Three lines spare a slow pigment

The limits assume a pigment that switches instantly charged the object-colour solid for the assumption that a reflectance can jump between nought and one at a wavelength. A pigment whose absorption edge takes eighty nanometres loses five per cent of the median direction’s reach and nearly a quarter of the tenth percentile’s, and the cost falls hardest on saturated greens. All of it was computed under daylight.

The solid is not a property of surfaces alone. Its boundary is where the largest weighted sum of a reflectance reaches, and the weights are the lamp’s spectrum times the matching functions — so a reflectance matters only where the lamp puts light. A lamp that is nearly dark between three narrow lines weights almost nothing between them, and a reflectance that cannot change quickly can do its changing where nothing is measured.

That is a prediction with a specific shape. Under a three-line lamp a slope limit should cost little until its transition width reaches the spacing of the lines, and then cost about what it costs under daylight. Both halves can be checked, and so can the claim that it is the lines, rather than anything else about the lamp, doing the work.

Most of the solid is spared, until the width reaches the lines’ spacing #

Under a three-emitter LED with lines near 455, 530 and 625 nanometres, a forty-nanometre transition limit costs more than a per cent of reach in 62 of the solid’s directions. Under daylight it costs that much in 183. At eighty nanometres — about the spacing of the lines — the LED costs 214 and daylight 218.

• The tenth-percentile direction keeps 0.981 of its reach under the LED at forty nanometres, against 0.938 under daylight.
• Daylight, a tungsten lamp and a phosphor white LED behave alike, within a few tens of directions at every width. The lines make the difference, not the colour temperature and not the presence of a spectral dip.
• The saturated green that is the hardest direction under daylight keeps 0.98 of its reach under the LED at forty nanometres and 0.76 under daylight; at eighty, 0.75 against 0.39.
• By 160 nanometres the LED costs slightly more than daylight, 230 directions against 221: once a transition spans two lines, concentrating the weight in lines no longer hides anything.
• Each lamp is measured against its own ideal solid, so the comparison is of how much of each lamp’s boundary a slow pigment reaches, not of which lamp’s boundary is larger.

Four lamps, one limit #

The census is the one the daylight essay used — 305 directions of the object-colour solid, with the black corner and the directions that barely clear it set aside — computed again under three more lamps, each against its own ideal boundary. The lamps are daylight, a tungsten lamp, a white LED built from a blue emitter and a broad phosphor, and an LED built from three narrow emitters.

The figure above counts, for each lamp and each transition width, what share of the directions lose more than a per cent. Three of the curves run together: daylight, tungsten and the phosphor LED cost between 157 and 183 directions at forty nanometres and between 208 and 218 at eighty. A smooth spectrum and a spectrum with one broad hump are, for this purpose, the same lamp.

The three-emitter LED’s curve is the outlier, and it lags rather than differing in kind. At twenty nanometres it costs 17 directions where daylight costs 49; at forty, 62 where daylight costs 183. Then it catches up abruptly — by eighty it has caught up almost entirely, and at 160 it costs a few more than daylight.

The tenth-percentile direction tells the same story with a size attached. Under the LED it holds at 0.98 through forty nanometres and then falls to 0.83 at eighty; under daylight it has already fallen to 0.94 at forty and reaches 0.77 at eighty. The phosphor LED and tungsten sit with daylight at every width. Up to eighty nanometres the lines keep the middle of the census closer to its ideal, and by 160 all four lamps sit near 0.4.

Why the smooth lamps are one lamp here #

Before the lines, it is worth being clear why daylight, a 2856 K tungsten lamp and a phosphor LED give nearly the same census, because they differ a great deal in colour temperature and in how well they render surfaces.

What the census is sensitive to is where the lamp’s power is not. A slope limit costs reach wherever the optimal reflectance has to switch quickly, and a switch costs something only if there is light on both sides of it to be reflected or absorbed. All three smooth lamps put appreciable power at every visible wavelength — tungsten less in the blue, the phosphor LED less in the cyan trough beside its pump — so every edge in every optimal reflectance falls somewhere lit, and blunting it costs. Their censuses differ by a few tens of directions, which is the size of the tilt each spectrum puts on the weighting, and this essay does not try to attribute those differences to particular features.

The phosphor LED is the instructive one of the three. It is sold as a structured spectrum, and a lamp is not a blackbody showed what that structure does to rendering — but its trough between pump and phosphor never falls near zero, so for this question it behaves as a smooth lamp. Structure that matters for rendering is not the structure that matters for a slope limit: the second needs gaps, not dips.

Why the lines’ spacing is the threshold #

The mechanism follows from what the support function weights, and one direction shows it clearly: the saturated green that is the hardest direction of the daylight census.

Under daylight the weighting has a sign at every wavelength, and the optimal reflectance follows it: one where the weighting is positive, nought where it is negative. In this direction the positive part is a band from 505 to 555 nanometres, about fifty wide. A reflectance that takes forty nanometres to rise and forty to fall can still reach one inside it only by cutting into the band’s edges, and every nanometre of edge it gives up is light in the band it fails to reflect or light outside it fails to absorb. It keeps 0.76 of its reach.

Under the LED the weighting is nearly zero except at three lines, and the sign at 490 or 580 nanometres barely matters because there is almost no light there to reflect. The optimal reflectance needs to be high across the green line and low across the blue and red lines, and those are 73 and 97 nanometres away from the green line’s centre. A forty-nanometre transition fits in either gap with room to spare, the reflectance reaches one across the whole green line and nought across the others, and the direction keeps 0.98 of its reach.

So the threshold is geometric. A slope limit costs little under a line lamp as long as a full transition fits between adjacent lines, and the LED’s closest pair is about 75 nanometres apart. It is the same quantity that decides how a display with narrow primaries behaves as a light source, seen from another side: a narrow primary buys a disagreement between observers because the gaps between its lines are where individual eyes differ, and here the gaps are where a pigment’s bluntness goes unpunished.

At eighty nanometres a transition no longer fits between the blue and green lines. The same green still gains from the lamp — it keeps 0.75 of its reach under the LED against 0.39 under daylight — because the green-to-red gap is wide enough for most of a swing and the blue line is weaker than the other two. But it has lost a quarter of its reach where at forty it lost a fiftieth, and across the census as a whole the advantage has almost gone: most directions now need a transition between two lines closer together than eighty nanometres, and those directions lose under the LED about what they lose under daylight.

The sorted census, before and after the threshold #

The sorted census shows where in the distribution the lines’ advantage sits, and how completely it disappears.

At forty nanometres the LED’s curve is above 0.99 for four fifths of the directions, and daylight’s for two fifths. The whole middle of the distribution is lifted: of the 171 directions that lose between one and ten per cent under daylight, 130 lose under one per cent under the LED. That middle is most of the solid’s boundary, and it is where saturated but not extreme colours sit — the part of the solid a paint range or a brand palette is usually specified in, rather than the extreme corners a limit is quoted for.

The forty-one that the lines do not rescue are worth a sentence, because they are not random. They are directions whose optimal reflectance needs a switch close to one of the LED’s lines rather than between two: the switch nearest a line sits a median 12.5 nanometres from it among the unrescued directions and 30.5 among the rescued ones, and 23 of the 41 switch within fifteen nanometres of a line against 5 of the 130. Where the edge falls on a line, the lamp’s weight sits on both sides of it just as daylight’s does, and a slow transition costs about the same under either lamp.

At eighty nanometres the two curves nearly coincide in their counts: 214 of the LED’s directions lose more than a per cent against daylight’s 218, and the tenth percentile keeps 0.83 against 0.77. What separation is left is in the tail, among directions whose switches happen to fall in the wider green-to-red gap. The rescue, for most of the solid, is over. An ordinary pigment’s edge is too slow for the gap it would need, and the lamp’s lines no longer give it anywhere to hide the transition.

What this changes, and what it does not #

Three things are worth taking from it, each with a qualification.

A saturated colour made with ordinary pigments is closer to its limit under a three-line lamp than under daylight. That is the practical content of the census between twenty and sixty nanometres. What a gamut costs found the corresponding fact for displays, where narrow primaries reach further at a price; here the narrow lines are in the light rather than in the display, and what they buy is forgiveness for a slow reflectance. It is a statement about each lamp’s own ideal solid, and the solids differ — no surface can be that colourful measured how much a bound over surfaces can move — so it does not say a pigment looks more saturated under an LED, only that less of what the LED could show is lost to the pigment’s bluntness.

The advantage is set by line spacing, not by line width or colour temperature. A lamp designer adding a fourth emitter to fill a gap would shorten the spacing and bring the threshold down, trading some of this effect for rendering — the same kind of trade a fourth primary is a design described for a display. A screen is a poor lamp argued the other half: the narrow lines that spare a slow pigment are the lines that render real surfaces badly, because a real surface’s reflectance is exactly the slow kind the lines let hide.

And the census cannot say what happens inside a line. A direction whose weighting changes sign within an emitter’s band asks for a switch over tens of nanometres or less, where a line lamp concentrates all its weight. Those directions exist in the sweep, but they are the ones that barely clear the black corner, where the census’s ratios are least reliable, and nothing here is claimed about them. Three numbers cannot see a line is the instrument’s version of the same blind spot: the structure that matters is too narrow for anything slow to follow, and too narrow for the census to measure well.

How the census was computed #

The solid under each lamp is the support function through the 1931 observer, over a Fibonacci sweep of 2,400 directions of which every sixth is taken, with directions whose ideal support is under five units — the black corner and the directions nearly tangent to it — set aside. That leaves 305 directions under daylight and 312 under the three-emitter LED. Each constrained support is a dynamic program over the 81-band grid with the reflectance quantised into 161 levels, so each transition width is imposed exactly, and each ratio is taken against the same lamp’s own ideal.

The three-emitter LED is a constructed spectrum of three Gaussian emitters centred at 455, 528 and 625 nanometres, which carry 97 per cent of its power within a tenth of its peak and leave under one per cent of the peak in the gaps between them. The phosphor LED is a blue emitter at 452 nanometres and a broad phosphor centred at 565; tungsten is CIE illuminant A. The direction drawn is the one whose ratio under the LED exceeds its ratio under daylight by the most, among directions both censuses keep.

What this leaves out #

The LED here has one set of line positions and widths. A lamp whose closest lines are nearer together would have its threshold lower, and one with broader lines would soften it; neither is computed.

The slope limit is the weakest statement of a pigment’s bluntness, as the daylight essay said. A real colorant’s band shape removes more, and under a line lamp the question is sharper, because what matters is the reflectance at three narrow places and the shape between them barely enters.

And the ratios are against each lamp’s own ideal. A viewer standing under the LED does not see its solid as a fraction of anything; they see colours, and what those colours look like is a separate question with its own answer.

Still open: where a fourth line puts the threshold #

The threshold is the spacing of the closest pair of lines, which makes a clean prediction for a four-emitter lamp. Adding a line halfway between blue and green would bring the closest spacing down to under forty nanometres, and the census should jump at about forty instead of about eighty — trading away most of the rescue.

The computation is this one with a fourth emitter swept across the gap, and the output worth having is the pair of curves it trades: directions spared against rendering lost, since a fourth line is usually added to render better. If the two curves cross, there is a lamp that renders acceptably and still spares slow pigments, and its line positions are the answer.

A bound over surfaces is a bound under a light #

The habit is about a bound that looks like a property of the objects measured.

The object-colour solid is described as the set of colours surfaces can have, as though surfaces carried it. They do not: it is the set of colours surfaces can have under a light, for an observer, and the light chooses which features of a reflectance count. A constraint on reflectances then costs whatever the light says it costs, and the same pigment is nearly unconstrained under one lamp and substantially constrained under another.

The move is to recompute a bound under a light with a very different structure before trusting its shape. Daylight, tungsten and a phosphor LED are all smooth for this purpose and agree; three lines are not smooth, and they cut the count of affected directions by two thirds in the middle of the range and then gave it all back.

The failure mode is to treat a bound computed under one smooth light as a property of the materials. It is a property of the materials and the light together, and the light’s contribution is largest exactly where it is least smooth.