Nonzero threshold loudness: A circular argument in Zwislocki (1965) and Moore, Glasberg, and Baer (1997).

Abstract

ANSI S3.4‐2005 updates the calculation of loudness to reflect empirical evidence that loudness is nonzero at detection threshold. Hellman [Acoustics Today (2007)], in reviewing the changes, cited the work of Zwislocki [Handbook of Mathematical Psychology (1965) Vol. III] and Moore et al. [J. Audio. Eng. Soc. 45 (1997)] as theoretical support. Zwislocki proposed that detection threshold reflects an internal noise that acts like an external masker, and that (using generalized notation here for clarity) the tone+noise loudness for an rms tone pressure x is a function f of the sum of noise contribution(s) c and a tone contribution g(x). The Zwislocki g(x) was zero at x=0 and increased monotonically thereafter. Altogether, tone+noise loudness is f(g(x)+c). Zwislocki then assumed that listeners can perceptually separate tone from noise. He subtracted noise loudness from tone+noise loudness to get tone loudness, [f(g(x)+c)−f(c)]. That exceeds zero for whatever x is deemed the tone‐detection threshold. Unfortunately, with g(x)>0 for x>0, a nonzero threshold‐tone‐loudness was predetermined. Moore et al., using auditory filter output power as x, produced a congruent equation for tone loudness in quiet, repeating the circularity. Circularity was inevitable, because detection threshold is defined using a percentage‐correct performance that indicates nonconstant loudness.