Page:Journal of the Optical Society of America, volume 30, number 12.pdf/31

Now if the first color be 5/5, and the second be $V/-C$, then: $V_{1}=5$, $C_{1}=5$, $V2=V$, $C_{2}=C$,^[1] and we have from Eq. [1]:

$a_{1}=C_{V}/(C_{V}+25);\qquad$$a_{2}=25/(CV+25).$

By substituting these values in Eq. [2], and eliminating $X_{n},Y_{n},Z_{n},$ through Eqs. [3] and [4], we may solve explicitly for$X_{V/-C}$, $Y_{V/-C}$, and $Z_{V/-C}$:

$X_{V/-C}={\frac {V}{5}}\left[{\frac {V+C}{5}}X_{5/0}-{\frac {C}{5}}X_{5/5}\right]$

$Y_{V/-C}={\frac {V}{5}}\left[{\frac {V+C}{5}}Y_{5/0}-{\frac {C}{5}}Y_{5/5}\right]\qquad \qquad [5]$

$Z_{V/-C}={\frac {V}{5}}\left[{\frac {V+C}{5}}Z_{5/0}-{\frac {C}{5}}Y_{5/5}\right]$

Equation [5] expresses the color $V/-C$ in terms of the neutral 5/0 and of the complementary 5/5 color. A similar derivation for the color $V/C$ in terms of the 5/5 color of the same hue shows that it is necessary only to change the algebraic sign of C, thus:

$X_{V/C}={\frac {V}{5}}\left[{\frac {V-C}{5}}X_{5/0}+{\frac {C}{5}}X_{5/5}\right]$

$Y_{V/C}={\frac {V}{5}}\left[{\frac {V-C}{5}}Y_{5/0}+{\frac {C}{5}}Y_{5/5}\right]\qquad \qquad [6]$

$Z_{V/C}={\frac {V}{5}}\left[{\frac {V-C}{5}}Z_{5/0}+{\frac {C}{5}}Y_{5/5}\right]$

From these relations trichromatic coefficients may be derived, as follows:

$x_{V/C}={\frac {(V/C-1)x_{5/0}+(y_{5/0})/y_{5/5})x_{5/5}}{(V/C-1)+y_{5/0}/y_{5/5}}},\quad \quad [7]$

$y_{V/C}={\frac {(V/C)y_{5/0}}{(V/C-1)+y_{5/0}/y_{5/5}}}.$

Note that in Eqs. [7], $V/C$ is the only parameter; it follows that all samples of a given hue and for which $V/C$ is a constant will have the same trichromatic coefficients $(x,y)$. For example, R 2/2, R 3/3, R 4/4, and R 7/7 will have the same $(x,y)$ values as R 5/5; and G 8/4 will have the same $(x,y)$ values as G 4/2. The convention of writing Munsell value as the numerator of a fractional form whose denominator is Munsell chroma thus takes on an added meaning from the disk-mixture rule given in the Atlas apparently not foreseen at the time the convention was originated.

For a given dominant wave-length, excitation purity is proportional to distance on the Maxwell triangle, which, in turn may be measured by its projection either onto the x axis or the y axis of the $(x,y)$ diagram, or both. The relation between excitation purity and Munsell chroma on the basis of the psychophysical relation given by Munsell may be written out from Eqs. [7] as follows:

${\frac {P_{e(V/C)}}{P_{e(5/5)}}}={\frac {y_{V/C}-y_{5/0}}{y_{5/5}-y_{5/0}}}={\frac {y_{5/0}}{y_{5/5}(V/C-1)+y_{5/0}}}.\qquad \qquad [8]$

This derivation holds only for $y_{5/5}$, different from $y_{5/0}$, but if these two are equal, the same result may be obtained from the x differences.

As might be anticipated from the psychophysical definition of chroma by disk mixture, the relation with colorimetric purity is even simpler. From the known relation between excitation purity and colorimetric purity (7, p. 59), which in present terms is:

$P_{e}={\frac {y_{\lambda }}{P_{e}}}$

$P_{e5/5}={\frac {y_{\lambda }}{y_{5/5}}}P_{e(5/5)}.\qquad \qquad [9]$

combined with Eq. [8], we obtain

$P_{e(V/C)}=CP_{e(5/5)}/V\qquad \qquad [10]$

$C=VP_{e(v/c)}/P_{e(5/5)}\qquad \qquad \quad [11]$

Equation [8] enables us to test the psychophysical nature of the Munsell system as exemplified by the samples measured in 1919 and 1926; that is, to see if the actual samples conform to this relation derived on the basis of disk mixture according to the Atlas instructions.

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1. Although the notation for a complementary color is taken as V/—C, the actual values of C for both the 5/5 and it) complementary color are positive in accord with Eq. [1].