Counting the steps for a loudness scale

Abstract

It is a century-old desire to construct a scale for loudness. Such a scale’s intervals would, on face value, be just-noticeable changes in loudness. Those changes correspond to empirical just-noticeable-differences (jnds) in RMS sound-pressure-level. Hence, a putative loudness scale is the cumulative number of jnds, counted upward from an auditory stimulus’s absolute detection threshold (i.e., the just-perceptible stimulus). Jnd-counting has been intermittently pursued for over a century. The historical method is mathematical integration, but integration inherently assumes infinitely fine jnds, an absurdity. A novel, discrete method of jnd-counting was pioneered for hearing by Dallos and Carhart, after a step-counting method developed for genetics by the polymath J. B. S. Haldane. Nonetheless, one element of the integration method remains necessary: an equation for jnd versus stimulus level. The literature offers such equations in hearing, vision, and the other senses. Unfortunately, Dallos and Carhart omitted crucial details from their exposition, and employed a unique approximation for the jnd, expressing it versus decibels above absolute detection threshold, rather than versus RMS sound-pressure-level. Perhaps consequently, the method lacked acceptance. Here, the method is clarified by generalizing through full details, and then applied to a variety of jnd equations. Non-constant loudness intervals can be accommodated, if desired.