Written in energy, the bands spare the green

The pigment decides the trade, not the band moved the four-emitter lamp’s trade between colour rendering and a slow pigment’s reach by changing what the pigment was. Modelled as a slope limit — a reflectance that may change no faster than a full swing in forty nanometres — the pigment lost 19 more directions of the object-colour solid under the cheapest lamp rendering at a fidelity of 88 than under the three emitters alone. Modelled as three absorption bands none narrower than forty nanometres, it lost 10. And a fourth band at 590 nanometres, which renders at 86, cost the band-shape pigment six directions.

That essay flagged the assumption its best result leaned on. Forty nanometres is a width in wavelength, and an absorption band is a spread of photon energies — a transition between two states broadened by vibration and environment, whose width is closer to a fixed number of electronvolts than to a fixed number of nanometres. The same energy spread is fewer nanometres in the blue and more in the red, because wavelength goes as the inverse of energy. A limit written in energy charges the reds had already found that writing a slope limit’s width in energy moves its losses redward. The worry was that the same change would widen the red-facing flanks of the bands whose edges pass 590, and take back the free amber.

The calculation is the same census with every band’s minimum width fixed at 0.158 electronvolts — exactly forty nanometres at 560, the middle of the spectrum — and converted to nanometres at the band’s own centre.

Fewer losses everywhere, and the amber untouched #

Written in energy, the band limit loses fewer of the solid’s directions under every one of the ten lamps: 20 against 23 under the three emitters, 26 against 33 under the lamps rendering best. The fourth band at 590 costs it six directions more than the three emitters, exactly what it cost in nanometres. And the price of raising the fidelity from 78.6 to 88 falls again, from ten directions to six — against nineteen for a slope limit.

• The prediction was that energy would charge the amber. It charges the red a little and spares the green a lot.
• The directions won back have their edges between 460 and 580 nanometres, 64 of their 68 edges pooled over five lamps. The few newly lost sit at 460 to 480 and 580 to 620.
• An edge at 560 is usually a band centred near 520, and in energy that band is 34.5 nanometres wide rather than forty.
• Each more physical model of the same bluntness charges rendering less: nineteen directions for a slope limit, ten for bands in nanometres, six for bands in energy.

What the two conventions allow #

The two conventions agree at 560 nanometres and nowhere else. Written in nanometres, a band may be no narrower than forty wherever it sits. Written in energy, it may be no narrower than 26 nanometres at 450, 32 at 500, 37 at 540, 43 at 580, 46 at 600 and 54 at 650. The ratio of the two is the square of the ratio of the wavelengths: a band at 450 is (450/560)² — two thirds — as wide as one at 560, in nanometres.

What a slow pigment is charged for is its edges, not its bands, and a sharp edge is bought with depth worked out how the two relate. A deep band’s flank rises from ten to ninety per cent reflectance over a distance set by its width and its absorbance; at an absorbance of twelve, the deepest this model allows, a forty-nanometre band draws an edge 21 nanometres wide. The dashed curves are that edge under each convention: a constant 21 in nanometres, and in energy from 14 at 450 to 29 at 650. A pigment written in energy has blue-green edges a third sharper and red edges a third blunter than the same pigment written in nanometres.

Every lamp, charged three ways #

On every row the three markers sit in the same order: the slope limit furthest right, bands in nanometres in the middle, bands in energy furthest left. The energy convention’s gain over the nanometre one is three directions under the three emitters, three at 540, four at 560, nine at 570, eleven at 580, three at 590, and seven under both lamps rendering at 88 or better. So the largest gains are under the lamps with a fourth band at 570 and 580 — the lamps that were dearest for both earlier models, because their fourth band lights the orange where the solid’s edges are dense.

That is the reverse of what the proposal feared, and it is worth being clear why the fear was reasonable. A lamp band at 580 lights exactly the stretch where a band-shape pigment’s red-facing flanks rise, and in energy those flanks are blunter than in nanometres. If the flanks at 580 belonged to bands centred at 580, they would be about 43 nanometres wide instead of forty, and the lamp would charge them more. They mostly do not.

Which directions change sides #

Pooled over the three emitters, the 580 and 590 lamps and the two lamps rendering at 88, 55 directions change sides: 43 are lost in nanometres and reached in energy, 12 the other way. The edges of the ones won back crowd between 460 and 580 nanometres — 64 of 68. Those newly lost put 17 of their 20 at the two ends, 460 to 500 in the violet-blue and 580 to 620 in the orange-red.

The ends are where the conventions differ most, and in opposite directions. At 600 a band in energy is 46 nanometres wide — blunter, and the orange-red directions pay for it. Those are the red charges a limit written in energy charges the reds predicted: eight edges’ worth, real, and outnumbered by the green by more than five to one. At 470 a band in energy is 28 wide, sharper, and the census still loses a few directions there. The likely reason, which this census does not isolate, is that the minimum width is a floor and not a gift: a direction whose best band in nanometres was wide and shallow, holding a stretch of the violet low with its wing, is not helped by permission to be narrower, and the search’s starts are built for the nanometre floor. Nine violet-blue edges against 38 in the green is the size of it either way.

What the histogram cannot show directly is the thing that settles the amber: an edge’s position is not its band’s position. A band’s red flank sits half its width and more above its centre, so an edge at 560 or 570 nanometres belongs to a band centred at 520 or 530, and in energy that band is narrower than forty. The edges a fourth band at 580 or 590 lights are, for the most part, the red flanks of green bands.

This direction wants a reflectance that is high everywhere except for a notch from 497 to 545 nanometres, straddling the green emitter. Both conventions put one absorption band near 518 to cover it. In nanometres the band may be no narrower than forty, and the best one has an absorbance of 4.4; in energy it may be 34, and the best one is 34.1 nanometres wide with an absorbance of 6.7. The narrower, deeper band’s flanks rise more steeply on both sides — through the blue-green gap and through the green-to-amber one — and the direction reaches 99.4 per cent of its ideal instead of 98.7. That is the whole mechanism in one picture: the gain is a green band made narrower, and it shows up at the amber because that is where the band’s red flank is.

The positions, and the price #

Both curves have the same shape: cheap at 540, dear at 570 and 580, falling back at 590 and 600. The energy curve is below at every position, and furthest below at 580, where it is eleven directions under. At 590 the two are three apart, and each sits six above its own three-emitter floor. The free amber survives the change of convention exactly: a fourth band at 590 renders at 86 for six directions whichever way the bands are written.

Three models of one bluntness, and three prices. A slope limit pays 19 directions to go from the three emitters to a fidelity of 88; bands in nanometres, 10; bands in energy, 6. Each step is a step towards what a colorant physically is — first from a rule about slopes to a sum of absorption bands, then from a width in the wrong unit to a width in the right one — and each step makes rendering cheaper for the pigment.

That ordering is the practical content. The limits assume a pigment that switches instantly introduced the slope limit as a floor on bluntness: no real colorant does better. As a model of a colorant’s cost under a lamp, it is the most pessimistic of the three, and a lamp designer who took its price at face value would be paying three times what real pigments charge.

Four censuses of one decision #

This is the fourth census of one lamp decision — where to put a fourth band in a three-emitter LED — and read together they say something none says alone.

Three lines spare a slow pigment found the three-emitter lamp itself kind to a slow pigment, because it is dark where a blunt reflectance does its changing. A fourth emitter spends the gap it fills found the fourth band taking that darkness back in proportion to how well it renders, with 580 the best renderer and among the dearest. The gap has to be dark, not the line narrow had already shown that what matters is the darkness between lines rather than their narrowness, which is why a band placed in a gap costs what it does. A lopsided band does not break the trade freed the band’s shape and moved the answer by a few directions. And the last two censuses changed the pigment instead and moved it by half, then by half again.

So the size of the trade was mostly a statement about the pigment model, and its shape was a statement about the lamp. Under every model the fourth band is cheap at 540, dear at 570 and 580, and cheaper again at 590 and 600; that pattern comes from where the band sits against the solid’s edges, and it survived every change of pigment. What changed was how much each position cost: a fourth band at 580 costs a slope limit 22 directions beyond the three emitters, bands in nanometres 15, bands in energy 7.

That is also why the best recommendation moved. With a slope limit, the cheapest rendering improvement was a band at 540, which renders two and a half points better. With bands in nanometres it was 590, rendering seven and a half better for six directions. With bands in energy, 580 — the best renderer of the single-band sweep, at 88 — costs seven directions and 590 costs six, so the difference between them nearly disappears and a designer can simply take the better renderer. The advice depended on the model at every step, and the most physical model gave the least restrictive advice.

What a lamp designer should take from it #

The free amber at 590 is not an artefact of the convention. It held with bands written in nanometres and it holds, to the direction, with bands written in energy. A designer adding a fourth band to a three-emitter lamp for rendering can put it at 590, render at 86, and cost the pigments in the room about six of the solid’s hundred and fifty directions — four per cent — whichever way their bands are described.

The dearest positions are dearer for crude models. A fourth band at 570 or 580 renders best, and it is exactly where the three models disagree most: 51 to 53 directions for a slope limit, 38 to 40 for bands in nanometres, 27 to 31 for bands in energy. A decision about those positions depends on the model more than any other, and the most physical model says they cost barely more than 590 does.

And the blue matters more than it looks. The directions energy won back were won in the green and blue-green, by bands narrower than forty nanometres. A pigment library whose blue and green colorants are narrow in wavelength — as dyes with a fixed energy width are — is better served by a four-emitter lamp than a model with a uniform width predicts, anywhere in the amber.

How the census was run #

Everything is as in the pigment decides the trade, not the band: the ten lamps; every twelfth direction of the Fibonacci sweep of the sphere, 152 to 157 clearing the boundary floor; three Gaussian absorption bands with absorbances up to twelve under Beer–Lambert; the same Nelder–Mead search from the same structured starts; a direction lost when its reach is below 99 per cent of its ideal. The one change is the floor on each band’s width, which is 0.158 electronvolts converted to nanometres at the band’s centre, $w = 0.158\,\mu^2 / 1239.84$. A band the search pushes outside the visible range is floored at the width it would have at the nearest end of the range, so an idle band far off the grid cannot acquire a width of thousands of nanometres and reach back into it.

What this leaves out #

A fixed energy width is also an idealisation. Real absorption bands broaden with temperature and environment, are often asymmetric with a longer tail towards higher energy, and in the red are sometimes vibronic progressions rather than single bands. Writing the width in energy is closer to the physics than writing it in wavelength; it is not the physics.

The reference point is a choice. The two conventions were made to agree at 560 nanometres. Agreeing at 500 would make every band in energy narrower than here and the gains larger; agreeing at 600 would make them wider and the gains smaller. The finding that the amber is free at 590 does not depend on the choice, because the edges there belong to bands near 520 — but the size of the overall gain does.

And the census is half density. Every twelfth direction, as in the essay before, so the counts are about half what a full sweep would give.

Still open: whether the ordering of the three models holds under daylight #

This census and the two before it compared pigment models under lamps built around three or four narrow emitters, and found each more physical model charging rendering less. A sharp edge is bought with depth compared the slope limit with bands in nanometres under daylight and found bands reaching further in most directions, but that is a statement about reach, not about what a change of lamp costs.

The calculation is the price of a change of lamp under all three models: from daylight to each of this collection’s broadband lamps — the incandescent, the phosphor LED — with the directions lost counted for each model under each. The prediction is that the ordering holds, slope dearest and energy cheapest, because it comes from the pigment’s edges and not from the lamp’s lines; and that the sizes shrink, because a broadband lamp lights every stretch of the spectrum a little and no single band’s flank is singled out. If the ordering reverses under a broadband lamp — if a slope limit charges less than bands there — then the ordering here is a property of line lamps, and the slope limit’s pessimism is specific to lamps with gaps.

Unit is part of the model #

The habit is about treating a unit as a modelling decision rather than as notation.

“Forty nanometres” and “0.158 electronvolts” are the same number at 560 nanometres and different constraints everywhere else, and the choice between them was made implicitly, by the grid the reflectances were tabulated on. A model written in the grid’s unit inherits the grid’s idea of what is uniform. Rewritten in the unit the physics uses, the constraint loosened where the edges that mattered were, and the prediction that it would tighten at the amber turned out to be about the wrong bands.

The failure mode is to predict a change of unit by where the numbers change most. The numbers changed most at the ends of the spectrum; the result changed most in the green, because that is where the bands were whose edges the lamp was lighting.