Every essay — page 30
What light is What the eye does Matching and measuring Difference and uniformity What the brain does What a scene does What a camera does Where the model breaks What it takes to deliver it
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Where the model breaks
Seventeen observers in 1931, an error in the blue that was never fully repaired, colour vision deficiency, and the display this page is being read on.
A slow pigment splits mercury's lines only if its swing fits between them
Under a clear mercury lamp a pigment whose reflectance may swing from nought to one over no less than forty nanometres loses 21 of the census's 148 directions, and every one of them asks it to reflect one of the lamp's green and yellow lines and absorb the other. The census samples the spectrum every five nanometres, which smears each six-nanometre line across a band, and the prediction was that a one-nanometre grid would let the pigment's cliff fall between the lines and cut the count by a third. It cuts it by nothing: 21 on either grid. Narrowing the lines to one and a half nanometres cuts it only to 17. What decides the losses is the lines' spacing, 31.5 nanometres, against the pigment's swing. With narrow lines a pigment whose swing is 30 nanometres loses no direction at all, and one whose swing is 34 loses fourteen; the step sits at the gap. The mercury lamp's cost to a slow pigment belongs to the lamp, not to how the census samples it.
A band pigment must be narrower than the gap it splits
Under a clear mercury lamp a pigment built from absorption bands no narrower than forty nanometres falls short in fifteen of the twenty-five directions that ask it to reflect the green line and absorb the yellow, or the reverse — the count an earlier search left short however deep it cut its cliff. The slope-limited pigment's losses turned out to be the lines' 31.5 nm spacing against its swing, and the prediction was that the band pigment's are the same arithmetic, falling to nought near a band thirty nanometres wide. Searched on a one-nanometre grid with the lines narrowed, the count falls at every narrower band — fifteen at forty, eleven at thirty-five, nine at thirty — and reaches nought at twenty-five, not thirty. Every loss already splits the pair and falls short on both lines at once: a Gaussian band deep enough to absorb one line keeps absorbing a little of the other. One band of depth twelve, placed as well as it can be, errs by 3.2 per cent at forty nanometres, 0.8 at thirty and 0.2 at twenty-five; doubling its depth buys less than narrowing it by five nanometres.
A steep side splits mercury's lines at forty nanometres
Under a clear mercury lamp a pigment built from Gaussian absorption bands no narrower than forty nanometres falls short in fifteen of the twenty-five directions that ask it to reflect the green line and absorb the yellow, or the reverse, because a band's tail reaches the line it should pass. Real dye bands are often lopsided, and the prediction was that a band whose one side is half the other's, the steep side free to face either way, loses at forty nanometres about as many directions as a symmetric band thirty wide — nine — and helps only the directions that pass the green. It loses one. At thirty-five and thirty it loses none. In every one of the twenty-five fits the absorbing band turns its steep side to the line it must pass, and both kinds of direction gain: none of fourteen passing the green is lost, one of eleven passing the yellow. A single band forty nanometres wide with a steep side splits the two lines within 0.37 per cent; a symmetric band must narrow to about twenty-seven to do as well.
A band must be nearly two to one before its steep side pays
Under a clear mercury lamp, a pigment of absorption bands forty nanometres wide loses fifteen or sixteen of the twenty-five directions that ask it to reflect the green line and absorb the yellow, or the reverse, and the same bands with one side a third of the width and the other two thirds lose one. The prediction for milder skews was that the losses fall in step with a single band's error — to about eight at sides of 0.55 and 0.45, three at 0.6 and 0.4 — so that a three-to-two skew already halves them. It does not. The census loses thirteen at 0.55, ten at 0.6, five at 0.65 and one at two to one: three directions recovered per twentieth of skew at first, twelve at the last. The losses track one band's error against the one-per-cent line the census counts by, and a single band crosses that line only near a skew of 0.62. A dye's modest natural asymmetry buys a few directions; the steep side pays only when it is close to half the other.
A band is only as wide as its steep side
Under a clear mercury lamp, a pigment of absorption bands forty nanometres wide must be skewed nearly two to one before it can split the green line from the yellow, and the prediction for bands five nanometres narrower was that a single band crosses one per cent of error near a skew of 0.57 and the census's losses fall to five or fewer by 0.6. The losses do fall to four by 0.6; the single band crosses one per cent at 0.555. But the forty-nanometre rule that read the losses off that one-per-cent line breaks at once, and what replaces it is simpler: a thirty-five-nanometre band at 0.6 and a forty-nanometre band at 0.65 have the same steep side, fourteen nanometres, and lose the same four directions by the same shortfalls to a hundredth of a point. Three widths and fourteen pigments fall on one staircase in the steep side alone. Searching every loss again from the milder skews' answers finds one direction the census missed, at both widths, and corrects the forty-nanometre counts to nine and four.
At one half-width a flank can lose nothing or nearly everything
Under a clear mercury lamp, a pigment of Gaussian absorption bands splits the green line from the yellow according to its steep side's half-width alone, and a Gaussian flank is fixed by its half-width — so the census could not say whether it measures the half-width or the absorbance left in the flank's foot at the line it should spare. The prediction was that the foot decides: at a half-width of fourteen nanometres, a Lorentzian steep side loses most of the twenty-five splitting directions and a sharper-footed one none. It does. A power-four Gaussian loses none, the Gaussian four, a squared hyperbolic secant nineteen and a Lorentzian twenty-two, in the order of their single bands' errors. But a single band's error is not the whole account: three profiles drawn at the half-widths that give a single band the same error — 7.73, fourteen and 39.1 nanometres — lose two, four and seven directions, and the widest of them loses directions the others keep, because its shoulder starts absorbing nearer the green.
A hundredth of Lorentzian undoes a Gaussian flank
Under a clear mercury lamp, a band pigment whose steep side is a Gaussian fourteen nanometres wide at half depth loses four of the twenty-five directions that split the green line from the yellow, and the same side drawn as a Lorentzian loses twenty-two. A real absorption band is usually a Voigt profile, a Gaussian core with a Lorentzian share in its wings, and the prediction was that a twentieth of Lorentzian doubles the Gaussian's losses and a fifth reaches nineteen. The direction is right and the scale is not. A hundredth of Lorentzian takes the pigment from four lost directions to eleven; a fiftieth to fourteen; a twentieth to eighteen; a fifth to twenty. The best-placed band leaves the passed line more than three half-widths from its centre, out in the wing, and there a wing one per cent Lorentzian carries twice the absorbance of the whole Gaussian. The count saturates near twenty while the worst direction's shortfall keeps growing, from 2.8 per cent at a hundredth to 51 with a full Lorentzian.
A narrower flank moves its wing in with it
Under a clear mercury lamp, a band pigment whose steep side is a Gaussian fourteen nanometres wide loses four of the twenty-five directions that split the green line from the yellow, and with a twentieth of Lorentzian in that side it loses eighteen, because the wing three half-widths out out-absorbs the whole Gaussian there. A Lorentzian wing falls as the square of the distance over its half-width, and the prediction was that narrowing the side to seven nanometres brings the twentieth-Lorentzian pigment back to four losses and to five nanometres to none. It loses thirteen at ten nanometres, twelve at seven and seven at five, while the Gaussian loses nothing at all three. The wing still decides: the narrower band does not stay where the wider one sat but moves fourteen nanometres in towards the line it must pass, so its wing there falls fourfold rather than eightfold.
A five-nanometre flank carries a fiftieth of Lorentzian
A band pigment splitting mercury's green and yellow lines with a steep side five nanometres wide loses none of the twenty-five splitting directions as a Gaussian and seven with a twentieth of Lorentzian in its flank, because the band slides in towards the passed line and carries its wing there. The prediction was that at five nanometres the count follows the single band's error, which grows in proportion to the share: about two lost at a fiftieth and none at a hundredth. A hundredth loses none, as predicted. The first search lost one direction at a fiftieth — and searched again from the best pigments found at the neighbouring shares, that direction reaches within two thousandths of the ideal: a fiftieth loses none either. The same second search recovers two of the twentieth's seven, leaving five. A sharp enough flank tolerates a fiftieth of Lorentzian where a fourteen-nanometre one lost fourteen directions to it.
Each direction falls short by one band's ratio
A band pigment asked to reflect mercury's green line and absorb its yellow, with a twentieth of Lorentzian in its steep side, loses eighteen of twenty-five splitting directions at a steep side fourteen nanometres wide, thirteen at ten, twelve at seven and five at five. A second search had lowered the five-nanometre count, and the prediction was that searching the seven- and ten-nanometre pigments again lowers each by one or two, leaving a count nearly linear in the half-width. Searched again, both lose exactly what they lost — no direction gains six tenths of a point. The uneven fall is real, and the reason is plainer than any search: every direction's shortfall falls with the half-width by the same factor as a single band's error, so the count is a fixed ranking of constants, one a direction, read against a line that moves.
The ranking reads six nanometres to the direction
Every direction a band pigment loses under mercury's lines falls short, as its steep side narrows, by the same factor as a single band's error — so each direction carries a constant, and the count at any half-width can be read off one band with no search at all. Read that way, a twentieth of Lorentzian loses nine directions at a six-nanometre steep side, three at four and a half and none at four. Searched, six nanometres loses nine: the same nine, every direction within 6.2 per cent of where the constants put it. Four loses none. Four and a half loses one, not three — the two the ranking added sit 0.005 and 0.011 of a point under the one-per-cent line. And at the narrowest widths the gentle directions begin to fall short by more than the ranking says, by up to 29 per cent, which is where the single band stops being the whole story.
Matching and measuring
CIE XYZ, chromaticity, and gamut — a system for predicting when two lights match, frequently mistaken for a system that predicts how they look.
