Sensation-growth equations for non-zero threshold sensation, evaluated using non-traditional, bounded Fechnerian integration, for Fechner's Law, Ekman's Law and for Ekman's Law, using 12 different Weber Fractions

An ongoing mystery in sensory science is how sensation magnitude F(I), such as loudness, increases with increasing stimulus intensity I. No credible, direct experimental measures exist. Nonetheless, F(I) can be inferred algebraically. Differences in sensation have empirical (but non-quantifiable) minimum sizes called just-noticeable sensation differences, ΔF, which correspond to empirically-measurable just-noticeable intensity differences, ΔI. The ΔIs presumably cumulate from an empirical stimulus-detection threshold Ith up to the intensity of interest, I. Likewise, corresponding ΔFs cumulate from the sensation at the stimulus-detection threshold, F(Ith), up to F(I). Regarding the ΔIs, however, it is unlikely that all of them will be known experimentally; the procedures are too lengthy. The customary approach, then, is to find ΔI bat a few widely-spaced intensities, and then use those ΔIs to interpolate all ΔIs using some smooth continuous function. The most popular of those functions isWeber's Law, ΔI/I=K. But that is often not even a credible approximation to the data. However, there are other equations for ΔI/I. Any such equation for ΔI/I can be combined with any equation for ΔF, through calculus, to altogether obtain F(I). Here, two assumptions for ΔF are considered: ΔF=B (Fechner's Law) and (ΔF/F)=g (Ekman's Law). The respective integrals involve lower bounds Ith and F(Ith). This stands in broad contrast to the literature, which heavily favors non-bounded integrals. We, hence, obtain 24 new, alternative equations for sensation magnitude F(I) (12 equations for (ΔI/I) × 2 equations for ΔF).