NASA Reference Publication 1001
 

Oyerall Loudness of Stead, 

Sounds According to The( '-I 

and Experiment coo 

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oo 

Walton L. Howes "= 

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NANASA STi FACILIT 

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NASA Reference Publication 1001
 

Overall Loudness of Steady 

Sounds According to Theory 

and Experiment 

Walton L. Howes 

Lewis Research Center 

Cleveland, Ohio 

Prepared for 

Aerospace Safety Research and Data Institute 

NASA Lewis Research Center 

NASA 

Scientific and Technical Information Branch 1979 

National Aeronautics and Space Administration 

Washington, DC 
Library of Congress Cataloging in Publication Data 

Howes, Walton L. 

Overall loudness of steady sounds accordng to theory and experiment. 

(NASA reference publication ; 1001) 

Bibliography: p. 

1- Auditory perception. 2. Noise-Physiological effect. 3. Sound-measurement. I. United States. Lewis Research Center, Cleveland. Aerospace Safety 

Research and Data Institute. II. Title. ill. Series: United States. National 

Aeronautics and Space Administration. NASA reference publication ; 1001. 

QP465.H68 612' :85'028 78-24563 

For sale by the Superintendent of Documents, 

U.S. Government Printing Office, Washington, D. C. 20402 

Stock Number OR-000-00690-0 
FOREWORD 

The growth of technology has been accompanied by numerous 

consequences harmful to the environment. Among these negative 

effects is noise. In recent years, as noise sources have become more 

prevalent and more intense, the general population, increasingly annoyed by noise, particularly from aircraft, trucks, and machinery, 

has exerted pressures for laws and regulations to establish allowable. 

limits on noise. As a consequence, the reduction of noise has become 

a significant cost of doing business. 

One of the problems of noise abatement and legal restrictions is 

that of quantitatively defining the level of a noise when it becomes 

annoying. There is no precise answer to this problem because the 

individual, his environment, and the exposure circumstances all 

strongly influence the result. However, annoyance is closely relatedto 

loudness, which is much less influenced by nonanditory effects. If 

two sounds are annoying, then the louder sound tends to be more 

annoying. Loudness provides a simpler specification than annoyance 

for noise abatement and legal purposes. 

For both abatement and legal purposes it is most important to be 

able to predict the loudness of sounds. The relationship between the 

sound and loudness is determined by the operation of &he intervening 

auditory system. Previous loudness calculation schemes are questionable because they hav6 incorporated this operation in a sketchy and 

dubious fashion. 

The purpose of this publication is to present a new quantitative, 

analytical ,model for the operation of the human auditory system as 

it pertains to loudness. The resulting theory provides a practical 

means for calculating the loudness of any steady sound from given 

information regarding the so-and. It is hoped that this new model 

of the auditory system will provide important new insights concerning the manner in which the loudness sensation develops and 

that the resulting loudness calculation scheme will become accepted 

as a replacement for existing schemes. 

iii 
CONTENTS
 

PAGE 

FOREWORD ---

INTRODUCTION ------------------------------------

OPERATION OF AUDITORY SYSTEM ----------------- 2 

Filtering --------------------------------------------------- 4
 

Neural Response --------------------------------------------- 5
 

Beats ---------------------------------------------------- 7 

OriticaZ ban id --------------------------------------------- 7 

Secondary tones -------------------------------------------- 8 

Channels and nveron bandwidts ------------------------------- 10 

-Auralharmonics anti combinationetones ------------------------- 10 

Loudness -------------------------------------------------- 11 

MODEL OF AUDITORY SYSTEM --------------------- 11 

LOUDNESS OF ANY STEADY SOUND: ANALYSIS --- 13 

Discrete Spectrum ------------------------------------------- 15 

Category A: SuberittcaZ bandspread --------------------------- 17 

Category B: Supercritical tone spacing ------------------------- 21 

Category C: Hypercritical tone spacing------------------------- 7 

Category AB: Suboritically spaced tones spread over spercritical 

balldwidt ---------------------------------------------- 27 

Category AC: Suberitically spaced tones spread over Ltyperoritical 

bandwidt ---------------------------------------------- 28 

Category BC: Sipercitically spaced tomes spread over hypercritical 

band vidtA ------------------------------------------------ 28 

Continuous Spectrum ---------------------------------------- 8 

Rubritical bandnidthi -------------------------------------- 28 

Supercritical badiidth -------------------------------------- 30 

HUypercritical-bandwidth ------------------------------------- 31 

Mixed Spectrum -----------------------------------------

LOUDNESS LEVEL -------------------------------- 81 

TRANSMITTANCE FUNCTIONS ---------------------- 82 

SIMPLE FORMUGLAS FOR LOUDNESS LEVEL -------- 37 

TEST CONDITIONS -------------------------------- 9 

V
 
PAGE 

LOUDNESS OF SPECIFIC SOUNDS WITH DISCRETE 

SPECTRA (COMPARISON OF MQDEL WITH EXPERIMENT) ------------------------------------- 40 

One Tone -------------------------------------------------------- 40 

Two Tones ------------------------------------------------ 42 

Von BQkaY's experiment (1929) --------------------------------- 43 

Fletcher anat Munson's emperitlzent (1988) ------------------------- 45 

Small andt hurlown's experiment (1954) ------------------------- 47 

D. H. Howee' experiment (1950) --------------------------------- 50 

Sckarf's experiment (1959) -------------------------------------- 53 

Three Tones ----------------------------------------------------- 57 

Baueh's experiment (1956) -------------------------------------- 57 

D. H. Howes' experiment (1950) --------------------------------- 62 

Scharf's experiment (1962) ------------------------------------- 65 

Four Tones ------------------------------------------------ 67 

D. H. Howes' experimcnt (1950) --------------------------------- 67 

Zwcker, E ottorp, and Stevens' experiment (1957) ----------------- 69 

Scharfs erperimcnt (1959) -------------------------------------- 73 

Five Tones ------------------------------------------------ 75 . H. Howea' experiment (1950) --------------------------------- 75 

Zwdiker, Fiottorp, and Stevens' experiment (1957) ------------------ 80 

Quietgch&'s experiment (1955) ----------------------------------- 80 

Six Tones ------------------------------------------------- 82 

D. H. Holwee' experiment (1950) --------------------------------- 82 

Seven Tones ----------------------------------------------------- 82 

D. . Howes' cxpcrimcnt (1950) --------------------------------- 82 

might Tones ----------------------------------------------------- 83 

D. H, Howes' experinent (1950) --------------------------------- 83 

Quietzsceh's experiment (1955) ------------------------------------ 84 

Ten Tones ------------------------------------------------------- 85 

Fletcher and Mmson's experiment (1938) ----------------------- 85 

Eleven Tones ---------------------------------------------------- 88 

D. H. Howes' emperiment (1950) --------------------------------- 88 

Fletcher and Munsowes experiment (1983) ------------------------ 89 

Twenty Tones or More ------------------------------------------ 91 

Quietzech's experiment (1955) ----------------------------------- 91 

Loudness as Function of Number of Tones Contained Within Fixed 

Frequency Band ----------------------------------------------- 92 

Scharf's experiment (1959) ------------------------------------- 92 

LOUDNESS OF SPECIFIC SOUNDS WITH CONTINUOUS SPECTRA '(COMAPISON OF MODEL 

WITH EXPERIMENT) ---------------------------- 93 

Effect of Bandwidth --------------------------------------------- 98 

Zivteker and Feltfkeler's experihent (1955) ---------------------- 93 

Niese's experiment (1961) --------------------------------------- 94 

Zwfceer, Flottorp, ant Stevens' experiment (1957) ---------------- 95 

Octave Bands ---------------------------------------------------- 96 

Robinson antI Whittle's experiment (1964) ----------------------- 96 

Creyner,Plenige, and Schivarzi's experinent (1959) ----------------- 99 

G. Jahn's experiment (1959) ------------------------------------ 101 

Stevens' experiment (1956) ------------------------------------- log 

vi 
PAGE 

Various Bands 103 

Zwtcker's experiment (1966) - 103 

Sums of Contiguous Bands ---------------------------------------- 104 

Stevens' experim ent (1956) .............................. ------- 104 

Sums of Noncontiguous Bands 106 

Stevens' expernzent (1956) - 106 

Effect of Bandwidth Near Loudness Threshold --------------------- 107 

Scharf's experiment (1959) ------------------------------------- 107 

White Noise --------------------------------------------------- 108 

Feldtkeler's experiment (1956) --------------------------------- 108 

Poiack's experiment (1951) ------------------------------------ 109 

Miller's experimeLt (1947) -------------------------------------- 110 

Br~ttain's experiment (1939) ------------------------------------ 112 

Hiese and Kilets experiment (1058) ---------------------------- 112 

Broadband Noises ----------------------------------------------- 114 

Liboke, Mittag, and Port's experiments (1964) -------------------- 114 

At. Jahn's experiment (1965) ------------------------------------ 115 

Warren's experiment (1973) ------------------------------------ 116 

Quietzsch's experiment (1955) ------------------------------- 116 

Corliss anZ Winzcrs experiment (1965) -------------------------- I 

Robinson and Bowster's experiment (1961) ----------------------- 118 

Copeland et al, experiment (1960) ------------------------------- 119 

LOUDNESS OF SPECIFIC SOUNDS WITH MIXED 

SPECTRA -------------------------------------- 119 

Liibcke, Mittag, and Port'Us Experiment (1964) --------------------- 120 

M- Jahn's Experiment (1965) ------------------------------------ 120 

Quietzsch's Experiment (195) ----------------------------------- 120 

Copeland et al. Experiment (1960) --------------------------------- 120 

RELATION BETWEEN PRESENT THEORY AND AWEIGHTED, SOUND-PRESSURE LEVEL ------------ 121 

DISCUSSION AND REMARKS ---------------------- 12 

SUMMARY OF RESULTS -------------------------- 2 

APPENDIXES 

A-SYMBOLS ------------------------------------ 127 

B-SPECTRAL REPRESENTATION OF OUTPUT 

POWER-SPECTRAL-DENSITY Y --------------- 131 

C-LOCATION OF CRITICAL BANDS --------------- 133 

D-CONCERNING BEATS OF MISTNED CONSONANCES -------------------------------------- 134 

E-PLANOHEREL'S THEOREM FOR LIMITED INTEGRATION TIME ------------------------------ 136 

F-EXTERNAL TRANSMITTANCE FUNCTION ------ 137 

REFERENCES ------------------------------------ 139 

vii 
INTRODUCTION 

If the attention of a listener is consciously directed toward perceiving sounds, then he is capable of estimating the relative magnitudes of pairs of sounds. The judged magnitude of a sound heard by 

a listener is called its loudness. By making quantitative judgments 

of the relative loudness of pure-tone stimuli at measured sound 

pressures and by adopting one tone and sound pressure as a reference, a quantitative relation between loudness and sound pressure 

can be established for pure tones. The resulting loudness. scale is a 

ratio scale because it is derived by judging ratios of loudnesses. 

Next, by subjectively equating the loudnesses of other sounds to 

that of the reference tone the loudnesses of the other sounds can be 

determined. 

('Analogous procedures have been adopted to evaluate the relative 

noisiness, annoyance, or other sensations induced by sounds. However, noisiness and annoyance, for example, involve the emotional 

state of -the listener. This implies that nonacoustic neural processes 

affect the judgment (ref. 1). These processes have not been evaluated. 

Moreover, noisiness and annoyance judgments should be quite variable because emotional states vary widely among individuals as well 

as for any given individual at different times (ref. 2). It is reasonable to expect, however, that a listener in a given emotional state 

will judge the louder of two similar sounds' to be noisier and more 

annoying as well. Thus, -the loudness scale may sometimes serve 

equally well as a noisiness, or annoyance, scale (ref. 3). For these 

reasons, noisiness and annoyance will not be considered further.) 

-This report is concerned with the loudness of steady sounds. 

Steady sounds are defined as those having statistical properties independent of time. For practical purposes a, sound can be regarded 

as steady if its statistics vary slowly over a few auditory integration periods, say over a second or so. With this relaxed definition the 

class of steady sounds is very important because it is very common. 

As will be shown, the loudness of a steady sound is not necessarily 

steady. 

The main purpose of this report is to propose and test a new psychophysiological model for predicting quantitatively the loudness of 

steady sounds. The model is termed a psychophysiological model 

because it is constructed from both psychoacoustic and physiological 

information. The present theory utilizes new ideas, particularly 

1
 
power summation and frequency interaction, to account for loudness in place of the older concepts of loudness summation and 

inhibition of discharges. Because of its almost complete departure 

from previous models (e.g., refs. 4, 5, 6, and 7), it is hoped that this 

new model will provide important new insights concerning the manner in which the loudness sensation develops in the auditory system. 

The proposed model was synthesized from data obtained using 

specific sound stimuli, that is, the development of the model proceeded from the specific to the general. However, the subsequent presentation will be in reverse order, that is, the general theory is 

presented and then analyzed to obtain specific results. In the process 

of analysis an attempt is made to show how the model is related to 

physiological and psyohoacoustic observations other than loudness. 

Loudness levels predicted using the model are compared, where 

possible, with available loudness judgment data, which are believed 

to consist of most of the published data. 

OPERATION OF AUDITORY SYSTEM 

The complete auditory system consists of a complicated transducer 

Which 'transmits the information in sound waves acoustically, then, 

mechanically, then hydrodynamically and, finally, electrodynamically 

to the brain (refs. 8 and 9). The fundamental equations of acoustics, 

solid dynamics, hydrodynamics and electrodynamics are well imown. 

However, the geometry of the auditory system is so complicated that 

a mathematical analysis of the complete physical system has not 

been achieved. Moreover, the processes involved in the hydrodynamic 

to electrodynamic conversion are not well understood. Because of 

these and other difficulties it is more feasible in predicting the loudness of sounds to consider what -the auditory system does rather than 

how it does it. The latter approach involves analysis of the physical 

system, whereas the former approach, adopted herein, involves, 

instead, an analysis of the operational characteristics of the system. 

A block diagram of the basic auditory system is shown in figure 1. 

In the outer, middle, and partly in the inner ear the wave is filtered 

and is physically represented by mechanical and hydrodynamical 

motions. In hair cells in the inner ear the mechanical motion becomes 

coded in the form of constant amplitude, impulsive, electric-potential 

discharges (up to several hundred per second in each active neuron) 

propagated (at speeds up to 100 m/s) along -auditory nerve fibers 

(approximately 30 000) toward the brain. In the mechanical-toelectrical transformation additional filtering probably occurs. In the 

auditory nerve the waveform can be depicted by the number of discharges occurring as a function of time. The discharges occur only 

in association with part of the waveform, so that u partial-wave 

2
 
ou~er aner d utory Brai se Io 

FGUrRE 1.-Operational diagram of auditory system. 

rectification of the waveform appears in the auditory nerve. In the 

brain, possibly by means of a metastabla chemical reaction, the 

energy: transmitted in the discharges gets summed over a short, run­ ning time interval, the auditory integration time (approximately 

0.1 s). The auditory integration time is the time taken to build up a steady response to a steady stimulus. The sum determines the intensity of the subjective sensation, that is, the loudness. 

Henceforth, for convenience in terminology only, quantities such 

as the effective sound pressure, ph~ysical power, and the neural dis­ charge rate will al be referred to the auditory integration time 

(0.1 s rather thlan 1.0 s). Thus, the effective sound pressure is the 

root-mean-square value over the auditory integration time; the 

physical power is the energy transmitted during the auditory integration time; the discharge rate is the number of discharges which 

occur during the auditory integration time, and so forth. 

In order to predict the loudness of any steady sound stimulus, consider, first, a previous model of the system (ref. 10) used to predict 

the loudness of a tone. 

Specifieally, it was proposed (ref. 10) on the basis of experimental 

data (ref. 11)e t, in response to atonal stimulus, the number of 

stimulated auditory nerve fibers is proportional to the effective sound 

pressure until all available fibers are stimulated. Experimentally, 

the discharge rate in each fiber is also approximately proportional to 

the effective sound pressure until the fiber discharge rate becomes 

saturated (ref. 12). At about 40 decibels above the loudness threshold, 

discharge rates in some fibers beg to saturate. However, other, 

previously 1)stimlated, fibers exceed their stimulation thresholds 

as the sound pressure increased, so that the total discharge rate 

continues to increase until all available fibers are stimlated and, 

then, saturated. The total discharge rate was expressed (ref. 10) as 

the product of the total number of stimulated fibers andthe dis­ as th, pessue sond isincras ttettldshrert 
charge rate per fiber averaged over all stimulated fibers. Consequently, as the sound pressure is increased, the number of stimulated 

fibers increases linearly, but the average discharge rate per fiber increases very little because some fibers have become saturated whereas 

others have just achieved their thresholds of stimulation. Hence, the 

total discharge rate is essentially proportional to the effective sound 

pressure. Since electric potential is proportional to electric energy, 

and all discharges are identical, the total di scharge rate is proportional to the total electric power transmitted by the auditory nerve. 

Therefore, -the total electric power is proportional -to the effective 

sound pressure of the stimulus, and the ear is found to be a nonlinear 

detector, specifically a squaring detector. 

It has long been suspected (e.g., refs. 13, 14, and 15, chap. 1B) on 

the basis of experimental evidence (refs. 16 and 1), that, except 

near threshold, the magnitude of the loudness sensation in response 

to a tone stimulus is proportional to the total discharge rate (total 

electric power) transmitted in the auditory nerve. (The fact that 

the discharges are summed implies that total current, which is proportional to discharge rate, is proportional to loudness.) As a consequence of'this observation, loudness is proportional to the effective 

sound pressure. Since the mean-square pressure is proportional to 

the average power of the stimulus, the, loudness must also be proportional to the square root of the stimulus power transmitted by the 

system. 

Experimental evidence (ref. 18) indicates -that for individual 

nerve fibers the dependence of discharge rate on sound pressure is 

uhe same for broadband noise as for a tone. If it is assumed that the 

dependence of the number of active fibers on sound pressure is also 

the same for both sounds, then it may be concluded from an analysis 

similar to that in reference 10 that, in general, loudness is propor­

.tional to the square root of the power transmitted by the auditory 

system. 

In order to predict the loudness of sounds more complicated than 

a lone tone, a broader description of the system is required. 

Filtering 

The auditory system is a bandpass filter. Psychoacoustic data 

(refs. 19 and 20) indicate that the filter is passive, and, of course, 

nonideal. From the experimental standpoint, it is convenient to analyze the filter into an external filter in series with an internal bandpass filter (ref. 21). The external filtering occurs in the outer ear, 

which precedes the eardrum (of. fig. 1). The characteristics of this 

filter depend on the orientation and extent of the sound source relative to the listener. The internal filter involves the entire auditory 

4
 
system succeeding the eardrum (of. fig. 1). This, filter is weakly 

dependent on the effective.sound pressure. 

The internal filter may be analyzed into a low-pass filter in series 

with a high-pass filter. The low-pass filter is probably associated 

primarily with the mechanical part of the auditory system (ref. 8), 

whereas the high-pass filter may be associated primarily with cochlear processes (particularly excitation of cochlear hair cells) involved in converting the signal 'to electrochemical form. 

Neural Response 

Mechanical vibrations of a region of the tectorial membrane 

within the inner car (ref. 8) produce excitation of hair cells and, 

hence, of the auditory neurons. Each neuron is specified by its "best," 

or "characteristic," frequency, that is, the frequency for which its 

stimulation threshold is lowest. However, each neuron may be excited 

by a broad range of stimulus frequencies. In figure 2 nominal bandwidths (adapted from ref. 12) of auditory nerve fibers in squirrel 

monkeys are shown. 

Consider a tonal stimulus. If the tone is sufficiently intense and its 

frequency lies within the transmission bandwidth of a given fiber, 

then the tone periodically stimulates discharges in the fiber (ref. 12). 

(Although a periodicity is commonly believed to exist only for 

stimulus frequencies less than 5000 Hz, a periodicity which may be 

associated with pitch actually exists at all frequencies (ref. 22).) 

At the lower amplitudes of stimulation 'the periodicity of the discharges tends to be statistically distributed at time intervals which 

level, 104 -

C90
 

103 

~102 

105 10
 1 3
 10 2 

10 1 

Fiber characteristic frequency fiber of auditory nerve as 

bandwidth one-half-power as parameter. FIGURE 2.-Approximate level with sound-pressure of its best frequency are much greater.) function aximum bandwidths ref. 12. 

(Adapted from data in 

5
 
are integer numbers of, times the periodicity of the stimulus (ref. 

23). At higher stimulus amplitudes a discharge tends to occur more 

often during successive stimulus periods. For a large number of 

realizations of the stimulus, the number of discharges as a function 

of the phase of the stimulus graphically resembles that part of the 

stimulus waveform which exceeds some constant amplitude (often 

zero in the auditory nerve, refs. 24 and 12). The number density is 

effectively zero throughout the rest of the period. Hence, for each 

fiber the waveform appears to be partial-wave rectified. The preponderance of data indicate that, when all fibers are considered, the 

waveform may be approximately half-wave rectified. For periodic 

stimuli the trace of the wavefonn from discharges in single units 

over many realizations of the stimulus might be approximated alternatively by discharges over many units for a single realization of 

the stimulus. This resembles an ergodic hypothesis. 

Stimulation by more than one tone yields a distribution of stochastically independent discharges which also tends to duplicate graphically part of the waveform (refs., 12 and 24). Similar results are 

expected for any other periodic sound. In fact, it seems reasonable 

to expect that rhe ,temporal history of the instantaneous number of 

discharges over all fibers traces out part of the waveform (suitably 

filtered) even if it is aperiodic. 

In the mechanical part of the auditory system the trace of the 

waveform is determined by the amplitude of mechanical motions, 

whereas in the electrical part of -the system the waveform is traced 

by the instantaneous number of discharges, which is proportional to 

total instantaneous power. Therefore, since power is proportional to 

the square of the amplitude, the squaring operation in the auditory 

system results simply from the transformation of the representation 

of the waveform from mechanical to electrical form ! 

Additional clues on how to model the auditory system are supplied by various observed psychoacoustic phenomena and their plausible explanation in terms of neural activity. Among possible significant 

phenomena are audible beats (refs. 25 and 26), critical bandwidths 

for loudness (refs. 27 and 28) and frequency resolution (ref. 29), 

secondary tones (ref. 30), harmonic stimuli (ref. 81), and aural 

harmonics and combination tones (e.g., refs. 15 and 32). The loudness features associated with the beats are claimed herein to be a 

consequence of squaring and of the limited auditory integration 

time; whereas those associated with critical bandwidths, secondary 

tones, and harmonic stimuli are proposed consequences of squaring 

and of the broad bandwidths of individual neurons. 

In order to explain beats and other psyehoacoustic phenomena in 

terms of neural activity, the assumption will be made that stimuli 

iohich excite the same netron interact whereas stimuli which do not 

6
 
excite the same neuron do not interact.This interaction has an important effect on loudness. 

Beats.-The loudness of two tones with comparable amplitudes and 

nearly equal frequencies, or of tones which are nearly harmonics of 

one another, oscillates because of interference between the tones (refs. 

26, 26, and 83 (pp. 14-16)). The loudness oscillations, equal in frequency to physical beats, are referred to as "audible beats." For 

nearly equal tone frequencies the oscillations are called "beats of imperfect unisons" (ref. 25) ; whereas for nearly harmonic tone frequencies they are called "beats of mistuned consonances" (ref. 26). 

Subjectively, since tones which produce beats of imperfect unisons 

are separated by less than a critical separation, they cannot beo resolved. (This defines the critical separation, which is less than onethird octave, except at the lower audible frequencies.) The consequence in the case of nearly equal tones, is an intertone-with frequency near that of the primaries-modulated at the beat frequency. 

As the separation of the primary tones is increased, the beat frequency increases; but the loudness of the beats vanishes, as will be 

shown mathematically. 

Two, or more, closely spaced tones stimulate the same nerve fibers 

because the fiber bandwidths are broad. Hence, the beats produced by 

interference of two, or more, tones are observed as fluctuations of the 

number of discharges at the beat frequency as these discharges trace 

out the rectified waveform of the combined stimuli (ref. 34). The 

period of the beats exceeds the auditory integration time. Loudness is 

presumed to be the subjective measure of the transmitted electric 

power corresponding to the discharge rate over all fibers, or, alternatively, of the equivalent transmitted stimulus power (ref. 10). If the 

auditory averaging time were infinite, the power contribution from 

the tone interaction would vanish. However, the averaging time is 

finite (0.1 s), so that a time-dependent interaction exists if the 

period of the beats is greater than, or approximately equal to, the 

integration time. 

As two closely spaced tones or harmonics are increasingly separated 

in frequency, the period of the beats becomes less than the auditory 

integration time. Consequently, the maximum and minimum average 

power transmitted by the auditory nerve throughout a running integration period tend to the same value, independent of the tone separation, as does the loudness (ref. 35). 

Criticalbandwidt.-Two critical bandwidths (ref. 36) are significant, namely the critical bandwidth for loudness (ref. 28) and the 

critical bandwidth for frequency resolution (ref. 29). The critical 

bandwidth for loudness about any given center frequency is the maximum sound bandwidth for which loudness is independent of bandwidth when the overall sound pressure is held constant. The critical 

7
 
bandwidth for frequency resolution about any given center frequency 

is the maximum tone separation for which two, or more, tones cannot 

be resolved psychoacoustically. The two critical bandwidth functions 

are similar, as shown in figure 3. As the respective critical bandwidths are exceeded by two, or more, tones, the loudness rises (refs. 

37 and 38), and the tones are resolved (ref. 29). The critical bandwidths indicate-an important transition in the neural coding. 

Secondary tonc.-In addition to the loudness increase and tone 

resolution, secondary tones other than the imposed primary tones may 

be heard as the critical bandwidth for frequency resolution is exceeded. The most prominent secondary tone is frequently the 

"periodicity pitch" (ref. 39) with frequency equal to the difference 

in frequency of the primary tones. If several primary tones are harmonically related, then the fundamental may be heard even if it is 

not included among the primaries. An audible missing fundamental 

has been called the "pitch of the residue" (refs. 31 and 40).

The following explanation for these phenomena is proposed. The 

pattern of neural discharges traces a filtered and rectified version of 

the acoustic stimulus waveform. This has been demonstrated from 

many realizations of periodic stimuli in single units (ref. 30). If two 

tones are closely spaced, the intertone is determined by the periodicity 

of the envelope of the instantaneous number of discharges. DisFrequency of first harmonic, wl2gi,Hz 

1 io' 3 102 1o4 1oi5 

Loudness (ref. 2) 

=- - Frequency resolution, multiple tones (ref. 29) Frequency resolution, two tones (ref. 29) M 

Region of pitch of residue, three tones 

(ref. 85) - -- - First harmonic ' / "--- -- C 

--- ,- -"/ / /

1 2' 02 

4 0

5
 101 102 103 10 8o 

Geomtetric-mean frequency. &j.124Hz 

Fx U E 3.--rittcal bandwidths and region of pitch of residue. 
charges occur in both positive and negative phases of the beat cycle. 

The beat frequency lies outside the transmission bands of the fibers. 

As the tone separation approaches the critical value the beat frequency may exceed the lower cutoff frequency of some fibers which also 

transmit the primaries. Then, fibers will be stimulated at approximately the periodicity of the beat, so that the beat will be transmitted 

as if it were a real tone and may be heard as 1. secondary tone. In this 

regime, due to the variation of frequbncy pass bands among fibers, a 

mixture of the primary tones, intertone, beats and secondary tones 

may combine to produce the sensations of "intermittence" or "roughness" (ref. 25). When the critical separation is exceeded, the primary 

tones and secondary tone are dominant (ref. 80). All three tones lie 

within the pass bands of the same fibers and are transmitted as audible tones. This accounts for the periodicity pitch. The pitch of the. 

residue is similarly explained when the stimulus consists of more than 

two tones. Since the auditory system is a squaring device, the secondary tone is heard as if stimulated by real acoustic power. The 

detection of the secondary tone should result from an increase in the 

overall discharge rate and a redistribution of the discharges in time 

to establish the periodicity of the secondary tone. An increase in 

loudness 

ment 

lower 

tion on 

the 

cutoff 

the 

should, 

critical 

frequency 

primary 

therefore, 

bandwidth 

tones. 

of those 

also 

This 

for 

nerqve 

be 

loudness 

comparison 

expected. 

fibers 

is 

which 

equal 

According 

is 

propagate

shown 

to 0 oharaoteristic 

to 

in 

this arguinformafigure 1, 

where equality of critical bandwidth and fiber cutoff frequency is 

denoted by the "ideal" curve. 

The boundary of the regime of a clear periodicity pitch obtained 

from a uniformly spaced, three-tone stimulus heard through earo
 

00
 

C -o 

CC, 

2 3 4
 101 10 10 10 105 

Estimated lower cutoff frequency of fiber 

FiGuRE 4.-Critical bandwidth as function of complete-lower-cutoff frequency 

of single auditory nerve fibers. (Data from refs. 12 and 42.) 

9 
phones is shown in figure 3 for comparison with the critical bandwidths previously discussed. The lowest audible periodicity pitch corresponds roughly to the critical bandwidth. (Of course, many of the 

physiologically significant frequencies are not precise because of 

physiological variations among and within individuals.) The highest 

audible periodicity pitch is less than the frequencies of the tones 

which produce it. The frequency range over which the periodicity 

pitch may be obtained is discussed in the next paragraph. 

Uhannel8 and neuron bandwidthl.-Since the critical bandwidths 

represent important transition bandwidths in neural coding, it is convenient to evaluate loudness by grouping all neurons into artificial 

channels having widths equal to the critical bandwidths for loudness. 

Each channel includes every neuron whose best frequency is within 

the critical bandwidth for that channel. In most channels the neuron 

bandwidths tend to be much greater than the channel, or critical, 

bandwidths (refs. 12 and 28). Thus, each channel may transmit a 

much broader range of frequencies than the channel bandwidth 

would imply. All frequencies transmitted in a channel may interact 

if they excite the same neurons. Thus, if tones transmitted in a ehannel are suberitically separated, they may generate audible beats; 

whereas if they are supercritically separated, they may generate secondary tones. The secondary tones can only be generated if the 

neuron bandwidths (fig. 2) exceed the critical bandwidth (fig. 8). 

Neuron bandwidths clearly exceed critical bandwidths for best frequencies in the range from 200 to 5000 hertz, approximately, at moderate sound pressures. These frequencies establish the mininmum and 

maximum limits, respectively, for which secondary tones can readily 

be heard. (See limits of pitch of the residue in fig. 8.) 

The preceding channels are "floating" in the sense that their midfrequencies or, alternatively, their cutoff frequencies are determined 

by the spectrum of the imposed sound. However, the bandwidth of a 

channel with a given midfrequency is effectively always the same in 

response to a steady sound. 

Aural harwwnies and combintion tones-These tones are assumed 

to be subjective consequences of mechanical nonlinearity, that is, of 

functional defects, of the auditory system in response to single tones 

or to multitone stimuli, respectively (ref. 41). This is in contrast with 

the squaring process, a nonlinear process fundamental in the operation and interpretation of the system. Although these tones may be 

audible, they are usually very weak relative to the primaries, beats, 

or secondary tones. f-ence, their contribution to overall loudness is 

likely to be negligible, except possibly at loudnesses near the threshold of feeling (ref. 32, oh. 7). The contribution of these tones to 

overall loudness will be neglected. 

10
 
Loudness 

The loudness of any sound is numerically expressed relative to that 

of some precisely defined sound which serves as a reference. Thus, in 

order to specify the loudness it is not necessary to know how much 

power is transmitted by the system. Only the ratio of transmitted 

powers, namely the ratio of the transmitted power of the test sound 

to that of the reference sound, need be determined. This can be found 

after the test sound and characteristics of the auditory system are 

specified mathematically. 

MODEL OF AUDITORY SYSTEM 

A nmodel of the auditory system previously proposed (ref. 10) for 

predicting the loudness of a tone will serve as the basic model to be 

further expanded for predicting the loudness of any steady sound. 

It is assumed that the sensation of loudness in response to a sound 

stimulus is measured by the power (energy per auditory integration 

time ) transmitted to the brain. Physiologically, the power is determined by the product of the total current (proportional to total 

discharge rate) and the potential, -which is the same for each discharge. Since the loudness sensation commences for some discharge 

rate exceeding zero (ref. 10), the loudness is proportional to the 

amount by which the electric power exceeds some threshold power. 

The proportionality constant is a reciprocal power (reference power), 

so that the loudness is a dimensionless quantity, and the power can 

always be expressed relative to the reference power in dimensionless 

form. 

In the basic model the electric power is proportional to the effective 

sound pressure. This results from the analysis of discharge rates in 

neurons (ref. 10). From the preceding discussion of the operation of 

the auditory system it is concluded that the extension of the basic 

model of the system in response to any steady sound stimulus must, 

also include filtering, channeling, squaring, and rectification. 

The complete filter, which is separable into an external filter in 

series with an internal filter, possesses known characteristics derived 

from experiment. The complete filter is a linear, passive, time-invariant, nonideal, band-pass type. Its input is the sound. Its output is 

determined from subjective judgments, and therefore, takes into 

account the entire auditory system. 

In the organ of Corti the overall signal is distributed among up to 

24 parallel channels spanning the audible frequency range (ref. 28). 

Each channel has the width of a critical band for loudness in the 

sense that it includes all auditory neurons with best frequencies lying 

11
 
within the critical band. Each channel should have a different frequency response. Nevertheless, since the specific response for each 

channel is unlnown, it will be assumed that the response associated 

with each channel is that of the complete system. This may be the 

main deficiency of the model. However, this is compensated for by 

assuming that only those channels within the limits of the sound 

spectrum transmit appreciable power. Since the transmitted power in 

each channel is to be specified relative to the same overall reference 

power, it may be assumed that the entire input is imposed in each 

channel. Thus, the signal tends to get multiplied by the number of 

channels which transmit it. In the limiting case of a tone, only one 

channel, that which includes the tone, is activated. The frequency 

response is precisely that for the tone. The resulting loudness is found 

to be that previously predicted for a tone (ref. 10), as will be shown. 

Since the neuron bandwidths exceed the associated critical bandwidths over most of the audible frequency range, signals over a wide 

range of frequencies may interact. In the squaring process involved in 

the transformation from a mechanical to an electrical signal, both 

self-interaction of signals at the same frequency and cross-interaction 

of signals at different frequencies occur in each channel. However, because of the expected decline in response in the channel to frequencies outside the associated critical band, appreciable cross-interaction 

between different frequencies will be assumed to be limited to frequency pairs where at least one frequency of the pair lies within the, 

critical band. 

Since all channels within the limits of the sound spectrum are 

assumed to transmit appreciable power, whereas the sound spectrum 

may vanish over entire intermediate critical bandwidths, "empty" 

channels may occur which transmit no power in the associated critical 

bands. However, the empty channels may transmit power due to the 

self-interaction of frequencies outside the empty critical band. As in 

the case of nonempty bands, it will be assumed that the cross-power 

contributions due to the cross-interaction of frequencies outside the 

empty critical bands are negligible. 

The pattern of neural discharges is partial-wave rectified. Specifically, it will be assumed that the pattern is half-wave rectified. 

This completes the expanded model which -now incorporates most 

of the previously discussed psychoacoustic and physiological phenomena. The subsequent mathematical representation of the model 

yields the magnitude of the loudness sensation .Z as some measurable 

function of the free-field pressure perturbation p of an arbitrary 

steady sound. 

12 
LOUDNESS OF ANY STEADY SOUND: ANALYSIS 

A 1-kilohertz tone imposed as progressive plane waves from the 

front at a free-field, sound-pressure level S of 40 decibels is defined 

to have a loudness 2 of one sone. (All symbols are defined in appendix A.) A sound x times as loud as the tone at one sone is said to have 

a loudness of x sones. The sound-pressure level S is defined by 

"effective," 

S=20 log C9)(decibels) (1) 

where 

P3= [ je':_ fl (2) 

The quantity 57i is the normalized root-mean-square, or normalized 

sound pressure over the averaging time T. The time 

T-0.1 second at suprathreshold loudness is the integration time of 

the auditory system, that is,the time taken to achieve a steady loudness sensation in response to a steady sound. The instantaneous sound 

pressure p is real, and t is the time. Finally, 

is a reference sound pressure near that at the threshold of loudness 

of a 1-kilohertz tone. 

According to the proposed model, the overall loudness _£ of any 

steady sound is given by 

where r, the total neural discharge rate (in 0.1 s) over all fibers, is 

proportional to the total electric power transmitted in the auditory 

nerve, o' is a constant, and the subscript t refers to the value at the 

loudness threshold. Since the electric power is proportional to -the 

square root of the stimulus power distribu.ted over all neural 

channels, 

1/2 - (]1/2 £ 

where 2W is the total number of transmission channels spanning the 

sound spectrum, (du) k is the relative power transmitted in each 

channel k to the brain, and Kis a constant (K=10-2 if P is in sones, 

as will be shown. Also, see ref. 10). The relative power d-,q in each 

channel is proportional to the square of the linearly filtered, instantaneous sound pressure integrated over 'the running auditory 

integration time T.The filtered output from each neural channel is 

given by 

q(') f (")p(t,'- t-")dt" (5) 

where A is the real impulse response of the auditory system, and q 

is the output. As a result of the transformation from mechanical to 

13
 
electrical form, the signal transmitted in the auditory nerve is proportional to q2. This yields the relative output power 

dflq( 5tjT)=(1Tp )fl, q2 (t1)dt" (6) 

involved in equation (4). In equation (6), s9qrd (0,t,T) is a 

specialization of the autocorrelation function dq(r,tT), in which r 

is a delay time. By virtue of equation (6), equation (4) becomes 

H ~1/2 

11/2 

where, in concordance with equation (6), 

= (1p 01 ) fl q2 (t')d'() 

at the loudness threshold. Equation (7) in combination with equation 

(5) yields the overall loudness in terms of measurable quantities. 

It may be more convenient to express the overall loudness in terms 

of spectral, rather than temporal, quantities. The output autocorrelation function is defined by 

d9q(r,t,T) - (1/2'pt) q(t')2(t'+r)dt' (9) 

and is the Fourier transform of the output power-speotral-density 

Wq, that is, 

s,q (rjt;T)= (1/2Y) f- 'Wq(w~t,7') ei""d& (10) 

where w is the angular frequency. Hence, 

cR.q(Ot,T) = (1/2r) f iVq(,T)de (11) 

If equation (9) is combined with the Fourier inverse of equation 

(10), then 

W (w,,T)= (1/Tpt,) f!e7-rdrfLq(t'),q(t'+r)dt' (12) 

This expression can be further transformed into the spectral form 

Wq,(w,t,fl) =(1/p 1 )H( 0))P(m) 

X (1/2w) f-_ H* (w")P*(w")sine [ (W- ')T/27r] 

X e-(w<"-(-M /-)d w,, (13)

as shown in appendix B, where 

is the Fourier transform of the impulse response A(t) and is called 

the frequency response, or system function; 

14 
P(6) =f" p(te Wdt (15) 

is the Fourier transform of the pressure history and is called the 

pressure spectral density; and the asterisk denotes the complex conjugate. Also, by definition, 

sine 6 = sin () (6 

Finally, the combination of equations (4) and (11) yields 

J2 Wx/ 2,)~f' [Vq(WAT)]L.dw} 

(Mn2/27r)_ fe" W~(wt,T) 2 ]cb }/ (17) 

where the 17,'s are given in spectral form by equation (13). This 

alternative form of equation (7) is the desired spectral representation of the overall loudness. 

The threshold, output spectral density 3Vt is defined by shifting 

the spectrum, without changing its shape, so that the contribution 

from every channel is subliminal, except for one or more, which 

corresponds to 'the loudness threshold. The threshold terms in equations (7) and (17) are significant only if the overall loudness is 

near the loudness threshold. These equations are defined only for 

y--0. The possibility that _>0 for a sound which is subliminal at 

every frequency is implicit in equations (7) and (17). 

In general, the sound spectrum may be partly discrete and partly 

continuous. The discrete part consists of tones whereas the continuous part might consist, for example, of random noise. Thus, it is 

sometimes convenient to treat the output spectral density IVq as if it 

were either discrete w,(o.), where n is an integer, or continuous 

W (w) depending, respectively, upon whether the discrete or the 

continuous components of the sound are dominant, but also upon 

the method of detemining the spectrum. Of course, a sound can be 

specified in terms of both wq(w.) and Wq(c(). 

Discrete Spectrum 

The discrete part of the sound consists of a number of tones, which 

can be represented by 

P)W A~eos (wt+) (18a) 

or 

where the amplitude A. is real, q,, is the phase angle of the tone denoted 'by the integer n, 

15 
-n n n n 

According to equation (5) the corresponding output is zHenaew t (19) 

Because the signal transmitted by the auditory nerve is most 

conveniently analyzed in channels, three categories of tone separation 

or bandspread appear relevant, as shown in figure 5. These are denoted as 

CategoryA: Suboriticalbandspread (fig. 5(a)) in which the overall tone separation is less than, or equal 'to, a critical bandwidth for 

loudness (approximately one-third octave, but greater at low frequencies). Beats of imperfect unisons fall in this category. 

W 

(a) $ ICritical, 

bandwidth 

w 

(b) I__.____ritical 

bandwidth 

W >5 

Critical 

bands 

4 Critical 0

bandwidth 

(a) Two tones snbcritically spaced. 

(b) Two tones supereritically spaced.
 

(c) Two tones hypercritically spaced.
 

'FiGURE 5,-Categories of tone spacing relative to channel bandwidths.
 

16 
Category B : Supereritica tone spacing (fig. 5 (b)) in which the 

overall tone spacing is greater than a critical bandwidth. This category includes beats of mistuned consonances, periodicity pitch, and 

the pitch of the residue. 

Category 0: Hyperenitca tone spacing (fig. 5(c)) in which the 

tone spacing is greater than the bandwidths of nerve fibers with 

characteristic frequencies intermediate to the tones, so that no tone 

interaction occurs. The theory for category B also includes this category implicitly. Overall loudness judgments of such widely spaced 

tones are very difficult, 

Various combinations of these basic categories occur. The bandspread of suberitically spaced tones may be supercritical (category 

AB) or hypercritical (category AC), and the bandspread of supercritically spaced tones might be hypercritical (category BC). 

The overall loudness P for each category is determined as follows: 

Category A: Subcritica bandspread.-Whenthe tone bandspread 

is less than a critical bandwidth, equation (7) reduces to 

Jj (i2/Tp2 ) f1 q2 (t')dtrJ].. (R2&- W't) (20) 

where ,yq is given by equation (8). By considering equation (19) 

the integral in equation (20) becomes 

c~~(l/2p2 ), fl 2 (t') dt' 

=(1/TPX,) fZl H*()HW a aeEorwndt 

-r(1/4Tpef) f' : H(w.)H(,.A,. 

Define the relative power 

where, in particular, 

w(Wm )-- (.W)=A',/2p, 

if , -n. Also, for brevity, let 

Then, 

+?&(/22') 5(,g_.)Nw,(H() .)E_. W(WntO

+H (,_,)H(T N ) f (_,,wn) 

=7 
The frequencies a,, -,w., and om+ , implicit in the E's can be generated only in those neurons which transmit periodic discharges corresponding with the frequencies w. and w,,as well. If c -w,, is less than 

the critical bandwidth, then r - w lies outside the pass band of those 

neurons which transmit w,, and w.. Discharges do not occur regularly 

with a periodicity corresponding to am(On, and this tone is not heard. 

Rather, w, - 4 is the frequency of oscillation of the envelope of the 

output power clue to the interaction of frequencies (o. and w.. On the 

other hand, except possibly at low frequencies w. and w,, and high 

stimulus amplitudes, the sum , ,+w, tends to lie outside the nominal 

transmission band of those neurons which best transmit corn and w. 

Thus, h..+c is transmitted very weakly, if at all. Therefore, those 

terms which involve w,+ c, will be neglected. 

Next, the complex, total cross-transmittance Q is defined according 

to 

Q m,) 'H(w.n)Hw) +iQjciwrn~oi) +re~n (92) 

where Q, is the real part, and Qj is the imaginary part, of Q. If 

WO=M n,Q (Cum , W) reduces to the transmittance

Q(e.) = JH(o.) 2 = Q,(0.)
 

which is real, 

The total transmittance Q consists of the product of an external 

part g preceding the eardrum and an internal part AV succeeding the 

eardrum, that is, 

Q~cor~t(n) 'f7o con) A'i (Wnttn) (23) 

or 

(ca)=2(,)SAf(.) 

if (m=o. In parallel with the definition of Q,

57(wm,(o) /:(wr)J/e (on) f17r(co,wn) +i2 (cnm,o+) (24) 

or 

-:(jJ/0(')I='r-")
 

if w.=w,. Also, 

JV(wr,wn) = W:(-,)J/(wn) =rA/r(rn,n) +icA(nz,a) (25) 

or 

(.)= IW/c( J,-) 66) 

if w =w,.The external and internal frequency responses 9,4(,) and 

,C'~,) respectively, which will be explicitly specified subsequently, satisfy the conditions 

18
 
in the case of the external response, and 

C4(W-() c'4( = c("1") C* (27)

in the case of the internal response. Therefore, 

whereas 

cA (W-,-) =J *(W - ) (c-O) =-C4 .2 (W.) --A' (WMIWO 

By neglecting the nontransmitted frequencies (o. + o,. and adopting 

the newly defined transmittance functions, it follows that 

x=o/__ Es.(m n Em +(cf)( wr, )mt t 

Xf'-1 

where T 'W .W"(,). Because a+ z'= 2 Rf; z for z complex, 

and because &?, is real, 

cf~q 9Jr(Ornmcn)Vw(omiwn) 

X (1/T) f o [A(won)Erjdt' 

or 

M=1 n=1 

x< (1/1T) fff T (Wr~Mo)cas (onW ) t'O+ 'rn 

+J1It(.m ,) sin[ (w -,A,) P+ Oz- ] }dt,' (28) 

An alternative form of the equation may be written by noting that 

leA (mon, ) I= [A( ,,,,.)+ ,(d.,.0,j) ]1/2 (29) 

and setting 

tan Bnhorno)/rcoco)(30) 

Then, 

c~q :4 E7 ICrOMnWn) IS)V (wnWM)I')(-W~~n) 

x (1/T) f'_cos[(wn-oi)t' +',-4- Om]dt' (81) 

19 
so that 

05 

X sinc[ (on- to.) T/jr]cos[(o,-,) (t-T12) +m€.-OOmn] 

(32) 

Therefore, the overall loudness of tones distributed over a subcritical 

frequency band is given by 

X sine (w.)lT/2r]cos [( o.) (t- T2) 

+O-no- 4-(iC'c~jqj) (38) 

Since, for subcritical bandspread, loudness is independent of bandspread if the overall sound pressure is held constant, it is reasonable 

to assume that, at the loudness threshold, 

?q = Q, (&),,)wt (00 (31) 

which is the lone tone result for the characteristic frequency WM/.2r, 

say the mean frequency of the tones. Because ,Rqt may be determinedby several tones it follows that several suberitically spaced tones, each 

of which is subliminal, may produce an audible sound. This possibility is reinforced by electrophysiological experiments (ref. 42). 

Except possibly at high frequencies, the transmittance functions 

7(im,n) and SY(wmw) tend to remain constant over a critical 

bandwidth, as will be shown. Hence, 

91r(WmjWn) 97r('9A)
 

and
 

I0VG, ,w)I'-aJfr(o*)
 

so that 

j.)z { xQr~a)~w ( sine[[ (*).-wi)V/2r 

x cos[(wm-+w) (t-T2) +€+-€.] }+- (,+jAq,)% (35) 

where 

,and, as before, w.2r is the mean frequency of the tones. 

The loudness may fluctuate slowly as audible beats if 

20 
where y and Eare integers. Beats obtained when y = c are called beats 

of imperfect unisons (ref. 26). These beats are predicted by the preceding loudness equations. The sine function accounts for the relative 

amplitude of the loudness oscillations as a function of WM-W., and tile 

sinusoidal dependence on the time t accounts for the time history of 

the oscillations. It will be shown in connection with two-tone loudness 

judgment tests by von B6kesy that beats might be detectable for 

IM- nI/2r--100 hertz, but are strong only if Iw-0n1/2r<10 

hertz. Beats for which y-- will be discussed in connection with more 

widely separated tones. 

Category B: Supercritical tone spaceing.-When the tone separation exceeds the critical bandwidth, that is, fdr supercritical tone 

spacing, the audible range is divided into adjacent channels. The 

channels are allowed to shift as a group in accordance with the spectrum of the sound stimulus. A method for locating channels is described in appendix C. The bandwidth of each channel is a critical 

bandwidth for loudness. The neurons associated with each channel 

have best frequencies within its associated band. However, except at 

the lowest and highest audible, frequencies, neuron excitation bandwidths exceed the associated channel bandwidth so that the power 

transmitted in a channel will be affected generally by tones outside 

the critical band (cf. figs. 3 and 6). The interaction of tones inside 

the critical band with those outside the critical band accounts for the 

secondary tones. 

Consider asound stimulus consisting of several tones located in adjacent critical bands. None of the tones lies in the same critical band. 

Each tone stimulates all channels whose neurons respond to the tone 

"2 High sound-pressure level 

Best frequency, Hz 

200 1000 4000 17.000 

Lowelsound-pressure 

IO1 

C2 

102 IO0 

Characteristic frequency 012gn Hz 

104 105 

FrGURF 6.-Nominal transmission curves for individual 

from ref. 12.) 

nerve fibers. (Adapted 
frequency. The output from a typical channel k may be represented 

by 

qk(t)= J 

where Jk-(w) is the frequency response of channel k to rotational frequency en. Since the Jk(w,) are presently indeterminate, they will be 

replaced by H(w.), as in the treatment of tones spread over a subcritical band (and as in other loudness calculation procedures). The 

total output from all channels can be approximated by considering 

only enough channels to encompass the bandwidth of the sound. As 

mentioned earlier, the replacement of J by H is expected to constitute 

the main weakness of the present model since it implies that the frequency re­ response is the same in each channel when, in fact, the 

sponse in each channel peaks in the neighborhood of the channel 

midfrequency (ref. 12). 

In conformity with the treatment of suberitically spaced tones, it 

is assumed that frequencies e,,+w, are not transmitted appreciably 

by the nervous system because, generally aw, and wm + w, do not all 

lie within the nominal transmission band of the same fibers. 

In general, in any channel k which includes a stimulus frequency, IN N

(g~q)n= : Y E5r(,t) JJ(moit)t(0m,wn) 

X (1/T) fcos [(o,-n) t'+ m -kn-mndt'L (36) 

as in the subcritical case, except that N tones are now spread supercritically over N critical bands. By assuming that only self-interactions of all tones and cross-interactions involving one tone in the 

channel are significant, equation (36) reduces to 

X(11T) (37) 

It is quite apparent from psychoacoustic experiments that, for 

supercritical tone spacing, since the secondary tones wn-w; are clearly heard, they must contribute appreciably to the overall loudness. In 

order to incorporate this contribution to the overall transmitted power, the tone interaction must be evaluated in more detail. 

If the difference frequency W,,-c is supereritical, it will be transmitted as if it were a real acoustic signal (ref. 30) and will be audible 

22
 
if it lies within the passbands of auditory neurons which, also transmit wm and wk. Whereas the period of beats is of the same order as, or 

greater than, the auditory integration time, the period of this difference tone is much less than the auditory integration time. Since 

orn-Wk is transmitted as an acoustic signal, neural discharges occur in 

only part, approximately one-half, of each cycle of oscillation (refs. 

12, 94,19,, and 80). This effect must now be considered in evaluating 

the preceding integral of the cosine. Over time T, 

IwrnoWJT> >2r 

Define 

At=27/w-)l <<T 

The active phase of each cycle of the wave at frequency Io,-wkI/27r 

during which discharges are generated is centered at the maximum 

value of cos[(wm-0k)t'+m-p-nm], that is, when 

= -_ 6

uuik t= 

0O - Wkj, 

Over each stimulus cycle, during which neural activity occurs in onehalf cycle, the normalized value of the integral in equation (37) is 

(1/At) ft. + At/4 P+ €- - dt 

tin: - At/4 

= (1/2) sine[ (m - w):eAt/4w] 

= (1/2) sine(1/9) 

In order to be consistent with the other terms in equation (37), which 

are treated as full-wave rectified, the preceding integral should be 

multiplied by 2. (The reference power is implicitly full-wave rectified.) Since there are a+l cycles of ,')-C throughout the period T, 

where a is an integer and 04c<1, it follows that over the auditory 

integration time 

(1IT)s:os(cmn)t+# Oml' 

(2/r) (o-+n)/At / 2/r 

because q< <a. Therefore, for each channel, equation (37) becomes GR Q(.),woow> 

+(4r)(lnio'~r,o.),IcVo,,moo, om) 

The corresponding overall result for all channels is obtained by 

appropriately combining the results from each channel. Tones cen23
 
tered in adjacent critical bands are a critical bandwidth apart. For 

this condition the preceding equation summed over any two adjacent 

bands must agree with the corresponding equation for two tones-subcritically separated by almost one critical bandwidth. Thus, 

+ }L, 

(1/2 M[=wI~o)+n= m)Qrwn'~m 

+(1/2) (Wk) W(0 + ) (wmiwk)}] 

where, in the first form, the subscript 7 denoting the channel has been 

replaced in the second form by the subscript k denoting the tone in 

channel k. The factor 1/2 is associated with the fact that two tones 

separated by a critical bandwidth are also approximately centered in 

adjacent critical bands. Thus, there is a significant overlapping of 

neurons appreciably stimulated by the two tones in the sense that the 

two adjacent channels in which the tones are centered are associated 

with a power transfer equal to that from one channel bounded by the 

tones. 

The newly introduced empirical quantity flm7 is a function of Wm 

0 and a monotonic function of - . Specifically, 

I =0 for adjacent tones separated by a critical bandwidth 

Pm,-+l as the separation of adjacent tones approaches two critical 

bandwidths, 

lmk=l for nonadjacent tones 

Thus P., differs from unity only for adjacent tones separated by one 

to two critical bandwidths, and then only in interaction terms containing m and k. Therefore, in general, the function #m7 can be set 

equal to unity. The only time its variation is likely to be of practical 

significance is in the case of sounds with bandwidths slightly supercritical and with nearly equal power in both critical bands. 

For adjacent tones-the function fl.kAO is related psychoacoustically to the onset of secondary tones and physiologically to the fraction 

of the total number of neural pathways which may transmit information on these tones. Thus, as the tone separation increases, the corresponding fl, tends to increase to unity. However, as this separation 

limit is greatly exceeded, p,!k might decline again as the number of 

fibers which transmit secondary tones declines. Also, the interaction 

transmittance function Q (c,,wk) must tend to vanish When hyper24
 
critical separation is reached since both frequencies, w., and wl, cannot, then, stimulate the same fibers. The loudness of hypercritically 

spaced tones will not be studied further. 

When the preceding equation is introduced in the loudness equation 

(4), it becomes 

2/[ 

+ 

+ +2(4/V)P2 , £7r (w1:wk) 14-/ S(wmwk) toW (&JrnOk) j}'

which is the overall loudness equation for tones supercritically spaced 

in adjacent critical bands, and where 

+
 

This threshold expression might seem too large, since o, (ok), alone, 

is a threshold value. In order to justify equation (39) consider the 

broad range of frequencies for which Q, (w) is constant. Suppose that - the threshold term in equation (38a) is (n9)1/ with no sum,and 

that R7t is given by equation (34), which applies for a subcritical 

hand of tones. Then, as the tone separation is made supercritical 

without changing the overall sound pressure, equation (38a) would 

indicate that the loudness rises. This is true for suprathreshold loudness. However, experimental data (ref. 43) clearly indicate that, near 

threshold, loudness does not rise but, in fact, may tend to decline. 

Both the suprathreshold and threshold loudness results are accounted 

for if it is assumed that the power R is determined in the same 

way down to the loudness threshold. Then, equation (39) is valid. Of 

course, the threshold term is important only for sounds near the 

loudness threshold. 

The preceding loudness equation applies for tones in adjacent 

critical bands. If the tone separation is supercritical but the tones do 

not lie in adjacent critical bands, then the equation must be modified 

to incorporate the fact that neurons characterizing intermediate, 

"empty" channels are effectively stimulated by the interacting tones. 

This might be accomplished by multiplying the contributions to 

25
 
occupied channels bounding the empty channels by a factor greater 

than unity in order to account for the empty channels. An alternative 

procedure is to relabel the sums so that they denote critical bands 

rather than tonet. Then, the preceding loudness equation becomes 

£'(Kz2 /2)[ Q()IV(K) + s {Q,(W)V(WI) 

+ (4/r)f6ILKT~~w)IVa~r ~,;n~ 

rH2 2 - K 7\=( 

where the subscripts icand ft refer to each of the H critical bands 

spanning the total bandwidth of the tones, ' (OG) 0 if the critical 

band is empty, and epr may differ from unity only in conjunction 

with interactions between adjacent critical bands. Also, y,(jcqt)x is 

expressed as in equation (39), with the reinterpretation just stated. 

Beats of mistuned consonances (ref. 26) represent a special case 

in which loudness fluctuations may occur even though [w- ,[> >0. 

Closely spaced tones produce audible beats. MIore, generally any tone 

pair which satisfies ywmtcwn may yield audible beats. Beats of mistuned consonances occur -when the integers y and/or c are greater 

than unity. The possible values of - and c which yield subjective beats are limited by the fact that the tones must interact. This implies that both tones must stimulate the same nerve fibers. W/hen the 

transmittance bandwidths for individual fibers are considered (ref.

12), it is concluded that possible absolute values of -yand c are probably limited to 5, or so. This conclusion is confirmedby psychoacoustic 

experiments (ref. 44), where the maximum number of audible harmonics was found to be 5 to 7. For , and/or e>1, in equation (37), 

(lIT)f'..T COS[ (Om ek'+ 0tn 4'A Omklldt' 

S(2/w,) sine [ (c - k)T/9v,] Cos {[ (yw- cw) 1-y] (t - T/2) I 

as shown in appendix D. Hence, 

) = 

+ (4/)tMES(onoa,) lcI(ot)Ik) (o,,,,) 

X sne[ (ywm- w) T1/7]cos{ [ ( - w)/-/] (t -)}}] 

-[j ' (j? A] (38b) 

replaces equation (38a) in the presence of beats of mistuned consonances. Equation (38b) reverts to equation (38a) in the absence of 

26
 
these beats, that is, if yo)-Cwt?0. Beats of mistuned consonances 

may be significant only if Iyw-cwkI <10 hertz, as in the case of beats 

of imperfect unisons. In figure 7 the upper limit of the frequency 

difference for detectable beats of imperfect unisons (ref. 25) is compared with the critical bandwidth (ref. 28). 

Category C: Hypercriticaltone spacing.-When the tones are sufficiently separated, Q(w,,w.)->, as will be shown; so that the tone 

interaction, and hence the periodicity pitch, vanishes in equation 

(40).
 

Category AB: Subcriticallyspaced tones pread over sbperorlioal 

bandwidth.-Subcritically spaced tones cannot be resolved. Supercritically spaced tones can be resolved. Threfore, the output power 

from subcritically spaced tones spread over a supercritical band may 

be evaluated from the standpoint of loudness by treating each critical 

band of tones as if it were a single effective tone and then treating the 

effective tones from all bands as if they were supereritically spaced 

tones. Thus, the output power in any given critical band, assuming 

unity transmittance, is 

to~~~ ()={_ wownsine[ (,-n)T/27r] 

xcosr (m-w.) (t-T/2) +mO-O1 } 

104-

Critical bandwidth 

(ref. 28) 

102
 

o ~~~for Maximum beatsf. tone 7)separation for beats (ref. 23) A 

3 5 1ot 10? 10 104 10

Geometric mean frequency. w.12g Hz 

FiouR 7.-Comparison of critical bandwidth for loudness and maximum tone 

separation for beats of imperfect unisons. 

21 
as in equation (35). However, the subscript urefers to a critical band. 

By using this expression to evaluate the effective power in each 

critical band, the overall loudness can subsequently be computed from 

equations (38) or (40). 

An elementary interpretation of the sums in equation (40) is as 

follows -wheneach critical band contains several tones. The tones contained in each of the critical bands 4 interact collectively in each 

neuron with those within a critical band x to produce additional relative output power, ascribed to secondary tones, from the neurons 

characterizing the xth band. All contributions to the sth band from the 

i bands act independently because they arise from statistically independent neural discharges. Therefore, the contributions of the bands 

to the transmitted power in the th band should tend to be additive in 

all fibers which are dynamically active. The same argument applies 

with respect to tones in each critical band xjtinteracting collectively 

with tones within the critical band It. Since the neuron bandwidths 

generally exceed the critical bandwidths the same argument can be 

used to account for power transmitted by fibers characterizing intermediate, empty, critical bands. 

CategoryAC: Subo2ritically spaced tones spread over hypercritical 

badwidtk.-The loudness is evaluated in the same manner as in 

category AB, by means of equations (88) or (40). 

critical

Continuous 

Category 

bandwidth.-The 

BC: 

Spectrum 

Supercritically 

loudness 

spaced 

is given 

tones 

by equation 

spread 

(40). 

over hyperThe loudness of sounds with continuous spectra may be deduced in 

the same manner as for tones. As in the case of sounds with discrete 

spectra, the appropriate expression for the loudness depends on the 

spectrum bandwidth. The bandwidth of the sound may be subcritical, 

supercritical, or hypercritical. 

Subcriticalhandidth.-Sounds with continuous spectra are usually specified in spectral, rather than temporal, form. The loudness of a 

subcritical band of noise expressed in spectral form is 

_ =( mdfW- 17q(W-,T)dw1]1-[(K-22w f- _Wa,(,,tT)dw]­

(41) 

by virtue of equation (17), where TV is given by equation (13). In 

order to evaluate TV, from spectral measurements, equation (13) 

must be approximated. Specifically, by assuming that H(n) and 

P(r) are constant in frequency intervals at least 20 hertz wide and 

don't vary by orders of magnitude within a few hertz, it is shown in 

appendix B that 

28
 
'Wq(07t7') D(J~)H(wv 1 121P(wU) 1I2 (42) 

where 

]-i - 0 0 >t >T 

is the rectangular function. Let 

W(caf,T) = -(4- )IP (,)12/Tp 2

re, (43) (T 


define the relative power-spectral-density of the sound, where, without loss of generality, the dependence of IV on t has been suppressed. 

(Note that TV (cT) has the dimension [seconds), whereas wa(in) -for 

tones is dimensionless (of. eq. (21)).), Then, the loudness is given by 

[()/2-r) 

- (0/24r) 

f 

fr_ 

(wW(eo,T)d w] 

Q(c)W(tT)dw]A (44) 

where 

¢ (+)--IH( I:(45) ) 

This formula for loudness can easily be related to experimental 

measurements. As defined in equations (1) and (2), the overall 

sound-pressure level is given by 

S=10 log[ (1/Tp2) ft_rp (t) dt']


However, as shown in appendix E, 

if the temporal integral is stationary. Hence, 

-- llog[(1/2r) f- W(jT)dw] 

An audio-spectrum analyzer indicates the sound-pressure level in 

frequency bands. Ideally, in the band denoted by a, 

S(o ,(0'4,) ,T)= 10 log [(1/27) f% W(,T)d] (47) 

where S is to be measured over an ideal subcritical bandwidth 

(4o) = (Om)b-1a 

geometrically centered at the rotational frequency 

Thus, the cutoff frequencies are ekplieitly given by 

29 
== $(A) I +[(A.)2+4.] $} 

By inverting equation (47) it follows that the relative power in 

the bandwidth (Aw) is given by 

CZ~a(ACJ)a~(1/2-7r) J'6 1V (w,T) do 

=antilogES(wa,(Aw) a)/10] 

where, for brevity, the variable T has been suppressed. Since 

(neglecting beats) loudness is found to be independent of bandwidth for sounds with suberitical bandwidth, the loudness of a 

sound with subcritical bandwidth may be determined from measurements over a critical band which contains the subcritical band, 

that is, 

Y~ [c2Q(NIL) flcr(W'g) ({ 2Q(O)A) [n2cr((OA) ] t)}% (48) 

where (Aci)g (suppressed) spans the uth critical band, which is assumed to contain all the noise. In equation (48), fl, is to be interpreted as the acoustic power imposed in time 2' in the optical 

bandwidth (so that (Aw), can be suppressed). Also, it has been 

assumed that Q is constant over the critical band. 

Supeicritica bazndwidth.-If the spectrum bandwidth is supercritical, then the relative output power can be predicted in a manner 

analogous to that in category AB for tones, except that the relative 

power is over critical bands rather than for discrete frequencies. 

The relative output power in any given critical band Mis given by 

where Q is assumed to be constant over the critical bandwidth. 

Equation (49) permits the calculation of (K2q),, from spectral-measurements of S. Since (92,) , can be calculated, the loudness can be 

calculated from 

£= K2/2)[Q(O)oro) 

+ : ('-BA)fQ(COA52(0 j) 

+ (4/7) 9,2K Lir (OigwX) I V (W/,Wx) [2Cr (wt Wr) ) 

-[K 2 (50) 

30
 
which follows from equation (40), and where 

or (wpwX')-- [fr (cg) Qc~~ (c)] 1/2 (51) 

The interpretation of the sums in equation (50) is similar to that 

previously given for the sums in equation (40). Signals associated 

with separate critical bands interact because the neuron bandwidths 

exceed the critical bandwidths. 

HyperoiticaZ bandwidth.-If the spectrum bandwidth is hypercritical, then the output power fa and overall loudness _2 are obtained as in the preceding case. 

Mixed Spectrum 

A mixed spectrum is defined as one consisting of both discrete and 

continuous components. For a sound with a mixed spectrum the 

output over a critical band is given by the sum of contributions from 

the discrete and continuous parts of the sound spectrum. If a broadband spectrum analyzer, say a one-third-octave analyzer, is used to 

analyze a sound with mixed spectrum, the tones will be unresolved. 

Then, the spectrum may be treated as if it were continuous. This 

might be satisfactory if the tones are not closely spaced. However, 

it should be noted that since one-third-octave bandwidths exceed 

critical bandwidths over a large portion of the audio-frequency 

range an instrument with narrower bandwidths would be preferred. 

LOUDNESS LEVEL 

It is more common to express the subjective intensity as a loudness level in phons rather than loudness in sones. In general, loudness /£ and loudness level L are related by 

L =20 log[ (J_/J(1 pz) + 1] 

where L=0 when _P=0 (ref 21). By definition, Pj=1 sone when 

L =40 phons. Hence, Xijrf10- 2 , with negligible error. Therefore, 

E = 20 log (101 P+ 1) (59-) 

or, for suprathreshold loudness (241), 

L-20 (log _+ 2) (53) 

For future use, the loudness-level difference between N tones and 

any one tone, or group of tones, is expressed by 

so that 

L(wco,... ,(o) L(ooi,...) +AL (55) 

31 
TRANSMITTANCE FUNCTIONS 

The transmittance function Q (w) may be evaluated explicitly 

from results of psychoacoustic studies (refs. 20, or 19, 45, and 

46). These studies show that Q, more particularly )AV, is a weak 

function of the sound pressure p, as well as of t. Specifically, 

Q(w,p) -7(w)J V(,pe) (56) 

where p,=p,(p,97) is the sound pressure at the eardrum (ref. 21). 

It is convenient to express 7, cA, and p, in decibel notation. Thus, 

by definition, 

2=10 log g (57) 

N=lOlog A (58) 

where T is -the external transmittance level, and N is the internal 

transmittance level. The sound pressure %t the eardrum p, corresponds to the sound-pressure level at the eardrum S., in conformity 

with equations (1) and (2). Also, S, is readily evaluated from 

S,=S+T 

The total -transmittance Q is derived from judged equal-loudness 

curves shown in figure 8 (raft 20). The external transmittance 

level T has been measured directly (refs. 46 and 46). The curves 

tabulated in table I and displayed in figure 9 approximate these 

results. Hence, the internal transmittance A, or its level Y, can be 

determined. 

For free-field listening the experimentally derived transmittance 

7 can be satisfactorily approximated by an empirical formula given 

in appendix F. The empirical formula and the experimental curves 

are in satisfactory agreement, as shown in figure 9. 

For listening through earphones the selected model of the outer 

ear consists of a larger diameter outer tube, representing the carphone and pinna, in series with a smaller diameter closed tube, 

representing the auditory meatus. For an assumed plane wave the 

extermal transmittance of this arrangement is indepeiident of frequency (ref. 33, pp. 154-156), so that it is sufficient to set 9T=1, or 

T=0)in this case. 

The internal-transmittunce-level function N( oS) can be found 

(ref. 21) from the equal-loudness curves shown in figure 8 by 

subtracting out the external-transmittance-level function T(w) for 

free-field listening (fig. 9). The resultant dashed curves are in 

fairly good agreement, especially for suprathreshold loudnesses, with 

equal loudness curves for earphone listening reported in reference 

82
 
Equal-loudness curves
 

--- Equal-loudness curves based on sound-pressure
 

level at eardrum S (estimated, see ref. 21)
 e 

Loudness level, 

L, 

phons 140 


140
 

120
 

00 

go 

E 60 zW 

40 TO 

20 20 

0o­

-20 i i h I I , 1, I I I ,1111 3 o4 15 101 1o 10

Frequency, w&2IfHz 

FmrmE 8,-Equal-loudness curves for plane waves incident from front. (Data 

from ref. 20.) 

___ Plane waves from front (ref. 19) 

0 - - Diffuse field (ref. 46) 

-. -0 Empirical formula (app. F)

-0) 

,, 2 -2011 11 

2 103 104 101 10 105 

Frequency, wlI2'Hz 

FIGURE 9,-External transmittance ,level function 2'(w). 

27. The experimentally derived internal transmittance N_/ can be 

satisfactorily approximated (ref. 21) by the formula 

cJV(W),S) {(a,)2 [1+ (CoW)2] 1 + (Wo/W=) 2] 

[1+ ( /o)2h-[1+ (W/W )]-h}G (59) 

3a 
TAiLE I.- xTEaNAL TnASmtiTTANCE L Em FUNCTION T 

External transmittance level, One-thirdT, dE octave-band 

midfrequency, Plane wave 

RZ incident frontally Diffuse field 

<160 0 0 

.5 

200 1 

160 .5 

1 

250 1.5 1 

315 2 1 

400 2 .5 

300 2 0 

630 1.5 -1 

800 .5 -2 

1000 0 -2,5 

1250 0 -2 

160 1.5 0
 

2000 4 8,5 

2500 17 7.5 

3150 10 11 

4 000 12 12.5 

5000 10 8.5 

6 30 6 2 

8000 0 -5 

10 000 2 -1 

12500 9 9 

16000 5 

20000 0 

which originates from the problem of a high-pass filter in series 

with a low-pass filter (ref. 47). In equation (59), 

w /2,r= 1000 hertz 

is taken as the center frequency; 

u./2r-18 000 hertz 

is the upper cutoff frequency, and 

Z/2 = antilogo (2.1761- 0.005706 S.) hertz 

is the lower cutoff frequency of the auditory system. The lower cutoff frequency is a weak function of S,. The experimental and analyticl transmittance levels N(mS,) compared in reference 21 are in 

good agreement. The analytical curves are shown in figure 10. (The 

standard A-weighting for sound-level meters is also shown in fig. 

10 for comparison. Clearly the A-weighting is in poor agreement 

with the standard transmittance curves for loudness (ref. 20), especially at low frequencies.) 
20 Present weighting
 . - A-weighting (American standard for
 

sound-level meters, Z24.3 -1944)'
 0 

Sound-pressure level 

-200 " -ateardrum, 120 20 So. dB 

. -40 40 

-60 ­

101 102 103 104 105 

Frequency, 0 i27 Hz 

FIGURE 10.-Internal transmittance level function N(w, S0). 

In summary, for free-field (wwithout earphones) listening, Q is 

given by equation (56), and the required formulas for g and 'N 

are given in appendix F and by equation (59), respectively. For 

listening through earphones, = 1, so that 

Q (o,Se) =A (WAS) (60) 

In addition to ithe total transittance Q(o) (where, for brevity, 

the dependence on S, has been suppressed), the cross-transmittance 

Q (Wm:on) is involved in those terms of the loudness equations which 

represent spectrum interactions. 

For free-field listening, in which ease the external transmittance 

g must be considered, the required real part of the external crosstransmittance £(0,.), that is, £7remoi), can be expressed by an 

empirical function given in appendix F. The dependence of' this 

function on ]w-w/2w for various geometric-mean frequencies 

Jmw/w/27r is shown in figure 11. It is apparent -that 

Hz 

1000 20 - 5000
 

m 2010000
 

0- --------------------- ----------------------------­

2 3 4
 I00 101 10 10 10


)12 ; Frequency difference, W. - Wn Hz

FIGTunE t-External cross-transmittance-level function T (w.,w.). 

85 
Tr (Wm~wn) r'rT won for Iwn rI/2w,2O0 hertz 

The required absolute value (of. eq. (38) or (40)) of the internal 

cross-transmibtaace sy(m, ) can be determined by comparing the 

relation A(uw,)=z' (-)( .) with equation (59) for 1(w ). 

It is apparent that equation (59) results if 

,-(.)= { (i l )[1+ (Wol I)2]1./Z[1+ ((dJW)2]3-/ 

x (1+iw,,lO (I+,,/ 1}, (61) 

since Jf(-w.)=Jf( ). From this it can be shown that 

IsV(o, )J-- (+ )(62) 

where 

P---(O/d){(1+ .~/ ) E(cn-co/nu)J + (1+'%.%/<) 1Q %j ~)/WZI 

(64) 

and 

d -{(I+%=W/ ,2)2+ [ ,,, -,,) ,,] }X (1_+ W( /0,,)'

+ (+ -l)/W:) 

(65b) 

Also 

c/If(wn.,wn) iS\/(m,o) Icos 6? 

",(wn,,) = jA/1.(wm, n)Ijsin 6 r 

where 

tan '--f3/a 

To a good approximation, 

I ( o, W) I= I9(vC) V - -(6) 

if Icu.I <<. This approximation is also valid if on< < (t.,wn) 

<<w. The dependene of j (Im )j on w -vi, for various geonaetric-mean frequencies is shown in figure 12. This function also 

equals IQ ( ) for listening with earphones. 

Since £(w) can often be approximated by 3( /-) and 

can often be approximated by sV(/w),it follows that 

(,,,,,) (V-) (66) 

if fom,,-,u<Ctn and Iw,,-w,.I/2 r<2 000 hertz. From figures 11 and 

12 it would appear that equation (66) is satisfactory if Kr-cI/ 2 r 

<2000 hertz, except at low frequencies. 

36
 
Hz 

100 

500 

0 - 1-O0000 

COC
 

10\
 

-20 . . ...... 

-
 - - -. - - - - - - - -­

-10 "
 

'20 

NN
 

, -10
 

-20 " 

101 102 iO3 104 

Frequency difference, (wm- WI[2g Hz 

(a) Sound-pressure level at eardrum S.=100 db. 

(b) Sound-pressure level at eardrum S.=80 db.
 

(e) Sound-pressure level at eardrum .=40db.
 

FIGUIME 12-Internal cross-transmittance-level function N(0,, w.).
 

SIMPLE FORMULAS FOR LOUDNESS LEVEL 

For large numbers of tones or broadband sounds the lengthy 

summations make the loudness formulas too cumbersome for manual 

calculation. A simple approximation for the loudness level would be 

desirable. For broadband, suprathreshold sounds which are not 

dominated by a signal at one frequency, each term of the sums in 

equation (40), say, contributes only a small amount to the overall 

loudness. Morever, the total crosstransmittance £7r (wgw,) 1,) (ogox) I 

is less than, but generally approximately equal to, the total transmnittance Q (Vouok) (cf. eq. (66)). Hence, (wgco) [wq(w w() ] 

37
 
for frequencies not too widely separated. In equation (40) the total 

value of the self-interaction terms is M w q () when there are no 

empty channels. The total value of the cross-interaction terms is 

There are M self-interaction terms and H(M-1) cross-interaction 

terms. The cross-interaction terms satisfy 

vwAo + (87r <2E'w~(w +Wq(WK) 

which follows from Schwarzs inequality by assuming that 8/%-2. 

Thus, at suprathreshold loudnesses, equation (40) may be approximated by 

Therefore, the loudness level is given by the simple formula 

L 2O log (10_f) 

<10 log wq(u) +10 log(- 1/2) (68) 

which - follows using equation (53), and where Kx10 2.If there are k 

empty bands, then 

and 

L 10 log Y, w('W') W' + 101log -(70) 

Similar formulas apply for sounds with continuous spectra, except 

that w is to be replaced- by 0. These formulas can easily be evaluated 

by manual calculations. Note that, for each critical band, wq= Qw, 

and that y,w is simply the total transmitted relative power. The 

quantity Y, w can be evaluated from spectral measurements in bandwidths equal to, or less than, critical bandwidths. As a last resort the 

quantity 10 log wq can be approximated by the A-weighted, overall, 

sound-pressure level B4. Then, 

LS=A2+10 log(M--L) (71) 

38 
This may not be a suffiiently good approximation. Specifically, since 

the A-weighting tends to underestimate the response (cf. figs. 9 and 

10), the loudness levels indicated by this formula may be too low. 

Other simplifications are possible. For equally intense tones, w (c) 

is the same for each tone. On the other hand, for equally loud tones, 

Q (wn)'w (t) 'is the same for each tone. Hence, for N equally loud 

tones spaced superoritically in adjacent critical bands (k=0), it follows from equation (38a) that 

.2((01,)2,. .. , ,, . .. , )/ _ (W,) (1/2) [N 2+ (4!v)N (Y- 1)]}'h 

=[(P1I17)[(1+1r/ )N-1])" (72) 

if N>I (ref. 48). If empty critical bands exist between the tones ( <5 

critical bands between tones), equation (40) yields the corresponding 

result 

S{(2N/.=) 1 (1 + 7r/ 1)N- 1 + Wk/43] ) (78) 

For equally intense, supercritically spaced tones the same formulas 

are valid if all the tones lie in the frequency range for which QUI. 

For listening through earphones the maximum number of tones 

which may satisfy this condition is about 15, since a maximum of 

about 15 supercritically spaced tones will span the frequency range 

for which Qr I. For free-field listening the maximum number of 

tones is less. 

TEST CONDITIONS 

Von Beksy (ref. 49) has listed some requirements for good psychological observations. These conditions include precisely defined 

stimuli, repeatable experimental conditions, an understanding by the 

listener of the particular phenomenon to be judged, and inhibition of 

irrelevant sensations. The listener should be unaware of what the experimenter regards as a "correct" judgment, say, of the loudness (ref. 

50). The experiment should be "blind" in the sense that neither the 

experimenter nor the listener should exercise immediate control over 

the stimulus. Each test should involve a single comparison between 

the test and reference sounds (ref. 51). Finally, because of the large 

errors inherent in psychoacoustic judgments the data must be averaged property (ref. 52). The importance of averaging sound power 

rather than sound-pressure level is well recognized in reading a meter 

(ref. 53, p. 181f.) but appears to have been generally overlooked in 

evaluating loudness judgments. 

Some of the preceding requirements have been violated in all psyehoacoustio studies. The magnitudes of the resulting biases are often 

89
 
unknown. However, in comparing experiments with theory, the divergence among the results of similar experiments, when compared 

with the theory, may disclose significant biases. 

LOUDNESS OF SPECIFIC SOUNDS WITH 

DISCRETE SPECTRA (COMPARISON OF 

MODEL WITH EXPERIMENT) 

In this section the loudness of specific sounds having discrete spectra is calculated and compared with psychoacoustic data, reported by 

various investigators. Sounds with discrete spectra consist of one or 

more tones. 

In the experiments to be discussed a variety of loudness comparison methods were employed and many empirical, loudness-calculation 

procedures were tested. These will not be discussed herein. This information is best obtained by consulting the original references. 

One Tone 

The sound consists of a single discrete frequency. Equation (18a) 

reduces to 

P() =A. cos(W't+ #i) 

Equation (38), or (85) , becomes 

f= [K2Q(ffll ) 1~i)' 1 [56ox'te~i] (74) 

where 

w~w,) =A~ 2p (75) 

according to equation (21). The quantity Kcan be evaluated by comparing this loudness equation with equation (29) in reference 10, 

which is 

_£=Hpro(S2-l) (76) 

in the present notation, where E is a function of the number of dynamically active and saturated nerve fibers, hence of the sound pressure and frequency. Since the loudness of any sound is specified in 

sones relative to that of a 1-kilohertz tone at one sone, equation (76) 

may be rewritten for any tone relative to a 1-kilohertz tone at one 

sone in the form 

iL(ia =LA[ Goi)d] II9oi)d -tfoi) ] (77) 

where 

X.[il(T0t ) ] { [ P(i,) ]/H[qfn 1 (/27r= 1000)] } 

X [1/iL=P (//27 1000) ] 

=1O-{H[iffp (,) /K[93L)=(ou/2'==1000)]} (78) 

40 
and 

9JJL-i (w/27r 1000) =102 

is the relative, effective sound pressure of a 1-kilohertz tone when its 

loudness is one sone. By comparing equations (74) to (78), it is evident that = [(Woi) i ((01) ]1/, and 

[Qr( =KE()(w,)1/K [-L_,(,w/-r=1O000) (79) 

so that 

x=10- 2 (80) 

Therefore, 

2-= 10-, IQ, (WOW (Wi) P4 - 10-, [Q, (COO) WC(,01) ] ( 

As shown in reference 10, with w, in the neighborhood of 1 kilohertz and with sound-pressure levels in the range 84 dB<S <90 dB, 

.U(10- . Therefore, Qr(e1 ) =1; and, since w>>t, 

Zj= 10-2,(-) 

follows from equation (81). Corresponding formulas for the entire 

acoustic amplitude range are derived in reference 10. For completeness, the loudness function given by equation (81) is compared in 

figure 13, as it was in reference 10, with psychoacoustic data reported 

by Warren. The agreement is excellent. 

For suprathreshold loudnesses the special result 2(w)=9(( ) 

P(oux) (cf. eq. (77)) resembles an input-output relation for a linear 

system. For more complex steady sounds it is, therefore, tempting to 

assume that the overall loudness is obtained by summing the loudness 

contributions associated with the individual frequency components 

comprising the sound. For example, Stevens, following others (ref. 

4), proposed several versions of such a procedure (refs. 5, 6, 54, and 

55) in which the individual loudness components were also unequally 

weighted in order to incorporate empirically the apparent "mutual 

inhibition" of neural discharges by different frequency components. 

A more careful scrutiny of the preceding specialization of equation 

(77) leads to doubt concerning the validity of loudness surmnation. 

Firstly, the nervous system is a physical system and obeys physical 

laws. The psychoaconstic quantities, specifically the loudness sensation in the present instance, are subjective, nonphysical consequences 

of physical processes in the nervous system. Input-output relations 

generally indicate the amplitude or the power transferred by a physical system. However, equation (77) relates sound pressure to loudness 

in a mixed physical-psychoacoustic system. Thus, equation (77) does 

not correspond to the usual input-output relation because equation 

(77) does not relate physical quantities nor does it directly indicate 

4:1
 
10 

-- Source 

/ Standard 102 


j Comparison (loudness 

judged equal to one- Warren (ref. 51) 

half loudness of j 

standard)
 

101 Present theory -- Fletcher (ref. 86) 

95 Percent
 

confidence If10-[ limits 

10-2
 

1Q3 I I I 

0 40 80 120 160
 

Sound-pressure level, S,dB
 

ErounE 13.-omparison of judged and calculated loudness for 1-kilohertz 

tone. 

amplitude or power transference. Loudness is an alternative expression of the relative overall output power of the physical system. 

Two Tones 

The sound consists of two discrete frequencies. Equation 18(a) 

reduces to 

p(t) =A, oos(ut+#i) +A 2 Cos(w1t+#2 ) 

For two suberitically spaced tones, equation (33) becomes 

42 
.=10-{ QG(,)'w (W) + Qr(,i2) (WO 

+ 2 9PJ (oi,,W)_ dV' (0,30 2) 1W (W1, 2 )

* sine [ (W2-w+) Ti2w] Cos [(o,2-coi) (t- T1/2) ++-+ 

-1o-o? (82) 

where 02-0. If the tone frequencies are not too low or too high (200 

Hz! (oil,e2)-92000 Hz) or if the two tones are closely spaced (say 

100 Hz, or less, apart at high frequencies), 

=10-[ QV (VW.m ) ( ((a) +w (w2) + 2w (1,o2) 

XsineE(o,-co)T/2r]cos[(o.-w) (t-T/2)+02-.ol, 

- 10- 2 (8) 

where, in the former case, Q,(Vwcos) =1. 

If the tone separation is supercritical or hypercritical, equation 

(40) becomes 

f7_ 10-2{ (1+k/) [Q(o)w (W) + Qr+7C)1 (a) ] 

+ (+/ -)~ £lr :(o,,w) IAV(o+,,)I (G,,+))) 

where the subscripts refer to tone numbers (rather -thancritical band 

numbers) ; k (not the subscript) denotes the number of empty, intermediate, critical bands; and P2.=P/32 is unity unless the tone separation is only slightly supereritical. 

loudness 

of 

shown 

Von 

two 

Bik~sy's 

in 

closely 

judgments 

figure 

spaced 

emperiment 

14 as 

reported 

tones 

a function 

with 

(1929).-The 

by von 

equal 

of the 

Bk6sy 

suprathreshold 

spacing 

predicted 

for listening 

for 

relative 

comparison 

amplitudes 

monaurally 

loudness 

with 

is 

­ through an earphone (ref. 35, p. 235). The theoretical curves (to be 

explained) were derived from equation (83). In the experimeit one 

tone was kept at a frequency :/2 r= 5 00 hertz, whereas the frequency 

4,2/ 2 r of the second tone was adjustable. The loudness of the combined 

tones was judged relative to the loudness of the 500-hertz tone alone. 

Von BWk6sy reported that the observations were difficult. Because the 

tones were closely spaced, beats were significant, thus the observational difficulty. Periodicity pitch was not important because, at the extreme separati6n, the tone spacing was only slightly supercritical, in 

which case k=0, and fl -.O, in equation (84). Nearly all the data 

were for a subcritical separation in the frequency range for uniform 

transmittance of the auditory system. Specifically, Q (wi) = Q (Vww2O) 

=Q (w2) =1. As a consequence of these conditions the loudness of the 

combined tones relative to the loudness of the 600 hertz tone is given 

by 

48 
2(flW2) /J((a) = v2{1+ sine[ (2 - ) 1'/2] ×cos[(( -oo.) (t-T1/2) +€52-€,])% 

which follows from equation (83). For to and w2 given, this function 

oscillates temporally between the limits 

V21+ sino(o- wO)T/27r] 1' 

and 

These relative loudness limits for T=0.1 second are shown in figure 

14. For larger values of T the central peak would be narrower and 

the fluctuations of the loudness limits would occur for smaller frequency intervals. Figure 14 illustrates that, for narrow spacing of the 

tones (:10 Hz), the listener tended to characterize the loudness of 

the tone combination by its maximum loudness. For greater tone 

spacing the judged relative loudness either fell within the limits or 

else, on the average, fell about 1.25 decibels below the limits set by the 

theory. Von Bekgsy reported that test results similar to those shown 

in figure 14 were obtained for w/2r = 200 and 2000 hertz and that the 

0 	 Experiment 

Loudness limits given by 

=Wt2 i;LsncN0a- &w)TI2ir])l12 

2 

where wil2ir =500 Hz. T=0.1 s 

6 2.0 

I00 

00 

0. 	 0 

-

- . 1.5
 

23 - :"-


Z0 

00 

3_0 40 40 0 0 0 00 


S 00 0 00 0 0
 

Frequency, oj212g Hz 

FIGuRE 14.-Itelative loudness of two equally intense tones as function of tone 

spacing. (Data from von 33ekdsy (ref. 35).) 

4 

0 
loudness decrease depended on the frequency difference, as in the 

present thory, and not on the relative frequency difference. Contrary 

to his expectation that the relative loudness should tend to unity as 

I(o - w,] is increased, the present theory and other tests (ref. 36) suggest that the loudness should remain constant, as in figure 14, and 

then begin to increase when the tone separation exceeds the critical bandwidth (115 Hz for V--p/2r=500 Hz). Von Bekesy's results as 

n2--4(, illustrate that a 6-decibel (not the usually accepted 10-dB) 

reduction in the sound-pressure level of a tone corresponds to halving 

its loudness (ref. 56). 

FletcherandMunson's experiment (1933).-Fromdata obtained by 

11 listeners with normal hearing, who listened through earphones, 

Fletcher and Munson compared the binaural loudness of pairs of 

equally loud tones with that of one of the tones alone (ref. 13). The 

tonal pairs were as follows: 

Pair Hz Lz 

1 1000 1100 0 

2 1000 2000 6 

3 1000 325 6 

The frequency separation of the first tone pair is somewhat less 

than the critical bandwidth (see fig. 3 or 7), so that equation (88) 

should be applicable in computing the loudness. The auditory system 

is equally sensible to both tones of this, and the second, pair. Therefore, both tones must be incorporated in the threshold term. For the 

first two pairs Q (0) = Q ( 2) - 1 (of. fig. 10), and Q (Wi,W2) = Q (VC/,2) 

= 1. Therefore, for each tone the given loudness level L equals its 

sound-pressure level S. 

For suprathreshold loudnesses it is convenient to set w (w) =w (w,) 

=w (W5 2) = 1 when the tones are equally intense and then to compute 

the ratio of the loudness of the tone combination to that of one of the 

tones. This procedure eliminates the necessity for calculating w(ft) 

if the loudness, rather than the loudness ratio, were evaluated. Considering equation (8a), the loudness ratio is given by 

for two suboritically spaced tones in the absence of beats. In phons 

this loudness ratio corresponds to a loudness-level difference AL given 

by 

AL =20 log[_(w,,w,)/jZf(w1 )] 8.0 phons 

which is to be added to the loudness level of the lone tone, that is, 

45
 
in order to obtain the loudness level L(0,)02) of the pair. 

The, judged and calculated loudness levels are compared in figure 

15. If the agreement between the judged and calculated values were 

perfect, the data, would fall along the diagonal straight line. Except 

near threshold (S, B dB; see fig. 8), the levels indicated by equation (83) average about 1 phon less than the judged values. It is noteworthy that, for 100 hertz separation of the tones, von B16ksy's data 

(ref. 35) yielded a loudness level about 1.5 phons less than that predicted using equation (83) whereas Fletcher and Munson's data 

yielded an average judged loudness level 1 phon greater than that 

predicted using equation (83). The difference between these results 

seems to imply that the principal difference between theory and experimnent may be due to systematic experimental error. 

The second pair of tones is harmonically related. The tone separation is supercritical, and the tones are separated by three empty critical bands. Thus, equation (84) applies with k= 3, and the Q's=1.For 

the results shown in figure 15 the absolute average difference between 

the calculated and judged loudness levels is about 1.5 phons. 

The final tone pair is 	also harmonically related. The tone separation is superoritical, and the tones are separated by six empty critical 

bands. For these tones, the transmittances are nonniform; Q (mi) 1, 

Q (W2) = 0.5 (cf. fig. 10) and Q (w1,w2) -1. Since the tones were equally loud, it follows that w (o) =w(w1 )/0.45. The loudness level pre100­

0 -- i 

-	 r0one frequencies, Separation 

o 	 Hz 

Wj 102 

21r 2vr 

0 1000 1100 Subcritical /10 0 :ll 20( } supercritical 100 

0 Calculated 20 40 6D) 8D 100 loudness level, phons p 

FIGURE 15.-omparison of judged and calculated loudness levels for two tones. 

(Data from Fletcher and Munson (ref. 18).) 

46 
dictel using equation (84) is compared in figure 15 with the judged 

value for one sound-pressure level. The difference between the calculated and judged loudness levels is 2 phons. 

Small and Thurlow'8 experimnnt (1954).-Small and Thurlow, 

using themselves as listeners, equated the loudness of pairs of tones 

imposed monaurally through earphones to the loudness of a 2000­

hertz comparison tone imposed on the other ear (ref. 57). The following pairs of tones were used: 

Pair 3Z Hz k 

1 400 1000 8 

2 1900 3100 2 

3 1900 4400 4 

4 1900 5100 5 

The primary tones were imposed at unequal intensities. Moreover, 

the transmittance function Q is not uniform for all frequencies involved. Finally, the data were displayed in terms of sensation levels 

S rather than sound-pressure levels. 

The sensation level S is the level of a sound above its loudness 

threshold. In general, 

5=S-C (85) 

where 

S 20 log(inpt) 

and 

C - 20 logfPt 

is the sound-pressure level of the sound at its loudness threshold. 

The sensation level of each tone alone is shown in figure 16. The 

sensation level of the lower frequency tone was held fixed in each set 

of tests while the sensation level of the higher frequency tone was adjusted. For all of the fixed-level (lower-frequency) tones, it happens 

that = . The sound-pressure levels of the adjusted (higher-frequency) tones can be determined from the sensation-level data. The 

sensation level of an equally loud, 1200-hertz tone is indicated by the 

ordinate in figure 16. Since, for a tone, 

L=N+S (86) 

(where N is the internal transmittance level, and the external transmittance level T = 0 when using earphones), and considering that the 

ordinate in figure 16 is S(m/27r=1200) and that S(ao/2r=1200) 

47
 
=S(w/27r=10o0)S(w/2 1o0O)=L(o/27r = 1oo) = L(w/27nc-1200), 

it follows from this approximation and equations (85) and (86) that 

Therefore, reconsidering equation (85), there results 

S(Wn) = (W/2r= 1200) -N(mn) (87) 

where the appropriate values of N(w,) 

Q( n) =o'V(n) are as follows: 

(of. fig. 10) and, hence, of 

H]z (ott) Q(,) 

1000 0 100 

3100 -1.5 .71 

4400 -8.0 .50 

5100 -&5 .45 

O Judged 

00 calculated 

S100 Hz 

0alone 

A (a-2)I 400Hz , 20oaI
 

0 o
 

QT 

S1000Hz H 

a olone/ 

90 

0 20 40 0 80 20 40 60 

sensation level. .'(eoI0 10o0), dR 

(a-0) -Reference 57, figure 7(a); (a-2) Reference 57, figure 7(h) 

listener A.M.S. listener W.R.T. 

(a-3) Reference 57, figure 8(a) (a-4) Reference 57, figure 8(b) 

listener A.M.S. listener W.R.T. 

(a) z/2wr, 400 hertz; co/ 2 r, 1000 hertz. 

FxoIns 16.-Comparison of judged and calculated sensation levels of unequally 

loud pairs of tones. (Data from Small and Thurlow (ref. 57).) 

48 

80 
0 Judged 

C Calculated 

80 

alone -

S40- 3100 Hz 

R alone 

20 0b-1) -2) I I

900 

S10-	 1900Hz _R. 8. 

, alone 

.o S-50 3100 Hz alone 

3 -I I I M -41) I I I 

0 20 40 60 80 100 0 20 

Sensation level, f(o127r= 3100), dB 

40 60 80 

(b-I) Reference 57, figure 9(a); (b-2) Reference 57. figure 9(b); 

listener A.M.S. listener W.R.T. 

(b-3) Reference 57, figure 10-(a); (b-4) Reference 57, figure 10(b); 

listener 	 A.M.S. listener W.RT. 

(b) 	 6i/2r, 1900 hertz; w,/2,r, 8100 hertz. 

FIGURE 16.-Continued. 

Thus, equation (87) can be used to compute the sound-pressure levels 

of the adjusted-level tones from the data in figure 16. The difference 

between the values of S(w.) found from equation (87) and those 

found from equation (85) using the abscissa values for S(,) in 

figure. 16 and published data on C(w.) (ref. 19 or fig. 8) is an indication of the inconsistency of the data (differences as great as 10 dB). 

Possibly this inconsistency is not unexpected since there is no guarantee that the listeners' hearing was equal in both ears. 

If w(o2)-O, then, for suprathreshold loudnesses, equation (84) 

reduces to 

£=' -2h(1+ k)(wivw*(603 J 

in which the quantity in brackets is excessive by an amount 

(1/9)&Q (o)w(.), unless 7c is set equal to zero. This has been done 

in figure 16, except where the sound-pressure levels of the two tones 

are within 1 decibel of each other. 

Among all data, except in the instance just cited, the average difference between calculated and judged levels appears to be at least as 

49
 
* Judged 

0 Calculated 80 _­

~~1900 Hz _ 

alone - .. 

,-4
 

40 

4400 Hz 

alone 

- I I I (c-2) 1 

Sensation level, 9(&127r= 4400), dB 

90 

-1900 HZ 

alone o 

00 0 

a o3-a-0- -a- -­ 50 00

0­

10 (G-3) I L 1I 14) 1 

0 20 40 60 80 0 20 O 40 

Sensation level, 9,((a127r = 5100), dlB 

(a-1) R~eference 57, figure 11l(a); (c-2) Reference 57, figure 11(b); 

listener A.M,S. listener WV.R.T. 

(c-3) Reference 57, figure 12 (a); (c--4) R~eference 57, figure 12(b); 

listener A.S. listener W.R.T. 

(C)W,12r, 1900 hertz; w../2 r,4400 hertz and wj2Tr, 1900 hertz; ./2r; 5100 hertz. 

PxmURr M6-oncluded. 

small as the average difference of the observations by the two 

observers. 

D9. H. Howes' empoiwnt (l950).-By listening monaurally 

through an eaphone, four trained listeners equated the. loudness of 

pairs of equally intense tones, and two trained listeners equated the 

loudness of pairs of equally loud tones to that of a 1-kilohertz referenos tone (ref. 58). The following equally intense component pairs 

were presented at sound-pressure levels of 40, 50, 60 70, 80, and 90 

deeibels for each.component: 

Separation, Oi/27, wa/2r Pair reels 33z lTZ 7C 

0
 

2 500 1420 2-150 2
 

1 250 1900 (1420, 2450) 


50
 
(The mel is a subjective unit of frequency or pitch. A 1000-hertz 

tone at a sound-pressure level of 4:0 decibels is defined as having a 

pitch of i000 mels. A 2000-mel tone has twice the pitch.) For pair 1, 

two values of 92/2r are given in the table becuse these frequencies 

were alternated on successive presentations of the pair. Loudness 

judgment data were presented separately for each sound-prelsure 

level and for each tone pair in order to demonstrate the effect of these 

quantities on loudness. However, since the average of loudness judgments by each of the two listeners differed by factors up to 3, the 

indicated nonlinear dependence of loudness on sound-pressure level 

will not be stressed. Thus, the data listed in table II, which contains 

results for various numbers of tones, will be averaged over all soundpressure levels. 

For 250-mel separation the tones were in adjacent critical bands, 

whereas for 500-reel separation the tones were separated by two 

empty critical bands. Since the tone separation was supercritical and 

the tone frequencies were in the flat response range of the internal 

auditory system, the predicted loudness of the tone pairs should be 

given by equation (84) with k=0 (250-mel separation), k=2 (500­

mel separation), and the Q's=l. For the equally intense tones at 

suprathreshold loudness it is simpler to determine the ratio of the 

loudness of the two tones relative to that of one of the tones and then 

express the result in terms of the loudness-level difference AL (ef. eq. 

(54)). The predicted and judged loudness-level differences AL are 

compared in table II, which includes both the average and median of 

the judged level differences. The judged and calculated loudness 

levels L are compared in figure 17. 

For the judgment data the average and median differences AL are 

6.5 phons for tone pair 1 (250-mel separation) and about 8.2 phons 

for tone pair 2 (500-mel tone separation), whereas the calculated 

values are 5.2 phons in the former case and 7.2 phons in the latter 

case. Thus, the average difference between theory and experiment is 

about 1 phon. 

D. H. Howes also presented data on judged loudness levels of equally intense tones centered at 1000 hertz, and with each tone at a soundpressure level of 60, 70, 80, or 90 decibels and spaced 950 mels apart. 

The resultant difference AL between the loudness level of the tone complexes and that of the central tone is shown in table III as a function 

of the number of tones and the sound-pressure level of each component. The loudness levels L are compared in figure 17. In table III 

the average and median level differences AL over all sound-pressure 

levels are also compared with the theoretical prediction. For two 

tones, with k = 1, the disagreement between calculation and judgment 

is about 1 phon for listener J. B1. and is negligible for listener D. R. 

51
 
TALE 11.-LouNESS OF EQUAL-INTENSITY ToNE COMPLEXES CENTERED AT 1900 HERTZ 

[Data from D. H. Howes (ref. 58).] 

(I) ) .,Number of tones 

d1 2 3 4 5 6 1 8 1 11 

Tone separation, rels 

250 500 250 500 250 ,.i 500 250 1500 250 1500 1250 1 250 

AL-=L (c,, .. )-L(), phons. 

40 7 8 14 13 18 19 20 21 23 22 26 29 

50 8 [o 15 15 19 20 23 25 25 23 28 81 

60 8 8 13 14 16 19 20 22 23 23 26 28 

70 6 $ 9 14 14 19 16 20 18 21 23 26 

so 5 7 9 12 12 16 14 18 17 19 20 23 

0 5 9 8 11 12 15 14 !16 16 17 19 21 

Average AL, phons 6.5 8.3 11.8 13.2 15.2 18 17.8 20.3 20.3 20.8 23.7 26.3 

Median AL, phons 6.5 8 11 13.5 15 19 18 20.5 20.5 21.5 24.5 27 

Calculated 5L, phons (approx.) 5.2 7.2 9.9 112.0 12.9 15 115.2 16.6 16,6 18.8 10.4 22.4 

- For equally intense tones. 
100
 

8O
 

0 ­

- 60 	 Equaly Equally Tone Tone frequencies, 

intense loud separation, Hz 

,/27 	 tones tones eels col W27r2 

40 -1420
 

0 V 250 iSod(24g 

> 500 1420 2450 

20- Listener

A-0 D.. 2IOH: {1420 

0 	 20 40 60 80 Inn
 

Calculated loudness level, L, phons
 

FIGURE 17.-Comparison of judged and calculated loudness levels for two tones, 

(Data from D. H. flowes (ref. 58).) 

Finally, 	similar loudness judgment data are presented in table IV 

for complexes of equally loud tones (rather than equally intense 

tones) centered at 1900 hertz and spaced 250, and then 500, mels 

apart at the same sound-pressure levels as before. The test data in 

table IV and figure 17 am presented in reference 58 as londnesses determined from an experimentally judged loudness function of questionable validity (cf. ref. 10). Thus, Howes' loudness data, have been 

reconverted to loudness levels using his table II in order to compute 

the loudness-level differences AL and their averages and medians, 

which are also listed in table IV herein. For two tones, with k = 0 

(250-mel spacing) or k=2 (600-mel spacing), the average difference 

between these results and theory is about 1.5 phons for tone pair 1 

(250-mel tone separation) and about 2 phons for tone pair 2 (500­

mel separation). 

Scharf's experiment (1959).-By binaural listening through earphones, eight to twelve listeners equated the loudness of a pair of 

equally loud tones to that of a reference tone held at a fixed loudness 

(ref. 43). By this procedure the loudness of each pair as a function of 

tone spacing was determined with constant loudness of each tone as 

a parameter. All judgments were made in the vicinity of the loudness 

threshold. The 1500-hertz comparison tone coincided with the geometric-mean frequency of the two tones. 

The judged loudness level of each tone and the equal-loudness 

curves of Robinson and Dadson (ref. 20 or fig. 8) modified (dashed 

58
 
TA33E IlI.-LouDNEss OF EQUAL-NTENSITY TONE CoMPLEXES CENTn9ED AT 1000 HERTZ WITH 250-Mm SPACING BETWEEN COrPONENTS 

[Data from I. H. loowes (ref. 58).] 

( ) - ) .... Listener 

d1I M. D.fl. 

Number of tones 

-AL-L(Iw, ... )-L(w), phons a 

60 8 15 18 24 25 29 8 13 17 21 20 24 

70 5 12 16 19 21 24 7 10 14 18 20 22 

80 8 10 16 13 18 20 6 9 15 13 16 17 

90 8 11 12 14 17 13 5 8 8 11 13 15 

Average AL, phons 7.2 12 15.5 18,8 20.2 22.8 6.5 10 13.5 15.8 17.2 19.5 

Median, AL, phons 8 11,5 16 18,5 19. 22 6.5 9.5 14.5 15.5 18 19.5 

Calculated AL, phons (approx.) 6.3 10,5 13.4 115.5 17.2 18.6 6.3 10.5 1 13.4 15.5 17.2 13.0 

For equally intense tones. 
TABLE, XV.-LouDrESsS Or EquALTx LOUD TONE CO rL1xES CENTsRns AT 1900 HERTZ 

[Data from T) H.fHowes (ref. 58).] 

,=S()=.... Number of tones 

dB 2 3 4 5 6 8 11 

Tone separation, mels 

250 1500 1250 1 500 1250 500 250 1500 250 1.00 250 1250 

) ..-L(), phons' 

30 8.5 10.5 12.0 I1,0 14,t 16.5 18.0 20.2 20.0 ---. 23.0 ---­

40 8.5 9.0 12,5 14.5 15,5 18.5 19.0 19,5 22.5 .... 23.0 ---­

50 7.0 8.5 13.0 14.0 17,0 20.0 20.0 22,0 23.0 28.5 30.0 33.0 

0 8.0 12.0 17.0 15.0 18.0 20.0 21.0 25.0 24,5 2T.U 28.5 34.0 

70 5.5 13.0 18.5 15.5 16,0 20.5 20.0 27.5 28.0 29.5 28.0 35.0 

80 4.5 7.5 9.5 11.5 12.0 14.5 13.5 20.5 16.0 21.0 20.5 27.0 

90 4.5 5.5 85 8.5 110 11.0 11.5 8.5 18.0 18.0 17.5 24.0 

Average AL, phons 6.6 94 12.8 12.9 14.9 17.3 17.6 20.5 20,3 24.9 24.8 80.6 

edian AL, phons 

alculated AL, phons (approx.) 

7.0 

5.2 

9.0 

17,2 

12.5 

9,9 

14.0 

12.0 

15.5 

12.9 

18.5 

15,0 

19.0 

15.2 

20.5 

16.0 

22,5 

10.0 

27.5 

18.8 

23.0 

I.4 

33.0 

22.4 

-For equally loud tones. 
portion in fig. 8) for earphone listening were used to determine the 

sound-pressure level of each tone at the eardrum. These values permitted the overall loudness level to be computed. The predicted overas a function of tone separation all loudness level of the tone pairs 

with sensation level (not shown) of the comparison tone as parameter are shown in figure 18. For each parameter value the results 

would fall on a horizontal line if the measurements and calculation 

procedure were ideal. In the figure the horizontal lines are nominal 

loudness level values of the pair. The dashed line at the bottom of the 

figure indicates the interquartile range of the original data, that is, 

the phon range containing the second and third quartiles of the data. 

It is evident that nearly all of the calculated loudness levels agree 

with the nominal loudness levels to within a difference less than the 

interquartile range. 

Nominal 

loudness 

level, 

phons 

O 40 

o 20
 

ZA 10
 

Plain symbols are duo adjusted
 

60 Tailed symbols are reference adjusted
 

Nominal expected value
 

---------- Interquartile range
 

-50 Critical bandwidth
 

0o0
 

C0
 

0
 

F 

C, 1 13 P 

- -------- 7 ..T !,---, 

10o12 103 U 

Tone spacing, Ac/2r, Hz 

F GIE I8.-Calculated loudness level of equally loud tone duo near threshold 

as function of tone spacing. Arithmetic-mean frequency, .. /2,, 1500 hertz. 

(Data from Seharf (ref. 43).) 

56 
Three Tones 

The sound consists of three discrete frequencies, so that 

p(t) =A~cos (W~t+.0,) +A 2 cos(Wot+ #2) +Aacos( 3t+ a) 

For three tones spaced suberitically, equation (33) becomes 

=10-2 {Q(i)'w (et) + Qr(W2 )w(W2) + Qr(w)w)

+ 2f?, (01,02) Ic,/(cl,w2) 1 (o1,w2) 

X sinc[ (02-) T/Q-r] cos ['(o 2- &) (t- T/2) + 2- ] 

+QE9r(wi11ws) I~,4 wa) j(Wi503 ) 

X sino[ (w- w,) T/2i]cos [ (w,- w,) (t- T/2) + € 3 - ofl 

+2-7(32,w3) 1,jt(W 2,Ows)[w(w)2,W3) 

Xsino[ (w3-wo) T/2]cos [ (cww) (t-T/2) +Oa-p2] }­

1 - 2 ,I (88) 

If the frequencies of three tones are not too low or too high, or if they 

are closely spaced, then all the Qrs in this equation may be replaced 

by Q.(W2), say, in analogy with the ease of two tones. 

For three supercritically spaced tones, equation (40) becomes 

L=102 [ w ) +Q((02)'W(02) +Q(&i0-W(&3)] 

+ (4/.) [p ,jyr(wi,o )IJMf( 1 ,e)Ijw(w 1,w) 

*+1r23s71o)21 V&c( Wa, O2 

V~Sr(2 +I% jgoo9) (/sw 

-1 -2 1 t ,(9 

where the subscripts now denote the tone number, and k denotes the 

total number of empty critical bands between the three tones. 

Bauch's experiment (1956).-Bauch and Zwicker equated the suprathreshold loudness of a triad of equally spaced tones, heard 

monaurally through an earphone, with that of the central tone alone 

(ref. 87). The results are displayed in figure 19 as loudness level of 

the triad when held at a fixed sound-pressure level as a function of 

the tone separation, which was varied from narrow (1 iz) to supercritical (up to 9 lI-z). However, the maximum overallseparation was 

not sufficient to achieve peak loudness. Loudness levels were judged 

for center frequencies w/27r=400, 500, 1000, and 4000 hertz, a variety 

of fixed overall sound-pressure levels ranging from 80 to 70 decibels, 

and for three ratios, 1/2, 1, 2, of the double amplitude of the outer 

frequencies to the amplitude of the center frequency. In addition, results were obtained for two phases of the center frequency which differed by w/2. 

67
 
0 	 Theory at critical separation Experiment 

Theory 

Critical 

bandwidth 78 Audiblebeats 


76 Phase angle,
 

74= - 0 -L.... Stot =70 dB 0
 

72 0-i- N Stot ll 	 C 

7C0 7d 2--­

5 	 Sto= 45 dB 

9! 44­

,2 	 Stot 4.5 dB -9 50 --- O w(col) = 

= w(2)= W( 3) 0 

49- 7-, I-Critical --­

46-
 bandwidth 

44() 	 _[. 110 100 1000 

Tone separation, 6012g Hz 

2 	 20 200 200 

Overall separation, AW27; Hz 

(a) Center frequency, 400 hertz.

(b) Center frequency, 500 hertz. 

FiGuBE 19.-oudness level of three tones of equal intensity as function of 

tone separation. (Data from Bauch (ref. 37).) 

Since all tone frequencies corresponded to the fiat response range 

of the auditory system, all the Q's in the appropriate loudness equations (88) and (89) can be set equal to unity. The selected toneamplitude ratios and corresponding acoustic power ratios are as 

follows: 

A,/A±_A3 /A2-	 1/4 1/2 1 

lw(o.) =-IV (Wa) /I (W) 1/16 1/4 1 

• *otf'Wo(0) 	 9/8 3/2 3 

where 

viWt=W(oi) + 0 2 ) +W(r 3 ) 

58 
0 	 Theory at critical separation 

Experiment 

Theory
 

76 - Audible beats Critical bandwidt4 

74 Phase angle, So od 

72 
 N W W( 3) 1 

70­

52

50 	 r Stot = 45 dB 

B 441 I 46- I ! " 1- 1 I I I l !, ! I ! l l 

44 

7j 

38 

" 35 	 Stot --30 dB , -9 34 "t____ [ 1 : ", w ... l = ( ]

1 
 10 	 100 mo00

Tone separation, 6w[2g Hz 

I 	 1 1
 

2 20 200 2000 

Overall separation. AwI Z Hz 

.(c) Center frequency, 1 kilohertz. 

FiGURE 19.-Concluded. 

From equation (88) it can be shown that, for a symmetric spectrum of three suprathreshold tones uniformly spaced over a subcritical frequency range, 

£/2 	 = (1 + (2/W± 0t) { 2 (0,,02) sinc[ (2 - W1) T/M] 

XCosL (W2- WI) (t - T/) + (01+ 03)/2] 

x cos[ (2€ 2 - 1 - 0pa)/21 + w(w) sine [2 (w - i)T/ 2 r] 

where 	 w~t t.ois the loudness of the triad when its total range is slightly less than 

a critical bandwidth. The loudness P fluctuates temporally at the rotational freqqeneies w2 - and 2(L2- o±). The maximum loudness is 

obtained if .¢ = == 0. For this value of the phase angles the 

loudness still fluctuates. However, a listener tends to classify the sab-­

59
 
76 

Audible beats Critical bandwidth 0 Theory at critical separation 

Phase angle. Experiment 

--- Theory Stot" 70dB 	 0 

72 	 0 N= 4wiw) =4 7w()2) 	 - 4 w-7l} 4 003)

1 SV,1t 11 ! I , , , I I ,I 

52 

050 

'. 0 Stot 45 dB
 

48w = 4 w(= 4 w(' 3)
 

-S 46 	 S..o=dde 

-, 

, 32 -­50: 111 1<2 iii I , il, 

..9 

5246 	 w(&)&,) 416() 4 	 (a) 0O Sit t 3=45 dB o

TI T I in i T r II!l I Iiil~l 

1210 100 l0 

Tone separation, 6wi2 Hz 

2 20 200 200 

overal separation, Ao12 Hz 

(a) Center frequency, 1 kilohertz. 

FxomLu& 	 20.-Loudness level of three tones of unequal intensity as function of 

tone separation. (Data from Bauch (ref. 37).) 

jewtive intensity of the sound according to its maximum loudness, 

which, relative to _2,,, is given by 

(1/Ia-,),.= (1+ (2/wl0 ,) {2Wto (,l,W2) sine[ (2 -a) /T12~] +,w (cal) sne [2 (W- ,)T/,] }) -

If the phase of the center frequency is shifted by w/2, so that 

1=03=O1 02=v/2, then
 

(1A/2er)max'= { + 2 ['a (Wi) /In,0e] sinc [2 (, - <t) T/2-] }S -


If the adjacent tones are spaced supercritically in adjacent critical 

bands, then, since the Q's=l, and w(oi) =w(ws), equation (89) re­

'duces to 

60 
0 Theory at critical separation 

-_ Experiment

-- Theory 

76 Audible beat-. Critical bandwidth 

74 ' -- Phase angle, Stot = 70 dB o 

72!- 0 - ww21 44=- 4ww / 

£68 

JQ52 

" 50 	 Stt 45dB 

= 48,-j 	 "(( )= } 4 o 48 	 w(w2) =4w(wq=4(w ( 3) 

46 	 "2" 

~44 

36 

o - fosbtSo dB 

32 

30 

10 	 100 100 

Toneseparation, 6w12n, Hz 

2 	 20 200 '2000 

Overall separation, AwI2g Hz 

(b) Center frequency, 4 kilohertz. 

FIGiuRE 20.-Concluded. 

£/I ={ (3/2) + (4 7rwtg) [w (m±) +9m (ew) ] } 

For lesser spacings the adjacent tones are spaced suboritically whereas the outer tones may still be spaced supereritically. Hence the 

method for category AB should be appropriate. However, because the 

detailed functional dependence of fl., on tone spacing is unknown, in 

the present instance, where the valte of an individual A,,,, may be 

highly significant, the loudness in the interval between supercritical 

overall spacing and critical individual spacing of the tones will not be 

evaluated. However, calculated levels for critical individual spacing 

can be computed. When more tones are involved, the value of an 

individual fl,, is not so crucial, since only a minimum number of frequency components is spaced slightly supercritically, in which case 

Pui,1, is most sensitive to spacing. 

The loudness levels given by the preceding formulas are compared in figures 19 (equally intense tones) and 20 (unequally intense tones) 

61
 
with the corresponding loudness judgmuent data for the various center 

frequencies and sound pressures. The theoretical loudness ratios have 

been converted to loudness-level differences AL and thence to loudness 

levels L by means of equations (54) and (55), respectively. In a1 

instances the predicted loudness levels in the subcritical tone-separation range are within 2 phons of the judged levels. IMoreover, the 

effect of the phase of the central tone is properly accounted for quantitatively. The small downward digression of the theoretical curves 

for p=ir/2 in the subcritical range is hardly significant. For critically separated tones the predicted loudness levels are within 2.5 phons 

of the judged values in all instances. 

D. H. Howes' experiment (1950).-Loudness levels of the following triads were judged as in the preceding two-tone experiment (ref. 

58). 

-2/2v, a/2r, Triad Spacing, COJ/2r, mels 11% Hz Hz 

1 250 1420 1900 2450 1 

2 500 1000 1900 3120 5 

The tones were equally intense and spaced supercritically. Overall 

loudness judgments were made with each tone at the, sound-pressure 

levels previously mentioned. The data and their comparison with predictions are presented in table II and figure 21. 

The loudness of each triad is given by equation (89). For each 

tone, Q-1 for practical purposes. For triad 1, To=l. For triad 2, 

k=5. Hence, the ratio of the loudness of triad 1 to that of the center 

tone is 

The corresponding loudness-level difference is 

AL = 9.9 phons 

For triad 2 the corresponding results are 

£(o1,~2,a)/L~c2)=3.98 

AL = 12.0 phons 

These results are listed in table IL. The average experimental level 

differences were AL - 11.3 phons for triad 1,'and AL = 13.2 phons for 

triad 2. These values disagree with the predictions by less than 1.5 

phons. 

For a triad of equally intense tones centered at 1000 hertz and 

spaced 250 mels apart, the average difference between calculated and 

judged loudness levels is about 1 phon for listener J. MW. and less than 

1 phon for listener D. H., as found from table III. 

62
 
120 

100 

o-80 

0 

-- 60 Equally Equally Tone Tone frequencies, 0 intense loud separation, Hz 

tones tones mels W1 02 1113

40) 

0 250 1420 1900 2450 

50D 1000 1900 3120 

2o o 1M4° 20 

Listener; J-M.- 250 670 1000 1420 

1I I I I
 

0 20 40 60 80 100 120
 

Calculated loudness level, [, phons
 

FiGU 21.-Comparison of judged and calculated loudness levels for three 

tones. (Data from D. H. Howes (ref. 58).) 

If equation (73) is used, for three equally loud tones centered at 

1900 hertz and spaced 250 mels (and then 500 nels) apart, the average difference between calculated and judged loudness levels is found 

to be about 2.5 phons for 250 mel spacing and about 1.5 phons for 500 

mel spacing (see table IV). 

I-owes also reported on an unequal loudness, three-tone experiment 

in which the loudness of three tones of unequal intensity was equated 

to that of three equally intense reference tones. Both triads were 

centered at 1900 hertz and spaced 250 mel, and then 500 Inels, apart. 

The loudness level of the central component of one triad was set at 

60 decibels, which equaled the loudness level of each component of 

the reference triad. The low frequency component of the former triad 

was set at a fixed attenuation below the level of the central component. Then the level of a high-frequency component was adjusted 

until the loudness of the triad equaled that of the reference triad. 

Since Q-1 for all frequencies concerned, and since _f (iW 2,wa) = 

.L[_(Ci)=-(w2) =-2(w,)],where the left side refers to the trio of 

unequally loud tones and the right side refers to the reference triad of 

equally loud tones, it can be shown using equation (89) that 

68
 
= 16 1+r (oil l 72 +16 

7r -r-7r W2 (+ 7c 

where (O~ is - 16.+ok+k2)4 w 1 

where w, isthe high frequency, and w is the center frequency. When 

expressed in decibels, this ratio yields a sound-pressure-level difference which is added to the 60-decibel level of the center frequency to 

obtain the predicted sound-pressure level SQ(w) of the high-frequency 

tone. 

In figure 22 the measured sound-pressure level of the high-frequency component is shown as a function of the attenuation, in decibels, of the low-frequency component. In the theoretical calculations, 

k=l for 250-mel spacing and k=5 for 500-meel spacing. The two 

theoretical curves differ by less than 1 decibel. Although the curves 

indicated by judgment data and by calculations are generally similar 

in shape, the experimental sound-pressure level of the high-frequency 

tone for equal loudness exceeds the theoretical value by up to 6 

decibels. Since the sound-pressure level S(c 3 ) increases by only a 

small amount, say 3 decibels, as the low frequency attenuation is 

changed from -10 to -40 decibels, it is possible that the disagreement between theory and experiment is primarily due to experimental 

error. It is difficult to imagine how a reduction of power in the lowPrediction Listener Tone Tone frequencies. C.H. 	 D.A. D.H. separation, Hz 

mels 0)1 &2 W3 

752- irU 21r 

--	 0 0 0 250 1420 1900 2450 

0, C, 500 1O0 1900 3120 

-"0
 0

4) 	 0 

.. 55 ­

0 -5 -10 -15 -20 -25 -30 -35 -40 

Attenuation level of low frequency component, S(wM) - S(w2), dB 

FiGuRE 22.-Sound-pressure levels of components of tone triad with loudness 

equaling that of 	similar triad of 60-decibel tones. Tone frequency w'/2, 

:1900 hertz; 2(w2 ), 	60 decibels. (Data from D. H. Howes (ref. 58).) 

61
 
- - frequency component from 1O to 10 4 that of the center frequency 

would require doubling the power of the (most intense) high-frequenoy tone, especially since the relative power in the low-frequency 

tone is so low to start with. 

S&ha'fs experiment (1962) .- Nine trained listeners equated the 

loudness of a triad of equally spaced tones to that of the center frequency (2000 Hz) alone (ref. 59). The intensities of the outer frequency tones were varied relative to that of the center tone, but the 

overall sound pressure of the triad, or of the comparison tone, was 

held constant. Judgments were made for overall separations of 280, 

1100, and 2200 hertz and a variety of sound-pressure levels. The first 

overall separation is approximately critical, and the other two separations are supercritical. Hence, equation (88) should predict the loudness if ]3 -wi/2=280 hertz, and equation (89) should be appropriate if IW3-<o1/2r= 1100 or 2200 hertz. 

For the suberiticaily separated triad with a symmetrical spectrum 

and the intensity of the outer tones less than, or equal to, that of the 

central tone, it was found from tests at overall sound-pressure levels 

of 20, 50, and 80 decibels that the loudness is independent of the 

intensity of the outer tones relative to that of the center tone. This 

result is confirmed by equation (88) since Y Q(W,) w (w,,) -constant 

for any given fixed overall sound pressure. 7=1 

For the corresponding supereritically spaced triads, the following 

conditions applied: 

Triad 1 2r, -z 3/2r, k

1 1450 2000 2550 1 

2 900 2000 2100 5 

For practical purposes, Q-1 for each tone. The tone frequencies are, 

nearly the same as those selected by D. H. Howes (ref. 58). As shownL 

in figure 23 the overall loudness rose as the intensities of the outer 

tones approached that of the center tone. 

It follows from equation (89) that the loudness of the superoritically spaced, symmetric triad relative to that of the triad when the 

center tone is strongly dominant is given by 

£ (W,,2),,a) /2 (0,,,0) 

( {.(3+7)+ 16[ o -_+[4(3+k)+- ]L( 8 

since w (w) = w (w). The curves given by this specialization of equation (89) are compared in figure 23 with the preceding data. The 

theoretical curves have been shifted so that they coincide with the 

65
 
Total sound12 pressure 

level, 

Stot. dB 

0 35 

) 65 

080 

- 4-2 

- (a) 

Theory II 

00 

al 

01 3 3 2 0 15 !0Z 

So ndpes eIv fcne tn iussun-rsur evlo I ue Ioe 

Theo00 

(b) 	 Triad 2; tone spacing 8w/9-7, 1100 hertz; overall tone spacing, Aw/2r, 2200 

hertz. 

FxoV 	 E 23.-Youdness of Lone triad with ymmetric spectrum centered at 2000 

hertz as function of attenuation of outer tones, (Data from Sharf (ref. 59).) 

nominal data our" given in reference 59 when the central tone intensity is dominant. 

The agreement betwen theory and experiment shown in figure 23 

appears only fair because of the expandled scale and the wide scatter 

ol the data. This scatter -resultsprimarily fromra systematic tendency 

for the loudness to rise more for the central range of the soundpressure level Sot, in agreement with other data (ref. 58 and tables II 

66
 
to IV). This effect is contained only implicitly in the fl.. The functional dependence of this effect cannot yet be explicitly predicted 

from the theory and, more importantly, is not present in data from 

all sources (e.g., ref. 13). 

In addition to loudness judgments of triads describing symmetric 

spectra, tests were also made of the same triads describing spectra 

with constant slope on a semi-logarithmic plot. 

It follows from equation (89) that the loudness of a triad of supercritically spaced tones with a constant sloped spectruin is given, relative.to the loudness for infinite slope (that of the outer tone), by 

(Wo) I I W(co,) w (vi) ( [i+I++ (0 2 )_J+ 

for negative slope and by 

(3+k7) [t (W,) + Wu'(W 2 ) + 

3 1w(W,) wZL(wa) . 

+ 3j7( W(s)}W(20W-+(wa)}/-J

for positive slope. 

The experimental and theoretical results are compared in figure 24, 

in which the reference sound-pressure level is set to zero when the 

spectrum slope is minus infinity. Considering the scatter of the data 

the judged and calculated results are in fairly good agreement. There 

is no evidence that the loudness of the triad depends on the algebraic 

sign of the spectrum slope. 

The reported interqnartile range of sound-pressure level variability averaged over all observers was 4 decibels for the experiments on 

symmetric spectra and 6.5 decibels for the experiments on sloped 

spectra. In the latter case the overall variability was sometimes as 

great as 20 decibels. 

Four Tones 

The explicit expressions given for the loudness of two- and threetone complexes should be sufficient to show how the loudness equations expand as the number of tones is increased. 

P. H. HUowes' experiment (1950).-Howes' studies (ref. 58) of 

four or more tones were similar to those already described. The following quartets were studied: 

67
 
To l Soundpressure
 

level, 

Stot, dB 

0 45 

0 60 

C,,, 0 80 

_5 Theory 

10 Theory, 

5L~ 

= (a) Is., Total sound-


C, pressure
 

04 level, 

Stot, dB 

0 45 

nl 60 

0 90 10 - Theory 

0 0 0 =heory
 -T

5 (b) 

5-0

I 1 I I I~O-Ter 

-30 -20 -10 0 10 20 30 40 50 

Spectrum slope, dB [ocave 

(a) Triad 1; overall tone spacing, Ac/2r, 550 hertz. 

(b) Triad 2; overall tone spacing, &i/2r, 1100 hertz. 

FiGuRE 24.-Loudness of tone triad centered at 2000 hertz as function of its 

spectrum slope. (Data from Scharf (ref. 59).) 

Qua Qatt Separation, Melo w/27, 11Z w_/2r, HEz %/2W' Ffz w/2r, "Uz

1 (1000) 1420 1900 2450 

1420 1900 2460 (3120) 2 

2 500 670 1420 2450 4000 8 

(Loudness judgments for 250-mel tone separation were made by 

alternating the first and second row of tones with the reference tone. 

Thus, the frequencies 1000 and 3120 Hz wers alternated on successive 

presentations.) Quartets of equally intense tones centered at 1000 

hertz and spaced 950 mels apart were also studied. 

For the tone quartet with 250-meel spaeing, 

22 (wl1co2,c&s3ai 4) /2(t~,) =4 

which corresponds to 

AL 12.9 phons 

68 
For 500-mel spacing the corresponding results are 

17 (CW1 ,W3 04) / P(uq) = 5.62 

AL= 15.0 phons 

In table II the predictions are compared with the judged loadness levels of the preceding tone quartets. For quartet 1 the difference 

between calculations and judgments is approximately 2 phons, whereas for quartet 2 the difference is approximately 3.5 phons. 

For equally intense tones centered at 1000 hertz and spaced 250 

mels apart, the disagreement between calculations and judgments is, 

as shown in table III, approximately 2.5 phons for listener J. Al. and 

0.5 phons for listener D. H. 

For equally loud tones centered at 1900 hertz (quartets 1 and 2 

previously tabulated) the disagreement between calculations and 

judgments shown in table IV is about 2 phons for 250-mel spacing 

and about 2.5 phons for 500-reel spacing. 

In all cases the judged and calculated loudness levels are compared 

graphically in figure 25. 

ZWicker, Flottorp, and Stevens' experiment (1957).-The loudness of quartets of uniformly spaced, equally intense tones was 

judged by comparison with a reference tone at the center frequency 

(ref. 38). Judgments were by 6 to 22, apparently naive, subjects 

utilizing binaural listening through earphones. Determinations of 

loudness were made as a function of tone spacing about the center frequencies 500, 1000, and 2000 hertz and for a fixed overall soundpressure level of 57.5 decibels. An additional similar set of tests was 

performed using a tone quartet centered at 1000 hertz and at three 

other sound-pressure levels. 

The tone spacing covered both subcritical and supereritical ranges. 

For the subcritical range the loudness of the quartet is given by 

equation (33), whereas for the supercritical range, equation (40) is 

appropriate. 

No information was given as to the phase of the tones at t=0. It 

is evident from equation (38) and from the results for two and three 

tones that phase relations have an important effect on loudness if 

adjacent tones are separated by less than 10 hertz. Since a listener 

tends to characterize a fluctuating sound by its maximum loudness, 

it is to be expected that the four-tone data shown in figures 26 and 27 

should tend to rise for overall tone spacing less than 30 hertz. Some 

rise appears in some of the data. However, if the rise is actually a 

systematic one it is certainly minimal, so that the initial phases of 

the tones may have been essentially random. The loudness is otherwise effectively independent of tone spacing for subcritical overall 

spacing, in agreement with equation (33). For supercritical overall 

spacing the loudness rises, as in the case of three tones. The calculated 

69
 
Equally 

intense 

tone 

Equally 

loud 

tones 

Tone 

separation, 

rels 

Tone frequendes, 

Hz 

W1 02 (03 04 

2T" T1r 2ir "9 

0 < 250 1000 

11420 

1420 

1900 

1900 

2450 

2450 

3120 

0 C> 500 670 1420 2450 4000 

Listener 

10 J.M.A 250 394 610 1000 1420t 

120 A D.H.S 670 1000 1420 .1900 

t Each row was presented alternately 

100­

0 ­

80 

: 60­

0, 

40­

20 

I I I I I 

0 20 40 60 80 100 120 

Calculated loudness level, l_ phons 

FIGTmE 25.-Comparison, of judged and calculated loudness levels for four tones. 

(Data from D. H. Howes (ref. 58).) 

and judged loudness levels are generally in good agreement. The rise 

is between 5 and 10 phons for the maximum spacing studied. These 

results may be compared with those of D. H. Howes (ref. 58). 

Some of the present results are for center frequencies of 1000 and 

2000 hertz with a maximum overall tone spacing of 1536 hertz (51P 

Hz between tones). D. I-. Howes' results included those for 1000­

and 1900-hertz center frequencies with an overall spacing up to 1230 

hertz in the former case and averaging 1576 hertz in the latter case. 

Thus, the given conditions in the two experiments are comparable. 

The loudness level increases, AL, of the quartets relative to the loudness level of one of the tones ar quite similar in the two experiments, 

as shown in table V in which the level increases can be compared 

with each other as well as with the calculated values. Among all 

70
 
0 Reference adjusted
 

70 0 'Quartet adjusted
 

------ Calculated values
 

60 --(a l Critical bandwidth
 

50 

2	 

50 I I 11 11 I i i I , I , I , rriI 

Critical bandwidth °1 	 ­ (b)
 

Z-	 50 

70­

- 60Critical bandwidth
 0 S(b) Cemerc-enreeny ,2, 0 hr.
 

o 	 (C) 

100 i0n 1 2 103 14 

Overall tone spacing, Amf27; HI 

(a) Geometric-mean frequency, w,/2w, 500 hertz. 

(b) Geometric-mean frequency, v,/2w, 3.000 hertz. 

(c) Geometric-mean frequency, (o'/2vr, 2000 hertz. 

FsunrE 26.-Loudness level of equally intense tone quartet as function of uniform tone spacing showing effect of arithmetic-mean frequency coa/2w. 

(Data from Zwicker, Flottorp, and Stevens (ref. S8).) 

o Reference adjusted 

o Quartet adjusted 

100 	 - --0- Calculated values
 

-Critical bandwidth
 

80 

S60 

40,. 

204
 

S401 

102 	 104

101 103

Overall tone spacing, AW27; H7 

YlGnm 27.-Loudness level of equally intense tone quartet as function of uniform tone spacing showing effect of sound-pressure level. Geometric-mean 

frequency, w,/2r, 1000 hertz. (Data. from Zwicker, lFlottorp, and 'Stevens 

(ref. 38).) 

71
 
TABLE V.-COMPARISON OF FIOUR-TNE LOUDNESS DATA OF D. H. HOWES (REF. 

58) AND OF ZWICKER, FLOTTORP, AND STEVENS (REF. 38) wiTH THEORY 

Total 

Overall sound- Expert- CaleuReference 

Center frequency, 

Hz 

tone 

spacing, 

Hz 

pressure

level, 

St10 

mental 

AL, 

phons 

lated 

AL,

phons 

dB 

D. 11. Howes 58 1900 1575 56 15 5 12.9 

Zwicker, Flottorp, 38 2000 1536 57.5 13.5 12.7 

and Stevens 

D. U. fowes 68 1000 1280 76 15.2' 13.4 

Zwicker, Flottorp, 38 1000 1536 71.5 14.2 14.0 

and Stevens 

appropriate comparisons the maximum disagreement is about 2.5 

phons. 

Zwicker, Flottorp, and Stevens (ref. 38) also reported on an experiment by Scharf who found that the overall loudness level of a 

quartet of suberitically spaced (150 Hz overall spacing), equal­ intensity tones is a linear function of the loudness level of a component, that is, the two loudness levels differ by a constant. This is 

in agreement with the present theory, as shown infigure 28. 

The loudness of irregularly spaced quartets of tones was also determined telative to that of a tone approximating the center frequency 

0 Calculated values 

0- Experiment, quartet adjusted 0, Experiment, reference 

adjusted 

£ 60 

40­

0­

020 40 60 80 

Loudness level of each component, LI.( ), phons 

IPYGURE 28.-Calculated loudness level of quartet of equally intense tones with 

suberitical overall spacing as function of loudness level of one tone. Gee­

.metric-mean frequency, o,/2y, 1000 hertz; overall tone spacing, Aw/2 r, 15.0 

hertz. (Data from Zwicker, Fiottorp, and Stevens (ref. 38).) 

72
 
of the quartet. Three basic quartets were studied, namely, one centered. at 500 hertz with an overall spacing of 390 hertz, one centered 

at 1000 hertz with an overall spacing of 770 hertz, and one centered 

at 1783 hertz with an overall spacing of 1053 hertz. In all cases the 

overall sound-pressure level of the quartet was 57.5 decibels. The 

results are shown in figure 29. For the ten irregularities tested about 

the 1000-hertz center frequency, the total calculated and experimental 

ranges of loudness variability were only 1.7 and 4.6 phons, respectively. In the other tests the ranges were the same or less. The order 

of loudness of the various irregularities are listed in figure 29. The 

judged and calculated orders of the quartets centered at 1000 hertz 

are in fair agreement, considering the restricted range of loudnesses. 

Perfect agreement of order is obtained for the other two quartets. 

Over all tests the predicted loudness levels are, on the average, about 

1.5 phons different than the test levels. 

Zwicker, Flottorp, and Stevens concluded that maximum loudness 

is achieved for uniform spacing of the tones. This is in essential 

agreement with the present theory according to which, for equal 

intensity tones, the power from each tone is summed if the tones are 

suberitically spaced overall and each pair is more than 10 hertz apart. 

Otherwise, if the overall spacing is supercritical the power interaction terms in equation (40) indicate that a significant increase in 

loudness can be expected. 

Scharf's experiment (1959).-Eight to twelve listeners listening 

binaurally through earphones equated the loudness of a quartet of 

equally loud tones near the loudness threshold to that of a reference 

tone held at a fixed loudness (ref, 43). The loudness of each tone as 

a function of tone spacing was determined with constant loudness of 

the combination as a parameter. The tones were apparently spaced 

in constant geometrical ratio of frequencies, and the reference-tone 

frequency was at the geometric-mean frequency of the quartet. Quartets with mean frequencies of 500, 1000, 2000, and 3000 hertz were 

studied. As shown in figure 80, most of the calculated loudness levels 

agree with the corresponding nominal levels (horizontal lines) to 

within the expected interquartile range obtained using tone pairs 

(of. fig. 18). One cause of the discrepancy between theory and experiment is the fact that the tones were specified according to their loudness rather than ound-pressure level. The sound-pressure levels were 

determined by interpolation of the Robinson-Dadson, equal-loudness 

curves (dashed curves in fig. 8) with the external transmittance excluded. Any error in the interpolation would tend to be magnified 

because, near threshold, the loudness is determined by the difference 

of nearly equal quantities. Thus, in figure 30 the deviation from the 

nominal loudness level tends to be greatest near threshold. 

73 
Sound-pressure Predicted loud- Order of Loudness 

Quarte level 00-Hz ref- ness level,[1 Ezerl- lereterence tone), phons mental Teat S. dB
 

I 60.6 62.8 1 
 1 

Itt 	 59.8 61.7 2 2 

'C Pri ' bandvidth 59.3 61.1 3 3 

Ita) I 

0 200 400 

Tone spacing (arbitrary Sound Predicted Order of loudness 

referrnce) pressure loudness 

303 500 

Frequency. Hz 

[ 

693 

level 

(I0ff-4z

reference 

level, , 

phons 

Ekperimental Teoretical 

tone). 

S. dB 

1 ' 63.8 63.9 1 a2 

2 	 1 ' _62.2 63.9 a2 
 a3

3 	 I ' 61.9 63.2 a5 a6 

4 	 6?.2 63.9 a3 a4 

61.9 63.2 a4 
 a5


I 	 1 60.8 63.1 8 87 

7 ' 	 59.2 62.3 10 9 

8 	 60,6 63 1 9 a8 

9 	 61.0 62.2 7 10 

' 0 	 61.1 63,9 6 a1 Critical bandwidth., Critical 

l'- / .9 ban .vidth --More than one combination has the 

! I I I I I I sameloudness. 

0 100 200 300 400 50) 600 700 800 Sound- Predicted Orer of loudness 

Tone spacing (arbitrary reference) pressure loudness 

level level, Experi- Theoret200-Hz L mental ical 

615 1000 1395 reference phons Frequency, Hz 	 tone, 

S. dB 

7I ' 64.5 63 7 1 j 

61.2 63 1 2 2 

3 1 61.0 62.2 3 3 Critical bandwidth­

(c) I I I I ! 

0 200 400 600 800 3000 120 

Tone spacing (arbitrary reference) 

I 1 1 

1257 1783 2)10 

Frequency. Hz 

(a) Arithmetic-mean frequency, w.m/27r, 500 hertz. 

(b) Arithmetic-mean frequency, wa/21r, 1000 hertz. 

(c) Arithmetic-mean frequency, a',n/2v*, 783 hertz. 

FIGURE 	 29.-Loudness of irregularly spaced quartets of equal intensity tones. 

(Data from Zwieker, Flottorp, and Stevens (ref. 38).) 

74 
Norninal Nominal 

loudness loudness 

lever, level, 

phons 	 phons
 

o 48 	 0 41 

3 38 0 31 

0 28 0 21 

A 18 n 11 

Plain symbols denote quartet adjusted Plain symbols denote quartet adjusted Tailed symbols denote reference adjusted Tailed symbols denote reference adjusted 

- Nominal expected value - Nominal, expected value 6¢ CriticalI bandwidth 	 , S 60 ciiab 	 ido 60 Critical bandwidth

4o 	 40 40

Mw30 ­ 3
 

20 - 20
 

.a) I o (b) I 

-911o 2 103 ll 10 103 

Overall tone spacing, Aw/2it Hz 

Nominal Nominal 

loudness loudness 

level, level, 

phons 	 phons
 

0 40 o 43 

0 30 0 33 

C> 20 0 23 

A 10 A 13 

Plain symbols denote quartet adjusted Plain symbols denote quartet adjusted 

Tailed symbols denote reference adjusted Tailed symbols denote reference adjusted 

- Nominal expected value -Nominal expected value 

Critial bandwtdth 60 Crc b 	 Critical bandwidth. 

= 30 - 40 -	 40 T 9 

O	 40. 10 -	 2S310 Ad 	 00 

101 1? 103 10 _ 01 I02 103 104 

Overall tone spacing, AwI2g4 H7 

(a) 	 Arithmetic-mean frequency, (b) Arithmetic-mean frequency,
 

ca.z/2r, 500 hertz. wa,,/2r, 1000 hertz­

(e) 	 Arithmetic-mean frequency, (d) Arithmetic-mean frequency,
 

t.,/2vr, 2000 hertz. 2000 3/2?r, hertz.


FIGURE 80.-Calculated ]oudness level of equally loud tone quartet near threshold as function of uniform tone spacing. (Data from Scharf (ref.43).) 

Five Tones 

D. H. 1owe8' eo peniment .(1950).The following quintets were 

studied with monaural earphone listening by four trained subjects 

75
 
listening to equally intense tones or two subjects listening to equally 

loud tones (ref. 58) 

Quintet Separation, reels Wi/2r, Hz 1/2r c/2, H n Htz Hz H/2r,2

1 250 1000 1420 1900 2450 8120 

2 500 2D4 1000 1900 3120 5100 

Similar measurements were made for a quintet of equally intense 

tones centered at 1000 hertz and spaced 250 mels apart. 

The judged and calculated loudness levels for equally intense tones 

are shown in tables II and IIL The results for equally loud tones 

are presented in table IV. All results are displayed graphically in 

figure 31. For equally intense tones centered at 1900 hertz the average disagreement between test and theory is about 2.5 phons for 

Equally Equally Tone Tone frequencies. 

intense loud separation. Hz 

tones tones mels 11 (d2 J3 04 05 

2iT 27- 2ir 27 27r 

0 V 250 100o 1420 1900 2450 3120 

o D 500 394 1000 1900 3120 5100 

Listener 

120 - ' J.M. - 250 394 670 1000 1420 1900 

A DH.J 

100 -

C 80­

4, 

20
 

0 20 40 60 80 100 120

Calculated loudness level, L. phons 

FGURE 31. -ComparSon of judgel and calculated loudness levels for five tones. 

(Data from D. H. Howes (ref. 58).) 

76 
TADLE VI-DATA FROM QUIRTzsOH'S EXPERIMENTS (1EF. 60) 

(a) Noise sources and their loudness levels 

Subjective! 

loudness 

Noise Identification of sound or sound source level, 

phons 

1 Murmuring, after passing through a wall. 74.5 

2 Busing truck in stand. 98 

3 Ventilator noise at exit of a 5-rn long channel. 88 

4 Murmuring (spectrum not changed). 60.5 

1 Large compressed air hammer in room. 103 

6 Murmuring, after passing through wall. 53.5 

7 Passing freight train at 12 m distance. 91.5 

8 Music (spectrum not changed). 60 

0 Lohmann auxiliary motorcycle in stand, measured 7 m 

to right rear. 02.5 

10 Aircraft motor noise inside test stand (level decreased 

by about 44 dB). 101 

11 1. Electrical machine. 105 

12 Four-cylinder motorcycle (BMW) in stand, measured at 

5 m distance in open, nonbuiltup area. 845 

13 Music, after passing through wall, 53 

14 Portable drill, drilling aluminum (closed room). 82 

15 Four-cylinder motorcycle (BMW) (see noise 12). 103 

16 Noises, after passing through wall. 49 

17 Noise from milling macbine, taken at 2 m distance. 83 

18 2. Electrical machine. 105.5 

19 Shaking of screws in metal box. 76 

20 3. Electrical machine (motor-generator). 104 

21 Noise at exit of aircraft motor test stand, at 1.5 m 

distance toward the side 92 

22 Impulses with noise (impulse rate 30 s-t). 78 

23 .Eight tone mixture. 80.5 

24 Noise, with uniformly decreasing spectrum. 61 

25 White noise (up to 10 kHz). 74.5 

26 Dropping of water on roofing felt with stone underlayment. 79.Z 

27 Screw machine with felt pad on wooden table. 82 a28 Hissing of compressed air escaping from nozzle. 82 

29 1. Automobile horn (fanfare type). 86 

30 Impulse noise (pulse rate 2250 s'). 69 

31 Hammer blows on iron plate. 73.5 

32 Tone mixture of 20 harmonic components with statistical 

phase relations to each other. 71 

33 2. Automobile horn. 86 

34 Noise, with steadily increasing spectrum. 66.5 

35 White noise, after passing through wall. 62.5 

'36 Buzzing noise. 87 

'37 .Electric bell. 87.5 

-Noises * 28, 36, and 37 are original sounds (not produced by loudspeakers). 

77
 
TABLE VI.-Continued. 

(b) Octave-band sound-pressure levels of 87 noises 

Geometricmean fre- Lower cut- Upper cutoff 

quency, off frequency, frequency, 

wz/2r, H Hz 

Hz 

71 50 100 

141 10 200 

283 200 400 

566 400 800
 

1131 800 1 600 

2262 1600 3200 

4525 3200 6 400 

9051 6400 12800 

Total 

sound- Geometric-mean frequency, w,/27, Hz 

Noise Pressure 7 4 8 6 11 26 55 95 

8 6 111 22 55 01
 level, 71 7 

dB3 Sound-pressure level, S, dB (in octave) 

1 75. 55 71,5 70.5 67 68.5 50 <27 

2 98.5 97 87.5 85.5 86.5 86.5 83.5 71 54 

3 88 70 69 77,5 77 74 74 6445 50 

A 55 81 43 49 52 45 40 26 

5 97.5 87.5 89.5 91 90 87 87.8 84 81.8 

6 53 33 49.5 48.5 45 86.5 28 <15 

7 86.5 82.5 81.5 75 73.5 74 74r5 65.5 53 

8 53.5 28.5 88 48 51.5 8486.5 21 

9 83 67 70.5 70.5 73 75.5 79.5 73 63 5 

10 925 81.5 85 81 84 88.5 845 82.5 77 

11 95 69 78 85.5 90.5 89.5 88 86 80.5 

12 77 73 71.5 71.5 67.5 68.5 55 49.5 44 

13 49 29.5 39.5 45.5 45 37.5 26 <16 

14 69.5 50.5 43.5 46 55.5 67 63.5 615 57.5 

15 98 87.5 95.5 94 86 82 71 71 69 

16 

ft? 

48.5 

70 

37 

50 

45,5 

48.5 

45 

59.5 

36,3 

01.5 

28.5 

64.5 

26 

66 

<18 

53.5 455 

18 95 75.5 79 90 92 86.5 87.5 79 74.5 

19 61.5 43.5 41 37 56.5 55 56 50.5 49.5 

20 94 83 90 89.5 865 84 75.5 79.5 73. 

21 94 94 78 61 58 56.5 57 58 56.5 

22 62 47.8 54.5 51 51.5 53 55.5 55.2 47 

23 64 56 53.5 55 55 55.5 54 54 58 

24 48.5 29 41 445 48 41 38.5 29 17 

25 57 37.5 40 44.5 46.5 47.5 50.5 51.5 51.5 

26 62.5 53 50.5 46 49 51 56 58 51 

27 65.5 60 55.5 52.5 51 59.5 5T 55 58 

28 75 35 34.5 36 39.3 41 52 66 74 

29 70.5 ---- 48 56.8 68.2 67.3 50.8 40 

78
 
TnrL VI.-Oontlnued. 

(b) Continued 

Total 

sound- Geometric-mean frequency, ,/2zr, EIz 

Noise lrevse 8 MiI14 560 1131 2262 9051

dB Sound-pressure level, B, dB (in octave) 

30 55.5 532 50 5 40 

81 54.5 41.5 44.5 34.5 40.5 48.5 49.5 49 40 

32 58 ---- 89 41.5 43.5 47.5 9.5 33 --­

38 70 ---- 53.5 46.5 47.5 70 58.5 48.5 

84 52.5 25 31 29.5 30.5 35 41 45.5 50 

85 47 41 41 40 38 34.5 86 36 34 

36 66.5 32.5 40 48 51 64 56 49 58. 

87 75.5 32.5 32.5 40.5 40 41 53 60 75 

(c) Sound-pressure levels of each component of noises 23, 29, 30, 32, and 38 

Number of Sound- Number of Soundthe Frequency, pressure the Frequency, pressure 

component Hz level, dB component lz level, dB 

Noise 23 18 7880 28 

71 56 19 7790 31 1 


2 142 58.5 20 8200 33 

3 288 55 21 8610 31 

4 560 15 22 9020 29 

5 1132 65.5 Noise 30 

6 2264 54 

7 4528 54 1 2250 53.2 

a ooso 2 4500 50.5 

3 6 750 45 

Noise 29 4 9000 39.5 

1 410 56.5 5 11250 24 

2 820 64 Noise 32 

8 1280 08 

4 1640 66 1 127 39 

5 2050 60 2 254 83 

6 2460 55 3 881 39.5 

7 2870 44 4 508 38 

8 3280 45 5 635 88,5 

9 3 690 38 6 762 40.5 

10 4100 O4 7 889 89 

11 4510 43 81 1016 89 

12 4920 42 9 1143 40.5 

18 5830 40 10 1270 40.5 

14 5740 26 11 1397 40 

15 6150 84 12 1524 40 

18 6560 81 13 10651 39 

17 6970 88 14 1778 89.5 

79 
TABLE Vt--Concluded. 

(c) Concluded 

Number of Sound- Number of Soundthe Frequency, pressure the Frequency, pressure 

component HZ level, dB component Hz level, dB 

15 1.905 39.5 11 3520 56 

16 2032 38 12 3840 43 

17 2159 39.5 13 4160 41 

at8 2286 87.5 14 4480 84 

19 2413 385 15 4800 33 

20 2540 38 16 5120 ----

Noise 33 

17 

18 

5440 

5 760 

40 

35 

1 820 53 19 6080 36 

2 640 46.5 20 6400 38 

3 960 42 21 6720 ---­

4 1280 31 22 7 040 40 

5 1600 44 23 7360 48.5 

6 1920 50 24 7 680 44.5 

7 2240 56 25 8000 39 

8 2 560 69 26 8320 34 

9 2880 65 27 8640 38 

10 3200 52 28 8960 A 3 

250-mei spacing and less than 4 phons for 500-mel spacing. For 

equally intense tones spaced 250 mels apart and centered at 1000 

hertz the difference between theory and experiment is about 3 phons 

for listener J. My.and is negligible for listener D. H. For equally loud 

tones the average disagreement is about 3 phons for 250-mel spacing 

land 4 phons for 500-mel spacing. 

Zwicker, Flottorp, and Steen8' experiment (1957).-Zwioker, 

Flottorp, and Stevens reported on data obtained by Scharf on the 

loudness levels of quintets of (assumed) uniformly spaced, equally 

intense tones as a function of the level of one component for earphone listening (ref. 98). In one case the overall spacing was 5000 

hertz with a geometric-mean frequency of 1800 hertz, whereas the 

other case involved an overall spacing of 1800 hertz with a geometricmean frequency of -50 hertz. The judged and calculated results 

are compared in figure 32. Except, at the lowest levels (<20 phons), 

the maximum absolute difference between the calculated and judged 

loudness levels of the quintets is less than 9 phons in the former case 

and 6 phons in the latter case. 

Quietzsch's experiment (1955).-At least 20 listeners with normal 

hearing compared the loudness of a tone quintet with that of a 

1-kilohertz tone under conditions of free-field, binaural listening in 

an anechoic chamber (ref. 60). The unequally intense tones consisted 

80
 
---- Calculated values 

0 Experiment, quintet adjusted

EL Experiment, reference adjusted 

120 

~100 

C-k 

__ , 80

400 

20 030 

1! I I I () i 

20 

l . 40 60 80 00 

0 20 40 60 80 0 0 

Loudness level of each component, L(w,) , phons 

(a) 	Geometric-mean frequency, w./2r, (b) Geometric-mean frequency, w./2r, 

1800 hertz; overall tone spacing, 1250 hertz; overall tone spacing, 

Aco/2 1r, 5000 hertz. A/,/2,r, 11800 hert, 

FIGURE 32.-Calculated loudness level of equally intense tone quintets with 

supercritical overall spacing as function of loudness level of one tone. (Data 

from Zwicker, Flottorp, and Stevens (ref. 38).) 

Sound Num- Funda100 -num- berof mental 

bet tones tI 

HZ 

90 - 29 22 410 

33be2 to23 	 s 71 

3220 127 

0 

52N 

/3 0	 0 5 0 25 

/ 0 From measurements of soundpressure level of each tone 

E From measurements of sound­ 50 pressure levels in octave 

bands 

40 I 1 I I 

40 50 60 70 80 90 

Calculated loudness level, L phons 

FIGURE 33.-Judged loudness level of multiple tones as function of calculated 

loudness level. (Data from Quietzsch (ref. 60).) 

81
 

100 
of an impulsive sound with a fundamental pulse rate of 2250 per 

second plus its first four harmonics. The frequencies and corresponding sound-pressure levels are listed in table VI(c). Actually, the 

sound-pressure levels were measured in two ways, namely by measuring the sound-pressure level of each tone and also by measuring the 

sound-pressure level of the quintet in octave bands. Of course, the 

former method is more appropriate, but the latter method may be 

more common, especially if it is not realized that the spectrum is discrete. As shown in figure 83, the disagreement between the judged and 

calculated loudness levels is less than 4 phons in both cases. As in 

other cases to follow, the levels calculated from octave-band data are 

greater (by as much as 4 phons) than those calculated from narrowband data. 

Six Tones 

D. H. Howes' experiment (1950).-The loudness of the following sextets of equally intense (four trained listeners), and then 

equally loud (two trained listeners), tones centered at 1000 hertz was 

determined, as'before, by monaural earphone listening (ref. 68): 

Separation, cc, w/2r,, 2, j2 
 w.127, wo/2r, co/2ir,

Sextet reels Hz Hz Hz Hz Hz Hz
 

1 250 4(670) 1000 1420 1900 2450 3120 

1000 1420 1900 2450 3120 1(4000) 

500 610 670 1420 2450 4000 6600 

*Frequencies in parentheses were alternated. 

Judgments were also made for a sextet of equally intense tones centered at 1000 hertz and spaced 250 mels apart. 

The various experimental results are listed in tables II, III, and 

IV and shown graphically in figure 34. For the tones centered at 

I900 hertz the disagreement between the calculated and judged resalts is, for the equally intense tones, less than 4 phons for 250-mel 

separation and about 2 phons for 500-mel separation. In the case of 

equally intense tones centered at 1000 hertz, the disagreement is about 

2.5 phons for listener J. M. and less than 0.5 phon for listener D. H. 

For the equally loud -tones the disagreement is about 5 phons for 

250-mel separation and about I phons for 500-mel separation. 

Seven Tones 

D. H. Howes' experiment (1950).-Only a septet of equally intense tones spaced in 250-mel increments about a 1000-hertz center 

frequency was considered (ref. 58). Experimental results and theo82
 
Equally Equally Tone Tone frequencies, intense loud separation, Hz 

tones tones Mels 01 2 3 ' 4 5 W6

2Er 2 - 27 2 - 2T 2T
 

0 v 250 (670 	 1000 1420 1900 2450 

1420 1900 2450 3120 4000 

o 500 160 7O 1420 2450 4000 6600 

Listener 

0 J.IM.' 250 J160 394 670 1000 1420 19 A D.H.J 394 670 1000 1420 1900 2450) 

120 - Frequencies in parentheses were alternated 

100 

C0 80 10 

40 

-II 

20
 

020 40 	 60 g 100 120 

Calculatedl oudness level, L, phons 

FIGURF 34 Co0mparlson of judged and calculated loudness levels for six tones. 

(Data from D- H, Howes (ref. 58).) 

retical predictions for these tones are compared in table III and 

figure 35. For listener J. Af. the average disagreement between theory 

and experiment is about 4 phons, whereas the corresponding difference is I phon for listener D. H. 

Eight Tones 

D. H. Howes' experiment (1950).-Only an octet of tones spaced 

250 	mels apart about a 1900-hertz center frequency was considered 

(ref. 58). The specific frequencies comprising the octet were (394), 

83
 
Equally 

intense 

Listener Tone frequencies, 

Hz 

tones 01 02 w3 04 '5 66 '7 

120 - A20 IJ-MJ 

Tir 

160 

2T 2 Tr 

394 670 1W 

2 " 

1420 

29¢r 

1900 

2i2450 

D.H.J 

100 

80 ,_C, 

Z 

60 

0 

20
 

20 40 60 80 00 120 

Calculated loudness level, L phons 

PieuRn 85.-Comparison of judged and calculated loudness levels for seven 

tones. (Data from D. H. ]Howes (ref. 58).) 

6.70) 1000 1420, 1900, 2450, 3120, 4000, and (5100) hertz, in which 

the frequencies in parentheses were alternated. 

Judgment results and corresponding- ealculations are compared in 

tables II and IV and in figure 36 for octets of equally intense and 

equally loud tones, respectively. For both complexes the average 

difference between calculated and judged loudness levels is about 

4.5 phions. 

Quietzshls expe~ime (1955).-As in the five-tone test, the loudness level of an oatet of tones was judged under conditions of binaura listening in an anechoic chamber (ref, 60). The equally intense 

tones consisted of a 71-hertz fundamental plus seven harmonics 

m=20, where = 1,2,3,. .... The frequency and precise sound-pressure 

level (ave. = 51.5 dB) of each tone are listed in table VI. The tone 

levels -were measured separately. Also, the level of the octet was 

measured in octave- bands. As shown in figure 33, the judged and 

calculated (narrowv band) loudness levels differ by about 3 phons. 

For the octave band measurements the disagreement is about I phon. 

844
 
Equally Equally Tone frequencies, 

intense loud Hz 

tones tones w1 &)2 &)3 c04 &J5 w7 OJ8 

2ir 2T TT 2 7r ST 

0 . (394 670 1000 1420 1900 2450 3120 4000j 

120 670 000 1420 190 2450 3120 4000 5100] 

tFrequencies in parentheses were alternated 

100 -V 89 

o ­ 0.8 

60
 

40
 

20
 

0 20 40 60 80 100 120
 

Calculated loudness level, L,phons 

FruRn 36.-Comparison of judged and calculated loudness levels for eight 

tones. Tone separation, 250 mels. (Data from D. H. Howes (ref. 58).) 

For 16 arbitrary noises and 24 listeners, the standard deviation of 

the judged loudness levels was 2.5 phons, whereas the maximum 

deviation from the mean for any individuar value of loudness level 

was 9.5 phons. Thus, the calculated value compares favorably with 

the test results. 

Ten Tones 

Fetacherand Mumon' experment (1933.-Eleven subjects listening binaurally through earphones judged loudness levels of ten tones 

by comparison with a 1-kilohertz tone (ref. 13). Data were reported 

for equally loud tones and for unequally loud tones with even numbered tones attenuated 5 decibels, and then 10 decibels, below the odd 

85
 
numbered tones, where the lowest frequency tone is designated as 

number 1. The following tone combinations were studied: 

Overall frequency Tone spacing, 

Decuplet range, Sw 2 , Aw/2w, Hz 

Hz 

Equally loud tones 

1 50 to 500 50 

2 1400 to 1895 55 

$ 100 to 1000 100 

4 8000 to 8900 100 

5 1100 to 3170 230 

6 260 to 2600 260 

7 530 to 5300 530 

L of even-numbered tones 5 dB less than
 

L of odd-numbered tones
 

8 1.725 to 222 55 

L of even-numbered tones 10 dB less than
 

L of odd-numbered tones
 

9 1725 to 2220 55 

The judged loudness levels are shown graphically in figure 37 for 

comparison with the theoretical predictions. 

In all examples the overall frequency range extended over at least 

two critical bands, Frequently two or more tones are included in one 

critical band. The only empty critical bands occurred with decuplet 

6, for which 7o = 1, and 7 for which c= 4. 

In order to predict the loudness of suberitically spaced tones the 

phase angles of the tones should be kmown at some precise time. 

Fletcher's experiment included subaritically spaced tones; but the 

phase angles were not determined. In the absence of these data and 

because of the large number of tones comprising the complex, it is 

sufficient to assume that the phase angles were uniformly distributed 

in the interval 0 to 2r radians. In equation (33) the expectation of 

o(,w n) for suboritically spaced tones is then zero if W , but 

remains unity if Wm-- (so that #m,4'n)- Hence, equation (33) reduces to 

- -2= 10-2IQ, (WO)W (Co) - 10-, (ClAt) 

if the phase angles of the tones are uniformly distributed. 

86 
120 

100 

80 

60 E) a 

40
 

u'40­

= (a) 

-. 20 

-J 	 (b) S 0I 	 I I I 

120 

~100­

-	 S80 2

]' 

60 

40­

(C)
 

"d) (20 


0 20 40 60 80 100 120 0 20 40 60 80 10 120 

Calculated loudness level, L,phons 

(a) 	 o,/2r, 50 hertz; w,1o/2r, 00 (b) 4.'/2r, 1400 hertz; c ./2r, 1895 

hertz; &a/2,r,50 hertz, hertz; &o/2r, 55 hertz. 

(c)ws/2r, 100 hertz; c /2r, 1000 (d) wd2w, 3000 hertz; Wa/2r, 3900 

hertz; 6o/27-, 100 hertz, hertz; 8w/2r, 100 hertz. 

FuRE 37.-Comparison of judged and calculated loudness levels for ten tones. 

Tones equally loud, except in (h)- (Data from Fletcher and Munson 

(ref. 13).) 

The probable error of the experimental overall loudness levels was 

claimed to be approximately ±2 decibels. However, as is evident 

from the data in reference 13, for decuplet 8, the variation of judged 

loudness for a given selected loudness was as great as 7 decibels between different tests involving the same set of tones. For suprathreshold, overall-loudness levels the median loudness-level increase 

AL was always within 1 decibel of the average increase. The average 

absolute difference between the calculated and judged loudness levels 

varies from about 1 to 6 phons for the various complexes. The difference between calculated and judged levels does not appear to be systematic and appears to be approximately the same magnitude as that 

87
 
120 

100 

80 

60 

420 20 -	 (0 o

120/ 

100 0I0 

60 

o Llodd) -L(Weven) +5 

20 00 []o L(Wod) - L(weven +10 

0 20 40 60 80 100 120 0 20 40 60 80 100 120 

Calculated loudness level, L, phons 

(e) 	 ,/2,r, 1100 hertz; cojo/2w, 3170 (f) ws/ 2 r, 260 hertz; eo/2r, 2600 

hertz; 8$/2r, 280 hertz, hertz; 8&/2,r,260 hertz. 

(g) 	 ,i/2wr, 530 hertz; "'a/2m, 5300 (h) wi/ 2

w, 1725 hertz; wa/2r, 2220 

hertz; 8/2r, 580 hertz, hertz; S"/2-, 55 hertz. 

F GuR 87.-Concluded. 

obtained using Fletcher and Munson's empirical formula, which was 

derived to fit these specific data. 

Eleven Tones 

D. H. Howes' experiment (1950) .- Eleven tones spaced 250 mels 

apart-about a 1900-hertz center frequency were considered (ref. 58). 

The specific frequencies comprising this complex were 160, 394, 60, 

1000, 1420, 1900, 2450, 8120, 4000, 5100, and 6600 hertz. 

The experimental and calculated loudness levels are compared in 

tables II and IV and figure 38. For the complex of equally intense 

tones-the average difference between theory and experiment is about 

4 phons whereas for the equally loud tone complex the average difference is about 9 phons. 

88 
120 

Equally Equally intense loud 

tones tones 

0 V 

W, P)2 

T 2Tr 

160 594 

Tone frequencies, 

Hz 

W03W5)4w5 &)6 W7 8 W9 

2r 2 2ir 2 2i 2r 2r 

670 100 1420 1900 2450 3120 400 

I0 Oil 

27 27T 

5100 6600 

100 

' Vt 

70 

o 

08200 

20 40 60 80 100 12U
 

Calculated loudness level. 1, phons 

Flzoxm, 38.-Oomparison of judged and calculated loudness levels for eleven 

tones. Tone separation, 250 rels (Data from D:. H. Howes (ref. 58).) 

Fletcher'and Mm/son's cope 'iment (1933), The loudness of the 

follo-wing eleven-tone complexes was judged by eleven subjects binaurally through earphones (ref. 13): 

Overall frequency 

rangeTone spacing, ' 

Undeeuplet ange, o/r 

HZ 
 Hz

Equally loud tones
 

2 1000 to 2000 100
 

2 1150 to 2270 112
 

8 1120 to 4520 8340 of even-numbered ones 5 d less than L 

ofudd-numberd tones 

4 57 to 627 57 5 T420 to 4020 60 

Lrof even-numbered tones 10 dB less than 

n aof odd-nebered tones 

6 57 to 627 57 

7 3420 to 4020 60 

89 
120 

100 

80 

60 

40 

20 

0 20 40 60 80 100 120 

120 

100 

s 80
 

~60
 

*S 40 ­

00
 

~20 

0 20 40 60 80 I00 120 

120 

1000 

80 

60 

0 

40 

20 0 

(c)I 	 I I I 

0 20 40 60 80 100 120 

Calculated loudness level, L,phons 

(a) 	wi/ 2 r, 1000 hertz; wn/2, 2000 hertz; (b) Oi/2r, 1150 hertz; w /2r, 2270 

&/2r, 100 hertz hertz; 6-2r, 112 hertz. 

(c) 	 kh/2#, 1120 hertz; wn/27t, 4520 hertz; 

8o/257, 340 hertz. 

FiGUR 39.-Comparison of judged and calculated loudness levels for eleven 

equally loud tones. (Data from-Fletcher and Munson (ref. 13).) 

90 
The experimental results and theoretical predictions are compared 

graphically in figures 39 and 40. There were no empty critical bands. 

As in the case of the 10 tone data, the probable error of the experimental overall loudness levels was claimed to be approximately ±2 

decibels. For suprathreshold overall-loudness levels the median loudness-level increase AL was within 1 phon of the average increase, 

except with undecuplet 7 where the median is about 3.5 phons less 

than the average value. The divergence between the median and 

average is accounted for by one test result. Undecuplet 7 happens to 

be one for which the overall bandwidth is subcritical. For the tone 

complex in this case, and also with undecuplet 5, the loudness level 

should be given approximately by the sound-pressure level corresponding to summation of the relative powers over all tones. Where 

the agreement between theory and experiment is poor, there is an 

implied inconsistency between the data of Fletcher and Munson and 

those of Zwicker and Feldtkeller (ref. 27), which were used to establish critical bandwidths. 

For suprathreshold tests the average absolute difference between the 

calculated and judged loudness levels varies from about 2 to 9 phons. 

The difference between theoretical and experimental results does not 

appear to be systematic. 

Twenty Tones 	or More 

Quiektse'sexperinent (1955)-Under conditions previously cited, 

the loudness levels of 20, 22, and 28 tones were judged (ref. 60). The 

o L(.odd) = L(Ceven) + 5 

o = L(wodd) L(weven) + 10 120 ­

100o 	 00 

80-	 0 

60 

"gA C0

X20 

(a) 	 (b) 

0 	 20 40 60 80 100 1200 20 40 60 801 0 120 

Calculateci loudness level, L, phons 

(a) 	 wi/2r, 57 hertz; wn/27r, 627 hertz; (b) w1/2r, 3420 hertz; w.i/2r, 4020 

Sw/ 2 r, 57 hertz. hertz; l2vr, 60 'hertz. 

FiGuRE 	40.-Comparison of judged and calculated loudness levels for eleven 

tones. (Data from Fletcher and Munson (ref. 13).) 

91 
20 tones consisted of a 127-hertz fundamental and its first 19 harmonies, all at the same sound-pressure level (approx. 39 dB) and in 

random phase. The 22 tones, generated by an automobile horn, had 

a 410-hertz fundamental plus the first 21 harmonics, all at different 

sound-pressure levels listed in table VI. The 28 tones, generated by 

another type of automobile horn, consisted of a 320-hertz fundamental plus its first 27 harmonics, all at different sound-pressure 

levels listed in table VI(c). The tone sound-pressure levels were 

measured separately and in octave bands. 

For the 20-tone stimulus, the disagreement between the judged and 

calculated loudness levels shown in figure 33 is 5.5 phons. The difference for the complex measured in octave bands is about 1.5 phons. 

For the 22-tone stimulus, the corresponding differences are 1.5 phons 

and 2 phons, respectively. For the 28-tone stimulus, the disagreements are 2.5 phons for the tone measurements and 3 phons for the 

octave measurements. As before, these results compare favorably with 

the standard deviation of 2.5 phons among 16 arbitrary noises and a 

maximum deviation from the mean of 9.5 phons for any individual 

value. 

Loudness as Function of Number of Tones Contained 

Within Fixed Frequency Band 

Seharj' experiment (1959).-Scharf studied the loudness of tone 

complexes as a function of the number of tones with the overall 

bandwidth, center frequency, and overall sound pressure held constant (ref. 61). _All tones were equally intense and located in the flat 

frequency response range of the auditory system. The loudness of the 

complexes was determined by comparison with a 1500-hertz tone 

using eight subjects listening binaurally through earphones. Complexes of 2, 3, 4, and 8 uniformly spaced tones centered at 1500 hertz 

were studied for overall bandwidths of either 175 hertz (subcritical) 

or 1600 hertz (supereritical) at nominal sound-pressure levels of 25, 

50, 75, and 90 decibels. 

It was found that, for subcritical bandwidth, the sound-pressure 

levels of the complex and comparison tone were equal for equal loudness, independent of the number of tones in the complex. For supercritical bandwidth the sound-pressure level of the comparison tone 

exceeded (by up to 11 dB, depending on the selected nominal soundpressure level) that of the complex when the loudnesses were equal. 

Again, the result was independent of the number of tones in the 

complex. 

Similar results were obtained for random noise. Specifically, the 

loudness of a pair and a quartet of equally intense tones with an 

overall spacing of 3400 hertz centered at 2600 hertz essentially 

92
 

C-2­
equalled that of -a white noise with similar bandwidth and center 

frequency when the overall sound pressure of the three complexes 

was the same. 

The preceding results for sabcritical and supercritical complexes 

are consistent with equations (83) and (40) (also see category AB 

for suberitically spaced tones spread over a supereritical band), respectively, which indicate that loudness depends on input power in 

each critical band and not on the number of tones producing that 

power. The condition of constant input power in each critical band 

was satisfied in the experiment because the tones were equally intense 

and uniformly spaced over an interval of uniform frequency response, and the overall sound-pressure level was held constant. 

LOUDNESS OF SPECIFIC SOUNDS WITH 

CONTINUOUS SPECTRA (COMPARISON 

OF MODEL WITH EXPERIMENT) 

In this section the loudness of specific sounds having continuous 

spectra is calculated and compared with psychoacoustic data reported by various investigators. Sounds with continuous spectra are 

here presumed to include no tones. In fact, however, tones may be 

present but are not evident either because the bandwidth of the 

spectrum analyzer was too broad to resolve them or because their 

spectral densities were less than that of the continuous spectrum. 

Effect of Bandwidth 

Zwioker and FeldtkeZler's eoperiment (1955).-The loudness level 

of band-limited white noise was judged by comparison with a 

1-Idlohertz tone as a function of bandwidth with the overall soundpressure level held constant (ref. 27). Judgments were made by several (possibly as many as seven) listeners utilizing monaural earphone listening. Judgments were made for several sound-pressure 

levels and band geometric-mean frequencies of 500, 1000, 2000, and 

4000 hertz. Both subcritical and supereritical bandwidths were considered. The judged and calculated (using eq. (50)) loudness levels 

are compared as a function of bandwidth in figure 11. Among all the 

data the maximum increase in loudness level for supereritical bandwidth exceeded the subcritical value by about 14 phons, although the 

sound-pressure level was held constant. However, near the loudness 

threshold, judged loudness tended to be independent of the noise 

bandwidth. The agreement between the judged and calculated loudness levels is generally excellent. The difference between judged and 

calculated levels exceeds 5 phons in only one instance. Levels calculated with and without the inclusion of the threshold term show 

93
 
that the threshold term is significant only for loudness levels less 

than 40 phons and nearly always results in improved agreement with 

the judged loudness level. Levels calculated using the approximation 

equation (70) are in good agreement with those calculated from 

equation (50), except near the loudness threshold when equation (50) 

clearly yields better agreement with judgment data. However, even 

near the loudness threshold, equation (70) yields slightly better 

agreement with judgment data than does equation (50) with the 

threshold term omitted. 

Niese's experiment (1961).-In comparing the loudness of bandlimited white noise by monaural and binaural listening through earphones, loudness judgments as a function of bandwidth were recorded 

Judgment 

0 Calculation, threshold term omitted (eq. 50)) 

O Calculation, threshold term included (eq. (50)) 

" 10 	 Approximation (eq. (70)) 

o -Critical bandwidth 

0 

-

0)0 

0 

o 9 	 Ia I I I ~f il! l I I II I l II I 

100 

- Critical bandwid 

to 0 

0 Ml I 1 11r 	 b ,I 	 1L L III, I I, Il~ 1[11), 

10 02 103 104 

Cveralnbandwidth, wi2d Hz 

(a) Noise geometric-mean frequency o,/2ws; 500 hertz. 

Mb INToise geometric-mean frequency ww2v7, l000 hertz. 

FIGuRF 	41.-Loudness level of band-limited, white noise as function of bandwidth. (Data from Zwieker aud Feldtkeller (ref. 27).) 

94
 
Judgment 

o Calculation, 	threshold term omitted eq. (50)) 

o Calculation, 	threshold term included(eq. (50)) 

o Approximation (eq. (70))
 

100- Critical bandwidth
 

43 83 

"T 60 

C 

d2 40­

0 (e 

-"Critical bandwidth 

€,80­

..60 ­

a, 

E 20­

(d) I 	 1111 1 , I J I I , i, DII 

3 101 	 0# 10 10 

Overall h12g bandwidtht Hz

(c) Noise geometric-mean frequency w,/21,, 2000 hertz. 

(d) 	 Noise geometric-mean frequency w,/21r, 4000 hertz. 

FiauRE 41.-Concluded. 

for both listening modes at a fixed overall sound-pressure level of 60 

decibels (ref. 62). Disregarding the judged loudness difference 

(which was not conclusive) between the two modes, the judged and 

calculated loudness levels are compared in figure -12 as a function of 

the noise bandwidth. The rmaximum disagreement is 5 phons. 

Zwicker, Flottorp, an~d Stenm ewpeuiwnt (1957).-The experiment of Zwicker. and Feldt'keller on loudness determination as a 

function of bandwidth at constant overall sound pressure was repeated for bands of noise with geometric-mean frequencies of 40, 

1420, ad 5200 hertz for binaural earphone listening by 12 listeners 

(ref. 38). However, loudness comparisons were made -with a 

95
 
70­ ritical bandwidth 

-o-- Calculation 

, 

50-

40 1, 

0 Experiment, monaural listening 0 Experiment, binaural listening 1 1 11 r r 1 ,I IIII 1l- I , I 

101 I0? 103 104 

Overall bandwidth, Aeo2i Hz 

FIGnmE 42.-Loudness level of band-limited, white noise at constant overall 

sound pressure as function of bandwidth. Geometric-mean frequency, 0,/27r, 

11000 hertz. (Data from Niese (ref. 62).) 

440-hertz tone or bands of noise rather than with a 1-kilohertz tone. 

Thus the results shown in figure 43 indicate the variation of loudness 

level as a function of bandwidth but not necessarily the absolute 

loudness levels. Although the variation of loudness level as a function of bandwidth is qualitatively similar to that found by Zwicker 

and Feldtkeller, the ,agreement between judgments and predictions 

for supereritical bandwidths is not quite as good as it was in that 

former experiment. The disagreement between theory and experiment is greatest at the maximum bandwidth and for the highest 

and lowest sound pressures. The judged levels often tend to be less 

than the predicted values whereas in the case of Zwicker and Feldtkeller's results there is no consistent systematic difference. This difference is especially important when comparing Zwicker and Feldtkeller's results for geometric-mean frequencies of 1000 and 2000 

hertz with those of Zwicker, Flottorp, and Stevens for 1420 hertz. 

Clearly the results of these two experiments are sufficiently inconsistent, say by up to 4 or 5 phons, that the disagreement between 

theory and experiment caimot be regarded as systematic, except 

possibly at very low and very high sound pressures. 

Octave Bands 

Robinson and Whittle's experiment (1964).-The loudness of 

octave bands of noise was judged by free-field exposure to plane 

waves incident from the front in an anechoic chamber (ref. 63). The 

test data were modified by the authors to correspond to diffuse waves. 

The number of listeners varied from 20 to 30 with normal hearing. 

Judgments were made as a function of sound-pressure level for eight 

96
 
El Judgment
 

a 00 Critical bandwidth Calculation
 

, - 80 

?-~60­

40 (a) I I lIlIlri I 

Comparison 

noise, 

Aw r, 

Hz 

0 210 

0 2300 

Plain symbols denote calculation 

120- Tailed symbols denote judgment 

'3 Critical bandwidth 

o 100­

, -8080 

-7 60­

40­

o ~(b)
 20 

120 -

Critical bandwidth 0 Calculation 

E ~100 Caclto 0 Judgment 

60 

C 40 I 

101 102 103 104
 

Overall bandwidth, AwI2z Hz 

(a) Geometric-mean frequency, o,,/27, 440 hertz. 

(b) Geometric-mean frequency, w,/27T, 1420 hertz. 

(c) Geometric-mean frequency, w/2, 5200,hertz. 

Fiouns 4.-Loudness level of band-limited, white noise as fnnction of bandwidth. (Data froih Zwicker, Flottorp, and Stevens (ref. 38).) 

97 
120 

80
 

60 

04
 

20
 

0 (a)r Z1IN{I) 1 

120 

n100­

80
 

60 

40 

0 

20 

0 20 40 60 80 100 120 0 20 40 60 80 100 120 

Calculated loudness level, L,phons 

(a) Geometric-mean frequency, w,/12r, (b) Geometric-mean frequency, 

58 hertz. o,/29r, r111 hertz. 

(c) Geometric-mean frequency, w,2r, (d) Geometric-mean frequency, 

274 hertz. o./2,r, 564 hertz. 

FIYMtYE 44.-Comparison of judged and calculated loudness levels for octave 

bands of noise. (Data from Robinson and Whittle (ref. 63).) 

octave bands with geometric-mean frequencies ranging from 58 to 

8000 hertz. 

The judged and calculated loudness levels are compared in figure 

44. For the octaves centered at 58, 111, 274 (the value in the.original 

text appears to be a misprint) and 3750 hertz the agreement between 

judged and calculated levels is excellent with differences no more 

than 2 or 3 phons. For the other octaves there is a systematic disagreement, as great-as 9 phons for the octave centered at 1950 hertz. 

Among other experiments a similar disagreement does not appear. 

Thus, it seems that the disagreement may result from a systematic 

experimental, error. In particular, for the octave centered at 940 

hertz the sound-pressure level is only about 1 decibel less than that 

associated with an equally loud 940 hertz tone. Since the correspond98
 
-- 

120 ­

100 -	 0 

O0
 

0 	 0 

80 	 00 

60 	 0 

0 0 6 0 

- 0 ­

100 c20

- 20 --00 

(80 IaI I I 

0 00000000 I ! I 1 I 0 

60 

100 

h 

0 o 0 @ 100 40 @0 

00 

0I0 10 

40 0 

20 

- I ) I I I I (I Ih 

0 20 40 60 	 80 100 120 0 20 40 60 80 100 120 

Calculated loudness level, L phons 

( GeOmetri-mean frequeny, c0,/27r, (f) Geometric-mean frequency, 940 hertz. ar,/2w, 1950 hertz. 

(g) 	 Geometric-mean frequency, ,/2cr, (h) Geometric-mean frequency, 

370 hertz. t kioe 8000 hertz, 

FIOUmE 44.-Concluded. 

ng octave bandwidth is 665 hertz, whereas the critical bandwidth 

is only 160 hertz, a considerably greater difference ihwsound-pressure 

levels for equal loudness should be expected. This further reinforces 

the suspicion that the observed loudness-level differences areattributable to experimental error. 

Cremecr, FLenge, and SoltwarsZ'a emperiment (1959).-A total of 

411 mostly naive listeners with normal hearing determined the loudness of octave hands of noise by comparison with a one-third-octave 

noise centered at 1 ilohertz (ref. 64). Listening was in the free 

field in a reverberant chamber. Thus, the sounds -were diffuse. Seven 

noises, mach one octave in bandwidth, -the lowest with a lower cutoff 

frequency of 100 hertz and the highest -with an upper cutoff frequency of 12800 hertz, were studied. Loudness judgments -were made 

99
 
---------------

Reference soundpressure level, 

S. dB 

o

0' so70 

A w0 

7 50 

100 - - Reference sound 

90
 

S60 

= C 

~50 

0­

1
 

2 103 4
 0


Geometric-mean frequency, 0u1

2, Hz 

FIGUz 4 5.-Calculated loudness level of judged equally loud octave bands of noise as function of geometric-mean frequency of octave. (Data from 

Oremer, Plenge, and Schwarzl (ref. 64).) 

and sound-pressure levels recorded for one-third-octavce reference 

noise levels from 50 to 90 decibels in 10 decibel steps. The calculated loudness level of each octave noise at each indicated 

overall-sound-pressure level is shown in figure 45. For each given reference level the corresponding calculated loudness levels of the 

octave noises should be the same for each center frequency since they 

were adjusted for equal judged loudness. The calculated loudness 

levels of the one-third-octave reference sounds are also shown in 

figure 45. (Actually, these values were estimated from equivalent calculations performed in connection with Zwicker and Feldtkeller's 

data shown in fig. 41.) The authors estimated t. standard deviation 

of about 3 decibels among the loudness judgments. With the excep­ tion of the calculations for the highest octave, all calcualated loudness 

levels corresponding to a given loudness lie within a maximum total 

range of 8 phons with a maximum disagreement with the reference 

loudness level of 5 phons. All calculations for 'the highest octave yield 

100
 
a low level, as much as 15.5 phons less than the reference level. 

Whether this disagreement results from experimental or theoretical 

errors is not completely clear, although the fact that the empirical 

external transmittance level is approximately 2 decibels less than 

the measured value at 9 kilohertz (fig. 9) accounts for part of the 

discrepancy. 

0. Jahn'sexpeiment (1959).-The loudness of noises with octave 

bandwidths was equated to that of a one-third-octave band of noise 

having a geometric-mean frequency of 1 kilohertz (ref. 65). Judgments were performed in the free field in a reverberant chamber, so 

that the sound was diffuse. Judgments were made for fixed intensity 

of the one-third-octave reference noise corresponding to measured 

loudness levels of 40 and 55 phons. For each loudness level, comparisons with octave noises having geometric-mean frequencies ranging from 100 to 12 800 hertz were made. In the former case, 10 listeners were used; whereas in the latter case, six listeners were used. 

Preliminary tests indicated that, for equal loudness, the soundpressure level of the reference one-third-octave noise should be 2 

decibels less than that of the 1-kilohertz tone. Also, for equal loudness 

at 1 dlohertz, the sound-pressure level for diffuse waves was found 

to be 3 decibels less than for plane waves Irom the front. (The sign 

Nominal Sound-pressure Sound source 

loudness level, level, 

L. S(Aw/12­

phons 113 octave), 

dB 

0 55 50 Diffuse 

7 40 35 Diffuse 

0 40 38 Plane waves 

- __r -55 phons 

" 40- -------- -- -40 phons 

3 121 14 10 

Geometric-mean frequency, w.12g, Hz 

Fi(+uRE 4t6.-Caleulated loudness level of judged equally loud octave bands of 

noise as function of geometrie-mean frequency of" eetave. ({Data from G. 

Jahn (rkf.0).) 

101 
of this difference is in opposition to that reported in ref. 46 and 

shown in fig. 9 herein.) These results were used to establish the 

loudness levels of the reference noises. The measured sound-pressure levels of the octave noises judged equally loud were used to calculate 

the loudness levels of the octave noises. For a given reference level 

the calculated loudness levels for all octaves should be the same. 

However, as shown in figure 46, this is not the case, except in the 

interval 200 to 2000 hertz. G. Jahn's data cover the same frequency 

range as those of Cremer, Plenge, and Schwarzl (ref. 64), which do 

yield a relatively constant loudness level, except at 9 kilohertz. 

Hence, the two sets of data are inconsistent, except near 10 kilohertz, 

where a similar falloff in calculated loudness level is obtained. However, both of these results tend to disagree with those of Robinson 

and Whittle at 8 kilohertz, where the calculated loudness level is 

greater, not less, than the judged level. 

If Robinson and Whittle's value (-2.5 dB at 1000 Hz) of the 

external-transmittance-level difference between diffuse and plane 

waves replaces the value (3 diB) assumed by G. Jahn, then the 

nominal levels (40 and 55 phons) shown in figure 46 must be raised 

by 5.5 phons. Consequently, the judged octave levels are in much 

better agreement with the nominal values over a much greater 

frequency range. 

Stevens' experiment (1956) .- Twelve listeners equated the loudness of octave bands of noise to a variety of reference, band-limited 

noises (ref. 5). Sometimes the octave noise was adjusted, and other 

times the reference noise was adjusted. Judgments were made in a 

diffuse sound field. The fixed sound was set at a nominal loudness 

level of 72 phons. Octave cutoff frequencies ranged from less than 

75 hertz to 9600 hertz. 

As shown in figure 47 the dependence of loudness on center rerquency appears similar to that obtained by G. Jam, and the effect 

of the choice of fixed sound is negligible in comparison with the 

frequency dependence. The similarity of the two frequency dependences is probably fortuitous since the maximum predicted loudness 

occurs at 1700-hertz center frequency according to Stevens' data 

and in the interval 3 to 5 kilohertz according to G. Jahn's data., 

Stevens' data exhibit good reproducibility but poor agreement with 

predicted loudness levels. Consideration of other data reported by 

Stevens in the same report suggests that this poor agreement might be expected. For example, semi-interquartile ranges up to 7 decibels 

and standard deviations up to 9 decibels were reported. Thus, the 

scatter of the data was relatively large. Also, it is reported that, for equal loudness, the sound-pressure levels of an octave band of 

noise with geometric-mean frequency at 850 hertz (cutoff frequencies, 

600 and 1200 Hz) and a 1000-hertz tone differ by only 1 decibel. 

102
 
Reference band, 

Hz 

o 5oo to 2000 

EU 100 to 2O 

o 1200 to 2400 

2400 to 4800 

9C - 3000 to 5000 

C, ~ 0 

3 Nomirinl loudness 

~-' 70 

8 g 

- level, L= 

73phons 

2 104
 101 10 103 105
 

Geometric-mean frequency, wvI2g Hz 

FIGURE 47.-Caleulated loudness level of judged equally loud octave bands of 

noise as function of geometric-mean frequency of octave and for various 

reference noises. (Data from Stevens (ref. 5).) 

The preponderance of other data and the present theory indicate 

that the difference should be considerably greater, say 3 'to 7 decibels. 

Various Bands 

Zwicker&' ex eriment (1966) .- The loudness of three different 

noises was judged by 12 naive listeners with normal hearing by freefield listening in an anechoic chamber (ref. 66). The three sounds 

Stevens' data exhibit good reproducibility but poor agreement with 

consisted of a one-octave, two-octave, and broadband noise which 

were equated in loudness to a 1-kilohertz tone. The spectra were measured with a one-third-octave, audio-spectrum analyzer. The octaveband noise at an overall sound-pressure level of 80 decibels was centered at 250 hertz. The 2-octave-band noise at an overall soundpressure level of %0 decibels was centered at 1 kilohertz. The 

broadband noise, imposed at an overall sound-pressure level of TO 

decibels, peaked near 200 hertz, and its spectral density possessed a 

2-decibel-per-octave slope at higher frequencies. 

The calculated and predicted loudness levels are compared in figure 48. In comparison with the calculated loudness levels, the judged 

values were low, although in at least one of the examples the mean 

variation of the judgments included the calculated value. (For the 

2-octave noise the published tabulated and graphical loudness levels 

disagree. The iormer is 77 phons whereas the latter is 78.5 phons. The tabulated value is shown in fig. 48 herein.) The result that the 

judged loudness of the one-octave and two-octave noises is low relative to the calculated values is inconsistent with the corresponding 

108
 
o Broadband, S=70dB. spectrum peak, wt2-= 20O Hz 

o Two oCave. S 70 dB, wo,/2r- 1000 Hz 

0 Octave. S= =0dB. wVJ2w 250 Hz

I Mean variation 

10) 

so 

-?60 

40­

20 

0 20 40 60 30 100 

Calculated loudness level. L.phons 

FIOURE 48.-Comparison of judged and calculated loudness levels for three 

noises. (Data from E.Zwicker (ref. 66).) 

results of Zwicker and Feldtkeller and of Robinson and Whittle, 

where the judged values tend to be in good agreement with the 

predicted values, as shown in figures 41(a) and 44(0), respectively. 

However, the present results tend to agree with those of Zwicker, 

Flottorp, and Stevens (fig. 48(a)), where the corresponding judged 

values are also lower thaii the predicted values. 

Sums of Contiguous Bands 

Stevens' experiment (1956).-Naive listeners (probably 12 in 

number) equated the loudness of bands of noise (usually octaves) to 

that of a comparison noise previously calibrated subjectively against 

a 1-kilohertz tone so as to determine loudness levels (ref. 5). The 

loudness of sums of contiguous groups of these noise bands was also 

judged in order to determine how the loudness of the-sum of bands 

of noise is related explicitly to the loudness of each band, not to the 

acoustic power in each band. However, from the loudness given for 

each band the corresponding loudness level can be read from a table 

(table III) given by Stevens. Next, the sound-preasure level at the 

eardrum can be approximated from the relation S0 (<)-L()­

Z(e), which applies precisely to a tone. Finally, using the approxi104
 
mate value of S,determined for each band, the loudness of a sum of 

contiguous noise bands may be calculated and the result compared 

with the judged value. This comparison is shown in figure 49. Except for the lowest (75 to 300 Hz) and highest (1420 to 10 000 Hz) 

bands of noise, the data generally lie within 3 phons of a line displaced about 8 phons from the ideal line, for which judged and calculated levels are equal. Since the displacement is independent of 

sound pressure and frequency, the most likely cause of the shift is 

experimental calibration error. For the lowest and highest bands an 

additional frequency calibration error may have existed. The low 

frequency problem is mentioned in the text, and serious nonuniformity of the earphone response above 2 kilohertz is evident in 

figure 2 of reference 5. Finally, median level values were reported 

rather than the higher levels corresponding to the average of power 

measurements. 

Cutoff frequencies,
 

Hz
 

Jl 0
2 

2g" 27r 

0 75 9600 

Cr 3O0 4800 

0 600 2400 

AS 75 300
 

V 300 2400 

140- rL 300 4800 

A 600 4800 

V 125 8000 

,q , 250 4000 

I> 500 2000 120 <3 67080 10000 660 , C3
 Q 1420 10 0)0 0:) 4,, 

80­

e) 670 660 55-0
 a, 80 - 000 

10 

4DO
 

no 

400 

V 

20- V 

20 40 60 80- 100 120 140

calculated loudness level, L,phons 

FIGURE 49.-Judged loudness level of contiguous bands of noise as function of 

calculated loudness level. (Data from Stevens (ref. 5, table I).) 

105 
Sums of Noncontiguous Bands 

Stevens' experiment (1956).-An experiment similar to that for 

contiguous bands of noise was performed using noncontiguous bands 

(usually octaves) of noise (ref. 5, table III). The loudness of the 

bands and of the sum of noncontiguous bands was judged by groups 

of 20 or 12 listeners with earphones. 

The comparison between the judged and calculated loudness levels 

of the sum of noncontiguous bands shown in figure 50 is very similar 

to that for contiguous bands in that the resultant data generally lie 

within 8 phons of a line axially displaced about 9 phons from the 

ideal line. As before, the displacement is attributed to experimental 

calibration error. 

Cutoff (W4- frequencies, Hz 

CJ3) -Q2.ji 

O 2400 12(X 12C0 600 

o 4800 2400 1200 600 

O 4800 2400 600 300 

6 4800 2400 30 75 

V 9600 4800 600 300 

120 t> 9600 48W 3W) 75 

100­

o- 80-­

60­

40­

240 

20 

0 20 40 60 80 10' 120 

Calculated loudness level, L, phons 

FiGRE 50--Ooiparison between judged and calculated. loudness levels of 

separated bands of noise. (Data from Stevens (ref. 5, table II).) 

106 
Effect of Bandwidth Near Loudness Threshold 

Schar 's expe? ernt (1959).-In addition to studies of the loudness of equally loud tone complexes as a function of tone spacing at 

constant overall loudness near the loudness threshold, similar studies 

were made of white noise near the loudness threshold (ref. 43). 

Bands of noise with geometric-mean frequencies of 425, 1370, and 

5050 hertz were studied as a function of bandwidth. The noise centered at -25 hertz was compared with a 425-hertz tone, that centered 

at 1370 hertz with a band of noise 2340 hertz wide, and that at 5050 

hertz -witha band of noise 1730 hertz wide. The center frequencies of 

the comparison noises were not mentioned. 

In the case of the tones, the loudness level of each of the tones 

was given relative to that of the reference tone rather than a 1-kilohertz tone. If the internal transmittance level at the reference frequenoy differed significantly from that at 1 kilohertz, then the given 

data values would have to be shifted by the amount of the referencelevel difference in order to compute proper values of the loudness 

level. The case of the four-tone data centered at 3000 hertz and nearest the loudness threshold was the only one for which the correction 

might have been required. However, the correction was not included 

in the calculations. 

In the case of the white noise a correction is clearly indicated for 

the data centered at 426 hertz (correction= ±3 dB) and nearest the 

loudness threshold as well as for all data centered at 5050 hertz. The 

correction was included in the loudness-level calculations for the 

noise centered at 425 hertz. The correction could not be incorporated 

in the calculations for the noise centered at 5050 hertz because the 

comparison sound was a broadband noise, and its center frequency 

was not specified. 

The calculated and nominal (horizontal lines) loudness levels are 

compared in figure 51, wherein the dashed lines signify the interquartile ranges. Except for two points at the lowest level, the loudness levels calculated for the noise centered at 425 hertz agree with 

the nominal values to within a difference less than the interquartile 

range. For the other two noises the agreement is not as good, especially at the broader bandwidths. It is likely that this disagreement 

results because the levels used in the calculations are relative to the 

comparison noise rather than a 1-kilohertz tone. With the latter 

reference the indicated sound-pressure levels would be greater. Near 

threshold the transmittance at the outer limits of the noise bands 

would, hence, be increased. Therefore, the calculated loudness levels 

for the broader bandwidths would be increased relative to the calculated levels for the narrower bandwidths, as desired. 

107
 
Nominal loudness 

level,
 phons 

" ,, 0 /5 iCalcu.lated Calculaed Nominal expected value 

II Z5values 

g40 - ---- Interquartlle range "= Critical
 

30 -bandwidth
 

~200 

10 

I P~tnT~tI 	 !IiQ.I 

Nominal loudness 

level. phons 

o 	 372 Plain symbolsdenote noise adjusted 2 Tailed symbols denole comparison o] 20 adjusted 

o - Nominal expected value 

A 4- Interquartile range 

50- 2
 

S -Critical bandwidth40- AR I 

30 8 q0 

o 	 2 

2 10- 8A _ i, . Ou 

Nominal loudness 

level, phons 

17211Calculated 

-3 17 values Nominal expected value 

3 0 61J .... Interquartile range 40- 2
 

40" CriticaI bandwidth30f o o o o___ 

20 	 CI 00
 

012 ia0 o 20 

Bandwidth. Aw127 Hz 

(a) Comparison, 425-hertz tone; geometric-mean frequency, o,/27, 425 hertz. 

(b) 	 Comparison, noise with 2840-hertz bandwidth; geometric-mean, frequency, 

w./27, 1870 hertz. 

(e) 	 Comparison, noise with 1780-hertz bandwidth; geometric-mean frequency, 

w,/27, 5050 hertz. 

FIGRE 	5z.-alculated loudness level of band of white noise near threshold 

as function of bandwidth. (Data from Scharf ,(ref. 43).) 

White Noise 

FeldtkA ller's eperiment (1956).-In connection with a comparison 

of loudness judgment data for repetitive impulses and a 1-kilohertz 

108
 
tone, some data for white noise in the frequency range from 0 to 

20 000 hertz were also displayed (ref. 67), apparently for earphone 

listening. The sound pressure level of the white noise as a function 

of that for an equally loud, 1-kilohertz tone was shown, so that the 

calculated loudness level of the noise as a function of that of the tone 

could be compared. The resultant comparison is shown in figure 62. 

The greatest disagreement between the judged and calculated loudness levels is 8 phons, which occurs in the middle of the studied 

sound-pressure level range. 

Pollack's eperinet (1951).-The loudness of white noise in an 

interval somewhat less than 60 to 5800 hertz was judged in comparison with a 1-kilohertz tone by an unspecified number of listeners 

using earphones (ref. 68). No information on the precise shape of 

the spectrum was given other than the fact that the earphone output 

was flat within _2 decibels over the range 60 to 5700 hertz. The 

judged and calculated loudness levels are compared in figure 58 for 

stimulus sound-pressure levels ranging from 10 to 110 decibels. The 

agreement for the highest sound-pressure levels is not especially 

good, with level differences up to 9 phoans and the calculated levels 

on the high side. However, in the middle range from 50 to 80 phons 

120 ­

100 ­ 0 

0 

0 

m 60­

0)0 

00 

40 0 

200 

I I I I I I 

0 20 40 60 80 100 120

Calculated loudness level, L,phons 

FrnUHE 52.-Comparison of judged and calculated loudness levels for white 

noise in frequency interval of 0 to 20 000 hertz. (Data from . Feldtkeller 

(ref. 67).) 

109 
140 

120­

0 

0 

100-

C0 

80­ -oo60 800 

2
 

020 	 40 60 so 100 120 All 

Calculated loudness level, L phons 

F m 53.-Comparison of judged and calculated loudness levels of broadband 

(60- to 5800-H~z) white noise. (Data from Pollackc (ref. 68) .) 

the agreement is good, with level differences generally 3 phons, or 

less. 

MRier's experinwnt (194'1).-The loudness of -white noise in an 

interval from 150 to '7000 hertz was judged in comparison with 0, 

1-Inl"ohertz tone by 12 listeners wearing earphones (ref. 69). The 

spectrumn of the noise was flat to within --t+5 decibels. The judged and 

calculatedl loudness levels are compared in figu.le 51 wherein it has 

been assumed that the judged loudness levels were based on indicated sensation levels of the reference tone. Otherwise judged and 

calculated levels would differ by 30 phons, the claimed difference 

between the sound-pressure level and sensation level of the noise. 

Also, the difference is almost the same as that (8 phons) found from 

Stevens' data (see fig. 49) taken iii the same laboratory using what 

appears to be similar equipment calibrated in a similar way. The 

precise explanation for what appears to be a calibration error is not 

evident. However, the error m~iy be accounted for by the fact that 

the f~oudness threshold for the i-kilohertz tone may have been well 

above the assumed reference pressure pj1=20"micropascals. 

110
 
120­

100 0 

-C 

a 

0 

80 

6 

60 

020 40 60 80 100 120 

Calculated loudness level, L. phons 

FIGURE 5,--omlarison of judged and calculated loudness levels for white 

noise in frequency interval 150 to 7000 hertz, (Data from M~iller (ref. 69) .) 

120­

200 

100­

•00 

60 I 1 I 40­

100 

Calculated loudness level, I-, phons 00 40-

Fiau 54.-Comparison of judged and calculated loudness levels for white 020 40 60 80 00 120 noise in frequency interval 40 to 75000 120hertz. (Data from Miritn (ef. 69)) 00 

200 0 

~ 0 20 40 6a0 02 CacftdcodeslveL hn 

FiuE5.Comai ofjde 0nacltdlodeslvl o ht 

nos nfeunyitra 0t 500hrz Dt rm=iti rf 0. 

II 
B~ittai's epwryment (1939) .- The loudness of white noise in an 

interval from 40 to 15 000 hertz was judged in comparison with a 

1-kdlohertz tone by 10 naive listeners in the free field (ref. 70). The 

spectrum was flat to within ± 10 decibels, based on the response of 

the sound production system. The judged and calculated loudness 

levels are compared in figure 55 for noise sound-pressure levels ranging from 19 to 84 decibels. The disagreement between the judged 

and calculated loudness levels is 10 phons or less in all samples and 

usually less than 4 phons in the intermediate range of loudness. 

Niese and Kdiher' experiment (1958).-Thirty-five listeners 

equated the loudness of bands of noise to that of the geometric-mean 

frequency of the band (ref. 71). In the first experiment, loudne~s 

was studied as a function of bandwidth with the acoustic power increasing in proportion to the bandwidth. Free-field listening was 

hsed. The judged and calculated results are shown in figure 56 in 

which the maximum bandwidth barely exceeds the critical bandwidths in any instance. The agreement of the judgments and calculations is good. 

The second experiment was similar, except that the overall sound 

pressure was held constant. This experiment was to determine 

whether amplitude-modulated and frequency-modulated noise would 

result in a different dependence on bandwidth. The results for freefield listening are shown in figure 57 and for earphone listening in 

figure.58. No difference is evident. However, for supercritical bandwidths the loudness rise reported by others is not observed. 

g Theory Experiment Geometric-mean 

S frequency. -= 'Z: 80 w 17r 

E EHz o.0 - -­

0 4000 

60­

a tt 50 

!
 100 10 0O 103 

Bandwidth, &121, Hz 

Fiaunm 56.-Comparison of judged and calculated loudness levels as function 

of bandwidth for various mean frequencies if acoustic power is doubled 

for each doubling of bandwidth. Free-field listening. (Data from Niese and 

KXhler (ref. 71),) 

112 
Theory Experiment Geometric-mean 90 frequency, 

w,i2, Hz 

80 --	 0 600 

7,0 

10 ~s70	 0
 01 10010 

Ban'vdth, AA2n Hz 

(a) Amplitude modulated noise, 

(b) F'requency modulated noise. 

P iG~ir, 57.-Comparison of judged and calculated loudness levels as function of 

bandwidth for various mean frequencies at constant overall sound pressure. 

Pree-field listening. (Data from 'Niese and 1181hler (ref, 71).) 

80 

Theory Experiment Geometric-mean 

90- 60 frequency, 

0 600 

E _0[ ~ ~ 1000 _ 

. 70 

100 lol 	 102 

Bandwidth, AwIZ. Hz 

(a) Amplitude modulated noise. 

(b) Frequency modulated noise. 

Feurm 	 58.-Comparison of judged and calculated loudness levels as function 

of bandwidth for various mean frequencies at constant overall sound pressure. 

Earphone listening (Data from Niese and MThler (ref. 71).) 

113 

10 
Broadband Noises 

Li oke, Mittag, and Port's eoperiments (1964).-These experiments (ref. 72) and the following experiment by f. Jahn (ref. 78) 

are especially important because the loudness of the same magnetictape recordings of machinery and other noises was judged idependently by three different groups at three different facilities by 

means of free-field and earphone listening. In Berlin, 10 subjects listened through earphones. In Stuttgart, 12 subjects with normal 

hearing made judgments under free-field conditions in an anechoic 

chamber. The imposed sound-pressure level of all sounds was about 

80 decibels. The sound-pressure-level spectrum was reported for onethird-octave bandwidths. Loudness levels were determined by equating the loudness of the test sound to that of a critical band of noise 

centered at 1 kilohertz, rather-than to a 1-kilohertz tone. The latter 

two sounds were assumed to be equally loud at the same sound 

pressure. 

The judged and calculated loudness levels are compared in table 

VII and in figure 59 (a). The disagreement between the two results 

is less than 3 phons for nearly all of the examples. Except for two 

of the noises, the maximum disagreement is less than 3.5 phons. Of 

the exceptions, the spectral data for the noise designated K24 may 

be erroneous since the tabulated data do not agree with the spectrum 

shown in the original report. (The original published data were 

selected from 89 noises.) The difference shown for the octave noise 

centered at 1 kilohertz is 5.5 phons, similar to the 6 phon difference 

obtained by Robinson and Whittle for a 940-hertz center frequency 

(fig. 44(e)). However, this agreement between experiments may be 

120 ­

0 Free field (Stuttgart)
 

0 Earphones (Berlin)
 

110
 

100 

0 '- Sound number
 

CC90 K24 (inref. 72)
 

" O'-Octave
 

)0090 110 120 80 90 100 110 

Calculated loudness level. L, phons 

(a) Noises. (b) Noises containing intense tones. 

FIGURE 59.-Judged loudness level as function of calculated loudness level of 

various engine noises (Data from Liibeke, Mittag, and Port (ref. 72).) 

114 

120 
TJAnEL YIL-LouDNss oF EeiNon NOISES
 

[Data from Liibeke, Mittag, and Port (ref. 72).]
 

Loudness Loudness 

Noise Expert- Calculated level dif- Experi- Calculated level difdesig- mental L, phons ference, mental L, phons ference, 

nation L,phons AL,phons L, phons AL, phons 

Stuttgart, free field Berlin, earphones 

P7 97 95.7 1.8 92 93.1 -1.1 

44 93 9.7 -.7 92 92.3 -. 3 

MIS 92 923 -. 3 92.5 92.1 .4 

D3 96 97.8 -1.8 91.5 92.8 -1,8 

D8 91 93.7 -2.7 905 91.8 -1.8 

D14 91.5 945 -8 0 91 98.2 -2.2 

Ki 98 96.4 1.6 94 02.9 '1.1 

K1O 95.5 94.3 1.2 93.5 92.6 .9 

Kn 95 93.1 t.9 93.5 91.7 1.8 

K19 97 96.8 .2 94 93A .6 

IC21 97 96.0 1.0 ............ 

124 96 103.8 -7.8 .... 

J5 95.5 93.9 1.6 ............ 

112 96 93.9 2.1 ............ 

Octave 86 91.6 - .6 ............ 

USN 96 96.8 -. 8 96 92.8 3.2 

Absolute average AL,phons 2.1 Absolute average AL, phons 11.8 

fortuitous since in other experiments a different result was obtained. 

The average difference between theory and experiment is 2.1 phons 

for free-field listening and 1.3 phons for earphone listening. 

The loudness levels of all the sounds studied were recalculated 

using the approximation equation (TO). The levels given by this 

equation usually were higher than those given by the "exact" equation (50). The maximum difference in loudness level indicated by 

the two equations was 1 phon. 

IV. Jahn's experimen (1965).-Twenty-eight listeners, some 

trained, some untrained, all with normal hearing, judged the loudness (ref. 73) of some of the same recorded engine noises studied by 

Liibcke, Mittag, and Port (ref. 72). The judgments were made in an 

anechoic chamber by equating the loudness of the engine noise to 

that of a superoritical, 200-hertz bandwidth noise centered geometrically at 1 kilohertz. In order to obtain judged loudness levels relative 

to a 1-kilohertz tone the measured levels were reduced by 2 phons. 

The consequent comparison between the judged and calculated loudness levels is as appears in figure 60. The maximum scatter of the 

results about an average is less than 2 phons; however, the average 

disagreement between the judgments and calculations is about 6.5 

phons, possibly due to a calibration error. 

115 
110 

O Noise without intense tones 

o Noise with intense tones (P6) 

~o 

a,Sgo70 80 90 100 9l0 

Calculated loudness level, L phons 

FIGURE 60.--Judged loudness ILev1 as function of alculated loudness level of 

various engine noises. (Data from MI Jahn (ref. 73).) 

The calculated results in -figure 60 were from data obtained by 

somnd-pressure level measurements in one-third-octave bainds. Calculations using data obtained from octave-band measurements yielded 

loudness levels from about 0.5 to 2 phonls higher, which illustrates 

the effect of insufficient spectral resolution. A similar lack of resolution of data measured in one-third-octaves might account for some 

of the residual difference between theory and experiment. 

Tar'ren'sexperimnent (1978).-A total of 600 naive listeners judged 

the sound-pressure level reduction necessary to reduce the loudness 

of broadband -white noise by one-half (ref. 74). Listening was 

through earphones or in the free field in a semi-reverberant room. 

Careful efforts were made to elimninate experimental biases. Over the 

sound-pressure-level range from 45 decibels through 90 decibels, 

halving the loudness was found to correspond to a sound-pressure 

level reduction of 6 decibels. This implies that loudness is proportional to the square root of the imposed acoustic power, in complete 

agoreement -with the presently proposed theory over the frequency

about range 200 for to -which 4000 the hertz auditory for listening transmittance through is uniform, say from earphones. 

QuietzsaCs experiment (1955).luThe loudness of 32 noises with 

continuous spectra was judged by aunco0flisteners in the free field 

(rrf. 60). Some tests (2 noises) were performed in an anecho 

chamber, whereas others (10 noises) were performed in a lecture hall. 

In the anehoic chamber the judgments were made relative to a 

tidlohertz tone whereas in the lecture hall he comparisons wit the 

1-kilohertz reference tone were made indirectly by comparison'with 

one-third-octave bandwidth noise with cutoff frequencies at 900 

116
 
and 1120 hertz. Most of the sounds listed in table VI were generated 

from magnetic tape recordings using loudspeakers, but three sounds 

(noises 28, 86, and 87) were generated by the original sources. The 

spectra of the noises were measured in octave bands. The resultant 

calculated loudness levels are compared in figure 61 with the judged 

values. The maximum scatter of the data about the ideal line is 11 

phons with a systematic shift of about 3 phons, which can be 

ascribed to the fact that the spectra were measured in octave bands 

and to averaging error. 

Gorlios and "Winzer'aexperinwnt (1966) .From magnetic-tape recordings, nine listeners equated the loudness of a tapping machine 

and of three women with steel-tipped shoe heels acting on various 

concrete-backed, hard floor surfaces to that of a narrowband noise 

centered at 1000 hertz and a warble tone produced by a 955- and a 

1005-hertz tone (ref. 75). The loudness judgments were performed in 

the free field ina reverberant chamber. Because of standing waves 

Noise number 

(see table VI (a) 

for identification) 

10- 0, Continuous spectrum in lecture hall 

o Continuous spectrum) /18 I O0
o0 /
 i' Originalsource inanechoic chamber 28 Ockl1 , DImpulsive so urce 

'0 Impuls150 05 

6 2 

S2 1 

90 3771 

70 27 2 

3 25 

5270 17 8090lOU11012 

0 02 

502 164 7 

40~2 14 I I 

Frnun ~ ~ 50 

~ ~ 

0

Caclae luneslee, ~phn 61-ugdluns ee ~ 01sfnto fcluae odeslvlo 

in-5 

varousons4Dt rmQuezc rf0. 

13116 
-- 

C 0 

80­ 6C5) 

0)/ 8 /6

70 

70 80 90 100 

Calculated loudness level, L,phons 

ExGuBE 62.-Judged loudness level as function of calculated loudness level. 

(Data from Corliss and Winzer (ref. 75).) 

in the reverberant chamber the two reference sounds were compared 

in loudness, to a 1-kilohertz tone by listening through earphones in 

order to determine the loudness levels of the tapping sounds. 

The judged and calculated levels are compared in figure 62. Both 

the judged and calculated levels occur essentially in the same order 

of relative loudness and are all grouped within 3 phons of a line 

paralleling the ideal line. However, the data are shifted approximately 16 phons from the ideal line, This shift may result partly 

from the fact that the sounds were impulsive rather than continuous. 

However, it was claimed that the impulsiveness was smoothed out by 

the room reverberance. More likely sources of error are calibration 

errors, especially in, using earphones, and the use of a warble-tone 

reference, because of the influence of its unsteadiness on subjective 

judgments. 

Robinson and Bowser's experiment (Y61).-Souncl-pressure levels corresponding to equal judged loudness were determined for five 

helicopters from magnetic tape recordings listened to by a total of 

570 listeners, ten at a time, in a "moderately" reverberant room (ref. 

76). Since the loudness of the noises was not compared with that of 

110 

-,.~JQ15 ---- 0 

S n 0 Average 

A B C D E 

Noise 

FIGURE 63.-Calculated loudness level for five helicopter noises judged equally 

loud. (Data from Robinson-and Bowsher (ref. 76).) 

118 
a 1-kilohertz tone, absolute loudness levels could not be determined 

experimentally. From the measured one-third-octave, sound-pressurelevel spectra the loudness levels were computed, with the results 

shown in figure 68. The calculated average sound-pressure level was 

lt0.8 phons with a maximum spread of about 4 phons among the 

individual values. 

Copeland et al. experiment (1960).-Recordings of five sounds of 

aircraft, or simulated aircraft, were played for 1578 listeners (up to 

20 at a time) (ref. 71). The sounds were imposed in diffuse form by 

means of multiple loudspeakers in an auditorium. Listeners judged 

the louder of two of the sounds imposed at fixed sound-pressure 

levels. From these data, the sound-pressure levels at which the sounds 

would apparently be equally loud could be estimated. No comparison 

with a reference tone was made which would permit absolute loudness levels to be determined. However, from the estimated soundpressure levels and corresponding spectra measured in octaves, loudness levels could be calculated. The results are shown in figure 64. 

Neglecting sound "E," which contains introduced pure-tone coniponents, the range of calculated loudness levels of the other four 

sounds was 4 phons. Ideally this range should, of course, be zero 

phons. 

LOUDNESS OF SPECIFIC 'SOUNDS WITH 

MIXED SPECTRA 

In some psychoacoustic studies it is common to treat equally intense, discrete tones and noises as if they produced quite different 

sensation levels obeying different empirical rules. In the present 

loudness theory this differentiation does not occur. Rather, response 

to both tones and noise appears as a natural consequence of speciali120­

0 No intense tones 

-30 Intense tones 

115- 0 

0) 0 

Average 110--


E; 0 

105 I 1 1 I! 

A B C D E 

Noise 

FiauRE 64.-Calculated loudness levels of aircraft noises subjectively judged 

equally loud. (Data from Copeland, et al. (ref. 77).) 

119 
zation of a general theory which implicitly incorporates both types 

of sound. The next few experiments illustrate the success of this 

approach when applied to noises containing intense pure-tone 

components. 

Liibcke, Mittag, and Port's Experiment (1964) 

Among the noises studied, five contained strong pure-tone components (ref. 72). The judged and calculated loudness levels of these 

sounds axe compared in table VIII and in figure 59(b). The maximum difference between these two levels for both free field and earphone listening is 1.8 phons. The average difference, is 1.0 phon for 

free field listening and 1.4 phons for earphone listening. 

M. Jahn's Experiment (1965) 

Of the noises studied by Liibcke, Mlittag, and Port (ref. 72) and 

then by M. Jahn (ref. 73), only one (noise P6) contained strong 

pure-tone components. For this sound the difference between the 

judged and calculated loudness level for free field listening was 4.5 

phons, as shown in figure 60. It should be recalled, however, that 

most of this difference may result from a calibration error. 

'Quietzsch's Experiment (1955) 

Some of the noises previously discussed would appear from the 

description of the source (table VI) and the exhibited spectra 

(ref. 60) to contain strong pure-tone contributions. Among these 

noises might be noises number 31 and 87. The difference between the 

judged and calculated loudness levels for these noises is 0 and 1.5 

phons, respectively, as shown in figure 61. 

Copeland et al. Experiment (1960) 

Two of the five sounds (D and E) evaluated by free-field listening 

(ref. 77) were identical except that sound E had a 3-kilohertz square 

wave added, which, however, hardly contributed to measured octave 

levels. Nevertheless, the sound-pressure level of sound E was judged 

to be 8.3 decibels less than that of sound D for equal loudness. Thus, 

for judged equal loudness the calculated loudness level of sound D 

was .3 phons greater than that of sound E, as shown in figure 64. 

Although this difference is not great, the total spread among the five 

sounds is 7.3 phons, which is considerably greater than that in the 

better controlled laboratory experiments of Liibcke, Mittag, and Port 

and of M. Jahn. 

120 
TABLE VItf.-LOUDNESS OF ENGINE NOISES CONTAINING STRONG PURE-ToNE 

COMPONENTS
 

tData from Lfbcke, Littag, and Port (ref. 72).]
 

Loudness Loudness 

Noise Experi- Calculated level dif- Experi- Calculated level difdesig- mental L, phons terenCe, mental L, phons ference, 

nation L, phons AL, phons 4, phons AL, phons 

Stuttgart, free field Berlin, earphones 

PG 95 95.3 -0.8 91.5 92.7 -1.2 

M7 93.5 92.A .1.1 91 92.5 -1.5 

G! 94 94.2 -. 2 94 93.0 1.0 

04 93 94-6 -1.6 95 93.3 1.7 

J11 91.5 92.7 1.8 ............
 

Absolute average AL, phons 1.0 Absolute average AL, phons 1.4 

RELATION BETWEEN PRESENT THEORY 

AND A-WEIGHTED, SOUND-PRESSURE LEVEL 

For many years the A-weighted, sound-pressure level has been generally regarded as the best, though unsatisfactory, physical measure 

of the subjective intensity level of a sound (e.g., ref. 78). As various 

alternative loudness prediction procedures are proposed and then fail, 

use of the A-weighted level is revived, especially because of its simplicity and the fact that sound-level meters provide an A-weighted 

reading. The A-weighted level is supposed to be in accordance with 

inversion of equal-loudness contours near 40 phons determined by 

modifying Fletcher and Munson's equal-loudness curves (ref. 13) to 

account for a diffuse source (ref. 53, p. 22). This weighting is derived 

from measurements using earphones. It does not take into account 

the dependence of the internal filter on sound pressure. When this 

factor and standard equal-loudness curves (ref. 20) were adopted, 

improved agreement with loudness judgments resulted (ref. 79). 

Although the calculation procedure in reference 79 appeared to yield 

loudness levels in good agreement with a limited set of judgment data 

it was subsquently found to disagree with a much larger body of 

judgment data included in this report, just as A-weighted soundpressure level, especially with the cited improvements, sometimes 

represents correctly the relative loudness of certain limited groups 

of similar sounds, but nevertheless completely fails to predict correctly the absolute loudness levels of steady sounds in general. 

It has been demonstrated that the theory proposed in this report 

does predict absolute values of loudness levels in accordance with 

judged values. Moreover, the present theory can be used to show 

when the A-weighted sound-pressure level provides useful information on loudness levels. Specifically, according to the present theory 

121
 
two supratireshold sounds with the same spectrmn shape possess a 

loudness ratio equal to the square root of the ratio of their spectral 

densities. This applies for spectra with subcritical or supercritical 

bandwidths since all values of w or W in all of the loudness equations 

are multiplied by the same constant in going from one sound to the 

other. The same relation also applies to the A-weighted, effective 

sound pressure when it is assumed to represent loudness. Thus, 

neglecting the weak effect of sound pressure on transmittance, the 

A-weighted soundtpressure levels will otherwise correctly yield the 

loudness level difference between two sounds which have the same 

spectrum shape. Moreover, if, in addition, the spectrum bandwidths 

are subcritical and beats are not significant, the A-weighted, soundpressure levels equal the true loudness levels to the extent that the 

A-weighting corresponds to the true weighting of the auditory 

system. However, in general, the present theory indicates that the 

A-weighted sound-pressure level is not simply related to loudness 

level and may fail more than 15 decibels (phons) below the loudness 

level (of. fig. 41). 

DISCUSSION AND REMARKS 

It is important to recognize that the present loudness calculations 

are based on an operational theory of the auditory system rather than 

on an empirical formula selected to fit some psychoacoustic data, as is 

often the case. The empirical aspects of the theory have been minimized so as to yield a precise loudness prediction minimally influenced by empirical functions, The theory predicts loudnesses in fairly 

good agreement with a large body of, but not all, loudness judgment 

data. However, where the predictions and judgments fail to agree, the 

judgment data usually conflict with other data which do agree with 

the theory. None of the data were taken by the author. Therefore, the 

theory is not biased in an attempt to fit the author's own data, nor is 

it designed to fit a specific limited set of data, from one source. 

Rather, in an effort to gain credibility the theory has been designed 

to account most simply for the maximum amount of physiological 

and psychoacoustic data while obeying physical principles. Of course, 

inconsistency of some of the data limits this possibility. 

The generally good agreement between theory and experiment 

(level differences less than $ to 5 phons, say) implies that biases imposed by the variety of psychoacoustic judgment techniques employed by the experimentalists and temporal fluctuatiops of the auditory system are not significant enough to preclude repeatability of 

results among various experiments. This is not to say that these effects 

are insignificant. They may, in fact, account for a large part of the 

variability of the data as well as much of the disagreement between 

122
 
theory and experiment. Since the calculations yield loudness levels 

relative to a fixed reference, namely a 1-kilohertz tone, calibration 

differences among different experiments should be exposed. A calibration error should appear as a constant difference between theory 

and experiment among all tests in a given experiment. Potential large 

calibration errors are indicated in a few experiments (of. figs. 49, 50, 

51, 60, and 62). 

The method of averaging raw judgmont data may introduce a bias 

in the results. Mean-squared pressures should be averaged rather than 

the decibel equivalents. In a. typical experiment the former average 

was found to yield a level more than 3 decibels greater than the latter 

average (ref. 52). This error alone might account for the residual 

difference between predicted and judged loudness levels in some experiments. Unfortunately the raw judgment data is normally not ineluded with published results, so that the appropriate averages cannot now be computed. 

Although the periodic phenomena (for example, phase locking of 

neural discharges and periodicity pitch) incorporated in the present 

theory are generally not observed for frequencies greater than about 

5 kilohertz, at higher frequencies discharge poriodicities nevertheless 

exist. These, in fact all discharge periodicities, may be associated with 

pitch rather than frequency (ref. 22). Hence, there is apparently no 

reason to expect that the basic mechanisms for loudness production 

are any different at high stimulus frequencies. 

A more refined analysis than the present one might involve transmittance of signals at all frequencies in all channels with different 

filtering of each frequency in each channel. Unfortunately the complexity of the filtering process for multiple tone stimuli in individual 

neurons (ref. #2) indicates that this approach might require a much 

more complicated analysis than the present one, but with little, if 

any, better agreement between calculated and judged loudness because 

of the variability of loudness judgments. 

Channels with associated critical bands containing frequencies included in the imposed sound spectrum are most strongly stimulated. 

As a,.result, the broader the bandwidth of a sound with a fixed overall sound pressure, the louder the sound will tend to be, at least until 

its spectrum extends into a frequency range of reduced transmittance. 

The broader bandwidth sound is louder because more channels are 

strongly stimulated. Since the acoustic frequency range includes 24 

channels, the loudness level may be increased by 14 phons, or even 

more, simply by broadening the sound bandwidth. The larger increases would be expected if the spectrum is broadened into- a frequency range of increased transmittance. An additional increase of 

up to 3 phons may result from tonal interaction. 

123 
The most effective loudness reduction for any sound should be obtained by attenuating the sound at frequencies associated with maximum output power lWq. 

It should be possible to construct on the basis of the present theory 

a loudness meter which would perform the necessary calculations electronically and indicate the loudness, or loudness level in real time. 

It seems quite possible that the present theory may be readily applicable for predicting the loudness of impulsive sounds as well as 

steady sounds. The question of whether critical bands are significant 

for impulsive sounds as well as for steady sounds is undecided. From 

some studies it is concluded that critical bands develop as a function 

of exposure time (ref. 80) whereas from other studies it is concluded 

that critical bands are not time dependent (ref. 81). In either case 

the present theory might be valid without major modifications. 

SUMMARY OF RESULTS 

From a study of the relation between any steady sound stimulus 

and its judged intensity, or loudness, the following results and conclusions emerged: 

1. Loudness is a measure of the electric power (defined over the 

auditory integration timeo0.1 s) transmitted by the auditory nerve 

toward the central nervous system in response to an acoustic stimulus. 

2. The auditory system is a squaring device in that the electric 

power transmitted by the auditory nerve is a function of the square 

root of the input sound power. The squaring results from transformation of the acoustic waveform from mechanical to electrical form. 

3. Sound stimuli which excite the same auditory neurons interact. 

Stimuli which do not excite the same neurons do not interact. The interaction affects loudness significantly. 

4. The maximum number of audible harmonics which may interact 

by exciting the same neurons is about 5. This number is about the 

same as the observed maximum number of harmonics (5 to 7) which 

may be heard in response to a, multitone stimulus. 

5. Audible beats resulting from closely spaced tones (beats of imperfect unisons) or from tones which are nearly harmonics of one 

another (beats of mistuned consonances) are a consequence of the 

squaring process and of the finiteness of the auditory integration 

time. The finite integration time leads to time-dependent, cross-power 

contributions from the interaction of the tones. The loudness of the 

beats tends to decline as the tone separation is increased because the 

integration time approaches and then exceeds the period of the beats. 

The predicted amplitude of beats as a function of frequency difference and phase difference is in good agreement with experimental 

data. 

124 
6.The nerve fibers comprising the auditory nerve may be classified, 

according to their best, or characteristic, frequencies into transmission channels having bandwidths equal to critical bandwidths for 

loudness. Individual fiber bandwidths generally exceed associated 

critical bandwidths. Thus, any signal at a given frequency may be 

transmitted in several channels in addition to the one characteristic 

channel which includes the signal frequency within its associated 

critical band. The interaction of signal frequencies associated with 

different characteristic channels may stimulate in each transmission 

channel a cross-power contribution corresponding to secondary tones 

which increase the overall loudness of the sound. 

7. The critical bandwidth for louchess tends to equal the frequency 

of onset of the secondary tone which, in turn, equals the characteristic 

lowest response frequency of neurons with best frequencies lying in 

the critical band. The regime of secondary tones is limited to the frequency range in which neuron bandwidths clearly exceed associated 

critical bandwidths. 

8. Sounds with a vanishing spectrum in intermediate critical bands 

stimulate neurons characterizing these "empty" bands as well as 

neurons characterizing critical bands corresponding to stimulus frequencies. Stimulation of neurons associated with the empty bands 

increases the loudness. 

9. Tones which are individually subliminal might, in combination, 

stimulate a loudness response if they lie within the same critical band. 

10. Minimum loudness for a given overall sound pressure occurs 

for sounds with a spectral bandwidth equal to a critical bandwidth 

for loudness, or slightly less. 

11. Except near the loudness threshold, for a. fixed overall sound 

pressure the loudness of a sound with supercritical bandwidth tends 

to a maximum as its bandwidth is increased and then declines as the 

bandwidth is increased further. The sound is louder for broader 

bandwidth because more channels are strongly stimulated. Since the 

acoustic frequency range includes 24 channels, the loudness level may 

be increased by 14 phons, or even more, by broadening the bandwidth. 

An additional increase of up to 3 phons may result from tonal interaction. The decline for very broad bandwidth occurs because the 

sound then extends into a frequency range of reduced system 

transmittance. 

12. Using published data from a large number of experiments involving sounds with discrete spectra (tones), continuous spectra 

(noise), and mixed spectra (noise plus tones), generally good agreement between judged and calculated loudness levels was found. 

Where considerable disagreement was encountered for a given experiment, results of another similar experiment usually indicated good 

agreement between predicted and judged loudness levels. 

125 
13. Von Bik6sy's loudness judgment tests of two equally intense 

tones of the same frequency show that a 6 decibel reduction in the 

sound-pressure level of a tone corresponds to halving its loudness, in 

agreement with several other studies as well as the present theory. 

14. A change of n decibels in the overall sound-pressure level of a 

suprathreshold sound -without changing its spectrum shape results in 

a change of approximately n phons in its loudness level. 

15. The A-weighted, sound-pressure-level difference approximately 

indicates the loudness level difference for sounds with the same spectrum. The A-weighted, sound-pressure level equals the loudness level 

for subcritical bandwidth sounds (without beats) in spectral regions 

where the signal is not attenuated relative to the system transmittance 

at 1 kilohertz. However, in general, A-weighted sound-pressure level 

is not simply related to loudness level, and the two levels may differ 

considerably, say by more than 15 decibels. 

126
 
APPENDIX A 

SYMBOLS 

A 

A 

a 

B 

0 

o 


o 


d 

) 

H(w) 

9/ ( ) 

h(t) 

d (<o) 

K 

zk 

X 

SQ 

Ii 

k'n 


L 

AL 

_ 

_zor 

I 

sound-pressure amplitude 

II- J,(2rw/)] si2 (,l/c) appendix F 2ro,/fc 

complex amplitude, (A/2) e

1i1 

[-2 - .(2rw/c)sln( o) -cos( ol/c) 1 

appendix F 

sound-pressure level at loudness threshold, dB 

function given by eq. (65a) 

speed of sound in air, appendix F 

function given by eq. (65b) 

frequency response of auditory system 

frequency response of internal auditory system 

frequency response of external auditory system 

impulse response of auditory system 

VT 

frequency response of channel k to rotational 

frequency w. 

first-order Bessel function of first kind, appendix F 

dimensional transmission factor 

modified Bessel function, appendix F 

dimensionless transmission factor 

dimensional constant, 0.5X1O- 3 Pa-' 

total number of empty critical bands 

integers 

loudness level, phons 

loudness level difference, eqs. (54) and (55) 

loudness, sones 

loudness of sound spanning a critical band, sones 

length of auditory meatus 

127 
2 

N 

AT 

c~f(wn) 

cat (LomO) 

P(,) 

n7 

p 

Prot 

Q (C) 

Q(wm)),) 

q"output" 

r 

r 

sW(t) 

IV.? 

Is 

SA 

S 

sinc, 

2' 

T 

7 ( 

t 

t. 

w (con) 

v ,, 

Wq (c) 

(MM) 

wtos 

0 

a 

pl 

total number of transmission channels spanning 

sound spectrum 

total number of tone stimuli in sound 

internal transmittance level 

internal transmittance 

internal cross-transmittance 

sound-pressurc spectral density, s 

normalized, "effective" sound-pressure defined 

over auditory integration time, eq. (2) 

instantaneous sound pressure 

reference effective sound pressure, 20 sPa 

total transmittance of auditory system 

total cross-transmittance of auditory system 

of auditory system 

total number of neural discharges over all 

neurons in 0.1 s 

radius of auditory meatus, appendix IT 

normalized autocorrelation function of output in 

a transmission channel 

real part 

sound-pressure level, dB 

A-weighted sound-pressure level, dB (A) 

sensation level, dB 

sinc(4)/wC, eq. (16) 

auditory integration time, 0.1 s 

external transmittance level 

external transmittance 

external cross-transmittance 

time 

characteristic time denoting a particular cycle 

of oscillation within auditory integration 

period 

9k4" - -0.k

dimensionless power of tone 

dimensionless cross-power of two tones 

"output" power spectral density, s 

dimensionless "output" power of tone 

dimensionless total power of tones 

complex quantity 

function given by eq. (63) 

function given by eq. (6) 

198 
PMn 

7, ,ur 

A,8 

t mn 

?I 

6 grn 

' 

K' 

A 

r 

*tai-c'(f/a/) 

C2 (,a) 2(C) 

2cr 

Qr(ogt) 

f29 

/2_r 

Wb/21r 

w,/2,r 

ro/2w 

&),124r 

oam/ 2 ,r 

W/2S 

Subscripts: 

OP 

e 


i 

7integer 

intertone coefficient for supercritically spaced 

tones 

positive integers 

difference 

Dirac delta function 

Kronxecker delta 

positive number, 0-----,v< 1 

tan-[/V .,) /,:)Vo,nl - constant in eq. (4) for londness, 10 I 

constant in fundamental eq. (3) for loudness 

positive number 

vT/2 

delay time 

phase angle 

dimensionless acoustic power in subcritical frequency band (Aw/2r). centered at frequency 

dimensionless acoustic power in critical band 

dimensionless acoustic cross-power between critical bands, eq. (51) 

dimensionless "output" power in subcritical frequency band 

rotational frequency 

lower cutoff frequency of ideal, spectrum-analyzer filter having subcritical bandwidth 

upper cutoff frequency of ideal, spectrum-analyzer filter having subcritical bandwidth 

center frequency of internal auditory system 

lower cutoff frequency of internal auditory 

system 

upper cutoff frequency of internal auditory 

system 

arithmetic-mean frequency 

geometric-mean frequency 

rectangular function 

in critical bandwidth 

at eardrum 

imaginary part 

denoting channel or tone in channel 

129 
L--I refers- to condition when loudness equals one 

sone 

min integer denoting tone 

maxs maximum 

r real part 

t loudness threshold 

a integer denoting subcritical frequency band 

J integer denoting critical band 

v,p integer denoting tone 

Superscript: 

complex conjugate 

180
 
APPENDIX B 

SPECTRAL REPRESENTATION OF OUTPUT 

POWER-SPECTRAL-DENSITY W 

The output power-spectral-density 

(wtT) = (l/Tp2 (Wq , )S1 e-wrr f g(t')q('tr)dt' (12) 

may be expressed as a function of frequency rather than of time by 

applying the time convolution theorem (ref. 82) to the convolution of 

q given by equation (5). Thus, 

= (l/2r) fW HW(w')P (t") e "&" 

= (1/1r) fd *(i")P*(w")e-f°"tdof 

where 

HZ -H*(0) and P(-) =P*(W) 

because 4(t) and p(t) are real. If the theorem is also applied to 

q(t'+r),then 

'Wq (&~tT)= (1/Tp"~ ) f0 tWrdr f' dt'j (1/27r) 

d ft, '
 7* (,,P)P*(of') e-d''a (1/2w) 

ft H(W)P) eW'+tI)dI11 

- (1/%7rTp2, )H(w)P(o) f' fit- e-i(W"-0,>VdV, 

because 

However, 

f t e( ,,"w) dtt _ sin[(f"- ( (6) W)._T//) )T/2]

so that 

74(wt,T) = (1/27rp0 )H(w)P(w) 

1sin[T12


f- H) P*f19 (j, 
Assume that H(') and P(WP) are constant in the range of 1' for 

which the integrand in this equation contributes appreciably to the 

integral. This range is approximately 

21"-wjT/2<9 

or 

2j1o'/w9v</,T,2 hertz 

since 

sin(ire) /ressinc s 

is small unless ICI <i.Let v l" -m. Then, 

jw,(tT) = (l/ 2 7rp IH(w) 121P(.)l2 fin 2)dv 2e) sine(vT12r)-'('- 1 

= (l/T.2)l1H(, jIIP(w)I sine $e-ff ' IT)

l - /2] d$ 

where 

Finally, by virtue of the integration shown in reference 83, 

-Wq(o,t,T) E '-') I 2P(w) 12/Tp2, 

where 

if ifI '-l< or Oczt<T El 

isthe rectangular function.
 

132
 
APPENDIX C 

LOCATION OF CRITICAL BANDS 

Since the loci of the mean frequencies and cutoff frequencies of the 

channels, or critical bands, depend on the sound spectrum, a method 

for choosing these loci must be established. Probably the most 

realistic procedure is to locate the geometric-mean frequency of a 

critical band so that the maximum possible spectral power falls within one critical band., Then, the loci of all other critical bands can be 

determined relative to this reference band using bandwidths found 

from figure 3, or 7. A simpler alternative is to locate the geometricmean frequency of the reference band at the frequency of the maximum spectral density of the sound. An even simpler alternative is to 

let the upper cutoff frequency of the uppermost critical band coincide with the upper cutoff frequency of the measured spectrum and 

then to determine the loci of all other critical bands relative to this 

reference band. This last procedure avoids the problem of choosing 

the reference band when the maximum spectral density occurs at 

several nonadjacent frequencies and was used, where appropriate, in 

all calculations reported herein. 

133
 
APPENDIX D 

CONCERNING BEATS OF MISTUNED 

CONSONANCES
 

Beats of mistuned consonances occur if 

where y and c are positive integers, at least one of which is not unity. 

Let 

where Aw k< <Ok. Then, 

cqW{0 k Ey) -l1wk+ (Am/)(M) 

Consider the integral of the cosine in equation (37). When equation (D) is introduced in this integral, it becomes 

)0a]I'(t+- ) 

=(lIT) f-V, cos +14p'+ jcos

where, for brevity, 

Over each cycle of (cly- 1) wk only one-half cycle includes discharges. 

Also, since (./.-1)wk> >A; /y, the terms involving Awmr are approximately constant over the cycle. Therefore, over each cycle of 

(cly- 1)wk,with period At, the contribution to the preceding integral 

becomes 

+ Co sn [ 1) + + os( A 

=(1/2) sine -R WkA/r]COS - t. 
where 

and 4 is a characteristic time denoting the particular cycle. Since 

27r
 

the integrated result becomes 

(1/r)Cos(
 

This is valid for each cycle of (ety -1) w7,. To be consistent with other 

terms in equation (37), which are treated as full-wave rectified, this 

result should be multiplied by 2. The result is, then, 

(2/7r os ( AW~mtrc 

In time T there are v+ 77 cycles of (cly- 1) w where a is an integer, 

0v<1, and a> >7. Therefore, the summation of the preceding expression over the entire intervalt - T-"t'< t yields 

- 2 'o AWX~ ' t 

t(O_+,)Atc 4 0 AOS'\ 

~,(+2),to f$T-c(t, - dt 

=(2/w)sine At,2 T cos[ Aam74 (t-T/2)]
 

where the sum may be approximated by the integral because
 

At,<<T. 
APPENDIX E 

PLANCHEREL'S THEOREM FOR LIMITED 

INTEGRATION TIME 

Define the runming-interval Fourier transform 

P(,5tT) =f:_ r(tP)e-t CU' 

- P(w+,,,T) ='+~~ fl" =,tf + 

so that 

f- IP(witT) 12dW= ft~pt)di~pt)t f ' ewW't')dw 

=27 ft pttf*(t) 8(t' t)dt 

=2wsz1_2p2 (') de 

because p (t) is real. Therefore, by virtue of equation (43), 

(1/2-)f- Wf(4.) dTV(l/Wp,1 )dws: (t') dt' (46) 

where 

136
 
APPENDIX F 

EXTERNAL TRANSMITTANCE FUNCTION 

The external ear resembles a closed, flanged tube. The transmittance function for a closed tube is lmown (ref. 84). Unfortunately, 

the theoretical result does not agree with experiment unless the 

former is modified. If the theoretical formula is squared, then 

9(-)= (A2 + 2) " (F ) 

where 

Qrw/c j in2 (wi/c) 

(2rc/o) J (2rw/c)sin(o2/o) -Cos (l/c)] 

J,is the first order Bessel function of the first kind,and K1 is a modified Bessel function defined by 

2 j n+1

where 2(2(.)n+ [(2t-i)I 1]32 

where 

(2n-1) 1!= (9ot-1) (2n-3)...5.3.1 

Also, r is the radius, Zis the length of the tube (auditory meatus), 

and c is the speed of sound in air. The empirical and experimental 

functions T(0 ) are compared in figure 9. The calculated results were 

obtained by assuming that 2r/o=2.035x10 - 5 second, and Z/o=4.542 

X10- 5 second. 

For free-field listening the real part of the external cross-transmittance , occurs in. those terms of the loudness equation 

which involve spectrum interactions. To evaluate L(w,*o), with A 

and B defined as before, let 

A-At,, B--B ifw 

A-A,,B-B,if'=,
 

137 
Let 

cq,:tt)(A -Ov)-

Then, 

£lr~w~wn)AmA,~+Bm~n

(A2 +B2) (A2 +B2) 

which reduces to equation (Fl) if m=n. 

138
 
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112 
85. RITSMA, B. J.: FdZStence Region of the Tonal Residue. 1. J. Acoust. Soc. 

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van Nostrand Co.. Inc., 1953, p. 271. 

143
 
I 	 Reran I Gornmoni Acession No 3 Rfle.ooitt% cutlog No
 

NASA RP -1001 I
 .No 	 e 

4- Title andSubwItile R 'elpot o'"t
 

OVERALL LOUDNESS OF STEADY SOUNDS ACCORDING October 1977
 

S Pealon O'wi...un CI.e
 TO THEORY AND EXPERIMENT 

7. Aott tll 	 S Perfornng Or.a'.,nn on Rport No 

Walton L. Howes 	 E-8342 t0 Week Unt No 

9 	 Per1nnun Kr~~imo ozead Addres

National Aeronautics and Space Adinistrati 0 n II Co.. o.Crane No 

Lewis Research Center 

Cleveland, Ohio 44135 13 Tp of Rfert lAd Ped CoverS 

12 Snon'srl A oCy Nme and AMiee" 	 Reference Pubication 

National Aeronautics and Space Administration 24 SoO loAeuo Cole
 

Washington, D C. 2054.0
 

I15&oipleetway Nots 

IS Alsac 

A mathematical theory for calculating the loudness of any steady sound from informabon on 

Its spectrum is constructed from physical principles and psychological and physiologzcal information on mammalian auditory systems. The theory involves filtering, channeling, 

squaring, half-wave rqctification, and time averaging of the signal. The theory accounts 

for critical bands for loudness, audbihty of sounds consisting of subliminal components, 

audible beats, periodicity pitch, and pitch of the residue These and other psychoacoustic 

phenomena are explained in terms of electrical activity in the peripheral nervous system. 

Simple approxunations for loudness are derived from the more exact formulas. Loudness 

predictions are compared with a wide varlety of published loudness jadgment data with considerable success. 

N2. Key Wor., (tSiotid by Autiio )lI 	 ID Oittri1tiion Suttespent 

Loudness loudness level, noise threshold, Unclassified - unlimited
 

threshold (perception), subliminal stmuli, STAR Category 53
 

tone, Psychoacousfics , auditory perception,
 

auditory stimuli, hea ing, sound perception;
 

Psychophysiology. ear, auditory nerve, bloelectricity, neurons, central nervous system,
 

brain, peripheral nervous system, electrophysiology, Acoustics: audiofrequeneies,
 noise (sound), noise Intensity, sound intensity,
 

noise measurement; sound pressure, Fourier
 

analysis Fourier transformation, tfnsfer r
 

functions bandwidth, beat frequencieS, hlrmonies, ourler series, power spectra, autocorrelation, broadband, frequency dlstnbuton, frequencies, frequency ranges, periodic
 

functions.
 

39 S'o C.,af (of lhi- reporl 20 secerty Classf(of T, page) 21 No of Pge free w 

Unclassified 	 Unclassified I 

FMrsale by the Nahorsj Techncil Inia fmal SejYce, Spinlheld Vaginia 22161

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