Figure 10.
The tau effect. (A) Three taps to the arm, at positions x1 = 0 cm, x2 = 3 cm, and x3 (variable), define two spatial intervals, l1 = 3 cm and l2 (variable), and two temporal intervals, t1 = 0.5 s and t2 (variable). Because t2 < t1, at some l2 > l1 the two intervals will be perceived to be of equal length (l2* = l1*). (B) At each of five t2 settings (identified at right of plots), Helson and King (1931) progressively increased l2 by shifting x3 along the arm in 0.5-cm increments. On each trial, the participant reported whether the second spatial interval was perceived to be shorter than, equal to, or longer than the first interval. To accurately estimate each participant’s point of subjective equality (PSE), we transformed these data into a two-alternative forced-choice format by distributing the participant’s “equal” responses evenly to the “shorter” and “longer” response categories. We then fit each participant’s transformed data (proportion “l2 is longer” responses) at each t2 setting with a Weibull psychometric function (blue curves). Each psychometric function provides a PSE (vertical line): the x3 at which the psychometric function intersected 0.5 (horizontal line), indicating that l2* = l1*. The PSE shifted progressively to the left as t2 was increased (note: when x3 = 6 cm, l2 actually does equal l1). The transformed data shown are from one participant (“Observer C”) in Helson and King (1931). (C) Trajectories for which l2* = l1*. Blue points: mean x3 that resulted in l2* = l1* among the six participants tested by Helson and King (1931), at each of the five t2 settings. Blue lines: ±1 SD. Red points: best-fit performance of the Bayesian low-speed observer (τ = 0.10 s).