The finish adds the room's own colour

A dark wall pays for a finish found two rules in one census of seventy-two painted rooms. The first-order rule is that the share of a room’s colour a gloss finish takes tracks how much light the wall returns and not how colourful it is: a rank correlation of −0.89 with the wall’s luminance factor and −0.04 with its chroma. That rule has a mechanism — the interface returns a fixed share of whatever arrives and the body returns what the pigment gives back, so a dark wall’s own return is small and the same interface return is a larger share of it.

The second-order rule does not. Inside a band of lightness, chroma still moves a room’s loss, and it moves it in opposite directions at the two ends: among the darkest walls a more saturated paint loses less, among the palest it loses more. The essay named a candidate and a test.

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 needs no new paints. Band width is one of the census’s three axes, so it already holds four families of equal width, and holding lightness still is a partial rank correlation.

The width does nothing, and the addition is not white #

With the wall’s luminance factor held, the band’s width orders the losses at a rank correlation of 0.01 over the census and never past 0.22 in any lightness group. The wall’s chroma runs from −0.90 among the darkest walls to +0.81 among the palest, crossing zero in the middle.

• The reversal survives a change of metric. Measured as distance from the lamp’s white on the 1976 chromaticity diagram, which has no lightness in it and no cube root, it runs from −0.89 to +0.61 and crosses in the same place.
• The four equal-width families each carry the whole effect, spanning at least half again in loss, and their ranges overlap.
• The light a gloss finish adds is not the lamp’s white. It carries 26 per cent of the room’s own colour among the darkest walls and 62 per cent among the palest.
• Among dark walls the loss is ordered by how coloured that addition is — a partial rank correlation of −0.98 — and hardly at all by how much of it there is.
• Among pale walls the colour term saturates and the amount takes over, at +0.58. The sign change is two mechanisms handing over, not one mechanism reversing.

Holding the lightness still #

The three quantities a paint varies in this census are its band’s centre, its band’s width and its peak reflectance, and they produce walls of the same chroma at very different lightnesses and walls of the same lightness at very different chromas. That was the design that separated the first-order candidates, and it separates these too.

A lightness group is a band of luminance factors rather than one value, so a correlation taken inside a group still has some lightness variation in it — enough to matter, since lightness is what orders the losses. The right instrument is a partial rank correlation: the correlation of the loss with a candidate, with the wall’s luminance factor held still by the usual first-order formula on the ranks.

It is worth saying what a partial correlation can and cannot do here. It removes the part of the candidate that is a monotone function of lightness, which is the confound; it does not remove a curved one, and the loss is a curved function of lightness. So a candidate that is related to lightness quadratically could survive this and be nothing. A tolerance has a grain is the neighbouring caution about a statistic whose shape is being read through a transform; the protection taken here is to repeat the whole thing in a second metric whose curvature is different, which the next section does.

The chroma line crosses zero and the width line does not go anywhere. Chroma runs −0.90, −0.47, −0.22, +0.71, +0.81 across the five groups. Width runs −0.07, −0.07, +0.16, −0.15, +0.22, which is scatter. Peak reflectance follows chroma at about two thirds the size, which is what it should do — a paint with a higher peak and the same lightness has a narrower band and a higher chroma, so the three are not independent, and only one of them is doing anything.

The second half of the prediction was that the reversal should vanish for a family of equal widths. All four families lie on the same curve and overlap each other, and each spans at least half again in loss on its own. If width were the variable, the four sets would be four curves; they are one, and the separation the figure draws is a separation on an axis that does not matter.

Not a property of the ruler #

Before looking for another mechanism it is worth ruling out the possibility that there is nothing to explain.

A second-order effect that only appears in one colour space is usually a property of the space. CIELAB’s chroma is a distance in a cube-rooted space with a lightness axis, and a fixed additive white moves a dark colour’s lightness far more than a pale one’s, so a reversal with lightness is exactly the shape a compression artefact would take.

Both cross zero between the third and fourth group. The u′v′ measure has no lightness in it at all — it is a chromaticity, and adding light of a given chromaticity moves a point along a straight line towards it — and it runs −0.89, −0.26, −0.25, +0.53, +0.61. The effect is smaller in u′v′ and it is the same effect.

So there is something to explain, and it is about the mixture rather than about how the mixture is measured. A bounce is a multiplication is the reminder that what makes room colour hard is that light arrives having already been somewhere; that is where the answer is.

What the interface actually returns #

The first-order account described the interface return as white — the lamp’s own colour, added to the wall’s. That is right for the first bounce and wrong for the room.

An interface returns a fixed share of whatever arrives at it. In a room, most of what arrives at a wall has already come off another wall, and every bounce multiplies by the paint’s reflectance. So the light the finish adds is the room’s ambient light, which is the lamp’s spectrum filtered through the walls one or more times — and how filtered it is depends on how much interreflection the room has, which depends on how much light the walls return.

Among the darkest walls the addition is 26 per cent as coloured as the room and 19 per cent of its light. Among the palest it is 62 per cent as coloured and 12 per cent of its light. The two move in opposite directions and both matter.

A dark room is nearly a single-bounce room: light arrives from the lamp, hits the wall once and is mostly absorbed, so the ambient the interface samples is still close to the lamp’s. A pale room is a multiple-bounce room, the ambient has been through the paint several times, and what the interface returns is nearly the wall’s own colour — which barely dilutes anything.

That is the same accounting a gloss finish takes colour out of the whole room set out for the first-order rule, read one term further along. The finish’s effect on a room is not a coating on a wall; it is a change to what every bounce multiplies, and a tenth of the return arriving white is where the size of that change was first measured. What this adds is that the colour of the interface return is a room quantity too, and it is the one that carries the second-order rule.

Two terms, changing places #

That gives two candidate orderings inside a lightness group and they do not act in the same place.

Among the darkest walls the colour of the addition orders the loss at −0.98 and the amount at +0.10. A more saturated dark paint colours its own ambient more, the interface returns something closer to the wall’s colour, and the room loses less. That is the first half of the reversal with a mechanism in it, and the mechanism is not about the band’s shape at all — it is about how many times the light has been through the paint.

Among the palest walls the colour term reads +0.08 and +0.60 and the amount reads +0.37 and +0.58. The addition is already most of the way to the wall’s colour, so making the paint more saturated cannot colour it much further; the term has run out of room. What is left is the amount, and a more saturated pale paint at the same lightness has a slightly larger interface share, so it loses more.

The sign of the reversal is therefore a handover rather than a reversal. One term is large and negative at the dark end and saturates; a second is small and positive throughout; their sum changes sign where the first runs out. That is a different kind of explanation from the one that was looked for — nobody was looking for two mechanisms — and it is the reason no single paint property orders the whole census.

Whether the second term deserves the name mechanism is worth a sentence, because it is thinner than the first. A more saturated paint at the same lightness returns its light over a narrower range of wavelengths, so at equal luminance factor it presents a slightly higher reflectance to the lamp where the lamp is strong, and the interface — which takes a share of what arrives rather than of what the body returns — gets a slightly larger share of a slightly brighter room. It is a small effect and the correlations that carry it are the census’s smallest. What makes it visible at the pale end is only that the term competing with it has stopped moving, and a gloss room looks less colourful than it measures is the standing warning about reading a residual as a result.

What was computed #

The room is the same radiosity ledger the earlier censuses used: a cube with one lamp, walls of one paint, the body return computed spectrally at every bounce and the interface return computed from a microfacet lobe at a stated roughness. The finish is the satin one, a roughness of 0.2.

The seventy-two paints are six band centres from 450 to 650 nanometres, four band widths from 15 to 100, and three peak reflectances of 0.3, 0.6 and 0.85, over a base reflectance of 0.03. The loss is one minus the glossy room’s chroma over the matt room’s, both read against the lamp’s white.

The added light is the glossy room’s total minus the matt room’s, band by band, which is the interface’s contribution and nothing else. Its colouredness is its distance from the lamp’s white on the 1976 diagram divided by the matt room’s, and its share is its luminance divided by the glossy room’s. Neither quantity was computed before; both fall out of a ledger the census already had.

Where this stops #

One room and one finish. The cube’s geometry sets how much interreflection there is, and a room with a window, a dark floor or furniture in it has less. The direction of the mechanism does not depend on the geometry — more interreflection means a more coloured ambient — but where the handover falls does, and a less enclosed room should put it at a higher lightness.

The partial correlation is first-order. It removes a monotone relation between the candidate and lightness and not a curved one, and the loss is a curved function of lightness. The u′v′ replication is some protection against that, since the curvature differs between the two metrics, and it is not a proof.

The paints are analytic bands over a small base. A real paint’s reflectance is not a Gaussian over 0.03, and in particular a real dark paint is rarely as spectrally selective as this family’s dark members, because the pigment loadings that make a paint dark also broaden its absorption. A sharp edge is bought with depth is the census that priced that relation from the other side; the consequence here is that the dark end of this family may be more colourful than a real fan deck’s dark end, which would put the handover at a lower lightness in practice than the third it sits at here.

And the groups are small. The palest group has nine rooms in it and the correlation quoted for it is +0.81, which is one paint away from +0.6. The claim the small groups are carrying is that the sign is positive at the pale end and negative at the dark end, which four groups agree about; nothing should be read into the exact values.

Still open: where the handover falls in a real room #

The two terms cross somewhere, and where they cross is the only number here a decorator could use. In this cube it is at a wall returning about a third of the light, which is a mid-tone.

The computation is this census run at several room geometries — the cube, a long corridor, a room with one wall replaced by a window — with the crossing recorded for each. The prediction is that it moves up as the enclosure falls, because less interreflection means a whiter ambient at every lightness, so the colour term saturates later. A room with a window should have no pale end at all.

The measurement that would test it needs a spectroradiometer and a repainted room, and the quantity to record is not the loss but the chromaticity of the ambient — the light arriving at a wall from the rest of the room, which a small integrating probe held against the wall reads directly. That is the quantity this essay says is doing the work, it has never been measured against paint colour, and the prediction is sharp: it should sit a fixed fraction of the way from the lamp’s white towards the wall’s colour, and the fraction should rise with the wall’s reflectance rather than with its chroma.

A mechanism proposed for a reversal should be checked against the census that found it #

The habit is about how far a candidate mechanism has to travel before it is tested.

The spectral-concentration account was a good guess. It names a real effect — the matching functions are steeper in some places than others — it predicts the right kind of thing, and it came with its own falsifier attached: it should vanish for a family of paints whose bands all have the same width. That is exactly how a candidate should be stated.

What it did not do is look at whether the family was already there. The census had four of them, because band width was one of the axes it was built on, and the test cost a partial rank correlation rather than a new set of paints. A prediction that asks for a new experiment sometimes asks for one that has already been run, and the reason to check is not thrift: the family already in hand is the one whose other variables are controlled the way the original census controlled them.

There is a second, smaller lesson in what replaced it. The account that works has two terms in it, and neither term alone orders the census: the colour of the addition explains the dark end and nothing else, the amount of it explains the pale end and little else. A search for the mechanism — one property of the paint, one correlation, one sign — could not have found it, because the thing being explained is a sum. A difference has no rate makes the neighbouring point about reading a single number off a quantity that is a composition of two, and this is the same shape: a correlation that changes sign is usually two things rather than one thing changing its mind.

The failure mode is proposing a test the existing data can already answer, and filing it as future work. The candidate then sits in a ledger as an open question rather than as a refuted one, and the next essay inherits it as a lead.