A probe at the wall prices the finish

The finish adds the room’s own colour found that a gloss finish does not add the lamp’s white to a room. It returns a share of whatever arrives at the wall, and what arrives has already bounced off the walls, so the finish adds the room’s ambient — and how coloured that ambient is carries the rule for which paints a finish costs most. The essay ended by naming the measurement that would test it in a real room. Not the loss, which needs the room painted twice, but the chromaticity of the ambient: the light arriving at a wall from the rest of the room, which a small integrating probe held flat against the wall reads directly.

It made a sharp prediction about that reading. The ambient “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.”

The computation below makes the reading in five rooms and seventy-two paints each. Two of the prediction’s three parts hold. The third does not, and its failing is the useful part.

What the probe reads #

The ambient at a painted wall lies on the line from the lamp’s white to the wall’s colour, to within a degree at the median. How far along it lies is set by the wall’s luminance factor, at a rank correlation of 0.95, and not by its chroma, at −0.03. It is not a fixed fraction: across seventy-two paints in one room it runs from 2 to 46 per cent. Read in the matt room, it orders what a satin finish costs better than the paint’s lightness does in every one of five rooms.

• The ambient’s direction from the white is the wall’s: the median angle between them is 0.41 degrees over the cube’s seventy-two paints, and the worst 6.9.
• In every room the fraction follows the wall’s lightness at 0.94 to 0.96 and its chroma at under a tenth either way.
• The fraction runs over more than a factor of ten in every room — from 0.019 to 0.46 in the cube and from 0.027 to 0.72 in the corridor.
• Within each room the probe orders the finish’s cost at −0.84 to −0.98, the paint’s lightness at −0.73 to −0.94, and the probe is the tighter in all five.
• Pooled over the four rooms where a finish always takes colour, the probe orders 288 rooms at −0.93 and lightness at −0.76. The low wide room, where a third of the paints gain colour from a finish, breaks it.

Where the ambient falls #

The figure is six walls in the closed cube, each painted with a saturated band centred somewhere from 450 to 650 nanometres, on the 1976 chromaticity diagram. The lamp’s white is in the middle. Each wall’s own colour under the lamp is an open circle, and the light arriving at that wall from the rest of the room — the lamp’s direct light plus everything the floor, ceiling, end walls and the opposite painted wall send it — is a filled one.

Every filled circle sits on the line from the white to its open circle. Over the full census of seventy-two paints the median angle between the two directions is 0.41 degrees. The largest departures are for the broadest bands at the ends of the spectrum — 6.9 degrees for a band a hundred nanometres wide centred at 650 — which is what light filtered twice by such a wall should do: a squared broad band has a slightly different centre of gravity from the band itself, so the twice-bounced part of the ambient points in a slightly different direction.

So a single number describes the ambient: how far along the line it has gone. For the six walls drawn, all at a peak reflectance of 0.85 and a width of 25 nanometres, it is between 5 and 19 per cent, largest for the green and yellow walls near the middle of the spectrum and smallest for the deep red.

That spread is already a hint about what sets it. The six walls have bands of the same shape; they differ in how much light they return, because a band at 530 nm sits under the eye’s peak sensitivity and a band at 650 nm does not.

Lightness sets it, chroma does not #

Against the wall’s luminance factor, the seventy-two fractions rise together at a rank correlation of 0.95, from 2 per cent for the darkest paints to 46 per cent for the palest. The mechanism is the one a dark wall pays for a finish found for the loss, and the one a tenth of the return arriving white first measured compounding: a wall that returns more light sends more of it round the room again, the opposite wall and the grey faces send some of that back, and the ambient accumulates the wall’s colour with each round.

Against the wall’s chroma there is no order at all: a rank correlation of −0.03. The most saturated walls in the census pull the ambient no further than the least saturated ones do, because saturation is how far the wall’s colour is from white, and the fraction is measured as a share of that distance. A saturated wall’s ambient moves further in absolute terms and exactly as far in proportion.

With lightness held still, chroma adds something small: a partial rank correlation of 0.17 to 0.28 across the rooms. At the same luminance factor, a more saturated paint is a narrower band with a higher peak, and a higher peak sends slightly more of its own colour round again. The first-order statement stands: the fraction is a property of how much light the wall returns. That half of the prediction holds, and in all five rooms.

Not a fixed fraction #

The other half does not. The prediction was a fixed fraction — one number per room, which would make the probe a way of measuring the room rather than the wall. In every room the fraction runs over more than a factor of ten across the paints.

Every room draws the same rising curve at a different slope. The corridor, whose long painted walls face each other across a short distance, rises furthest: 72 per cent for its palest walls. The closed cube reaches 46. The rooms with one and two walls open reach 35 and 28, because light that would have come back coloured leaves through the opening. The low wide room reaches only 22: its painted walls are narrow bands between a large grey floor and ceiling, and most of what arrives at them has come off grey.

So the probe’s reading is a property of the wall and the room together. That is less convenient than the prediction, which would have let one reading characterise a room for every paint. It is more useful for the thing a specifier actually wants to know, because it is exactly the quantity that decides what a finish will cost that wall in that room.

The reading prices the finish #

The loss a satin finish causes — one minus the satin room’s chroma over the matt room’s — is ordered by the wall’s luminance factor at about −0.9 in a closed cube, which was the first-order rule of the census. The probe’s reading contains the luminance factor and something more: how the room’s shape and openings return that light to the wall.

In every room the probe orders the cost more tightly than the paint’s lightness. In the cube, −0.97 against −0.89; in the corridor, −0.97 against −0.87; with one wall open, −0.98 against −0.92; with two, −0.98 against −0.94; in the low room, −0.84 against −0.73. The margin is largest in the rooms where lightness does worst, which is the pattern of a measurement carrying information the paint’s specification leaves out.

The explanation is the chain the finish adds the room’s own colour laid out. A satin finish returns a share of the arriving light; how much colour that return takes from the room depends on how coloured the arriving light is compared with the room; and the probe reads how coloured the arriving light is. The paint’s luminance factor predicts that reading well in a closed cube and less well elsewhere, because it knows nothing of the room.

Across rooms, and the room that breaks it #

Within one room, any monotone function of the wall’s lightness would order the paints about as well as lightness does. The harder test is across rooms, where the same paint costs different amounts in different rooms and a paint property cannot tell them apart.

Pooled over the cube, the corridor and the two opened rooms — 288 painted rooms — the probe orders the finish’s cost at −0.93 and the paint’s lightness at −0.76. On the left, the four rooms’ paints spread into four bands at each lightness, one per room, and lightness cannot say which band a room is in. On the right, the bands close up onto one curve: the probe has read the room as well as the paint.

The low wide room does not join it. Its probe readings are small — its walls see mostly grey — and its losses are smaller still, and for 21 of its 72 paints negative: a satin finish there makes the room more colourful, by as much as 14 per cent. An open room hands over sooner found the same gain and left its mechanism open. Pooled with the low room, the probe orders all 360 rooms at −0.52 and lightness at −0.59, so the probe loses its lead. Within the low room it still orders the loss better than lightness does. What it cannot do is put a room where a finish adds colour on the same scale as rooms where a finish takes it, because the probe reads how coloured the light arriving at the wall is, and in the low room the light that matters is the light leaving the grey faces, which no probe on the wall sees.

How to read a probe #

The measurement the earlier essay proposed is a practical one, and the computation says what to do with it.

Hold the probe flat against the painted wall, facing into the room, in the room as it will be lit. It integrates everything arriving at the wall, which is what a finish on that wall will hand back. A reading taken elsewhere — at the centre of the room, facing the lamp — reads a different mixture, and nothing here says how well that one would do.

Read it as a fraction, not a colour. The number that carries the prediction is how far the ambient has gone from the lamp’s white towards the wall’s own colour, and that needs two further readings: the lamp’s white, from the probe facing the lamp with the wall behind it shaded, and the wall’s colour, from a spectrophotometer on the paint. All three are chromaticities, and the fraction is a ratio of two distances on the diagram.

Expect a small fraction to mean a large cost. In the closed cube, a wall whose ambient has gone 5 per cent of the way towards it loses around a seventh of its room’s colour to a satin finish; one at 40 per cent, around a thirtieth. That is the same direction as the paint’s lightness says, sharpened by the room — and in a room with little painted area and a great deal of grey, like the low room, the reading should be treated as ordering that room’s paints and nothing more.

A gloss room looks less colourful than it measures is the standing caveat on all of these numbers: they are what the room’s light does, and a viewer adapted to the room loses more than the light does, by a factor that what an adapted viewer loses is set by the wall found depends on the wall’s colour.

How the ambient was computed #

The rooms and paints are those of the handover census: seventy-two paints from six band centres, four widths and three peak reflectances on the two side walls of five rooms, grey elsewhere, lit by daylight from the ceiling, solved by the directional transport solver matt and at a satin roughness of 0.2. The ambient at a painted wall is the sum, over every other face, of the radiance that face sends towards the wall times the cosine-weighted solid angle it subtends there — including the ceiling’s own emission, since the lamp’s direct light is part of what a probe reads.

Chromaticities are CIE 1976 u′v′ under the 1931 observer. The lamp’s white is the lamp’s own spectrum; the wall’s colour is its reflectance times the lamp’s spectrum. The fraction is the ambient’s distance from the white divided by the wall’s; the angle is between the two directions from the white. Losses and rank correlations are computed as in the census that found the reversal.

What this leaves out #

A probe is modelled as reading the whole hemisphere with a cosine response. A real integrating probe has an angular response that falls off near grazing incidence, and the light arriving at a wall from the adjacent floor and ceiling comes in at grazing angles; a probe with a poor cosine response would under-read them and over-read the opposite wall, pushing the fraction up.

The rooms are five boxes with two painted walls and a ceiling lamp. A real room’s ambient includes daylight from windows, light from other lamps and light from furniture, each with its own colour; a room with two lights has no white is the reminder that the fraction then has no single white to be measured from.

And the prediction is about the finish the census used, a satin roughness of 0.2. A glossier finish returns more of its light into a narrow lobe in the mirror direction, and what that lobe hands back is the light arriving from one direction, not the hemisphere a probe integrates. For a gloss finish, the right probe may be a narrow one, aimed along the mirror direction from where the room is viewed.

Still open: whether a narrow probe prices a gloss finish #

A satin finish spreads its return widely enough that the whole hemisphere of arriving light matters, which is why a cosine-weighted probe prices it. A gloss finish sends most of its return into a lobe — a lobe takes colour out of a bounce is where that lobe’s colour was first priced — and a viewer sees the part of the lobe that points at them: the reflection of whatever lies in the mirror direction from their position.

The prediction is that as the finish gets glossier, the probe that prices it narrows, from the whole hemisphere at a satin finish to a cone around the mirror direction at a gloss one, and that at a high gloss the room’s cost is decided by what one particular face of the room looks like from the wall. The computation is this census at several roughnesses, down to where the boundary belongs to the quadrature found the directional solver stops resolving a lobe, with the ambient read two ways — integrated over the hemisphere and through a cone around the mirror direction from the viewing position — and the rank correlation of each with the loss recorded at each roughness. Where the two cross is the finish at which the probe a specifier should hold changes.

A fixed number would have been less use #

The habit is about what to want from a prediction that fails.

The prediction said the ambient would sit a fixed fraction of the way towards the wall’s colour — a single number per room, set by the room’s shape. It sits a fraction that varies tenfold across the paints of one room, and the prediction was wrong. But a fixed fraction would have been a property of the room alone, and the quantity that decides what a finish costs is a property of the room and the paint together. The variation that falsified the prediction is exactly what makes the reading worth taking.

The failure mode is to discard a measurement because it did not come out as the simple number predicted. A reading that varies where the prediction said it would be constant has found a dependence, and the useful question is then what the dependence tracks. Here it tracked the thing a specifier pays for, more closely than the specification of the paint.