A dark wall pays for a finish

A satin finish takes about a tenth of the colour out of a room. A finish adds colour only to a daylight meter established the size, once each room’s colour was measured against the white of the lamp it is lit by: between 7.7 and 12.9 per cent across six wall colours and six lamps. The lamp moved the number by a point or two, and the wall moved it by five.

That leaves the obvious question of what about a wall decides its share, and there is an obvious answer. The interface of a glossy paint reflects a little of the light that reaches it before it enters the pigment, and that light carries the lamp’s spectrum — a tenth of the return arriving white measured it at about nine per cent. A fixed amount of white dilutes a weak colour by a larger fraction than a strong one, so a paint that makes a pale, weakly coloured room should lose a larger share than a paint that makes a vivid one.

It is a good explanation and it has a testable shape: under one lamp, the loss should follow how colourful the wall is. The test needs paints that separate colour from lightness, because in any single family of paints the two move together.

Lightness orders the losses and chroma does not #

Over seventy-two paints under daylight, the share of a room’s colour a satin finish takes runs from 2.5 to 15.5 per cent. It follows the wall’s luminance factor — how much of the lamp’s light the wall returns — at a rank correlation of −0.89. It follows the wall’s own chroma at −0.04.

• The least colourful wall of the seventy-two and the most colourful, at chroma 16 and 117, lose exactly the same 10.0 per cent.
• Two rooms of the same matt chroma, about 25, lose 14.7 and 2.5 per cent: a narrow dark blue band and a broad pale green one. The darker wall loses nearly six times as much.
• At every hue and every band width, a paler peak loses less, without exception across twenty-four rows.
• Held at one lightness, chroma still moves the share by a few points, and in opposite directions among dark walls and pale ones — a second-order effect with no settled explanation here.
• Every room is brighter glossy, by 13 to 25 per cent, and the ordering of that gain by lightness is perfect: a rank correlation of −1.00.

Seventy-two paints built to separate two things #

A wall in this room is painted with a reflectance that has a single Gaussian band on a low base: a peak at some wavelength, a width, and a height. The three dimensions are what make the test possible.

• Six hues, with the band centred at 450, 490, 530, 570, 610 and 650 nanometres — blue through deep red.
• Four widths, 15, 25, 50 and 100 nanometres. A narrow band is a saturated, dark paint; a broad one returns light across much of the spectrum and is paler and less colourful.
• Three peak heights, 0.33, 0.63 and 0.88. A taller band returns more light at its own wavelengths, and at a narrow width it is also more colourful.

That last point is the lever. Raising the peak of a narrow band makes the wall both paler and more colourful at once; widening a tall band makes it paler and less colourful. So the census contains walls of the same chroma — about 24 — returning 4 and 79 per cent of the light, a factor of nineteen, and walls returning the same 30 per cent of the light at chromas of 16 and 117, a factor of seven — which is the arrangement a correlation needs if it is to say which of two properties is doing the work.

The first of the two figures above plots the losses against luminance factor, and they fall along it — steeply among the darkest walls, where walls returning under six per cent of the light lose between 11.4 and 15.5 per cent of the room’s colour, and more gently as the walls get paler. This figure plots the same seventy-two losses against chroma, and nothing falls. Walls with a chroma between 60 and 70 lose anything from 5.2 to 14.7 per cent, a factor of nearly three, and the two ends of the chroma axis lose the same share to a decimal place.

The two plots are the whole test, and it is worth being clear about what a correlation of −0.04 means here. It does not mean chroma has no effect; the census is not designed to show that and later sections find a small one. It means that across paints as a whole, knowing how colourful a wall is says nothing about how much of its room’s colour a finish will take, and knowing how light it is says most of it.

The room’s own matt chroma — the quantity the dilution explanation actually names — sits between the two, at −0.51. That is what a confounded variable looks like: a colourful room tends to have walls that return a lot of light, because a tall band returns more of both, so the room’s chroma inherits a share of lightness’s correlation without carrying any of the mechanism. A gloss finish takes colour out of the whole room kept the ledger this quantity comes from, and on one paint it could not have told the two apart.

Two rooms that are equally colourful #

The sharpest single comparison in the census is the pair of paints whose rooms are equally colourful matt and whose losses differ most.

The first paint is a narrow blue band peaking at a third — a dark, saturated wall that returns 3.6 per cent of daylight. The second is a broad green band peaking at nearly nine tenths — a pale, washed-out wall that returns 79 per cent. On a colour card the first is vivid and the second is a greenish off-white.

Painted on two walls of the same room, they make rooms of the same matt chroma, 24.2 and 25.1, because the dark wall’s strong colour contributes little light to a room dominated by its grey faces and the pale wall’s weak colour contributes a lot. By the dilution account the two rooms should lose about the same share.

The dark wall’s room loses 14.7 per cent and the pale wall’s 2.5. No story about weak colours produces that, since the weakly coloured wall is the one that barely loses. A story about dark walls produces it directly.

Why the dark wall pays #

The mechanism is the one a lobe takes colour out of a bounce described, with the right variable in it.

A glossy wall returns light by two routes. The interface reflects a share of whatever arrives, set by the paint’s refractive index and the angles, and that share is the same for a dark paint as for a pale one: it depends on the surface, not on the pigment behind it. The body returns what the pigment gives back, which is the wall’s luminance factor. A pale wall’s body return is large and a dark wall’s is small.

So the same white interface return is a small addition to a pale wall’s light and a large one to a dark wall’s. On the palest wall in the census the body returns about four fifths of the light and the interface adds a few per cent more; on the darkest, the body returns under four per cent and the interface return is of the same order. What leaves a dark glossy wall is substantially white, whatever colour the pigment is, and the room’s colour, which is a sum over what leaves each face, loses a correspondingly larger share.

Every step that makes the wall paler lowers the loss, whichever way the step is taken. Across a row the band gets taller and the wall paler and more colourful; down a column the band gets wider and the wall paler and less colourful. Both directions lower the share, from 14.2 per cent for the narrowest, shortest band to 2.5 for the widest and tallest. Chroma goes up along the rows and down along the columns, and the loss goes down along both. It is the same census as the scatter, read on one hue so that the two directions can be seen separately.

The dark wall pays at every finish #

A single finish could hide a crossing, so the comparison is worth running across the whole range.

The darkest paint loses more at every finish, and the gap widens as the finish sharpens: 4.9 per cent against 1.7 at a roughness of 0.8, and 16.5 against 3.5 at 0.15, the glossiest finish the directional solver resolves. The pale wall’s curve flattens past a roughness of 0.3, where the interface light a sharper lobe sends into the room stops being a noticeable share of the wall’s large body return. The dark wall’s curve is still climbing at the end of the range, and the boundary belongs to the quadrature is why the range ends there rather than at a high gloss where it would be steeper still.

This is also the direction in which the practical stakes rise. A decorator choosing a gloss finish for a dark, saturated accent wall is choosing the combination that loses most — the one that looks richest on a matt sample card and gives up the largest share of itself in a room.

Chroma returns as a second-order effect, in both directions #

Lightness orders the losses closely but not perfectly, and the residual is where chroma turns out to matter after all.

The groups step down with lightness, from 10.3–15.5 per cent among walls returning under a tenth of the light to 2.5–6.8 among walls returning more than half. That is the first-order result again. Inside each group the losses still spread by five or six points, and that spread correlates with chroma.

It does so in opposite directions. Among the darkest walls the more colourful lose less, at −0.84; among walls returning between 35 and 55 per cent of the light the more colourful lose more, at +0.71. The middle groups sit between. At one lightness, a darker-looking wall that happens to be more saturated keeps more of its room’s colour, and a pale wall that happens to be more saturated keeps less.

This essay does not explain the reversal, and it is worth saying so rather than reaching for a story. One candidate is where the dark walls’ light is concentrated: a narrow, tall band returns its light at wavelengths where the observer’s matching functions are steep, which would make its colour robust to a neutral addition in a way a broad band’s is not. That is a hypothesis, not a measurement, and the census has two walls per cell of the group table rather than the dozens a test of it would need. What the figure establishes is only the order of the effects — lightness sets the level, and chroma moves a room within its level by a few points, not always the same way.

The other half of the ledger #

The quantity that moves the other way is the room’s brightness, and it obeys the same rule without exception.

Every room is brighter glossy, by between 12.9 and 24.8 per cent, and the ordering is perfect: the rank correlation between the gain and the wall’s luminance factor is −1.00. A dark wall’s room brightens most, because the interface adds about the same light to every wall and that light is a larger share of what a dark wall contributes.

The two results are one result. The interface light is a larger share of a dark wall’s return, so it adds more brightness and more white to that wall’s contribution — the room gets lighter and less colourful together, by the most, when the wall was dark to begin with. The darkest wall anybody sells found the same shape for a different question, where the worst change of light a room could produce came from a dark wall rather than a saturated one: in both, the property everybody would nominate turns out to be a proxy for lightness.

What a specifier can take from it #

Three things, and the first corrects advice that is easy to find.

Do not choose a finish by how saturated the paint is. The intuition that gloss makes a colour richer is right about a sample held under a lamp at the specular angle, where the interface reflection is aimed away from the eye and the body colour looks deep — gloss changes the measurement is the instrument’s version of that. In a room, where the interface light is sent everywhere, a gloss finish on a dark wall takes more of the room’s colour than it takes from a pale one, and saturation does not enter.

Expect a dark accent wall to lose the most. The narrow, dark paints at the top of the scatter lose between eleven and sixteen per cent of their room’s colour at a satin finish and more at a higher gloss, against two or three per cent for a pale wall at the bottom. A room with one dark satin wall and three pale matt ones is paying for its finish almost entirely at the dark wall.

And a room’s measured colour loss is a poor guide to what a person will see. Everything above is the room’s light measured against its lamp’s white. A viewer adapted to the room loses more, and what an adapted viewer loses is set by the wall finds that the extra is governed by hue rather than lightness — so the two readings disagree about which walls pay most.

How the census was computed #

The room is the unit cube used throughout the directional work: a daylight lamp in the ceiling, two opposite walls painted with the stated reflectance — a Gaussian band of the stated centre, width and peak on a base of 0.03 — the other four faces a neutral 0.5, every face carrying a microfacet lobe of refractive index 1.5 at the stated roughness, and the transport solved directionally in eighty-one wavelength bands.

The loss is one minus the chroma of the glossy room’s total reflected flux over the matt room’s, with the lamp’s own emission removed from the ceiling first and each flux normalised to its own luminance, measured in CIELAB against the lamp’s white. The wall’s luminance factor is the luminance of its reflectance under the lamp over the lamp’s own; its chroma is the CIELAB chroma of the same product. Correlations are Spearman’s, over all seventy-two paints unless a subset is named.

What this leaves out #

The census is one lamp, and a different lamp moves every wall’s luminance factor. The dark-wall rule survives that: across the six lamps of the earlier measurement the losses on one wall follow the luminance factor under each lamp at −0.94, on a green wall and again on a red one. But it has not been run over seventy-two paints under each lamp, and the within-group reversal in particular has been seen under daylight only.

The walls are single Gaussian bands. A real paint’s reflectance has several features and a scattering base that does not reach 0.03, and a real dark paint is usually dark because it absorbs broadly rather than because it reflects a narrow band. That would place most real dark paints towards the broad, low-peak corner of the census, which the scatter says loses heavily — but the corner is sampled by a handful of paints.

And the room is one geometry with two coloured faces. More coloured faces put more of the room’s light through the coloured walls, which should sharpen the rule, and a room whose grey faces are themselves glossy adds interface light that is neutral from the start. Neither is computed here.

Still open: whether the reversal inside a lightness has a mechanism #

The first-order rule has a physical explanation that follows from the energy accounting. The second-order one does not yet.

Among dark walls, a more saturated paint loses less; among pale walls, more. The candidate is spectral concentration: a narrow tall band puts the wall’s return where the colour-matching functions are steep, so a neutral addition moves its chromaticity less than it moves a broad band’s. If that is right, the reversal should track the band’s width inside each lightness group rather than its chroma, and it should vanish for a family of paints whose bands all have the same width.

The test is a census with lightness held and width varied independently of chroma — possible by shifting the base rather than the peak — and it needs about four times as many paints per group as this one has. It would say whether chroma is a second variable or a second proxy.

A proxy is a correlation that has not been separated #

The habit is about a property that everybody would name and that turns out to be standing in for another.

Saturation and lightness move together in almost any family of paints, of lamps, of inks: a band that returns more light usually returns more colour with it. So almost any measurement made across a natural family will find the effect correlating with both, and the one with the more intuitive mechanism will be credited. The dilution story was intuitive, and it was quoted here on the strength of a table in which it could not have been distinguished from the lightness story.

The move is to build a family in which the two are not tied — here, by varying width and height separately, so that one lever raises chroma and the other lowers it while both raise lightness — and then to ask which of the two the effect follows. When both levers move the effect the same way, the variable they share is the one doing the work.

The failure mode is a mechanism accepted because it would produce the correlation, without a family in which a competing mechanism would produce a different one. A mechanism that predicts a correlation is not evidence until the correlation has been given a chance to come out another way.