A limit written in energy charges the reds

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, by forbidding it to change faster than one unit per stated number of nanometres. At forty nanometres the median direction loses 1.4 per cent of its reach and the tenth percentile six; the greens pay most.

That essay said, in passing, that the limit is written the simplest way: the same number of nanometres at every wavelength. Nothing about a colorant is uniform in nanometres. An absorption band is a transition between energy levels, broadened by the molecule’s vibrations and its surroundings, and to a first approximation its width is a spread of energies. A spread of fixed energy covers a number of nanometres that grows as the square of the wavelength — the same band is 2.4 times as wide at 700 nanometres as at 450.

So the limit has a second natural form, and the two forms disagree about the ends of the spectrum. The question is whether that changes what the limit costs, and the answer turns out to be that it changes very little about how much and a good deal about who.

The same limit, redistributed #

Written as a fixed spread of energy that equals forty nanometres at 550, the limit costs more than a per cent of reach in 489 of the solid’s 913 directions, against 544 when it is written in nanometres. The median direction keeps 0.989 against 0.986. And the reds lose more — 8.0 per cent against 6.3 — while the greens, yellows, blues and purples lose less.

• At eighty nanometres the effect is larger in both directions: the tenth percentile keeps 0.81 against 0.77, and the reds lose 26.7 per cent against 22.0.
• The greens still pay most either way, 9.7 against 11.1 at forty and 32.8 against 36.0 at eighty.
• A direction’s share is set by where its optimal reflectance switches. Of the 543 directions whose edges lie on average below 550 nanometres, 525 lose less when the limit is written in energy; of the 130 whose edges lie above, 103 lose more.
• The limit in energy is sharper than the nanometre version below 550 and blunter above, 21 nanometres wide at 400 and 67 at 700 — so the rule is the arithmetic of the conversion, not a property of any hue.

Two ways to write one sharpness #

The figure above draws both forms of a forty-nanometre limit across the spectrum. Written in nanometres it is a horizontal line. Written in energy it is forty at the reference wavelength of 550 and rises as the square of the wavelength, from 21 nanometres at 400 to 67 at 700, in the steps the calculation’s quantisation actually imposes.

The choice of 550 as the reference is itself a choice, and it is the one that makes the comparison fair: it is near the middle of the visible range and near the peak of the luminance weighting, so the two limits agree where most of a colour’s luminance comes from. A reference at 500 would make the energy-written limit blunter everywhere above 500 and the comparison would mostly measure that; a reference at 600 would do the opposite. With the reference in the middle, the two limits have about the same average sharpness across the spectrum, and what differs is the distribution.

Neither form is a colorant. A brand colour is an ink is the reminder that a specified colour is made of a particular material with a particular spectrum, and both forms here are stand-ins for the edge such a material would have. Five nanometres is a choice made the corresponding point about the wavelength grid the whole calculation runs on: a representation chosen for convenience carries an assumption about which parts of the spectrum deserve equal treatment. Writing the limit in nanometres assumes a blue edge and a red edge of equal sharpness are equally hard to make. Writing it in energy assumes they are equally hard when the red is nearly twice as wide.

The census barely moves #

The first question is the total.

The pairs of curves nearly coincide. The medians agree to within a third of a point up to forty nanometres and part by 2.8 points at 160; the tenth percentiles by one point at forty and four at eighty. At no width does the energy-written limit keep less of the solid, and from twenty nanometres up it keeps slightly more; the count of directions losing more than a per cent falls from 544 to 489 at forty, while barely changing at eighty and 160.

That the energy-written limit is kinder overall follows from where the solid’s difficult edges are. Among the directions that lose anything to a slope limit, the average wavelength at which the optimal reflectance switches has a median of 520 nanometres, and four in five lie between 490 and 570 — the region where the matching functions cross each other most often. Below 550 the energy-written limit is the sharper of the two, so a slightly sharper limit where most edges are outweighs a much blunter one where few are.

So a reader who wanted to know how much of the solid a blunt pigment gives up can use either form and be within a few points. The difference is not in the total.

The bill moves along the spectrum #

It is in which colours pay.

At forty nanometres the reds’ loss rises from 6.3 per cent to 8.0 and the cyans’ from 1.2 to 1.4; every other sector’s falls. The greens go from 11.1 to 9.7, the purples from 3.2 to 2.4, the yellows from 1.8 to 1.4, the blues from 1.5 to 1.2. The largest change in proportion is the purples’, a quarter less; the largest in points is the reds’, nearly two.

The reds and the cyans are an odd-looking pair — opposite hues — and the pairing is the first hint of the rule. An optimal saturated red is, in five cases of six, a long-pass reflectance — nought below an edge and one above it — with the edge at a median of 583 nanometres. An optimal saturated cyan is almost always the complement, one below an edge and nought above it, with the edge at a median of 573. Both are single edges placed above 550, where the energy-written limit is blunter than the nanometre one, and both pay more. The greens have two edges each, at a median of 535 between them; the yellows, blues and purples switch at medians between 505 and 513. All of those sit below 550, and all of them pay less.

The rule is the edge #

The hue sectors are a summary of something that can be measured direction by direction.

Left of 550 nanometres the points sit below the line and right of it they mostly sit above. Of the 543 directions whose optimal edges average below 550, 525 lose less when the limit is written in energy. Of the 130 whose edges average above, 103 lose more. The 27 exceptions on the right are directions with two edges, one on each side of 550, whose average happens to fall above it while the sharper side does most of the work.

That is the whole mechanism, and it is arithmetic rather than colour science. The energy-written limit is blunter than the nanometre limit exactly above the reference wavelength, so any direction whose reflectance has to switch there is charged more, and any direction that switches below it is charged less. A colour’s share of the bill follows where its edges are, not what it is called — which is why red and cyan, opposite in hue and alike in edge, move together.

One red, followed down #

A single direction shows what the blunter edge does to the reflectance itself.

The ideal reflectance is nought below 595 nanometres and one above, a long-pass edge that makes a saturated orange-red of lightness 51 and chroma 123. Written in nanometres, a forty-nanometre limit lets the reflectance rise over forty nanometres centred on that edge, and the direction keeps 0.89 of its reach. Written in energy, the limit at 600 nanometres is 47 nanometres wide, the rise is correspondingly slower, and the direction keeps 0.85.

Four points of reach for seven nanometres of edge is a steep price, and the reason is where the edge sits. A red’s edge falls on the long-wavelength flank of the luminance and red-channel weighting, where a small shift in where the reflectance rises moves a large share of the light the colour depends on. The same seven nanometres at an edge near 470 would cost almost nothing, because there is little weight to move.

At eighty nanometres the gap widens to nine points. The nanometre-written limit keeps 0.66 of the red’s reach and the energy-written limit, 100 nanometres wide at the edge, keeps 0.57. The pattern holds across the sectors: at eighty the reds lose 26.7 per cent in energy against 22.0 in nanometres, while the purples fall from 12.3 to 9.1 and the greens from 36.0 to 32.8.

The eighty-nanometre sectors confirm that the redistribution grows with the limit rather than changing character: every sector moves in the same direction it moved at forty, and by more. At an ordinary pigment’s sharpness a saturated red gives up about a quarter of its reach if pigment edges are uniform in energy, and a fifth if they are uniform in nanometres, and nothing in the calculation can say which assumption is closer without measured colorants.

What this changes for a colourist #

Two things, and one of them is a warning about the earlier result.

The ranking of which colours are hardest to make does not change, and the gaps between them do. An extended ink set is the place this bites: the fifth ink buys a corner found an orange ink’s whole contribution landing in its own region of the gamut, and whether an orange pigment’s edge near 580 nanometres is charged like a red’s or like a yellow’s is exactly the difference between the two forms. Greens remain the costliest direction under both forms, then reds, then purples. But the ratio between greens and reds falls from 1.8 at forty nanometres in the nanometre form to 1.2 in the energy form. A specification process that sets tolerances by how far a colour sits from its theoretical limit would treat reds noticeably more gently under one assumption than the other.

And the daylight census’s hue ranking should be read with the form of its limit attached. The limits assume a pigment that switches instantly reported the reds as losing about half what the greens lose. That is true of a limit written in nanometres. In energy the reds lose four fifths of what the greens lose. The qualitative claim — greens pay most — survives; the quantitative one depends on an assumption about chemistry that the census cannot check.

The lamp’s structure enters the same way it did in three lines spare a slow pigment: what the limit costs depends on where the light is. Nothing here computes the energy form under that lamp, but the arithmetic makes a prediction worth recording — a blunter limit near the red emitter than near the blue one should end that essay’s rescue sooner for directions that switch between the green and red lines than for directions that switch between blue and green.

How the census was computed #

The solid is the support function under D65 through the 1931 observer, over a Fibonacci sweep of 2,400 directions, of which every second is taken and those whose ideal support is under five units — the black corner and the directions nearly tangent to it — are set aside, leaving 913. Each constrained support is a dynamic program over the 81-band grid with the reflectance quantised into 161 levels. For the nanometre form the step limit is the same at every band. For the energy form the transition width at each band is the stated width times the square of the band’s wavelength over 550, turned into a whole number of levels, so the steps drawn in the first figure are the ones imposed.

A hue sector counts the directions whose ideal boundary colour has a CIELAB chroma of at least 20 and a hue angle inside the sector. A direction’s edges are the wavelengths at which its weighting changes sign; the comparison of edges counts only directions that lose at least a thousandth of their reach to the nanometre form.

What this leaves out #

The energy form is a first approximation, and so is the idea that a pigment’s reflectance edge inherits its absorption band’s width. A pigment in a binder scatters as well as absorbs, and paint is not a filter is the argument that its reflectance is not a simple function of its absorption; the edge of a scattering layer can be sharper or blunter than the band that produces it. Real absorption bands are not all of equal width in energy either — some chemistries are sharper in the blue, some in the red — and a band’s two sides are rarely symmetric. The census is the same dynamic program with a different step at each band, so any measured profile of sharpness against wavelength can be substituted directly, and that substitution is the useful next measurement. A notch a pigment cannot cut put a floor under how narrow a band a molecule can produce; stated as a function of wavelength, that floor is exactly the kind of profile the program would take.

The slope limit remains the weakest statement of a colorant’s bluntness in either form. It allows triangular and trapezoidal reflectances that no dye produces, so both forms understate the cost; what this essay establishes is how the cost moves between them, not its size.

And the reference wavelength is a choice. It was put at 550 to keep the two forms’ average sharpness close; moving it moves the boundary between the colours that pay more and the colours that pay less by exactly the amount it is moved.

Still open: a measured sharpness against wavelength #

The rule makes a prediction that measured colorants could test directly. If a library of pigment reflectances shows edges that are, on average, of equal width in energy, then the census in energy is the one to quote, and the reds are the second-costliest colours by a much smaller margin than the nanometre census says. If the edges are of equal width in nanometres, the original census stands.

The measurement is the width of each colorant’s steepest edge, recorded against the wavelength at which it falls — a table a pigment manufacturer already has in the form of its reflectance curves. A straight line through those points in a plot of width against the square of wavelength would say energy; a flat line would say nanometres; and anything else would be the per-wavelength limit to substitute into the program as it stands. The colour is in the thickness adds a complication a real table would have to record: the same colorant at a greater thickness has steeper edges, so the width of an edge is a property of the colorant and its loading together, and the table needs both columns.

A unit is an assumption about what is equal #

The habit is about the unit a constraint is written in.

A constraint of “no faster than forty nanometres” looks like a single number, and it is also a claim that a nanometre at 450 and a nanometre at 700 are equal amounts of whatever the constraint is about. For a grid that is harmless. For a physical process whose natural variable is energy, it silently makes the red end of the spectrum sharper than the chemistry and the blue end blunter.

The move is to rewrite the constraint in the variable the process actually runs on, with the two forms agreeing at a reference point, and to see what moves. Here the total barely moved and the distribution moved a great deal — which is the common outcome, and the reason a single summary number is a poor check on whether a change of unit mattered.

The failure mode is to verify a change of units by its effect on the total, find it small, and conclude the unit was immaterial. A redistribution that leaves the total unchanged is exactly the change a total cannot see.