MODELING THE RELATION BETWEEN THE INTENSITY JUST-NOTICEABLE DIFFERENCE AND LOUDNESS FOR PURE-TONES AND WIDE-BAND NOISE

A classical problem in auditory theory is the relation between the lou dness L(I) and the intensity just-noticeable difference (JND) Delta I( I). The intensity JND is frequently expressed in terms of the Weber fr action defined by J(I)=Delta I/I because it is anticipated that this r atio 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. Furt hermore it remains unexplained why Weber's law holds for wide-band sti muli. 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 (L) over tilde(I), namely the loudness is L(I)=E (L) over tilde(I), wh ile the variance of the single trial loudness is sigma(L)(2)=E((L) ove r tilde-L)(2). The JND is given by the signal detection assumption Del ta L=d'sigma(L), where we define the loudness JND Delta L(I) as the ch ange in loudness corresponding to Delta I(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-S75 (1928)] Delta I(I) data to the Fletcher and Munson [J. Acoust. Soc. Am. 5, 82-108 (1933)] loudnes s growth data. We then make the same comparison for Miller's [J. Acous t. Soc. Am. 19, 609-519 (1947)] wideband noise JND and loudness match data. Based on this comparison, we show empirically that Delta L(L)pro portional to(l/p), where p=2 below approximate to 5 sones and is 1 abo ve. Since Delta L(I) is proportional to sigma(L), when p=2 the statist ics of the single-trial loudness (L) over tilde are Poisson-like, name ly sigma(L)(2) proportional to L. This is consistent with the idea tha t the pure tone loudness code is based a neural discharge rare (not th e auditory nerve). Furthermore, when p=1 (above about 5 sones), the in ternal loudness signal-to-noise ratio is constant. It is concluded tha t Ekman's law (Delta L/L is constant) is true, rather than Weber's law , in this loudness range. One of tile main contributions of this paper is its attempt to integrate Fletcher's neural excitation pattern mode l of loudness and signal detection theory. (C) 1997 Acoustical Society of America. [S0001-4966(97)02411-9].