Sound Loudness According to New Studies

A. V. Rabinovich, Moscow

Contents: I. Judgment of loudness. II. Minimal perceptible change in loudness. III. Fatigability of hearing. IV. Loudness level. V. Judgment of changes in loudness. VI. Loudness of complex sound. VII. Theory of loudness. VIII. Loudness of very brief sounds. IX. Measurement of loudness.

The solution of a number of problems in sound engineering—such as improving sound in radio and sound cinema, the design of concert halls, theaters, and auditoriums, and eliminating the harmful influence of noises on the human organism—is closely connected with the study of the loudness of sound. Wherever auditory perception is involved, it is necessary to take into account the complex dependence that exists between the magnitude of the sensation and the corresponding physical factors.

The question of the loudness of sound has recently received much attention in the specialized foreign literature. In a number of works the meaning of the word “loudness” is clarified, and the dependence of loudness on physical and physiological factors is established.

Unfortunately, translations of these works have not been published in the Soviet literature, with very rare exceptions. There are also no review articles. I therefore considered it timely to present, in a brief article, the most interesting of the new experimental and theoretical works on loudness, systematizing them as far as possible and critically comparing them.

I. Judgment of Loudness

By the loudness of a sound we understand the degree or magnitude of the auditory sensation. A judgment of loudness is quantitative in character. It may appear: 1) as a judgment of the equality or inequality of loudness, and 2) as a judgment of a change in loudness by some amount.

The first kind of judgment is simpler than the second. Hearing, for example, two different sounds, we shall always say either that one is louder than the other, or that they are equal in loudness. The second kind of judgment consists in establishing how much (or how many times) one sound is louder than another. Such a judgment may also be expressed with respect to a single sound that is changing in strength.

In ordinary everyday conditions this judgment is reduced to very approximate determinations: “considerably louder,” “a little louder,” and so on. But in music, for example, changes in loudness are more or less specified: there exists a whole series of gradations having special designations—very soft (pianissimo, pp), soft (piano, p), moderately loud (mezzoforte, mf), loud (forte, f), very loud (fortissimo, ff), and others. Of course, these gradations have no numerical expression, since they are terms of art, but a quantitative judgment about a change in loudness is present.

Loudness is a function not only of the intensity of sound, as is usually assumed, but also of frequency: sounds of the same intensity but of different frequency are not equal in loudness. At the same frequency, however, loudness is indeed determined only by the intensity of the sound (if one disregards the phenomenon of auditory fatigue).

The dependence of loudness on frequency and on the intensity of sound is complex. The experimental determination of this dependence is also complex, since in a person’s judgment of loudness the numerical element is absent and it is extremely difficult to establish a unit of measurement. In contemporary works, three methods have emerged for determining the indicated dependence.

The first method, which is also the first historically, consists in measuring increments of sound intensity corresponding to minimal perceptible changes in loudness (the difference threshold). However, from such measurements one cannot directly determine the dependence of loudness on the intensity and frequency of the tone; it can be established only indirectly, by making a known assumption about the summation of elementary increments of sensation (the Weber–Fechner law).

The second method consists in comparing the loudness of two tones of different frequency and establishing the equality of their loudness. This method, based on the first type of judgment about loudness, in itself likewise does not make it possible to establish a complete picture of the dependence of loudness on the frequency and intensity of the tone, determining only the difference in the character of the increase in loudness as the intensity of the sound increases at different frequencies.

Finally, the third method consists in the direct evaluation of the change in loudness that has occurred. With the aid of this method one can establish the complete dependence of the loudness of sound on both intensity and frequency. Experimentally, however, it is the most difficult and is not entirely free from contingencies of a subjective and methodological order. Therefore the results obtained should be treated with caution, and, for control, compared with results obtained by other methods.

II. Minimal Perceptible Change in Loudness

The first method has already been sufficiently covered in the specialized literature. Therefore we shall touch on it only briefly, insofar as this is necessary-

necessary, in the interests of the further exposition, to dwell mainly on the most recent critical and experimental works on this question.

The Weber–Fechner psychophysical law, common to all sense organs in the form given to it by Fechner, states: sensation is proportional to the logarithm of the stimulus. As applied to auditory sensations this law may be formulated as follows: the loudness of a sound is proportional to the logarithm of the relative intensity of the sound above the threshold of audibility.

\[ L = K \log \frac{I}{i}, \tag{1} \]

where \(L\) is the loudness of the sound, \(I\) is the intensity of the sound, \(i\) is the intensity of the sound at the threshold of audibility for the given frequency, and \(K\) is the coefficient of proportionality.

Since in our conception there is no “natural” unit of loudness of sound,* the quantity \(L\) can be determined in conventional units. Consequently, the coefficient \(K\) is an arbitrary quantity. Using common logarithms and taking, for convenience, \(K = 10\), we obtain:

\[ L = 10 \log \frac{I}{i}. \tag{2} \]

In this formula \(L\) will be expressed in units called decibels.**

As was already said above, the experiments on the basis of which the Weber–Fechner law was derived consisted in determining the minimal distinguishable increments of sensation. In this case it was established that the ratio of the increment of sound intensity \(\Delta I\), necessary to obtain the indicated increment of sensation, to the initial sound intensity \(I\) is a constant quantity for every initial sound intensity:

\[ \Delta L = \frac{\Delta I}{I} = \mathrm{const}. \tag{3} \]

On the basis of this differential dependence Fechner obtained the dependence of the magnitude of sensation on the stimulus by integrating equation (3), which leads to the law (1) indicated above.

The Weber–Fechner law was tested for hearing by Riesz\(^1\) and Knudsen\(^2\). These two investigators, using different methods, obtained somewhat different results. In one respect, however, the results unquestionably agree: the quantity \(\frac{\Delta I}{I}\) is not constant and depends on the initial sound intensity \(I\), namely: with an increase in intensity

* As an example, let us point out that such a natural unit of pitch is the octave.

** The unit bel and its \(1/10\) part—the decibel—were introduced in 1928 on the initiative of the American school of physicists and engineers working in the Bell laboratories.

the sound \(\frac{\Delta I}{I}\) decreases to a certain limit and only above this limit becomes constant.

In Figs. 1 and 2 are given the curves obtained by the investigators mentioned. On the ordinate axis in both figures is plotted the minimum perceptible relative increment of sound intensity \(\frac{\Delta I}{I}\); on the abscissa axis in Fig. 2—the sound intensity above the threshold of audibility in decibels (sensation level), and in Fig. 1—the frequency of the tone under investigation.

Fig. 1.

The curves of Fig. 1 correspond to sensation levels of 5, 10, 20, 30, 40, and 60 decibels.

Lazarev\(^3\) gives a correction to the Weber–Fechner law, based on the ionic theory of excitation proposed by him and confirmed by experimental material:

\[ \Delta L=\frac{\Delta I}{I+\beta^3}=\mathrm{const}, \tag{4} \]

where \(\beta\) is a small quantity characterizing the proper sound of the cochlea, caused by thermal irritation. It is easy to show that the ratio \(\frac{\Delta I}{I}\) decreases as \(I\) increases, tending at large values of \(I\) to a constant limit. Near the threshold, however, the ratio \(\frac{\Delta I}{I}\) increases. Bekesy\(^4\) explains the observed phenomenon as follows.

A certain stimulus, exciting a number of nerve cells, produces a definite sensation. A stronger stimulus brings new nerve cells into action, and only when brought into action

although at least one new nerve cell is involved, loudness increases. To excite a single nerve cell there is always required one and the same relative increment of the stimulus.

However, a person cannot always distinguish the addition of one excited nerve cell: very often, under the influence of subjective fluctuations of loudness or under the influence of extraneous interfering circumstances, the impression of a change in loudness occurs only when an entire group (two, three, four) of cells has come into action. This usually happens with weak sounds.

Thus the difference threshold is not an unchanging, physiologically determined quantity and can be reduced by means of

Fig. 2.

exercises or concentration of attention. The limit of the difference threshold is the minimum relative increment of intensity necessary for the excitation of one nerve cell; the difference threshold cannot be smaller than this quantity.

The loudness of a sound is determined by the number of excited nerve cells. Therefore it is impossible to derive a law of change of loudness by integrating the observed difference thresholds, since they do not always correspond to the true \(\Delta I/I\) necessary for the excitation of one cell. This, evidently, was the weak point in Fechner’s construction, which established, by integration, a logarithmic dependence between the intensity and the loudness of sound (see above).

Returning to the results of the works of Knudsen and Riesz, let us note that in Knudsen no dependence of the difference threshold on frequency is observed, whereas in Riesz such a dependence is present. However, as we shall see below, the frequency dependence established by Riesz is precisely the opposite of that obtained when comparing the loudness of sounds of different frequencies (the works of Kingsbury and others).

Bekesy subjected Knudsen and Riss’s work to criticism and, from his own experiments, established the absence of dependence of the threshold of discrimination of sound intensity on frequency. However, Bekesy does not think that the result obtained contradicts the observed fact of the dependence of loudness on frequency: he believes that very small changes may be independent of frequency, whereas comparatively large changes depend on frequency. The reason for this, in Bekesy’s opinion, should be sought in the nonlinear distortions introduced by the ear itself.

In Fig. 3 are given the experimental data obtained by Bekesy. He made measurements only at two levels: at 20 decibels (upper line) and at 40 decibels (lower line).

Fig. 3.

Bekesy’s data confirm the data of Knudsen and Riss in the respect that for weak sounds the threshold of discrimination is greater than for strong ones. The explanation of this phenomenon has already been given above.

Thus, as a result of a series of works on determining the increment of sound intensity corresponding to the minimal perceptible increment of loudness, the following may be considered established:

1) the relative increment of intensity \(\frac{\Delta I}{I}\) has its greatest value for the weakest sounds (at the threshold of audibility), then decreases with increasing sound intensity, and, beginning from a level of 40–45 decibels, remains unchanged; 2) the relative increment at a given level above the threshold is the same for all frequencies (opposite results are given only by Riss’s work); 3) integration of the increments does not lead to a dependence of loudness on sound intensity.

Although the Weber–Fechner law in its usual formulation is apparently inapplicable to auditory sensations, the replacement of sound intensity by its logarithm has become widely used in those cases where the perception of sound is concerned. This is explained, first, by the fact that the logarithmic scale corresponds much more closely to our judgment of loudness than the linear one, and, second, by the fact that working with logarithms is simpler and more illustrative than with long-

expressed by figures in an enormous range of the order of \(10^{14}\), expressing the strength of sound.

The expression of sound strength in logarithmic units—decibels (see p. 647)—is called the “sound-strength level.”*

III. Fatigue of Hearing

Before turning to the second method of determining the dependence of loudness on sound strength, it is necessary to touch upon the question of fatigue of hearing.

The phenomenon consists in the fact that the sensation of the loudness of a sound depends on the preceding sound stimulus. Under the action of a loud sound the auditory apparatus becomes fatigued (adapts),

Fig. 4.

and all sounds perceived anew seem more muffled than they would if there had been no preceding fatigue. We have used the expression: “sounds seem more quiet.” But, in essence, since loudness cannot be determined objectively, one should simply say that the loudness of sounds decreases when there is preceding fatigue. In other words, loudness is a function not only of strength and frequency, but also of the state of the auditory apparatus at the moment when loudness is determined. Therefore, when we speak of the dependence of loudness on strength and frequency, it should always be understood that the matter concerns loudness for an unfatigued ear.

The adaptation of hearing was studied by Lazarev\(^{7}\), and the observed facts were explained from the standpoint of the ionic theory of excitation.

Recently this question was investigated in detail by Bekesy\(^{8}\). In Fig. 4 is shown the dependence he obtained of the degree

* We give the exact definition proposed by Fletcher and adopted in 1933 by the Commission on Acoustical Measurements and Terminology of the American Standards Association. The sound-strength level is the number of decibels above a certain conventional level, determined by an effective pressure \(P = 0.000204\) bar at \(20^\circ\)C and an atmospheric pressure of 76 cm. This level corresponds to the threshold of audibility of a sinusoidal tone of 1000 hertz.

fatigue on the duration of action of the fatiguing tone (800 hertz) for sounds of different intensity.

The experiment was carried out as follows: the fatiguing tone, with a frequency of 800 hertz, was delivered to one ear by means of an earphone. Immediately after the tone ceased, a comparison tone was delivered to the other ear—a brief sound impulse (0.2 sec.). The subject indicated whether the loudness of the impulse was equal to the loudness of the fatiguing tone. The experiment was repeated with changes in the intensity of the comparison tone until the subject no longer gave a positive answer to the question of equality of loudnesses. In Fig. 4 the abscissa gives the duration of action of the fatiguing tone, and the ordinate gives the difference between the levels of intensity of the fatiguing tone and the comparison tone. From the drawing it is clear that the intensity of the comparison tone decreases with increasing fatigue time, and this means that the loudness of the fatiguing tone itself decreases, while its intensity remains unchanged. The upper curve was obtained for a fatiguing tone having a sensation level* of 80, the middle one—94, and the lower one—108 decibels (2, 10, and 50 bar). Thus we see that, with increasing sound intensity, the degree of fatigue continuously increases.

Fig. 5.

After the cessation of the action of the fatiguing tone, the ear does not at once, but only gradually, return to its normal state. As is seen from Fig. 5, where the abscissa gives the time elapsed after the cessation of an 800-hertz tone, of sensation level 94 decibels, that had lasted 2 min., and the ordinate gives the perceived loudness, expressed through the sensation level of the comparison tone, the effect of fatigue may be considered (for the given conditions) to have almost ceased after 7–8 sec.

* By sensation level is meant the ratio, expressed in decibels, of the intensity of a given sound to the intensity above the threshold of audibility for the same frequency, i.e., the level of intensity above the threshold of audibility. For a tone of 1000 hertz the sensation level coincides with the intensity level.

The effect of fatigue (adaptation) is strongly manifested when there is a sudden change in the strength of a sounding tone. The perceived change in loudness is then considerably greater than is due to the difference between the levels of sound intensity before and after the change, i.e., than in the case of comparing two sounds separated by a pause. A similar “contrast effect” is clearly seen in Figs. 6 and 7*.

Fig. 6.

Along the ordinate axis is plotted the level of sensation of the fatiguing tone (dashed line) and of the comparison tone (solid line)**. After 120 sec. from the beginning of the action of the fatiguing tone (94 decibels, 800 hertz), its amplitude in Fig. 6 decreases, and in Fig. 7 increases twofold, i.e., the level of sensation changes by 6 decibels. The observed change in loudness (solid line) corresponds to a considerably greater change in the level of sensation, especially when the amplitude is decreased.

Fig. 7.

* Figs. 6 and 7 have been compiled by me in a somewhat modified form, in comparison with the corresponding figures of Békésy.

** The inscriptions on the figures themselves are made using the terms explained in the note on p. 654.

The contrast effect must be taken into account in developing a methodology for studying the loudness of sound (see p. 661).

IV. Loudness Level

The ear’s different sensitivity to sounds of different frequency is determined by the curve of the threshold of audibility. From this curve it becomes clear that, at the same sound-pressure level, sounds are not equally loud. However, on the basis of the Weber–Fechner law one might suppose that sounds will be equally loud at the same level of intensity above the threshold (i.e., at the same sensation level). Kingsbury’s experiments proved the untenability of this assumption. The equal-loudness curves established by Kingsbury have recently been checked by Fletcher and Munson1. The discrepancies in the results of the two studies are comparatively small. We give (Fig. 8) the curves of Fletcher and Munson, as based on more substantial experimental material.

Fig. 8.

Along the ordinate axis is plotted the sensation level of sounds of different frequencies that are equal in loudness with respect to other sounds on the same curve. It is customary to equate the loudness of sounds of different frequencies to the loudness of a tone at 1000 hertz. In this way one can compare the loudnesses of any sounds by expressing each of them through the sensation level of a tone at 1000 hertz. The expression of the loudness of any sound through the sensation level of an equally loud tone at 1000 hertz is denoted by the term loudness level.2

The loudness level determines the magnitude of loudness, but does not give an idea of the character of the variation of the magnitude itself. The decibel scale is a functional, not a natural, scale of loudness*.

In our figure the curves of equal loudness are drawn at intervals of 10 decibels, i.e., at loudness levels of 10, 20, 30, etc., decibels. These curves converge more closely in the region of low frequencies, run almost parallel and at the greatest distances in the region of middle frequencies (from 1000 to 3000 hertz), and again converge somewhat in the region of high frequencies. From this it may be concluded that the loudness of sounds of low and extreme high frequencies, when the sensation level is increased, grows more rapidly than the loudness of sounds of middle frequencies. This is clearly shown in Figs. 9a and 9b, taken from the same experimental study by Fletcher and Munson. Along the abscissa is plotted the intensity level, and along the ordinate—the loudness level of tones of different frequency. The frequency is taken as a parameter and is indicated on the figures themselves.

Yanovsky\(^{11}\), on the basis of Kingsbury’s experimental curves and also on material from his own observations, gave a formula for the approximate calculation of the loudness level for a given frequency and effective pressure. We give this formula:

\[ L = C \log \frac{P}{P_0}, \]

where \(L\) is the loudness level, \(P\) is the effective pressure, \(P_0\) is the effective pressure corresponding to the threshold of audibility and, consequently, being a function of frequency, and \(C\) is a constant which is a function of frequency.

\[ C = 19.9 + 10(3.1 - \lg F)^2, \]

\[ \lg P_0 = (3.361 - \lg F)^2 - 3.63. \]

The curves constructed according to the indicated formula are sufficiently close to the observed curves of equal loudness.

V. Judgment of Changes in Loudness

Both of the above-mentioned methods of studying the loudness of sound—the method of determining the smallest perceptible changes and the method of comparison—give, as has already been said, only indirect indications of the dependence of the loudness of sound on its intensity and frequency.

Establishing the dependence, i.e., establishing the natural scale of loudness, is evidently possible only on the basis of direct estimation by the subjects.

The idea that a person is capable of numerically estimating loudness seems absurd at first glance. However, a number of works have proved the complete possibility of this.

* Just as, for example, barometric pressure is a functional scale of height above sea level.

Fig. 9a.

Fig. 9b.

For the first time,* Gam and Parkinson raised the question in this form.^12 The authors, presenting to the subject successively two sounds of the same frequency but of different loudness, required him to determine, in percentages, the loudness of the second sound in relation to the first. However, such a task proved too difficult for the subjects, and the authors turned to the evaluation of more easily perceived tasks, namely: it was required to determine a doubling, tripling, or quintupling of loudness, or a decrease in loudness by a factor of two, three, or five. In this case the initial sound was compared successively with 8–9 sounds of different loudness, among which the subject was allowed to choose the sound whose loudness most nearly corresponded to the specified ratio. However, even this method of comparison must be acknowledged as somewhat complicated.

Laird, Taylor, and Wille^13 proceeded more simply. They gave the subject the initial sound and then, after a pause, the sound to be compared, the loudness of the latter being changed according to the subject’s instructions until the correct loudness ratio, from the subject’s point of view, was established. The authors of this work took a twofold increase in loudness and its decrease in the ratios \(3/4\), \(1/2\), and \(1/4\).

In approximately the same way, experiments were carried out by the authors of the third work—Geiger and Firestone^14—except that the subjects were given the opportunity to select for themselves the loudness of the sound being compared. An ingenious device excluded in this case the possibility of any prejudiced approach on the part of the subjects. Geiger and Firestone had the subjects estimate increases and decreases in loudness by factors of 2, 4, 10, and 100. The results obtained by the authors of the three works mentioned, although they differ among themselves, are basically similar, and this makes it possible to draw general conclusions of practical and scientific interest.

Gam and Parkinson, on the basis of their experiments, derived the following.

1. Man is characterized by judgments about changes in loudness by a certain number of times, but not by judgments about loudness as a magnitude that can be added or subtracted.

2. Loudness is a function of the level of sound intensity, expressed by the approximate formula:

\[ y = e^{0.076x}, \]

where \(y\) is the loudness of the sound, expressed in conventional units, and \(x\) is the sensation level. This dependence is shown in Fig. 10.**

1. It is impossible to establish a relation between judgments about changes in loudness and frequency, since the observed deviations lie within the limits of experimental errors.

* The indicated paper was delayed in the editorial office of the journal of the American Acoustical Society and was printed after the work of Laird, Taylor, and Wille.

** Gam and Parkinson derived this dependence for levels considerably above the threshold of audibility. Therefore, the zero point on the abscissa axis of Fig. 10 corresponds not to the threshold of audibility, but to an arbitrary level not below 35 decibels.

Laird, Taylor, and Wille did not give their observations so complete a form as the authors of the first article mentioned. Therefore, for comparison with other works, I had to derive, on the basis of the data they obtained, an approximate curve, which, together with the curve derived in the same way from the data of Geiger and Firestone, is shown in Figs. 11 and 12.

Laird, Taylor, and Wille also did not succeed in discovering a dependence between the change in loudness and frequency, although they conducted careful experiments with frequencies of 64, 256, 1024, and 4096 cycles. In this respect the work of Geiger and Firestone is different. They took only two, but sufficiently different, frequencies—1000 and 60 cycles—in order thus to verify the relation, derived by Kingsbury, between frequency and the rate of increase of loudness. The results they obtained generally confirm this relation: the loudness of a tone at 60 cycles increases faster than the loudness of a tone at 1000 cycles for the same increase in sound intensity. We shall discuss the significance of this observation below.

Fig. 10.

Fig. 11.

Turning to the comparative graph (Fig. 11), we see that all the experimental curves deviate in one and the same direction from the Weber–Fechner law. The curves of Gemma-Parkinson (2) and Geiger-Firestone (1) almost coincide with each other; the curve of Laird-Taylor

* For a more precise comparison there is not a sufficient amount of experimental material.

Ville (3) passes somewhat lower. In any case it may be said that loudness increases considerably faster than follows from the Weber–Fechner law (4).

If loudness is plotted on a logarithmic scale (Fig. 12), then the curves of Gema-Parkinson and Laird-Taylor-Ville turn into straight lines.

The first curve is also straightened to a lesser degree. It may therefore be assumed, with a known approximation, that over a certain interval there exists a linear dependence between the logarithm of loudness and the logarithm of the sound intensity.

In what follows, in analyzing the theory of Fletcher and Munson, we shall give an analytical expression for this dependence.

As we have already said above, only Geiger and Firestone established the dependence of the change in loudness on frequency for one and the same change in sound intensity. Does it follow from this that the data of the other two works contradict the observed convergence of the equal-loudness curves in the region of low frequencies?

Fig. 12.

To clarify this question it is of interest to become acquainted with the work of Riesz[^15], who compared the data of Gema-Parkinson and Laird-Taylor-Ville with his own observations of the minimum perceptible increment in loudness (see p. 648). Summing the relative increments \(\left(\frac{\Delta I}{I}\right)\), whose magnitude changes with a change in sound intensity, and assuming that for one and the same frequency loudness is proportional to the number of relative increments (or, in Riesz’s terminology, distinguishable steps) above the threshold of audibility, Riesz derived the dependence of loudness on sound intensity shown in Fig. 13*.

Like Gema-Parkinson and Laird-Taylor-Ville, Riesz finds that an increase or decrease in loudness when the sound intensity changes follows the same law at all frequencies. However, there is a fundamental difference between the two works mentioned and Riesz’s work. The dependence derived by Ge—

* The curve was constructed by me on the basis of Riesz’s data.

…by Mom and Parkinson, as well as the dependence derived by Laird, Taylor, and Wille, are expressed, on a logarithmic loudness scale, by a straight line; this means that, for a given change in

Fig. 13.

the level of sound intensity, loudness changes by the same number of times, whatever the initial level of sensation may be. The dependence derived by Rees, however, is expressed by a curve whose slope decreases as the level of sensation increases. Consequently, according to Rees’s data, the loudness of quiet sounds grows faster than that of loud sounds for the same increase in the level of sensation.

Fig. 14.

It is quite clear that establishing a linear dependence between the logarithm of loudness and the level of sensation, the same for all frequencies, contradicts the difference in the rate of growth of loudness of high and low sounds. Establishing a nonlinear dependence, though one identical for all frequencies, means precisely a nonidentical rate of growth of loudness of sounds of different frequency, since equally loud sounds have different levels of sensation. The schematic Fig. 14 explains what has been said.

The areas of the squares symbolize the magnitude of loudness. Two equally loud sounds, for one and the same increment of the level of sensa—

S changed in loudness to different degrees: the loudness of the low sound increased more. This occurred because the low sound, equally loud with the high one, had a lower sensation level, and according to Riess’s curve, at a lower sensation level the loudness, for one and the same increment in sensation level, increases more rapidly.

Riess’s conception, which we have only just considered in part, taken as a whole is distinguished by a certain complexity and cannot claim to provide a satisfactory explanation of the question of changes in loudness. Riess believes that his results agree with the experimental data of Laird, Taylor, and Wille. However, the agreement is more or less satisfactory only when the change in loudness is estimated as twofold.

The causes of the discrepancies among the results of the three studies investigating the immediate judgment of the loudness of a sound should be sought in differences in the experimental methods, chiefly in the part connected with ear fatigue. With fatigue there appears a contrast effect, which consists in the fact that the judgment of a change in loudness becomes exaggerated in comparison with the judgment of an unfatigued ear. Thus a judgment, for example, of a doubling of loudness, with a prolonged initial sound or with the absence of a sufficient pause between the initial and the compared sounds, occurs with a smaller increment in sound intensity than in the case in which conditions of comparison without fatigue were provided—a short initial sound and an appropriate pause.

If one also takes into account that, for the same fatigue, the contrast effect is more significant when the intensity of the sound is decreased (Figs. 6 and 7) than when it is increased, then it is easy to understand that failure to take ear fatigue into account in developing the experimental method must entail, as shown in the work of Rzhevkin and Rabinovich*, very substantial discrepancies in the results obtained.

Summarizing all that has been said above, we arrive at the following conclusions:

1. A reliable judgment about a change in loudness by a certain small number of times is possible (by 2, 3, 4).

2. A change in loudness by a definite number of times corresponds to a definite (approximately) change in the level of intensity, the same for every initial level.

3. The change in the level of sound intensity corresponding to a given change in loudness is smaller than follows from the Weber–Fechner law.

4. The dependence of the change in loudness on frequency has not yet been definitively established; however, taking into account the studies comparing the loudness of sounds of different frequency (the Kingsbury and Fletcher–Munson curves), one must suppose that such a dependence exists.

* The work is in press.

VI. Loudness of a Complex Sound

In speaking of the dependence of the loudness of a sound on intensity and frequency, we have so far had in mind only the loudness of simple sinusoidal tones. Experience shows that the loudness of a complex sound is determined not only by the intensity and frequency of its components, but also by certain contributing factors that depend on the relationship between the individual components. These contributing factors are mutual masking and beats. Without describing the sufficiently well-known phenomena of masking and beats,^16 we shall concern ourselves only with the connection between these phenomena and the loudness of a sound.

If one sound completely masks another sound sounding simultaneously with it, then, obviously, the loudness of the masked sound will be equal to zero. If, however, the weaker sound is barely audible against the background of the stronger one, then the loudness of the former will be very small, in any case less than if it were not partially masked (Yanovskii^11).

The degree of masking is some function of the difference in frequency of two tones. With sufficient separation in pitch, masking does not occur and, consequently, the loudness of each of the tones remains unchanged; when the tones are brought closer together, however, masking occurs, and the loudness of the masked sound gradually decreases—such is the idea developed by Fletcher and Munson.^10

However, much in this question still remains unclear and unverified experimentally. First, the decrease in loudness under masking cannot be verified by comparison with a standard loudness, since we can compare only the loudness of the entire complex of sounds, and not that of its separate components.

Second, if one allows a gradual decrease in loudness under masking, it follows from this that the presence of the masking tone affects not only the degree of loudness, but also the rate of change of loudness when the intensity of the sound changes. For example, take two sounds with frequencies \(F_1\) and \(F_2\) and sensation levels \(S_1\) and \(S_2\) decibels, and suppose that their frequencies and sensation level are such that the first, lower sound completely masks the second. Gradually increasing \(S_2\), we shall achieve the point at which the sounds become equal in loudness. Let this occur at the level \(S'_2\) decibels. By assumption, at the level \(S_2\) the loudness of the second sound was equal to zero; at the level \(S'_2\) decibels it reached the same magnitude as it would have at this level without the masking tone, since the masking action of the low tone has ceased. Consequently, a change of the sensation level by \((S'_2 - S_2)\) decibels produced the same change in loudness in the presence of the masking tone as would have been produced by a change of \((S'_2 - O)\) decibels in its absence.

Third, it has not been experimentally established whether the loudness of the masking tone changes under complete masking; in other words, whether masked sounds do not increase the loudness of the masking tone.

of the entire complex, perceived in the given case as a single masking tone.

The effect of beats on the loudness of a sound is manifested in its periodic change during beats. At a low beat frequency, the fluctuations of loudness are heard quite distinctly; at a high frequency they are perceived as roughness of the sound. In both cases they introduce into the perception of loudness a certain new element, which is not evaluated quantitatively.

The loudness of a complex sound was experimentally studied by Yanovsky[^11], and then by Fletcher and Munson1. However, the results obtained in the two works differ considerably from one another. According to Yanovsky, the maximum loudness of a two-tone sound is obtained when two tones of identical or nearly identical pitch are combined. If the tones are equally loud, then the loudness level is increased by an amount from 6 to 11 decibels. As the difference in pitch between the tones is increased, the loudness of the complex gradually decreases, and at a large separation the increase in loudness level barely reaches 1 decibel. According to Fletcher and Munson it turns out, as we shall see from what follows, exactly the reverse, i.e. the maximum loudness of a two-tone sound will be reached at the greatest separation of the components in pitch.

Fig. 15.

The results obtained by Fletcher and Munson for the loudness of a set of several tones, sufficiently separated from one another in pitch so that there would be no influence of masking or beats, are given in Fig. 15. The upper curve refers to a set of 10 equally loud components, the lower—to 2 equally loud components. For convenience of study, Fletcher and Munson took a set of tones of equal loudness, which was established by comparing each component separately with a 1000-cycle tone.

To study the influence of masking on the loudness of a complex sound, Fletcher and Munson carried out another series of experiments, the results of which are given in Fig. 16. The frequency difference \(\Delta F\) between each two neighboring components was kept constant. The sound consisted of 10 equally loud components, with the frequency of the lowest component being 1000 cycles. The first curve from the top was obtained for \(\Delta F = 340\) cycles, the second—for \(\Delta F = 230\) cycles, the third for \(\Delta F = 112\) cycles, and the fourth—for \(\Delta F = 50\) cycles.

From the experiments described, the following two conclusions may be drawn:

1. The loudness of a complex sound increases with an increase in the number of components*.

2. Bringing the components closer together in pitch entails a gradual decrease in the loudness of the complex sound.

The last conclusion indirectly confirms the proposition expressed by Fletcher and Munson concerning the gradual decrease in the loudness of a masked sound. We say “indirectly,” since in order to pass from the experiments described to this proposition it is necessary first to assume that loudness has an additive property (see below).

VII. Theory of Loudness

The theoretical substantiation of the experimentally established dependence of loudness on the strength of the sound and on frequency (and, for a complex sound, on the strength and frequency of its components) has been given to one degree or another by the authors of investigations, but not in the form of a coherent theory, only in the form of separate observations and assumptions. At the same time, for the systematization of all the material, an explanation of the physical and physiological mechanism of the perception of loudness is very important.

Fig. 16.

In this respect, the article by Fletcher and Munson, cited several times above by us, deserves the greatest attention; in it an attempt was made for the first time to give a complete theory of the loudness of sound. The authors develop the following hypothesis**: the loudness of a sound is proportional to the number of nerve impulses passing through the auditory nerve to the brain per unit time. Loudness has an additive property; in other words, when several sounds are perceived simultaneously, the loudness of the entire complex is equal to the sum of the loudnesses of the components.

Proceeding from this hypothesis, it becomes possible, for the study of the character of the change in loudness when the strength of the sound is changed, to use—

* It is interesting to note that, as a result of the superposition of several sounds lying below the threshold of audibility, an audible sound may arise, i.e., one lying above the threshold.

** Partly this hypothesis had already been expressed by Fletcher in one of his preceding works.

be guided not only by a direct judgment of loudness, but also by the measurement of the loudness of complex sounds.

Fig. 15 (see above) gives us an experimentally established relation between the loudness level of a complex made up of several (2 and 10) equally loud sounds and the loudness level of each of them separately. The relation between the loudness of the complex and the loudness of an individual component is established by the basic hypothesis. From this it is already comparatively easy to establish the dependence of loudness on loudness level, i.e., ultimately, on the frequency and intensity of the sound.

Denoting by \(N\) a number proportional to the number of nerve impulses passing through the auditory nerve per unit time, i.e., the loudness of the sound in arbitrary units, we may write:

\[ N = G(L), \]

where \(G\) is some function of the loudness level \(L\).

The dependence of the loudness level on the intensity and frequency of the sound is given by the equal-loudness curves (Fig. 8). The upper curve of Fig. 15 refers to a complex sound consisting of 10 equally loud components, the lower one to 2 equally loud components. Here we shall have the following dependences:

\[ G(L) = 10G(L_k), \]

\[ G(L) = 2G(L_k), \]

where \(L\) is the loudness level of the entire complex and \(L_k\) is the loudness level of each separate component.

The dependence of loudness on loudness level derived on the basis of these experiments is given in Fig. 17. Loudness is plotted in arbitrary units on a logarithmic scale. This curve is very close to the curve given by Gem and Parkinson (Fig. 12, 2), and also to the curve derived by us on the basis of the observations of Geiger and Firestone (12, 1). In the range from 40 to 100 decibels the dependence between the logarithm of loudness and the loudness level is close to linear.

The loudness of a complex sound according to Fletcher and Munson is determined by the formula:

\[ N = \sum_{k=1}^{k=n} b_k G(L_k), \]

where \(n\) is the number of components, \(L_k\) is the loudness level of an individual component, and \(b_k\) is a coefficient indicating the decrease in the loudness of the component due to masking by other components.

Making a series of assumptions that simplify the problem, the authors ultimately arrive at a rather complicated formula for \(b_k\), which we give:

\[ b_k = [(250 + F_k - F_m)1000]\,10^{(L_k - L_m)T} Q(F_k + 30 \log F_k - 95), \]

where \(F_k\) is the frequency of the given component of the complex sound, \(F_m\) is the frequency

of the masking component, $L_k$ and $L_m$ are the corresponding loudness levels, $T$ is the degree of masking, given graphically on the basis of experimental data, $Q$ is a certain function of $F_k$ and $F_m$, given graphically on the basis of experimental data, and $P_k$ is the loudness level of the given component.

The following assumptions underlie this formula:

1. If the components are far from one another in pitch, then there is no masking effect and $b_k = 1$.

Fig. 17.

1. If the components are close in frequency, then

a) the lower component masks the upper one (i.e., the masking effect is taken into account only with respect to higher sounds);

b) only the single nearest lower component exerts a masking effect on the given component; the effect of all the others is neglected;

c) the degree of masking is proportional to the frequency difference of the components.

1. If the components are very close in frequency, then one must speak not of the addition of loudnesses and masking, but of the addition of energies.

Thus, when the loudness of a complex sound is taken into account, the energies of a group of very close components are summed, and this group is regarded as a single component. The concept of “very close” components was found from experience.

For frequencies below 2000 hertz, components within a band of 100 hertz may be regarded as very close; for frequencies from 2000 to 4000 hertz—in a band of 200 hertz; for frequencies from 4000 to 8000 hertz—in a band of 400 hertz; and for frequencies above 8000 hertz—in a band of 800 hertz.

The calculations of the loudness of a complex sound made by Fletcher and Munson according to the formula derived above agreed very closely with the observed values.

VIII. Loudness of Very Brief Sounds

This question had to be singled out and placed at the end because, in order to understand it, preliminary acquaintance with all the material set forth above is necessary. Bekesy\(^{8}\) compared the loudness of a very brief sound (beginning with hundredths of a second) with the loudness of a control sound pulse of the same frequency and a duration of 0.2 sec., and determined that, as the duration of a very brief sound increases, its loudness increases until it reaches the full loudness (determined by the intensity and frequency of the given tone).

Fig. 18.

Fig. 19.

In Fig. 18, where the duration of the tone under study is plotted along the abscissa axis, and the sensation level of the control pulse along the ordinate axis, the experimentally determined Bekesy dependence is shown. The result obtained at the same time indicates that

the loudness of any sound does not immediately, but only gradually, reach its full magnitude (after which it again begins to decrease under the influence of fatigue—see p. 651). This character of the phenomenon is observed independently of the frequency of the tone.

In another of his works Békésy⁵, investigating the same question, established the exact dependence between the duration and loudness of a very brief sound. It is shown in Fig. 19, where along the ordinate axis is plotted the logarithm of the ratio of the effective pressure of a very brief sound \(P\) to the effective pressure of an equally loud control impulse \(P_0\), and along the abscissa axis—the logarithm of the ratio of the duration of the control impulse \(t_0\) to the duration of the very brief sound \(t\).

From the drawing it follows:

\[ 2\lg \frac{P}{P_0}=\lg \frac{t_0}{t}, \tag{1} \]

whence

\[ \left(\frac{P}{P_0}\right)^2=\frac{t_0}{t}=\frac{I}{I_0}, \]

where \(I\) and \(I_0\) are the corresponding sound intensities; consequently,

\[ It=I_0t_0. \tag{2} \]

From this relation one may derive the supposition that the loudness of a very brief sound is determined by the product of the sound intensity and the duration of its sounding, i.e. by the magnitude of the total sound energy.

It is interesting to note that a similar dependence, analogous to the law of photochemical action, has long since been established for brief visual sensations.

For convenience in determining the loudness level of a very brief sound at a specified loudness level of a sound of the same duration, formula (1) may be given a somewhat different form:

\[ 20\lg_{10}\frac{P}{p}=20\lg_{10}\frac{P_0}{p}+10\lg_{10}\frac{t_0}{t}, \]

where \(p\) is the threshold value of the effective pressure for the given frequency. Substituting, instead of \(20\lg_{10}\frac{P}{p}\) and \(20\lg\frac{P_0}{p}\), the corresponding sensation levels, we obtain:

\[ S=S_0+10\lg_{10}\frac{t_0}{t}. \]

It is quite clear that if the sensation level of a very brief sound \(S\) is greater by \(10\lg_{10}\frac{t_0}{t}\) than the sensation level of the equally loud control impulse \(S_0\), then, at an equal sensation level, the very brief sound will have a loudness level \(L\) by the same amount

(that is, by \(10\lg_{10}\dfrac{t_0}{t}\)) less than the loudness level of the control impulse \(L_0\), i.e.

\[ L_0 = L + 10\lg_{10}\frac{t_0}{t}. \tag{3} \]

In Békésy’s experiments \(t_0 = 0.1\) sec.

The same dependence was obtained by Støydel[^18]. In Fig. 20 are given curves derived by him on the basis of extensive experimental work. Along the abscissa is plotted the duration of a very short sound; along the ordinate, the sensation level of the control impulse. The curves correspond to sensation levels of the very short sound of 92, 82, and 72 decibels. Full loudness is already attained at a sound duration of 0.1–0.15 sec. (according to Békésy, at 0.20 sec., Fig. 18). Formula (3), which follows from Békésy’s curves, also satisfies Støydel’s curves quite well.

Fig. 20.

A somewhat different form is taken by the dependence derived by Lifshitz[^19]. He studied the loudness of short sounds consisting of a series of clicks following one another with a definite frequency, and established the following:

\[ L = L_a + 10\lg_{10}N, \]

where \(L\) is the loudness level of the series, \(L_a\) is the loudness level of a single click, and \(N\) is the number of clicks in the series. Taking into account that \(N=\dfrac{t_0}{t_a}\), where \(t_0\) is the duration of the series and \(t_a\) is the duration of a click (more precisely, the interval of time from one click to the next), the above formula may be given a form extremely reminiscent of Békésy’s formula (3):

\[ L_0 = L_a + 10\lg\frac{t_0}{t_a}. \tag{4} \]

The repetition frequency of the clicks does not affect the total loudness of the series; only the number of clicks perceived by the ear is important (provided, of course, that the strength of the clicks remains unchanged). In other words, the ear integrates the energy of the individual impulses of the entire series.

Passing from clicks to short sounds whose intensity \(I\) is ...

becomes a continuous function of \(t\), Lifshits gave the dependence observed by him the following final form:

\[ L=\lg\int_{t_1}^{t_2} I\,dt. \tag{5} \]

If the threshold value of sound is regarded as zero loudness, then, obviously, in this case as well, with a considerable decrease in the duration of sounding, there will be an increase in sound intensity in order to preserve unchanged loudness (i.e., the threshold of audibility).

Belikov’s experimental work \({}^{20}\) on this question confirmed the dependence established by Lazarev on the basis of the ionic theory of excitation proposed by him:

\[ It=a+bt, \]

where \(I\) is the sound intensity at the threshold of audibility, \(t\) is the time of sounding, and \(a\) and \(b\) are constants.

Fig. 21.

The loudness of a click was studied by Stoydel \({}^{18}\) in the investigation mentioned above. He established that the loudness of a click depends on its form, i.e., on the form of the rise and fall of sound pressure during the click, and formulated his observations as follows: the human ear reacts only to a change in pressure, integrating the effective pressure over time during a very short interval of time \(\tau = 0.3\) milliseconds.

In Fig. 21 three equally loud clicks of different form are given. Stoydel extends this proposition to all sounds in general, considering that during each oscillation the ear integrates changes in pressure over the time interval \(\tau\), with the following conditions:

1) if \(\tau < T\) (the period of oscillation), then the beginning of the integrated section of the curve is established in such a way that the integral is greatest (Fig. 22);

2) if \(\tau > T\), i.e., if during the integrated interval several oscillations occur, then the ear integrates only one complete oscillation (Fig. 22).

It is not difficult to see that, with an increase in frequency up to 2000–3000 hertz, the integral will increase, after which it will begin to decrease. This corresponds approximately to the curve of sensitivity of the ear. Stoydel considers that the proposition derived by him applies to levels from 50 to 100 decibels.

IX. Measurement of Loudness

The measurement of loudness, as this is understood at the present time, is reduced merely to determining the loudness level, i.e., to comparing the loudness of the given sound with the loudness of a sinusoidal tone at 1000 Hz. As was said above (see the note on pp. 652 and 654), for a sinusoidal tone at 1000 Hz the magnitudes of the loudness level, sensation level, and intensity level coincide. Thus, by comparing some loudness with the loudness of a tone at 1000 Hz, we obtain an expression of this loudness in decibels through the intensity level of the 1000-Hz tone—this is the normal scale of loudness level adopted in all countries.

Fig. 22.

For establishing the normal loudness scale and measuring the loudness of pure tones, the circuit is used (Fig. 23) consisting of two generators, two attenuators, and a telephone receiver.

Fig. 23.

[In the diagram: “Generator 1000 Hz”; “Attenuator I”; “Variable-frequency generator”; “Attenuator II”.]

An attenuator (A. Gund\(^{21}\)) is a device that gives any prescribed weakening of the sound intensity while maintaining an unchanged wave resistance (usually \(600 \Omega\) of purely ohmic resistance). In Fig. 24 a circuit is given for an attenuator consisting of separate sections, switched in alternately by a special three-pole switch (at points \(A\) and \(B\)). Each section gives an attenuation by a definite number of decibels. An attenuator giving a total attenuation of 100 decibels in steps of one decibel consists of 19 sections. The first group of 9 sections gives attenuations from 1 to 9 decibels; the second group of 10 sections—attenuations from 10 to 100 decibels. By switching in subse-

Fig. 24.

sufficiently two sections, one from each group, one can obtain any specified attenuation.

To establish the loudness level it is necessary to determine experimentally which position of the switches of attenuator I corresponds to the threshold of audibility for normal hearing at the given generator power.

Measurements with an attenuator encounter a fundamental difficulty, consisting in the dependence of the resistance of the telephone on frequency. To avoid this difficulty, one usually connects at the output of the attenuator a resistance equal to the characteristic impedance of the attenuator, taking from a small part of it a branch of current to the telephone. In this way the resistance at the output remains practically constant. In the circuit shown, however, there is a large loss of power. This drawback is absent in the circuit (Fig. 25) proposed by S. Rzhevkin.* The circuit consists of a parallel combination of a resistance \(R\) and a capacitance \(C\), inserted in the telephone circuit. Such values of \(R\) and \(C\) can be selected that the total resistance will be practically constant and independent of frequency, while purely ohmic. Transformer \(U\) brings the resistance to the value of the characteristic impedance of the attenuator.

Fig. 25.

Measurement of the loudness of sound coming from outside is carried out with the aid of a phonometer, sometimes also called an audiometer. The phonometer is an electroacoustic instrument giving a sound of constant frequency and timbre and variable loudness. The scale of the phonometer indicates the loudness level in decibels and is usually graduated in steps of one or two decibels over the range from 0 to 100 decibels.

Fig. 26.

The phonometer essentially consists of three parts: a sound source, a potentiometer, and a telephone receiver. As the sound source there is usually used a buzzer, or a tuning fork with an interrupter, more rarely a tube generator, since it is desirable that the phonometer be portable. Figure 26 gives the schematic diagram of the Barkhausen phonometer. The phonometer is calibrated according to the normal loudness scale—

* The work is being prepared for publication.

ness. In Fig. 27, as an example, the calibration curve (upper line) is given for the 3-A audiometer, very widespread in America, which gives a noise tone whose timbre is close to speech. The abscissa axis gives the audiometer steps; the ordinate axis gives the loudness level. For comparison, the same figure gives a curve (lower line) of the dependence of loudness level on sensation level for speech. The telephone receiver of the phonometer is arranged in such a way that, when it is pressed to the ear, a small slit remains between it and the auricle, into which the measured sound arriving from outside can penetrate. The method of working with the phonometer is as follows: placing the telephone tightly to one ear and closing the other, one changes the loudness of the sound of the phonometer until the impression of equality of loudnesses occurs. Very often it is difficult to set an equal loudness on the phonometer, but it is comparatively easy to set a greater or lesser loudness. Then the loudness of the sound being determined can be established by interpolation between two readings of the phonometer, of which one is slightly louder and the other slightly quieter than the sound being determined. One may also use the masking method. The loudness of the phonometer sound is varied until it is established at the threshold of masking by the measured sound, i.e., until the moment when the phonometer sound will be barely audible against the background of the measured sound. The readings of the phonometer scale by this method give the so-called masked loudness. In order to establish the loudness of the measured sound, it is therefore necessary to have an experimentally determined conversion scale.

Fig. 27.

The degree of accuracy of phonometer measurements was studied by the Japanese authors Obata and Morita²² on the noise of the subway in Tokyo. They arrived at the following conclusions:

1. In order to measure noise loudness correctly, one must acquire a certain amount of experience. 2. To increase the accuracy of measurement, one must practice specifically on the given noise. 3. The probable error of observation (the mean deviation from the obtained means) is, for experienced observers, less than 1.5 decibels.

As may be concluded from the work of Obata and Morita, the accuracy of phonometer measurement is sufficiently high.

The loudness of sound can also be measured with the aid of a tuning fork equipped with a mechanical hammer, which always delivers a blow of the same force to the tuning fork. This method was described by Davison²³ and improved by Stowell²⁴. When one wishes to measure the loudness of some sound (with a tuning fork one can measure only sounds arriving from outside at both ears), the tuning fork is set into vibration and brought as close as possible to the ear. Simultaneously with the blow on the tuning fork, a stopwatch is started, stopping it at the moment when the sound of the tuning fork, while dying away, begins to be masked by the sound being measured. If the decay of the tuning fork has been measured beforehand, then from the time elapsed from the beginning of the tuning fork’s sounding to the moment of masking one can judge the loudness level of the sound being measured. In the instrument constructed by Stowell, the stopwatch is mounted on the instrument itself and begins to operate automatically at the moment of the blow.

In conclusion, let us mention the “automatic phonometre” constructed by Stößel, which determines loudness without the participation of the human ear. The instrument is built on the principle, already set forth above, of summing the effective pressure over a short interval of time (see p. 670). Judging from Stößel’s own experiments, the readings of his instrument are very close to the mean readings of the subjects tested. In view of the great complexity of the instrument, we do not give its description in the present article.

References

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1. Fletcher and Munson. ↩↩

2. According to the definition of the American Standards Association, the loudness level of any sound is the intensity level of an equally loud tone at 1000 hertz. As was said above, for a tone at 1000 hertz the sensation level and the intensity level coincide. ↩