A catalogue is not a vocabulary

A colour order system exists so that a person can find a colour. That is the whole design brief, and it has one structural consequence: the samples have to sit on a regular lattice. Even steps of lightness, even steps of chroma, even steps of hue — because a book whose pages are irregular is a book nobody can navigate.

Naming is not regular in anything. Put the two together and they come apart in a way that is measurable, unflattering to both, and completely invisible from inside either.

The claim #

The two ways people organise colour are a lattice and a vocabulary, and they do not fit each other.

Three measurements say so:

• The chips divide unequally by a factor of three. Grey takes 198 of 1,320, purple 181, white 165; blue takes 66 and red 68. The lattice gave every region the same density of samples and the names did not ask for that.
• A fifth of the moves in the lattice are category changes. Of 3,339 one-step moves between adjacent chips, 718 — 21.5 per cent — land on a different name. So the neighbour relation the book is organised by and the neighbour relation a speaker uses cross each other constantly.
• And neither of them is the metric. The names do not follow it and neither does the lattice: a regular step in chroma is not a regular step in colour difference, which is what the previous rung of this ladder was about.

Three regular structures, no two of which agree.

What a colour order system is for #

Worth being precise about, because the criticism only lands if the purpose is stated fairly.

A system like Munsell, NCS or a printer’s swatch book is a set of physical samples arranged so that a person can (a) find one that matches something in front of them, (b) name its position in a way another person can look up, and © interpolate between neighbouring samples with some confidence about what lies between.

All three need regularity. Finding by search needs a predictable ordering; naming a position needs coordinates; interpolating needs neighbours that are close in whatever sense the user cares about. A lattice delivers all three and nothing else does.

What it does not deliver is any relationship to what the chips are called, and there is no reason it should — that was never on the list.

What was computed, and how #

The lattice is built at ten units of L*, ten units of C* and ten degrees of hue, from L* 20 to 90 and out to C* 90, with the achromatic column collapsed to one chip per lightness because a chroma of zero has no hue. Chips outside the sRGB gamut are dropped, which leaves 1,320.

Each chip is handed to the naming partition and the counts totted up. That is the census.

The crossing rate is a separate pass over the same chips. For each chip, three neighbours are looked up — one step lighter, one step more chromatic, one step round in hue — and the pair counted as a crossing if the two names differ. Missing neighbours, at the edges of the gamut and at the achromatic column, are skipped rather than counted as anything, which makes 3,339 comparisons out of a possible 3,960.

Both numbers are re-measured with the centroids displaced by the five units they are quoted to, in both directions, and the ordering and the factor survive. That is the whole method: nothing is fitted, one lattice and eleven points.

Two decisions inside it are worth defending, because either could be argued the other way and both change the numbers.

The step sizes are equal in CIELAB units rather than in the book’s own units. A real system’s steps are equal in its scale — Munsell’s chroma steps are Munsell chroma, which is not C* — so a lattice built here is a lattice a book would approximately produce rather than the one any particular book has. Making the steps finer moves the counts up in proportion and leaves the ratio between names almost exactly where it is, which is the property that matters: the census is about shape, not about how many chips a publisher chose to print.

And the crossing rate counts three neighbours rather than six. Each chip is compared with the one above it in lightness, the one outward in chroma and the one round in hue, and not with their opposites, because counting both directions counts every pair twice and reports the same fraction. What it does mean is that the rate is a property of the lattice’s edges rather than of its chips, which is the right object: a person moving through a book moves along edges.

Where the disagreement bites #

Finding a colour by name is a search, not a lookup. A person asked for “a mid orange” has 69 chips to choose between, distributed over five lightness levels and several chroma levels, and there is nothing in the book’s ordering that groups them — orange’s chips are wherever orange’s territory happens to fall across the pages. A person asked for a grey has three times as many and they are all in one column, which is the only case where the two structures coincide.

Interpolating across a boundary is not the same operation as interpolating inside one. The book promises that the chip between two chips looks between them, and it does. What it cannot promise is that the chip between a chip called green and a chip called blue will be called either.

And a specification written in names is not a specification. This is where it costs money. A brand colour is an ink precisely because “the blue” is not a specification — but the reason it is not a specification is usually given as vagueness, and the measurement here is sharper than that. Blue’s territory contains 66 chips of this lattice, which are spread over ΔE00 distances of tens of units. The word is not vague; it is coarse, by a stated amount, and the amount is computable.

The distinction matters because the two failures have different repairs. Vagueness is fixed by asking the client to be more careful. Coarseness is not fixed by care at all: a word that covers 66 chips covers them for everybody, including the person who chose it, and the only repair is to stop using words — which is what a colour order system’s coordinates and an ink’s spectral definition both are. Every step in the history of colour specification has been a step away from names, and this census is the size of the problem they were stepping away from.

There is a fourth consequence and it runs the other way. A lattice cannot tell a client what a chip will be called. Somebody choosing a corporate colour from a book is choosing a coordinate and buying a word, and the word arrives later, from other people, and cannot be read off the page. Two chips one step apart in the book — a difference the client can see and has deliberately chosen between — will in about a fifth of cases be given different names by everybody who ever sees the result.

Reach is the outer measure of a name. The inner one is how far two colours have to move before a speaker stops giving them the same word, and it is the quantity a catalogue’s step size should have been compared against.

The two measurements agree about how big a name is #

The crossing rate and the naming radii are computed by different passes over different objects, and they can be made to answer the same question — which is the check neither provides on its own.

A crossing rate is a mean free path in disguise. If a step of ten CIELAB units changes the name 21.5 per cent of the time, then a walk through the lattice travels

$$\frac{10}{0.215} \approx 46\ \text{CIELAB units}$$

on average before the name changes. That is the average chord a name is crossed on, over all three directions and all eleven names at once.

The radii figure says the same names reach between ten and twenty-nine units from their centroids — diameters of twenty to fifty-eight. Forty-six sits inside that range, near the top, which is where a chord-through-a-cell average should sit relative to a centroid-to-nearest-boundary minimum. Two passes, two objects, one answer.

That makes the slack figure quotable in a single sentence. A colour specification given as a word carries about forty-six units of freedom along a random direction, against the one unit a supplier is held to — a factor of forty-six, computed two ways, and the number that says why every step in the history of colour specification has been a step away from names.

And a third of the census is the lattice’s own shape #

The factor of three between the biggest name and the smallest is the headline, and part of it belongs to the lattice rather than to the vocabulary.

The lattice is regular in chroma and hue, which means it is polar, and a polar lattice does not sample space evenly. Thirty-six hues at a chroma of ten are packed around a circle sixty-three units in circumference; the same thirty-six at a chroma of ninety are spread over five hundred and sixty-five. Chips near the neutral axis are nine times denser in space than chips at the rim.

So a name that hugs the axis collects chips out of proportion to its territory, and two of the three largest do exactly that: grey at 198 and white at 165 are low-chroma names living where the lattice is densest. The three smallest — blue at 66, red at 68, orange at 69 — are saturated names living where it is sparsest.

Purple, at 181, is the exception that shows the two effects are separable: it is not an axis-hugging name, and its count is a hue-span effect rather than a sampling one.

The essay’s own control already separates them and is reported too gently. The territory figure counts by volume over a Cartesian lattice, and its shares are described as “similar in ordering and not identical”. The difference between the two censuses is the polar over-sampling, so the volume census is the one that measures the vocabulary and the chip census is the one that measures what a book would print. Both are worth having — a publisher deciding how many pages to give to grey wants the first — and only one of them is a fact about English.

Where the model stops #

The lattice here is an idealisation and a real book is worse. Munsell’s actual sheets have ragged outer edges, because the space is not a cylinder and the maximum chroma available at a given hue and value depends on what pigment exists — a fact the previous rung is about. That raggedness would make the census more uneven rather than less, so the factor of three quoted here is a floor.

The names are English and the catalogue is not naming anything. Munsell notations are coordinates, not words, and that is a deliberate feature: the system was built to avoid names precisely because names were known to be unreliable. So this essay is not accusing a colour order system of failing to be a vocabulary. It is measuring the size of the gap that the system’s designers were right to build around.

And a chip is not a colour. Every chip in this lattice is a point in CIELAB under D65 for the standard observer. A physical chip is a reflectance, and two physical chips that match under the book’s stated illuminant need not match under another — which adds a whole dimension of disagreement that this computation, working in colorimetric coordinates throughout, cannot see.

The generalisation #

The sentence worth carrying is: regularity is a property of a coordinate system, and there are at least three coordinate systems here that all claim it.

A colour order system is regular in its own steps. A colour space claims to be regular in perceived difference. A vocabulary is regular in nothing and does not claim to be. Each is a legitimate way of imposing structure on the same set of stimuli, and the three cross each other everywhere.

The surprising connection is with ink limits and process control. A press operator matching to a swatch book is working in the lattice; a specification holding them to ΔE00 1.5 is working in the metric; and the client who will accept or reject the sheet is working in the vocabulary, with a resolution ten times coarser than either. All three parties are being reasonable and no two of them are measuring the same thing — which is a better explanation of a great many colour disputes than anybody’s carelessness.

Who found it, and when #

Munsell published his system in 1905, and the crucial decision was made then: the coordinates are hue, value and chroma, deliberately independent, deliberately numerical, and deliberately not names. He had been a painting teacher, the vocabulary he was replacing was the trade’s — vermilion, ultramarine, a good strong pink — and the replacement was built because those words could not be handed from one person to another.

The 1943 renotation re-fitted the system against visual spacing experiments and is the version still in use. It is a fit to judgements, which is what makes Munsell value a measurement of people rather than a definition, and it is why this site’s own value function is quoted rather than derived.

NCS took the opposite route in 1979, coordinatising by resemblance to six elementary colours rather than by even spacing — which is the closest any commercial system has come to building the vocabulary in. It is a different lattice, and it does not remove the mismatch: it moves it, because the elementary hues and the basic terms are different sets.

And Berlin and Kay’s eleven terms arrived in 1969, sixty-four years after Munsell, into a field that had spent that time building systems designed around the assumption that colour names were not worth formalising. Both judgements were correct and nobody has reconciled them.

What the pictures cannot show #

They cannot show a page. A swatch book is a physical object with an order, and the census here is a count with no order in it at all: the figure sorts by how many chips a name has, which is not how any book is arranged. What a reader would need to see the argument properly is the lattice laid out as pages with the naming boundaries drawn across them, and those boundaries would run diagonally through every sheet.

And they cannot show the chips. Every one of the 1,320 is drawn here as a count. Drawing them as patches would need a display that can show all of them, and a lattice reaching C* 90 is well outside sRGB at most hues — the reason the census is 1,320 chips rather than 2,880 is that the rest were dropped for exactly that reason.

Where the ladder goes next #

The nearest unfinished piece is the ordering question. This essay counts chips per name and says nothing about how those chips are distributed through the book — whether a name’s chips are contiguous in the lattice or scattered. Contiguity is computable from the same census and would say whether a person searching for “a green” is turning consecutive pages or hunting.

The second is the physical half. Every chip here is a colorimetric point; a real chip is a reflectance with an illuminant dependence, and two chips in the same book that match under its stated light need not match under a shop’s. The census would look different again under a fluorescent tube, and by an amount the lamp essays can compute.

And the third is the one that would need data this site has not got: whether a system built around the vocabulary — NCS, or something like it — has a lower crossing rate than a system built around even spacing. It is one census against another, the arithmetic is the arithmetic above, and the answer would say whether the two structures can be reconciled at all or merely traded off.