A template is mostly its tail

Four curves at the same peak with the same full width at half maximum, and the observers built from them are between 0.90 and 1.74 units from the standard one. Something other than peak and width is doing the work.

The claim #

Hold a pigment template’s peak and its width and vary only which side its half-maximum reaches further, and the observers built from it are ordered by that number.

• The four templates are width-matched by construction. Both caricatures are given Govardovskii’s own α-band full width at half maximum in wavenumber, measured rather than assumed.
• They differ in one thing. Tail ratios of 1.19, 1.16, 0.97 and 0.79, where above one means a longer reach towards the short wavelengths.
• A real pigment is above one, and the reason is physical: the short-wave tail is vibrational structure in the same molecule every visual pigment is built round.
• The observers are ordered by it. 0.90 and 0.95 for the two above 1.1; 1.12 and 1.74 for the two below one.
• And the ordering survives the fit. Each template gets its own least-squares matrix to tristimulus values, so the caricatures are not being charged for a mismatch.

What is held and what is varied #

The comparison is only worth anything if the templates differ in one property, so it is worth being explicit about the four things that are held.

The peak. All four are evaluated at the same three wavelengths — 566, 541 and 441 nanometres — because a nomogram takes the peak as an argument and the argument is not what is being varied.

The width. Govardovskii’s α-band at 566 nm has a measured full width at half maximum, taken in wavenumber because that is the variable a pigment’s width is roughly constant in — which is the same reason a spectral quantity is usually plotted against wavenumber in physics and against wavelength in colour science, a mismatch of conventions that turns up in several places here. Both Gaussians are given that width. This is the step that makes the comparison about shape: a narrower caricature would separate wavelengths more finely and would look better than it is, and a wider one worse.

Everything downstream. The same optical density through Beer–Lambert, so a cone’s sensitivity is broader than its pigment’s absorbance; the same lens and macular pigment at the population’s median values; and a cone-to-tristimulus matrix refitted per template, so that each gets the best matrix available to it rather than being charged for a seam.

What is varied is the shape between the half-maximum points and outside them.

Measuring an asymmetry #

A pigment’s α-band is not symmetric. It rises steeply on the long-wavelength side and falls slowly on the short-wavelength side, so plotted against wavelength it has a long tail towards the blue.

The measure used here is the simplest one that captures it: how far the curve reaches below its peak at half maximum, divided by how far it reaches above. Above one is a long short-wave tail; exactly one is symmetric; below one is the reverse.

The two published nomograms and the β-band-free version sit between 1.15 and 1.19 and cost between 0.90 and 0.95. The two Gaussians sit below one and cost 1.12 and 1.74.

Within the real group the ordering by tail does not predict the ordering by cost — Lamb has the largest tail ratio and the lowest cost, Govardovskii the middle ratio and the highest — and the differences there are five per cent, which is the noise floor of this comparison. Between the groups it does, and the differences are 20 and 80 per cent.

So the honest statement is a threshold rather than a gradient: having a real pigment’s asymmetry is worth a great deal and the exact amount of it is worth almost nothing. Which is the same shape as the collection’s finding about published adaptation transforms — a group of serious candidates clustered close together, and a large gap to anything outside the group.

Why a wavenumber Gaussian has the tail on the wrong side #

The worst caricature is the one a chemist would draw, and its failure is instructive rather than embarrassing.

An absorption band is a transition between two energy levels, energy is proportional to wavenumber, and a band broadened by a sum of many small independent effects is Gaussian in energy. So a Gaussian in wavenumber is the null model for an absorption band and is what a first approximation should be.

Converting to wavelength inverts and stretches. A constant step in wavenumber is a step in wavelength proportional to λ², so the long-wavelength side of the band is stretched and the short-wavelength side compressed — which puts the long tail on the red side, exactly opposite to a visual pigment.

The tail ratio measures this precisely: 0.79 at 566 nm, against a symmetric-in-wavelength Gaussian’s 0.97 and a real pigment’s 1.16. And it explains why the wavenumber Gaussian is the worst of the four rather than merely a different wrong: it is not symmetric where a pigment is asymmetric, it is asymmetric in the opposite direction, so it is twice as far from the target as a symmetric curve is.

Where the asymmetry comes from #

The short-wave tail of a visual pigment’s α-band is not a fitted detail. It is vibrational structure.

The absorbing molecule is 11-cis-retinal, the same in every visual pigment there is; what the surrounding protein does is shift the energy of the electronic transition, which is why one formula with one argument fits pigments across the animal kingdom. Transitions to excited vibrational levels of the excited electronic state cost more energy than the pure electronic transition, so they absorb at shorter wavelengths, and their intensities fall off in a characteristic way that produces a tail.

That gives the finding here a cause rather than a correlation, which is worth having because a correlation across four constructed curves is exactly the kind of thing that would not survive a fifth. The one property of a pigment template that decides the observer built from it is the one property with a physical origin, and a caricature that gets the peak and the width right and the vibrational structure wrong is a caricature of the wrong thing.

It also explains why the two real nomograms differ so little. Lamb’s and Govardovskii’s coefficients were fitted to different and overlapping sets of measurements of the same molecule in different proteins, so they are two fits to one shape. The shape is what matters and both have it.

Why the short flank matters so much #

There is one more step to explain, because a tail is a small part of a curve and it is producing an eighty per cent effect.

The three cone sensitivities overlap enormously — the long and medium pigments peak 25 nanometres apart and their curves are 110 nanometres wide, so most of what each catches, the other catches too. The information in a colour signal is in the differences between three heavily correlated numbers, and a difference between two nearly identical curves is dominated by wherever they are least identical.

For the long and medium cones, that is the flanks. Their peaks are close and their far tails are both near zero; the region where they differ most in ratio is the short-wavelength side, where one is falling from its peak and the other is already well down.

So the tail is a small part of each curve and a large part of the difference between them. Change the tail’s shape and the long-medium opponent signal changes disproportionately — which is the same amplification this collection meets whenever a difference of two large similar numbers is taken.

What the finding is worth to somebody not building a population #

The result generalises past this collection’s own machinery, and there are three places it lands.

Modelling a species nobody has measured. The commonest use of a nomogram is exactly this: an animal’s pigment peak is known from a behavioural or electrophysiological measurement and the whole curve is wanted. The finding says the choice among published nomograms hardly matters and that a Gaussian approximation does — which is worth knowing, since a Gaussian is what a quick script produces and a caricature that gets the peak right looks convincing.

Modelling an individual. Human long-wave pigments are polymorphic, with a common variant about four nanometres from another, and the variation is one of the five this collection’s population carries. Predicting what a carrier of one variant sees means evaluating a template at a shifted peak, and the template’s tail is what carries the shift into a colour difference.

And reading a published cone fundamental. The CIE’s physiological fundamentals are themselves derived rather than measured directly — built from colour-matching data, dichromat matches and a template — so a reader taking them as ground truth is taking a template’s asymmetry as ground truth one step removed. That the standard functions are a construction rather than a measurement is a point this collection makes elsewhere, and the template is one of the constructions in it.

The one place the ordering breaks #

Inside the group of real nomograms the tail ratio does not predict the cost, and it is worth not smoothing over.

Lamb has the largest tail ratio at 1.19 and the lowest cost at 0.901. Govardovskii’s α-band alone has the smallest at 1.15 and a cost of 0.935. Full Govardovskii is between them on tail at 1.16 and highest on cost at 0.952.

If the tail ratio were the whole story, the three would be ordered 1.19, 1.16, 1.15 against 0.901, 0.935, 0.952 — and they are not, because the α-only version is out of order.

What separates them is the second band, which the tail ratio at half maximum does not see at all: it sits far below the half-maximum crossing, so removing it changes the ratio by 0.01 and changes the observer by 0.017 ΔE2000. The measure is blind to a feature that matters, which is a limitation of the measure rather than a failure of the finding, and it is why the finding is stated as a threshold between two groups rather than as a gradient within one.

It also names the next question. A feature the shape measure cannot see, worth about a third of what the choice of nomogram is worth, sitting in the ultraviolet where the lens has nearly closed — that is a coefficient worth its own dial.

What a width-matched control is worth #

The comparison rests on one construction decision and it is worth defending separately, because getting it wrong is the easiest way to produce this essay’s conclusion without earning it.

A caricature that is narrower than the template it stands in for will always look worse, and for a reason that has nothing to do with shape: three narrower cones overlap less, so they separate wavelengths more finely, and the observer built from them is a different instrument rather than a worse version of the same one. A caricature that is wider will look worse too, for the mirror reason. Either mistake gives a large number and no information.

So both Gaussians are given Govardovskii’s own α-band full width at half maximum, measured from the curve at 566 nm rather than quoted, and in wavenumber rather than in wavelength because that is the variable a pigment’s width is roughly constant in. The measured widths that come back — 118.3 and 118.4 nm against the nomograms’ 113.8 to 114.4 — are within four per cent, and the residual four per cent is the asymmetry itself moving the half-maximum crossings rather than a failure of the matching.

The one place the matching does not hold is the short-wave cone, where a fixed width in wavenumber becomes a much narrower band in wavelength: 71.6 nm against the nomograms’ 88 to 91. That confound is named in the section below rather than repaired, because repairing it means giving the caricature a different width per cone, and a template that is not one formula evaluated at three peaks is not a template.

Where the model stops #

The tail ratio is a crude measure of asymmetry: two numbers off the half-maximum crossings, which says nothing about the shape between them or beyond them. A moment-based measure would be better and would be harder to read; the crude one is enough to order four curves and would not be enough to order forty.

The comparison is at one peak. The tail ratios at the medium and short peaks are similar for the nomograms — 1.16 and 1.23 for Govardovskii — and quite different for the caricatures, because a fixed width in wavenumber is a much narrower band in wavelength at 441 nm than at 566. So the short-wave cone’s caricature is wrong in width as well as in shape, and the eighty per cent figure is not entirely asymmetry.

That confound is real and is worth stating rather than repairing. Repairing it means matching the width per cone, which makes the caricature a family of three unrelated curves rather than one template evaluated three times — and a template that is not one formula is not a template, so the confound is a property of the comparison being between templates at all.

Nought point nine, and one inversion #

The ordering claim can be given a coefficient rather than a description. Across the five rows the rank correlation between the tail ratio and the observer’s cost is +0.90, with a single inverted pair — Govardovskii’s α-band-only version and the full one, the two the essay already names as being out of order.

That is the useful summary because it separates the two claims. The correlation across the whole set is strong; the correlation within the real group is negative and is one swapped pair out of three. So the observers are ordered by it holds as a statement about the five and does not hold as a statement about the three, and the coefficient says exactly how much of it survives.

It also prices the inversion. Moving one pair in a set of five costs 0.10 of a Spearman; the same inversion inside a set of three would cost 0.5. The measure looks better than it is because the two caricatures are so far out, and a version of this comparison with four real nomograms and no caricatures would report a correlation near zero.

The dek says two and four times; the table says 1.2 and 1.9 #

The essay’s opening summary and its own arithmetic give different figures for the same comparison.

The three real nomograms average 0.929 and span 5.7 per cent. Against that mean the wavelength Gaussian costs 1.20 times and the wavenumber Gaussian 1.87 — the 20 and 80 per cent the body quotes. Against the cheapest real nomogram they are 1.24 and 1.93; against the dearest, 1.17 and 1.83.

No pairing of the numbers gives two and four times. The dek’s figures are about a factor of two above what the table supports, and the body’s are the ones to keep — which matters because the essay’s conclusion is a threshold rather than a gradient, and a threshold’s usefulness depends on how tall it is. Eighty per cent is a large penalty for getting a curve’s asymmetry backwards. It is not a factor of four.

The width confound cannot explain the ordering #

The essay names its own confound and leaves it unrepaired: the caricatures are width-matched at the long cone and not at the short one, so the eighty per cent figure is not entirely asymmetry. The two caricatures’ width errors settle how much of it could be.

At the long cone the matching holds to 4 per cent — 118.3 and 118.4 against 113.8 to 114.4. At the short cone the wavenumber Gaussian is 20 per cent narrower than the nomograms, 71.6 against 88 to 91.

The wavelength Gaussian does not share that error. A Gaussian of fixed nanometre width would be 118.3 at 441 nm, which is 32 per cent wider than the nomograms; one scaled with its peak would be 92.2, which is within 3 per cent of them. Either way its short-cone width error is the opposite sign to the wavenumber Gaussian’s, or absent.

That settles the confound without repairing it. The two caricatures err in opposite directions on width and in the same direction on tail, and their costs order with their tails. If width were driving the result the two would not be on the same side of the real group, and they are — 1.115 and 1.741 against 0.929. The width can inflate the wavenumber Gaussian’s penalty and it cannot produce the ordering, because the ordering has the two caricatures together and the width has them apart.

So the eighty per cent is not entirely asymmetry and the threshold is. That is a weaker claim than the essay’s and it is the one its own construction supports, and it is enough for the finding it is offered for: a template that gets the tail backwards builds a worse observer than one that gets it right, by an amount much larger than the difference between two serious nomograms.

Who found it, and when #

Dartnall’s 1953 observation is that visual pigment spectra superimpose when the wavelength axis is scaled by the peak, and the asymmetry is in his nomogram from the beginning; every refinement since has kept it because every measurement shows it.

The vibrational explanation is standard photochemistry and is not specific to vision. What is specific is the size of the consequence, and the consequence is measurable here only because this collection builds an observer out of a template rather than tabulating one. A treatment that starts from a tabulated fundamental cannot ask this question at all, which is the same limitation that makes a tabulated fundamental unable to support a population.

Where the ladder goes next #

One part of the template has not been varied and is not a shape: a second, smaller absorption band below 400 nanometres, published at 0.26 of the main band’s height, sitting where the lens has almost stopped transmitting. It is the one coefficient in the whole template that can be dialled rather than swapped, and dialling it produces a curve with no interior optimum.