MODIFICATIONS TO THE POWER FUNCTION FOR LOUDNESS Søren uus 1,2 and Mary Florentine 1,3 1 Institute for Hearing, Speech, and Language 2 Communications and Digital Signal Processing Center, ECE Dept. (440 DA) 3 Dept. of Speech-Language Pathology and Audiology (133 FR) Northeastern University, 360 Huntington Avenue, oston, MA 02115, U.S.A. E-mail: buus@neu.edu, florentin@neu.edu Abstract 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 5- and 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 uus 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 ecause 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. ased on this premise, uus 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 (arks) and for ten-tone complexes with frequency separations of 1 and 2 arks.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 d 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 d SL are as loud as a single tone at 4 d SL. This indicates a local exponent of log(4)/0.4=1.5. Ten-tone complexes with a component level of 0 d SL are as loud as a pure tone at 7 d SL. This indicates a local exponent of log()/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 arks (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 Pure Tone Level [d SL] 70 60 50 40 30 20 L1 Quiet f = 1 ark f = 2 arks f = 4 and 6 arks 0-5 5 15 25 ten-tone complex J four-tone complex ten-tone complex J four-tone complex J f = 4 arks F f = 6 arks -5 5 15 25-5 5 15 25 35 Component Level in Complex [d SL] Figure 1. Representative loudness matches between 1.6-kHz pure tones and tone complexes centered near 1.6 khz. Unfilled symbols show data obtained when the adaptive procedure varied the level of the tone complex and filled symbols show data obtained when the adaptive procedure varied the level of the pure tone. The solid lines show polynomial functions fitted to the data. The error bars show plus and minus one standard error of the mean calculated across three repetitions of each loudness match. The vertical dashed line indicates threshold for the individual components in a tone complex. The dotted lines indicate equal SLs for the pure tone and the components of the tone complex.
components and that setting the components to equal SLs makes them approximately equally loud. L1 s data for the 4- and 6-ark frequency separations show that a four-tone complex with a component level of 20 d SL is as loud as a pure tone at 46 d SL. This indicates that the local exponent averaged across levels from 20 to 46 d SL is about log(4)/2.6=0.23. Similar results were obtained for the other listeners. To determine individual listeners loudness functions more precisely, a simple model of loudness summation was applied to the data for each listener. It showed that all the data could be accounted if the loudness of a single tone, N t, was described as follows: s s SL Nt( SL)= k 1+ snrth 1 (1) where k is a scaling factor that does not affect predictions of loudness matches, snr th is the signal-to-noise ratio for a tone at threshold (estimated within the critical band centered on the frequency of the tone), SL is the sensation level of the tone (in d), s - is the asymptotic local exponent of the loudness function at low levels, and s is the asymptotic local exponent at high levels. (Note that the asymptotic local exponents may not be reached within the range of levels encompassed by the data.) This function is similar to Zwislocki s (1965) modified power function, except that the parameter s - has been added to allow the low-level slope of the function to vary. The average of the loudness functions obtained by fitting the model to loudness-matching data for six listeners with normal hearing is well represented by Eq. (1) when snr th =0.25, s - =1.4, and s =0.11. The thick solid line in Fig. 3 shows the resulting loudness function, which agrees with data obtained in a variety of studies and appears to provide a good description of the loudness of pure tones with levels up to 40 d SL (see uus et al., 1998). The local exponent is about 1.3 at 0 d SL and 0.19 at 40 d SL. The latter local exponent is somewhat smaller than that of about 0.3 generally used to describe the growth of loudness at moderate at high levels (Hellman, 1991), but it is consistent with loudness functions derived from loudness matches between long and short tones, as discussed below. s The loudness function at moderate and high levels Several recent studies from our laboratory indicate that the loudness function may be derived from loudness matches between short and long tones (e.g., Florentine et al., 1996; for review, see uus, 1999). The derivation is based on the assumption that the loudness ratio between equal-spl long and short tones is independent of SPL. This assumption seems well supported, except perhaps at low levels (Florentine et al., this volume). Typical data and the loudness function derived from them are shown in Fig. 2 (uus et al., 1999). The left panel shows that the amount of temporal integration defined as the level difference between equally loud short and long tones varies non-monotonically with level and is largest at moderate levels. For 5- and 200-ms tones it is about 14 d at 8 d SL, increases to about 24 d at moderate SLs, and decreases again to about 12 d around 0 d SPL. The right panel shows the loudness functions for 5- and 200-ms tones derived from the temporal-integration data. As indicated by the arrows, the ratio between the two loudness functions is constant, but the slope is forced to vary to make the horizontal distance consistent with the temporal-integration data. The loudness functions are steep when the amount of temporal integration is small and are shallow when the amount of temporal integration is large. Thus, the finding that temporal integration is largest at moderate levels indicates that loudness may grow more slowly at moderate levels than at low and high levels. Nevertheless, the difference between Zwislocki s (1965) standard loudness function and those in Fig. 2 is quite modest and is well within the error of measurement.
0 L 5 ms - L 200 ms [d] 30 20 14 d 24 d Long varied Short varied Loudness [Sones] 1 0.1 14 d x 4.2 24 d x 4.2 0 0 20 40 60 80 0 120 Level of 200-ms Tone [d SPL] 0 20 40 60 80 0 120 Tone Level [d SPL] Figure 2. The left panel shows the level difference between equally loud 5- and 200-ms tones, averaged across six normal listeners. The right panel shows the loudness functions derived from the amount of temporal integration (see text for further explanation). Relation between loudness and basilar-membrane input-output function The slow growth of loudness at moderate levels is not widely recognized, but it is consistent with loudness functions derived from loudness matches between tones and tone complexes, as noted earlier. It is also apparent in the loudness function obtained by Fletcher and 50000 Munson (1933). Finally, it appears consistent with measurements of basilarmembrane input-output functions (e.g., Ruggero et al. (1997) 00 khz 000 Ruggero et al., 1997), as demonstrated in 9 khz 8 khz 7 khz 0 Fig. 3. Thus, there is considerable support for the idea that loudness functions for pure tones deviate from simple power 00 functions, even at moderate and high 1 levels. Although the deviation is quite modest, taking it into account simplifies 0.1 the explanation of how the level difference 0 between equally loud sounds of different 0.01 bandwidths or different durations varies with level. As discussed further below, it uus et al. (1998) 0.001 also helps establish a relatively simple Florentine et al. (1998) relationship between basilar-membrane 0.0001 5 mechanics and the growth of loudness. M Velocity [µm/s] 0 20 40 60 80 0 120 Tone Level [d SPL] Loudness [sones] Figure 3. Comparison between loudness and basilar-membrane input-output functions. The symbols show basilar-membrane vibration amplitudes measured at and below the best frequency (left ordinate). The lines show loudness functions obtained from spectral (solid) and temporal (dashed) integration (right ordinate). Note that the two loudness functions overlap between 23 and 43 d SPL. Figure 3 compares loudness functions obtained from loudness matches between tones and tone complexes (uus et al., 1998) and between 5- and 200-ms tones (Florentine et al., 1998) with basilarmembrane input-output functions measured by Ruggero et al. (1997). The figure shows input-output functions for frequencies at and below the best frequency of khz. As the sound level increases above 60 d SPL, the largest
response is produced by frequencies below the best frequency owing to the basalward migration of the traveling-wave envelope at high SPLs. Presumably, the -khz tone produces a similarly large response at a place somewhat basal to the place of measurement. ecause the place at which the vibration occurs is likely to be relatively unimportant for the loudness produced by some stimulus, a first approximation to the relation between loudness and basilarmembrane vibration reasonably can be thought to reflect the maximal vibration amplitude that a given tone produces anywhere along the basilar membrane. For measurements taken at a fixed place, however, we may consider the maximal vibration amplitude produced by any frequency, which ought to be equivalent owing to the trading of frequency and place. Viewed in this manner, Ruggero et al. s (1997) data are consistent with the loudness function being considerably less compressive at high than at moderate SPLs, but note that three log units of basilar-membrane vibration (left ordinate) match six log units of loudness (right ordinate). Thus, the good agreement between the basilar-membrane data and the loudness functions, at least up to 60 d SPL, indicates that loudness is approximately proportional to the square of the maximal basilar-membrane vibration amplitude. Above 60 d SPL, it appears that loudness may grow somewhat slower than the square of the basilar-membrane amplitude, but it is also possible that the deviation between the loudness function and the square of the basilarmembrane vibration amplitude may reflect differences between humans and chinchillas. Schlauch et al. (1998) suggested that the quadratic relationship between basilar-membrane velocity and loudness might reflect quadratic input-output functions of the inner hair cells. However, the quadratic input-output functions of inner hair cells hold only over a limited range of low levels and, as discussed below, there is reason to believe that the quadratic non-linearity occurs central to the auditory nerve. ecause loudness almost certainly reflects an integration of neural activity across time and frequency, considering the relation between the number of auditory-nerve firings produced by equal-spl stimuli of different durations ought to provide some information about how loudness is coded in the auditory system. To this end, one may consider whether integration of the auditory-nerve activity as it decays due to auditory-nerve adaptation (Young and Sachs, 1973; Westerman and Smith, 1984; see uus and Florentine, 1992) is compatible with the effect of duration on loudness. ecause the decay is approximately independent of SPL, such a model predicts that the loudness ratio between equal-spl long and short tones should be the same at all SPLs. This prediction is in qualitative agreement with the data, but the predicted ratios are much larger than those obtained. As shown in Fig. 4, however, integrating the square of the decaying auditory-nerve activity yields results that closely match the effect of duration on relative loudness. This finding indicates that a 1 quadratic non-linearity is likely to occur after the auditory nerve but before temporal integration. This suggestion agrees with Zeng and Shannon s (1994) finding that loudness in electrical hearing indicates the existence of a central expansive non-linearity. Although Florentine et al. (1996) they suggested that the non-linearity might be uus et al. (1997) exponential, their data may also be uus et al. (1999) compatible with a quadratic non-linearity. Relative loudness Integrate squared count 0.1 1 0 300 Equivalent Rectangular Duration [ms] Figure 4. Relative loudness as a function of duration. The symbols show data from three recent studies. The solid line shows the result of integrating the square of the auditory-nerve response as it decays due to adaptation. These findings point to a relatively simple model of loudness coding. The compressiveness of the loudness function appears to reflect primarily the strong compression that results from the non-linear basilar-membrane mechanics. ecause loudness is approximately proportional to the square of the basilar-membrane vibration amplitude at least up to about 60 d
SPL a quadratic non-linearity appears to intercede between the basilar membrane and the loudness code. The finding that loudness is approximately proportional to the square of the activity in the auditory nerve integrated over time indicates that the quadratic non-linearity occurs beyond the auditory nerve. Given the quadratic relationship between basilar-membrane vibration amplitude and loudness, it follows that intensity coding in the auditory nerve must produce a response that is approximately proportional to the basilar-membrane amplitude, at least up to 60 d SPL. At higher levels, the auditory-nerve response may become a somewhat compressive function of the basilar-membrane amplitude, but it is also possible that the basilarmembrane input-output function at high SPLs is somewhat more compressive in humans than in chinchillas. Further research is needed to clarify this question. Acknowledgment This work was supported by NIH/NIDCD grant R01DC02241. References uus, S. (1999). Loudness functions derived from measurements of temporal and spectral integration of loudness. In: A. N. Rasmussen, P. A. Osterhammel, T. Andersen and T. Poulsen (Eds.), Auditory models and nonlinear hearing instruments, 135-188. Taastrup, Denmark.: GN ReSound. uus, S. and Florentine, M. (1992). Possible relation of auditory-nerve adaptation to slow improvement in level discrimination with increasing duration. In: Y. Cazals, L. Démany and K. Horner (Eds.), Auditory Physiology and Perception, 279-288. New York: Pergamon. uus, S., Florentine, M. and Poulsen, T. (1999). Temporal integration of loudness in listeners with hearing losses of primarily cochlear origin. J. Acoust. Soc. Am., 5, 3464-3480. uus, S., Müsch, H. and Florentine, M. (1998). On loudness at threshold. J. Acoust. Soc. Am., 4, 399-4. Fletcher, H. and Munson, W. A. (1933). Loudness, its definition, measurement and calculation. J. Acoust. Soc. Am., 5, 82-8. Fletcher, H. and Steinberg, J. C. (1924). The dependence of the loudness of a complex sound upon the energy in the various frequency regions of the sound. Phys. Rev., 24, 306-317. Florentine, M., uus, S. and Poulsen, T. (1996). Temporal integration of loudness as a function of level. J. Acoust. Soc. Am., 99, 1633-1644. Florentine, M., uus, S. and Robinson, M. (1998). Temporal integration of loudness under partial masking. J. Acoust. Soc. Am., 4, 999-07. Hellman, R. P. (1991). Loudness scaling by magnitude scaling: Implications for intensity coding. In: G. A. Gescheider and S. J. olanowski (Eds.), Ratio Scaling of Psychological Magnitude: In Honor of the Memory of S. S. Stevens, 215-228. Hillsdale, NJ: Erlbaum. Hellman, R. P. and Zwislocki, J. J. (1961). Some factors affecting the estimation of loudness. J. Acoust. Soc. Am., 33, 687-694. Ruggero, M. A., Rich, N. C., Recio, A., Narayan, S. S. and Robles, L. (1997). asilar-membrane responses to tones at the base of the chinchilla cochlea. J. Acoust. Soc. Am., 1, 2151-2163. Schlauch, R. S., DiGiovanni, J. J. and Ries, D. T. (1998). asilar membrane nonlinearity and loudness. J. Acoust. Soc. Am., 3, 20-2020. Westerman, L. A. and Smith, R. L. (1984). Rapid and short-term adaptation in auditory nerve responses. Hear. Res., 15, 249-260. Young, E. D. and Sachs, M.. (1973). Recovery from sound exposure in auditory nerve fibers. J. Acoust. Soc. Am., 54, 1535-1543. Zeng, F.-G. and Shannon, R. V. (1994). Loudness-coding mechanisms inferred from electric stimulation of the human auditory system. Science, 264, 564-566. Zwislocki, J. J. (1965). Analysis of some auditory characteristics. In: R. D. Luce, R. R. ush and E. Galanter (Eds.), Handbook of Mathematical Psychology, 1-97. New York: Wiley.