one 2' 1 itself.Briefdescriptions
of theexperiments
onwhich
stimulusrelative to another.Not only is the variable thesetablesare basedare givenin the Appendix.
TA•.v.I. Tones--decibelreductionsrequiredfor half loudness.
Method
Year and investigator
1930
Richardson
1932
Laird, Taylor, and
Frequency
and Ross
1024
Decibels re 0.0002 dyne/cm•
and
10--20
subjects
ME
11
CS
10
ME
53-54
51
54-55
18
35-37
13-16
32-39
13-16
20--30
30-40
40--50
50--60
60--70
70--80
80--90
90--100
100-110
24.0
110--
12
5.0
13.5
15.1
19.0
19.0
19.5
20.5
20.0
22.0
10.0
7.2
9.0
7.0
11.0
8.2
10.0
10.0
7.8
9.6
8.8
11.3
7.6
6.7
7.6
Wille
1932
Ham
and Parkinson
350
1000
2500
ME
Warble
tone 500-1000
ME
ME
Warble
Warble
Warble
Warble
tone
tone
tone
tone
CS
CS
CS
CS
7.9
9.4
9.4
9.0
7.8
11.3
1933
1934
Geiger and Firestone
Churcher, King, and
Davies
1936
260•500
500-1000
260---500
500-1000
10.8
9.0
2.1
3.6
3.1
4.1
4.3
3.5
60
1000
1000
adj
31
adj
adj
44 (mean)
44 (median)
800
800
adj
adj
34
30
8.0
7.0
11
11
9.8
8.8
9.2
9.4
6.8
9.1
9.0
9.3
8.9
7.8
9.5
8.6
3.5
7.5
7.0
5.9
9.4
8.8
5.8
9.8
9.5
7.5
11.7
10.0
8.5
10.0
8.0
10.0
12.0
15.0
12.8
13.5
12.3
13,2
11.4
14.0
16.0
10.1
11.0
10.5
Rschevkin
and
Rabinovitch
1 ooo
1000
CS
CS
1951
Pollack
1000
adj
7
3.2
2.5
3.0
4.1
6.2
6.4
7.9
8.0
8.1
8.9
1952
Garner
1000
adj
18
6.3
9.4
13.0
t6.4
17.9
18.0
17.3
17.4
16.6
14.6
5.0
11.6
12.8
11.6
10.6
10.8
10.3
7.4
10.2
13.2
15.6
16.7
17.8
17.9
17.7
16.3
6.9
10.2
7.5
9.4
9.8
8.5
9.3
9.0
9.5
9.5
9.0
9.5
8.7
8.1
10.8
11.1
11.2
12.0
12.0
10.0
12.0
10.3
1953
Robinson
1000
cs
25
1954
Garner
1000
adj
18
P.A.L.
Stevens and
Herrnstein
1024
CS
20
Stevens
Stevens
1000
ME
Poulton
1000
1000
1000
1000
adj 22, 33 (mean db)
adj 22, 33 (mean sones)
adj 22, 33 (median)
ME
8 (numerical estimates)
Poulton
1000
ME
1000
ME
and Poulton
14-17
8
(fractional estimates)
32
9.0
7.8
7.4
13.0
13.5
17.5
13.0
6I. Pollack,J. Acoust.Soc.Am.24, 158-162(1952);T. Somerville,
BBC Quart.3, 11.-16(1949).
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06:45:44
MEASUREMENT
OF
LOUDNESS
819
TABLEII. Tones--decibelincreaserequiredfor twice loudness.
Year and investigator
1932
Ham
Frequency
Method and
subjects
20-30
30-40
40-50
Decibels re 0.0002 dyne/cm•
50-60
60-70
70-80
80-90
90-100
100--110 110-
and Parkinson
Warble
Warble
1932
Geiger and Firestone
1936
Rschevkin
tone 500-1000
tone 250•500
60
1000
1000
and Rabinovitch
CS
19
CS 40-47
adj
adj
adj
8.5
9.5
12.8
12.8
9.8
7.2
6.1
5.3
7.0
6.1
5.3
7.1
6.5
6.6
9.1
8.2
4.8
7.5
7.5
31
44 (mean)
44 (median)
5.6
10.6
9.5
5.3
8.4
8.0
18.5
16.3
17.2
15.5
16.7
10.8
14.0
9.0
11.3
10.8
9.5
10.8
7.5
6.5
10.3
10.3
1000
CS
11
1000
CS
11
1951
Pollack
1000
adj
7
3.5
6.3
9.9
9.7
9.3
17.5
17.2
17.0
17.0
14.6
1953
Robinson
1000
CS
25
P.A.L.
Poulton
1000
ME
8
P.A.L.
Poulton
1000
ME
32
11.0
In the tables the values recorded are the decibel re-
ductionsrequired to producehalf loudnessor double
loudness.Since not all experimentersused the same
standard levels, the various columnsin the tables are
madeto representintervalsof 10 db, and if the standard
usedfell anywherewithin a given 10-db interval it is
recordedin the appropriatecolumn.The justification
for this groupingof resultsrestson the fact that a small
changein the level of the standardstimulusproduces
a negligiblechangein the decibeldifferencerequiredfor
12.0
12.0
11.7
7.5
8.5
7.3
be used to strike somesort of averageand thereby
determinethe decibelchangecorresponding
to a loudness ratio of 2: !. If we could decide which data are
mosttrustworthywe couldweightthem heaviest.Or we
couldusesomeother weighting,basedperhapson the
numberof subjectsused.Of course,after workingon
this problemfor a coupleof years,and after usingthe
variousmethods,I find it difficultnot to form strong
opinionson this questionof which data to trust, but
opinionsare alwaysliable to be wrong.
a loudness ratio of 2' 1.
Sincethe data in Figs. ! and 2 are obviouslyskewed,
Thesetables,then, providean inventoryof most of perhapsthe safestobjectiveprocedureis to consider
what has been done about loudness.The next problem the mediansof the columnsas the most representative
is, what do we concludefrom thesedata?
values.The medianis usuallya goodstatisticto useon
skeweddata, provided the number of casesis not too
Treatment
of the Data
small.
An important questionthat must be decidedat the
The entriesof Tables I and II have been plotted in
outset
is whetherto combinethe data for halvingand
Figs.! and 2. Eachpoint represents
a tableentry, except
doubling.
At low intensities,half loudnessseemsto
that the mean values have not been plotted when
medianswere available and the data for 60 cps have requirea smallerdecibeldifferencethan twice loudness.
At high intensitiesthe reverseis true. But as we have
been omitted.
Theseplotsare encouraging
or discouraging,
depend- already noted, there are biasing forces that tend to
ing on one'slevel of aspirationin thesematters.Never- producethis kind of result. If we assumethat the effects
theless,these are the data, and we must now try to of theseforcesare roughlyequal and opposite,we can
decidewhat they mean. Many possibleschemescould argue that data on halving and doubling should be
TABLEIII. Noise--decibel reductionrequiredfor half loudness.
Year and investigator
Kind of noise
Method and
subjects
1932
Laird, Taylor, and Wille
audiometer buzz
CS
10
1932
Ham
recorded
ME
54
1933
Geiger and Firestone
and Parkinson
noise
40 tone complex
20-30
30-40
40-50
9.3
13.0
15.5
adj 31
(mean)
(median)
4.3
3.9
4.9
5.0
3.2
1951
Pollack
white
noise
adj
7
3.2
1953
Robinson
white
noise
CS
25
7.3
white
noise
P.A.L.J.C.
Stevens
P.A.L. Poulton and Stevens
Decibels re 0.0002 dyne/cm•
50-60
60-70
70-80
80-90
17.5
19.0
18.5
20.5
6.5
8.5
8.0
10.1
7.0
7.0
10.4
10.0
9.3
9.0
11.0
12.0
4.7
5.5
6.7
8.7
8.7
9.6
10.0
12.5
90-100
100--110
21.5
26.0
8.7
7.4
110-
8.3
12.3
12.4
ME
10
white noise
adj
16 (mean db)
6.7
7.1
8.6
white noise
adj
16 (median)
6.0
6.5
7.7
9.4
10.0
10.0
10.0
9.1
9.4
10.0
8.6
8.9
8.5
10.0
8.8
8.6
7.9
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06:45:44
820
S.
S.
STEVENS
TABLEIV. Noise--decibel increaserequiredfor twice loudness.
Year and investigator
Method and
subjects
Kind of noise
1932
Laird, Taylor, and
1933
Geiger and
10-20
20-30
30-40
40-50
14.3
16.0
20.0
19.5
6.2
5.8
5.0
audiometer buzz
CS
10
40 tone complex
adj
adj
31 (mean)
31 (median)
Decibels re 0.0002 dyne/cm •50-60
60-70
70-80
80-90
17.5
19.0
15.0
14.5
90-100 100-110
110-
Wille
Firestone
1951
Pollack
white noise
adj
7
1953
Robinson
white
CS
25
P.A.L.J.C.
P.A.L.
Stevens
Poulton and
noise
5.5
2.7
12.7
5.7
6.0
8.5
7.8
5.0
5.5
7.3
7.5
8.5
7.9
6.2
2.3
4.7
7.7
9.6
13.6
14.5
12.3
11.3
engine noise
CS
25
white noise
ME
10 (mean)
13.1
ME
10 (median)
white noise
adj
16 (mean db)
8.4
adj
16 (median)
7.4
12.0
9.0
9.0
11.0
11.0
9.0
8.7
9.5
8.9
8.8
9.4
Stevens
4.9
4.8
8.1
8.0
8.1
7.7
7.0
6.9
7.9
7.8
6.1
6.5
8.9
9.4
combined.
7 It then turns out that to a fair approxima- the fact that the arithmetic mean is 10.9 db, whichis
tion the decibeldifferencecorresponding
to a loudness higher than the median. The standarddeviation is 3.9
ratio of 2' 1 is constantthroughoutthe whole intensity db. The rangeon either sideof the medianthat includes
range.
If we acceptthis notion,it followsthat loudnesscan
be describedby a powerfunctionof intensity,and we
can write a simpleformula for it. Furthermore,if we
know that loudnessis proportionalto intensityraisedto
a (fractional) powerwe can, in principle,usebisection
data to determinethe exponent.For if loudnesses
have
beenadjustedso that La--L2=L2--L•, and if L= ki n,
then, cancelingthe k's, Ia•--I2=I2--I•
•.
50% of the cases
is 2.4db (the semi-interquartile
range).
Figure 3 showsa histogramof all the data for the
halving and doublingof tones.The valuesare grouped
by intervals of 0.5 db.
Since in Tables I and II some experimentersare
representedby more entries than are other experi-
menters,it might be instructiveto determineone representativevalue for each experimenterby taking the
medianof all his values.When this is done,it turns out
Since we assumethe three intensities Ix, 12 and 13 that the median of these medians is 10.3 db. If we omit
were measuredin the experiment, the value of n is the singleand somewhatquestionablevalue enteredfor
determinable•at least by a processof cut and try. Richardsonand Ross,this median becomes10.0 db.
Some of the difficultiesthat attend bisectionexperiMany other proceduresfor treating these data can
ments will be discussed later.
readily be thoughtof, but againstany givenprocedure
Our problemnow is to decidewhat decibeldifference reasonableobjectionscan probably be urged. There
corresponds
to a ratio of 2' ! in loudness
for tonesin the appearsto be no escapefrom the fact that a certain
vicinity of 1000 cps.The mediansof the valuesin the degreeof arbitrary judgment must attend our choice
tables suggestthat this decibel differenceis in the of a loudnessscalefor our typical "standardobserver."
With this arbitrarinessfully in mind, I shouldlike to
vicinity of 10 db. Actually the medianof all the 178
valuesin Tables I and II taken togetherturns out to be proposethat for the 1000-cycletonewe acceptthe value
exactly10.0db. The skewness
of thesedata is shownby of 10.0 db as the intensity ratio corresponding
to a
loudnessratio of 2:1. The exponentof the powerfunction then becomeslogx02 and the formula for loudness
ß
:
I
TONESO MEOIANS
OFCOLUMNS
becomesL= kI ø'3,whereI refersto energyflux density
g
..
•.
and is assumedto be proportionalto the squareof the
g
....
:'
o
,.--o
.ff..o
-':-o
•
soundpressure.
In termsof soundpressure
p the formula
•
ß
ß
• O
is L-k'p ø.6.If we measureI in unitsequalto the referencelevel of 10-•6 watt per cm2and defineL= ! sone
,
,
,
,
,
,
,
,
ß
HALF
,
LOUONESS
ß
e
ß
ß
.
when 1-40
e
e
•
ß
ß
db--then
k becomes 0.06 and we have
L=0.06I ø-3.Or, if we measure the stimulus as the
e
e
number of decibels N above the reference level we can
write
Fro. 1. The decibelreductionsrequired to producehalf loudness.Each point representsan entry from Table I, except that
mean values are omitted
when medians are available.
The data
for 60 cps are omitted, and the two rows of entries for Poulton
involving eight subjectshave been combined.The opencircle to
the right of each column indicatesthe median of the column.
ßE. C. Poulton and S.S. Stevens,J. Acoust.S•. Am. 27, 329331 (1955).
logL = 0.03N-- 1.2.
This is a simpleformulato use,but for mostpractical
work it will probably sufficeto rememberthat the
loudness
of a 1000-cycle
tonegoesup by a factorof two
for each 10 db-increase in the stimulus.
It will be noted that the chief difference between this
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06:45:44
MEASUREMENT
OF
LOUDNESS
revised sone scale and those previously constructed
concerns
the slopeof the functionat levelsbelowabout
40 db. Over this rangethe old scaleswere steeper,and
their steepnessincreasedas the thresholdwas ap-
821
TONES ß HALVING AND DOUr. INC
-
MEDIANß IO.Odb
MEAN '
IO.9
__
db
proached.
This steepness
is consistent
with the results
of experiments
on the halvingof loudness,
but if we
confineour attention to halving alone we are viewing
6
7
B
9
I0
II
12
13
14
15
16
I?
18
19
20 21
•.•
93
24
DECIBEL CHANGE REQUIRED FOR 2:1 LOUDNESS RATIO
only onesideof the coin.The outcomeof experiments
on the doublingof loudnesssuggests
that the loudness Fro. 3. Histogram of the resultsfor the halving and doubling
functionbecomes
lesssteepnear threshold,rather than of the loudnessof tones.The data of Figs. 1 and 2 are combined,
more steep.Sincehalvingand doublingappearto be making 178 entries.
subjectto a biasthat affectsthemin opposite
directions,
it seems reasonable to let the loudness scale be detersupportto the conclusionthat a loudnessratio of 2:1
corresponds
to an intensity ratio of 10 db. I used a
minedby a compromise
betweentheresultsof thesetwo
procedures.
The simplepower-function
loudness
scaleis variety of procedures,including the making of direct
loudnessestimatesover a range as great as 90 db at a
sucha compromise.
singlesitting.
LOUDNESS
AT THRESHOLD
Figure 4 showssomeof the resultsobtainedwhen the
Fromtheforegoing
equationwe seethat at the thresh- estimates were made relative to standard intensities of
old of hearing,loudnessis not zero.If we assumethat medium level. The median of the estimates of 18 subto inthe thresholdis equivalentto the referencelevel, then jectsshowsthat the loudnessratio corresponding
at threshold L--0.06 sone. That the threshold loudness
tensitiesseparatedby 90 db is approximately1000to 1.
The slopesof the straightlinesin Fig. 4 are suchthat
10 db corresponds
to a 2:1 loudnessratio. The fact that
,
,
ß
,
ß
TWICE
0
MEDIANS
LOUDNESS
TONES
ß
ße
•
ß
ß
ee
*o ;o
I O
I
standardchosen.In other words,the slopeof the function obtainedis alteredwhen the standardis shifted,so
that slopescan be obtainedthat are both greaterand
lessthan that of the straight lines in Fig. 4.
ß
ß
I
80-10
i
•-40
i
40-50
,•3UND
i
•10'-•0
i
JO'"'70
PRESSURE: LEVEl.
i
70-80
IN
i
80-•0
i
90-100
the data in Fig. 4 fall slightly below this loudnessfunction at the low intensitiesand above at the high intensities has been shown to be due to the level of the
ß
eeO
eø eeO I
0
OF COLUMNS
ß
IO0-11O
DECIBELS
I00
I000
Fro. 2. The decibel increasesrequired to produce twice loudness.Each point representsan entry from Table II, except that
mean valuesare omitted when mediansare available, and the data
for 60 cpsare omitted. The opencircle to the right of eachcolumn
representsthe median of the column.
,Olr
should turn out to be a small fraction
entirely unreasonable,provided loudness is fundamentally quantal in nature--provided,in other words,
loudnessis not infinitely divisible.What this meansis
simply that if a sound is heard at all, its loudnessis
finite. As Robinson8suggests,
when intensity crosses
the
threshold,loudnesscomeson with a jump.
Perhapsit will provepossibleto obtaina moredirect
estimate of the sonevalue of a just audible tone, but
until that is done the thresholdloudnessgiven by the
formulais probablyas goodas any.
SO ME
FURTHER
Y t 'oo
of a sone is not
EXPERIMENTS
z
•
Lo
•o.•
t
/o
•
•
,7"L.........
• /
•u•l•
],o
•
qLo
e
<
0.1
After the foregoingsummaryof the loudnessexperi•0
40
•
•
IO0
120
•UND
PRESSURE LEVEL IN DECIBELS
ments was prepared,I undertooksomeadditional experimentson the magnitudeestimationof the loudness Fro. 4. Magnitude estimations relative to a fixed standard
of 1000-cycletones?Theseexperimentsgive additional stimulus.Upper curve: the loudnessof the standard(80 db) was
a D. W. Robinson,Acustica3, 344-358 (1953).
9 S.S. Stevens,"The direct estimation of sensory magnitudes:loudness," Am. J. Psychol.(to be published).
called10. Lower curve:the loudnessof the standard(90 db) was
called10. The pointsare the mediansof 36 loudness
judgmentstwo by each of 18 subjects.The vertical lines mark the inter-
quartile ranges.
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06:45:44
822
S.
S.
STEVENS
In another experimentI dispensedwith a standard tensities,and to eachintensitythe subjecthasassigned
stimulusand merely presentedto each of 26 subjects a numberrepresentinghis perceptionof the loudness.
a seriesof eight intensitiesspaced10 db apart from 40 The variability isfairly large,but the medianjudgments
to 110 db. The order of the tones was different for each
fall close to the loudness function and confirm our con-
subjectand eachtonewaspresentedtwice.All but one clusionthat loudnessis a powerfunctionof intensity.
of the subjects
hadpreviously
madeloudness
judgments
NOISE
in othertypesof experiments.
The instructions
wereas
follows:
As shownin Tables III and IV, the data for noiseare
I am goingto give you a seriesof tonesof different not as numerousas those for tones. Although these
intensities.Your task is to tell me how loud they sound tablesincluderesultsonothertypesof noise,in trying to
by assigning
numbersto them.To turn on the toneyou analyzethe data we shallconsideronly thosefor white
simplypressthe key. You may pressit as oftenas you noise. From thesevalues for white noise,plotted in
Fig. 6, it wouldappearthat the decibelchangerequired
like.
When you hear the first tone, give its loudnessa for a loudnessratio of 2' 1 may be somewhatlessthan
10 db. The median of all the values for white noise is
number--anynumberyouthink appropriate.I will then
8.5
db and the meanis 8.4 db. I might point out, parentell youwhento turn onthe nexttone,to whichyouwill
thetically,that thesevaluesare strikinglysimilarto the
alsogive a number.
to produce
Try to make the ratios betweenthe numbersyou decibelchangeI havefoundto be necessary
a
2'
1
difference
in
the
apparent
brightness
of a white
assignto the differenttonescorrespond
to the ratiosbe-
light.
On the basisof the evidencein Fig. 6 we might well
concludethat, with white noise, the decibeldifference
hear it.
requiredfor a 2' 1 ratio in loudnessis relativelyinvariIn orderto combinethe resultsfor the differentsubjects ant with intensity.This is the conclusion
we reachedfor
it wasnecessary
to bring the estimatesinto coincidence pure tones.As we have seen,it is an attractive conclutween the loudnesses
of the tones.In other words,try to
make the numberproportionalto the loudness,as you
at a givenlevel (80 db) by multiplyingby an appropriate factor--a different factor for each subject. The
WHITE:
median
estimates
werethencomputed
for eachlevel.
The results are shown in Fig. 5. The straight line in
:o
z
Fig. 5 hastheslopeof theloudness
function(10db- 2:1
•
loudness).
This experimentgoesabout as far as it is possibleto
o
ioo
I
I
I
!
I
ß
o
ß
ooo
o
ß
o
o
o
•
=•
ß HALVING'
0 DOUBLING
oo
•
,o
•
ß
.'
ß
o
ß
e•
o
go towardgettingsubjectsto makedirectquantitative
judgmentsof theloudness
of sounds.
Here wehavedone
nothing more than presenta "random" seriesof in!
,'
NOISE
I
80 10-20 •0-•0
i
i
i
I
i
I
I
I
•0-40 40-50 50-60 60-70 70-80 80-90 90-100I00-110
SOUND
PRESSURE
LEVEL
IN
DECIBELS
Fro. 6. The median decibelvaluesfor the halving (filledcircles)
and doubling (open circles)of white noise. The data are from
I
Tables
III
and IV.
--NO
DESIGNATED
STANDARD
sion,becauseit entailsa simplepower-functionrelation
between loudnessand intensity. Furthermore, if the
z
loudness of tones and the loudness of noise are both
•- •o
- /I
0.1
•,'•'• *•_
,: , , ,•ou:.,o;
o.,-,..,'•-,•ø,'
/
.)O
40
60
SOUND PRESSURE
80
LEVEL
IO0
120
IN DECIBELS
powerfunctionsof intensity,then the ratio of thesetwo
loudnesses
is itself a powerfunctionof intensity.
But herewe run into trouble, for this last conclusion
doesnot agreewith the resultsof experimentsin which
subjectshave been asked to equate the loudnessof a
tone to the loudnessof a white noise. These experiments•øare not themselvesin particularly goodagreement, mainly becausethe task of equatingthe loudness
of a pure tone to the loudnessof a white noiseis difficult.
The difficulties are not unlike those encountered in the
frustratingart of heterochromatic
photomerry.Subjects
Fro. 5. Magnitude estimates of loudnessmade with no fixed seemto disagreemorewhenthey try to match the loudstandard.The eight differentintensitieswerepresentedin irregular nessof a tone to that of a noisethen when they try to
order, and the subject estimated the loudnessby numbersof his
own choosing.These numberswere multiplied by appropriate
•0F. H. Brittain, J. Acoust.Soc. Am. 11, 113-117 (1939); H.
factors to make each subject'sestimatesat 80 db average10. Fletcher and W. A. Munson, J. Acoust Soc.Am. 9, 1-10 (1937);
The points represent medians of the resulting values and the G. A. Miller, J. Acoust. Soc. Am. 19, 609-619 (1947); D. W.
vertical lines representthe interquartile ranges.
RobinsonS;I. Pollack?5
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06:45:44
MEASUREMENT
set one tone to half the loudness of another.
OF
LOUDNESS
823
But all the
experimentsrelating tonesto noiseseemto concuron
one point, namely, that the ratio betweenthe loudness
of tonesand noiseis not a powerfunctionof intensity.
Figure 7 showsthe data as I am able to read them
from publishedgraphs. The curvesrepresentthe difference in decibelsbetween the level of a 1000-cycle
tone and that of a white noise that matches the tone in
loudness.Why are thesecurvessofar apart? Is it owing
to differencesin the methodsusedto measurethe noise,
Fro. 8. Resultswhen the subjectadjuststhe loudnessof a
or to the large variability amongsubjects,or to both?
whitenoiseto equalthat of a 1000-cycle
tone.Eachpointis the
Whatever the reasonfor thesediscrepancies,
all five of medianof 24 adjustments--two
by eachof twelvesubjects.The
the curvesare somewhatsimilar in that they are more verticallinesmark the interquartileranges.
or less concave downward.
Three
of the curves show a
a general rule that subjectsoverestimatean intense
variablerelativeto an intensestandard--andthat they
do the opposite at low levels. Consistent with this
generalrule, when the subjectadjustsa stimulusto be
equal to an intensestandardhe tendsto undershoot,
and he tends to overshoot when the standard is weak.
Thus, in the extreme case,when the standard borders
on painful, the subjectdoesnot err in the directionof
•,/ FLETCHER
AND
MUNSON
TONE--NOISE_.
LOUDNESS
,•
'
•
SOUND PRESSURE
'
•
settingthe variabletoo high.As we shouldexpect,he
MATCHES
'
,'oo
LEVEL OF NOISE
Fro. 7. The results of five experientsin which,lthe loudness
errs in the directionof settingit too low.
We can predict, therefore, that it makes a difference
whether the variable stimulus is the tone or the noise.
of a tone was matched to the loudness of a noise. All but Fletcher
A convincingdemonstrationof this differencewas ob-
and Munsonusedan approximatelywhite noise.In theseexperimentsit appearsthat the variable stimuluswas the tone.
tainedin a preliminaryexperiment
on a dozensubjects
usingthe methodof adjustment.Onetime the subject
adjustedthe tone, and anothertime he adjustedthe
definitemaximum.The curvesare not the straight lines
they would have to be if, for both tones and noise,
loudnesswerea powerfunctionof intensity.
It is clear from Fig. 5 that, at low levels,a given increasein the stimulusproducesa greaterchangein the
loudness of a white noise than in the loudness of a tone.
This fact seemscertain.Apparently,whenwe start from
below thresholdand increasethe intensity of a white
noise,the amount of energy surpassingthe threshold
increasesvery rapidly for a time. Sinceall the energy
that exceedsthe thresholdpresumablycontributesto
loudness,it is not unreasonablethat the loudnessof
white noise should grow rapidly over the first few
decibelsabovethreshold.This rapid growthof loudness
is not inconsistent with at least some of the data in
Tables III and IV--those
of Pollack and of Poulton
and Stevens.
noise.His two types of settingswere inconsistentwith
eachother at the high intensitylevels,and they were
equally inconsistent,in the oppositedirection,at the
low intensity levels.The discrepancies
at 30 and at 100
db were of the order of 5 db.
Followingthis encouraging
outcome,I improvedthe
apparatusandundertookto testanotherdozensubjects,
usingthe noiseas the variable.The subjectuseda "sone
potentiometer"to adjust the intensity of a 10-kc band
of white noisefed througha pair of PDR-10 "extended
range"earphones.
The response
of theseearphoneswas
approximately"flat" up to 7500 cps and down about
10 db at 10 kc. A rotary switchpresentedthe tone and
the noisealternately. Each stimuluspresentationlasted
about 1.3 sec, with a silent interval of about 0.5 sec.
The levels of tone and noise were measured with
a
Ballantine electronicvoltmeter (model 300), and for
our presentpurposethe sound-pressure
level is consideredthe samewhen the voltage readingis the same.
(Actually, when measuringwhite noisethe Ballantine
read about ! db lower than a thermocouple.)The onset
and declineof the stimuli weregovernedby an electronic
On the other hand, the fact that someof the curves
in Fig. 7 turn down at the higher intensitiessuggests
that over the upper rangethe loudnessof white noise
growslessrapidly with intensitythan doesthe loudness
of a 1000-cycletone. But in TablesI to IV there is no
evidencefor a lessrapidgrowthin the loudness
of noise.
Obviouslywe face a discrepancythat needsexplaining. switch set for a 5-msec rise-time.
The resultsare shownin Fig. 8. We seethat when the
A possibleexplanationcan be foundin the fact that
noise
is varied there is no tendency for the curve to
in the experiments
underlyingthe curvesin Fig. 7 the
variable stimulus seems to have been the tone and not
turn down at the high intensities.In this respect,the
with the evidencein
the noise.As we havepreviouslynoted,it appearsto be data in Fig. 8 are moreconsistent
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06:45:44
824
S.
S.
STEVENS
Tables I to IV than are the data in Fig. 7. This is an usedsonepotentiometersinsteadof decibelattenuators
for this purpose.
encouragingfact.
It is a distressingfact that my resultsdo not agree
Nevertheless,the problemof decidingon the exact
form of a standard loudness scale for white noise can with thoseobtainedby Garner.The intervalsproduced
probablynot be settledon the basisof our present by Garner's18 subjectsare morenearly equal in decibel
knowledge.
At any rate,my ownimpression
isthat there sizethan are the intervalsproducedby my 49 subjects.
are still too many looseends and inconsistencies.
It I cannot prove it, but I rather suspectthat Garner's
appears,however,that at the low intensities
the scale data are somewhatbiasedby the fact that the subjects
for noisewill prove to be decidedlysteeperthan the couldjudge the time it took for the motor to drive the
scalefor the 1000-cycletone, and that at the higher attenuator through the 40-db range (80 seconds)and
intensitiesthe loudnessscale for noise will have ap- that their fractional judgmentswere perhapsunconproximately
thesameslopeastheloudness
scalefor the sciouslyinfluencedby the judgmentof fractionalvalues
of this time interval. An additional,but slight,sourceof
1000-cycletone.
Plotted against a commonscaleof sound-pressurebias lies in the fact that Garner averageddecibels.The
level, the two loudnessscaleswill crosseach other, bisectionexperimentis onein whichaveragingby sones
probablyin the vicinityof 30 or 40 db. This crossing has a demonstrableadvantage.4
In anotherexperiment,I tried to checkthe validity
point is one of the importantitemsthat needto be
of the results by having the experimenterreset the
pinneddown.
The principlementionedaboveregardingthe neces- variableintensitiesto variouslevelsand then requiring
sity of usingboththe toneandthenoiseasthevariable 25 subjectsto estimatethe relative spacingbetweenthe
by means of a set of adjustable
in a loudnessbalanceappearsto have wide generality. resultingloudnesses
An experimenton loudnessbalanceshoulditself be markersona steelbar. The subjectsetthe spacingof the
balanced.
Thisprinciplesuggests
that, in thedetermina- markers to representhis perceptionof the spacingof
The data obtainedagreequite precisely
tion of equalloudness
contours,eachof the frequencies the loudnesses.
to be comparedshouldbe made to serveas both the with the soneaveragesof the data obtainedunder the
fixed stimulus and the variable stimulus. In measuremethod of adjustment.The bar and markersconstitute
ments in which this balancedprocedurehas not been a rather usefuldevicefor this purpose.
One of the first things I discoveredin these studies
followed we have reason to expect that systematic
errors may be present--errorsthat are presumably was the disconcertingfact that the point at which a
minimal near the middle of the loudnessrangeand that subjectbisectsa 40-db interval dependson the orderin
which he listens to the tones. For example,when he
increaseas we go to very faint or very loud tones.
hears them in the ascendingorder from faintest to
BISECTION
EXPERIMENTS
loudesthe may judge the midpoint to be about 32 db
Let usnowreturn to the problemof the bisection(or from the bottom of the range.When he hearsthem in
orderhe may judgethe midpointto be about
equisection)
of loudnessintervalsin orderto seehow descending
27
db
from
the bottom. In an appendixto his main exwell the experimentalresultsagreewith thoseof experiperiment,
Garner
verified this order effect,but in what
ments on ratio determination.Two major studies of
bisection have been made, one by GarnerTMand the orderor ordershis subjectspressedthe keysin the main
other by me (unpublished).In both experimentsthe experimentis not known.
Becauseof this order effect, I found it necessaryto
subjectsatbeforea setof five keyswhichhe pressed
in
control
the order in which the stimuli were presented.
orderto producethe tones.The intensitiesof the tones
producedby the two end keyswerefixed,say, 40 db For the part of the experimentthat we shall consider
apart,andthe subjectadjustedtheintensities
produced here, the subjectwasrequiredto useboththe ascending
orderand to try to adjust the loudnesses
by the intermediatekeysin orderto dividethe 40-dbin- and descending
so
that
the
intervals
soundedequal in both directions.
terval into four equal-appearing
stepsin loudness.
Garner'ssubjectsadjustedthe intermediatetonesby
TABLE V. Sample results obtained by subjects who set one
throwing switchesthat controlledmotor-drivenat- tone to the midpoint of the loudnessinterval between two other
tenuators(2 sec/db). Before the beginningof a given tones. The predicted values were calculated on the assumption
the decibelchangecorrespondingto a 2:1 loudnessratio is
run, all the variable toneswere set at one or the other that
10 db. All values are in decibels re standard reference level.
end of the 40-db range.In mostof my experimentsthe
subjectsimplyturnedattenuators(novisibledials)and
Interval
Midpoint
Midpoint
Number of
bisected
obtained
predicted
subjects
before each run the attenuators were randomly posi16-56
42.5
46.9
7
tioned.My three attenuatorshad differentsizedsteps36-76
65.3
66.8
12
1.5 db for the lower quarter point, 1 db for the mid56-96
86.2
86.8
45
point, and 0.5 db for the upper (threequarter)point.
Actually,I nowbelieveit wouldhavebeenwiserto have
90-120
109.6
111.7
12
101-130
111.5-140
119.3
130.6
121.8
131.8
10
12
n W. R. Garner, J. Acoust.Soc.Am. 26, 73-88 (1954).
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06:45:44
MEASUREMENT
OF
LOUDNESS
825
It ispossible,
of course,to controlthe orderof presenta- at least 140 db the loudnessof a 1000-cycletonegrows
tion of the tones,but it is not possible,unfortunately, as a power function of intensity.
to controlthe degreeto whichthe subjectpaysattention POSSIBLE ENGINEERING
RULE FOR MEASURING
to the two orders.He may or may not chooseto ignore
THE
LOUDNESS
OF NOISE
one of them.
The determinationof a scaleto expressthe loudness
Table V givessomeof the resultsobtained,together
judgmentsof a typical listenerconfrontedwith a 1000with the valuesthat wouldbe predictedon the assumption that a loudnessratio of 2' 1 corresponds
to a decibel cycletoneis only a first steptowardthe developmentof
a procedurefor measuringthe loudnessof soundsin
changeof 10db.It turnsout that themidpointto which
general.In principlewe can usethe 1000-cycletone as a
the subjectsadjustedthe attenuatoris slightlylower
yardstickand evaluatean unknownnoiseby finding
thanthepredicted
midpoint.Thissameeffectwasfound the level of the 1000-cycletone that matchesthe unby Garnerwhenhe comparedhis bisectiondata with known noise in loudness. But such loudness balances
the fractionationdata obtainedfrom the samesubjects.
Sincebisectionand ratio determination do not quite betweentone and noiseare difficult to perform in the
field. For one thing, in order to avoid systematicbiases
agree,we must either choosewhich methodwe will we need to make the noise as well as the tone serve as
trust,or elserejectthemboth.Garnerdecidedto accept the variable stimulus.
the verdict of bisection and to correct the ratio deter-
What we need, therefore,is a reasonable,workable
minationsaccordingly.
I wouldpreferto do theopposite.
procedurefor transformingsoundlevel measurements
My mistrustof the bisectiondata is basedon two coninto sones.The procedures
t5 now in vogue give fair
siderations.One is the rather largedifferences
obtained
results
but
they
are
not
particularly
simple.In addition
as a function of the order in which the tones are preto assuminga loudnessfunctionfor the 1000-cycletone
sented.The other is the old question
t2whetherthe subthat is significantlysteeperthan the present evidence
jectisableto bisectanintervalwithoutbeinginfluenced
indicatesit oughtto be, they makethe further assumpby theratiosaswellasthe differences
betweenthe loud- tion that the total loudness of a wide-band noise can be
nesses.Does he set the middle tone b sothat a-b-b-c,
found by adding togetherthe loudnesses
of certain of
or doeshe set if so that a/b-b/c? Or, doeshe comproits componentbands (octave bands or mel bands).
misebetweenthesetwo criteria and therebyproducea
This assumedadditivity certainly needsa more thorsettingof b that is closerto the half-waypointbetween oughvalidation than it has yet received.
a and c in decibels than it would otherwise be?
Perhapsa better way to approachthe problemof the
What lookslike a tendencyamongsomesubjectsto
loudnessof noiseis first of all to break the probleminto
slipoverfromanintervaltowarda ratiojudgmentseems
two separateparts.We may then ask (1) how doesthe
to be particularlystrongwhenthe lowertone of the
loudnessof noise depend on the over-all level, and
interval is within about 20 db of threshold. t3
In view of the difficultiesthat attend bisectionjudg- (2) how doesit dependon the frequencycomposition?
It appearsnot unlikelythat the first of thesequestions
ments,the agreementshownin Table V betweenthe
canbe givena simpleanswer.As moreand more results
obtainedand the predictedbisections
might be conaccumulate,I becomeincreasinglyconvincedthat all
sideredreasonablysatisfactory.There is a similarly
continuousnoisesof engineeringinterest follow to a
goodagreement
betweenthe bisections
we wouldpregoodapproximationthe samesimplerule.This rule says
dict and thoserecentlyreportedby BlackTM
who used
that, for all levels above about 50 db, the loudnessL
a noisewith a spectrumthat sloped6 db per octave.
changesby a factor of two when the over-all level N
SinceBlack'ssubjectslistenedto the noisesonly in
changesby 10 db. In other words, for a constant
ascending
order, the midpointobtainedtends to be
spectrumand for N> 50 db
higherthan the midpointpredicted.This is just what
we shouldexpect.
The data in Table V also demonstrate another im-
logL=O.O3Nq-S
where S is the spectrumparameter.
portantfact, namely,that loudness
continues
to grow
The value of S will dependupon the make-up of the
as a powerfunctionof intensityup to at least140db. spectrum(includingits phaserelations)and will there-
At this level the stimuli, under binaural listening,are
fore needto be evaluatedempirically.As we have seen,
truly painful.Oneof a pair of earphones
burnedout at the value of S for a 1000-cycletone is -1.2. In Figs. 7
this level! If, as somehave supposed,
loudnessreaches and 8, we seethat above 50 db a white noiseis as loud
a ceilingabovewhichit ceases
to grow,thisupperlimit as a 1000-cycletone when the noiseis roughly 10 db
mustbe beyond140db. It appearsthat overa rangeof lower than the tone. Hence for white noise the value of
•' S.S.
Stevens and H. Davis, Hearing, Its Psychologyand
Physiology
(JohnWiley andSons,Inc., New York, 1938).
•aNewman,Volkmann,and Stevens,Am. J. Psychol.49, 134137 (1937).
14John W. Black, "Control of the soundpressurelevel of
voice,"Joint ReportNM 001 104500.42,U.S. Naval Schoolof
AviationMedicine,Pensacola,
Florida (February15, 1955).
the spectrumparameterS is approximately-0.9. We
obtain this result by a formula that givesthe value of
S from the results of matching the 1000-cycletone to
•5Beranek, Marshall, Cudworth, and Peterson,J. Acoust. Soc.
Am. 23, 261-269 (1951); F. Mintz and F. G. Tyzzer, J. Acoust.
Soc.Am. 24, 80-82 (1952).
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06:45:44
826
S.
S.
STEVENS
the noise in loudness. If N•000 is the level of the tone in
Over the rangeof levelsexploredthe resultsfrom 21
decibelsand Ns is the level of the noisein decibels,the subjectsusingthe methodof adjustmentfor halving
formula becomesS=O.03(N•ooo-N,,)--l.2. The values and doublinggive a loudnessfunctionvery much like
of S andN will dependof courseuponthe characteristics the oneI haveproposed.Aboveabout 50 db the slope
of the meter. Provided, however,the meter is properly is approximately10 db for a 2' 1 loudness
ratio, and becalibratedto read the sound-pressure
level of the 1000- low about 50 db the resultsfor halving and doubling
cycletone, the loudnessL computedby theseformulas showtheusualdivergence:
halvinggivesa steeperslope;
is invariant with meter characteristics(expectfor cer- doublinggivesa flatter slope.
tain nonlinearities).
Quietzschalso studied37 widely different types of
In principlethe valuesof S for otherspectracanbe noiseby meansof (1) loudnessbalancesagainsta 1000obtained by a similar procedure:a loudnessbalance cycletone,(2) soundlevelmeasurements
with twotypes
made againsta 1000-cycletone. The optimal level for of meters,and (3) measurementsbasedon the summamakingthis loudnessbalanceis in the vicinity of 70 db tion of the loudnessin separateoctavebands.Among
(the comfortablelevel for listening) becausein this other things,Quietzschproposescertain correctionsto
regionthe bias in loudnessmatchingis minimal. By make the summationtechnique[procedure(3)• agree
followingthis procedurewe couldreadilydeterminethe more closelywith the resultsof direct loudnessbalance
value of S for a variety of representativespectraof [procedure(1)•.
engineeringinterest.
From the resultsof Quietzsch'sprocedures(1) and
Alternatively, we could perhapswork out a method (2) we can determine the value of what I have called
for evaluatingS from an octave-or a mel-bandanalysis the spectrumparameterS for his variousnoisesand for
of the noise. But we would first have to determine the
histwo typesof meters.Unfortunately,in balancingthe
properrule for combiningthe loudnesses
of the separate tone against the noise he varied only the tone, but
bands.This may provemoredifficultthan an evaluation despitethis possiblesourceof bias we learn many inof S for representativespectra by a direct loudness structive facts from his experiments.Applying our
balance.In any case,I believe that the equation here formula for S, we note that when he usesa standard
proposedwill lead to a more valid assessment
of the soundlevel meter (DIN-Lautst•rkemesser) the value
loudnessof noisethan the proceduresnow in use.
of S rangesfrom very near - 1.2 (the valueof S for the
For many purposeswe do not needto know the value 1000-cycletone)to about --0.5, with an averagevalueof
of S. In order to determine by what ratio we have about --0.85. When he usesa particularpeak-reading
changedthe loudnessof a noisewe needto measureonly meter (Ger•uschspannungsmesser)
with a frequency
the sound-pressure
level beforeand after, providedthe characteristicresemblingthe human threshold, the
spectrumremainsconstant.For example,a 20-db re- valueof S rangesfrom -- 1.3 to --0.74, with an average
duction in the over-all level of a noise will decrease its
value of about --1.05. These figuressuggestthat the
loudnessto 25% of the original value. Only when a peak-readingmeterhasan advantagein that it indicates
significantchangeis made in the spectrumwould we a more nearly invariant spectrumparameter.
need to worry about S, and then we would face the
Ideally, what we would like to have is a meter whose
empiricalproblemof evaluatingS either by meansof a readingswould always lead to a value of S--1.2,
loudnessbalance,or by a properrule for the combining whichwouldmeanthat it wouldgivethe samereading
of loudnesses
in separatefrequencybands.
for a 1000-cycletoneas for a noise,providedthe noise
This schemedoesnot depend,of course,upon the use and the tonewereequallyloud. If we wereto limit our
of the 1000-cycletone as a referencestandard, and it concernto levels above 50 or 60 db, couldwe combine
may ultimately prove better to chooseas the reference in a singledevicea set of characteristics
(frequency
sound a spectrum that can be more easily balanced response,peak indication, time constant, etc.) that
againstother noises.In the meantime,the potentiality would allow us to measure the loudness level of noise
of the simplifiedprocedurebasedon the power-function with reasonable
accuracy?This is an old question,I
relation betweenloudnessand intensity invites serious realize,but onethat may needre-examination.
If such
consideration.The procedurehas the advantagethat it a devicecouldbe constructed
it wouldthenbe a simple
explicitly separatesthe problem of the dependenceof matter to calibrateits scaleto readdirectlyin sones.
In the meantime it would seem advisable to use the
loudnesson intensity from the problemof its dependenceon frequencycomposition.
equationlogL=O.O3N-t-S,and to evaluateS eitherby
As this paper goesto.pressthere appearsan interest- directloudness
balanceor by an improved"summation
ing new study by Quietzsch which is relevant to our method."
presentconcern.Quietzschcarriedout experimentson
APPENDIX.
DESCRIPTION
OF THE EXPERIMENTS
the halving and doublingof loudness(1000 cps,30 to
80 db) and on various proceduresfor measuringthe
AlthoughMerkeP7 (1889) appearsto have beenthe
loudness of different kinds of noise.
•6G. Quietzsch,Acustica$, 49-66 (1955).
firstto performan experiment
in whichthesubjectwas
•7j. Merkel, Phil. Stud. $, 499-557 (1889).
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06:45:44
MEASUREMENT
OI•
LOUDNESS
827
requiredto estimatethe ratio betweentwo loudnesses, Ham and Parkinson
• producedtonesby meansof a
it was not until the developmentof the modern elec- loudspeakerand had groups of untrained subjects
tronic art that systematic studies could be made to (collegestudents)estimatethe percentageof the louddeterminethe dependenceof loudnesson stimulusin- nessthat remainedafter a tone had beenreducedby a
tensity. Merkel dropped balls from various heights givennumberof decibels.This is oneform of the method
onto a soundingblock in order to produce different we have called magnitudeestimation (ME). From the
sound intensities.
curves plotted from these percentageestimatesit is
Richardsonand Ross•s (1930) used electricalequip- possibleto determine the decibelreductionrequired for
ment to producetonesfor this purpose.The observer half loudness.For this interpolation procedure, the
heard tonesof two different intensities,and he was re- logarithm of the estimatewas plotted against decibels.
quired to rate the loudnessof one of the tones as a The resultingplot is reasonablyrectilinear, and the
fraction or multiple of the loudnessof the other. Al- fitting of a curve to the data is not difficult. This curve
though their results were not reported in acoustical was usedto estimatethe half loudnessand the quarter
quantities,from their statedvaluesof telephonecurrent loudnessreductions.Somearbitrary judgment is necesit can be determinedthat the averagereductionin in- sarily involvedin this process,but it probablyprovides
tensity neededto reducethe loudnessby a factor of two a fair estimateof the half-loudnesspoint.
was about 12 db. This is the value entered in Table I.
In the experimentsof Ham and Parkinson some
Laird, Taylor, and Wille•9 (1932) were among the degreeof biaswasprobablyintroducedby the fact that
first to make fractional estimates of loudness over a wide
the comparisonstimuli were always presentedin derangeof intensities.For their tonal stimuli they useda scendingorder, instead of in a random order.
Ham and Parkinson also used another method to
2-A audiometerwhich providesstimulusintensitiesin
5-db steps.Their procedurewasa simplifiedmethodof determine the decibel difference for ratios of 2:1--a
constantstimuli. Unfortunately,their resultsare sofar methodof constantstimuli (CS). The subjectwasgiven
out of line with thoseof mostotherexperimentersthat a standard and eight comparisonlevels and was retheir data have beendeliberatelyignoredby thosewho quired to pick the one of the eight whoseloudnesswas
have tried to constructloudnessscales.An attempt was nearestto half (or double) loudness.They also used
made by Stevensand Herrnstein to repeat thesemeas- ratios of «, {, 3, and 5 and obtainedjudgmentsthorurementswith a 2-A audiometer,usingpresumablythe oughly consistentwith thoseobtainedfor « and 2. The
same procedure.Their results,shownin Table I, are judgments for half and double loudnessare entered
quite at variancewith thoseof Laird, Taylor, and Wille. directly in the tables. An interpolation between the
Later on, Stevens,Rogers,and Herrnstein2øtried de- values for « and { and for 3 and 5 has been made to
liberately to alter the subjects'judgmentsin the direc- give a value for -} and for 4, from which additional
tion of larger decibelreductionsfor half loudness.They estimatesof -} and 2 have been obtained.
presentedcomparisontonesat fainter levelsthan those
Geiger and Firestone5 useda method of adjustment
used by Stevensand Herrnstein and they altered the in which the subjectswere allowedto set the levelsof
scheduleby whichthe successive
comparisontoneswere the comparisontones by means of a potentiometer
chosen.This procedurewas fairly successful
with sub- (characteristicnot stated). Their 44 subjectsset the
jects who had never judged loudnessbefore, but it comparisontone to variousfractionsand multiples' «,
made much lessdifferenceto subjectswho had served -}, •o, 1/100, 2, 4, 10, 100.They computedboth means
previouslyin a loudnessexperiment.Apparently, by and medians for their data, so that we have here a
presentingfirst a standardand then a batch of properly chanceto comparethesetwo measures.
Exceptwhenthe
chosencomparisonstimuliwell belowthe standard,the adjustmentsare beingmadeto the moreextremeratios,
subjectcanbe inducedto chooseoneof theselow values the mediansare nearly alwayslessthan the means.
as being half as loud. Garner2• has also demonstrated
Churcher,King, and Davies•a allowedtheir subjects
this effect.The comparisontonespresentedby Stevens to adjust an attenuatorto reducethe comparisontone
and Herrnsteinwerelimited to those5, 10, 15, 20, and 25 to half loudness.
Having foundthe half-loudness
point,
db below the standard, becausethese reductionswere the subject proceededfrom there to make a second
all that turned out to be necessary.Of course,if the fractionation, and so on. Thus a curve was obtained for
subjecthad said that a 25-db reductionwas louder than eachsubject,and finally the decibelvaluescorrespondhalf, the comparisonwould have been lowered still ing to the various fractionswere averaged.They refurther. In the later experimentthe rangeof comparison peated the procedurefor judgmentsof -} loudnessand
levels was extended to 40 db below that of the standard.
obtainedquite consistentresults.
Rschevkinand Rabinovitch
•4usedtoneslastingonly
•8L. F. Richardsonand J. S. Ross,J. Gen. Psychol.3, 288-306
(1930).
•9Laird, Taylor, and Wille, Jr., J. Acoust.Soc.Am. 3, 393-401
(1932).
•0Stevens, Rogers, and Herrnstein, J. Acoust. Soc. Am. 27,
326-328 (1955).
• W. R. Garner, J. Exptl. Psychol.48, 218-224 (1954).
• L. B. Ham and J. S. Parkinson,J. Acoust. Soc.Am. 3, 511534 (1932).
•a Churcher, King, and Davies, J. inst. Elec. Engr. (London)
75, 401-446 (1934).
34S. N. Rschevkinand A. V. Rabinovitch, Rev. Acoust. 5,
183-200 (1936).
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06:45:44
828
S.
S.
STEVENS
0.4 secondand determinedthe ratios «, -}, 2, and 4.
They arguethat the useof longertonesfatiguesthe ear
and leadsto spuriousresults.Otherwisetheir procedure
was not unlike that of Laird, Taylor, and Wille. They
say that their resultsagreewith thoseof Laird, Taylor,
and Wille, but this seemshardly to be the case.
When Rschevkin and Rabinovitch lengthened the
tones to 4 seconds,the average reduction needed to
producehalf loudnessdecreased
from 11.0db to 9.5 db.
But sincethe authorsprefer the shorterpresentation,
the values recordedin Table I are the larger onesthey
A possiblesystematicbias may be presentowing to
the fact that the spacingof the comparisonstimuliwas
in decibelstepsinsteadof in equal-loudness
steps.And
since it has been demonstrated
that the outcome of the
method of constantstimuli is particularly sensitiveto
the rangeof the comparisonstimulipresented,we must
considerthe possibilitythat this "context" factor may
have played a role in Robinson'sexperiment.
The resultsof certain experiments,either finishedor
in progressat the Psycho-Acoustic
Laboratory, are included in the tables as P.A.L. entries. These experiobtained with the shorter tones.
ments have been undertakento explorethe effectsof
Pollack25 used a method of adjustment with a re- variousprocedureson loudnessjudgmentsand the relatively small numberof subjects.His are the smallest sults obtained have been incidental to this main purvaluesyet reportedfor the decibelreductionrequiredfor pose.Nevertheless,theseresultsneed to be considered
half loudness. Since it is difficult to read values from
in any decisionrelatingto the choiceof a loudnessscale.
The results of Stevens and Herrnstein with the 2-A
the graphspublishedin his paper,the valuesenteredin
the tablesare thosekindly suppliedme by Dr. Pollack. audiometerhave already been referredto.
Garner•øusedthe methodof adjustmentto obtain 20
The experimentby Stevens
øusingmagnitudeestimasettingsat each level by each of 18 subjects.His sub- tion was an early attempt to seehow far it might be
subjectsadjusteda decibelattenuator(1-dbsteps).His possibleto go toward getting the subjectsto make abis the most thoroughstatisticalanalysisof an experi- solute numerical judgments of loudness.Three standment of this sort. Unfortunately, the data, as he says, ardswereused:70, 100,and 120db sound-pressure
level
"disagreemarkedly with data previouslyreported," (binaural earphonelistening).The standardwas preand we are left wonderingwhy. His secondexperiment,n sentedonly onceand then 8 to 10 comparisontones
employing18 other subjectswho made 10 settingsat spaced5 db apart werepresentedin randomorder.The
eachlevel, alsousedthe methodof adjustment,but here subjectwasaskedsimply, "If the standardwere called
tones?"
the subject threw a switch to control a motor-driven 100,what wouldyou call eachof the comparison
attenuator which moved at the rate of 1 db every 2 This procedurewas repeated twice at each level. The
seconds.In this secondexperimentthe decibelreduc- logarithms of the median estimates give fairly recti~
tions requiredfor half loudnessare smallerat the low linear functionswhenplotted againstdecibelsand from
levels and larger at the high levels than in the first these plots the « and { loudnesspoints have been deexperiment.The data recordedin Table I were kindly termined.
suppliedme by Dr. Garner, sincethey were not tabled
The experimentof J. C. Stevens(reportedin reference
9) with white noisewas similarto the foregoingexcept
in his paper.
The large decibelreductionsreportedby Garner are that the standard was presentedbefore each of the
probably due to a number of factors. His subjects comparisonstimuli. In part of this experiment the
adjusteda decibelattenuator insteadof a sonepotenti- stimuli were spaced5 db apart belowthe standard;in
ometerwhoseattenuation,asa functionof angularturn, anotherpart they werespacedmorenearlyproportional
is more nearly proportionalto loudness.And insteadof to equal-loudnessintervals. These different spacings
averaging in sones,or computinga median, Garner madeno significantdifferencein the resultingmagnitude
averagedthe decibelvalues themselves.If he had used estimations(Table III).
Another variation involved the use of a faint standard
a sonepotentiometerand averagedin sones,the decibel
reductionswould probably have turned out smaller. (45 db) which was assignedthe value 1. The subjects
Also, the fact that the subjectsworkedfor fairly long then assignednumbers to a random seriesof louder
periodsat a stretchprobablyaccentuatedthe distortion comparisonnoises(Table IV). The functionsdeterdue to whateverbiasingfactorswere present.
minedby this ascendingprocedureare not significantly
Robinson's
s experimentwas perhapsthe most thor- different (in the middleof the intensityrange)from the
ough study yet made with the method of constant functions obtained when the standard is called 100.
stimuli. The stimuluswas presentedby a loudspeaker Other evidence indicates that at the extremes of the
to a subjectseated1 meter in front of it. Tones and
intensityrange thesetwo proceduresgive differentrenoiseswere used, and results were obtained for the
suits--in the directionone would predict. At high inratios «, x--•,2, and 10. These resultsare internally
tensities
the ascendingprocedure(standardcalled 1)
quite consistent,except for the usual discrepancybegives
a
steeper
loudness
functionthan doesthe descendtweenthe data for halving and doublingat low loudness
ing procedure(standardcalled100). The reverseis true
levels.
•5I. Pollack, J. Acoust.Soc.Am. 23, 654-657 (1951).
20W. R. Garner,J. Acoust.Soc.Am. 24, 153-157 (1952).
for low intensities.
In the experimentby Stevensand Poultona using33
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06:45:44
MEASUREMENT
OF
LOUDNESS
829
subjects,an attempt wasmade to seewhat peoplewho
had neverbeforemadequantitativeloudnessjudgments
would do when askedto adjust one tone to a fraction of
the loudnessof a standard (90 db). The controlusedwas
a gangedpotentiometer,which reducedthe level about
called 100. Groupsof 8 subjectseachjudged, as their
first attempt ever made, a differentcomparisonlevel:
6, 10, 20, or 40 db below the standard.The mediansof
these first estimates were 62.5, 50.0, 29.5, and 12.0,
respectively.From the pooledestimatesof the group
15 db when turned from full-on to half-on and about 12
ratingson all sevencomparisonlevelsultimately used,
db when turned from half-on to quarter-on. In other half-loudnessand quarter-loudness
values were deterwords,it wasmadeto approximatea sonepotentiometer. mined, as shownin Table I. Also shownare values for
The level was so adjusted that when this potenti- similar tests with a standard of 80 db. Since these latter
ometerwasfull on, the comparisontone was at 94 db, values were out of line with expectations,Poulton
i.e., 4 db more than the standard.
satisfiedhis curiosity a month later by repeating the
On their first attempts, eleven subjectsset to -} test on eight of the subjects.Sevenof the eight then
loudness,elevento « loudness,and elevento -• loudness. gave lower estimates,but thesehave not been entered
Thosewhosetto r}and -• thenwent on to set to the other in Table I. What went wrong the first time is still a
fractions.These other settingswere quite consistent mystery--as it often remainsin theseexperiments.
with the first. On the first judgmentsthe mean reducThe same 32 subjectsalso made multiple estimates
tionswere'-•= 4.7 db; «= 9.3 db; -}= 19.0 db.
relative to standards(40 and 60 db) which were called
In Table I are entered the decibel means, the sone 1. The results are shown in Table II.
means,and the mediansof the half-loudness
judgments
Poulton and Stevens
a had a group of 16 previously
and of half the values of the quarter-loudness
judg- unpracticedsubjectsadjust the sonepotentiometerso
ments.
In this same experimenta study was made of the
influence
of the characteristics
of the control
device
that the loudness of a white noise was half and twice
that of various standards. The means and medians of
the resultsare shownin Tables III and IV. (Twenty
othersubjectswerelater addedto the groupmaking36
subjectsin all.)
A word shouldperhapsbe saidabout anotherP.A.L.
experimentthat is only about half finished.It beganas
a rather ambitiousproject, employingthe method of
adjustmentfor the ratios«, -•, ½-•,1/100, 2, 4, 10, 100,
usedby thesubjects.Onegroupof elevensubjectsmade
their first settingswith a decibelattenuatorand, a week
later, switchedto a sonepotentiometer.Another group
of eleven subjectsdid the reverse.For both groups
combined,the effect of using the sonepotentiometer
was to reduce the decibel differencerequired for half
loudnessby 2.7 db.
Using the methodof magnitudeestimation,Poulton
did a variety of experiments.The purposeof one of
them (reportedin reference9) wasto explorethe effect
of the particularnumbersusedby the subjectsto report
their judgments.Eight experiencedsubjectssometimes
were told that the standard (100 db) was 100 and that
the comparisons
were to be judgedaccordingly.Sometimesthey weretold the standardwas 1 and that they
were to say what fractionalpart of 1 seemedto represent the comparisontone. These two proceduresgave
very similarresults,ascanbe seenfrom the interpolated
estimatesof half loudnessand quarterloudnessentered
in Table I. These subjectsalso rated louder tones
relative to a standard(60 db) called1, and theseresults
cendingand descending.Sixteen subjectswere to go
through this array of tasks in different orders, all
counterbalancedin accordancewith the precepts of
modernstatisticaldesign.It lookednice on paper, but
we have run into two difficulties.We are finding in
practice that a few of the subjects"can't take it."
They are requiredto shift their criterionso drastically
and so often that consistency
becomesalmostimpossible. Some of them complainof losing confidence,and
their resultsshow it. The seconddifficulty has arisen
from the fact that we unwiselyset someof the levelsso
ßthat the subjecthad to use the extremelow end of the
sonepotentiometerwherea very small turn of the knob
makesa large changein the loudnessratio. Apparently,
were used to estimate
if we are to determine how loud different tones sound to
twice and four times loudness
and also the method of magnitude estimation both as-
(Table II).
people,we must be careful to set for them simple,unIn another test with magnitude estimation, 32 in- confusingtasks,and we must stopour experimentsbeexperienced
subjectshearda 100-dbstandardwhichwas fore the listenersbecomeuncriticalin their judgments.
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