Review of Philosophy and Psychology

, Volume 2, Issue 4, pp 601–628

Where Do the Unique Hues Come from?


    • Department of PhilosophyBrown University

DOI: 10.1007/s13164-011-0050-7

Cite this article as:
Broackes, J. Rev.Phil.Psych. (2011) 2: 601. doi:10.1007/s13164-011-0050-7


Where are we to look for the unique hues? Out in the world? In the eye? In more central processing? 1. There are difficulties looking for the structure of the unique hues in simple combinations of cone-response functions like (L − M) and (S − (L + M)): such functions may fit pretty well the early physiological processing, but they don’t correspond to the structure of unique hues. It may seem more promising to look to, e.g., Hurvich & Jameson’s ‘chromatic response functions’; but these report on psychophysical behaviour, not on underlying physiology. So ‘opponent processing’ isn’t any particular help on the unique hues—and even physiology in general seems not to have come up with any good correlate or explanation. 2. Wright (Review of Philosophy and Psychology 2: 1–17, 2011) looks in a different place: to (a) magnitude of total visual response that a stimulus light provokes, the maxima and minima of which, he thinks, give us the boundaries of the main hue categories (Wright connects these with Thornton’s ‘prime’ and ‘antiprime’ colours in an illuminant: Journal of the Optical Society of America 61 (1971): 1155–1163); and (b) the ratio of chromatic to achromatic response, the maxima and minima of which, he suggests, give us the focal points of the unique hues. The suggestions are extremely interesting; but the desired correspondences have some counterexamples; and where they hold, one could wonder how much they depend upon the particular choice of functions to measure (a) and (b); and one could hope for more of an explanatory linkage between the sets of items in question. 3. Could the unique hues come from, so to speak, the external world? White and black can easily be defined as particular kinds of reflectance. What of the standard four unique hues? Variation in kinds of sunlight and skylight coincides well with variation along a line from unique yellow to unique blue (cf. Shepard 1992, Mollon 2006). If we wanted something to calibrate our standards for unique yellow and blue as the lens of the eye changes with age, and despite interpersonal cone differences, this would be a good basis—and there are several ways this can be extended to surface colours. But is there any essential connection between these things: is there any rationale why the light of the sun and the sky should be counted as unique hued? An answer may be: because in our environment, these illuminants are as close to white (or the natural illuminant colour) as you can get—to see things tinged with sunlight or skylight should be to see them minimally tinged with any alien colour. Whereas other hues in an illuminant would be treated as tinging with a more alien colour the thing seen.

There is something special about red, green, yellow and blue: people find that there is a particular variety of each of these colours that seems in a sense unmixed. There are no purples that do not look to have some red and some blue in them, no turquoises that do not in some way seem bluish and also greenish; but there are reds that don’t look in any way bluish or yellowish and yellows that seem to contain no hint of either red or green. These are unique red (uR) and unique yellow (uY)—and in a similar way, there seems to be a unique green and a unique blue. And it seems to be true that, with only a limited amount of instruction and discomfort, people can work with a vocabulary of just these four colours, together with ‘white’ and ‘black’ (or notions like saturation and lightness or brightness), to describe the full range of colours presented in experimental situations: talking of some spectral colour, for example, as 20% Green and 80%Yellow or 42% Blue and 58% Red. (See e.g. Hering 1878; and, on colour naming with ‘red’, ‘green’, ‘yellow’ and ‘blue’ (RGYB) as hue terms, Boynton 1979, 210–11, and, e.g., Abramov and Gordon 2005. For the Swedish Natural Colour System using the terms RGYB and ‘white’ and ‘black’, see, e.g., Hunt 1987 §4.7, 1998 §7.7.) There is a significant amount of inter-personal variation in the placing of the unique hues, especially with green (see below, and e.g. Kuehni 2004) and there are questions about how culturally widespread or well-characterized the phenomena are (e.g. Saunders and van Brakel 1997; Jameson and D’Andrade 1997); but there are striking phenomena here to be explained (for another of them, see Kuehni et al. 2010), however they should be characterized in detail (cf. Broackes 1997). And yet, in a physical spectrum there is continuous variation in hue, and there seems nothing physically special about any particular spectral colours to single them out as more basic or ‘unmixed’ than the others, or to make them particularly suitable to use in a toolkit for describing the other colours. So where do the unique hues come from? What makes them special in the ways they are? I shall look at three suggestions: one that traces them to the physiology of the opponent processes in the retina and beyond (cf. Boynton 1979, 207–15, and many others); secondly, a recent suggestion by Wayne Wright (2011), who develops what we might call information-theoretic considerations concerning retinal response; before considering some more ‘ecological’ factors (proposed also by Shepard 1992 and Mollon 2006). There is no essential rivalry between the three kinds of approach—ecological ‘embedding’ may affect physiology (in long-term evolutionary change, medium-term development and short-term adaptation) and information-theoretic considerations may apply at all of these scales and levels. So it may be that one would ideally have explanations of the phenomenological and functional phenomena which drew on considerations of all three types. But I shall be arguing that, as a matter of fact, it is hard as yet to see firm success in the detail of the first two kinds of proposal, while the third shows the promise of being not just accidentally associated with the phenomena but also linked by some interesting rationale with them.

1 Opponent-Processing Models and the Unique Hues

Suppose we start with the kind of simplified model of opponent processing that is well-known to students of colour perception. Taking L, M and S as the outputs of long-, medium- and short-wave cones, Redness-vs-Greenness would be signalled by an (L − M) channel, Yellowness-vs-Blueness by an ((L + M) − S) channel, and an achromatic signal by (L + M) (e.g. Boynton 1979, 211–13; Hardin 1988, 34–36; cf. also Palmer 1999, 108–114).1 That there are some such mechanisms in early visual processing seems to have good experimental support in what has been reported of opponent processing in ganglion cells of the retina and in the parvo- and koniocellular layers of the lateral geniculate nucleus (De Valois et al. 1966; De Valois and De Valois 1975; Derrington et al. 1984; Dacey and Packer 2003)—and it might seem attractive to look for a basis for the unique hues in functions like these. (For some philosophers’ usage of these functions, see, e.g., Hardin 1988, 34–40, and Byrne and Hilbert 2003.) On one view, unique yellow would occur where the (L − M) channel was in equilibrium (signally neither red nor green) and the ((L + M) − S) channel was positive (or even, one might hope, at a maximum); we would have unique green where the ((L + M) − S) channel was in balance (signalling neither yellow nor blue) and the (L − M) channel was negative (or again, perhaps, at a maximum); and so on.

Figure 1 shows how badly this proposal actually would fit with the known layout of regions of red, green, yellow and blue in the spectrum as identified by normal observers. Functions like this may be good models of the processing in early stages of the visual system; but it would be a mistake to look to them for the structure of perceived colours, and in particular the unique hues.
Fig. 1

Illustrative examples of functions to describe opponent processes in early visual processing: (L − M) and ((L + M) − S), and some variant functions with different ‘weighting’ of the inputs; they should be compared with (at bottom of the Figure) representative examples of unique hues actually reported. Note how the functions (and the actual processing in the retina or LGN which corresponds to them) stand in no direct correspondence with the structure of experienced colours. If one supposed that opponent processes like these really yielded the unique hues, then unique yellow ‘ought’ to occur at ~550 nm (which is actually a yellowish green) and unique green at 484 nm (which is close to unique blue!); and there would be no unique blue in the spectrum at all, since the (L − M) function doesn’t cross the horizontal axis a second time. (All spectral blues ‘should’ therefore be greenish, and unique blue be at best asymptotically approached toward the short-wave end of the spectrum.) Note how, by varying the ‘weights’ on the inputs, one can generate large changes in the maxima and minima of the resulting opponent function (marked with the λmax and λmin in nm), and the zero-crossing (if any); but with an (L − kM) function one will not get a function that crosses the horizontal axis twice. Labels “Red”, “Green”, etc., are given as a reminder of what, e.g., a (L − kM) channel on a simple interpretation ‘ought’ to be signalling: part of the point of the diagram is to show how unsuccessful such an interpretation is. The (L − M) function, for example, simply divides the spectrum into one zone (below 550 nm) and another (above); whereas a realistic function would need to count the middle region (from uB~476 to uY~576) as greenish, and the light of wavelengths both below and above the region as reddish. (Graphs: my calculation using L, M and S hypothesized cone response-functions, i.e. König fundamentals, derived from CIE X, Y, and Z functions: Wyszecki and Stiles 1967/1982, 607. More accurate functions have been devised by Stockman and Sharpe (1999), but the König fundamentals fit precisely with the Jameson & Hurvich functions which we shall soon discuss; and the general theoretical points emerge in almost exactly the same way whichever set of fundamentals we use. Unique hues: Jordan and Mollon 1995; Webster et al. 2000a, b; Webster et al. 2002; and other investigations reported in Kuehni 2004 and Philipona and O’Regan 2006. Bars (at bottom of Figure) give an idea of range of reported interpersonal variation in placing of unique hues; bold filled circles (accompanied by a figure for wavelength in nm) indicate representative means or focal points: in the case of green, two points are marked corresponding to the wide differences between focal figures reported in different studies (512 nm in Jordan and Mollon 1995; and ~540 nm in Webster et al. 2002): some find it plausible that unique green is ‘bimodal’ (e.g. Volbrecht et al. 1997), others find no evidence of this (Mollon and Jordan 1997). Most researchers place red either near the end of the spectrum or as an extraspectral colour (e.g. complementary to ~495 nm); but Webster et al., rather exceptionally, report 605 nm, using computer-printed hue palettes. I am aiming here only to give an impression of the range of reports, not to adjudicate in any way on the disagreements among them.)

The trouble is that the proposed functions do not yield unique hues in the places where they actually occur (as inspection of the figure reveals: related points have been made by Abramov and Gordon 1994; Mollon and Jordan 1997; Mollon 2006; MacLeod 2010 §6.2, and others)—or even yield the right structure: if we took functions like these as our guide, for example, all the blues in the spectrum would be greenish, and there would be no unique blue, let alone any violet.2

If we consider the actual operation of opponent cells in the retina or early stages of visual processing, it may be more plausible to focus not on arithmetical differences, but ratios of inputs: e.g., not (L − M) but (L / M). This, one might notice, quickly removes one kind of indeterminacy: if we choose functions of the form (L / kM), then their maxima and minima will occur at the same wavelength, regardless of variation in the choice of k. But of course there are other ways in which plausible functions may vary, illustrated below (Figs. 2 and 3): and maxima and minima—and even the general shape of a graph—will change with small variation in the construction of the function. (I mention this here because it will be relevant to the question, in considering Wright’s and Thornton’s ideas, of how impressed we should be if we find some one particular difference or quotient formula which turns out to fit nicely with the structure and position of the unique hues—given that there are many other related functions that turn out not to do so. If there were some special evidence that the ‘successful’ formula was actually realized in our visual system, it might have a special claim on us; but if it is merely one among many, any of which might be, and quite some variety of which in fact probably are in some approximate way physiologically realized, then it is not clear how strong its claim can be as a definite explanation.) And it should be obvious that the (L / M) function and its kin are nowhere close to capturing redness-vs-greenness: if we took (L / M) to do so, we would have a maximum signal for ‘green’ indeed at 461 nm—which is actually a somewhat reddish blue. (Cf. Mollon and Jordan 1997, 382, referring to Mollon and Estévez 1988; MacLeod 2010, 160.) And (though the graph is cut off at −6.0 loge units) the function supposedly indicating ‘Yellow’ continues to decrease (i.e. supposedly signalling increasing Yellowness) all the way to the redmost end of the spectrum.3
Fig. 2

Illustrative examples of functions that may be at work early visual processing, of the form Lk / M and Sk / (L + M). (My calculation, using, as L and M, hypothesized König fundamentals as defined in Wyszecki and Stiles 1967/1982, 607.)
Fig. 3

Illustrative examples of functions that may be at work in early visual processing: Logarithms (to base e) of functions in Fig. 2. Wavelengths at which a function reaches a maximum or minimum (λmax or λmin) are marked with figures in nm. Note again (as in Fig. 1) how changes in the constant k move these maxima and minima (and larger changes in k have much larger effects than are shown here). Note how little of the structure of unique hues is evident in such functions—indeed between about 465 and 665 nm (which ranges from violet-blue, through B, G, Y to orange-red, and contains three unique hues), these YB functions are all monotonically decreasing and the RG functions all monotonically increasing. It may be interesting that two of the RG functions have a significant rise at short wavelengths: those who want an RG channel to have an additional input from S- as well as L- and M-cones in order to signal the redness in violet part of the spectrum might note that log difference functions like these can achieve the same effect. With the Stockman and Sharpe 1999 functions, there are differences in the numerical values (and the lower part of the Sk / (L + M) graphs is smooth even at values well below −4 loge units), but the basic structure of the graphs is much the same; the λmin of (L / M) occurs at ~461.4 nm (compared with ~461 nm on the König fundamentals of Wyszecki & Stiles). I have inserted labels to mark the colours that one might at first sight hope such functions as these to be indicating; it should be obvious on inspection how poorly they would do so. (My calculations on basis of König fundamentals defined as in Wyszecki and Stiles 1967/1982, 607.)

What about the kinds of function that Jameson and Hurvich used (as in Fig. 4 below), which were supposed to capture the actual structure of our colour space? (See also Hurvich and Jameson 1957, 389 and 392.) Do these yield any better chance of explaining the unique hues?
Fig. 4

‘Chromatic response functions’ for the standard CIE observer (drawn from functions specified in Hurvich and Jameson 1955, 602; see also 603). Hurvich and Jameson stated the functions in terms of the CIE 1931 XYZ functions, proposing 1.0 X − 1.0 Y and 0.4 Y − 0.4 Z respectively as their RG and YB functions. If we combine that with the definitions of the König fundamentals L, M and S in terms of the X, Y and Z functions, then we can derive the stated specification of the Hurvich & Jameson RG and YB functions in terms of L, M and S (Wyszecki and Stiles 1967/1982, 604–608; see also 457–8, 643–4.)

I shall make three points, first, about what this diagram really says—and how little we can hope for from it in the way of an explanation of the unique hues; secondly, about how, read carefully, it actually contains some implausible or anomalous features; thirdly, about how one could adjust the diagram if one wanted—but how little this would help with the kind of explanation we are hoping for.

First, then, is the point that the facts reported with these functions (whether strictly accurately or not) are entirely psychophysical: Jameson and Hurvich and many others talk of these functions as specifying the ‘chromatic response’ of the visual system to any particular spectral stimulus; but what they actually describe is, more exactly, the amounts of light needed in a ‘hue-cancellation’ experiment to ‘cancel’ the apparent redness, greenness, yellowness or blueness of a given light. The procedure is illustrated below (Fig. 5).
Fig. 5

CIE 1931 x, y chromaticity diagram to illustrate Jameson & Hurvich’s ‘hue-cancellation’ procedure. Left: We may cancel the ‘yellowness’ in, e.g., the six lights YG525, YG550, Y575, YR600, YR650, and (y)R700 by mixing each with a suitable quantity of B477: the chromaticity will shift in the direction of B477 (along a dotted line), and the resulting mixture (if the proportions are chosen properly) will lie on the straight line (bold) that runs from Jameson & Hurvich’s ‘unique green’ through white (marked Wh) towards what they must take as unique red (on the ‘purple line’ running from B400 to R700, at approximately the point, here marked uR). According to Jameson & Hurvich’s YB function (as in Fig. 4), the amounts of B477 needed to cancel the yellowness in one unit of light at each of those six wavelengths, should be in the ratio of approximately 0.294, 0.395, 0.365, 0.252, 0.043 and 0.0016. I have confirmed the result theoretically, by calculating CIE X, Y, Z tristimulus values for the resulting mixtures and mapping the x, y chromaticity values. Mixtures need to be in the right absolute quantities (so the mixtures do not move too far along the dotted line, or not far enough): it turns out that for 1 unit of the target light of wavelength λ, one needs 3.07 YBHJ(λ) units of B477 to ‘cancel’ the yellowness in the way illustrated here (where YBHJ(λ) is the value of the YBHJ function at wavelength λ). The effect of mixing just 1.0 YBHJ(λ) units of B477 is to ‘cancel’ just part of the yellowness in the stimulus—as shown by the marker a little less that half-way along the dotted line away from the position of the original stimulus. Right: Similar procedure, modelling the cancellation of the ‘redness’ in, for example, YR600, YR650, R700, B475, RB450 and RB400 by mixing with each a suitable quantity of G498, in accordance with Jameson & Hurvich’s RG function. In this case what is illustrated is the calculated result of mixing with 1 unit of the target light with 3.5 RGHJ(λ) units of G498: the resulting mixtures do indeed lie on a straight line running between unique blue477 and unique yellow578. (Mixing merely 1.0 RGHJ(λ) units of G498 results in a smaller amount of ‘cancelling’, as illustrated by the marker a shorter distance along the dotted line.) Similar diagrams can be drawn to illustrate ‘cancellation’ of blueness with Y578 and of greenness with R700. (For colorimetric methods, see e.g. Hunt 1987/1998, chs. 2 and 3.)

It should be obvious that the functions are not descriptions of any physiological process. (Incidentally, think how ambitious it would be to claim physiological evidence of channels in the visual system fitting at all closely such functions as 0.3410L + 0.0615M − 0.7130S and 1.6645L − 2.2301M − 0.3676S.) A fortiori they cannot serve as physiological explanations of the chosen positions of the unique hues. One sometimes hears people explaining a diagram like this, saying: ‘At 477 nm the RG opponent system is in equilibrium, and the Blue-Yellow system is firing strongly on the blue side: hence you get unique blue.’4 This has the air of being an explanation—and so it would be, if we had evidence of a Blue-Yellow physiological system of the kind described. But the physiological systems actually found are very various—there are some that approximate to the (L − M) and (S − (L + M)), or (L / M) and (S / (L + M)) functions we investigated earlier; but, as we have seen, those do not closely correspond to the structure of the unique hues; and functions like Hurvich’s and Jameson’s have not been found realized by physiologists. (See also Mollon and Jordan 1997, 382–3.) To the extent that graphs like this are accurate, they report or systematize performance at psychophysical hue-cancellation tasks, without explaining how or why it has the character it does.

But are they accurate? It should be noticed that the Hurvich & Jameson functions imply unique green at an unusually low wavelength: 498 nm, instead of the more usual reports in the 510–540 nm range. (We are told that Observer H and Observer J in Hurvich’s and Jameson’s 1955 studies did indeed choose 490 nm and 500 nm respectively as their unique green (Jameson and Hurvich 1955, 549). But those are exceptional figures: 490 nm is outside the full range of variation found among the 100 subjects in Volbrecht et al. 1997, the 51 subjects in Webster et al. 2000b, the 175 subjects in the printed surface colour experiments of Webster et al. 2002, and the 105 subjects in the monitor experiment of Webster et al. 2002—though it is just within the range recorded for the 97 subjects of Jordan and Mollon (1997) (which runs from 487 to 557 nm). I wonder if Hurvich and Jameson may have been influenced by the convenience that would follow from having a figure for unique green that made it approximately complementary to a plausible unique red: G548 and even G512 are clearly (by definition!) not complementary to the kinds of red that actually get chosen as unique—such as 495c (i.e. a blue-red mixture complementary to 495 nm), or even R700.) In any case, to get a hue-cancellation graph that was decently applicable to the population at large—rather than the special observers who contributed to the original experiment—, one would need the zero-crossing of the YB function to occur in something in the region of 512 or 540 nm.

Could one easily modify the Hurvich & Jameson functions? With a little adjustment of the coefficients in a function of the form L + k1M + k2S one can certainly move the zero-crossing of the YB function from 498 nm to the right (as one can see from the graphs in Fig. 1); but the obvious way to do so—while maintaining a peak (for ‘yellow’) in the right-hand part of the graph and a trough (for ‘blue’) in the left-hand part—is to reduce the coefficient on M, while making some compensatory increase in the coefficient on L to prevent the peak for ‘yellow’ from declining too much. 0.9L − 0.6488M − 0.7130S would give us a zero-crossing at 512 nm and 0.9L − 0.8366M − 0.7130S would give us a zero-crossing at 540 nm. (The required change in the second case from Hurvich and Jameson’s functions is huge: at 540 nm, the Hurvich and Jameson YB function is close to its maximum, and instead we want it to be at zero!) We should note that both of these functions are now of the form ((L − k1M) − k2S) rather than ((L + k1M) − k2S) (where k1, k2 > 0)—so the M-cones are now pulling against the L-cones rather than with them—, which has not been widely championed before. (And this would clearly rule out much chance for such a channel to be an expected inheritance from a phylogenetically earlier dichromat system that the evidence suggests our ancestors had (see e.g. Regan et al. 2001) before the differentiation of L and M cones.) So, one can cook up linear functions of L, M and S which at least have zero-crossings in the places desired. But I have no evidence that any such functions are in fact realized in the human visual system—and the shape of them (I shall save the illustration for another occasion) would anyway fail to fit with other aspects of the structure of colour space. At this point one should ask two questions: whether too much emphasis has been placed upon linear functions, as opposed to other easily enough devisable functions of L, M and S; and secondly, whether there is any imperative for there to be processes in the brain that map to the structure of the hues with anything like the neat correspondence that is being sought when people look for functions like these, physiologically realized in the brain. I must leave those questions for another occasion. But it should be clear that opponent functioning, either as now found physiologically, or as mathematically invented with simple difference or quotient functions, does not seem capable of offering us an explanation of the position and character of the unique hues.

2 Wayne Wright on the Unique Hues

Wright (2011) has two main proposals, which I will take up in the two graphs that follow. One is to derive the boundaries between the various hue-category regions from the ‘primes’ and ‘anti-primes’ that W. A. Thornton (1971) had found to be (respectively) either particularly good or particularly bad as illuminant wavelengths to use as components in fluorescent lighting. (Wright derives these ‘primes’ and ‘anti-primes’ themselves from the points in the spectrum which, we might say, have a particularly large or particularly small impact on the visual system, or (more precisely) which are maxima and minima of the function Norm(e1(λ), e2(λ), e3(λ)), to be explained in a moment.). The e1, e2, e3 functions are designed on the principles of Jozef Cohen (2001) as a set of orthogonal basis functions that can be taken as components of the L, M and S cone-response functions: they are illustrated in Fig. 6. They can be thought of as giving us units for a system of description that maximally disentangles information that is highly entangled if stated in terms of the L, M and S functions. For example, the L and M functions have much the same shape and they peak at a separation of less than 30 nm. Hence, if we suppose we already have an L signal (on the subject of a range of stimuli in the world), then having in addition an M signal provides additional information, but in a form that entangles it closely with our existing information. On the other hand, having (along with, e.g. an L or M or (L + M) signal) a difference signal of the form (k1L − k2M), we will have a much more sensitive indicator of wavelength changes in the regions where the L and M functions are close together. And that is just the kind of thing that the e3 function gives us in its right-hand half, above about 530 nm: the e3 function is (given that the S-function is almost zero in this region) almost equivalent to −0.36676 L + 0.573022 M. The same information can be stated in terms of L, M, S and e1, e2, e3 units; but in an environment where there is noise or a tendency to inaccuracy in information-transfer (and where limits on the range of real-world stimuli mean much of the LMS space is never in practice used at all), a statement in the latter terms will in a sense make better use of limited bandwidth. (For further details, see Wright 2011 §2 and Cohen 2001.) One unit of, e.g., 500 nm light will have L, M and S values of (0.2890, 0.4278, 0.1228); it will have e1, e2, e3 values of (−0.0687, 0.0733, 0.1140); and the norm of the latter set of values can be defined as the Euclidean distance of that point in e1, e2, e3 space from the origin (0, 0, 0), i.e. \( {\sqrt {{\left( {{\left( { - 0.0687} \right)}^{2} + {\left( {0.0733} \right)}^{2} + {\left( {0.1140} \right)}^{2} } \right)}} } \), i.e. 0.1520, which can also be written as || –0.0687, 0.0733, 0.1140 ||. For different wavelengths λ (and derivatively, by summation or integration, for any combination of different wavelengths of light coming from any visible object) there will be values of e1(λ), e2(λ), e3(λ) and, in turn, the norm, || e1(λ), e2(λ), e3(λ) ||. And this norm can be thought of (though one might ask precisely how accurately) as a measure of what we might call the total amount of visual response to the stimulus in question. One reason for doubt is that, of course, there is also a perfectly well-defined measure Norm(L(λ), M(λ), S(λ)), which has, however, a rather different graph from that of Norm(e1(λ), e2(λ), e3(λ)); and there will also be the norms of many other three-dimensional transformations of LMS values: and the question therefore arises, which, if any, of these sets of functions ‘really’ models the actual activity of the visual system, and which norm, if any, of them might be taken as ‘really’ the proper measure of ‘total visual response’, and in which way.
Fig. 6

Central elements of Wright’s explanation of the unique hues. Approximately: there are prime and antiprime colours (Thornton 1971’s best and worst components in a 3-component illuminant), marked at the top of the diagram; these primes and antiprimes can be explained as coinciding with the maxima or minima of the function ||(e1(λ), e2(λ), e3(λ))|| immediately below, which can be taken as a measure of something like total amount of visual impact of light of wavelength λ—that measure, in turn, being defined as the norm of values of the three functions e1(λ), e2(λ), and e3(λ), lower down in the diagram—which themselves are a special kind of set of maximally independent basis functions spanning the space of the cone-response functions L, M and S given (in reduced form) in the middle of the graph. On Wright’s view, the layout of primes and antiprimes, thus explained, is in turn an explanation of the layout of unique hues (illustrated at the bottom of the diagram): the primes and antiprimes give the boundaries of the main hue-categories. See main text for details. For comparison with Wright’s e1 function, the CIE function Y(λ) (equivalent to V(λ)), as a rough measure of luminance, is also illustrated (reflected in the x-axis and reduced to −0.2 Y(λ)). (L, M, S cone-sensitivity functions: Stockman and Sharpe 1999 (and <>. I am grateful to Wayne Wright for letting me know his exact definition of the e1, e2 and e3 functions, defined in terms of the Stockman & Sharpe cone-fundamentals. CIE Y(λ) function: e.g. Wyszecki and Stiles 1967/1982, 725–735.)

Wright’s second main idea is that the key wavelengths of ‘focal’ or ‘unique’ examples within a colour category are those for which the ratio of chromatic to achromatic response (defined in a particular way) is at a maximum or a minimum. I shall take the two main ideas in turn.

Wright’s account of the unique hues takes off from some remarkable general claims in William Thornton. Thornton’s ideas were initially made in relation to what might seem a specialized question—which wavelengths of light are best or worst as components in fluorescent lighting—but they are in fact of much more general significance. There are two sometimes conflicting desiderata in a fluorescent lamp: good colour-rendering and high luminance. Aiming at a fair balance between them—good colour-rendering at not much less than maximum attainable luminance—, Thornton found that, for a three-component fluorescent tube, 490 nm and 570 nm were particularly bad components, and 450, 540 and 610 nm particularly good. To explain this, Thornton effectively developed (1) the obvious enough point that different wavelengths of light can differ in what we might call general visual efficacy (the eye is diminishingly sensitive to the extremes of the spectrum, for example, and differingly sensitive to wavelengths in the middle); and (2) the more unfamiliar point that different wavelengths of light may have differing degrees of what we might call colour distinctiveness. (Yellow around 575 nm is good on general visual efficacy, and bad on colour distinctiveness: it is relatively bright, but (according to Fig. 9 in Thornton (1971)) one watt of power at 575 nm added to a white light will bring about much less of a chromaticity shift than, e.g., one watt at 610 nm.) Thornton (1971, 1160–61) comes to many of the same ideas from several different directions, finding more or less the same set of primes and antiprimes showing up again in different areas. He defines a function R (x + y + z) (‘the vector distance from the white point to the periphery of the color diagram’ (1971, 116))—which could be taken as a measure of what we might call general colour-effectiveness. (It cannot, I think, be quite the same as either the general visual efficacy or colour distinctiveness that I mentioned earlier. The function is similar in general form to the Norm (e1, e2, e3) function in Wright’s Fig. 6) and has similar λmax and λmin, which correspond well with positions of the primes and antiprimes. And this lends support to the idea that the primes are the colours that occur at wavelengths that are maxima of general colour-efficacy, and antiprimes are colours that occur at minima.

To explain his ‘primes’ at about 450, 540 and 610 nm, Thornton (1971, 1161) also examines difference functions, defined in terms of the CIE XYZ functions, which have some attraction: he introduces Z − X − Y (or strictly \( \overline z \left( \lambda \right) - \overline x \left( \lambda \right) - \overline y \left( \lambda \right) \)) as, approximately (I omit some details), a measure of something like blueness or ‘bluminosity’ of a stimulus, Y − X − Z (for greenness or ‘verdinosity’), and X − Y − Z (for redness or ‘rubinosity’). And it turns out that these difference functions peak at 446, 534 and 612 nm respectively (by my calculation: Thornton’s Fig. 11 illustrates the general results)—and those maxima correspond nicely to the experimentally discovered wavelengths of ‘prime’ illuminant components, at 450, 540 and 610 nm. Thornton takes these functions as indicators of the colour distinctiveness of stimuli: so in a way, his explanation of the successfulness of the prime wavelengths is that they have (more than other wavelengths) very distinctive effects upon the visual system.

Thornton’s explanation in terms of his XYZ difference functions seems vulnerable: if you change the definition of functions like these a little (changing the coefficients on the component functions L, M and S) then the peaks and zero-crossings of the difference formulae can change very substantially—and they will no longer neatly fit with the positions of the experimentally-identified primes.5 (Fig. 1 especially illustrates the way such differences can make big differences to the graphs; see also Figs. 2 and 3.) If there were some particular physiological or functional rationale to the exact coefficients in the definitions of the CIE X, Y and Z functions, then perhaps one would have a secure explanation of the primes, in terms of that rationale; but in the absence of such a rationale, one must ask whether the agreement is some kind of coincidence. Similarly, in Wright’s account, one must ask whether it was something of an accident that the norm of Wright’s three functions of chromatic response e1, e2, and e3 is—to the extent it is—a good indicator of the breadth and limits of the various unique hue categories.

How good are the correspondences that Thornton and Wright offer us? We may distinguish (A) the wavelengths that are particularly good or bad in a three-component fluorescent lamp; (B) the peaks and troughs of Wright’s Norm (e1, e2, e3) function; and (C) the border regions of the hue categories. The correspondences between (B) and (C) may seem good (though I shall examine them further below). But there are interesting discrepancies between (A) and (B). What Wright calls his ‘primes’ are not quite the same as Thornton’s primes in (1971). (The reported wavelengths do indeed, as Wright says (2011, 8), ‘vary slightly’ across Thornton’s publications: more on this later.) Wright’s prime in the 440–445 nm region indeed corresponds well to a maximum of the Norm (e1, e2, e3) function. (It was chosen as such.6) But what Thornton actually states as being a prime in this region is not in the 440–445 nm region nm but 450 nm (Thornton 1971, 1155 (Abstract), 1156–1157, etc.)—which is noticeably to the right of the maximum of the norm function. And, much more importantly, it seems—if you look at Thornton’s original experimental data, especially Figs. 2, 3 and 4 in his (1971)—that 460 nm would be an even better ‘prime’ than 440–445 nm or 450 nm on the original experimental grounds: as a component of fluorescent illumination, 460 nm seems (in Fig. 2) to have a higher Color Rendering Index (CRI) than nearby wavelengths, while (in Fig. 3) it is hardly if at all lower in luminosity. And in the graph that links the two dimensions of luminosity and CRI together (Fig. 4), it is clearly 460 nm that is in the position of being either closest or very close to closest to the top left of the diagram—which is Thornton’s rough criterion for overall excellence as an illuminant here. But 460 nm is clearly nowhere near a maximum or minimum of the Norm function. Of course Thornton’s original experimental data might themselves be put into question. But at least prima facie, it seems we have a clear counter-example to the linkage between (A) and (B).

There is a second counter-example. Thornton 1971 has an antiprime at 575 nm (rather than 570 nm, which Wright prefers): but 575 nm is certainly not (as Wright would like) a colour-region border (presumably between Y and G). On the contrary, 575 nm either itself is, or is very close to being, the position of unique yellow itself. So one might well wonder whether the link between (A) Thompson’s primes and (C) the layout of the unique hues was a secure one.

One might say, however, that these are relatively minor worries: perhaps Wright could abandon any connection with Thompson’s original list of primes, and merely stick to the connection between (B) and (C). So let us examine the alleged correspondence there.

Let us start by looking at the correspondences between the peaks and troughs of the norm function and the boundaries of the hue categories. Let us take the peaks first, at 605, 535 and 440 nm. The first is certainly in good correspondence with what we might take as a boundary between the yellow and red categories, the wavelength taken to be 50%–50% yellow-red. The second peak, however, presents difficulties: 535 nm, at least initially, looks to be not a boundary, but something close to a focal point for a unique hue, namely green! (Unique Green, as we have seen above, has been placed at 512 and at ~540 nm.) Wright has a special account of green—which presents it as something like the combination of two sub-categories, green and green-yellow. But still it is odd: if there is some, so to speak, general pressure for primes to define boundary regions (e.g. on the ground that they are points at which rate of change of hue with wavelength becomes zero—whereas it seems that near the focal point of a hue-category, people are specially sensitive and ‘choosy’ about colour-differences (more on this below)), then it would seem at least odd that 540 nm could end up being (even if only as a result of a combining of categories) both a natural boundary wavelength and a (newly-adopted) focal point. There is much of interest in Wright’s suggestions about green; but if they are adopted, it is still not clear that they solve our puzzlement with this category. As for the third peak: 440 nm, we are told, marks a boundary between blue and purple. But is it clear that there is a boundary—or even a border region—at all between blue and purple in the same way as there is, say, between blue and green, or green and yellow? On the journey from unique blue to unique green, one may talk of a hue that seems 50% blue, 50% green—and we may treat this as a boundary between those categories. But on the journey from blue to purple, does there come a time where the hue seems 50% blue 50% purple? One might use that form of description, but that is not quite the same thing as describing the blue-content of the new hue as 50%. Purple itself can be described as having a blue content (perhaps, for focal purple 50% blue 50% red). But in that case, could a hue that was 50% blue 50% purple also be described as 75% blue 25% red? If so, why should that hue be counted specially as a category boundary—and be specially linked with a region where the rate of change of colour-response with wavelength was low? (The matter is highly contested, but I also have my doubts about treating purple as a unique hue in the way that red, green, yellow and blue are—since there is no purple that does not seem to contain other hues (namely, blue and red). And while one can certainly talk of seeing a purple tinge in something, it is not clear that one talks of the amounts of purple contained in a hue in the same way as one does with, say, blue or red: the violet part of the spectrum seems tinged with red, I would say, rather than tinged with purple.) Usually—if we try to think through the present proposal—as we approach a boundary (a dotted line in my graph, identified by a prime or antiprime) the hue that has been reducing approaches a contribution level of 50%; but in this case, blue would only be able to reduce to, say, something like 75%. So one wonders in what way exactly the theory can treat the primes as all marking a boundary in hue space—the present case is one where the boundary would at best be of quite a different kind from what is elsewhere required.

So, of the three primes that are supposed to correspond to hue-category boundaries, 610 nm seems fine; 535 nm (or 540 nm) is clearly an anomaly—it is more of a focal point than a boundary—; and as for 440 nm, it will not be a 50%–50% boundary where the blue-content reduces to 50%.

It is not clear, therefore, that the correspondences that Wright wants are actually very securely found. If they are, then a further question arises: what significance they may have. A correspondence between a mathematical function and the division of colour space into hue categories is not in itself an explanation of the latter if the mathematical function is merely accidentally in agreement with it. What we would like is, first, some kind of rationale, explaining how it was that the organization of the primes should connect with the organization of hues. It was indeed in this spirit, that I suggested (when I saw an earlier version of this paper) that there might be a line of connection: it might go, I thought, via the idea that a zero rate of change of hue with wavelength was just the kind of thing that we should expect towards the boundary of a category, rather than in the centre. So it might be expected that peaks and troughs of something like the norm function should correspond to category boundaries. At the edges of categories, one might expect lower than normal sensitivity to small difference in wavelength.

But I have to admit I am not sure how well my own suggestion stands up to further examination. We are not ordinarily aware of change of wavelength as such, so how would an organism know—or how would it even be significantly affected—if the rate of change of hue with wavelength in the region of a particular wavelength was or was not slow? A connection with something we have control over (e.g. in the processes of colour mixing) could very well be important to us; but it is hard to see how a connection with something we cannot control (or, at least, without special help from modern physics cannot control), and which we are not aware of as such, could be of any relevant significance.

If the basis of the segmentation of the hue categories lies in the norm of the three e-functions, then it is important to know a little more about these e-functions. Might the correspondence of the Norm(e1, e2, e3) function with the unique hues, such as it is, be an accidental artefact of the character of these functions e1, e2, and e3? If we had used slightly different functions, would we have got a significantly different outcome? One just might answer yes in a way to the last question, and still say there was something specially appropriate about using the e-functions rather than others that were subtly different; but given that there is no claim that the e-functions are actually realized in the retina or elsewhere in the visual system, one wonders, so to speak, why the visual system would, so to speak, care about the e-functions and their norm. An answer might be available, in terms of their being a measure of certain magnitudes (including e.g., ‘general colour-efficacy’) that were themselves specially salient in the visual system; but then something special would, I think, have to be done to establish this.

Lastly, it is perhaps worth mentioning that there is a danger that the explanation of the prime and antiprimes in terms of the rather specially defined e1, e2, e3 functions may be undercut. In Thornton’s 1999 article he comes at more or less the same set of three prime colours once again—at 450, 533 and 611 nm. The prime qualification for being a prime colour is not (as in 1971) maximal efficiency as a component in a three-component fluorescent light but now maximal efficiency as a primary to use in colorimetric colour-matching. It turns out, for example, that if one chooses a higher wavelength like R645 as one’s red primary (along with reasonable choices for the other two)—as Stiles & Burch did in the experiments that fed into the definition of the CIE 1964 10°Standard Observer—, then it takes relatively large quantities of it to match other wavelengths in the orange part of the spectrum: three watts of this R645 are needed, for example, (along with other appropriate components) to match one watt of R600; and similarly, if one takes a lower wavelength like YG565 as the long-wave primary (Thornton 1999, Fig. 26), it too will be a relatively weak stimulus (and three watts or more of it are needed in mixtures to match just one-watt stimuli in the 595–620 nm region). 611 nm turns out (according to Thornton’s own empirical studies) to be a maximally efficient red primary (to use in combination with 533 and 450 nm)—while, on the 1964 CIE data, 600 nm would be best (along with 538 and 446 nm). Thornton’s explanation for this is in the 1999 article of remarkable simplicity: it is not that special colour-difference and vector-distance formulae have maxima at those prime wavelengths; nor (as Wright proposes) that maxima and minima of the norm of the specially-devised e1, e2, e3 functions occur at those points. Thornton’s explanation is simply that the ‘true spectral sensitivities of the normal human visual system’ (Thornton 1999, 153) have maxima at 450, 533 and 611 nm. (Thornton proposes his sensitivity functions in his Fig. 27, and derives the antiprimes at 490 and 570 nm as crossing-points of the ‘skirts’ of the three graphs.) If this is right, then linking this with Wright’s proposal, one might propose Thornton’s ‘true spectral sensitivities’ as giving us the boundaries between the unique hues regions too.

On a first understanding, one might think Thornton was proposing his new ‘spectral sensitivity’ functions as being cone-response functions. He compares them (1999, 153 and 140) with Helmholtz’s functions for the ‘excitation of the three kinds of fibers’—which would now be thought of as an early attempt to give cone-response fundamentals. As such, the new Thornton functions would be strange: to give L-cones a maximum at 611 nm would be a huge departure from received opinion. The Stockman and Sharpe functions give them a maximum at 570 nm; a lot of work has been done on the basis that the difference between the λmax for L- and for M-cones is relatively small (~27 nm on the Stockman & Sharpe functions) and it would be interestingly disruptive to conclude that in fact it was 68 nm instead. But I can see on reflection that that is not the way Thornton wishes to be understood.7 I am not sure how promising the chances are of there being a physical realization of the sensitivities that Thornton proposes elsewhere in the visual system. But if there were any such things, then Wright’s account would have a rival: to the extent that the position of the primes and anti-primes was a guide at all to the boundaries of the hue-categories, the explanation might be not in terms of e-functions and their norm, but in terms of the sensitivities of Thornton’s processes ‘deep’ in the visual system.

What would be the explanatory connection between the factors Wright mentions and the ‘uniqueness’ or primacy of the kinds of light that are chosen as ‘unique’ hues? There are certain features that make a particular hue (I mean here a point hue, like that of, say, 576 nm, not a hue category or range, like that of, e.g., 570 to 611 nm) count as not just any old yellow, but unique yellow: (a) its looking maximally unmixed, or its looking to contain no other hue (i.e. no hue other than itself), and (b) its forming one of a collection of hues (typically taken to be four in number) that can be said to be ‘in’ other hues and which together are sufficient for characterizing all hues whatever. A boundary or border colour like, perhaps, orange611 would presumably have a rather opposite character: (a′) its looking maximally mixed, and (b′) its not being the kind of hue that one thinks one finds ‘in’ other hues. (One might say in clarification of (a′): spectral hues lie in one dimension, on a line (curving round, perhaps, via the purples, to form a closed curve like a circle), and none looks to contain three hues; so looking maximally mixed can for them only mean looking to contain two hues in maximal confusion—with minimum dominance and clarity from just one of them: that is, it would need to be something like a 50%–50% mixture. And (on (b′)): if one were to claim to find, e.g., both teal (GB) and chartreuse (YG) ‘in’ a presented hue (say G540), one would presumably thereby be committed to claiming that the components of the components were in it too, i.e. green and blue and yellow and green. And perhaps we can now see why that is something that seems ruled out: for it is hard to see how we could find both yellow and blue ‘in’ at least ordinary examples of any colour--let alone find the two of them in a colour with, additionally, green.)

Let us move now to the other part of Wright’s proposal: his account of the focal points of the hue categories, which I illustrate in Fig. 7.
Fig. 7

Some candidate measures for the ratio Chromatic/Achromatic response. Wright uses what I shall call C1 (i.e. || (e2, e3) || / || e1 ||): it has minima and maxima at 479, 511 and 573 nm, which correspond well to focal points (or unique hue positions) for blue, green and yellow. But suppose we try function C2 (i.e. || (e2, e3) || / Y)—using, instead of e1, the CIE spectral luminous efficiency function V(λ) or Y(λ), which certainly has its problems but is a better measure of brightness (and, presumably, achromatic response) than e1 is: then we gain a new maximum at 420 nm (corresponding to nothing in the realm of unique hues), we lose a true minimum in the region of unique blue (though we keep a noticeable change of direction at ~480-485 nm), and there is absolutely nothing to link with unique green. (The C1 minimum at 573 nm does at least shift only negligibly to 572 nm.) Suppose we try measure C3 (i.e. || RGHJ, YBHJ || / Y), which uses the Hurvich and Jameson RG and YB functions (as in Fig. 4 above): then we have minima at 495 nm (which is much too high to have anything to do with unique blue) and 582 nm (which is too high for unique yellow); there is nothing obvious to link with unique green (though there is a very gentle maximum at 514 nm); and two new changes of direction show up at 409 and 433 nm, both of which, again, have no evident connection with unique hues. I hold no brief for any of these measures; but it is not obvious that C2 and C3 are intuitively any worse candidates than C1 for being a measure of Chromatic/Achromatic response. So the question arises, how accidental it may have been that the e functions issue in the (indeed notable) correspondence. If we were looking for a measure of chromatic/achromatic response, the function C1 would not be an obvious choice (because the e1 function is so very dissimilar from measures of luminance, especially for lights in the blue-green part of the spectrum); but it may have other recommendations under other headings. (CIE functions: Hunt 1987/1998 chs. 2–3, appendices 2, 3. For Hurvich & Jameson functions, see e.g. Wyszecki and Stiles 1967/1982, 457–8, 643–4.)

The main difficulty here is that there seem to be many alternative functions one could use as a measure of the intended magnitudes—and they do not point to the same key wavelengths as Wright would like us to find.

The e1 function is surely not (as Wright wants it to be) a good measure of Achromatic response: an established measure of luminance (the psychophysical analogue of brightness) such as the CIE Y (or V) function (or some improved analogue of it like Stockman & Sharpe’s V*(λ)) would surely be a better measure of Achromatic response.8 But suppose we started with some such function as Y(λ) or V*(λ) as our measure of achromatic response and then (for our chromatic response functions) asked, which two other functions would be maximally independent of this and of each other, then would suitable candidates issue in a measure of chromatic/achromatic response that had Wright’s desired outcome? (That is: e1, e2, e3, defined on Jozef Cohen’s principles, may have a certain information-theoretic excellence to them. But they pay a price for that excellence: none of the three functions is going to correspond closely to brightness or luminance. So I am supposing now, that we might attempt a project modified from Cohen’s, which would hope to be for certain purposes more realistic.) Now, using such new functions, would we get a ratio chromatic/achromatic response that indicated, in the way Wright wants, the position of the unique hues? Or would it look more like measure C3 (in my Fig. 7), which would be the analogue of this within Hurvich & Jameson’s system—which clearly gives us no help with the position of the unique hues? It seems quite possible that the result would not be a corroboration of the proposal Wright has made.

What is more, probing more deeply, one might ask, what would be the rationale for thinking there was any connection between uniqueness and having a specially high or a specially low ratio of chromatic to achromatic response? But this is not to speak against Wright’s proposals, it is to say that we do not know, yet, I suspect, how much truth there is in them.

3 Sun and Sky and the External World

So far we have looked, so to speak, into the eye or brain to find the unique hues. A rival suggestion would be that we should look out into the world. It would obviously be a mistake to suggest that the character of the eye has absolutely nothing to do with the groupings we make: the fact that a certain set of metameric, physically different, colours look (in a certain illuminant) the same shade of slightly bluish green is determined by the response patterns of the cones of the eye, which issue in more or less the same pattern of firing for all of the colours in that set. But if we ask, about the many confusion classes of physical stimuli each of which constitutes a distinguishable shade for a normal trichromat, which of them are the ones that count as in one sense or other ‘unique’, it could be that the answer was importantly influenced —perhaps even principally determined—by environmental or ecological factors outside the eye.

Let me mention two pieces of evidence that suggest a role for ecological facts. First, the lens of the eye changes over time; but the location of the unique hues by an individual over time does not vary similarly (Schefrin and Werner 1990; Werner and Schefrin 1993). Secondly, there are many varieties of L and M cones (with different λmax) found in the population, and there is wide variation in the ratios of numbers of L and M cones between persons (as there is also in ratios of numbers of L and M cones across different parts of one person’s retina): but the positioning of unique yellow, at least, is subject to extremely little variation in the location. (Abramov and Gordon 2005, 2151 (with refs.); on reported variation in uY, see e.g. Kuehni 2004, 160.)

Abramov & Gordon ask what the process might be that ‘tunes everyone’s uY essentially to the same spectral locus under all conditions’—but they do not give a firm answer (2005, 2151). (They consider a role for a ‘Grey World’ hypothesis—the idea that the visual system might adapt over time by using the assumption that that the world it is presented with is, on average, grey—but they register it as having problems.) But there is, I think, one important fact that may play a important role, that has been stressed by Roger Shepard (1992) and Mollon (2006). The illumination from the sun and the sky varies constantly with the height of the sun in the sky, the amount of cloud and other factors. The light of a glowing log fire, or even from a tungsten filament light-bulb is certainly more orange, and the chromaticity graph of blackbody radiation at increasingly low temperatures curves round to approach the line of spectral hues at ~600 nm and above (for radiators at ~1200 K and lower temperatures: see e.g. Judd 1952, 207). But variation in actual sunlight and skylight turns out to lie almost exactly on a line joining unique yellow to unique blue (see Fig. 8). So if one wanted a constant paradigm yellow and blue for ‘retuning’ the visual system, then the yellow of the sun and the blue of the sky would be good and constantly available.
Fig. 8

CIE x, y chromaticity diagram for representative light sources. Note that natural light (except for low temperature sources) varies mostly along something very close to a straight line between unique blue and unique yellow (~476 nm and ~576 nm). (My graph from data in Wyszecki and Stiles 1967/1982, 28 & 8–9.)

There is a host of further issues that I cannot consider here. In some cases, it seems that the range of interpersonal variation in choice of unique hues is smaller when the testing is with surfaces than with spectral lights or colours on a computer monitor. (Studies have reported, e.g., a range of variation of 6 or 11 nm for unique blue with surfaces, but as much as 37 and 55 nm with monitor and spectral colours: Kuehni 2004, 161–2.) Is there a reason why surface colours should be less subject to interpersonal variation? In some cases there may be an easy theoretical explanation. Take white and black—which are not among Hering’s four ‘basic colours’ (Grundfarben), but, which, with them, make up his six ‘basic sensations’ (Grundempfindungen) (Hering 1878, (pt. 6 [1875]) §§41–42). There may be disagreement over what to take as a perfect white light: the CIE has, for various purposes, specified standard illuminants of various colour temperatures—and some of the lights that by one standard count as white, by another will count as yellowish, and by a third as bluish. There is nothing very surprising about this: it is a feature of our natural environment that illuminants change substantially, even within the ranges of what is quite ‘normal’ variation, and, while we can say that it would definitely be abnormal to take a 60 W incandescent bulb as a paradigm of excellent white light, there is no obvious right or wrong about calling in some context D65 white and D55 slightly yellowish, or calling in another context D55 white and D65 slightly bluish—and indeed over the years of colour television the standards have changed, generally becoming slightly bluer. But there is no place for similar disagreement over what would be a perfect white surface: it would ideally be a diffuse reflector of 100% of incident light evenly throughout the visible spectrum. (Fresh snow in practice reflects only about 80%—and less at wavelengths above 550 nm. Smoked MgO, which for a long time was a standard, reflects about 97–98% of the visible spectrum, and slightly more in the green than toward the extremes. (Wyszecki and Stiles 1967/1982, 57, 63).) And, as Mollon has pointed out (2006, 306) there are ways in which—even while the illumination may vary—a white surface will reveal itself to the viewer: white surfaces are (if we set aside fluorescent surfaces) the lightest of surfaces, and (provided there is only one illuminant falling on them) there will be no variation in their chromaticity across the surface (unlike coloured surfaces, which will exhibit, because of specular reflection, a range of chromaticities lying on a line between the chromaticity of the surface and the chromaticity of the illuminant). We should hardly be surprised, therefore, that there is less variation in the ‘placing’ of pure white as a surface colour than in talk of a neutral or white light. Similarly with black: we understand black as ideally the colour of a surface that would reflect 0% of incident light throughout the visible spectrum. (In practice many blacks only reach down to about 5%, though certain cloth materials and flat black paint can reflect as little as 0.5% to 1.0% in the visible spectrum—though more in the infra-red.) And again, one can see that variations in the character of illuminants or in the exact retinal equipment of viewers should not be expected to pose much of an obstacle to the stable recognition of the colour.

But can we develop similar accounts of unique red, yellow, green and blue? Even if we have stabilizing paradigms for yellow and blue light, can they help us with surfaces? Here is a start. One might crudely take the colour of direct sunlight as fixing unique yellow for lights, and then extend that designation also to those surfaces that (in some normal illuminant) reflect light of the same chromaticity, or a desaturated version of it. And one could do something similar with blue. But there may be more sophisticated ways in which the specialness of sun- and skylight could influence our choice of unique yellow and blue for surfaces. When the illuminant changes chromatically, e.g. becoming bluer, then some objects in the scene will lose in luminance, others gain; those that gain most are those with the same colour as the illuminant, namely (in this case) blue. (Golz and MacLeod 2002; Broackes 2010 §12.6.) So it may be the case that—irrespective of many variations in individual retinal equipment—there are ways for viewers to identify unique blue in surfaces on the basis of those surfaces’ behaviour under environmentally prominent changes of illumination. A unique blue surface would be one that enjoyed (compared with other surfaces in the scene) the largest proportionate increase in luminance when the light changed from direct sunlight to skylight. A unique yellow surface would be one that enjoyed the largest proportionate increase in luminance when the light changed, conversely, from skylight to direct sunlight. It would be interesting, therefore, to make an experimental test and see whether there is indeed reduced interpersonal variation in the placing of unique yellow and blue when the testing takes place under conditions of ‘natural lighting’ that includes both sunlight and skylight: those factors might provide some stabilization for the subjects’ choices.

I do not at the moment see how to extend such an account to red and green. The suggestion of Shepard (1992, 505) will, I think, not work. He points out that the many and various kinds of natural daylight can be treated as linear combinations of just three basis functions S0, S1 and S2, adopted by the CIE in 1971—as illustrated in Fig. 9 and explained in the caption. S0 can be thought of as a neutral white, the mean of the various natural forms of daylight. Shepard’s proposal is to take the S1 function as giving us the blue–yellow axis and the S2 function as giving us the red–green axis. As it turns out, the first suggestion is pretty much accurate; but as Fig. 10 shows, the second is not.
Fig. 9

Basis functions from which CIE Illuminants matching natural daylight can be constructed. Direct Sunlight of 5000 K, with a lack of blue (registered by the negative coefficient on S1), for example, can expressed as S0 − 1.040S1 + 0.367S2; a more ordinary Overcast sky of 7000 K is S0 − 0.070S1 − 0.757S2 and the deep blue sky of 10000 K (with a positive coefficient on S1) is S0 + 1.003S1 − 0.369S2. (Technically, S0(λ) is the Mean relative spectral radiant power distribution, and S1(λ) and S2(λ) are the first two eigenvectors used in the CIE method of calculating daylight illuminants.) Shepard (1992) would like S1 to correspond to blue vs. yellow (slightly oddly specified as ‘Blue/Violet (vs. Yellow)’ (my emphasis) in Shepard’s Figure 13.2) and S2 to correspond to red (vs. green). The correspondence is very poor in the case of S2 (see Fig. 10 below). The function S2′ (λ) is my own alternative construction, defined as −0.1S1(λ) + 0.9S2(λ): it corresponds better with red vs. green, but it has its own problems (see Fig. 10 below). (My graph from values as in Wyszecki and Stiles 1967/1982, 762; or Hunt 1987, 192–95; 1998, 268–71. The functions go back to Judd et al. 1964.)
Fig. 10

CIE x, y chromaticity diagram, showing the character of the functions S1 and S2. S0 (the mean of the desired range of illuminants, close to a light of 7160 K) appears at the point of intersection of the various lines near the centre of the diagram; and the various lines indicate the results of adding or subtracting from S0 different amounts of the S1 or S2 function. The S1 function does indeed (as Shepard proposes), correspond to blue and the negation of S1 to yellow: varying the S1 component in a mixture with S0 moves S0 approximately in the direction of unique blue (~476 nm) or unique yellow (~576 nm). But the addition of positive (or negative) amounts of the S2 component turns out to move the neutral S0 in the direction of orange592 (or conversely greenish-blue487)—nowhere near the positions of unique Red and unique Green. I have defined an alternative S2’ function (explained in Fig. 9 and caption): this comes closer to pointing in the directions of unique Red and Green—but the red is too purple, and the green too bluish, and correcting either of those problems would exacerbate the other one. There is a general difficulty for anyone looking to take a single natural phenomenon to fix both unique red and unique green: there is no single function that (by being added or subtracted from a neutral) will span the space from unique Red to unique Green—since unique Red and Green are not complementaries at all. (Arrows indicate illustrative positions for unique red and green: they do not lie on a straight line.) Also marked are the chromaticities of some standard light sources, as also in Fig. 8: they show how well S0 and S1 on their own do at capturing the varying types of natural light, at least in the 5000–15000 K range, even without a third component such as S2. (Graph from my own calculations, from S-functions defined as in Fig. 9: Wyszecki and Stiles 1967/1982, 762; Hunt 1987, 192–95; 1998, 268–71.)

If we cannot fix unique red and green by a suggestion like Shepard’s, what other resources might we use? It would be nice to build on the suggestion of Mollon and Jordan (1997, 388–90), that the interpersonal variation in the placing of unique green in lights may connect with interpersonal variation in iris coloration: those who place unique green at < 510 nm tend to have lighter irises than those who place unique green at longer wavelengths. Mollon’s and Jordan’s suggested explanation is that the subjects (whether they have light or dark coloration) may fix their judgments on surface colours—coordinating their application of unique green to, e.g., leaves that are ‘neither glaucous nor yellowish’ (390); given that darkness of iris is at least often taken as an index of increased (presumably brown) pigmentation in the fundus of the eye, we must expect that those same leaf samples will then match in hue different wavelengths of monochromatic light in the differently pigmented subjects—the light-pigmented perceivers would find the leaves matched with lower wavelength green lights, while the darker-pigmented subjects would find that they matched with higher wavelength green lights. The suggestions is attractive; but it leaves open the question, how suitable choices of, for example, leaf for unique green might be environmentally or ecologically set.

Any account of the unique hues faces the demanding task of showing not just that there is a correlation between some favoured set of features (in the eye, in the brain, or in the world) and the unique hues—but also that there is a connection between them: in particular, one would hope for something in the former features that might actually contribute to the ‘uniqueness’—the status of being ‘unique’, whatever that might involve—of the colours picked out or favoured by the relevant set of features. Now, is there anything in the suggestions that I have made about yellow and blue (and, perhaps differently, black and white) that might explain their status as ‘unique’—that is, I think, (a) their seeming (to the extent they do) pure or unmixed and (b) their seeming a kind of primary element, of which (with others) the whole range of yet other colours are composed? I think there may be. If unique yellow and blue are set by the illuminant of our environment, it is hardly surprising, I think, that they should seem in some sense ‘pure’ and rather minimally coloured. The illuminant colour is in a sense something to be ‘discounted’ or ‘compensated for’ (as well as, rather differently, ‘taken account of’) in our perception. If the colour of sunlight has to be in some sense regularly taken as a kind of neutral, despite its being also yellowish, then it perhaps is not surprising if that particular kind of yellow comes to be taken as, not nothing, but only a minimal darkening or variation from white. And similarly with blue.

That might seem to leave a problem with the other part of task: to show the connection of these colours with feature (b). If the colour is, so to speak, ‘neutralized’, then would it be an element in terms of which to describe the colouration in more complex ways of other things? A series of, so to speak very neutral colours, would look a poor group out of which to make very non-neutral darkenings. The solution is, I think, to recognize that highly saturated yellows may be much more yellow than direct sunlight, while having the same hue. They may have a chromaticity that differs from white or neutral in the same direction as direct sunlight does, but by a much greater extent. (The chromaticity of direct sunlight of 5500 K differs from a Planckian radiator of 7160 K by a distance of only about 0.0404 (in units of the CIE x, y chromaticity diagram), but Cadmium Yellow pigment (not exactly unique yellow, but close to it in hue) illuminated with 7160 K light differs from white under the same illuminant by more than five times the distance.) And conversely, the blue of the sky is bright but (except at very high altitudes) not very saturated; and certain plant colours—and of course paints—are much more saturated; one can identify them as of the same hue, but much more saturated. So the colours of these two illuminants give us both a kind of neutrality (in the relatively unsaturated cases) and untingedness and purity, and (in the more saturated cases of the same hue) a kind of primary characterfulness. I am talking here of course in somewhat imagistic terms, and wish I were in a position to develop the points more exactly.

What of white and black? It is entirely appropriate that white should have the character of ‘uniqueness’. White surfaces (a) look in some sense particularly pure and unmixed: they reflect incident light more or less without darkening it or adding any chromatic tinge to it. And a surface that reflects close to 100% of incident light regardless of wavelength will also be a surface that maximally shows any tincture: white paper, for example, will be excellent for showing the faintest coloration in watercolour painted on it—whereas, dark green paper, or light blue, for example, would not. Black is rather different: it actually does not have the status of (a) looking pure and unmixed—on the contrary, there is a sense in which black may be said to look maximally darkened, as if by the combination of all colours. And this impression can be said to be appropriate, in that whereas a red surface absorbs (so it may be) green light falling on it, and a blue one absorbs (so it may be) yellow light falling on it, a black surface absorbs all light whatever falling on it—apparently doing all the things that each of the other colours do individually. How do such ‘unique’ colours (b) constitute elements out of which other more complex colours are formed? Well, any surface colour will darken incident light in a manner that—at a rough level of description—places it somewhere between the darkening effects of white and of black, reflecting on average (if we as yet make no distinction between different parts of the spectrum) between 100% and 0% of incident light. What of the other four elements? Yellow and blue we have already identified, in their most saturated forms as something like ‘super-illuminant colours’: a real unique yellow is more saturated than the sun, a good unique blue more saturated than the sky. If each has only minimal complexity to it, then that would seem also to make them serve well as our elements, if we are treating other colours as composed. (There are reasons for taking the elements themselves to be uncomposed, if that is possible—as I explained above, with reference to the difficulty in taking, say, teal and chartreuse as ‘components’ of a green.) There is much here that needs further development and examination. If it is not yet the account that we need, I hope that it may still have merit as being in some sense the kind of account we would like to have: one that actually makes some connection between whatever fixes the wavelengths of the ‘unique’ hues and the fact of those hues’ actually having the status of being unique, in the rather complex sense that I have tried to unpack here.

There is a proposal that I have not mentioned so far: that of Philipona and O’Regan (2006), which derives the position of the unique hues from the interaction between the kinds of illuminant in our environment, the kinds of surface we encounter, and the cones sensitivities of the eye: the derivation is mathematically complex and I must leave it for another time. (For some discussion, see Johnson and Wright 2008 and Philipona and O’Regan 2008.) But I have a hope that the operative features of the unique hues as they show up in Philipona & O’Regan’s derivations will turn out to connect with the operative features of the hues in the more impressionistic and phenomenological account I have attempted here.


I shall later be talking of the functions used in the 1931 colorimetric system of the CIE (Commission Internationale de l’Éclairage). The functions \( \overline x \left( \lambda \right) \), \( \overline y \left( \lambda \right) \) and \( \overline z \left( \lambda \right) \) yield X, Y and Z values for a stimulus wavelength λ: they can be thought of as colour-matching functions giving the amounts of supersaturated red, green and blue lights that would be needed to match 1 unit of spectral light of wavelength λ presented in a 2° field. They are linear transformations of RGB functions that gave the amounts of real red, green and blue primaries in similar matching; for various technical reasons, the transformed functions have been found more useful than the original RGB functions relating to real primaries. For further details, see Wright 1944/1969, ch. 4; Hunt 1987 & 1998, ch. 2. There are weaknesses with the functions, some of which are corrected in the CIE 1964 colour matching functions for 10° fields, and there are now the improved cone-fundamentals of Stockman and Sharpe 1999. But the CIE 1931 system remains a standard one and useful if its limitations are recognized. The chromaticity of a stimulus may be specified by giving the proportions of X, Y, and Z needed to match it, which can be specified by x, y and z, where \( x = X/\left( {X + Y + Z} \right) \), \( y = Y/\left( {X + Y + Z} \right) \) and \( z = Z/\left( {X + Y + Z} \right) \). And since \( x + y + z = {1} \), one needs to specify only the x and y values for any given stimulus: as illustrated in the CIE x, y chromaticity diagram, e.g., as used in Figs. 5 and 8 below. For further details, see, e.g., Hunt 1987 & 1998, ch. 3.


The point is not new and should have been kept in mind even in the 1960s. Judd 1949 presents (as a development of G. E. Müller’s theory, with similarities also to Schrödinger 1925) an extraordinarily clear-sighted development of a ‘zone’ theory: the three photosensitive substances of the Young-Helmholtz theory feed into a second stage of ‘retinal sensory processes’, characterized by functions of the form a1L − a2M and b1L + b2M − b3S (1949, 4), much as in Fig. 1 above; which in turn feed into a third ‘optic-nerve fiber’ stage, characterized by functions equivalent (except for scaling factors) to those that became famous as Hurvich and Jameson’s ‘chromatic response functions’. (Judd (1949, 10) has an RG function equivalent to 1.0X − 1.0Y, and a YB function equivalent to 0.3168Y − 0.3168Z, whereas Hurvich and Jameson have 1.0X − 1.0Y and 0.4Y − 0.4Z (1955, 602, acknowledging their use of Judd 1951).) There are ideas in Judd’s paper that have not stood the test of time, such as Müller’s conception of deuteranopia as involving a failure of the RG-sense of the optic nerve, and his talk of these processes as (presumably chemical) changes in various ‘substances’ rather a matter of electrical firing rates in cells. But it is impressive work—and it is interesting that Judd has no tendency to talk of the second-stage retinal processes as relating to ‘red-green’ and ‘yellow-blue’ variation; rather he thinks of the a1L − a2M process as involving ‘yellowish red’ vs. ‘bluish green’, and the b1L + b2M − b3S process as giving ‘greenish yellow’ vs. ‘reddish blue’.


For discussion of a different kind of candidate neural basis for the unique hues, see Conway et al. 2007, Stoughton and Conway 2008, and Mollon 2009, which I am sorry not to be able to consider further here.


‘At several spectral loci, one or the other of the opponent curves crosses the zero line. This means that the corresponding opponent system is in neutral balance at that point. From left to right, the first of these … occurs where the red-green curve crosses the axis at 475 nm. At 475 nm, the yellow-blue system is negative and, hence, signals blue. So here redness, greenness, and yellowness are all zero. What the subject sees is a blue without any other chromatic component—a unique blue …’ (Hardin 1988, 37, my emphasis, except on ‘unique’.) This looks very much like a reference to an underlying physiological mechanism (‘the corresponding opponent system’) which is supposedly giving rise to the phenomenal character of the experience. (In general, Hardin believes, ‘The resemblances and differences of the colors are grounded not just in the physical properties of objects but even more in the biological makeup of the animal that perceives them.’ 1988, 7.) But graphs like these are reports of cancellation experiments (or of theoretical modelling of ‘cancellation’ on the basis of CIE 1931 colorimetric functions), not of discovered physiological processes. Hurvich & Jameson, and Hardin, of course knew perfectly well that such graphs as Fig. 2 are, in the first instance, graphs of psychophysical performance: their belief was that that psychophysical performance could be taken as an indication of corresponding physiological mechanisms that could then be cited as ‘explanations’ of the behaviour. But it was entirely speculative in their day, and it remains entirely speculative today, that there are physiological mechanisms of the kind they have in mind. It is no doubt true that character of brain and eye are, in some sense and some ways, responsible for the character of our colour experience; but that there is for each region in the visual field some particular channel representing degree of redness in line with positive values of a function defined as 1.6645L − 2.2301M − 0.3676S, is not something we are told by the physiologists, and not something we need to believe for a priori reasons about mind-brain type correspondence either. This is the basis of the general response I have to Byrne and Hilbert’s 2003 argument in favour of taking redness to be definable as (approximately) the external cause of some particular type of physiological response: they take it that, even if the (L − M) function doesn’t quite work, there will be a better function, physiologically realized, to plug into their theory instead. But it is, I think, entirely unnecessary, and quite possibly false, to assume there will be a single type of physiological response that is a measure, quite generally, of degree of redness (present when looking at a reddish yellow, a reddish blue, a pure red,. etc., and with a metric suitably corresponding to degree of experienced redness) and that it is the core of what underlies all our experiences of different shades and degrees of red. See also Broackes 2003 on Byrne and Hilbert 2003.


It might seem that functions with simple definitions like X − Y − Z at least have a simple naturalness to recommend them. But of course the X, Y and Z functions of the CIE are themselves defined functions, defined originally as linear transformations of certain colour-matching functions. Reexpressed in terms of hypothesized L, M and S fundamentals (the König fundamentals in Wyszecki & Stiles 1967/1982, 606), the simple-looking X − Y − Z, −X + Y − Z, and −X − Y + Z emerge with a little linear algebra as equivalent to: (1.6645L − 2.2301M − 1.4149S), (−1.6645L + 2.2301M − 2.1501S) and (−3.3695L + 1.9225M + 1.4149S). Is it plausible that functions with these coefficients have any physiological realization in the visual system? In the absence of any physiological realization, what explanatory power are we entitled to think they have?


The ‘zero-crossings from Fig. 8 will be used for prime and anti-prime colors’ (Wright 2011, 10): i.e. the values of λ for which d(|| e1(λ), e2(λ), e3(λ) ||) / dλ = 0, which will be the values of λ for which || e1(λ), e2(λ), e3(λ) || is at a maximum or a minimum. In the printed paper, Wright (working to the nearest 5 nm) gives these values as 445, 535, and 605 nm (for primes) and 490 and 570 nm (for anti-primes), and in an earlier version he gave 440 nm as the value of the first prime. By my own calculation (following the definitions he has supplied me with for the e-functions), the maxima and minima of his norm function are at 442, 539, and 604 nm and at 490 and 571 nm (working in 1 nm steps)—so in the main text I hope my talk of Wright placing his first prime in the ‘440–445 nm region’ is accurate enough for present purposes.


I thank Wright for pointing this out to me. Thornton’s sensitivity functions are to be understood as giving the sensitivities not of cones, but of three channels ‘deep in the normal human visual system’ (1999, 153). Whether there actually are any such ‘cortical signal’ channels meeting Thornton’s specifications deep in the visual system seems to me rather doubtful, and I am not sure that they are in any way needed to explain the colour-matching phenomena that fascinate him. A TV camera with sensors having maxima at 442, 543 and 570 nm and a sensitivity profile like the cones would also show just the same patterns of colour-matching (with, e.g., 1 unit of R600–610 light being much more efficient as a primary than R645 or YG565): there was no need for any appeal to processes with a special maximum at 600–610 nm ‘deeper’ in the visual system.


Wright tells us (Wright 2011, 5; citing Romney and Chiao 2009, 10378) that his e1 function correlates well with the CIE Y (or V(λ)) function: the correlation between the e1 function and the CIE L* function as applied to Munsell chips is as high as 0.9992 (where L* is defined in terms of Y/Yn, and Yn is the Y for a reference white). This may be true, but it is far from being a sign that the e1 function is a good measure of luminance. I have done a quick back-of-an-envelope type calculation of the values of the e1 and the CIE Y function for 15 pigments varying through the colour wheel, as illuminated in D55. (I have used the same pigments as in §12.6 of Broackes 2010, sampling the relevant functions at 20 nm intervals between 400 and 700 nm.) I have normalized the two functions to have an output of 100 for an ideal white. The correlation is indeed high (r = 0.9924); but the proportionate discrepancy between the two functions can in many cases still be very large: the discrepancies between the two functions are even higher if the illuminant is a bit bluer, e.g. D75. (The correlation falls to 0.9874, and the values of the e1 function for these pigments in D75 are in two cases 64% and 32% higher than the Y value.) The obvious conclusion is that we should not be misled by the ‘correlation’ of these two functions being above 0.99 (for some domains of application): that is perfectly compatible with the two functions having obvious and important discrepancies over significant parts of even those same domains. Of course, the Y(λ) function has its own problems as a measure of luminance and the Stockman & Sharpe V*(λ) function is surely an improvement on it. But the e1(λ) function diverges dramatically from the V*(λ) function too in application to real-world yellows and blues: e.g., the e1 value for Cobalt Blue in D55 is 43% above its V* value. There is no reason why, if one is making a free choice of orthogonal basis functions in terms of which to decompose the L, M and S functions, one should expect any of the functions to correspond particularly closely with luminance or brightness; conversely, since brightness is something with a fair degree of salience in the visual system, we have at least some reason to think that other functions than these e-functions are realized in the human visual system, and functions like the e1, e2, e3 functions may well not be.



Many thanks to Wayne Wright for his original paper and for continuing exchange of views on these points. I have benefited from his paper in revising the paper for publication. I am grateful to Steven Yamamoto for comments on an earlier draft and to the editors of this journal for encouraging a debate on this contentious issue.

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© Springer Science+Business Media B.V. 2011