just noticeable differences for intensity and their relation to loudness

Just noticeable differences for intensity and their relation to loudness

Jennifer H. Johnson

Communication Sciences and Disorders, Syracuse University, South Crouse Avenue, Syracuse, New York 13244-2280

Christopher W. Turner Communication Sciences and Disorders & Institute for Sensory Research, Syracuse University, South Crouse Avenue, Syracuse, New York 13244-2280

Jozef O. Zwislocki

Institute for Sensory Research, Merrill Lane, Syracuse, New York 13244-5290

Robert H. Margolis University of Minnesota, Department of Otolaryngology, Box 356 UMHC, Minneapolis, Minnesota 55455

(Received 18 July 1990; revised 8 January 1992; accepted 8 October 1992)

The present study examines the relation between the form of the loudness function and the size of the intensity just noticeable difference (jnd). The hypothesis that equal loudnesses at any given sound frequency yield equal-intensity jnd's was examined. In addition, Hellman et al.'s [J. Acoust. Sec. Am. 82, 448-453 (1987) ] experiment, which showed that jnd's are independent of the slope of the loudness function was replicated. Threshold shifts and altered loudness-balance functions for 1-kHz tones were produced by using backgrounds of narrow- or wideband noise. The two types of background noise produced intersecting points on loudness- balance functions at which intensity jnd's were obtained. Intensity jnd's were also obtained at equal-loudness levels (corresponding to 30, 40, 50, and 60 dB SL in the unmasked ear) under each of the two noise conditions and in quiet. The results indicate that tones of equal loudness produce approximately equal jnd's and that there is no apparent relation between the slope of the loudness-balance functions and the size of the intensity jnd.

PACS numbers: 43.66.Cb, 43.66.Fe, 43.66.Ba [LDB]

INTRODUCTION

Recently, there has been a renewed interest in relating loudness and differential intensity sensitivity (jnd's). Per- haps the earliest attempt to relate these two types of psy- cheacoustic data was proposed by Fechner, who stated that the sensation of loudness increases by a constant amount, each time the stimulus intensity is increased by ajnd (AI/I). Thus, according to Fechner, the total subjective magnitude (loudness) of a stimulus could be determined by counting the jnd's and the greater the slope of the loudness function, the smaller the jnd should be. In contrast, Stevens ( 1975, p. 182) stated that "the jnd is not the derivative of the magnitude function for loudness" and provided a demonstration of this in Stevens and Davis (1938). Miller (1947) directly examined the relation of jnd's to loudness. Subjects compared the loudness of broadband noise to the loudness of a 1- kHz tone in order to obtain a quantification of the growth of loudness for the noise stimulus. To do so, he scaled the noise and jnd's in terms of sones. Neither a constant nor a simple relation between loudness and the number of jnd's was found. Miller (1947) therefore concluded that jnd's did not correspond to equal-loudness increments.

Zwicker (1963) and Hellman (1970) have shown that narrow-band masking noise produces steeper loudness func-

tiens for a pure tone than does wideband noise. Taking ad- vantage of this finding, Hellman et al. (1987) directly tested the issue of whether or not the intensityjnd era pure tone in a noise background is related to the slope of the loudness function. Using seven normal-hearing subjects, loudness curves were obtained in quiet, and in backgrounds of narrow- and wideband noise. The loudness-level curves intersected to

produce points of simultaneously equal SPLs and loudnesses in which only the slopes of the functions were different. In- tensity jnd's were determined at these intersection points. The authors postulated that, if the jnd's depend on the slope of the loudness functions, then, the narrow-band condition should yield a smaller jnd than the wideband condition. Their results indicated, however, that the jnd's were essentially the same, thereby suggesting that thejnd's are independent of the slope of loudness functions. A possible criticism of Hellman et al.'s (1987) article is that their conclusion is based on group means, and a substantial intersubject variability may have obscured existing differences. In one sec- tion of the present study, we have employed the technique of Hellman et al. (1987) and have investigated the relation between the loudness-function slope and the intensity jnd in individual subjects.

The "proportional jnd" theory of loudness matching, originally described by Riesz (1933) and later elaborated on by Lim et al. (1977), also proposes a relation between jnd's

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and loudness. This theory is a modification of the Fechner- ian concept implying that the jnd's are related to the slope of loudness functions. According to Riesz's theory, two stimuli are judged to be equally loud only if their intensities divide the dynamic ranges of the two stimuli proportionately in terms of jnd's. The difference between the proportional jnd theory and Fechner's theory is that the jnd count is normal- ized with respect to the total number ofjnd's in the dynamic range.

Using hearing-impaired subjects, Florentine et al. (1979) evaluated the predictions of Riesz's proportional jnd theory. The hearing-impaired subjects included four with cochlear hearing loss and one with a vestibular schwannoma. Actual loudness matching curves from the vestibular schwannoma subject and the cochlear-impaired subjects were consistent with the curves predicted according to the "proportional jnd" theory. Their results also indicated that jnd's of each of the normal- and the hearing-impaired ears when compared at equal loudness levels, but not at equal sound-pressure levels (SPLs) nor at equal SLs for all subjects.

Houtsma et al. (1980) further evaluated the propor- tionaljnd theory using masking noise to modify the loudness functions. They performed careful loudness-matching experiments as a basis for their computations. The authors mentioned that if the noise and tone were gated on and off together, subjects had difficulty in judging the loudness of the tone alone. Instead, they obtained their binaural matches by presenting a pulsed tone alone to one ear and a pulsed tone in a continuous noise background to the opposite ear. Data of two of the five subjects were not included in the results because their performance was too erratic. The results from the remaining three subjects seemed to indicate that the relationship between intensity resolution and loudness equality can be successfully described by the "proportional jnd" theory.

A different type of relationship between loudness and jnd's was proposed by Zwislocki and Jordan ( 1985, 1986), on the basis of an earlier Ph.D. dissertation of Jordan

(1962), in which he tested Zwislocki's hypothesis that, at any one sound frequency, intensity jnd's are equal when the loudnesses are equal. Jordan's data were obtained on 26 subjects with normal hearing and 14 subjects with monaural sensorineural hearing loss. Intensity discrimination was determined using amplitude-modulated tones and a modified method of limits. A similar method was used to obtain loud-

ness matching curves. Jordan's data supported the hypothesis that the jnd's were approximately equal in normal and pathological ears when loudnesses were equal, except at very low SLs. Since loudness curves in pathological ears had steeper slopes than in normal ears, and since the jnd's were approximately equal in normal and pathological ears at equal loudnesses, thejnd's appeared to be independent of the slope of loudness functions. Jordan's (1962) study had two shortcomings, however. First, the comparison between normal and hearing-impaired subjects was based upon group data. Although Jordan included distributions of individual data that showed no statistical difference between normal

and impaired ears, no direct comparisons were made within

individual subjects. Second, the jnd's were not determined with a criterion-free method and the amplitude modulation technique used to determine jnd's may not have yielded data consistent with the more classical pulsed-tone methods.

Nevertheless, Schlauch and Wier (1987) confirmed Jordan's results, apparently without any knowledge of his work. They used adaptive forced-choice procedures for both loudness matching and intensity jnd's and evaluated their results in terms of both individual and group performance. They used both monaural masking and cochlear pathology as modifiers of loudness-level functions. They found the jnd's to be sufficiently correlated with loudness level to be able to construct theoretical loudness-level functions in

rough agreement with the empirical ones. The agreement was best at higher SPLs and worst at the lowest ones, in agreement with Jordan's findings. Interestingly, as in Jor- dan's study, the jnd's were relatively higher in the ears with the steeper loudness growth, contrary to Fechner's theory.

Results deviating from those of Jordan (1962; also Zwislocki and Jordan, 1986) and of Schlauch and Wier (1987) were obtained by Rankovic et al. (1988). They examined the relation between loudness and intensity discrimination in four normal-hearing subjects in quiet and in the presence of broadband noise. Intensity discrimination data were collected with gated and continuous pedestals. Masked and unmasked loudness-balance functions were obtained by means of two methods, the method of adjustment and an interleaved 2AFC procedure (Jesteadt, 1980). According to their data, equally loud tones did not yield equal jnd's at several midlevel intensities in the gated pedestal condition. In the continuous-pedestal condition, the equal loudness jnd's for masked and unmasked tones were different at all sound intensities, except the highest. Contrary to Jordan (1986) and Schlauch and Wier (1987), these authors concluded that equal loudnesses do not yield equal intensity jnd's when the slopes of the loudness-balance functions are different. With respect to the results of Hellman et al. (1987), in which loudness equality was associated with ap- proximate jnd equality, as in Jordan's and Schlauch and Wier's results, they suggested that this outcome may have been coincidental, since SPLs were also equal. However, a large number of studies indicate that equal SPLs do not gen- erally produce equal intensity jnd's (Jordan, 1962; Floren- tine et al., 1979; Houtsma et al., 1980; Schlauch and Wier, 1987; Turner et al., 1989). Therefore, not the loudness equality but rather the SPL equality appears to be coincidental in Hellman et al. study.

One possible reason for the disagreement of the Ranko- vic et al.'s results with those of Jordan, Schlauch, and Wier, Hellman et al. is the stimulus timing employed in their loudness matching experiments. They gated the 1-kHz signal simultaneously with the masking noise, a procedure that may have made it difficult for the subjects to judge the loudness of the signal separately from the noise. As we mentioned pre- viously, Houtsma et al. (1980) found this procedure unsatis- factory, and our experience agrees with theirs. However, Rankovic et al. reported that none of their subjects reported experiencing this difficulty. An asymmetrical loudness- matching procedure could be an additional reason. The stan-

984 J. Acoust. Soc. Am., Vol. 93, No. 2, February 1993 Johnson otaL: jnd's for intensity 984


dard reference tone was always presented to the masked ear and the comparison tone to the unmasked ear. However, Schlauch and Wier (1987) used a similarly asymmetrical procedure.

It should be clear from this Introduction that the quan- titative relationship between intensity jnd's and loudness- or loudness-level functions is still controversial. We have two

alternative theories--the equal-loudness, equal-jnd, theory and the proportional-jnd theory. Rankovic et al. were able to fit their results to the latter theory but not to the former. However, Rankovic et al.'s results are in disagreement with those of Jordan, Wier, and Hellman et al. We also have the experiments of Houtsma et al. oriented toward the proportional jnd theory, but not testing the equal-loudness, equal- jnd theory. We are uncertain if the relationship depends on the method of determining the jnd's. Jordan's experiments were based on amplitude modulation; in Houtsma et al.'s and Hellman et al.'s experiments, the pedestal and increment were gated simultaneously. Do similar relationships prevail with a continuous pedestal that produces the smallest jnd's and the smallest variability? Rankovic et al. used both the gated- and continuous-pedestal paradigms obtain- ing different results with each; Turner et al. (1989) found that both paradigms produce essentially the same results above 20 dB SL, except for a multiplicative constant. In addition, Jordan et al.'s and Hellman et al.'s results were evaluated on group rather than individual basis, and Schlauch and Wier and Rankovic et al. used unbalanced loudness-match-

ing procedures. Therefore, there are some experimental gaps that need to be filled before the issue of the relationship between intensity jnd's and loudness functions can be settled.

The present paper addresses several of the outstanding issues. In one set of experiments, the study of Hellman et al. on the independence of the jnd's of the slope of the loudness functions is extended to a continuous-pedestal paradigm and to an additional SPL, and individual results rather than group results are presented. In an additional set of experiments, closely related to the experiments of Rankovic et al., both the equal-loudness, equal-jnd, and the proportional-jnd theories are tested with the continuous-pedestal paradigm and a loudness-matching procedure.

I. METHODS

A. Subjects

Three graduate students (JK, SS, and MS) with normal hearing participated as subjects in the loudness-balance and intensity discrimination measurements. The subjects were between 23-26 years of age. They were paid for their partici- pation. Due to a change of residence in the middle of the study, subject MS was only able to participate in the first part of the experiments, in which jnd's were obtained at intersection points of loudness-balance functions.

B. Loudness balances

1. Stimuli and Instrumentation

The 1-kHz tonal stimuli were presented alternately to each ear via TDH-49 supra-aural earphones seated within MX-41/AR cushions. The subjects were instructed to at-

tend to the loudness of tones presented in quiet to the right ear, and in a background of narrow- or wideband noise to the left ear. The duration of each stimulus was 1-s with a 1.5-s

interstimulus interval and a 4-s interval between pairs. In the noise conditions, the 1-kHz tonal stimulus presented to the left ear was temporally centered within a 2-s noise burst. Rise-fall times of tonal and noise stimuli were 25 ms.

The noise was mixed with the tonal stimulus and always presented to the left ear. A 750-Hz-wide band of noise centered at 1000 Hz was used to obtain the wideband noise loud-

ness-balance functions. In the narrow-band conditions, the noise-generator output was filtered by a laboratory-built, passive filter with - 3-dB cutoffs at 910 and 1125 Hz. Out- side this 215-Hz narrow band, the noise fell off at a rate greater than 200 dB per octave. For each subject we chose to create a 55-dB threshold shift with the narrow-band noise

and a 47-dB threshold shift with the wideband noise, as was done by Hellman et al. (1987). These levels were successful in creating two loudness-balance functions of different slopes that intersected in a moderate intensity range. In addition, a second noise level was employed for the narrow-band noise condition to yield an additional intersection point with the wideband noise loudness-balance function for each subject. (See Figs. 1-3 for exact noise levels for each subject and condition. )

2. Procedures

A variation of the interleaved adaptive procedure described by Jesteadt (1980) was used to obtain loudness-balance measures. The second stimulus of the pair was always the variable stimulus. In all conditions, loudness-balance measures were first obtained by varying the stimulus in the left, masked ear, then by varying the level in the right, unmasked ear. The subject was instructed to respond "louder" when the second stimulus of the pair appeared louder, and "softer" when it appeared softer. The stimuli in two simulta- neous adaptive runs (ascending and descending) were presented randomly. The ascending and descending runs, each consisted of 20 trials. The initial level of the variable stimulus

of the ascending run was 20 dB below the presentation level of the fixed stimulus and the initial level of the variable stim-

ulus in the descending run was 20 dB above the level of the fixed stimulus. The variable stimulus level was decreased

after each "louder" response, and increased after each "softer" response. The initial step size was 6 dB and, after the first. reversal ("louder" response following a "softer" response, or vice versa), was decreased to 2 dB. For each run, the first two reversals were discarded and the mean of the

remaining reversals was calculated. The means of the ascending and descending runs were averaged to obtain the level required for equal loudness with the reference ear. The right-ear stimulus levels were 20, 40, 60, 80 SL or other levels required to define the loudness-balance functions with ade- quate resolution.

985 J. Acoust. Soc. Am., Vol. 93, No. 2, February 1993 Johnson eta/.: jnd's for intensity 985


C. Intensity discrimination 1. Stimuli and Instrumentation

The 1-kHz tonal stimuli were presented monaurally to the left ear according to a continuous-pedestal paradigm (Turner et al., 1989) in quiet, and in a background of narrow- or wideband noise, the same as in the loudness-balance experiments. All intensity increments were generated by the in-phase addition of the increment tone to the pedestal tone. A computer-controlled attenuator determined the level of the increment tones. The increment tones were 1200 ms in

duration and were added to the pedestal during one randomly chosen interval of a 2AFC paradigm. Each trial consisted of two 1200-ms observation intervals separated by 400 ms. All signals had 25-ms rise-fall times except for the in-phase increment, which had a 10-ms rise-fall. The pedestal tone began 400 ms before the first listening interval, stayed on throughout both listening intervals, and was turned off 400 ms after the second listening interval. The masking noise, when present, was presented continuously throughout the entire run. All signals were presented via TDH-49 supra- aural earphones seated within MX-41/AR cushions.

We used long duration signals for the jnd experiments to roughly coincide with the signal duration used in the loudness balances. In preliminary experiments, we noticed that the noise maskers, particularly the narrow-band noise, contained amplitude fluctuations that could confuse the subjects listening for an intensity increment when the signal duration was too short. In addition, although it would have been more consistent with the loudness-balance portion of the experiments to use pedestals gated simultaneously with the increment signals, we chose to employ the continuous- pedestal paradigm for the practical reason that the continuous-pedestal has been shown to yield substantially less within-subject variability than the gated-pedestal paradigm (Turner et al., 1989). In that study, the two stimulus paradigms produced parallel functions ofjnd's plotted as a function of pedestal level above 20 dB SL, with the continuous- pedestal paradigm yielding considerably smaller and less variable jnd's. It should also be pointed out that, according to a large number of studies (for review, see Scharf, 1978), loudness remains invariant for stimulus durations exceeding 200 ms.

2. Procedures

The jnd value were obtained at the two intersection points of the loudness-balance functions. In two subjects (SS and JK), jnd's were also measured outside the intersection points at four sets of equal-loudness levels (30, 40, 50, and 60 dB SL in the absence of masking) for the three conditions: tone in quiet, in narrowband, and in wideband noise. At the intersection points, the tones were judged to be equally loud at the same SPLs, whereas, the slopes of the loudness-balance functions were different. At each of the four additional

sets of equal-loudness levels, loudness equality did not coincide with SPL equality.

After each response, feedback was provided by a light corresponding to the interval which contained the correct r4sponse. A two-down, one-up adaptive procedure control-

ling the level of the in-phase increment in 2-dB steps (Levitt, 1971 ) was used to track the 71% correct point on the psy- chometric function. The procedure continued until 14 reversals occurred. The last ten reversals were averaged to represent the jnd for that run. After a sufficient number of practice runs to insure asymptotic performance, each subject completed a minimum of six runs in each condition. Runs in which the standard deviations of reversals were greater than 4.0 dB were not included in the averages and additional data points were obtained. Final jnd values for each condition were calculated by taking the arithmetic mean of the last six runs. The order of conditions was randomized for each sub-

ject.

II. RESULTS

A. Loudness balances

Figures 1-3 show the final loudness-balance functions obtained in quiet, narrow-band- and wideband noise on the three subjects. Each loudness-balance function shows the means of pairs of values (left, masked ear reference; and

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SPL IN LEFT EAR

FIG. 1. Loudness-balance functions obtained on subject JK in quiet, in wideband noise at 77 dB SPL and in narrow-band noise at 80 dB SPL (upper panel) and 75 dB SPL (lower panel). The intersection points marked with arrows on the abscissas correspond to a tone of 93 dB SPL (upper panel) and 86 dB SPL (lower panel) in the left, masked ear.

986 J. Acoust. Soc. Am., Vol. 93, No. 2, February 1993 Johnson eta/.' jnd's for intensity 986


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FIG. 2. Loudness-balance functions obtained on subject SS in quiet, in wideband noise at 70 dB SPL and in narrow-band noise at 70 dB SPL (upper panel) and 67 dB SPL (lower panel). The intersection points marked with arrows on the abscissas correspond to a tone of 81 dB SPL (upper panel) and 67 dB SPL (lower panel) in the left, masked ear.

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0 , I • I , I , I , I , 0 20 40 60 A80 1 O0 120

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FIG. 3. Loudness-balance functions obtained on subject MS in quiet, in wideband noise at 75 dB SPL and in narrow-band noise at 72 dB SPL (upper panel) and 68 dB SPL (lower panel). The intersection points marked with arrows on the abscissas correspond to a tone of 81 dB SPL (upper panel) and 75 dB SPL (lower panel) in the left, masked ear.

right, unmasked ear reference). The solid lines joining the data points in each figure, therefore, connect the averaged values of the stimulus level in the left, masked ear, required for equal loudness when the right, unmasked ear, served as reference and of the stimulus level in the right, unmasked ear, required for equal loudness when the left, masked ear, served as reference. The average deviation between the two estimates for the narrow-band noise condition across sub-

jects, calculated as the Euclidean distance in dB, was 3.27 dB. The average deviation for the wideband noise conditions across subjects was 3.19 dB, and for the quiet conditions across subjects, 0.96 dB.

The typical loudness-balance function in quiet (Figs. 1- 3) has a slope of approximately 1, while the narrow- and wideband noise functions have slopes greater than 1 (i.e., loudness recruitment). The loudness-balance functions in the narrow- and wideband noise cross at a point where the tones are judged to be of equal loudness at the same SPL. The two different levels of the narrow-band noise yielded two

different loudness-balance functions for each subject, resulting in a higher and lower intersection point. Thus each subject has two panels plotted for loudness-balance functions, showing the intersection point for the wideband noise function with one of the narrow-band functions.

B. Intensity discrimination

For purposes of comparison with previous research, jnd's obtained on two subjects with a continuous-pedestal paradigm in quiet across a range of sensation levels are plotted in Fig. 4. The jnd values expressed in terms of 10 log [ (I + AI)/I] are consistent with those reported by Turner et al. (1989) for the same paradigm and also with the data of Zwislocki and Jordan (1986) obtained by means of amplitude modulation of a pure tone. Note, however, the substantial intersubject differences.



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0.0 a I • I , I • I • 0 20 40 60 80 100

I(dB SL)

FIG. 4. Intensity jnd's obtained by means of the continuous-pedestal paradigm in quiet as a function of sensation level. The individual results of two subjects are shown.

C. jnd's as a function of loudness

One way to examine the assertion jnd's are independent of the slope of the loudness-balance function is to compare the jnd's obtained in a background of narrow-band noise with the jnd's obtained in a background of wideband noise at each of the intersection points of the corresponding loudness-balance functions shown in Figs. 1-3. This is done in Fig. $. Error bars representing plus and minus one standard deviation across the six individual determinations of each

jnd are also included in the plot. If the jnd's in the two noise conditions were equal at the intersection points of the loudness-balance functions, each point in Fig. $ would lie on a line with a slope of one and passing through the origin, as

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FIG. 5. The jnd's obtained on all subjects at the intersection points in a background of narrow- and wideband noise. Error bars indicate plus and minus one standard deviation calculated across the final six runs used to

determine the final data point. The dashed line indicates jnd independence of the type of noise.

988 J. Acoust. Soc. Am., Vol. 93, No. 2, February 1993

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LEFT EAR (dB SPL)

FIG. 6. Illustration of four equal-loudness levels for subject JK. Each equal- loudness level intersects with the loudness-balance functions at three points. The right-hand scale on the ordinate represents the equal-loudness levels in the dB SL with respect to the threshold of the right, unmasked, ear. The arrow on the abscissa marks the level in the left ear of the intersection of the two loudness-balance functions.

indicated by the dashed line. Out of six empirical points five lie close to this line. The regression line for all points has a slope of 0.71 and intercepts the ordinate at the value 0.39. These values are not significantly different from the theoretical line at the 95% level of confidence (Weisberg, 1980), although there seems to be a tendency for the jnd values obtained in the NBN condition, which produced the steeper slope of the function, to be slightly larger than in the WBN condition, contrary to the Fechnerian prediction. Hellman et al.'s data appear to reveal the same tendency.

Although the regression line of the six points is not significantly different from that line, the associated confidence intervals are large due to the small number of points and large standard deviations. Therefore, acceptance of the null hypothesis that the jnd's are the same for equal loudnesses under the two noise conditions is not entirely convincing. As a consequence, we performed additional jnd measurements on two subjects (JK and SS), who were still available, at equal-loudness levels in the no-noise, NBN, and WBN conditions outside the intersection points.

Figure 6 illustrates as an example the derivation of the 30, 40, 50, and 60 dB SL sets of equal-loudness levels for subject JK corresponding to the lower panel of Fig. 1. The SPL in the left ear is plotted on the abscissa, and the cor- respnding SPL in the right ear is plotted on the ordinate. Points of equal loudness for subject SS were derived in the same manner from the data in the lower panel of Fig. 2.

The resulting equal loudness jnd's for the two subjects are plotted in Figs. 7 and 8 as functions of loudness level in the unmasked ear). The jnd's at the intersection points of the wide- and narrow-band noise loudness-balance functions are

also included.

It should be clear from the figures that the jnd's and the slopes of the loudness-balance functions are not correlated. Remember that the smallest slope was associated with the quiet condition and the highest with the NBN one, placing the slope for the WBN condition in between. In Fig. 7, the jnd's for the quiet- and NBN conditions (smallest and lar-

Johnson eta/.' jnd's for intensity 988


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JK WBN

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NBN

0.0 , I , I • I i I i 0 20 40 60 80 100

LOUDNESS LEVEL IN dB

FIG. 7. Equal-loudness jnd's as functions of loudness level for subject JK. The jnd's for the four sets of equal-loudness levels and the two intersection points are plotted.

gest slope, respectively) are practically equal; the jnd's for the intermediate slope associated with the WBN condition tend to be somewhat lower. In Fig. 8, the jnd's for the quiet and WBN conditions are practically identical except at the loudness level of 40 dB. The jnd's for the NBN condition (greatest slope) are larger at the two highest loudness levels than for the other two conditions. Interestingly, for this subject, the loudness-balance functions for WBN and NBN were practically identical at these loudness levels, as is evi- dent in Fig. 2. Therefore, the jnd difference cannot be as- cribed to either a slope or a loudness difference. It should be realized that each set of three data points in Figs. 7 and 8 was obtained at an equal-loudness level for all three conditions, and that jnd independence of the slope of the loudness-balance functions also means that equal-loudness levels produced roughly equal jnd's, independent of the slope and SPL.

These subjective impressions are supported by a repeat- ed-measures ANOVA which examined the factors of condi-

tion (quiet, narrow- and wideband noise), loudness level, and subject. Both the effects of level [F(3,80)= 68.0, p • 0.001 ] and subject [F(1,80) = 66.5,p • 0.001 ] were sig-

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FIG. 8. Equal-loudness jnd's as functions of loudness level for subject SS. The jnd's for the four sets of equal-loudness levels and the two intersection points are plotted.

nificant; however, of particular relevance to the issue of the relation between loudness and the jnd, the main effect of condition was not significant [F(2,80) ---- 2.24, p • 0.11 ]. In addition, there was no significant interaction between subjects and condition [F(2,80) = 1.26, p y 0.29 ], demonstrat- ing that the jnd's under each condition, when plotted as a function of equal loudness, were not significantly different from one another regardless of subject. However, due to the small number of subjects in this experiment, a power analy- sis of the sensitivity of our ANOVA yielded a value too low to permit us to decisively accept the null hypothesis that equal loudnesses results in equal jnd's. Further research in- volving additional subjects is necessary to conclusively ad- dress this issue.

III. DISCUSSION

The results of this study are consistent with the results of Hellman et al. (1987) in concluding that jnd's measured at points of equal loudness and equal SPL, but lying on loudness functions with different slopes, are not dependent upon the slope of the loudness function. They extend Hellman et al.'s study to a larger range of parameter values and to relationships within individual listeners. Our results also sup- port Zwislocki's hypothesis first tested by Jordan (1962) and confirmed independently by Schlauch and Wier (1987) that equal loudnesses are associated with equal intensity jnd's, irrespective of SPL or SL, when the sound frequency is kept constant. In other words, as paradoxical as this may appear, the jnd's appear to be coupled directly to loudness rather than SPL or SL.

The data of the present study can also be evaluated in the framework of the proportional jnd model, as formulated by Lim et al. (1977) and tested most systematically by Houtsma et al. (1980). The present study'sjnd's are used to predict the loudness balance functions, which can then be compared to the loudness balance data determined experimentally. The lower and upper limits of the predicted loudness functions are taken from the range of experimentally obtained jnd's in our study, which correspond to the loudness levels of the signal level in quiet between 30 and 60 dB SL (see Figs. 7 and 8). Thejnd's were converted to predicted loudness functions over this range according to the method described in Houtsma et al. ( 1980); in other words, equally loud tones were predicted when the tones' intensities divided the dynamic range into proportional numbers ofjnd's. The four panels of Fig. 9 display the loudness balances predicted in this fashion, between the tone-in-quiet versus the tone in either wide- or narrow-band noise (dashed lines). In addition, on each plot, the actual loudness balances made by the subjects (from Figs. 1 and 2) are shown by the filled symbols.

At least within the restricted SPLs allowed legitimately by our data, the predicted loudness-balance curves for subject JK agree well with the empirical data. The agreement is considerably less good for subject SS. Interestingly, the data of subject JK also agree better with the equal-loudness, equal-jnd theory than those of subject SS. It seems, therefore, that the proportional-jnd and the equal-loudness, equal-jnd theories may not be mutually exclusive, at least to



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FIG. 9. Loudness-balance functions predicted from the present study's jnd data according to the "proportional jnd" theory. The four panels display the loudness balances between the tone in quiet versus the tone in narrow- or wideband noise for two subjects. The dashed lines are the loudness balances predicted from the jnd's, while the filled symbols represent the actual loudness balances obtained experimentally.

the first order of approximation. One of the reviewers of this manuscript has summarized for us the general circum- stances under which the two theories yield similar predictions. This occurs when the two test sounds to be compared both show the near-miss to Weber's law for their jnd's, for example, tones presented in quiet, tones presented in a wideband noise, or tones presented in a narrow-band noise. To the extent that our data are consistent with the proportional jnd theory, there is a relation between jnd's and the slope of the loudness function, although not in the manner originally described by Fechner.

ACKNOWLEDGMENTS

This research was supported in part by Grant NS24255 from the National Institutes of Health.

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