Modifications to the Power Function for Loudness

This paper reviews recent findings on the form of the loudness function for mid-frequency tones. Loudness matches between pure tones and tone complexes with equal-SL components show that loudness at threshold exceeds zero and that the loudness function approaches a power function with asymptotic exponent (re intensity) somewhat greater than unity at levels well below threshold. They also show that the local exponent at moderate SLs is about 0.2, somewhat lower than the exponent of about 0.3 that is usually assumed to describe the growth of loudness at moderate and high levels. This finding indicates that loudness is likely to deviate from a simple power function and grow more slowly at moderate than at high levels. Measurements of temporal integration of loudness (i.e., the level difference between equally loud 5and 200-ms tones) show that the amount of temporal integration varies nonmonotonically with level and is largest at moderate levels. Loudness functions derived from these measurements under the assumption that the loudness ratio between equal-SPL long and short tones is independent of SPL are shallower at moderate levels than at low and high levels. The local exponent at moderate levels is similar to that obtained from loudness matches between tones and tone complexes. The finding that the loudness function is shallower at moderate than at low and high levels agrees with basilar-membrane input-output functions. Loudness appears to be approximately proportional to the square of the basilar-membrane velocity, at least at low and moderate levels. Examination of the loudness ratios obtained between long and short tones of various durations indicates that a second-order non-linearity is likely to occur between the auditory nerve and the site of temporal integration. The loudness function for a mid-frequency tone at moderate and high levels is well described by a power function of the tone’s intensity (e.g., Hellman, 1991). The exponent is about 0.3. (Note that this and all other exponents stated in the present paper presume that the stimulus is measured in terms of its intensity; if the stimulus is measured in terms of its sound pressure, the exponents are twice those stated here.) At low levels, the loudness function deviates from a simple power function. A large number of studies show that the local exponent [i.e., the slope of log(loudness) as a function of log(intensity)] increases as the tone approaches threshold, but considerable disagreement exists about the form of the loudness function at low levels (for review, see Buus et al., 1998). The present paper reviews some recent studies that offer new insights into how the loudness function for tones deviates from a simple power function. The loudness function at low and moderate levels Because the true loudness function must be additive (Fletcher and Steinberg, 1924), a tone complex consisting of n equally loud components ought to be n times as loud as each component by itself—provided that the components do not mask one another. Accordingly, the level difference between individual components of such a tone complex and an equally loud pure tone ought to show the increase in level necessary to increase the loudness of a tone n times. Based on this premise, Buus et al. (1998) attempted to determine the form of the loudness function at low and moderate levels from loudness matches between pure tones and tone complexes. The components of the complexes were set to equal sensation level (SL) to approximate equal loudness (cf. Hellman and Zwislocki, 1961). To control for possible effects of mutual masking among the components and to assess the consistency of the listeners’ judgments, loudness matches were obtained for four-tone complexes with frequency separations, ∆fs, of 1, 2, 4, and 6 critical bands (Barks) and for ten-tone complexes with frequency separations of 1 and 2 Barks.μ Figure 1 shows results for a representative listener, L1. Several interesting findings are apparent. First, complexes in which the components are below their individual thresholds are consistently matched with pure tones that are a few dB above threshold. This shows that four or ten subthreshold components combine to produce a loudness greater than zero. Thus, the data for L1 show unequivocally that the loudness at and even below threshold must be greater than zero. Second, the loudness matches near threshold indicate that the local exponent of L1’s loudness function just above threshold is somewhat larger than unity. Four-tone complexes with each component set to 0 dB SL are as loud as a single tone at 4 dB SL. This indicates a local exponent of log(4)/0.4=1.5. Ten-tone complexes with a component level of 0 dB SL are as loud as a pure tone at 7 dB SL. This indicates a local exponent of log(10)/0.7=1.4, quite consistent with the estimate obtained from the four-tone complexes. Third, estimates of the local exponent at moderate SLs can also be readily obtained from the data. The four-tone complexes with ∆fs of 4 and 6 Barks (right panel) yield almost identical results. This indicates that masking is unlikely to occur among components with these large ∆fs. It also indicates that the loudness of the complex does not depend critically on the frequencies of the individual