Effects of aural combination tones on the loudness of a pure tone masked by an inharmonic pure-tone

J. Acoust. Soc. Jpn. (E) 18, 1 (1997) Effects of aural combination tones on the loudness of a pure tone masked by an inharmonic pure-tone Kenji Ozawa, YOiti Suzuki, Hisashi Uematsu, and Toshio Sone Research Institute of Electrical Communication & Graduate School of Information Science, Tohoku University, 2-1-1, Katahira, Aoba-ku, Sendai, 980-77 Japan (Received 8 January 1996) The loudness of a 1,600-Hz pure-tone signal partially masked by a 1,000-Hz, 70-dB SPL masker is experimentally examined using the loudness matching method. As for two of the four subjects who participated in the experiment, the loudness of the signal for signal levels higher than 45 db SPL was found to be enhanced in spite of the masking. This phenomenon disappeared in the presence of one-third-octave band noise whose center frequency was 2,500 Hz. These results suggest that judgments of the loudness are affected by aural combination tones when the masker used is a pure tone. Five loudness functions proposed earlier were tested to determine if they can explain the loudness of a pure tone partially masked by another pure tone. As a result, the formula using the driven firing rate function of the auditory sensory receptor shows good agreement with some of the data. Keywords: Partially masked loudness, Loudness function, Aural combination tone, Loudness enhancement, Pure-tone masker PACS number: 43. 66. Cb, 43. 66. Dc, 43. 66. Ki 1. INTRODUCTION The loudness of a pure tone is one of the most important psychoacoustic characters. Many researchers have therefore tried to determine the general form of the loudness function for a tone. The results are well summarized for example by Scharf (1978) or Scharf and Houtsma (1986), but briefly reviewing, Stevens (1959) argued that the loudness scale showed a power function. For sound-pressure levels above 40 db, the function is given by (1) where V is the loudness of a pure tone, I is the intensity of the pure tone, k is a constant, and the exponent n is approximately 0.3. In the region near threshold, however, the experimental data departs from the power function (Scharf and Stevens, 1961; Hellman and Zwislocki, 1961). This discrepancy is more evident when the tone is partially masked by noise. Even if the external noise is absent, the physiological noise can be regarded as a kind of masking noise to determine the hearing threshold. If so, the origin of the discrepancy from the simple power function of Eq. (1) is always attributable to a masking noise (Lochner and Burger, 1961). When the tone is partially masked, its loudness is referred to as a partially masked loudness or simply a masked loudness. Though a function to describe the masked loudness of a pure tone was proposed by Lochner and Burger (1961), it did not agree with all the subjective data obtained in other studies (Hellman and Zwislocki, 1964). Moreover, it is reported that a narrowerband noise masker produced a steeper loudness function for a pure tone than a wide-band noise masker (Hellman, 1970). Besides Lochner and Burger's function, several forms of the function have been proposed to describe the masked loudness of a

J. Acoust. Soc. Jpn. (E) 18, 1 (1997) tone (Scharf and Stevens, 1961; Zwicker, 1958, 1963; Zwicker and Scharf, 1965; Pavel and Iverson, 1981). All of the above studies, however, investigated the loudness of a tone masked by noise. Thus, it is not clear whether the loudness functions are applicable to the loudness of a tone masked by another pure-tone. We often hear tonal sound with harmonics in our daily life, such as that of a musical instrument or vowels in speech. It is natural to imagine that mutual masking between components of the sound takes place. Therefore, it is fundamentally important to establish the loudness function of a tone partially masked by another pure-tone in order to understand the perception of a complex sound consisting of harmonic tones. There are a few reports on the loudness of a tone masked by a pure-tone masker in which the general form of the loudness function was not considered. For example, Lamore (1977 b) investigated the loudness of the higher frequency (f) component of an octave complex with the low frequency (f) component being a masker. From the results, he determined the "loudness enhancement" phenomenon in which the loudness of the f2 component was enhanced when it was masked by the f1 component. However, the loudness function was not determined. To examine the general form of the loudness function of a tone (signal) partially masked by another tone (masker), it would be best to deal with two tones in harmonic relation because many sounds consist of harmonic components. However, as a nonlinearity exists in the auditory system, the frequencies of a signal and one of the harmonics of a masker, possibly produced by the nonlinearity of the auditory system, are identical if the masker is a subharmonic of the signal. In such a case a vector summation between the signal and the harmonic occurs, so that the loudness of the signal tone varies as a function of its phase difference (Lamore, 1975, 1977 a, b). Therefore, the general form of the loudness function becomes complicated. Regarding the above consideration, the present study investigates the loudness of a pure tone masked by an inharmonic masker as the first step towards understanding the comprehensive form of the loudness function of a pure tone masked by another pure-tone. 2. LOUDNESS-MATCHING EXPERIMEMT BETWEEN AN UNMASKED TONE AND A TONE PARTIALLY MASKED BY AN INHARMONIC TONE 2.1 Method and Apparatus A loudness matching experiment between a partially masked tone and an unmasked comparison tone was carried out using the method of adjustment. The frequencies of both the signal (masked tone) and the comparison tone were 1,600 Hz, and the frequency of the masker was 1,000 Hz, with its level fixed at 70 db SPL. Stimuli were presented monaurally via a headphone (YAMAHA YHD-3) to the right ear of the subject whilst seated in a soundproof chamber. Figure 1 shows the time pattern of the stimuli; each burst has rise- and decay-times of 200 ms with raised-cosine ramps to avoid any audible clicks. The subjects were asked to vary the level of the comparison tone, until it was perceived to have the same loudness as the signal. When the subject judged that the comparison tone was softer/louder than the signal, the level of the comparison tone was increased/decreased by as much as 1 db for the signal levels below 50 db SPL, or 2 db for the levels above 50 db SPL. If the subject did not respond, the level of the comparison tone in the following presentation was kept as in the present one. The initial value of the comparison-tone level was set to be obviously softer than the signal. For each experimental condition, levels of the signal in terms of PSE (Point of Subjective Equality) were examined four times: two times with the time pattern shown in Fig. 1, and two times with a time pattern which had the order of the comparison tone and the signal with the masker reversed. These four adjustments were averaged to obtain a result. Fig. 1 Temporal pattern of a pair of stimuli.

K. OZAWA et al.: LOUDNESS OF A PURE TONE MASKED BY ANOTHER PURE-TONE Four male subjects, 22 to 28 years of age, participated in the experiments. They had no history of auditory diseases, and the minimum audible pressure of each was checked as being normal using a Bekesy-type audiometer from 100 to 10 khz. 2.2 Results Figure 2 shows the results the abscissa is the level of the signal, while the ordinate represents the PSE level of the matched comparison tone. Vertical bars represent the 95% confidence intervals with the model t-distribution. The hearing thresholds of the comparison tone and the masked threshold of the signal were also determined by the method of tracking using the same apparatus as in the loudness-matching experiment. The tracking was started with a downward sequence, in which the level of the signal was decreased in 1-dB steps. Levels at the reversals between upward- and downward-sequences were recorded 20 times, and the last 18 levels were averaged to obtain the threshold data. The thresholds are plotted in Fig. 2 as the lowest point of the data for each subject. If the presence of the masker had no effects on the loudness of the signal, all of the plots would lie precisely upon the diagonal in the figure. The thresholds and the PSE's for low signal levels are lower than the diagonal because of the partial masking of the signal by the masker. As the signal level increases, the PSE's asymptotically approach the diagonal. This shows that recruitment occurs not only with a noise masker but also with a puretone masker. However, the data points lie above the diagonal for signal levels above 45 db SPL. This indicates that the loudness of the signal is "enhanced" while it is masked. This tendency is statistically significant beyond the 0.05 level for subjects 1 and 3. 2.3 Discussion An enhancement in loudness of a pure tone masked by a subharmonic tone was reported by Lamore (1977 b). He observed the enhancement for a wide signal level range, from near masked threshold to high levels. In the present results with an inharmonic-tone masker, however, the enhancement is observed only above 45 db SPL. Similar Sub.1 Sub.3 Sub.2 Sub.4 Fig. 2 Loudness-matching data between an unmasked 1,600-Hz tone and a 1,600-Hz tone masked by a 1,000-Hz tone at 70 db SPL. The ordinate represents the matched level of the unmasked tone as a PSE (Point of Subjective Equality). The vertical bars indicate 95% confidence intervals. 11

J. Acoust. Soc. Jpn. (E) 18, 1 (1997) phenomenon to the present results have also been reported for a noise masker (Sakai and Inoue, 1965). They found this phenomenon only when the frequencies of the signal and the comparison tone were different, whereas for this experiment the frequencies were identical. These differences in the experimental conditions suggest that both the enhancement phenomena observed by Lamore (1977 b) and Sakai and Inoue (1965) cannot be directly compared with the present results. In this loudness-matching experiment, the level of the comparison tone is varied. Hellman and Zwislocki (1964) showed the effect of the procedure on the results in their loudness-matching experiments, i.e., the loudness of the signal was judged to be louder when the level of the comparison tone was varied than when the level of the signal was varied. However, the effect was not so affective as to invoke the "enhancement" phenomenon. For this experiment, the results show that PSE's at signal levels above 40 db SPL increased rapidly for subjects 1, 3, and 4. In addition, all of the subjects reported that they were able to hear tones whose pitches were lower than the masker or higher than the signal at the levels where "enhancement" was observed. Incidentally, it is well known that the nonlinearity of the auditory system produces distortion products with frequencies lower and/or higher than those of acoustical input. From these facts, it can be therefore assumed that the present "enhancement" phenomenon was caused by the aural distortion products produced by the masker and the signal. To confirm this assumption, the subsequent experiments were carried out. 3. LOUDNESS-MATCHING EXPERIMENTS IN THE PRESENCE OF BACKGROUND NOISE 3.1 Method and Apparatus The masker tone of f1 (1,000 Hz) and the signal tone of f2 (1,600 Hz) would create at least quadratic and cubic aural-distortions at 2f1-f2; (400 Hz), f2-f1 (600 Hz), 2f1 (2,000 Hz), 2f2-f1 (2,200 Hz), f1+f2 (2,600 Hz), 3f1 (3,000 Hz), 2f2 (3,200 Hz), 2f1+f2 (3,600 Hz), 2f2 (4,200 Hz), and 3f2(4,800 Hz). Although it is impossible to precisely tell which of them affects judgment of loudness, only the quadratic aural-combination tones, i.e., f2-f1 and f1+ f2 components, are taken into consideration at the present time. This is because the aural combination tones are more predominant than the quadratic and cubic aural-harmonics at 2f1, 2f2, 3f1, and 3f2 (Zwicker and Fastl, 1990) and because the quadratic aural-combination tone at f2-f1, is expected to be superior to the cubic aural-combination tone at for the frequency ratio f2/f1 =1.6 used here (Plomp, 1965 ; Goldstein, 1967). To mask each of the quadratic aural-combination tones, one-thirdoctave band noise, of which the center frequency was either 630 Hz or 2,500 Hz, was adopted. The former will be hereafter referred to as N630, and the latter as N2500. They were fed from a white noise generator (Briiel & Kjxr 1402) via a one-thirdoctave band pass filter (Bruel & Kjar 1614). Their spectral level at the center frequency was set at 51.5 db below the masker in order to mask the aural combination tones (Clack et al., 1972). The experimental procedure was the same as in the previous loudness-matching experiment except for the presence of continuous background noise during the experiments. The experiments were carried out first in the presence of N630, then N2500. 3.2 Results The PSE's under the conditions with N630 and N2500 are shown in Fig. 3(a) and (b), respectively (the results under the condition without noise are taken from Fig. 2). To avoid confusion, the 95% confidence intervals are shown only for the results obtained under the conditions with the noises. The hearing thresholds changed slightly in almost all of the conditions, since background noise (N630 or N2500) was present for this experiment. Obviously, shifts in the hearing threshold would affect the loudness of both the signal tone and the comparison tone in the presence of background noise. However, considering the loudness function for a pure tone partially masked by noise (e.g., Lochner and Burger, 1961), the effect would be negligible when the level of the tone is about 10 db higher than the hearing threshold. Therefore, we assume that the results under the three conditions of background noise are comparable with each other except for levels near the hearing thresholds. Though PSE's seem to decrease slightly when N630 was present, the rapid increments in PSE's observed under the condition without noise still remain for subjects 1, 3 and 4. When N2600 was present, however, the rapid increments in PSE's under the condition without noise disappeared. Furthermore, the 12

K. OZAWA et al.: LOUDNESS OF A PURE TONE MASKED BY ANOTHER PURE-TONE Sub.1 Sub.3 Sub.2 Sub.4 (a) Data under the conditions without noise and with N630. Sub.1 Sub.3 Sub.2 Sub.4 (b) Data under the conditions without noise and with N2500. Fig. 3 Loudness matching data for individual subjects under the three conditions of background noises. Both N630 and N2500 are one-third-octave band noises. The center frequency of N630 is 630 Hz, while that of N2500 is 2,500 Hz. The vertical bars indicate 95% confidence intervals for the data with the noises. The data without noise are taken from Fig. 2. 13

J. Acoust. Soc. Jpn. (E) 18, 1 (1997) "enhancement" disappeared for subject 1, and was no longer statistically significant for subject 3. These results revealed that aural combination tones at frequencies near 2,500 Hz affected the loudness of the pure tone. The experiments were carried out in order of experimental conditions without noise, with N630, and with N2500. The authors consider that the order of the experimental conditions hardly affected the results, because the PSE's for signal levels below 45 db SPL are very similar among the experimental conditions for all the subjects including subjects 1 and 3 who exhibited clear disappearance of the enhancement when the masker was added. 3.3 Discussion The results support the assumption that the aural combination tones affect the loudness of a pure tone masked by an inharmonic tone so that the loudness of the masked pure tone was judged to be greater. In other words, part of the loudness of an aural combination tone is included in the loudness of the pure tone when being partially masked by another pure-tone. Returning to subject 3, the tendency of "enhancement" still remains, though it is no more statistically significant. There are two possible explanations for this. One is that the judgments were somewhat biased by the masker ; i.e., this subject would have judged the loudness of the signal including a part of the loudness of the masker. The other is that the distortion products other than that masked by the background noises were predominant, so that their effects could not be completely eliminated by the noises given here. Anyway, the tendency of subject 3 cannot be fully interpreted from the present data. Incidentally, all the subjects reported that at near threshold levels it was difficult to distinguish the loudness of the signal tone from that of the complex tone consisting of the signal and the masker, irrespective of the presence of the background noise. This seems to show that a more advanced procedure than the simple loudness-balance would be required to obtain an exact loudness-function under various conditions. 4. CONSIDERATION ON THE FORM OF THE LOUDNESS FUNCTION OF A TONE PARTIALLY MASKED BY AN INHARMONIC TONE As stated in the introduction, several equations have been proposed to estimate the loudness of a pure tone partially masked by noise. All of them were originally introduced to explain the loudness of a sound partially masked by wide-band noise. Among them, an equation proposed by Pavel and Iverson (1981) requires the level of the masking noise. It is therefore difficult to apply the equation to the present case where a pure tone is masked by another pure-tone. Conversely, equations proposed by Scharf and Stevens (1961), Lochner and Burger (1961), and Zwicker (1958, 1963), and Zwicker and Scharf (1965) seem to be applicable to the present condition because they only require the hearing threshold of the signal in the presence of the masker. The complete forms of the applicable equations are as follows: the equation proposed by Scharf and Stevens (1961) is given by (2) where I is the intensity of the tone, and I0 is the intensity of the pure tone at threshold. This function states that the masked loudness of a pure tone is given by a power function of its "unmasked intensity." Lochner and Burger (1961) proposed (3) where the exponent n is 0.27 for a 1,000-Hz puretone. This function shows that the masked loudness of a pure tone is found by subtracting the magnitude of the hearing threshold denoted in loudness from that in the absence of any masker. This equation implies that the partially masked loudness is an unmasked part of the original loudness and this idea fits well with the additivity of loudness. Lochner and Burger (1961) showed that the loudness of a 1-kHz pure tone at 40 db SPL (I=10-8 W/ m2) is 1.09 and 1.00 sone by Eqs. (2) and (3), respectively, if the threshold is 0 db SPL (I0=10-12 W/m2) and k=157.7. Zwicker (1958, 1963), and Zwicker and Scharf (1965) derived the loudness of a tone as an integral of specific loudness over the critical-band rate. The specific loudness ƒõ' of a partially masked tone is given by 14

K. OZAWA et al.: LOUDNESS OF A PURE TONE MASKED BY ANOTHER PURE-TONE where E is the excitation evoked by the tone, E0, is the excitation at threshold, and Er is the excitation corresponding to the reference intensity Ir=10-12 W/m2. The exponent n is 0.23 when the tone is not masked. It becomes larger if the tone is partially masked. At the signal frequency, the ratios Er/E0 and E/E0 are equal to Ir/I0 and I/I0, respectively. At other frequencies, excitations are determined by slopes of thresholds for pure tones masked by narrow-band noise. If the threshold is 0 db SPL, the specific loudness of 1-kHz pure tone at 40 db SPL is 0.444 soneg/bark at that frequency, and the loudness is calculated as being 1.12 soneg by a computational program (Paulus and Zwicker, 1972) for determining the excitations at frequencies other than 1 khz and integrating the specific loudness over the critical-band rate. In addition to these three functions, some other functions have been proposed to describe the loudness of an unmasked pure tone. Attneave (1962), Curtis et al. (1968), Curtis and Fox (1969), and Zwislocki (1983) argued a two-stage scaling model as a general form of the power function, i.e., Eq. (1). (4) In the model, the psychological process involved in judgment of loudness is divided into two parts. The first part transforms the intensity of a pure tone into a sensation magnitude. Then this sensation magnitude is transformed into loudness via a number-assigning process. Although the model adequately explains the loudness additivity (Zwislocki, 1983), it does not refer to the discrepancy near threshold of the loudness from the power function. Here it is assumed that the function can be expanded to describe this discrepancy. For example, if the sensation magnitude is given by the form of Eq. (3), then loudness ƒõ is given by where a and b are dimensional constants, and the product of n and m is known to be approximately 0.3. For example, the average numerical values for subjects seen in the literature (Zwislocki, 1983) are n=0.34 and m=1.08, respectively. If the constant abm is assumed to be 908.9, the loudness of a tone at 40 db SPL is 1.0 with these values of n and m and the threshold of 0 db SPL. Hellman and Hellman (1975) represented the loudness function of a pure tone using the driven (5) firing rate function of the auditory sensory receptor. They showed that the function agrees well with the loudness of a pure tone in the absence of masking sound. We assume that the function is also applicable to the partially masked loudness by slightly modifying the function as where I is the intensity of the tone, Io is the intensity of the pure tone at threshold in the presence of the masker, Ir is the reference intensity of 10-12 W/m2, C is a normalization constant, R0 is the firing rate in neurons in the absence of an external stimulus, Rm is the saturation firing rate, and the term Rm[1-R0/ the driven rate function. Clearly ƒõ=0 when I= because the driven rate should equal to zero. The ratio R0/Rm is approximately 0.1 (Hellman and Hellman, 1975). If the term CRm is 9.24 ~10-2 and (6) n is 0.27, the loudness of a tone at 40 db SPL is 1.0 when the threshold is 0 db SPL. We have tested these five equations to determine if they can explain the loudness of a pure tone partially masked by another pure-tone or not. Figure 4 shows a comparison between the data obtained in the presence of N2500 with the estimated values from Eqs. (2) through (6). It can be seen that the estimation from Eq. (2) is independent of the exponent term of n because the terms are canceled between the masked tone and the comparison tone. For Eq. (3), the exponent n is fixed at 0.27 (Lochner and Burger, 1961). Alternatively for Eqs. (4), (5), and (6), the parameter values which provide the best fit for each subject are determined by minimizing the mean-squares error along the ordinate. Regarding Eq. (4), ƒõ' of the signal is compared with that of the comparison tone directly without taking its integral over the critical-band rate. This means that we assume that the excitation patterns of the signal and the comparison tone are similar. Here, the exponent n in Eq. (4) for the signal is assumed to be greater than or equal to 0.23 because Zwicker (1963) stated that the exponent of the power function became greater when it was partially masked. Estimation from Eq. (5) is also independent of the exponent term of m, so that only the term of n was changed to fit the equation to the data. For Eq. (6), two parameters were changed, i.e., the ratio of 15

J. Acoust. Soc. Jpn. (E) 18, 1 (1997) Sub.1 Sub.3 Sub.2 Sub.4 Fig. 4 Comparison of earlier proposed loudness-functions with the loudness-matching data obtained in the presence of N2500. The vertical bars indicate 95% confidence intervals for the data with N2500. the spontaneous firing rate (R0M) to the saturation firing rate (RmM) for the masked tone, and the ratio of the saturation firing rate for the unmasked comparison tone (R,mQ) to that for the masked tone Table 1 shows the values of the parameters of Eqs. (4), (5), and (6) used to minimize the mean-squares error. For Eq. (4), differences in the values among the four subjects are small and the values are equal to or greater than 0.23 which well agree with Zwicker's (1963) results. As for Eq. (5), the difference in the values are greater than the individual differences from 0.23 to 0.43 reported by Zwislocki (1983). For Eq. (6), although the values are different among the subjects, it is difficult to discuss the validity of the values because there is insufficient data on the driven firing rate function masked by a pure-tone masker in the human auditory system. Though some data are available for some animals with different maskers, the data indicate quite a wide range of values. For example, for a single cat cochlear fiber masked by a wide-band noise masker, the ratio R0/ Rm varies from 0.05 to 0.5, and the ratio RmQ/RmM changes from 0.8 to 3.3 depending on the level of the masker (Evans, 1974). Thus, the authors consider the values shown in Table 1 Fitted values of parameters in Eqs. (4), (5), and (6). Superscripts Q, M of the parameters in Eq. (6) represent the condition of the absence and the presence of the pure-tone masker, respectively. Table 1 for Eq. (6) to be within the range of possibility. As shown in Fig. 4, Eqs. (2), (3), and (5) cannot explain the tendency of "enhancement" which is seen in the data of subject 3, whichever values are assigned to the parameters. As for the other subjects, the equations also fail to satisfactorily explain the masked loudness of the pure tone, even though the large individual values are assigned. Eqations (4) and (6), on the other hand, can explain the enhancement. For Eq. (4), when the exponent n of the partially masked tone is estimated to be greater than that of the unmasked tone, the 16

K. OZAWA et al.: LOUDNESS OF A PURE TONE MASKED BY ANOTHER PURE-TONE estimated PSE's exceed the diagonal for higher levels. For Eq. (6), the loudness of the partially masked tone can exceed the diagonal if it is assumed that the saturation firing rate Rm increases and the ratio R0/Rm decreases when the tone is partially masked as shown in Table 1. We are not confident as to the validity of this assumption as mentioned. above ; if the saturation firing rate Rm, is kept constant against the induction of the masker sound, Eq. (6) cannot then explain the enhancement. Some explanations are possible for the discrepancies between the data and the equations. For example, 1. The aural distortion products unmasked by either N630 or N2500 affect the function. 2. The general form of the loudness functions of a tone partially masked by another pure-tone is essentially different from that of the tone partially masked by noise. This implies that the mode of loudness perception is different between the two conditions. It is not guaranteed that the effect of the other aural distortion products is suppressed to be negligibly small by the background noise used here. If the effect of aural distortion products other than those that are masked by the noises still remains, it is interesting that Eq. (6) can explain the "enhancement" of the loudness when Rm in the presence of the masker is set to be greater than that without the masker. We can consider that the growth of Rn, in the presence of the masker is due to the contribution of not only the masker but also aural distortion products other than that masked by the background noise. When we review the data closely, the rapid increment in PSE's around 40 db is found in the data of subjects 1, 3, and 4. This might suggest that this increment is caused by other aural distortion products. Equation (6) seems to be the most promising one at present to explain the loudness of a tone partially masked by another tone. We cannot immediately conclude here, however, that Eq. (6) is the general formula which can explain the masked loudness of a pure tone. The equation has so many degrees of freedom that it provides a good fit even without reasonable modeling of the system. In any case, further experiments in the presence of wideband noise at low sound-pressure level might be useful in order to investigate this possibility. There are two possible explanations as to why a pure-tone masker and wide-band noise lead to different forms of the loudness functions. One is related to the correlation between the pure-tone signal and the masker. When a pure tone is masked by noise, there is little correlation between them. When a pure tone is masked by another tone, there can be some correlation. Thus the mode of detecting the pure-tone signal from the masker could be different depending on whether the masker is noise or a pure tone. This would then make the loudness function with a correlated masker different from that with an uncorrelated one. From the psychoacoustical point of view, the correlation between the signal and the masker might exist in similarity in timbre perceived by a subject. The similarity would have made it difficult for the subjects to separate the loudness of the signal from that of the complex sound consisting of the signal and the masker, as claimed by the subjects. Such a difference in the difficulty in the loudness matching between the cases with a pure-tone masker and that with a noise masker might cause biases in the loudness judgment of the signal. The other explanation is that the stage in the hearing system used to perceive the loudness of a signal partially masked by a tone may be different from the stage for a signal masked by noise. The analysis in the latter case is carried out within a single auditory filter such as a critical band filter. Conversely, the signal and the pure-tone masker used in the present experiments would not be included in a single critical band. Therefore, a different analysis than that referring to a signal detection in a single critical band would have operated in order to perceive the loudness of the signal that is masked by another pure-tone existed far enough apart from the signal beyond the critical band. Experiments using a signal and a masker that are included in the critical band might be useful to confirm this assumption. 5. CONCLUSION In this paper, the loudness function of a 1,600-Hz 70-dB SPL pure-tone masker has been examined. When the level of the signal tone was low, its loudness decreased because it was partially masked by the pure-tone masker. For two of the four subjects, its loudness was "enhanced" although it was masked at signal levels above 45 db SPL. The "enhancement," however, disappeared in the pres- 17

J. Acoust. Soc. Jim. (E) 18, 1 (1997) ence of the one-third-octave background noise whose center frequency was 2,500 Hz and the level at the center frequency was 51.5 db below the masker level. This suggests that the aural combination tone at f1+f2 affected the loudness judgments of the signal so that the loudness of the combination tone is included in that of the signal. An attempt was made to explain the data obtained in the presence of the background noise by fitting the loudness functions presented earlier. As a result, the formula that was originally proposed by Hellman and Hellman (1975) shows good agreement with some of the data. However, it cannot be asserted that the formula gives the general form of the loudness function of a tone partially masked by another pure-tone. ACKNOWLEDGEMENT This study was, partially supported by a Grant-in- Aid for Scientific Research (No. 07750071) by the Ministry of Education, Science, Sport and Culture of Japan and the Sound Technology Promotion Foundation. REFERENCES Attneave, F. (1962). "Perception and related areas," in Psychology: A Study of a Science, Vol. 4, S. Koch, Ed. (McGraw Hill, New York), pp. 619-659. Clack, T. D., Erdreich, J., and Knighton, R. W. (1972). "Aural harmonics: The monaural phase effects at 1500 Hz, 2000 Hz, and 2500 Hz observed in tone-on-tone masking when f1=1000 Hz," J. Acoust. Soc. Am. 52, 536-541. Curtis, D. W., Attneave, F., and Harrington, T. L. (1968). "A test of a two -stage model of magnitude judgment," Percept. Psychophys. 3, 25-31. Curtis, D. W. and Fox, B. E. (1969). "Direct quantitative judgments of sums and a two-stage model for psychophysical judgments," Percept. Psychophys. 5, 89-93. Evans, E. F. (1974). "Auditory frequency selectivity and the cochlear nerve," Facts and Models in Hearing, E. Zwicker and E. Terhardt, Eds. (Springer-Verlag, Berlin), pp. 118-129. Goldstein, J. L. (1967). "Auditory nonlinearity," J. Acoust. Soc. Am. 41, 676-689. Hellman, R. P. and Zwislocki, J. (1961). "Some factors affecting the estimation of loudness," J. Acoust. Soc. Am. 33, 687-694. Hellman, R. P. and Zwislocki, J. (1964). "Loudness function of a 1000-cps tone in the presence of a masking noise," J. Acoust. Soc. Am. 36, 1618-1627. Hellman, R. P. (1970). "Effect of noise bandwidth on the loudness of a 1000-Hz tone," J. Acoust. Soc. Am. 48, 500-504. Hellman, W. S. and Hellman, R. P. (1975). "Relation of the loudness function to the intensity characteristic of the ear," J. Acoust. Soc. Am. 57, 188-192. Lamore, P. J. J. (1975). "Perception of two-tone octave complexes," Acustica 34, 1-14. Lamore, P. J. J. (1977 a). "Pitch and masked threshold in octave complexes in relation to interaction phenomena in two-tone stimuli in general," Acustica 37, 249-257. Lamore, P. J. J. (1977 b). "Investigation of two-tone interaction in octave complexes with the help of the pulsationthreshold method," Acustica 39, 7-15. Lochner, J. P. A. and Burger, J. F. (1961). "Form of the loudness function in the presence of masking noise," J. Acoust. Soc. Am. 33, 1705-1707. Pavel, M. and Iverson, G. J. (1981). "Invarient characteristics of partial masking: Implications for mathematical models," J. Acoust. Soc. Am. 69, 1126-1131. Paulus, E. and Zwicker, E. (1972). "Programme zur automatischen Bestimmung der Lautheit aus Terzpegeln oder Frequenzgruppenpegeln," Acustica 27, 253-266. Plomp, R. (1965). "Detectability threshold for combination tones," J. Acoust. Soc. Am. 37, 1110-1123. Sakai, H. and Inoue, T. (1965). "Note on the masked loudness and the critical band," Proc. Autumn Meet. Acoust. Soc. Jpn., 43-44 (in Japanese). Scharf, B. and Stevens, J. C. (1961). "The form of the loudness function near threshold," Proc. Int. Congr. Acoust., 3rd, Stuttgart, 1959, L. Cremer, Ed. (Elsevier Publishing Co., Inc., Amsterdam), pp. 80-82. Scharf, B. (1978). "Loudness," in Handbook of Perception IV, Hearing, E. C. Carterette and M. P. Freidman, Eds. (Academic Press, New York), Chap. 6, pp. 187-242. Scharf, B. and Houtsma, A. J. M. (1986). "Audition II: loudness, pitch, localization, aural distortion, pathology," in Handbook of Perception and Human Performance I, Sensory Processes and Perception, K. R. Boff, L. Kaufman, and J. P. Thomas, Eds. (John Wiley & Sons, New York), Chap. 15. Stevens, S. S. (1959). "On the validity of the loudness scale," J. Acoust. Soc. Am. 31, 995-1003. Zwicker, E. (1958). "Uber psychologische und methodische Grundlagen der Lautheit," Acustica 8, 237-258. Zwicker, E. (1963). "Uber die Lautheit von ungedrosselten und gedrosselten Schallen," Acustica 13, 194-211. Zwicker, E. and Fastl, H. (1990). Psychoacoustics (Springer-Verlag, Berlin), Chap. 14. Zwicker, E. and Scharf, B. (1965). "A model of loudness summation," Psychol. Rev. 72, 3-26. Zwislocki, J. J. (1983). "Group and individual relations between sensation magnitudes and their numerical estimates," Percept. Psychophys. 33, 460-468. 18