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OCTOBER, ][933
J.
A.
S.
A.
VOLUME V
Loudness, Its Definition, Measurement and Calculation
H^RVE¾FLETCHER^St) W. A. MVNSO•, Bell TelephoneLaboratories
(ReceivedAugust 28, 1933)
INTRODUCTION
ear receiving it, and the physiological and
psychologicalconditionsof the listener. In most
engineeringproblemswe are interested mainly in
describe the magnitude of an auditory senthe effect upon a typical observer who is in a
sation. Although we use the terms "very loud,"
typical condition for listening.
"loud," "moderately loud," "soft" and "very
In a paper during 1921 one of us suggested
soft," corresponding to the musical notations
using the number of decibelsabove threshold as a
if, f, mf, p, and pp, to define the magnitude, it is
measure of loudnessand some experimental data
evident that these terms are not at all precise
were presentedon this basis. As more data were
and depend upon the experience, the auditory accumulated it was evident that such a basis for
acuity, and the customs of the persons using
defining loudnessmust be abandoned.
them. If loudness depended only upon the inIn 1924 in a paper by Steinberg and Fletcher1
tensity of the soundwave producingthe loudness,
somedata were given which showedthe effectsof
then measurements of the physical intensity
eliminating certain frequency bands upon the
would definitely determine the loudnessas sensed
loudnessof the sound. By using such data as a
by a typical individual and therefore could be
basis, a mathematical formula was given for
used as a precisemeans of defining it. However,
calculating the loudness lossesof a sound being
no such simple relation exists.
transmitted to the ear, due to changes in the
The magnitude of an auditory sensation, that
transmissionsystem. The formula was limited in
is, the loudness of the sound, is probably deits application to the particular soundsstudied,
pendent upon the total number of nerve impulses
namely, speechand a soundwhich was generated
that reach the brain per second along the
by an electrical buzzer and called the test tone.
auditory tract. It is evident that these auditory
In 1925 Steinberg2 developed a formula for
phenomena are dependent not alone upon the
calculating the loudnessof any complex sound.
intensity of the sound but also upon their
The results computed by this formula agreed
physical composition. For example, if a person with the data which were then available. Howlistened to a flute and then to a bassdrum placed
ever, as more data have accumulated it has been
at such distancesthat the soundscoming from
found to be inadequate. Since that time conthe two instruments are judged to be equally
siderably more information concerning the
loud, then, the intensity of the sound at the ear
mechanism of hearing has been discovered and
produced by the bass drum would be many
the technique in making loudnessmeasurements
times that producedby the flute.
has advanced. Also more powerful methods for
OUDNESS
isapsychological
term
Used
to
If the composition of the sound, that is, its
producing complex tones of any known compowave form, is held constant, but its intensity at sition are now available. For these reasons and
the ear of the listener varied, then the loudness
because of the demand for a loudness formula of
produced will be the same for the same intensity
general application, especially in connection with
only if the same or an equivalent ear is receiving
noise measurements, the whole subject was
the sound and also only if the listener is in the
reviewed by the Bell Telephone Laboratories and
same psychologicaland physiologicalconditions,
with reference to fatigue, attention, alertness,
• H. Fletcher and J. C. Steinberg, Loudnessof a Co•nplex
etc. Therefore, in order to determine the loudness Sound, Phys. Rev. 24, 306 (1924).
produced, it is necessaryto define the intensity
• J. C. Steinberg, The Loudnessof a Sound and Its
of the sound,its physical composition, the kind of Physical Stimulus, Phys. Rev. 26, 507 (1925).
82
LOUDNESS,
ITS
DEFINITION,
MEASUREMENT
AND
CALCULATION
83
centimeter. In a plane or spherical progressive
sound wave in air, this intensity correspondsto a
root-mean-squarepressurep givenby the formula
the work reported in the present paper undertaken. This work has resulted in better experimental methods for determining the loudness
level of any sustained complex sound and a
formula which gives calculated results in agreement with the great variety of loudness data
wherep is expressedin bars,I[ is the heightof the
which are now available.
barometer in centimeters, and 7' is the absolute
p=o.ooo207[-(tœ/76)(273/T)•:'
(2)
temperature. At a temperature of 20øC and a
pressureof 76 cm of Hg, p = 0.000204 bar.
DEFINITIONS
The subject matter which follows necessitates Intensity level
of terms which have often
The intensity level of a soundis the numberof
been applied in very inexact ways in the past. db above the reference intensity.
Becauseof the increasein interest and activity in
this field, it became desirable to obtain a general Reference tone
agreement concerningthe meaning of the terms
A planeor sphericalsoundwave havingonly a
which are most frequently used. The following single frequency of 1000 cyclesper secondshall
definitions are taken from recent proposals of be usedas the referencefor loudnesscomparisons.
the use of a number
the sectional
committee
on Acoustical
Measure-
ments and Terminology of the American Standards Association
and the terms
have
been used
with these meaningsthroughout the paper.
Note: One practical way to obtain a plane or
sphericalwave is to use a small source,and to
have the head of the observer at least one meter
distant from the source, with the external conditions such titat reflected waves are negligible as
Sound intensity
The sound intensity of a sound field in a
specifieddirection at a point is the sound energy
transmitted per unit of time in the specified
direction through a unit area normal to this
direction at the point.
In the case of a plane or spherical free pro-
comparedwith the original wave at the head of
the observer.
Loudness
level
The loudnesslevel of any sound shall be the
intensity level of the equally loud referencetone
at the position where the listener's head is to be
gressivewave having the effectivesoundpressure
placed.
P (bars), the velocity of propagationc (cm per
sec.) in a medium of density p (grams per cubic Manner of listening to the sound
cm), the intensity in the direction of propagation
In observing the loudness of the reference
is given by
sound, the observer shall face the source, which
J=P2/pc (ergsper sec.per sq. cm).
(1)
This same relation can often be used in practice
with sufficientaccuracyto calculatethe intensity
at a point near the sourcewith only a pressure
measurement. In more complicated sound fields
the results given by this relation may differ
greatly from the actual intensity.
When dealing with a plane or a spherical
progressive •vave it will be understood that the
should be small, and listen with both ears at a
position so that the distancefrom the sourceto a
line joining the two ears is one meter.
The value of the intensity level of the equally
loud reference..sound depends upon the manner
of listening to the unknown sound and also to the
standard of reference.The manner of listening to
the unknown sound may be considered as part of
the characteristics
of that
sound. The
manner
of
listening to the reference sound is as specified
intensity is taken in the direction of propagation above.
of the wave.
Loudness has been briefly
Reference intensity
defined as the
magnitude of an auditory sensation, and more
The reference intensity for intensity level will be said about this later, but it will be seen
comparisonsshall be 10 la watts per square from the above definitions that the loudness level
84
HARVEY
FLETCHER
of any sound is obtained by adjusting the
intensity level of the reference tone until it
sounds equally loud as judged by a typical
listener. The only way of determining a typical
listener is to use a number of observers who have
normal hearing to make the judgment tests. The
typical listener, as used in this sense,would then
give the sameresultsas the averageobtained by a
large number of suchobservers.
A pure tone having a frequencyof 1000 cycles
per secondwas chosenfor the referencetone for
the followingreasons:(1) it is simpleto define, (2)
it is sometimes used as a standard of reference for
pitch, (3) its use makes the mathematical formulae more simple, (4) its range of auditory
intensities (from the threshold of hearing to the
threshold of feeling) is as large and usually larger
than for any other type of sound, and (5) its
frequency is in the mid-range of audible frequencies.
There
has been considerable
discussion con-
cerning the choice of the reference or zero for
loudnesslevels. In many ways the threshold of
hearing intensity for a 1000-cycle tone seemsa
logicalchoice.However, variationsin this threshold intensity arisedependingupon the individual,
his age, the manner of listening, the method of
presentingthe tone to the listener, etc. For this
reason no attempt was made to choose the
reference intensity as equal to the average
threshold of a given group listening in a prescribedway. Rather, an intensity of the reference
tone in air of 10-• watts per square centimeter
was chosen as the reference intensity because it
AND
W.
A.
MUNSON
tone is the same as its loudness level L and is
given by
10 log Jrq- 100,
(3)
where -/r is its sound intensity in microwatts per
squarecentimeter.
The intensity level of any other soundis given
by
/• = 10 log J+ 100,
(4)
where J is its sound intensity, but the loudness
level of such a soundis a complicatedfunction of
the intensitiesand frequenciesof its components.
However, it will be seen from the experimental
data given later that for a considerablerange of
frequenciesand intensitiesthe intensity level and
loudness level for pure tones are approximately
equal.
With the reference levels adopted here, all
values of loudness level which are positive
indicate a sound which can be heard by the
referenceobserver and those which are negative
indicate a sound which cannot be heard by such
an observer.
It is frequently more convenient to use two
matched head receivers for introducing the
reference tone into the two ears. This can be done
provided they are calibrated against the condition described above. This consists in finding
by a series of listening tests by a number of
observersthe electrical power W• in the receivers
which producesthe same loudnessas a level 31
of the reference tone. The intensity level •,. of an
open air reference tone equivalent to that
producedin the receiver for any other power W,
in the receivers is then given by
was a simple number which was convenient as a
referencefor computation work, and at the same
•=/•q-10
log (W•/W•).
(5)
time it is in the range of threshold measurements
Or, since the intensity level •,. of the reference
obtained when listening in the standard method
describedabove. This referenceintensity corre- tone is its loudness level L, we have
spondsto the threshold intensity of an observer
L= 10 log W•+C,.,
(6)
who might be designated a reference observer.
An examination of a large series of measure- where C• is a constant of the receivers.
ments on the threshold of hearing indicates that
In determining loudnesslevels by comparison
such a reference observer has a hearing which is with a reference tone there are two general classes
slightly more acute than the average of a large
of sound for which
measurements
are desired:
group. For those who have been thinking in
terms of microwatts it is easy to remember that
(1) those which are steady, such as a musical
tone, or the hum from machinery, (2) those
which are varying in loudnesssuch as the noise
watt per square centimeter. When using these from the street, conversational speech, music,
definitions the intensity level 3,. of the reference etc. In this paper we have confinedour discussion
this reference
level is 100 db below one micro-
LOUDNESS,
ITS
DEFINITION,
MEASUREMENT
to sourceswhich are steady and the method of
specifyingsuch sourceswill now be given.
A steady soundcan be representedby a finite
number of pure tones called components. Since
changes in phase produce only second order
effectsupon the loudnesslevel it is only necessary
to specify the magnitude and frequency of the
components? The magnitudes of the components at the listening position where the
loudness level is desired are given by the intensity levels •, 32, '" 3•, '" •3• of each component at that position. In case the sound is
conducted to the ears by telephone receivers or
AND
CALCULATION
85
componenttoward the total loudnesssensation
depends not only upon the properties of this
component but also upon the properties of the
other components in the combination. The
answer to the problem of finding a method of
calculating the loudnesslevel lies in determining
the nature of the ear and brain as measuring
instruments in evaluating the magnitude of an
auditory sensation.
One can readily estimate roughly the magnitude of an auditory sensation;for example, one
can tell whether the sound is soft or loud. There
have been many theories to account for this
change in loudness.One that seemsvery reaqonbe known such that if this component were able to us is that the loudnessexperienced is
acting separately it would produce the same dependent upon the total number of nerve
loudnessfor typical observersas a tone of the impulsesper secondgoing to the brain along all
same pitch coming from a sourceat one meter's the fibres that are excited. Although such an
distanceand producingan intensity level of/•s-. assumptionis not necessaryfor deriving the
In addition to the frequency and magnitude of formulafor calculatingloudnessit aidsin making
the components of a sound it is necessary to the meaning of the quantities involved more
know the position and orientation of the head definite.
with respectto the source,and alsowhether one
Let usconsitler,then, a complextone having n
or two ears are used in listening. The monaural componentseach of which is specifiedby a value
type of listening is important in telephone use of intensity level • and of frequencyft,. Let N be
and the binaural type when listeningdirectly to a a number which measuresthe magnitude of the
tubes, then a value [1• for each component must
sound source in air. Unless otherwise stated, the
auditory sensation produced when a typical
individual listens to a pure tone. Since by
condition where the listener faces the source and
definitionthemagnitudeof an auditorysensation
is
uses both ears, or uses head telephone receivers theloudness,
thenN is theloudness
of this simple
tone. Loudness as used here must not be confused
which produce an equivalent result.
with loudness
level.The latter is measuredby the
FORMULATION OF THE EMPIRICAL TIlEOR'/
FOR
intensity of the equally loud referencetone and is
CALCULATING THE LOUDNESS LEVEL OF
expressedin decibels while the former will be
discussionand data which follow apply to the
A STEADY
COMPLEX
TONE
expressedin units related to loudness levels in a
manner to be: developed. If we accept the
complextone is the sum of the intensitiesof the assumptionmentioned above, N is proportional
individual components. Similarly, in finding a to the number of nerve impulses per second
method of calculating the loudness level of a reaching the brain along all the excited nerve
complex tone one would naturally try to find fibers when the typical observer listens to a
simple tone.
numbers which could be related to each comLet the dependencyof the loudnessN uponthe
ponent in such a way that the sum of such
frequency
f and the intensity • for a simpletone
numbers will be related in the same way to the
be
represented
by
equally loud reference tone. Such efforts have
failed because the amount contributed by any
N = S(f, 3),
(7)
It is well known that the intensity of a
* Recent work by ChaDin and Firestone indicates that
at very high levels these second order effects become large
and c•annotbe neglected.
a K. E. Chltpin and F. A. Firestone, Interferenceof Sub-
jettire liarmonies,J. Aeons.Soc.Am. 4, 1•6A (1933).
where G is a function a hich is determined by any
pair of valuesoff and•1.For the referencetone,f
is 1000 and/S i• equal to the loudnesslevel L, so
a determinationof the relationexpressed
in Eq.
86
HARVEY
FLETCHER
AND
W.
A.
MUNSON
(7) for the reference tone gives the desired This transformationlookssimplebut it is a very
important one since instead of having to deIf now a simple tone is put into combination termine a different function for every comwith other simple tones to form a complex tone, ponent, we now have to determine a single
relation between loudness and loudness level.
its loudness contribution, that is, its contribution
functiondependingonly upon the propertiesof
toward the total sensation, ;;'ill in general be
the reference
somewhat less because of the interference
tone and as stated
above
this
of the
function is the relationshipbetween loudnessand
other components. For example, if the other loudnesslevel. And sincethe frequencyis always
components are much louder and in the same 1000 this function is dependentonly upon the
frequencyregion the loudnessof the simpletone singlevariable, the intensity level.
in such a combination will be zero. Let 1-b be
This formula has no practical value unlesswe
the fractional
reduction
in loudness because of
can determine b• and G in terms of quantities
which can be obtained by physical measurecontribution of this component toward the loud- ments. It ;;,ill be shown that experimental measnessof the complextone. It will be seenthat b by urementsof the loudnesslevels L and Le upon
definition always remains between 0 and unity. simple and complextones of a properly chosen
It depends not only upon the frequency and structure have yielded resultswhich have enabled
intensity of the simple tone under discussionbut us to find the dependenceof b and G upon the
also upon the frequenciesand intensities of the frequenciesand intensities of the components.
other components.It will be shown later that this When b and G are known, then the more general
dependencecan be determinedfrom experimental function G(f, •) can be obtained from Eq. (9),
measurements.
and the experimentalvaluesof L•. corresponding
The subscript k will be used when f and • tore and t36.
correspondto the frequencyand intensity level of
the kth componentof the complextone, and the DETERMINATION OF THE RELATION BETWEEN
Lk, ft AND /•
subscriptr usedwhenf is 1000cyclesper second.
The "loudness level" L by definition, is the
Thisreiation
canbeobtained
fromexperiintensity level of the reference tone when it is mental measurements of the loudness levels of
adjusted so it and the complex tone sound pare tones. Such measurements;;'ere made by
equally loud. Then
Kingsbury• which covereda rangein frequency
and intensity limited by instrumentalities then
Nr= G(1000,L) -- E
=E
(8) available. Using the experimentaltechnique
k 1
k--1
describedin AppendixA, we have againobtained
the loudness levels of pure tones, this time
Now let the referencetone be adjusted so that it
coveringpractically the ;;-holeaudible range.*
soundsequally loud successivelyto simple tones
All of the data on loudness
levelsbothfor pure
correspondingin frequency and intensity to each
and alsocomplextonestaken in our laboratory
componentof the complextone.
which are discussedin this paper have been
Designate the experimental values thus detaken with telephone receivers on the ears. It
termined as L•, L•, Ls, ßßßL•., ß- ßL•. Then from
has
been explained previously how telephone
the definition
of these values
receivers may be used to introduce the reference
its being in such a combination. Then bN is the
N• = G(1000, L•) = G(f•,/•),
(9)
tone into the ears at known loudness levels to
obtain the loudnesslevelsof other soundsby a
sincefor a singletone beis unity. On substituting loudness balance. If the receivers are also used
the values from (9) into (8) there results the for producing the sounds whose loudness levels
fundamental equation for calculating the loud- are being determined, then an additional calinessof a complex tone
• B. A. Kingsbury,A Direct Comparison
of lite LmMness
of Pure Tones,Phys. Rev. 29, 588 (1927).
G(1000,L) = Y•.b•G(1000,L•).
(10)
* See AppendixB for a comparisonwith t(ingsbury's
results.
LOUDNESS,
ITS
DEFINITION,
MEASUREMENT
•ND
CALCUL•,TION
87
bration, which will be explained later, is neces- trolled, it is obviously impractical to measure
sary if it is desiredto know the intensity levelsof directly the threshold level by using a large
the sounds.
group of obser•-ershaving normal hearing. For
The experimental data for determining the most purposesit is •nore convenient to measure
relation between Lk and fk are given in Table I in the intensity levels fi•, t•, .-- •, etc., directly
terms of voltage levels. (Voltage level = 20 log l', rather than have them related in any way to the
where
[' is the ean.f. across the receivers in
threshold of hearing.
In order to reduce the data in Table I to those
volts.) The pairs of values in each double column
give the voltage levels of the referencetone and which one would obtain if the observers were
the pure tone having the frequency indicated at listeningto a freewave and facing the source,we
the top of the column when the two tonescoming must obtain a field calibration of the telephone
from the head receivers were judged to be receivers used in the loudnesscomparisons.The
equally loud when usingthe techniquedescribed calibration for the referencetone frequency has
in Appendix A. For example, in the second been explained previously and the equation
column it will be seenthat for the 125-cycletone
/• =/•+ 10 log (W,/gh)
(5)
when the voltage is +9.8 db above 1 volt then
the voltage level for the reference tone must be
derived for the relation between the intensity
4.4 db below 1 volt for equality of loudness.The
of the referencetone and the electrical power
bottom set of numbers in each column gives the
threshold values for this group of observers.
Each voltage level in Table I is the median of
297 observations representing the combined
results
of eleven
observers.
The
method
of
in the
receivers.
The
calibration
consisted
of
findingby meansof loudnessbalancesa power
in the receiverswhich producesa tone equal in
loudness to that of a free wave having an
intensity level/•1.
obtaining these is explainedin Appendix A also.
For soundsother than the 1000-cyclereference
The standard deviation was computed and it was
tone a relation similar to Eq. (5) can be derived,
found to he somewhat larger for tests in which
namely,
the tone differed most in frequency from the
•=01'-}-•0 log (W/W1),
reference tone. The probable error of the combined result as computed in the usual way was where • and [[h are correspondingvalues found
between 1 and 2 db. Since deviations of any one
from loudness balances for each frequency or
observer's results from his own average are less
complex wave form of interest. If, as is usually
than the deviations of his average from the assumed, a linear relation exists between • and 10
average of the group, it would be necessaryto log W, then determinations of/• and I,V1 at one
increase the size of the group if values more
level are sufficient and it follows that a change in
representative of the average normal ear were the power level of _X decibels will produce a
desired.
The data shown in Table I can be reduced to
the number of decibels above threshold if we
corresponding change of ,• decibels in the
intensity of the sound generated. Obviously the
receivers
must
not
be overloaded
or this
as-
accept the values of this crew as the reference sumption will not be valid. Rather than depend
threshold values. However, we have already upon the existenceof a linear relation between
adopted a value for the 1000-cyclereferencezero. and 10 log 1V with no confirming data, the
As will be shown, our crew obtained a threshold
for the reference tone which is 3 db above the
reference
level chosen.
It is not only more convenient but also more
reliable to relate the data to a calibration
of the
receivers in terms of physical measurements of
receivers used in this investigation were calibrated at two widely separatedlevels.
Referring again to Table I, the data are
expressed in terms of voltage levels instead of
power levels. If, as was the case with our
receivers, the electrical impedance is essentially a
the sound intensity rather than to the threshold
constant, Eq. (11) can be put in the form:
values. Except in experimental work where the
intensity of the sound can be definitely con02)
log (17 lr,)
T^BLE I. Voltagelevels(db) for loudnessequality.
Refer-
62
Refer-
250
Refer-
500
Refer-
2000 Refer-
4000 Refer-
5650 Refer-
8000 Refer-
11,300 Refer-
16,00•
ence
c.p.s.
ence
c.p.s.
ence
c.p.s.
ence
c.p.s.
ence
c.p.s.
ence
c.p.s.
ence
c.p.s.
ence
c.p.s.
ence
c.p.s.
ence
c.p.s.
--12.2
--17.2
--19.2
--15.7
--21.2
+
+
+
+
+
-- 4.4
--10.2
--13.3
--18.6
--23.2
+
+
----
-- 2.9
-- 3.7
-- 5.2
-- 6.7
--12.2
+
+
+
---
-- 2.2
-- 4.2
-- 6.2
-- 7.2
--12.2
+
q----
-- 2.2
-- 1.7
-- 3.2
--18.2
--21.2
-- 1.2
-- 1.5
-- 1.3
--13.4
--17.2
-- 1.7
-- 1.2
--23.2
--24.7
--44.7
-- 0.7
-- 1.5
--17.3
--16.7
--35.2
-- 3.7
-- 7.2
--28.2
--30.2
--53.2
+ 0.8
-- 1.2
--19.2
--19.2
--38.2
--10.9
--12.2
--26.2
--27.1
--46.2
-- 4.3
--12.0
--24.0
--24.3
--38.3
--20.2
--22.2
--38.2
--38.7
--55.2
+ 1.7
+ 1.8
--20.2
--20.3
--34.2
--50.2
--66.2
--77.2
--85.2
q- 1.:
--13.:
--28.:
--38.:
--56.2
--67.2
--68.7
--97.2
--15.2
--20.2
--20.3
--30.3
--88.8
--46.5
--90.2
--68.3
--43.7
--63.7
--64.2
--83.2
--42.2
--61.0
--61.2
--80.2
--61.2
--64.2
--78.2
--78.2
--57.5'
--57.3
--77.3
--80.2
--83.7
--78.0
--81.7
--77.3
-- 108.1
-- 102.6
--108.3
--101.7
--27.2
--32.2
--33.2
--41.2
--35.4
9.8
5.8
2.8
2.6
0.8
-- 0.2
-- 7.2
-- 7.2
--10.2
--10.4
--27.9
--31.0
--35.2
--40.7
--66.6
--108.1
--39.8
--109.3
--113.1
--42.4 --108.3
--38.5 --113.1
125 Refer-
9.8
7.9
0.8
3.2
5.2
--12.3
--14.2
--15.2
--23.6
--35.0
--25.5
--32.2
--32.2
--52.5
--72.9
6.6
5.7
5.8
2.2
2.2
--18.3
--22.2
--23.2
--40.4
--56.3
--21.2
--21.7
--32.2
--34.2
--41.7
--22.3
--21.9
--30.2
--31.2
--41.9
--108.3 --39.5 --108.1 --62.8 --108.1 --86.7 --108.3--60.7 --108.3
--63.5 --113.1
5.8
6.8
2.0
2.3
8.2
--22.2
--40.2
--41.2
--42.2
--59.2
--108.3
--18.3
--35.2
--35.3
--35.4•
--54.2
--47.7
--63.2
--65.2
--77.7
--80.2
--105.2 --108.3
--35.1
--54.2
--54.5
--72.2
--72.5
--54.7
--72.2
--72.7
--85.2
--92.7
--39.1
--58.2
--58.1
--71.2
--78.1
--48.2
--64.2
--70.2
--76.2
--82.6
--38.2
--52.2
--56.2
--72.2
--76.2
--55.7
--72.2
--78.7
--88.2
--90.7
--34.2
--52.2
--58.1
--72.2
--77.1
--104.6
99.7 --108.3--109.0 --108.3--105.7 --108.3--101.9 --108.3--108.1!--108.3 --93.7
--86.4 --109.3 -- 103.4 -- 109.3 --108.9 -- 109.3 --102.0 --109.3 -- 99.3 --109.3 --103.1 --109.3
--86.3 --113.1 --103.(1 --113.1 --111.4 --113.1 --108.1 --113.1 --102.3 --113.1 --106.• --113.1
--94.6
--93.7 --109.3
--57.:
TABLE II. Field calibrationof telephonereceivers.
Frequency c.p.s.
Voltage level (20 log V0
Intensity level (•t)
C• =•20 log V•
Thresholdvoltage level (20 log Vo)
Thresholdintensity level (•5o)
Co=t•o- 20 log Vo
Diff. = C•- Co
60
-- 13.0
+79.3
92.3
-48.0
+49.3
97.3
-5.0
120
--26.2
+71.0
97.2
--61.8
+33.7
95.5
1.7
240
480
960
1920
3850
7800
10,500
--38.5
--47.0
--48.2
--42.3
--36.3
--34.0
5400
--39.1
--32.4
--
+67.4
+63.8
+65.3
+64.0
+62.2
+65.5
+74.0
+78.6
+75.0
105.9
110.8
113.5
106.3
98.5
99.5
113.1
-- 86.2
-- 105.4
-- 110.7
-- 109.0
-- 104.0
--97.1
-- 100.5
+19.7
+8.4
+5.4
--0.9
--4.2
+2.7
+10.6
105.9
0
113.8
--3.0
116.1
--2.6
108.1
--1.8
99.8
--1.3
99.8
--0.3
Ilia
+2.0
111.0
--102.0
+16.1
118.1
--7.1
15,000
6.4
81.4
-- 74.0
+22.0
96.0
--14.6
>
LOUDNESS,
ITS
DEFINITION,
MEASUREMENT
AND
CALCULATION
89
fi = 20 log 1'+ C
or
tl=20 log V+C,
(13)
can be applied to our receiverswith considerable
where V is the voltage acrossthe receiversand C
confidence.
is a constant
The constant C determined at the high level
was determined with greater accuracy than at
the threshold. For this reasononly the values for
the higher level were used for the calibration
curve. Also in these tests only four receiverswere
used while in the loudnesstests eight receivers
of the receivers
to be determined
from a calibration giving correspondingvalues of
th and 20 log V•. The calibration will now be
described.
By usingthe soundstageand the techniqueof
measuring field pressures described by Sivian
and White • and by using the technique for
making loudness measurements described in
Appendix A, the following measurements were
made. An electrical voltage V• was placed across
were used. The difference between the efficiency
of the former four and the latter eight receivers
was determined by measurementson an artificial
ear. The figuresgiven in Table II were corrected
by this difference.The resultingcalibrationcurve
level producedwas the same at each frequency. is that given in Fig. 1. It should be pointed out
the two head receivers
The observer
listened
such that
the loudness
to the tone in these head
receivers and then after 1« seconds silence
listened to the tone from the loud speaker
producing a free wave of the same frequency.
The voltage level acrossthe loud speaker necessary to produce a tone equally loud to the tone
from the head receivers was obtained using the
procedure described in Appendix A. The free
wave intensity level fi• correspondingto this
voltage level was measured in the manner described in Sivian and White's paper. Threshold
values both for the head receivers and the loud
speaker were also observed. In these tests eleven
observers
were
used.
The
results
obtained
are
given in Table II. In the second rmv values of 20
FIG. 1. Field calibration
of loudness balance receivers.*
(Calibration made at L=60
db.)
here that such a calibration curve on a single
individual
would
show considerable
deviations
from this aw.•rage curve. These deviations are
real, that is, they are due to the sizesand shapes
log V•, the voltage level, are given. The intensity of the ear canals.
levels, fi•, of the free wave which sounded
We can now express the data in Table I in
equally loud are given in the third row. In the terms of field intensity levels. To do this, the
fourth row the values of the constant C, the
data in each double column were plotted and a
calibration we are seeking,are given. The voltage smooth curve drawn through the observed
level added to this constantgivesthe equivalent points. The resulting curves give the relation
free wave intensity level. In the fifth, sixth and between voltage levels of the pure tones for
seventh rows, similar values are given which
equality of loudness.From the calibration curve
were
bottom
determined
row
the
at the threshold
differences
in
level.
the
In the
constants
of the
receivers
these levels
are converted
to
intensity levels by a simple shift in the axes of
determined at the two levels are given. The fact
coordinates. Since the intensity level of the
that the difference is no larger than the probable
reference tone is by definition the "loudness
error is very significant. It means that through- level," these shifted curves will represent the
out this wide range there is a linear relationship
between the equivalent field intensity levels, •,
* The ordinatesrepresentthe intensity level in db of a
and the voltage levels, 20 log V, so that the free wave in air which, when listened to with both ears in
formula (13)
the standard manner, is as loud as a tone of the same
• L. J. Sivian and S. D. White, Minimum Audible Sound
Fields, J. Acous. Soc. Am. 4, 288 (1933).
frequency heard from the two head receivers used in the
tests when an e.m.f. of one volt is applied to the receiver
ternfinals.
90
HARVEY
FLETCHER
AND
W.
A.
MUNSON
loudnesslevel of pure tonesin terms of intensity Table I except for the threshold values. The
levels. The resulting curves for the ten tones results of separate determinations by the crew
tested are given in Figs. 2A to 2J. Each point on used in these loudness tests at different times are
these curves correspondsto a pair of values in given by the circles.The points representedby
(*) are the valuesadoptedby Sivian and White.
It will be seen that most of the threshold points
are slightlyabovethe zerowe have chosen.This
meansthat our zerocorresponds
to the thresholds
of observerswho are slightly more acute than the
average.
From these curves the loudness level contours
can be drawn. The first set of loudness level
contoursare plotted with levels above reference
threshold as ordinates. For example, the zero
loudness level contour corresponds to points
where the curves of Figs. 2A to 2J intersect the
abscissa axis. The
number
of db above these
points is plotted as the ordinate in the loudness
level contours shown in Fig. 3. From a con-
40
o
100
120
0
20
40
60
iNTENSiTY
LEVEL-DB
i
4C
80
•00
120
O
, 4ooo•
TONE
FIG. 3. Loudness level contours.
sideration of the nature of the hearing mechanism
we believe that
o
2o
40
6o
6o
O
Ioo
•2o
o
2o
40
INTœN•TYLEVEL-DB
60
•o
1oo
12o
60
80
too
120
H
11300
TJNE
0
20
40
60
!
80
Ioo
120
0
INTEN$4TY LEVEL-DB
20
40
j
FIG. 2. (A to J) loudnesslevels of pure tones.
these curves should be smooth.
These curves, therefore, represent the best set of
smooth curves which we could draw through the
observed points. After the smoothing process,
the curves in Figs. 2A to 2J xverethen adjusted to
correspond.The curvesshownin thesefiguresare
such adjusted curves.
In Fig. 4 a similar set of loudnesslevel contours
is shown using intensity levels as ordinates.
There are good reasonss for believing that the
peculiar shape of these contours for frequencies
above 1000 c.p.s. is due to diffraction around the
head of the observer
as he faces the source of
sound. It was for this reason that the smoothing
LOUDNESS,
ITS
DEFINITION,
MEASUREMENT
AND
CALCULATION
91
processwas done with the curves plotted with
the level above threshold
as the ordinate.
From these:loudnesslevel contours, the curves
shownin Figs. 5A and 5B were obtained.They
show the loudnesslevel rs. intensity level with
frequency as a parameter. They are convenient
to use for calculation purposes.
Fro. 4. Loudness
It is interesting to note that through a large
part of the practicalrangefor tonesof frequencies
froIn 300 c.p.s. to 4000 c.p.s. the loudnesslevel is
approximately equal to the intensity level. From
these curves, it is possibleto obtain any value of
L• in terms of •a and f•.:.
On Fig. 4 the 120-db loudnesslevel contour has
been marked "Feeling." The data publishedby
R. R. Riesz• on the threshold of feeling indicate
that this contour is very closeto the feeling point
throughout the frequency range where data have
level contours.
120
been taken.
DETERMINATION
OF TIlE LOUDNESS FUNCTION
G
In the section "Formulation of the Empirical
Theory for Calculatingthe Loudnessof a Steady
Complex Tone," the fundamental equation for
calculating the loudnesslevel of a complex tone
was derived, namely,
G(1000, L) = Z b,:G(1000,Zt:).
(10)
k=l
tf the type of complextone can be chosenso that
b•.is unity and also so that the values of L• for
each compommt are equal, then the fundamental
equation for calculating loudnessbecomes
ioo
G(L) = nG(L•),
where n is the number of components. Since we
are always dealing in this sectionwith G(1000, L)
or G(1000, L;:), the 1000 is left out in the above
nomenclature.. If experimental measurementsot'
L correspondingto values of Lt. are taken for a
tone fulfilling the above conditions throughout
the audible :range, the function G can be determined. If we accept the thory that, when two
simple tones widely separated in frequency, act
upon the ear: the nerve terminals stimulated by
• •o
o
-I-020-•O
(14)
0
10
•O
30
40
50
INTENSITY
•0
70
•0
90
100
110
LEVœL-D8
FIG. 5. (A and B) loudnesslevels of pure tones.
120
• R. R. Riesz, The Relationship BetweenLoudnessand the
Minimum PerceptibleIncrementof Intensity, J. Acous.Soc.
Am. 4, 211 (1933).
92
HARVEY
FLETCHER
AND
W.
A.
MUNSON
In this curve the ordinates give the loudness
levels when one ear is used while the abscissae
give the correspondingloudnesslevels for the
same intensity level of the tone when both ears
are usedfor listening. If binaural versesmonaural
loudness data actually fit into this scheme of
calculationthesepointsshouldbe representedby
G(y)= «S(x).
o-looo 2000
AND
Any one of these curves which was accurately
determined would be sufficient to completely
determine
o
20
40
LOUDNESS
60
LEVEL
80
OF EACH
lOO
12o
the function
G.
For example,considerthe curve for two tones.
It is evident that it is only necessaryto deal with
14o
relative
COMPONENT-OB
Fro. 6. Complextonesbavlng componentswidely separated
in frequency.
values of G so that
we can choose one
value arbitrarily. The value of G(0) was chosen
equal to unity. Therefore,
G(0) = 1,
each are at different portions of the basilar
membrane,then we would expectthe interference
of the loudnessof one upon that of the other
would be negligible. Consequently, for such a
combinationb is unity. Measurementswere made
upon two such tones, the two componentsbeing
equally loud, the first having frequenciesof 1000
and 2000 cycles and the second,frequenciesof
125 and 1000 cycles. The observed points are
shown along the secondcurve from the top of
G(yo)= 2G(O)= 2
where y0 correspondsto x = 0,
G(y• = 2G(x•) = 2a(yo) = 4
where yl correspondsto x• =y0,
G(y•.)= 2G(x2)= 2G(y•) = 8
where y2 correspondsto x•=y•,
G(yk) = 2G(x•) = 2G(yk_•)= 2
wherey• correspondsto x• = y•_•.
Fig. 6. The abscissae
give the loudnesslevel Lk of In this way a set of valuesfor G can be obtained.
each componentand the ordinates the loudness A smooth curve connecting all such calculated
level L of the two components combined. The
equation G(y)=2G(x) should represent these
data.
Similar
measurements
were made with
a
complextone having 10 components,all equally
loud. The method of generating such tones is
describedin Appendix C. The results are shown
by the points along the top curve of Fig. 6. The
equation G(y)=lOG(x) should representthese
data except at high levels where bk is not unity.
moo
There is probably a complete separation be-
tween stimulated patchesof nerve endingswhen
the first component is introduced into one ear
and the second component into the other ear.
In this casethe same or different frequenciescan
be used. Since it
is easier to make
• •o
õ
20
loudness
balances when the same kind of sound is used,
measurements were made (1) with 125-cycle
tones (2) with 1000-cycle tones and (3) with
4000-cycletones.The resultsare shownon Fig. 7.
/
0
20
40
60
LOUDNESS LEVEL,
•0
100
120
140
BOTH EARS-DB
FIG. 7. Relation between loudness levels listening with
one ear and with both ears.
LOUDNESS,
ITS
DEFINITION,
MEASUREMEN'['
TABLE III.
AND
CALCULATION
93
Valuesof G(L•).
L
0
1
2
3
4
5
6
7
8
9
-- 10
0
10
20
30
40
50
60
70
80
90
100
110
120
0.015
1.00
13.9
97.5
360
975
2200
4350
7950
17100
38000
88000
215000
556000
0.025
1.40
17.2
113
405
1060
2350
4640
8510
18400
41500
97000
235000
609000
0.04
1.90
21.4
131
455
1155
2510
4950
9130
19800
45000
106000
260000
668000
0.06
2.51
26.6
151
505
1250
2680
5250
9850
21400
49000
116000
288000
732000
0.09
3.40
32.6
173
555
1360
2880
5560
10600
23100
53000
126000
316000
800000
0.14
4.43
39.3
197
615
1500
3080
5870
11400
25000
57000
138000
346000
875000
(I.22
5.70
4;'.5
222
675
1640
3310
6240
12400
27200
62000
150000
380000
956000
0.32
7.08
57.5
252
740
1780
3560
6620
13500
29600
67500
164000
418000
1047000
0.45
9.00
69.5
287
810
1920
3820
7020
14600
32200
74000
180000
460000
1150000
0.70
11.2
82.5
324
890
2070
4070
7440
15800
35000
81000
197000
506000
1266000
points will enable one to find any value of G(x)
to give a to•e which can be heard. When the
for a given value of x. In a similar way setsof
componentsare all in the high pitch range and
all equally loud, each componentmay be from
values
can
be obtained
from
the
other
two
experimental curves. Instead of using any one of
6 to 8 db below the threshold
the curves alone the values of G were chosen to
tion will still be audible. When they are all in
the low pitch range they may be only 2 or 3 db
below the threshold.The closeness
of packingof
the components also influences the threshold.
For example, if the ten components are all
within a 100-cycle band each one may be down
10 db. It will be shownthat the formula proposed
best fit all three sets of data, taking into account
the fact that the observedpoints for the 10-tone
data might be low at the higher levels where b
would be less than unity. The values for the
function which were finally adopted are given in
Table
III.
From
these
values
the
three
solid
curves of Figs. 6 and 7 were calculated by the
equations
and the cmnbina-
above can be made to take care of these
vari-
ations in the 'threshold.
There is still another method which might be
S(y) = -}G(x). usedfor determining this loudnessfunction G(L),
provided one's judgment as to the magnitude of
The fit of the three sets of data is sufficiently
an auditory sensation can be relied upon. If a
good, we think, to justify the point of view taken
person were asked to judge when the loudnessof
in developingthe formula. The calculated points
a sound was reduced to one-half it might be
for the 10-component tones agree with the
expected that he would base his judgment on
observed ones when the proper value of b•, is
the experience of the decreasein loudnesswhen
introduced into the formula. In this connection
going from the condition of listening with both
it is important to emphasizethat in calculating
ears to that of listening with one ear. Or, if the
the loudnesslevel of a complex tone under the
magnitude of the sensation is the number of
condition of listening with one ear instead of
nerve dischargesreaching the brain per second,
two, a factor of 21must be placed in front of then when this has decreased to one-half, he
the summationof Eq. (10). This will be explained might be able to say that the loudnesshas dein greater detail later. The values of G for nega- creased one-half.
tive values of L were chosenafter consideringall
In any case, if it is assumed that an observer
the data on the thresholdvaluesof the complex can judge when the magnitude of the auditory
tones studied. These data will be given with the sensation, that is, the loudness, is reduced to
S(y) = 10G(x),
G(y): 2G(x),
other loudness data on complex tones. It is
interesting to note here that the threshold data
show that 10 pure tones, which are below the
one-half, then the value of the loudness function
G can be computed from such measurements.
Several different
research workers
have made
threshold when soundedseparately, will combine such measurements. The measurements are some-
94
HARVEY
FLETCHER
AND
W.
A.
MUNSON
what in conflict at the present time so that they
did not in any way influence the choice of the
calculated and observed results of data taken by
loudness function.
taken from Tables la, lb, 2a, 2b, 3a and 3b of
their paper. The calculation is very simple.
Rather
we used the loudness
function given in Table III to calculate what
such observations should give. A comparison of
the calculated and observed results is given
below. In Table IV is shown a comparison of
Ham
and Parkinson. 7 The observed values were
From
the
the number
loudness
350 cycles
Fractional
reduction in loudness
S
L
G
Cal. %
Obs. %
74.0
70.4
67.7
64.0
59.0
54.0
44.0
34.0
85
82
79
75
70
65
53
41
25,000
19,800
15,800
11,400
7,950
5,870
2,680
1,100
100
79
63
46
32
24
11
4
100.0
83.0
67.0
49.0
35.0
26.0
15.0
8.0
59.5
57.7
55.0
49.0
44.0
39.0
34.0
71
69
66
59
47
41
8,510
7,440
6,240
4,070
2,680
1,780
1,060
100
87
73
48
31
21
12
I00.0
92.0
77.0
57.0
38.0
25.0
13.0
24.0
29
53
86.0
82.4
79.7
76.0
71.0
66.0
56.0
46.0
86
82
80
76
71
66
56
46
56.0
54.2
56
54
324
1000 cycles
27,200
19,800
17,100
12,400
8,510
6,420
3,310
1,640
4
6.0
i00.0
68.0
53.0
41.0
26.0
20.0
13.0
8.0
100
87
76
100.0
93.4
74.6
51.5
52
3,310
2,880
2,510
48.8
49
2,070
62
55.0
46.0
41.0
46
41
1,640
1,060
49
32
40.9
24.5
36.0
36
20
10.8
74.0
70.4
69
64
2500 cycles
7,440
5,560
67.7
64.0
59.0
54.0
62
58
53
48
4,950
3,820
2,680
1,920
67
51
36
26
68.1
44.0
34.0
39
30
890
360
12
5
13.0
6.7
44.0
42.2
39.5
36.8
34.0
29.0
24.0
39
37
36
33
30
26
21
890
740
675
505
360
222
113
100
83
76
57
41
25
13
675
i00
75
above
threshold
is determined
from
S
the
reduction
in the loudness function
for the correspondingvalues of L. The agreement
between
observed
and
calculated
results
is re-
markably good. However, the agreement with
the data of Laird, Taylor and Wille is very poor,
as is shown by Table V. The calculation was
made only for the 1024-cycle tone. The observed
data were taken from Table VII of the paper by
Laird, Taylor and Wille. a As shown in Table V
the calculation
of the
level
for one-fourth
re-
duction in loudness agrees better with the observed data correspondingto one-half reduction
in loudness.
T^BLE V. Comparisonof calculatedand observed
fractional
loudness(Laird, Taylor and Wille).
Cal. level
Original
loudness
level
100
73
63
46
31
24
12
6
L
curvesof Fig. 3. The fractional reduction is just
the fractional
T^nLE IV. Comparisonof calculatedand observed
fractional
loudness(Ham and Parkinson).
of decibels
level
100
90
80
7O
60
50
40
30
20
10
Level for « loudness
reduction
Cal.
Obs.
92
82
71
58
50
42
33
25
16
7
76.O
68.0
60.0
49.5
40.5
31.0
21.0
14.9
6.5
5.0
for ¬
loudness
reduction
84
73
60
48
41
34
27
20
13
4
Firestone and Geiger reported some prelimi~
nary values which were in closer agreement with
those obtained by Parkinson and Ham, but their
completed paper has not yet been published.ø
100.0
Because
of the lackof agreement
of observed
86.4
data
of this sort we concluded
that
it could
not
49.5 beusedforinfluencing
thechoice
of thevalues
32.8
23.3
100.0
94.6
82.2
61.1
46.0
27.8
14.9
of the loudness function adopted and shown in
7 L. B. Ham and J. S. Parkinson,Loudnessand Intensity
Relations, J. Acous. Soc. Am. 3, 511 (1932).
• Laird, Taylor and Wille, The Apparent Reduction in
Loudness,J. Acous. Soc. Am. 3, 393 (1932).
9 This paper is now available. P. H. Geiger and F. A.
Firestone, The Estimation of Fractional Loudness,J. Acous.
Soc. Am. 5, 25 (1933).
LOUDNESS,
ITS
DEFINITION,
MEASUREMEN'['
Table III. It is to be hoped that more data of
this type will be taken until there is a better
agreement between observed results of different
observers. It should be emphasized here that
changesof the level above threshold corresponding to any fixed increase or decreasein loudness
will, according to the theory outlined in this
paper, depend upon the frequency of the tone
when using pure tones, or upon its structure
when using complex tones.
DETERMINATION
OF THE
FORMULA
FOR
AND
CAI.
CUI•ATION
95
which is unity. It is thus seen that the fundamental of a series of tones will always have a
value of b• equal to unity.
For the casewhen the maskingcomponentand
the kth component have the same loudness,the
function representing b• will be considerably
simplified, particularly if it were also found to
be independentof f• and only dependent upon
the differenos between fk and j;,•. From the
theory of hearing one xvould expect that this
would be approximately true for the following
reasons:
CALCULATING bk
The distance in millimeters between the po-
Having now determined the function G for all sitions of maximum response on the basilar
values of L or Lk we can proceedto find methods membrane for the two components is more
of calculatingb•. Its value is evidently dependent nearly proportional to differences in pitch than
upon the frequencyand intensity of all the other
components present as well as upon the component being considered.For practical computations, simplifying assumptions can be .made. In
most cases the reduction of b• from unity is
principally due to the adjacent component on
the side of the lower pitch. This is due to the
fact that a tone masks another tone of higher
pitch very .much more than one of lower pitch.
For example, in most casesa tone which is 100
cycles higher than the masking tone would be
masked when it is reduced
to differencesin frequency. However, the peaks
are sharpest in the high frequency regionswhere
the
distances
on
the
basilar
membrane
for
a
given/xf are smallest.Also, in the low frequency
region where the distances for a given /xf are
largest, these. peaks are broadest. These two
factors
tend
to make
the interference
between
two components having a fixed difference in
frequency approximately the same regardlessof
their position on the frequency scale. However,
it would be extraordinary if these two factors
25 db below the level
of the masking tone, whereas a tone 100 cycles
lower in frequency will be masked only when it
is reduced
from 40 to 60 db below the level of
the masking tone. It will therefore be assumed
that the neighboring component on the side of
lower pitch which causes the greatest masking
will account for all the reduction in bk. Designating thi's component with the subscript m,
meaning the masking component,then we have
bx. expressed as a function of the following
variables.
bk= B(f•., f,,,, S•., S,•),
(15)
wheref is the frequencyand S is the level above
threshold.
For the case when the level of the kth
componentis T db below the level of the masking
component, where T is just sufficient for the
component to be masked, then the value of b
would be equal to zero. Also• it is reasonable to
assumethat when the masking component is at
a level somewhat
less than
T db below the kth
component, the latter will have a value of b•
FIG. 8. Loudnesslevelsof complextones having ten equally
loud components 50 cycles apart.
96
HARVEY
FLETCHER
AND
W.
A.
MUNSON
just balanced. To test this point three complex
tones having ten components with a common
zXf of 50 cycles were tested for loudness. The
first had frequencies of 50-100-150...500, the
second 1400-1450...
3450...3900.
The
1900, and the third 3400results
of
these
tests
are
shown in Fig. 8. The abscissaegive the loudness
level of each component and the ordinates the
u
60
o•
measured loudness level of the combined tone.
,.-d,
so
Similar results were obtained with a complex u,40
tone having ten componentsof equal loudness
and a commonfrequencydifferenceof 100 cycles.
J
o FREQUENCYDIFFERENCE- 340
The results are shown in Fig. 9. It will be seen
that although the points correspondingto the
different frequency ranges lie approximately
/
upon the same curve through the middle range,
0_,
ø
o"P
x
there are consistentdeparturesat both the high
and low intensities.If we choosethe frequencyof
FIG. 10. Loudnesslevels of complextones having ten
the components largely in the middle range then equally loud componentswith a fundamentalfrequency
this factor b will be dependent only upon zXf of 1000 c.p.s.
>
and Lk.
To determine the value of b for this range in 56 cycles per second.The fundamental for each
terms of 6f and Lk, a seriesof loudnessmeasure- tonewascloseto 1000cycles.The ten-component
ments was made upon complex tones having ten toneshaving frequencieswhich are multiplesof
components with a common difference in fre- 530 was included in this series. The results of
quency zXf and all having a common loudness loudnessbalancesare shownby the points in
level L•. The valuesof zXfwere 340, 230, 112 and Fig. 10.
By taking all the data as a whole, the curves
were consideredto give the best fit. The values
of b were calculated
from
these curves as
follows:
Accordingto the assumptions
made above,the
componentof lowest pitch in the seriesof componentsalwayshas a value of b•.equal to unity.
Thereforefor the seriesof 10components
having
a common loudness level L•, the value of L is
related to Lk by
G(L) = (1+9b•)G(LD
or by solving for b•.,
b•= (1/9)[-G(L)/G(Lk)- 1].
(16)
The values of b•.can be computed from this
equation from the observed values of L and L•
20
-10
0
10
20
L0U0NES$
30
LEVEL
40
50
60
70
80
90
100
OF EACH COMPONENT-DB
FzG.9. Loudnesslevelsof complextoneshaving ten equally
loud components 100 cycles apart.
by usingthe valuesof G given in Table III. Because of the difficulty in obtaining accurate
valuesof L and L• suchcomputedvaluesof bk
will be rather inaccurate.Consequently,considerablefreedomis left in choosinga simple
LOUDNESS,
ITS
DEFINITION,
MEASUREMENT
5.0
AND
CALCULATION
97
agreewith thesetwo setsof data, Q was made to
depend upon
x=fi+30
4.0
log f-95
instead of L•.
It was found when using this function of/5 and
f as an abscissaand the same ordinates as in
Fig. 10, a value of Q was obtained which gives
just as good a fit for the data of Fig. 10 and also
gives a better fit for the data of Figs. 8 and 9.
Other much more complicated factors were tried
3.0
X 2.0
1.0
to make
0
0
20
40
LOUONE$1•
60
LEVEL
80
100
120
OF COMPONENT'OB
FIG. 11. Loudness factor Q.
Table
formula which will represent the results. When
the values of b/, derived in this way were plotted
with bk as ordinates and zXfas abscissaeand Lk
as a variable parameter then the resulting graphs
were a series of straight lines going through the
common point (-250, 0) but having slopes depending upon L•,. Consequently the following
formula
bk= [-(250+/xf)/lOOO-]Q(L•.)
(17)
will representthe results.The quantity Af is the
common difference in frequency between the
components,Lk the loudnesslevel of each component, and Q a function depending upon Lk.
The results indicated that Q could be represented
by the curve in Fig. 11.
Also the condition must be imposed upon this
equation that b is always taken as unity whenever
the calculation gives values greater than unity.
The solid curves shown in Fig. 10 are actually
calculated curves using these equations, so the
comparison of these curves with the observed
points gives an indication of how well this equation fits the data. For this seriesof tones Q could
be made to depend upon fi• rather than L• and
approximately the same results would be obtained since fi• and L• are nearly equal in this
range of frequencies.However, for tones having
low intensities and low frequencies, fik will be
the observed and calculated
results
shownin thesetwo figurescomeinto better agreement but none were more satisfactory than the
simpleprocedureoutlinedabove. For purposeof
calculation the values of Q are tabulated in
VI.
•I'^]•L• VI. Valuesof Q(X).
X
0
0
10
20
30
40
5.00
3.82
2.64
1.60
1.09
0.90
0.88
50
60
70 0.90
80 1.04
90 1.27
100 1.51
1
4.88
3.70
2.52
1.53
1.06
0.90
0.88
2
3
4
5
6
7
8
9
4.76
3.58
;!.40
1.47
1.03
0.89
0.88
4.64
3.46
2.28
1.40
1.01
0.89
0.88
4.53
3.35
2.16
1.35
0.99
0.88
0.88
4.41
4.20
4.17
4.05
3.94
3.33
3.11
2.99
2.87
2.76
1.95
1.25
0.95
0.88
0.88
0.97
1.17
1.41
1.64
1.85 1.76 1.68
1.20 1.16 1.13
0.94 0.92 0.91
0.88 0.88 0.88
0.89 0.89 0.90
0.99 1.00 1.02
1.19 1.22 1.24
1.44 1.46 1.48
1.67 1.69 1.71
0.91 (}.92 0.93
1.06 1.08 1.10
1.29 1.31 1.34
1.53 1.55 1.58
Note: X=•t.+30
2.05
1.30
0.97
0.88
0.88
0.94 0.96
1.13 1.15
1.36 1.39
1.60 1.62
log fz.-95.
There are reasonsbased upon the mechanicsof
hearing for treating componentswhich are very
closetogether by a separatemethod. When they
are close together the combination must act as
thoughthe energywere all in a singlecomponent,
sincethe componentsact uponapproximatelythe
same set of nerve terminals.
For this reason it
seemslogicalro combinethem by the energylaw
and treat the combinationas a singlefrequency.
That some such procedureis necessaryis shown
from the absurdities into which one is led when
one tries to make Eq. (17) applicable to all cases.
For example, if 100 components were crowded
into a 1000-cyclespaceabout a 1000-cycletone,
then it is obvious that
the combination
should
much larger than L7• and consequently Q will be
sound about 20 db louder. But according to
smaller and hence the calculated
Eq. (10) to make this true for valuesof L• greater
than 45, bkmust be chosenas 0.036. Similarly, for
10 tones thus crowded together L-L• must be
loudness smaller.
The results in Figs. 8 and 9 are just contrary to
this. To make the calculated
and observed
results
98
HARVEY
FLETCHER
about 10 db. and therefore be=0.13 and then for
two such tones L-L•
must be 3 db and the cor-
respondingvalue of b• = 0.26. These three values
must belong to the same condition zXf=10. It is
evident then that the formulae for b given by
Eq. (17) will lead to very erroneousresults for
AND
W.
A.
MUNSON
z•L = L•--L•.
(20)
Also let this differencebe T when L• is adjusted
so that the masking componentjust masks the
componentk. Then the function for calculating b
must satisfy the following conditions:
such components.
b•=[-(250+z_Xf)/lOOO-]Q when ,XL=0,
In order to cover suchcasesit was necessaryto
b•=0
when•L=-T.
group together all componentswithin a certain
frequencyband and treat them as a singlecomAlso the following condition when L• is larger
ponent. Since there was no definite criterion for
than L•, must be satisfied,namely, be= 1 when
determining accurately what theselimiting bands
should be, several were tried and ones selected
which gave the best agreement between computed and observed results. The following band
widths were finally chosen:
For frequencies below 2000 cycles, the band
width is 100 cycles; for frequenciesbetween 2000
and 4000 cycles,the band width is 200 cycles;for
frequenciesbetween 4000 and 8000 cycles, the
band width is 400 cycles; and for frequenciesbetween 8000 and 16,000 cycles, the band width is
800 cycles. If there are k componentswithin one
of these limiting bands, the intensity I taken for
the equivalent single frequency component is
given by
I= E I• = • 10•mø.
(18)
A frequencymust be assignedto the combination.
It seemsreasonableto assigna weighted value of
f given by the equation
f=•.
f•,Ie/I=Z
f•10a•/lø,/E 10ateø. (19)
Only a small error will be introduced if the midfrequency of such bands be taken as the fre-
6L=
some value somewhat smaller than +T.
The value of T can be obtained from masking
curves. An examination
of these data indicates
that to a good approximation the value of T is
dependentupon the singlevariable f•.--2f,,. A
curve showing the relation between T and this
variable is shown in Fig. 12. It will be seen that
I
I for CASE
WHϥ
f)fm
VALU•:S
a•' af-fm-f -•fm
FIG. 12. Valuesof the maskingT.
for most practical casesthe value of T is 25. It
cannot be claimed that the curve of Fig. 12 is an
accurate representation of the masking data, but
quency of an equivalent componentexcept for it is sufficientlyaccuratefor the purposeof loudthe band of lowest frequency. Below 125 cyclesit nesscalculation since rather large changesin T
is important that the frequencyand intensity of will producea very slight changein the final caleach component be known, since in this region culated loudness level.
Data were taken in an effort to determine how
the loudnesslevel Le changesvery rapidly with
both changes in intensity and frequency. However, if the intensity for this band is lower than
that for other bands, it will contribute little to the
total loudness so that only a small error will be
introduced by a wrong choice of frequency for
this function depended upon aXLbut it was not
possibleto obtain sufficientaccuracyin the experimental results. The difference between the
resultant loudness level when half the tones are
down so as not to contribute to loudness and
the band.
when these are equal is not more than 4 or 5 db,
This then gives a method of calculating be which is not much more than the observational
when the adjacent componentsare equal in loud- errors in such results.
ness.When they are not equal let us define the
difference/XL by
A series of tests were made with tones similar
to those used to obtain the resultsshownin Figs.
LOUDNESS,
ITS
DEFINITION,
MEASUREMENT
AND
CALCULATION
99
zXL=-25, the most probable value of T. For
/xf= 100 and Q = 0.88 we will obtain the smallest
was made in which every other component was value of b/• without applying the 2•L factor,
down 10 db. Although these data were not usedin namely, 0.31. Then when using this factor as
determining the function described above, it was given above, all values of b•,:will be unity for
useful as a check on the final equations derived values of AL greater than 12 db.
Several more complicated functions of zXLwere
for calculating the loudnessof tones of this sort.
The factor finally chosenfor representing the tried but no•e of them gave results showing a
dependenceof bk upon /XLis 10 aL/v.This factor better agreement with the experimental values
is unity for AL=0, fulfilling the first condition than the function chosen above.
8 and 9 except that every other component was
down in loudness level 5 db. Also a second series
mentioned
above.
It
is 0.10 instead
of zero for
The formula for calculation of b•:then becomes
b• = [(250+f• -f,•)/lOOO•lO(•-•"')/rO(3•:+30
where
f• is the frequency of the component expressedin cyclesper second,
f,,•is the frequencyof the maskingcomponent
expressedin cycles per second,
L•. is the loudnesslevel of the kth component
when soundingalone,
L• is the loudnesslevel of the masking tone,
Q is a function depending upon the intensity
level fi• and the frequencyf•; of each
ponent and is given in Table VI as a function of x=fi•+30 logf•-95,
T is the masking and is given by the curve of
(21)
whereb•.is given by Eq. (21). If the valuesoff•
and • are measureddirectly then corresponding
valuesof L• can be found from Fig. 5. Having
these values, the masking component can be
found either by inspection or better by trial in
Eq. (21). That componentwhose values of L,•,
f,• and T introduced into this equation gives the
smallestvalue of b• is the maskingcomponent.
The values of G and Q can be found from
Tables III and VI from the correspondingvalues
of L•-, ilk, and f•. If all thesevalues are now introduced into Eq. (10), the resulting value of the
summation is.the loudnessof the complextone.
The loudnesslevel L correspondingto it is found
from Table
Fig. 12.
logf•- 95)
I[I.
If it is desired to know the loudness obtained if
It is important to remember that b• can neverbe
greater than unity so that all calculatedvalues
greaterthan this mu.•tbereplacedwith valuesequal
to unity. Also all components within the limiting
frequencybandsmust be groupedtogetheras indicatedabove.It is very helpful to remember that
any component for which the loudnesslevel is
12 db below the kth component, that is, the one
for xvhichb is being calculated, need not be considered as possibly being the masking component. If all the components preceding the kth
are in this classthen b• is unity.
these limitations
the formula
for calculat-
ing the loudnesslevel L of a steady complex tone
having n components is
G(L) = 5• b•G(L•),
will be obtained
if the summation
indicated
in
Eq. (10) is divided by 2. Practically the same result will
be obtained
in most instances
if the
loudnesslevel L• for each component when listened to ;vith one ear instead
of both ears is in-
sertedin Eq. (10). (G(Lx.)for one ear listeningis
equal to one--halfG(L•) for listening with both
ears for the samevalue of the intensity level of the
component.) If two complex tones are listened to,
one in one ear and one in the other, it would be
expected that the combined loudness would be
the sum of the two loudness values calculated for
RECAPITULATION
With
the typical listener used only one ear, the result
(10)
each ear as though no soundwere in the opposite
ear, although this has not been confirmed by experimental trial. In fact, the loudness reduction
factor b• has been derived from data taken with
both ears only, so strictly speaking, its use is
limited to this type of listening.
100
HARVEY
FLETCHER
AND
To illustrate the method of using the formula
the loudnessof two complex tones will be calcu-
W.
A.
MUNSON
sections,the results of a large number of tests are
given here, including those from which the
lated. The first may represent the hum from a formula was derived. In Tables VII to XIII, the
dynamo. Its componentsare given in the table first column shows the frequency range over
which the componentsof the tones were distribof computations.
uted, the figuresbeing the frequenciesof the first
Computations
and last components. Several tones having two
componentswere tested, but as the tables indicate, the majority of the tones had ten com1
60
50
3
3
1.0
ponents. Becauseof a misunderstandingin the
2
180
45
25
191
1.0
ZbeGle= 1009
designof the apparatus for generatingthe latter
3
300
40
30
360
1.0
4
540
30
27
252
1.0
L=
40
tones, a number of them contained eleven com5
1200
25
25
197
1.0
ponents, so for purposesof identification, these
are placed in a separate group. In the second
column
of the tables, next to the frequency range
The first step is to find from Fig. 5 the values
of
the
tones,
the frequency difference (•f) beof L• fromf, and •t•. Then the loudness
valuesG,
tween
adjacent
components is given. The reare found from Table III. Since the values of L
mainder
of
the
data
pertains to the loudnesslevels
are low'and the frequency separationfairly large,
of
the
tones.
Opposite
L• are given the common
one familiar with these functions would readily
loudness
levels
to
which
all the componentsof the
see that the values of b would be unity and a
tone
were
adjusted
for
a particular test, and in
computationwould verify it so that the sum of
the G values gives the total loudness1009. This
the next line the results of the test, that is, the
corresponds
to a loudness
levelof 40.
observedloudnesslevels (Lo•,,.), are given. Directly beneath each observed value, the calculated loudnesslevels (L•.)
are shown. The three
The second tone calculated is this same hum
amplified30 db. It better illustratesthe useof
the formula.
Computations
k f•
•
L•
1
60
2 180
3 300
4 540
5 1200
80
75
70
60
55
69
72
69
60
55
G• f,, L,•(301ogf•-95)Q
7440
9130 60 69
7440 180 72
4350 300 69
3080 540 60
---28
--21
--13
-- 3
-0.91
0.91
0.94
0.89
b
bXG
1.00
0.41
0.27
0.23
0.61
7440
3740
2010
1000
1880
loudness G= 16070
loudness level L = 79 db
associated values of L•, Loh,., and L•.•t½.in each
column representthe data for one completetest.
For example, in Table VIII, the first tone is
described as having ten components, and for the
first test showneach componentwas adjusted to
have a loudnesslevel (L•) of 67 db. The resultsof
the test gave an observed loudnesslevel (Lo•,.)
of 83 db for the ten componentsacting together,
and the calculated loudnesslevel (L•.,lo.) of this
same tone was 81 db. The probable error of the
observed results in the tables is approximately
4-2 db.
The loudness level of the combined tones is
only 7 db abovethe loudness
levelof the second
component.If only one ear is usedin listening,
the loudness of this tone is one-half, correspond-
In the next seriesof data, adjacent components
T•BLE VII. Two componenttones(AL=0).
Loudnesslevels (db)
Frequency range
ing to a loudness
levelof 70 db.
L•
Lo•,•.
Leale.
83
87
87
63
68
68
43
47
47
23
28
28
2
2
4
000
Lt:
Lon•.
Le•c.
83
89
91
63
71
74
43
4o
52
23
28
28
--1
2
l
875
L&
Lobs.
' Leale.
84
92
92
1000-1100
COMPARISONOF OBSERVED AND CALCULATED
RESULTS OXi THE LOUDNESS LEVELS OF
COl•IPLEX
TONES
ßIn order to show the agreementbetweenobserved loudnesslevels and levels calculated by
meansof the formula developediv the preceding
1000-2000
125-1000
LOUDNESS,
ITS
DEFINITI()N,
MEASUREMENT
TABLE VIII.
CALCULATION
Loudnesslevels (db)
50 500
Lk
Lo•,•.
L•d•.
67
83
81
54
68
72
33
47
53
21
38
39
11
2:0
2:4
-- 1
2
8
50 500
Lk
Lobs.
L•ale.
78
92
91
61
73
77
41
53
60
23
42
42
13
2:5
27
--1
2
8
L•
78
69
50
16
6
Lobs.
L•::m•.
94
93
82
83
62
65
32
31
22
17
2
0
L•
Lobs.
Le=•.
57
68
73
37
50
52
20
34
36
3
2
5
84
95
100
64
83
83
43
59
68
24
41
47
2
2
12
84
94
100
L•
Lo•s.
81
93
64
82
43
65
23
49
13
33
-4
Lcale.
98
83
68
45
27
0
1400
1895
1400-1895
L,
Lobs.
L•,=m,.
100-1000
100
1000
-1
83
63
43
23
95
79
59
43
2
L•.•d•.
99
82
68
45
9
L•
Lo•,•.
Leal½.
83
100
100
63
82
80
43
59
60
23
32
38
78
99
95
59
81
77
1100-3170
L•
Lobs.
L½•t0.
79
100
100
60
81
83
41
65
64
17
33
34
7
22
18
--4
2
3
260-2600
L•
Lobs.
Leale.
79
97
100
62
82
85
42
65
68
23
44
45
13
28
27
- 2
2
5
3100
3900
L•
530
5300
530
5300
- 7
2
0
53
43
25
82
61
43
17
-- 2
83
82
73
72
52
48
105
108
90
89
73
72
40
34
2
5
Lk
Lobs.
61
89
41
69
21
45
--3
2
Lea•½.
89
70
42
4
In thefollowing
setoftests(Tables
XII and
XIII) the differencein loudnesslevel of adjacent
componentswas 10 db.
The next data are the results of tests made on
the complex tone generated
by the Western
Electric No. 3A audiometer. When analyzed, this
tone was found to have the voltage level specXIV.
27
38
42
75
loudness level given opposite L•, and the even
numbered components were 5 db lower. (Tables
X and XI.)
Table
48
62
65
100
101
had a difference in loudness level of 5 db, that is,
in
24
44
47
Lob•.
Lcale.
the first, third, fifth, etc., components had the
shown
43
63
68
When
the
r.m.s.
2
2
12
3
L•
1000
64
80
83
2
Lob•.
100
101
Ten componenttones(zXL=0).
Frequency range
trum
AND
voltage acrossthe receiversused was unity, that
is, zero voltage level, then the separate componentshad the voltage levelsgiven in this table.
Adding to the voltage levels the calibration constant for the receivers used in making the loud-
nesstests giw•s the values of fi for zero voltage
level acrossthe receivers.The values of • for any
other voltage level are obtained by addition of
the level desired.
Tests were made on the audiometer tone with
the same receivers* that were used with the other
complex tones, but in addition, data were avail* See calibration shownin Fig. 1.
102
HARVEY
FLETCHER
AND
W.
TABLE IX. Elevencomponent
tones(/•L=0).
Frequency range
1000-2000
1000-2000
1150-2270
100
112
MUNSON
TABLE XIII.
Loudness levels (db)
100
A.
Elevencomponenttones(•XL= 10 db).
Frequency range
af
57
L•
84 64 43 24
Loba. 97 83 65 43
Lc•le. 103 84 64 45
--1
2
7
57-627
Lk
84 64 43 24
Lobs. 99 82 65 42
Leale. 103 84 64 45
1
2
11
3420-4020
Lk
Lobs.
Lc•ic.
79 60 40 20
99 78 62 41
10 --5
25
2
98 81 61 40
23
Lk
Lobs.
Le•lc.
77 62 42 22
102 86 66 46
101 88 69 44
60
Loudnesslevels (db)
L•
Lob•.
80
88
L•lc.
90
L•
Lobs.
L•x•.
62
70
42
53
27 17
40 27
76 59 45 30
81 62 42 27
100 70 50 33
94 75 53 37
17
26
27
2
2
8
--4
2
0
TABLE
XIV. Voltage
level
spectrum
ofNo.3A
1
audiometer tone.
1120-4520
340
7 --7
2
20
19 - 1
Frequency Voltagelevel
-----
2.1
5.4
4.7
5.9
2128
2280
2432
2584
--11.4
-- 16.9
--14.1
-- 16.2
760
912
1064
1216
1368
1520
1672
1824
1976
---------
4.6
6.8
6.0
8.1
7.6
9.1
10.0
9.9
14.1
2736
2880
3040
3192
3344
3496
3648
3800
3952
-- 17.4
-17.5
--20.0
--19.4
-- 22.7
--23.7
-- 25.6
-- 24.6
--26.8
TABLE X. Ten componenttones(•L = 5 db).
Frequency range
Loudnesslevels (db)
1725-2220
55
L•
82 62 43 27
Loba. 101 73 54 38
Lcalc. 95 76 56 40
17
30
30
--6
2
-1
55
L•
Lobs.
Lc•t•.
12
22
22
-2
1725-2220
80 62 42 22
94 66 50 33
93 76 54 35
2
4
Frequency Voltage
level
152
304
456
608
given instead of voltage levels,so in utilizing it
here, it was necessaryto assumethat the thresh-
TABLEXI. Elevencomponent
tones(AL = 5 db).
old levels of the new and old tests were the same.
Frequency range
Loudnesslevels (db)
57 627
57
L•
79
Lobs. 91
Lc,,lc. 90
3420-4020
60
L•
Loba.
Lcfii.
76
95
89
61 41
73 56
76 59
26
41
43
16
28
28
61
77
75
25
33
36
15
25
26
42
55
54
1
2
8
Computations were made at the levels tested
experimentally and a comparisonof observedand
calculated
results is shown in Table XV.
-9
TABLE XV.
2
--4
A. Recent
r.m.s.
level
Lobs.
Leale.
TABLE XII. Ten cmnponent
tones(zXL= 10 db).
Frequency range
•f
1725-2220
55
1725-2220
55
Loudnesslevels (db)
tests on No. 3A audiometer
tone.
volt.
--38 --55 --59 --70 --75 --78 --80 --87 --89 --100 --102
95
85
79
61
56
41
42
28
22
2
2
89
74
71
57
49
44
40
:28
B. P•evious tests on No. 3A audiometer
L•
79 59
40
19
9
--5
Lobs.
Leale.
95
91
7l
73
54
51
33
31
22
17
2
--1
Lk
79
Lob•. 89
LcMc. 92
61
67
75
41
48
53
27
37
39
17
27
28
--1
2
4
r.m.s. volt. level
Loba.
LcMc.
25
7
4
tone.
+10
118
9
103
--40
77
--49
69
--6(I
6l
--69
50
--91
2
119
103
82
73
56
41
6
The agreement of observed and calculated resuits is poor for some of the tests, but the close
agreement in the recent data at loxvlevels and in
the previous data at high levels indicates that the
able on tests made about six years ago using a observed results are not as accurate as could be
differenttype of receiver.This latter type of re- desired. Because of the labor involved these tests
ceiver was recalibrated (Fig. 13) and computa- have not been repeated.
tions made for both the old and new tests. In the
At the time the tests were made severalyears
older set of data, levels above threshold were
ago on the No. 3A audiometer tone, the reduc-
LOUDNESS,
ITS
DEFINITION,
MEASUREMENT
AND
CALCULATION
103
,:
FiG. 14. (A to D) Iouduesslevel reduction tests on the No.
3A audiometer
FIG. 13. Calibration
of receivers for tests on the No. 3A
audiometer tone
tone.
testing the method of measuringand calculating
loudness levels.
rio,1 in loudnesslevel which takes place when
In view of the complexnature of the problem
this computation method cannot be considered
ternfined. As this can be readily calculatedwith fully developedin all its details and as more acthe formula developedhere, a comparisonof ob- curate data accumulatesit may be necessaryto
served and calculated results will be shown. In
change the formula for b. Also at the higher levels
Fig. 14A, the ordinate is the reduction in loud- someattention must be given to phasedifferences
ness level resulting when a No. 3A audiometer between the components.However, we feel that
tone having a loudnesslevel of 42 db was changed the form of the equation is fundamentally correct
by the insertion of a filter which eliminated all of and the loudness function, G, corresponds to
the componentsabove or below the frequency in- somethingreal in the mechanismof hearing. The
dicated on the abscissa. The observed data are the
present values given for G may be modified
plotted points and the smooth curvesare calcu- slightly, but we think that they will not be
lated results. A similar comparison is shown in radically changed.
Figs. 14B, C and D for other levels.
A study of the loudness of complex sounds
This completesthe data which are available on which are not steady, such as speechand sounds
steady complextones. It is to be hoped that others of varying duration, is in progressat the present
will find the field of sufficientimportanceto war- time and the resultswill be reported in a second
rant obtaining additional data for improving and paper on this subject.
certain components are eliminated was also de-
APPENDIX
A measurelncnt
A.
EXPERIMENTAL
•IETHOD
OF •IEASURING
of the loudness leYel of a sound consists
THE LOUDNEqS LEYEL
OF A •TEADV
•OUND
method of averaf/e error, constant stimuli, etc., but also in
of listeningalternately to the soundand to the 1000-cycle important experinIentaldetails suchas the control of noise
reference tone and adjusting the latter until the two are conditions and fatigue effects. In some instancesuniqne
equally loud. If the intensity level of the reference tone is
L decibels when this condition is reached, the sound is
devices have been used to facilitate a read_v comparison
of sounds. One of these, the alternation phonolueter,•ø
said to have
introducesinto the comparisonimportant factors sut•has
a loudness
level
of L
decibels.
When
the
characterof the soundbeingmeasureddiffersonly slightly
froin that of the referencetone, the comparisonis easilyand
the duration
time of the sounds and the effect of transient
quickly made, but for other soundsthe numerousfactors
which enter into a judglnent of equality of loudnessbecome
conditions. The merits of a particular method will delx-nd
upon the circumstancesunder which it is to be used.The
one to be described here was developed for an extensive
important,
series of laboratory tests.
and an experimental method should be used
which will yield results typical of the average normal ear
To dctermine when two sounds are eqnally loud it iq
and nornmlphysiologicaland psychological
conditions.
A variety of methodshave been proposedto accomplish
•0 D. Mackenzie, RelativeSensitivityof theF•ar at Different
this, differing not only in generalclassification,that is, the Levelsof Lo•td•tes•,Phys. Rev. 20, 331 (1922).
104
HARVEY
FLETCHER
AND
necessaryto rely upon the judgment of an observer, and
this involves of course, not only the ear mechanism, but
also associatedmental processes,and effectively hnbeds the
problem in a variety of psychological factors. Thesc
difficulties are enhanced by the large variations found in
the judgments of different observers, necessitating an
investigation conducted on a statistical basis. The method
of constant stimuli, wherein the observer listens to fixed
levels of the two sounds and estimates which sound is the
louder, seemedbest adapted to control the many factors
involved,when usingseveralobserverssimultaneously.By
meansof this method,an observer'spart in the test can be
readily limited to an indication of his loudnessjudgment.
This is essential as it was found that manipulation of
apparatus controls, even though they were not calibrated,
or participation in any way other than as a iudge of loudness values, introduced undesirable factors which were
aggravated by continued use of the same observers over
a long period of time. Control of fatigue, memory effects,
and the associationof an observer'sjudgments with the
results of the tests or with the iudgments of Other observerscould be rigidly maintained with this method, as
will be seen from the detailed explanation of the experi-
W.
control
the
reference
tone
and
the
sounds to
be
whether
the reference
measured.Vacuum tube oscillatorswere usedto generate
pure tones, and for complex tones and other sounds,
suitablesourceswere substituted.By meansof the voltage
measuringcircuit and the attenuator, the voltage level
(voltagelevel= 20 log V) impressedupon the terminalsof
the receivers, could be determined. For example, the
or softer than
the
booth.
The typical recording chart shown in Fig. 16 contains
the results of three observerstesting a 12S-cycletone at
three different levels. Two marks were used for recording
125 c.p.s. Pure Tone Test No. 4 Crew No. 1. 1000 c.p.s.
Voltage Level (db)
Obs. 4-6
4-2
-2
-6
-10
CK
4-
+
4-
4-
4-
0
0
0
c.p.s. AS
+
+
d-
+
0
0
0
0
0
Volt.
DH
level= CK
4-9.8db AS
DE[
CK
AS
DH
4+
44+
44-
4+
44+
4+
0
+
40
+
40
0
+
q0
+
0
0
0
+
0
40
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
--4
--8
--12
125
125
-14-18-22
-26
0
--16--20--24--28--32
CK
4-
+
4-
4-
0
4-
4-
0
0
c.p.s. AS
4-
4-
4-
4-
+
0
0
0
0
h+
4+
+
+
+
+
+
4h+
+
+
4+
hh+
+
+
4+
h0
+
+
0
0
44+
+
+
+
0
+
40
+
0
0
0
40
h0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
--15
--19
--23
--27
--31
--35
--39
--43
--47
h+
+
0
+
+
h+
+
+
44-
h+
0
+
+
0
h+
+
4+
4-
+
0
0
+
0
0
0
0
+
+
+
0
0
0
0
0
0
4-
0
0
0
0
0
0
0
0
+
+
4-
+
44-
o
0
0
+
0
0
+
+
0
+
+
0
0
0
4-
0
0
0
DH
CK
AS
DH
CK
AS
DH
125 CK
c.p.s. AS
for loudness balances.
tone was louder
other sound and indicate their opinions by operating the
switches.The levels were then changed and the procednre
repeated. The results of the tests were recorded outside the
Volt.
level=
-3.2db
FIG. 15. Circuit
MUNSON
of one second.After a pause of one secondthis sequence
was repeated, and then they were required to estimate
mental procedure.
The circuit shown in Fig. 15 was employed to generate
and
A.
Volt.
level=
-14.2
db
DH
CK
AS
DH
CK
AS
DH
Fm.
16. Loudness
balance
0
0
0
0
0
0
0
data sheet.
attenuator, which was calibrated in decibels, was set so
the observers'judgments,a cipherindicatingthe 125-cycle
that the voltage measuringset indicated 1 volt was being
tone to be the louder, and a plus sign denoting the reference
impressed upon the receiver. Then the difference between
tone to be the louder of the two. No equal judgments
this settingand any other settingis the voltagelevel. To
were permitted. The figures at the head of each column
give the voltage level of the reference tone impressed
upon the receivers, that is, the number of decibels from 1
volt, plus if above and minus if below, and thoseat the side
are similar values for the tone being tested. Successivetests
were chosenat random from the twenty-seven possible
obtain the intensity level of the sound we must know the
calibration
of the receivers.
The observers were seated in a sound-proof booth and
were required only to listen and then operate a simple
switch. These switcheswere provided at each position and
were arranged so that the operations of one observer could
not be seenby another. This was necessaryto prevent the
judgmentsof oneobserverfrom influencingthoseof another
observer.First they heardthe soundbeing tested,and immediately afterwards the referencetone, each for a period
combinationsof levelsshown,thus reducingthe possibility
of memory effects. The levels were selected so the observers listened
to reference
tones which
were louder and
softer than the tone being tested and the median of their
judgments was taken as the point of equal loudness.
LOUDNESS,
ITS
DEFINITION,
MEASUREMENT
The data on this recordingchart, combinedx•ith a
similar number of observationsby the rest of the crew,
•ND
C•LCULtkTION
105
gradually.
Relays
operating
in thefeedback
circuits
of the
vacuumtube oscillatorsand in the grid circuitsof anq)lifiers
(a total of elevenobservers)
are shownin graphicalform performedthis operation.The period of grox•thand
0.1 secondas ,own on the
in Fig.17.Thearrowindicates
themedian
levelat which decaywas approximately
typical oscillogram
in Fig. I9. \Vith thesealex
icesthe
vo,'r^Gd
L•&-'
7
:_-c__
0 ! e'Ec
FIG. 17. Percentof obser•ations estimating 1000-cycletone
to be louder than 125-cycle tone.
the 1000-c.xcle
reference,
in theopinionof thisgroupof
observers,
sounded
equallyloudto the 12S-cycle
tone.
The testingmethodadoptedwasinfluenced
by efforts
to minimizefatigueeffects,both mentaland physical.
Mentalfatigueandprobable
changes
in theattitudeof an
DECAY
FXG.19. Growth and decayof 1000-cyclereferencetone.
observer
duringthe progress
of a longseriesof testswere
detected
by keeping
a recordof the spreadof eachob- transient effects were reducedanti yet the sotrodsseemed
server'sresttits.As longas the spreadwas normalit was
assumed
that the fatigue,if present,wassmall.The tests
were conductedon a time schedulewhich limited the
observersto five minutesof continuoustesting, during
whichtimeapproximately
fifteenobserYations
weremade.
The maximum number of observationspermitted in one
day was 150.
To avoid fatiguingthe ear the soundsto whichthe
to start and stop instantaneously
unlessattention was
called to the effect. -k motor-driven commutator operated
the relayswhichstartedand stoppedthe sountis
in proper
sequence,
and sxGtched
the receixers
fromthe reference
tone circuit to the sound under test.
The customaryroutine measurements
to insure the
propervoltagelevelsimpressed
uponthe receivers
were
madewith the measuringcircuit shownschematicallyin
Fig. 15. Duringthe progress
of the testsvoltagemeasureseqnence
illustrated
onFig.18.Theduration
timeofeach mentswere made frequentlyand later correlatedwith
observerslistened were of short duration and in the
measurementsof the correspondingfield sound pressures.
Thresholdmeasurements
were madebeforeanti after the
loudness
tests.They weretakenon the samecircuit used
for the loudnesstests (Fig. 15) by turning off the 1000-
cycleoscillator
andslox•
ly attenuating
theothertouebelow
FIG. 18. Time sequence
for loudness
comparisons.
thresholdand then raisingthe level until it again became
audible.The observers
signalledwhenthey couldno longer
hearthe toneand thenagainwhenit wasjust amlible.The
soundhadto belongenough
to attainfull loudness
andyet average
ofthesetwoconditions
wastakenasthethreshold.
notsufficiently
longto fatiguetheear.The refereuce
tone
An analysisof the harmonicsgeneratedby the refollowedthe x soundat a time intervalshortenoughto ceiversand other apparatuswas madeto be sureof the
permita readycomparison,
andyet not be subject
to purityof the tonesreaching
the ear.The receivers
were
fatigueby prolonging
thestimulation
withoutanadequate of theelectrodynamic
typeand • erefoundto produceoverrestperiod.At highlevelsit wasfoundthat a tonere- tonesof the orderof 50 decibelsbelowthe fundamental.At
quiresnearly0.3 second
to reachfull loudness
antiif the veryhighlexels,distortionfromthe filterswasgreater
sustainedfor longerperiodsthan one second,thereis than from the receivers,but in all casesthe loudness
level
dangerof fatiguingthe ear.u
of anyovertone
was20decibels
or morebelowthatof the
To avoidtheobjectionable
transients
whichoccurx•hen fundamental.Experiencewith complextoneshas shown
soundsare interruptedsuddenlyat high levels,the con- that under thesecomlitionsthe contribution of the overtrollingcircuitwasdesigned
to startandstopthesounds tones to the total loudnessis insignificant.
n G. v. Bekesy,Theoryof IIearing, Phys.Zeits.30,
115 (1929).
The methodof measuringloudnesslevel which is de.
scribedherehasbeenusedon a largevariety of soundsand
foundto give satisfactoryrest,Its.
106
HARVEY
APPENDIX
B.
FLETCHER
AND
COMPARISON OF DATA
listened to the tones with both ears in the tests reported
here, while a single receiver was used by Kingsbury.
Also, it is intportant to remember that the level of the
tonesusedin the experimentswasexpressed
as the number
of db above the averagethresholdcurrent obtained with a
single receiver. For both of these reasons a direct comparison of the results cannot be made. However, in the
course of our work two sets of experiments were made
which give results that make it possibleto reduce Kingsbury's data so that it nray be compared directly with that
reported in this paper.
In the first set of experiments it was found that if a
typical observerlistened with both ears and esti•nated that
two tones, the referencetone and a tone of different frequency, appeared equally loud, then, nraking a similar
comparisonusing one ear (the voltages on the receiver
remaining unchanged)he would still estimate that the two
tones were equally loud. The results upon which this
TAnLE XVI.
XVI.
A.
MUNSON
ON 1HE LOUDNESS LEVELS OF PURE TONES
A comparisonof the present loudnessdata with that reported previouslyby B. A. Kingsbury4 would be desirable
and in the event of agreement, would lend support to the
general application of the results as representativeof the
average ear. It will be remembered that the observers
conclusion is based are shown in Table
W.
when the soundsare comingdirectly to the earsfrom a free
wave.
This result is in agreement with the point of view
adoptedin developingthe formula for calculatingloudness.
When listeningwith oneinsteadof two ears,the loudnessof
the referencetone and also that of the tone being compared
are reducedto one-half. Consequently,if they were equally
loud when listeningwith two earsthey mustbe equally loud
when listening with one ear. The secondset of data is
concerned with differences in the threshold when listening
with one ear versuslistening with two ears.
It is well knownthat for any individualthe two earshave
different acuity. Consequently,when listening with both
ears the threshold is determined principally by the better
ear. The curve in Fig. 20 shows the difference in the
•o
so
•oo
zoo
500
•ooo
zooo
sooo
,oooo
zoooo
In the first
F•o. 20. Differencein acuity betweenthe best ear and the
average of both ears.
Comparisonof one and two-ear
loudness balances.
A. Reference
Frecluency,
c.p.s.
Voltage level
difference*
62
--0.5
i25
0
tone voltage level
250
500
2000
4000
6000
8000
10,000
+1.0
--1.0
--0.5
--0.5
4-0.5
--3.0
--3.0
B. Other
Ref. tone
volt. level
--32rib
reference
62 c.p.s.
Volt. level
difference*
tone levels
2000 c.p.s.
Ref. tone
Volt. level
volt. level
difference*
-20
--34
--57
+0.5
+0.2
+2.0
-- 3
-22
--41
0.0
+0.3
--0.8
--68
--0.5
--60
--0.8
--7 ½
)
--6.2
threshold level between the average of the better of an
observer'sears and the average of eli the ears. The. circles
representdata taken on the observersusedin our loudness
tests while the crosses represent data taken from an
analysisof 80 audiogramsof personswith normal hearing.
If the difference in acuity when listening with one ear rs.
listeningwith two ears is determinedentirely by the better
ear, then the curve shown gives this difference.However,
someexperimentaltests which we made on one ear acuity
vs. two ear acuity showed the latter to be slightly greater
than for the better ear alone, but the small magnitudes
involved and the difficulty of avoiding psychologicaleffects
causeda probable error of the same order of magnitude as
* Differences are in (lb, positive values indicating a
higher voltage for the one ear balance.
row are shown the frequenciesof the tones tested. Under
these frcquenciea are ahown the differences in db of the
voitage levels on the receivers obtained when listening by
the two methods, the voltage level of the referencetone
being constant at 32 db down frmn I volt. Under the
caption "Other ReferenceTone Levels" similar figuresfor
frequenciesof 62 c.p.s. and 2000 c.p.s. and for the levels of
the referencetone indicated are given. It will be seen that
these
differences
are well
within
the
observational
error.
Consequently, the conclusion mentioned above seems to
be justified. This is an important conclusionand although
the data are confined to tests made with receivers on the
ear it would be expectedthat a similar relation would hold
Fro. 21. Loudnesslevels of pure tones--A comparisonwith
Kingsbury's data.
LOUDNESS, ITS DEFINITION,
MEASUREMENT A\'D CALCULATION 107
the qualitybeingmeasured.
At the higherfrequencies If now we add to the values for the level above threshold
anamountcorresponding
to thedifferwherelargedifferences
are usuallypresent
the acuityis givenby Kingsbury
deternfinedentirely by the better ear.
From valuesof the loudnessfunctionG, onecan readily
calcnlatewhat the differencein acuity when usingone z,s.
two earsshouldbe. Sucha calculationindicatesthat when
the two earshavethesameacuity,thenwhenlisteningwith
both earsthe thresholdvaluesare about 2 db lowerthan
encesshownby the curveof Fig. 20, then the resulting
valuesshouldbe directly comparableto our data on the
basisof decibelsabove threshold. Comparisonsof his data
on this basisaith thosereportedin this paperare shownin
Fig. 21. The solidcontourlineqaredrawnthroughpoints
taken from Table I and the dotted contour linc• taken
data.It willbeseenthat thetxvosetsof
whenlistening
with oneear.Thissmalldifference
would fromKingsbury's
data are in goodagreementbetween100and 2000cycles
above and belowthesepoints.
We are nowin a positionto compare
the dataof Kings- but divergesomexxhat
are slightly greaterthan would be
burywiththose
shown
in TableI. Thedatain TableI can The discrepancies
fromexperimental
errors,but mightbeexplained
be convertedinto decibelsabovethresholdby subtracting expected
accountfor the difficultyin t%ing to measureit.
ofa slightamount
ofnoise
duringthreshold
theaverage
threshold
valuein eachcolumn
frmnanyother bythepresence
deternfinations.
number in the same column.
APPENDIXC. OPTICALTONE (;ENER•,TOR
OF COMPLEX.
•V•,VEFORMS
For the loudnesstests in which the referencetone was
compared
with a complex
tonehavingcomponents
of
specified
loudness
levelsand frequencies,
the toneswere
listened
to bymeans
of headreceivers
asbefore;
thecircuit
shownin Fig. 15 remaining
the sameexcepting
for the
vacuumtube oscillatormarked "x Frequency."This was
replaced
by a complex
tonegenerator
dexiscdby E. C.
Wenteof the Bell TelephoneLaboratories.The generator
is shownschematicallyin Fig. 22.
The desiredwave form wasaccuratelydrawn on a large
scaleand then transferredphotographicalIsto the glass
diskdesignated
asD in thediagram.
Thedisk,drivenbya
motor,rotatedbetween
the lampL anda photoelectric
cell C, producing
a fluctuating
light sourcewhichwas
directed
by a suitable
opticalsystem
uponthe plateof
the cell.The voltagegenerated
wasalnplifiedandattenuated as in the case of the pure tones.
FIG. 22. Schematic of optical tone generator.
FIG. 23. Ten disk optical tone generator.
108
HARVEY
FLETCHER
The relative magnitudes of the components were of
course fixed by the form of the wave inscribed upon the
disk, but this was modified when desired, by the insertion
of elements in the electrical circuit which gave the desired
characteristic. Greater flexibility in the control of the
amplitude of the componentswas obtained by inscribing
each componenton a separate disk with a complete optical
system and cell for each. Frequency and phase relations
were maintained by mounting all of the disks on a single
shaft. Such a generator having ten disks is shown in
Fig. 23.
AND
W.
A.
MUNSON
An analysis of the voltage output of the optical tone
generators showed an average error for the amplitude of
the componentsof about 4-0.5 db, which was probably the
linfit of accuracyof the measuringinstrument. Undesired
harmonicsdue to the disk being off center or inaccuracies
in the wave form were removed by filters in the electrical
circuit.
All of the tests on complex tones describedin this paper
were made with the optical tone generator excepting the
audiometer, and two tone tests. For the latter tests, two
vacuum
tube oscillators
were used as a source.