A classical problem in auditory theory is the relation between the loudness L(I) and the intensity just-noticeable difference (JND) ΔI(I). The intensity JND is frequently expressed in terms of the Weber fraction defined by J(I)≡ΔI/I because it is anticipated that this ratio should be a constant (i.e., Weber’s law). Unfortunately, J(I) is not a constant for the most elementary case of the pure tone JND. Furthermore it remains unexplained why Weber’s law holds for wide-band stimuli. We explore this problem and related issues. The loudness and the intensity JND are defined in terms of the first and second moments of a proposed random decision variable called the single-trial loudness L̃(I), namely the loudness is L(I)≡EL̃(I), while the variance of the single trial loudness is σL2≡E(L̃−L)2. The JND is given by the signal detection assumption ΔL=d′σL, where we define the loudness JND ΔL(I) as the change in loudness corresponding to ΔI(I). Inspired by Hellman and Hellman’s recent theory [J. Acoust. Soc. Am. 87, 1255–1271 (1990)], we compare the Riesz [Phys. Rev. 31, 867–875 (1928)] ΔI(I) data to the Fletcher and Munson [J. Acoust. Soc. Am. 5, 82–108 (1933)] loudness growth data. We then make the same comparison for Miller’s [J. Acoust. Soc. Am. 19, 609–619 (1947)] wideband noise JND and loudness match data. Based on this comparison, we show empirically that ΔL(L)∝L1/p, where p=2 below ≈5 sones and is 1 above. Since ΔL(I) is proportional to σL, when p=2 the statistics of the single-trial loudness L̃ are Poisson-like, namely σL2∝L. This is consistent with the idea that the pure tone loudness code is based a neural discharge rate (not the auditory nerve). Furthermore, when p=1 (above about 5 sones), the internal loudness signal-to-noise ratio is constant. It is concluded that Ekman’s law (ΔL/L is constant) is true, rather than Weber’s law, in this loudness range. One of the main contributions of this paper is its attempt to integrate Fletcher’s neural excitation pattern model of loudness and signal detection theory.
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Allen et al. (1997) studied this question.