William S. Hellman

Intensity discrimination as the driving force for loudness. Application to pure tones in quiet

Loudness functions and their associated neural-count functions are derived for steady-state tones at 250 and 1000 Hz from measurements of intensity discrimination obtained under gated and continuous conditions. The calculations are based on a multichannel generalization of the McGill-Goldberg counting model [J. Acoust. Soc. Am. 44, 576-581 (1968)]. Using the data for just noticeable differences (jnd) in intensity as input, the generalized version gives an integral relation between the neural-count function N(x) and the intensity-jnd function, where x = I/I0 and I0 is the reference intensity. Loudness functions are generated through the prescription L(x) = AN(x)--B. To determine how the detailed shapes of the intensity-jnd functions affect the form of the loudness function within the model, integration was performed over the intensity-jnd functions with and without a power-function approximation. Over a range of intensity levels from 20-95 dB, the best agreement between the calculated and measured loudness functions is obtained from the unaltered intensity-jnd functions. Consistent with psychophysical evidence and several models of intensity coding, the results predict that the output of the whole auditory nerve is unnecessary to maintain the large dynamic range observed for loudness and intensity discrimination.

Relation of the loudness function to the intensity characteristic of the ear

A representation of the pure−tone loudness function given by Ψ = xϑF (x), where x is the ratio of stimulus to threshold intensity, is discussed. The form of F (x) is obtained from the empirical loudness curves at 1000 and 250 Hz. The structure of F (x) is shown to be in quantitative agreement with known steady−state driven firing rate activity of eighth−nerve units. We suggest that F (x) is proportional to the driven firing rate function RD(x) of the auditory sensory receptor. A specific choice is made for the algebraic form of RD(x) by identifying it with the driven rate function in a sensory receptor model proposed recently by Zwislocki. Within the model, a three−parameter fit to the loudness function is obtained. It is shown that similar results can be obtained for vibrotaction.Subject Classification: 65.35, 65.50.

Revisiting relations between loudness and intensity discrimination.

A comparison is made between the variation of delta Ljnd with L (loudness), based on the beat-detection data of Riesz at 1 kHz [Phys. Rev. 31, 867-875 (1928)], and analogous relations obtained from a cross section of studies. Data analysis shows that only beat detection exhibits the degree of level-dependent variation in slope relating log (delta Ljnd) to log (L) described in a recent paper by Allen and Neely [J. Acoust. Soc. Am. 102, 3628-3646 (1997)]. Moreover, the slope variation determined from beat detection is not dependent on the detailed shape of the loudness function. The results imply that Allen and Neelys strong conclusions about the dependence of delta Ljnd on L are too tightly coupled to Rieszs methodology to be generally applicable.

Frequency‐time dispersion products revisited

The interaction of bandwidth and duration is an important aspect of signal choice in psychophysical experiments. The traditional measures yield frequency‐time dispersion products which have a lower bound of 0.25. Gabor [J. Inst. Elect. Eng. Part III 93, 429–457 (1946)] argued that the measures for the standard deviation and mean in the frequency domain do not yield formulas in agreement with intuition when applied to the calculation of frequency dispersions for real signals f(t). Gabor’s prescription for correcting the problem was to limit the integrations in frequency statistic calculations to the positive frequency domain and, for consistency, use the analytic signal corresponding to f(t) to calculate time dispersions. A more recent statement of this prescription can be found in L. Cohen [Time‐Frequency Analysis, (Prentice Hall, 1995)]. However the elimination of the negative frequencies introduces difficulties which have not been fully addressed in the literature. It is shown here, that unless the Four...

Derivation of intensity‐discrimination functions from loudness functions

Intensity‐discrimination functions are computed from their concomitant loudness functions in a procedure where the only adjustable parameter sets the scale to match the measured intensity‐discrimination data. The equation connecting loudness to intensity discrimination is determined from monaural loudness‐ and intensity‐discrimination measurements at 1 kHz. Predictions of intensity‐discrimination functions are then computed for a low‐frequency tone masked by an adjacent high‐pass noise, for broadband noise, and for high‐frequency tones at 12.5 and 16 kHz. Results show that the derived intensity‐discrimination functions closely capture the overall shape of the experimental data. A U‐shaped function is obtained for the partially masked low‐frequency tone, whereas a function approximating Weber’s law is observed for broadband noise and also at 16 kHz. The good agreement between the calculated and measured functions implies that the form of the intensity‐discrimination functions is strongly dependent on local...

Relation of long‐term adaptation to the loudness‐function slope

The decline in loudness over time, defined as simple loudness adaptation, is modeled by introducing temporal factors into the nonadapted loudness function. This modification generates effective time‐dependent intensity and threshold shifts. Model predictions for several frequencies are in accord with recent psychophysical measures [Hellman et al., J. Acoust. Soc. Am. 101, 2176–2185 (1997)]. In contrast to the time constant description of adaptation, the present method produces a direct connection to properties of the loudness function. A consequence of this connection is the demonstration that the increase in loudness adaptation with decreases in SL is linearly related to the loudness‐function slope in log–log coordinates. Moreover, the good agreement between the measured and predicted loudness‐adaptation data implies a consistency and stability in the use of numbers for loudness judgments.

On the effects of a subharmonic masker on the loudness of a pure tone

Loudness matching functions between a 1‐kHz tone in quiet and in the presence of a 70 dB(SPL) 0.5‐kHz subharmonic masker were studied to determine the alterations in loudness due to intensity shifts at the stimulus frequency arising from the generation of the second harmonic of the masker. Such effects are understood to be due to a quadratic nonlinearity occurring in the inner ear transduction process. In the experiment, masked thresholds were measured at 30° phase angle intervals of the stimulus. Matching functions were obtained for phase conditions corresponding to minimum and maximum thresholds. Both the shape of the matching functions and their relative displacement from each other were consistent with the predictions of the vector summation model [Clack et al., J. Acoust. Soc. Am. 52, 536–541 (1972); Schubert, J. Acoust. Soc. Am. 45, 790–791 (1969)]. Assuming that the threshold shifts were due to constructive and destructive interference between the 1‐kHz tone and the second harmonic of the masker, a...

Deriving the Weber fraction from loudness functions.

In a previous paper [W. S. Hellman and R. P. Hellman, J. Acoust. Soc. Am. 91, 2380(A) (1992)], Weber fractions for intensity discrimination were derived from their concomitant pure tone loudness functions in normal hearing. The calculational procedure employed a generalized McGill–Goldberg model. This work extends these findings to more frequencies, to broadband noise, to tones masked by high‐pass noise, and to forward masking. Weber functions generated by the model are compared to empirical data for the various experimental conditions and stimuli. In all cases, the calculated Weber functions capture the overall shape of the measured intensity‐jnd data over a wide stimulus range. Not only does the model produce the near‐miss relation for pure tones in quiet, it also predicts Weber’s law for broadband noise, a rising characteristic in the Weber function above 60 dB SPL for a tone in high‐pass noise, and the midlevel hump observed in forward masking. These results show that there is sufficient information i...

Predicting intensity discrimination from power‐law modifications of the loudness function

Several modifications of Stevens’ power law have been introduced to account for the steepening of the loudness function below about 40 dB SPL. The relation between intensity discrimination (ΔI/I) and loudness is determined for three proposed modifications: (a) Ls=K(I−I0)n, (b) Ls=K(In−I0n), and (c) Ls=k([I+cI0]n−[cI0]n), where I0 is the sound intensity at threshold. Below 15 dB SPL Eqs. (a) and (b) generate a rapidly decreasing function for intensity discrimination, whereas the function derived from Eq. (c) rises with a slope of −1 in log–log coordinates. The slope of the measured intensity‐discrimination function is compatible with results predicted for pure tones from Eq. (c) but not with those predicted from Eqs. (a) and (b). [Work supported in part by NIH.]

Can the loudness function determine the shape of the Weber function

A recent model relating measures of loudness and intensity discrimination [W. S. Hellman and R. P. Hellman, J. Acoust. Soc. Am. 87, 1255–1265 (1990)] is shown to reveal that the near threshold slope of +1 obtained for the loudness function requires the near threshold slope of the Weber function for intensity discrimination to be −1. This result is consistent with psychophysical intensity discrimination measurements for pure tones and broadband noise, as well as with theoretical considerations which predict that the Weber function should have a slope of −1 as the detection threshold is approached [N. F. Viemeister, Auditory Function, 213–241 (1988)]. The model also generates the overall shape of the Weber function over a suprathreshold range of sound pressure levels up to 80 dB. Empirical examples are provided. [Partially supported by the Rehab. R. & D. Service of the VA.]