Page:Journal of the Optical Society of America, volume 30, number 12.pdf/30

direction of the departures from dominant wavelength constancy indicated in Fig. 5 for those hues is such as to accentuate the hue difference ascribable to the Bezold-Brücke phenomenon instead of compensate for it. Although detailed analysis by comparison with the Newhall (15) data has yet to be made, preliminary comparisons indicate that in general the departures shown in Fig. 5 are such as to make the samples of more nearly constant hue than they would be by keeping dominant wave-length constant.

On the other hand it is clear from Fig. 4 that most of the hues may be represented by lines, whether straight or slightly curved, which pass somewhere between the neutral gray point represented by Illuminant C and that represented by the disk mixture of the five principal hues. It is particularly noticeable that for the greens and complementary red-purples, and for the purples and complementary green-yellows, straight lines passing through the illuminant point do not properly represent the respective data, but that straight lines running through the point of disk mixture (DM) represent them more closely. The data for the yellows and complementary purple-blues may be represented by straight lines passing through either of the two neutral points, which themselves differ nearly in the yellow to purple-blue direction. Further discussion of the relative significance of these two neutral points is given in the following section.

Relation between Munsell chroma and colrimetric purity

In the constant hue charts of the Munsell Atlas we find statements such as the following, which is taken from the red and blue-green chart: “Any chosen steps of red and blue-green upon this chart may be balanced by noting their symbols: Thus light blue-green (BG 8/3) balances dark red (R 2/3) when the areas are inversely as the product of the symbols, viz.—six parts of light blue-green and twenty-four parts 7 of dark red.” Similar statements with illustrations may be found on the other charts.^[1] The general rule is stated by Tyler and Hardy (11) as follows: “When two complementary colors occupy areas on a Maxwell disk which are inversely proportional to the product of value by chroma, a neutral gray results.” The consequences of this psychophysical relation may be evaluated by the well-known laws of additive combination of colors by Maxwell disks (7, p. 30). This evaluation has been made easier than might have been supposed by writing out, in accord with suggestions by Dr. Judd hereby gladly acknowledged, the equations which by this relation connect the tristimulus specifications of color $V/C$ (Munsell value, $V$, Munsell chroma, $C$) with those of color 5/5 of the same hue and of color 5/0 (Munsell N 5/).

The first step is to derive the equations for the complementary color $V/-C$. Let the tristimulus specifications of the two complementary colors be $(X_{1},Y_{1},Z_{1})$ and $(X_{2},Y_{2},Z_{2})$, respectively, and let $(X_{n},Y_{n},Z_{n})$ be those of the neutral gray resulting from their combination in the proportions, $a_{1}$ and $a_{2}$, respectively. From the Munsell psychophysical relation:

$a_{1}=C_{2}V_{2}/(C_{1}V_{1}+C_{2}V_{2}),\qquad \qquad \quad$,

and

$a_{2}=C_{1}V_{1}/(C_{1}V_{1}+C_{2}V_{2}).\qquad \qquad [1]$

From the laws of additive color combination (7, p. 30):

$a_{1}{X_{1}},+a_{2}{X_{2}}={X_{n}}$

$a_{1}{Y_{1}}+a_{2}{Y_{2}}={Y_{n}}$

$a_{1}{Z_{1}}+a_{2}{Z_{2}}={Z_{n}}.\qquad \qquad [2]$

From the definition of Munsell value, we have the relation:

$Y_{V/C}=V^{2}/100,\qquad \qquad$

and, in particular,

$Y_{V/C}=Y_{5/5}=25/100.\qquad \qquad [3]$

And from the fact that all Munsell neutral grays are taken as having identical trichromatic coefficients, we may write:

$X_{n}/X_{5/0}=Y_{n}/Y{5/0}=Z_{n}/Z{5/0}.\qquad \qquad [4]$

---

1. The authors have been unable to find any statement made by A. H. Munsell which definitely says that the color standards of the Atlas were chosen on the basis of this relation. After the system was partially completed, he found that colors used in areas inversely proportional to the product of their V and C numbers, often gave a neutral. Evidently he believed this relation to apply more rigorously than it may, for he notes it in several places. But the system was already well established, with chroma relations on a single value level being tested by disk mixture, before he found that this $V\times C$ relation seemed to exist.