The Measurement of Loudness

TLDR

Intensity is assumed to be proportional to the square of sound pressure. The paper reviews evidence on the relationship between loudness and stimulus intensity. The authors model loudness as a power function of intensity, yielding L = kI^0.3, and propose log L = 0.03N + S for continuous noises above 50 dB. The study finds that loudness follows L = kI^0.3, that a 2:1 loudness ratio corresponds to a 10 dB difference across the audible range, and that white noise grows faster than a 1000‑cycle tone at low levels but becomes proportional above about 50 dB.

Abstract

This paper reviews the available evidence (published and unpublished) on the relation between loudness and stimulus intensity. The evidence suggests that for the typical listener the loudness L of a 1000-cycle tone can be approximated by a power function of intensity I, of which the exponent is log102. The equation is: L = kI0.3. Intensity here is assumed to be proportional to the square of the sound pressure. In terms of sones, where 1 sone is the loudness produced by a tone at 40 db above the standard reference level, the equation for loudness L as a function of the number of decibels N becomes: logL=0.03N−1.2. Otherwise said, a loudness ratio of 2:1 is produced by a pair of stimuli that differ by 10 db, and this relation appears to hold over the entire range of audible intensities. At low levels of intensity, the loudness of white noise grows more rapidly than the loudness of a 1000-cycle tone, but above the level of approximately 50 db the two loudnesses remain more nearly proportional. The suggestion is made that for all levels greater than 50 db the loudness of continuous noises may be calculated from the equation: logL=0.03N+S, where S is a spectrum parameter to be determined empirically.