A Mathematical Exploration of the Mystery of Loudness Adaptation

Abstract

Loudness adaptation, or the decrease in perceived loudness of a steady, prolonged tone is rather a mysterious phenomenon. When measured by one technique (utilizing both ears), loudness of an extended tone will decrease by as much as 35 decibels; when measured by another technique (using only a single ear), loudness does not adapt at all regardless of how long the tone persists. The mystery is even more intriguing. When loudness adaptation does occur, the fractional reduction in the loudness of a tone (adaptation dB/sound level of extended tone dB SL) provides a good measure of the Stevens exponent, n, for loudness, an exponent which depends on sound frequency. When we analyze mathematically the two methods for measuring loudness adaptation, the reason for the apparent difference in adaptation emerges. Moreover, we derive the approximate equation showing that n equals fractional adaptation, and a method for improving the derivation of n from adaptation data. These results, derived mathematically, compare well with measured data, both our own and those obtained from the literature.

Simultaneous dichotic loudness balance (SDLB): Why loudness “fatigues” with two ears but not with one

“…After all, a convincing set of concepts needs to arise first. The present author knows of only one attempt at a mathematical model (Norwich, 2010), which subsequently proved incompetent (Nizami, 2015(Nizami, , 2017, partly due to its critical lack of an adequate conceptual base. Furthermore, an overriding concern has not yet been addressed, in the form of an important limitation of mathematical models generally.…”

Advances in Computer Science and Engineering

“…Checking current position. computational biology, because they always started their derivations using the same base equation, whose utility was justified through curve-fitting to various kinds of empirical data (see Norwich, 1993). Norwich et al intended the scope of their theory to be vast, and so the present paper focuses on just one particular set of derivations.…”

Norwich's Entropy Theory: how not to go from abstract to actual

“…The entropy theory also produced Teghtsoonian's law (Norwich, 1987a; repeated in Norwich, 1991c), which relates Stevens's power exponent to stimulus range (Teghtsoonian, 1971); Pieron's law for reaction time (Norwich et al , 1989; repeated in Norwich, 1991c, and in Wong and Norwich, 1996); Bloch's law, which states that stimulus duration and intensity trade off to produce a constant effect (Norwich, 1981a; Wong and Figueiredo, 2002); the “threshold law for odorants” (Norwich, 1991c); the origin of otoacoustic emissions (Norwich, 1982, 1983); Riesz's law for auditory Weber fractions (McConville et al , 1991; repeated in Wong and Norwich, 1993, 1995, 1996, in Norwich and Wong, 1997b and in Wong and Figueiredo, 2002); Weber fractions for auditory frequency (Wong and Norwich, 1993); the Stevens's law exponent for taste (Norwich, 1984a); the total number of just‐noticeable‐differences “making up a stimulus” (Norwich, 1984a, p. 275; repeated in McConville et al , 1991, p. 171 and in Wong and Norwich, 1995, p. 3766); equal‐loudness contours (Wong and Norwich, 1995; repeated in Norwich and Wong, 1997b); and the dependence of auditory detection threshold upon stimulus duration (Wong and Figueiredo, 2002). Why loudness adapts for both ears but not for one alone was also addressed (Norwich, 2010a) along with “phantom limb” pain in amputees, and vertigo (Norwich, 2010b). No other model matches even a small fraction of all of these claims.…”