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

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.