The lamp outweighs the pigment model

The object-colour solid is every colour a surface can have under a lamp, and its boundary is drawn by ideal pigments that switch between absorbing everything and reflecting everything at one or two wavelengths. No real colorant switches instantly. That bluntness has been modelled three ways, each more physical than the last. The limits assume a pigment that switches instantly introduced a slope limit: reflectance may change by no more than a stated amount per nanometre. A sharp edge is bought with depth replaced it with absorption bands no narrower than forty nanometres, and found that a dense enough band draws a sharper edge than its width suggests. Written in energy, the bands spare the green fixed the bands’ minimum width in energy rather than wavelength, as an absorption band’s physics suggests.

Each census counted the directions of the solid a model cannot reach to within one per cent of the ideal, and each ran under lamps built from three or four narrow emitters. The ordering was the same every time: slope dearest, bands in nanometres next, bands in energy cheapest. The last essay asked whether that ordering is a property of the pigments or of lamps with gaps, and predicted that it holds under the collection’s broadband lamps, with every count smaller, because a broadband lamp lights every stretch of the spectrum a little and no single band’s flank is singled out.

The order holds; the sizes go the other way #

Under daylight, tungsten and a white LED, the slope limit loses more directions than bands in nanometres, and bands in nanometres more than bands in energy — the same order as under three narrow emitters. But every model loses far more under a broadband lamp, not fewer: under daylight 89, 73 and 70 of 154 directions against 31, 23 and 20 under the emitters, between 2.4 and 3.5 times as many. The losses are shallow — no model under any lamp falls more than 2.5 per cent short of the ideal on average. And the lamp outweighs the model: under daylight the three models differ by nineteen directions, while moving any one of them from the emitters to daylight costs it at least fifty.

• The ordering is the pigments’. It survives every lamp, broad or narrow.
• The prediction’s sizes are backwards. A lamp with gaps spares a smooth pigment; a lamp without them charges it.
• Which lamp matters more than which model, under every lamp and for every model.
• Under tungsten the three models nearly agree, and under the white LED they differ most.

Four lamps #

The three broadband lamps are the collection’s standard ones. CIE D65 is average daylight, a bumpy but continuous spectrum across the visible range. CIE A is a tungsten lamp, a Planckian radiator at 2856 K — the curve blackbody and the colour of temperature derives — whose power rises smoothly and steadily from violet to red. The white LED is a blue pump near 452 nanometres with one broad yellow phosphor, the most common lamp indoors; it has a dip near 480 but no gap. The fourth is the lamp the earlier censuses used, three narrow emitters in the blue, green and red, with dark stretches between them.

The census is otherwise exactly the earlier ones’. The same directions through the solid, 148 to 157 of them depending on which the lamp leaves meaningful; the same ideal against which each model’s reach is measured; the same slope-limit program and the same band searches from the same starts. Under the three emitters it reproduces the counts the earlier essays published — 31 for the slope limit, 23 for bands in nanometres, 20 for bands in energy — which is the check that nothing else has changed.

What each model loses under each lamp #

The order is the same under every lamp. Under daylight the slope limit loses 89 directions, bands in nanometres 73 and bands in energy 70; under tungsten 77, 69 and 66; under the white LED 83, 61 and 54; under the emitters 31, 23 and 20. In every row the slope limit is dearest and energy cheapest, and bands in nanometres sit between. Whatever makes the band models cheaper than the slope limit — a dense band’s edge being sharper than its width, and an energy-fixed width being narrower in the blue where many edges fall — is a property of the models, and it does not depend on the lamp.

Everything else does. Three lines spare a slow pigment found this for the slope limit alone, on a finer set of directions: a lamp of narrow lines measures reflectance only where its lines are, so a pigment can do its slow changing in the dark between them, and the lamp stops helping once the pigment’s transitions are as wide as the lines’ spacing. The three models here all benefit from the same spareness, and the broadband lamps take it away from all three.

Among the broadband lamps, tungsten is the cheapest for the slope limit and daylight the dearest for every model. Tungsten’s power at 450 nanometres is about a quarter of its power at 600, so the blue, where the eye’s sensitivities overlap most steeply and a pigment’s edge moves the colour most, is the stretch it lights least; the slope limit, whose losses gather in the blue-green, is spared a dozen directions there against daylight. Daylight lights the blue as strongly as the red. The white LED is in between: its pump puts strong light in the blue, but its phosphor leaves a dip near 480 nanometres, and the band models, which can put a sharp edge where the dip is, take advantage of it that the slope limit cannot.

How many times more #

Every broadband lamp charges every model between 2.4 and 3.5 times what the emitters do. Under daylight the slope limit is charged 2.9 times, bands in nanometres 3.2 and bands in energy 3.5; under tungsten 2.5, 3.0 and 3.3; under the LED 2.7 for all three.

So the band models, which gain most over the slope limit under lines, lose most of that advantage under daylight and tungsten. Under the emitters bands in energy lose 35 per cent fewer directions than the slope limit; under daylight 21 per cent fewer; under tungsten 14. The lines were doing part of the band models’ work: a band can put its steep edge exactly where an emitter’s flank sits, and between emitters its softness costs nothing. Under a lamp with no gaps a band has to be sharp everywhere a direction needs an edge, and a forty-nanometre band is not.

Where the lost directions’ edges fall #

Each direction of the solid is reached ideally by a pigment that switches at one or two wavelengths, and where those edges fall says which part of the spectrum a model fails in. Under the three emitters, the directions bands in energy lose have their edges between 480 and 580 nanometres — near the green emitter’s flanks, where a band would have to switch between two lines closer together than its width allows. Under daylight there are more than three times as many such edges, from 460 to 640 nanometres, gathered near 480 and near 560.

The 480-nanometre cluster is the blue-green, where daylight is strong and the eye’s three sensitivities cross most steeply, so a small shift in an edge changes the colour most; a limit written in energy charges the reds found the energy convention narrowing the bands there, which is why bands in energy lose fewest. The 560 cluster is the yellow-green, where a white LED’s phosphor and daylight both peak.

Many losses, each small #

A lost direction is lost by a little. Averaged over every direction of the census, the slope limit under daylight falls 2.5 per cent short of the ideal reach, the band models under 1.9; under tungsten 2.4 and 1.5; under the LED 2.2 and 1.3; under the emitters all three under one. A direction counts as lost at one per cent; under daylight half of the directions bands in energy lose are lost by under two per cent, and nine in ten by under nine.

This matters for reading the counts. The broadband lamps’ larger counts are not a collapse of what a real pigment can reach; they are many directions pushed just across the one-per-cent line. The pigment decides the trade, not the band priced a lamp’s rendering in lost directions, and a lamp that lights every wavelength charges a smooth pigment a little in many directions rather than a lot in a few.

The same directions, two lamps #

Direction by direction, daylight is usually harder. Of the 154 directions present under both lamps, 98 lie above the diagonal — the band model falls further short under daylight than under the emitters — and 36 sit at the origin, reached fully under both. Twenty lie below it, directions daylight makes easier than the emitters do — a seventh of the census, and the reason the emitters’ advantage is a count rather than a rule. The emitters’ advantage is not a few directions they happen to favour; it is a general slack in what a pigment is asked to do, because between the lines nothing is asked at all.

The lamp outweighs the model #

Under every lamp, choosing a different pigment model changes the count less than choosing a different lamp. Under daylight the three models span 89 to 70, nineteen directions; moving any one of them from the emitters to daylight costs it at least fifty. Under tungsten the models span eleven, under the LED twenty-nine, under the emitters eleven.

Tungsten makes the three models nearly one: 77, 69 and 66. A Planckian radiator’s power changes slowly and monotonically, and it weights the blue, where the models differ most, least of any lamp here; the directions whose edges fall in the blue-green, where bands in energy have their advantage, are the directions tungsten barely lights. The LED separates them most, 83 against 54: its strong narrow-ish pump at 452 nanometres is a feature a band model can exploit and a slope limit cannot, a small version of what the emitters do.

So the question the earlier censuses kept asking — which model of a pigment should a lamp designer believe? — has an answer that depends less on the model than it seemed. For a lamp with lines it matters by a third; for daylight by a fifth; for tungsten by a seventh. The lamp matters by a factor of three.

How the census was run #

The directions are every twelfth of the collection’s sweep of the object-colour solid under D65, kept when the ideal reach under the lamp is above the census’s floor. Under each lamp — CIE D65 and CIE A as tabulated, the collection’s white LED (pump at 452 nanometres and 22 wide, phosphor at 565 and 118 wide, 62 per cent converted) and its three narrow emitters — each direction’s ideal reach is the step pigment’s, the slope limit’s is the best reflectance the collection’s slope-limit program finds when no transition may be faster than forty nanometres from nothing to full, and each band model’s is the best of up to three Gaussian absorption bands found by Nelder–Mead from the earlier censuses’ starts, with widths floored at forty nanometres or at 0.158 electronvolts. A model loses a direction when its reach is under 0.99 of the ideal; the shortfall is one minus that ratio, floored at nought, averaged over directions. Edges are the ideal pigment’s switching wavelengths for each lost direction.

What this leaves out #

The lamps are three broadband lamps. A fluorescent tube has lines on a broad bed and would sit between the LED and the emitters; a daylight at a different colour temperature would move the blue weighting that separates tungsten from D65. The ordering should survive both, and the counts would move with how much of the spectrum each lamp leaves dark.

The pigment models are three. A real colorant library, measured, would replace them — a sharp edge is bought with depth left that open — and the question here would become whether real colorants lose directions as bands in energy do.

One per cent is a line, not a law. The counts are sensitive to it, the shortfalls are not, and the ordering holds for both.

At a coarser line #

Counting at one per cent treats a direction lost by one per cent like one lost by ten, and the models’ mean shortfalls under the broadband lamps are close together. Counted at two, three and five per cent, the slope limit’s disadvantage grows and the band models converge. At three per cent, under daylight the slope limit loses 39 directions and the band models 24 and 20; under tungsten 43, 19 and 18; under the LED 33, 14 and 14. The slope limit now loses about twice what either band model does under every broadband lamp, where at one per cent it lost a fifth to a half more; and the two band models, which differed by up to seven directions at one per cent, differ by at most four.

At five per cent the band models are within a direction of each other under every broadband lamp, and the slope limit still loses about twice as many. At ten per cent all three models lose about half a dozen directions under each broadband lamp and one under the emitters, and the distinction disappears into the handful of directions no smooth pigment can reach at all.

So the slope limit’s losses are deeper than the band models’, and the band models’ losses are the same losses, nearly, whichever width convention they use. For a lamp designer the practical reading is that either band model will do and the slope limit will overstate what a lamp costs, by a factor that grows with how large a shortfall the designer cares about.

Still open: whether a tube’s lines spare a pigment as an emitter’s do #

A fluorescent tube is both kinds of lamp at once: narrow mercury lines standing on broad phosphor bands. The lines might spare a slow pigment the way the emitters do, or the bed under them might charge it the way daylight does.

The calculation is this census under the collection’s tube and under a tube whose lines carry twice the share of its power, with the three models counted at one and three per cent. The prediction is that the tube sits much nearer the broadband lamps than the emitters — about two thirds of daylight’s count for each model — because its phosphor bed lights the gaps between its lines, and a pigment’s slowness is charged wherever there is light; and that doubling the lines’ share moves it only a quarter of the way towards the emitters. If so, the spareness the emitters grant is a property of darkness between lines, not of lines, and a lamp designer who wants it has to take the bed away.

A ranking that holds is not a size that holds #

The habit is about what a robust ordering does and does not carry.

The three models have kept one order under every lamp tried, and that is a real finding about them. It was easy to read the order as a statement about how much the choice of model matters, since the models were always compared under one lamp at a time. Put under several lamps together, the order stays and its importance shrinks: the lamp moves each model three times further than the models differ from each other.

The failure mode is to measure a difference under fixed conditions and then treat it as the largest difference in play. The pigment model’s effect was measured carefully under lamps with gaps. The gaps were a larger effect, and they were the thing held fixed.