The band below four hundred

There is a feature of every observer on this site that no account of colour vision this collection has read mentions, and it sits below four hundred nanometres.

The claim #

A visual pigment has a second absorption band in the ultraviolet, this collection’s template carries it at 0.26 of the main band’s height, and it is worth about a third of what the choice between two published nomograms is worth.

• It is real. The β-band is a property of the retinal chromophore and appears in measured pigment spectra across the animal kingdom.
• It sits where the eye has almost closed. For the long-wave cone the band peaks at about 367 nanometres, and the human lens transmits very little there.
• Removing it moves the median observer 0.017 ΔE2000 closer to the standard — against 0.052 for the difference between Govardovskii and Lamb.
• The sweep is monotone. Every increase from 0 to 0.52 moves the observer further away, with no interior optimum anywhere.
• And the confusion points feel it. Removing the band moves the protan copunctal point by 0.008 in chromaticity and the deutan point by 0.049.

What the band is #

A visual pigment’s absorbance against wavelength has two humps. The α-band is the one everybody draws: the main absorption, peaking at the pigment’s λmax, and the thing a nomogram is usually taken to be about. The β-band is smaller, sits well to the short-wavelength side, and is a second electronic transition of the same molecule.

Where it sits is determined by the α-band’s peak, and in Govardovskii’s formulation the relationship is close to linear: the secondary band’s own peak is about 189 + 0.315 λmax nanometres. For the long-wave cone at 566 that is 367 nm; for the short-wave cone at 441 it is 328.

Its height is 0.26 of the α-band’s peak in the published formula, which is not a small fraction. A quarter of a pigment’s peak absorbance in the near ultraviolet would be a substantial contribution to a visual response, if anything got there.

Why it nearly does not matter, and why it matters at all #

Almost nothing does get there, and the reason is the lens.

The human crystalline lens absorbs strongly below about 400 nanometres, and increasingly so with age; a lens at the population’s median age transmits a small fraction of 370 nm light. The macular pigment absorbs in the same region for a different reason. So the β-band’s contribution to a cone’s sensitivity at the cornea is the band’s height multiplied by a transmittance close to zero.

That is why the band is not in the picture anybody draws of colour vision, and why removing it changes the median observer by only 0.017 ΔE2000.

It is not zero, and the residue has two sources. The lens’s absorption is a shoulder rather than a wall, so there is transmission at 390 and 400 nanometres where the β-band is still rising towards its peak. And the collection’s own machinery is evaluated on a wavelength grid that begins at 380 nanometres, so the band’s upper flank is inside the domain even though its peak is not.

So the band’s effect is entirely in its overlap with the lens’s shoulder, which is a statement about a coincidence of two curves rather than about either — and is exactly the kind of thing that would change under a different lens model. The lens model here has an age argument with a measured range, and the band’s contribution grows for a young eye and shrinks for an old one.

The sweep #

The band’s amplitude is the one coefficient in this template that is a number rather than a shape, so it can be dialled where every other choice has to be swapped.

Monotone, over the whole range, with no interior optimum. Every increase in the band moves this collection’s median observer further from the 1931 standard observer, and the published value is unremarkable on the curve.

The temptation is obvious and should be named in order to be refused. If a coefficient can be moved and moving it improves agreement with a standard, why not move it?

Because agreement with the 1931 observer is not what a nomogram is fitted to. Govardovskii’s formula was fitted to microspectrophotometric measurements of individual receptors, in a large number of species, using a spectrophotometer rather than a person. The 1931 functions are the average of what seventeen people did with three knobs. The two are measurements of different things by different methods, and the residual between them is a mismatch of kinds — which is what the two simplifications in the previous essay established.

A coefficient whose measured value is not the one that best fits an unrelated agreement is a coefficient to leave where the measurement put it. The monotone curve with no optimum is what that situation looks like, and finding one is a reason to stop rather than to tune.

What it moves that is not an average #

An average over twenty-four reflectances is a blunt instrument, and the band’s effect on it is small. Two other quantities feel it more sharply.

The confusion points. A dichromat’s copunctal point is the null space of the two cone rows that remain, so it is derived from the fitted cone-to-tristimulus matrix and moves whenever the matrix does. Removing the β-band moves the protan point by 0.0077 in chromaticity, the tritan point by 0.0425 and the deutan point by 0.0493.

Those are not small on the scale the points are used at, and a confusion point is what every simulation of colour vision deficiency on this site is built from. The deutan point sits at about (1.43, −0.59), well outside the diagram, and a move of 0.05 out there is a rotation of the whole confusion line family by a visible amount.

And the short-wave cone. The band’s position scales with the peak, so for the S cone at 441 nm the secondary band sits at 328 — further into the ultraviolet, where the lens transmits even less. But the S cone’s α-band already reaches down towards 380, so its two bands are closer together and the total absorbance in the near ultraviolet is higher. Removing the band narrows the S cone’s full width at half maximum from 90.8 to 87.9 nanometres, which is a three per cent change in the width of the most poorly-constrained of the three curves.

The sweep has a shape, and the shape says more than the monotonicity #

Monotone with no interior optimum is the conclusion drawn from the nine rows, and the rows support a sharper one.

A quadratic through them fits to a root-mean-square of 0.00025 — the width of the table’s own rounding — and reads 0.9352 + 0.0462 x + 0.0741 x², with x the band’s amplitude. Two things follow that the word monotone does not carry.

The curve does not flatten at zero. Its slope there is 0.0462 per unit of band, so the left-hand end of the sweep is not a shoulder approaching a minimum; it is a wall the sweep ran into. The fitted parabola’s own vertex sits at x = −0.31, which is a negative absorption band and therefore not a place. The absence of an interior optimum is not a near miss — the nearest thing to an optimum is outside physics by more than the published value is inside it.

And the marginal cost rises. Going from 0.26 to 0.52 costs 0.0268 against the first 0.26’s 0.0168, so the second half of the sweep is 1.6 times as expensive as the first, and the slope at the published value is 0.0847 — nearly double the slope at zero. A collection tuning this coefficient towards the 1931 observer would find the ground getting steeper as it went, which is the opposite of the flattening that signals an approach to a fitted value.

Both readings say the same thing in different words: the site’s own agreement metric has no opinion about this coefficient other than “less”, and a metric whose only opinion is a direction is a metric with nothing to say.

The point that moves furthest is not the point that matters most #

The three confusion points are ranked here by how far they move — deutan 0.0493, tritan 0.0425, protan 0.0077 — and that ranking is not the one a simulation feels.

A copunctal point is the vertex of a family of lines, so what a simulation inherits is not the point’s displacement but the angle the lines through it swing by. That angle is the displacement divided by the distance from the point to the colours being simulated, and the three points sit at wildly different distances: the deutan point is 1.43 units from equal-energy white, the protan 0.42 and the tritan 0.37.

The ranking reverses. The deutan point moves 6.4 times as far as the protan one and turns its lines only 1.9 times as much, because it is three and a half times further away — and the tritan point, which this essay’s claim list does not mention at all, turns its lines three times harder than the deutan point does.

That is the quantity the essay’s own next sentence reaches for when it says a move of 0.05 out at (1.43, −0.59) is a rotation of the whole confusion line family by a visible amount. It is two degrees. The visible rotation is on the tritan family, and it is six and a half.

The reason is worth keeping, because it applies to every copunctal number this collection publishes. A distant vertex is a stable one. The deutan point sits far outside the diagram precisely because the two cone rows that define it are nearly parallel, and that same near-parallelism is what makes its coordinates jump about under any change at all — while the lines it generates, which are nearly parallel to each other, barely move. So a large displacement of the deutan point is the expected symptom of a small change, and quoting it as the largest effect reports the instrument’s own sensitivity rather than the band’s.

What would make it matter #

Three changes would take the band from a small term to a large one, and all three are situations this collection either models or could.

A young lens. The lens yellows steadily from about twenty, so a twenty-year-old receives roughly three times the 440 nm light an eighty-year-old does and proportionally more below 400. Sweeping the population’s age variate towards its young end raises the band’s contribution.

An aphakic eye. A person whose lens has been removed and not replaced, or replaced with a plastic one without a UV filter, sees into the near ultraviolet — which is not a hypothetical and was how ultraviolet vision in humans was established.

And a source with ultraviolet in it. A brightened paper is a fluorescent object, and A fluorescent lamp emits mercury lines at 365 nm, and a paper with an optical brightener in it is absorbing exactly there in order to re-emit in the blue. The collection has a whole set of essays about what brighteners do, and every one of them is computed with a template whose secondary band is in the same part of the spectrum.

That last is the one worth flagging. The brightener work integrates a source’s ultraviolet against a paper’s absorption and then against an observer, and the observer’s own ultraviolet sensitivity has never been varied in it. The number here — 0.017 ΔE2000 on a median observer under D65, which has little ultraviolet — is a lower bound on what it would be under a source that has some.

A band nobody mentions is a band nobody has checked #

The band’s obscurity is worth a paragraph of its own, because obscurity is the property that makes a coefficient dangerous rather than its size.

Every treatment of colour vision this collection has consulted draws a cone sensitivity as a single hump. That is correct as a picture of a sensitivity at the cornea, because the lens has removed the second hump. It is not correct as a picture of a pigment, and the two are used interchangeably in almost every account.

The consequence is that anybody computing a cone response from a nomogram inherits the band, and anybody reasoning about cone responses from a textbook figure does not. The two will agree about everything under an ordinary light and disagree under an ultraviolet one, which is precisely the case where somebody would go to a nomogram rather than to a table.

That is the shape of defect this collection keeps finding, and it has a name here: a choice made once, inside machinery, invisible in the output, and consequential only in the cases somebody would reach for the machinery to handle. The audit’s contribution is not the 0.017; it is that the number exists at all and can now be quoted.

The same four templates can be put through a different peak, and the middle-wavelength pigment is where the secondary band sits closest to the visible edge.

Why a coefficient like this is the last to be checked #

The band’s obscurity has a structural cause worth naming, because the same cause protects several other coefficients in this collection.

A number gets checked when somebody has a reason to vary it, and a reason to vary it usually comes from a disagreement. A lens model gets checked because two people’s lenses differ and the difference is visible in their matches. A test set’s saturation gets checked because two charts give different answers and somebody has to reconcile them. The β-band produces no disagreement: it is the same in every observer this collection builds, it acts in a part of the spectrum almost nothing reaches, and no two published treatments differ about it because most of them do not mention it.

So it sits in a category of input that is invisible to every ordinary reason for looking: a shared constant with a small effect in a region nobody measures. The only instrument that finds one is a deliberate sweep of everything, which is expensive and is why it took fourteen rounds.

The general lesson is about where to point the next sweep. A collection’s most-audited numbers are the ones its own arguments disagree about, and its least-audited are the ones every argument shares — which is exactly backwards from where the risk is, since a shared constant’s error is systematic and a disputed one’s is at least bounded by the dispute.

Where the model stops #

The band’s amplitude is one number in a formula and the formula has others: the band’s centre, its width, and the way both scale with the peak. None of those is varied here, and each is a fitted coefficient with the same status as the amplitude.

The lens model is a stated transmittance with an age argument, fitted to published measurements, and the band’s whole effect lives in the overlap between two curves — so the 0.017 carries the lens model’s uncertainty as well as the template’s, and the two are not independent.

And the wavelength grid stops at 380 nanometres, which is a decision made in the first weeks of this collection and never revisited. The band peaks at 367 for the long-wave cone, which is outside the grid entirely: the collection is seeing only the band’s upper flank, and the number here is what that flank is worth rather than what the band is worth. Extending the grid is the obvious repair and it is not free — every spectral quantity here is defined on that grid, so moving its lower bound changes every integral in the collection by a small amount, and the change would have to be measured before it could be believed.

The second figure names the reason the band escaped notice inside this collection rather than outside it. Almost every number the population work publishes is a spread — how far apart two members are, how wide the cloud of confusion points is, how much a lamp ranking moves between observers. A term that shifts every member equally cancels out of all of them.

The distance from the standard observer is the one published quantity that is not a spread, and it is the one the band shows up in. That is a general property worth stating: a systematic error is invisible to every statistic computed within a population and visible only in a comparison against something outside it.

Who found it, and when #

The β-band is old. It appears in absorption spectra of rhodopsin from the earliest careful measurements and is understood as a second electronic transition of the retinal chromophore — the cis band, in the photochemical literature, distinguished from the main α-band by its origin rather than by its position.

Its inclusion in a nomogram is Govardovskii’s contribution in 2000; earlier formulations including Lamb’s give the α-band alone. That the band matters at all for human colour vision is not usually asserted, and this collection’s number — 0.017 ΔE2000 on the median observer, a third of what choosing a different nomogram is worth — is consistent with the general silence.

What is worth carrying is the shape of the sweep rather than its size. A monotone curve with no interior optimum, on a coefficient with a measured value, is a specific piece of evidence: it says the collection’s own machinery has no opinion about where the coefficient should sit, and therefore no business tuning it. Finding an optimum would have been the worrying outcome.

Where the ladder goes next #

Two phases have declined to give the appearance model’s surround a continuous control, on the grounds that the frames would be too heavy — and neither phase measured the weight. A deferral made twice on an unmeasured number is the shape of decision this collection spends most of its essays arguing against, made about the collection itself.