Colour Science & Perception
what the eye settles, and what it cannot
What light is

Which end to buy

Refining a five-nanometre grid to one buys a daylight calculation 0.05 ΔE₀₀ and widening its range buys 0.54. Under a fluorescent tube the same two purchases are worth 0.83 and 0.0001. The ranking reverses completely, and what decides it is one length compared against one other length.

Assumes Two ends and one is empty, Where the grid starts and The index is a choice too.

14 min read 6 figures Computed, not quotedThe instrument is the readerThree numbers

Three tabulation decisions have now been measured apart. A collection with limited attention has to choose between them, and the choice has an answer that is short, mechanical, and different for two rooms in the same building.

The two tabulation choices over forty-two surfaces, under a 6500 K thermal radiator. Each column is one choice, measured over a family of forty-two analytic reflectances rather than on a single example: an absorption band of stated centre, width and depth. The four marks are the smallest, the median, the ninety-fifth percentile and the largest cost in ΔE₀₀, logarithmically. Under a smooth light the range is worth 9.1 times the step at the median, so a collection wanting one repair should widen its range rather than refine its step — and under a fluorescent tube the ranking reverses outright.
Fig. 1 The step and the range over forty-two surfaces under a 6500 K radiator, as four percentiles each. Under this light the range’s median is nine times the step’s and its worst case thirty times.

The claim

Which tabulation defect is worth repairing is decided by one comparison: the width of the light’s narrowest feature against the tabulation step. Nothing else in the calculation enters.

  • When the feature is much wider than the step, the sum is nearly exact, the aliasing is zero, and everything that is left is the range. Widen the range.
  • When the feature is comparable with the step, the sum is still nearly exact — better than any interpolation of it — but the answer has become fragile to the grid’s origin. Filter, do not refine.
  • When the feature is much narrower than the step, no arithmetic on the table helps at all and the range is irrelevant. The repair is upstream, at the instrument.
  • And the ratio is not a property of the collection. It is a property of each light, so a body of work computing under several lights is in several regimes at once.

Two numbers, per light

The measurement is the two decisions on one footing, over the same forty-two analytic surfaces, so that neither is represented by an example.

light step at 5 nm, median range 380–780, median ratio
a 6500 K radiator 0.060 0.542 9.1
tungsten at 2856 K 0.011 0.071 6.3
a white LED 0.0002 0.0008 3.7
a fluorescent tube 0.834 0.00013 0.00016

The first three rows say the same thing at three different magnitudes: under a smooth light the range costs several times the step, and both are small. The fourth row says the opposite by a factor of six thousand.

There is no averaging over that. A collection that computes some figures under daylight and some under a fluorescent tube has one defect in one half of its work and a different defect in the other, and a single sentence about “the wavelength resolution” cannot describe both.

One comparison decides it. Every result of this section reduces to a ratio between two lengths: the narrowest feature the light has, divided by the tabulation step.

ratio what happens what to repair
above about 4 sum nearly exact, origin irrelevant the range
about 1 to 4 sum exact, interpolation harmful, origin matters a little the slit, or nothing
below about 1 sum meaningless, origin decisive the instrument

The three-emitter LED sits at 3.6 and its direct five-nanometre sum is exact to five parts in ten million, while every interpolation of that same table is worse — the middle band’s characteristic behaviour. The fluorescent tube sits at 0.24 and nothing helps. Daylight and tungsten have no narrow feature at all and the ratio is unbounded.

What makes the ratio rather than the step the governing quantity is that both errors it separates are about representation rather than about accuracy. A step is fine enough when it can hold what the light has; a step held against a light with nothing in it is fine at any size, and a step held against a mercury line is coarse at any size a table is published at.

How much the answer moves when the 5-nanometre grid is slid through one cell. Each bar is the spread of one light's colour across five grid origins, all at the same 5-nanometre step, in ΔE₀₀. A smooth light barely moves, and what movement it has is the end cells rather than the sampling. The fluorescent tube moves by 3.18 units and the laser projector by 35.0, because their emission lines are narrower than the step and whether a sample lands on one is a coincidence of arithmetic. This is the measurement that separates a quadrature error from an aliasing error, and no average over origins can substitute for it.
Fig. 2 The spread across five grid origins, per light. The three bands are visible here too: the top band barely moves, the middle band does not move at all, and the bottom band moves by more than any tolerance.

A reader who wants one diagnostic rather than a ratio can use that figure directly. Slide the grid and see what happens, on the light the work is actually done under, with the sample it is actually done on. A movement below a hundredth of a unit puts the calculation in the top band; a movement in the units puts it in the bottom one; no movement at all, with the direct sum already exact, is the middle. It is three lines of code and it requires no knowledge of the light’s spectral structure, which is often unavailable anyway.

The ratio is the explanation and the origin sweep is the test, and it is worth having both. An explanation predicts what a change will do; a test says where the calculation stands today. Neither substitutes for the other, and a great deal of numerical practice consists of tests with no explanation attached — which is why the answers so often surprise the people who ran them.

Why refining is the wrong instinct in two of three regimes

Refining the step is the repair everybody reaches for, and it is the right one in none of the three bands.

In the top band it is nearly useless. Going from five nanometres to one takes daylight’s median from 0.060 to 0.012 — a factor of five — while the range still costs 0.542, so the calculation has bought a fifth of a tenth of what it left on the table. Worse, most of that factor of five is not the sampling: halving the two end weights buys most of the same improvement for one line of arithmetic and no resampling.

In the middle band refining is actively risky, because a table read finely has to be interpolated and interpolation in that band is harmful. The direct sum was exploiting a cancellation that the interpolation breaks.

In the bottom band refining does nothing until it passes the linewidth, which for a mercury line is fifty times finer than any published spectral data. Between the two there is no convergence at all, and the sequence of errors is not even monotone.

What a tabulation step costs, by light, on vermilion. The horizontal axis is the tabulation step in nanometres, from one to twenty; the vertical is how far the resulting colour is from the same integral taken at a tenth of a nanometre over the same range, in ΔE₀₀, on a logarithmic scale. Each line is one light. The three with no feature narrower than the step fall smoothly and stay below a tenth of a unit at five nanometres, which is the grid used throughout. The fluorescent tube and the laser projector do not fall at all: their lines are narrower than any step drawn here, so the answer depends on where the samples land rather than on how many there are. The sample is held at vermilion throughout.
Fig. 3 Six lights against tabulation step. The three that fall smoothly are the top band, and the two that do not fall at all are the bottom one; the flat near-zero line between them is the middle band, which is exact at every step drawn.

What the middle band looks like from inside

The middle band is where most modern lighting sits and it deserves a closer description, because its behaviour is the least intuitive of the three.

A light whose narrowest feature is two to four times the step is nearly band-limited. The tabulation almost determines it, and the small amount it does not determine is exactly the amount that the numerator and denominator of a tristimulus ratio get wrong in the same way. So the direct sum is not merely adequate; it is better than the same calculation done at any finer resolution reachable from the same table.

That is a strange property for a numerical method to have and it is worth stating in its uncomfortable form. A calculation in this band is more accurate than a calculation that uses more of the data, and the reason is that the extra data are invented. The information ceiling is the table, and every operation that appears to raise it lowers the accuracy instead.

The practical rule that follows is short: in this band, sum the table as it arrived and do not resample it. If two tables must be multiplied and their intervals differ, coarsen the finer one by averaging rather than refining the coarser one by interpolating — the first is a filter and the second is a fabrication.

The rectangle sum against the trapezoid sum, under a 6500 K thermal radiator. Colorimetry's summation is the rectangle rule at the tabulated points. The trapezoid rule differs from it by exactly one thing — half a cell at each end of the range — and the gap between these two lines is therefore that term and nothing else. At five nanometres it is a factor of 14.3, which means the number everybody calls a sampling error is mostly a truncation error wearing the step's clothes. On a light whose lines are narrower than the step the two rules agree to three decimal places, because there the error really is the sampling.
Fig. 4 The rectangle and trapezoid sums under a 6500 K radiator. The gap between them is the end-cell term, and on this light it is most of what the five-nanometre error is.

That figure is the reason the top band’s repair is not the step. Five times finer sampling buys a factor of five; halving two weights buys a factor of fourteen, on the same eighty-one numbers, with no new data of any kind. A repair that is available for free is not usually the one anybody plans for, and it should be the first one taken.

The remaining question in the top band is then whether the range is worth its own repair after the end cells have been halved, and the answer is yes by a wide margin. Daylight’s median falls from 0.060 to about 0.004 under the rule change, while the range stays at 0.542 — so the ratio between the two decisions goes from nine to about a hundred and thirty, and the range becomes the only thing left worth doing.

What a collection should do with this

Applied to this collection, the procedure gives a clear and slightly deflating answer.

Almost everything here is computed under daylight reconstructions, blackbodies and the CIE illuminants, all of which are firmly in the top band. So its tabulation defect is the range, its step is fine, and the aliasing is zero on the great majority of its figures. That agrees with what the ledger has been saying for two rounds, and it is now measured against the alternative rather than asserted.

The minority that is not in the top band is the lamp work: the discharge lamps, the laser primaries, the narrowband displays. Those are in the bottom band, their range cost is exactly zero, and their repair is not available in software at all — it is the slit an instrument has and a formula does not.

So the collection owes two different repairs to two different parts of itself, and neither of them is the one that would have been guessed. The step, which is the only tabulation parameter it has ever discussed, is not on the list.

The two tabulation choices over forty-two surfaces, under a fluorescent tube, mercury lines on a phosphor bed. Each column is one choice, measured over a family of forty-two analytic reflectances rather than on a single example: an absorption band of stated centre, width and depth. The four marks are the smallest, the median, the ninety-fifth percentile and the largest cost in ΔE₀₀, logarithmically. Under a smooth light the range is worth 0.0 times the step at the median, so a collection wanting one repair should widen its range rather than refine its step — and under a fluorescent tube the ranking reverses outright.
Fig. 5 The same two choices under a fluorescent tube. The step’s median is 0.83 and the range’s is 0.00013, which is the ranking of the previous figure inverted by a factor of six thousand.

The cost of the repairs, which is not symmetric either

Having decided which defect to repair, the second question is what each repair costs, and the three are not comparable.

Refining a step costs computation and nothing else. It makes no new claim about anything. A calculation at one nanometre instead of five is five times slower and is otherwise the same calculation.

Widening a range costs a modelling commitment. Sixteen new rows have to be filled in for every reflectance in the collection, and a reflectance below 380 nanometres is a guess rather than a measurement. That is why the repair has stayed outstanding: the arithmetic is trivial and the physics is not.

Fixing an aliasing fault costs the instrument. It cannot be done downstream at all, and a collection computing from formulae has no instrument to fix. Its options are to compute at a resolution finer than the linewidth — a factor of fifty in cost on every affected figure — or to say in the caption that the source is being read at the grid’s own points.

That asymmetry is the reason a purely numerical account of “which defect is largest” is not a decision procedure. A dial has a price and so does a repair, and the largest error is not always the one worth buying out.

Where the procedure would fail

A decision rule stated this confidently should be given the case it gets wrong, and there is one.

A sample with structure narrower than the step would put a calculation in the bottom band without the light having anything to do with it. Nothing in the rule above would catch it, because the rule looks at the light. Reflectances are smooth for physical reasons — a pigment absorbs over a band of a hundred nanometres — so this is rare, and it is not impossible: an interference filter, a structural colour, a laser-line notch filter, a Fabry–Pérot etalon in an instrument’s own path.

The general form of the rule is therefore about the product, not the light: the narrowest feature of R(λ)S(λ) against the step. For every case in this collection that is the light’s narrowest feature, because the samples are all far smoother. Stating it as a property of the light is a simplification that should be undone the moment a structured sample appears.

What was computed, and how

The two columns are computed over the same forty-two surfaces, and each is measured against a reference that holds the other decision fixed: a step against a tenth-nanometre sum over the same range, a range against a full-range sum at the same tenth of a nanometre. Mixing them was the first version’s error and it inverted one of the two conclusions.

Percentiles rather than means, for the reason this collection established about a mean and its set: a distribution spanning a factor of twenty is not summarised by its average, and the average of the range column over the family is 0.66 against a median of 0.54 and a maximum of 2.02.

The forty-two surfaces are seven band centres by three widths by two depths, which is a grid rather than a sample of anything. It is stated as such: the family is a way of turning one example into a distribution, and it is not a claim about what surfaces exist in the world. It is the same construction the adaptation census rests on, used for the same reason and carrying the same caveat: a declaration rather than a sample.

One last asymmetry decides how a body of work should be organised rather than repaired. A step and an origin are properties of a calculation and can be changed for one figure without touching another; a range is a property of every table in the system at once, because a reflectance array of ninety-seven entries cannot be multiplied by an observer array of eighty-one. So the range is the only one of the three that cannot be repaired incrementally, and a collection that widens it widens it everywhere on the same afternoon or not at all.

That is why the sensible order of work is the reverse of the order of magnitude. Take the free repair first, because it is local. Take the origin sweep second, because it is a diagnostic rather than a change. Take the range last, because it is a commitment — and take it knowing that the commitment includes an extrapolation for every sample in the collection, which is a piece of physics rather than a piece of arithmetic.

Where the model stops

The three bands have boundaries at about one and about four, and those numbers are read off six lights rather than derived. A proper treatment would compute the light’s spectral bandwidth directly — the width of its autocorrelation, or the point where its Fourier transform falls below the observer’s — and the boundaries would then be theorems rather than observations.

The ratio also assumes the tabulation and the slit have the same width, which is how instruments are built and not how formulae are evaluated. For a calculation with no slit at all the middle band is narrower and the bottom band begins earlier, which is exactly the collection’s own position.

And the sample family is analytic and smooth by construction, so the failure case named above cannot occur in any measurement in this essay. It is named because it is the rule’s boundary, not because it was tested.

The two tabulation choices over forty-two surfaces, under a tungsten lamp at 2856 K. Each column is one choice, measured over a family of forty-two analytic reflectances rather than on a single example: an absorption band of stated centre, width and depth. The four marks are the smallest, the median, the ninety-fifth percentile and the largest cost in ΔE₀₀, logarithmically. Under a smooth light the range is worth 6.3 times the step at the median, so a collection wanting one repair should widen its range rather than refine its step — and under a fluorescent tube the ranking reverses outright.
Fig. 6 The two choices under a tungsten lamp. Both are smaller than under daylight and the ranking is unchanged, which is what a band boundary looks like from well inside one side of it.

Three lights, three figures, two rankings. That is the honest summary of the whole section, and it is worth resisting the urge to average it into one recommendation. The literature’s habit is to publish a single guidance number — five nanometres is adequate, ten nanometres is adequate for smooth sources — and every such number is a statement about a band without saying which band it belongs to.

The generalisation

The habit is about reducing a multi-parameter decision to a ratio.

Three decisions with three mechanisms look like a three-dimensional problem and are not, because all three are answering one question — whether the index can hold what is being indexed. Once that is seen, the parameters collapse onto a single dimensionless number, and a dimensionless number is a thing a practitioner can carry in their head.

The failure mode is to keep the three separate and to rank them by measured magnitude on whatever example is to hand. That produces a ranking that is correct for the example and reverses on the next one, which is precisely what the two tables in this essay do. A ranking that reverses is a ranking with a hidden variable in it, and finding the variable is more useful than measuring the ranking more carefully.

Who found it, and when

The dimensionless ratio here is Nyquist’s condition with the names changed, and the three bands are the standard picture of a sampled system: oversampled, critically sampled, aliased. None of that is specific to colour.

What is specific is which band each light falls in, and that has changed within living memory. Incandescent lighting is unconditionally in the top band, and it was nearly all the artificial light there was until the 1940s. Fluorescent lighting put a large fraction of the world’s interiors into the bottom band. Solid-state lighting has put most of it into the middle band, where the direct sum is exact and every instinct about refining is wrong. The tabulation standards were written for the first of those three.

Where the ladder goes next

The decision procedure rests on a cancellation that has been quoted three times and never examined: the fact that the grid appears twice in every tristimulus value and that the two errors largely divide out. That is what makes the middle band exact, and it has a condition on it that is worth stating precisely.

Where this goes next

What each end of the 380–780 nanometre range costs, by light. Two bars per light, on a logarithmic axis: the upper is what extending the range down to 300 nanometres moves the answer, the lower what extending it up to 830 does. The asymmetry is the whole figure. A thermal source has about a fifth of its power outside this collection's range and almost all of it at the long end, where the observer is already zero; what costs money is the short end, where the observer is small but not zero and daylight is still strong. A light with no ultraviolet — an LED lamp, a laser — pays nothing at either end, which is the pairing again: a range only costs what the light puts in it. Two ends and one is empty Extending this collection's wavelength range down to 300 nanometres moves a red pigment under daylight by 0.502 ΔE₀₀. Extending it up to 830 moves the same colour by 0.00015. A fifth of a thermal source's power lies outside the range and almost none of its colour does, and confusing those two shares is how a range gets argued about instead of measured. How much the answer moves when the 5-nanometre grid is slid through one cell. Each bar is the spread of one light's colour across five grid origins, all at the same 5-nanometre step, in ΔE₀₀. A smooth light barely moves, and what movement it has is the end cells rather than the sampling. The fluorescent tube moves by 3.18 units and the laser projector by 35.0, because their emission lines are narrower than the step and whether a sample lands on one is a coincidence of arithmetic. This is the measurement that separates a quadrature error from an aliasing error, and no average over origins can substitute for it. Where the grid starts Holding the step at five nanometres and sliding the grid's origin through one cell moves a fluorescent tube's computed colour by 3.18 ΔE₀₀ and a laser projector's by 35.0. Refining the step does not fix it and averaging over origins hides it. It is the one tabulation fault with no smooth error to cancel against. Every figure a surround dial would apply to, weighed. The eleven generators in the appearance family, by the size of the drawing each emits — which is what a dial multiplies, because every frame carries the figure's whole body. The dashed rules are the thresholds the dial policy here is written in: under 3 KB gets thirteen stops, under 6 gets nine, under 10 gets seven, and above about 12 KB nothing gets a dial at all. The median body here is 2.25 KB and 4 of the eleven are inside the most generous tier, against a policy written when the distribution had a median of 4.0 KB. Two phases deferred a surround dial on the grounds that appearance swatches are not small. The exception is unique-hues at 14.1 KB, which is over the ceiling that says it should have no dial — and it has one, on chroma, declared in the same file as the ceiling. A dial has a price Two rounds declined to give the appearance model's surround a continuous control, both times because the frames would be too heavy, and neither time was the weight taken. It is 2.25 kilobytes at the median — the light end of these figures rather than the heavy end — and one figure is already breaking the rule the deferral was made under.

What this makes readable

Essays that name this one as a prerequisite.

Named alongside this one

Essays reaching for the same objects. Nobody chose these; they are what the index of named objects makes visible.

What links here

The 8 essays that link to this one and share the most of its objects, of 9 that link here.

The objects this essay names

Each one links to every other essay that touches it.

AliasingAuditMeasurement errorQuadratureSpecificationSpectral structureTest setTrade-offWavelength gridWorst case