## Abstract

From larger–smaller judgments of color differences, compared visually two at a time, the perceived sizes may be evaluated on an interval scale. Given numbers *B* so evaluated, and such that *B* is linearly connected to some power *p*, of the physical measure *D* (such as distance on any chromaticity diagram) of the differences, the additive constant *K** _{br}* such that the numbers

*B*+

*K*

*are expressed on a ratio scale may be found from judgments of the ratio of sizes of pairs of differences. To evaluate*

_{br}*p*, it is sufficient to observe the three differences, 12, 23, and 13 between the pairs of three colors, 1, 2, 3, forming a geodesic series, and chosen so that

*B*

_{12}is not much different from

*B*

_{23}. The scale formed by the numbers (

*B*+

*K*

*)*

_{br}^{1/}

*is additive if the*

^{p}*D*scale is additive. If the largest of the color differences judged exceeds the smallest by a factor not greater than 3, a close approximation to the (

*B*+

*K*

*)*

_{br}^{1/}

*scale may be found without evaluating*

^{p}*K*

*by ratio judgments. This approximation is based on the empirical discovery that scales based on the additivity condition: (*

_{br}*B*

_{12}+

*K*

*)*

_{bd}^{1/}

*+(*

^{p}*B*

_{13}+

*K*

*)*

_{bd}^{1/}

*=(*

^{p}*B*

_{13}+

*K*

*)*

_{bd}^{1/}

*, though it implies that*

^{p}*K*

*depends strongly on*

_{bd}*p*, are essentially identical regardless of the choice of

*p*between 1 and ${\scriptstyle \frac{1}{3}}$. It is sufficient therefore, to derive the additive scale by setting

*p*=1, and computing

*K*

*as*

_{bd}*B*

_{13}−

*B*

_{12}−

*B*

_{23}.

© 1967 Optical Society of America

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