colormaking attributes

 
This page addresses a single issue: how can we describe color experience? Because color occurs in the mind but is a response to light in the world, separate color descriptions are necessary for the external, physical light stimulus and the subjective color perception.

The techniques of photometry allow description of the intensity of a light stimulus as it appears to human vision, and colorimetry translates the stimulus into a color specification. A standard tool used in either approach is the spectrophotometric curve, which shows the exact mixture of light wavelengths emitted from a light source or reflected from a surface.

The standard photometric units provide a useful framework for learning about the geometry of light — how light is defined or measured as it propagates through space, is reflected from surfaces, and registers in optical systems such as a camera or the eye. Many photographers are aware of these basic photometric principles, as they help to judge the photographic demands imposed by contrasts in light intensity. But painters, especially landscape painters, can profit from a clear understanding of how light behaves.

Color experience, the subjective side of color, is described by three colormaking attributes — (1) brightness/lightness, (2) hue and (3) hue purity (colorfulness, chroma or saturation). These permit a sufficient and reliable description of isolated color areas under simple viewing conditions.

Physical color measurements and subjective color descriptions are only correlated, in the sense that one approximates but does not define the other. Fundamentally color depends on context, and context can dramatically change the appearance of lights and surfaces.

Painting is a form of description, and traditional methods of color mixing and color terminology developed among European painters as practical equivalents to the colormaking attributes. This page concludes by explaining these painting methods in the context of modern color description.

 
measuring light & color
 
The first requirement is a method to describe exactly the radiant power of the external light stimulus that creates the perception of light and color. This task is accomplished by three related methods.  

Radiometry is the measurement of radiant power or energy within that part of the electromagnetic spectrum that is optical, meaning it is refracted by glass or can be focused by a lens. This includes microwave, infrared, visible and ultraviolet wavelengths approximately in the range of 1 millimeter to 100 nanometers (10-3 to 10-7 meters, or frequencies of 3 x 1011 to 3 x 1016 Hz). Radiometry excludes radio waves, xrays and gamma rays.

The standard radiometric device is a vacuum glass bulb with a wheel of paddles inside, each paddle painted black on one side and white on the other; the wheel rotates when exposed to heat, light or an ultraviolet lamp.  

In radiometry, electromagnetic power is measured in watts (joules per second), which can in turn be converted into other units of energy or power. Actinometry is electromagnetic power measured in number of photons per second. Radiometry provides the fundamental link between our visual sense and the physics of matter and energy.

(A comment on terminology. Energy is the potential to cause a change in matter, for example a change in its structure, temperature, location, speed or direction of movement; it is measured in joules, roughly the amount of energy required to raise an apple 1 meter off the ground. Power is energy emitted within a fixed time interval, equivalent to the potential speed or rate at which a change in matter occurs; it is measured in watts (joules per second). Intensity is the quantity of power radiating into a fixed solid angle or projected area of space.)
 

the daylight spectral power distribution
the spectrum measured in physical units (power in watts) relative to the value at 555 nm; data from ASTM G173 (2003)

 
The fundamental radiometric description of a light source is its spectral power distribution or SPD, which shows precisely the amount of electromagnetic energy emitted per second at each wavelength interval (or frequency interval). Shown above, for example, is the spectral power distribution of the sun as viewed from outer space and average noon daylight (sunlight plus skylight) at the earth's surface; the peak power in both distributions is at around 470 nm.

These curves are often expressed as the power at each wavelength relative to the power at a standard wavelength, shown here for terrestrial radiation at 555 nm (visual "yellow green"). (Note that this peak shifts into the infrared when graphed against wavenumber, a measure of frequency.) This produces a relative spectral power distribution, or relative SPD.  

Photometry is radiometry adapted to represent a single attribute: the average brightness of light as perceived by the human eye. This is done by weighting the power at each wavelength by how strongly that light stimulates the photoreceptors in the eye, then summing the weighted values to get the total visible energy.

The weighting transforms radiometric watts into photometric lumens, the units of visible electromagnetic power. Lumens do not measure brightness specifically, because brightness is a visual sensation that depends on luminance contrast (for example, the full moon appears brighter at night than during the day). Lumens simply measure that part of the total radiant power that produce a sensation in the eye. Your light bulb is rated in watts, because that is how much energy the bulb consumes per second; the light from the bulb is rated in lumens, because that is how much of the energy can visibly brighten your world. The ratio between them is the efficiency of the light source: tungsten light bulbs yield roughly 15 lumens per watt (and lots of invisible heat); energy conserving fluorescent lights produce around 60 lumens per watt (and very little heat).
 

color
vision

measuring light & color
radiometry
photometry
colorimetry

the spatial geometry of light

the colormaking attributes
brightness/lightness
hue
hue purity
optimal color stimuli
are three attributes enough?

painting saturation & value
how to judge saturation
lightness, chroma & saturation
the painters' "broken colors"

These photometric weights define the luminous efficacy of each wavelength, and they combine as the photopic luminous efficiency function, the light adapted sensitivity of the cones (diagram, right). Wavelengths outside the visible range, typically stated as 400 nm to 700 nm, negligibly affect the eye and are usually ignored.

The photopic sensitivity curve is scaled so that 1 watt of radiant flux at a wavelength of 555 nm ("yellow green" light) equals a luminous flux of 683 lumens (diagram, right). (This odd number was chosen to provide continuity with the inherited, historical measures of light — as emitted from a single burning candle or lamp, or through an aperture the width of a pencil lead placed over white hot platinum.) The photopic curve then determines the proportional weights used to convert energy at other wavelengths into light.
 

the daylight spectral luminance distribution
the spectrum measured in perceptual units (luminance in lumens), relative to the value at 555 nm

 
Here is the daylight spectral power distribution weighted by luminous efficacy to show the photopic luminous intensity in lumens. The curve is again standardized on a peak luminance of 555 nm ("yellow green"). As the diagram shows, the eye responds to only 17% of the total radiation available at the earth's surface (if we assume a 100% response at 555 nm).

A second curve is available to describe the dark adapted visual sensitivity of the rods — the scotopic luminous efficiency function (diagram, above right). The 507 nm scotopic peak sensitivity is shifted toward the short wavelength side of the photopic efficiency curve; it is scaled so that it matches the sensitivity of the cones at the photopic peak wavelength. This raises the scotopic peak luminous efficacy up to 1700 lumens per watt: the same radiant power, under scotopic viewing conditions, appears roughly three times as bright.

In fact, the peak scotopic sensitivity is over 120 times greater than the photopic sensitivity, if measured as the minimum quantity of light necessary to produce a visible stimulus — not the 3 times greater implied by the photometric scaling. And the point where scotopic and photopic luminous efficacies have equal light sensitivity is actually in the "red" wavelengths, around 640 nm. Thus, the lumen is a different psychophysical unit under photopic, mesopic or scotopic light levels, and it generally understates the luminous efficacy of very dim light stimuli.  

Colorimetry is the measurement of color stimuli using photometric techniques. It does this by weighting the spectral power distribution of a light or surface using three different luminous efficacy curves — either standard colormatching functions or the L, M and S cone sensitivity curves. These values are then used to triangulate or calculate the color of the stimulus when viewed as an isolated patch; the values are also summed to get the color brightness. These techniques are explained in later sections on colorimetry and the CIELAB color model.  

The fundamental photometric description of the light stimulus is called a spectrophotometric curve, which describes the relative quantity of light (lumens or photon counts) as a proportion of some standard or maximum quantity across the visible wavelengths (typically 380 to 750 nm, or 400 to 700 nm). These curves come in three flavors:

• a spectral emittance curve describes the light emitted by sources such as the sun or artificial lights. The quantity of light emitted at each wavelength is expressed as a proportion of the quantity of light emitted at the most luminous wavelength, or at an arbitrary standard wavelength (usually 555 nm or 560 nm).

• a spectral transmittance curve curve shows at each wavelength the light that is passed through or transmitted by the medium as a proportion of the light incident on its opposite surface.

• a spectral reflectance curve shows at each wavelength the light that is reflected (not absorbed) by a surface as a proportion of the light incident on the surface.

Because prints and paintings are essentially surfaces, the spectral reflectance curve is the standard method to describe the color creating characteristics of inks or paints on paper.
 

reflectance curves and cone outputs for
titanium white (PW6) and ivory black (PBk9)

normalized cone spectral curves from Vos, 1978 and Werner, 1982

 
The two examples above show the reflectance curves for the most basic surface colors: black and white. The horizontal dimension identifies specific light wavelengths in the visible spectrum (symbolized in the diagram as spectrum colors). The height of the curve shows the proportion (from 0% to 100%) of the incident light that is reflected by the surface at each wavelength.

The reflectance curve eliminates any effect from variations in the illuminance or intensity of the light source: a surface that reflects 50% of moonlight will reflect 50% of sunlight too. The curve is also the same regardless of the color of the light source, provided only that all visible wavelengths are present in the light in some amount (though measurement is most accurate using a "white" light standard). When interpreting a reflectance curve, assume it represents the surface color as viewed under an equal energy illuminant or "pure" white light, which contains all visible wavelengths in equal amounts.

The difference between the reflectance curves for white and black paints shows that the lightness of a paint is proportional to the average height of the reflectance curve. However this proportion is not easy to determine from the curve itself, because lightness has a curvilinear relationship to reflectance; for example, the graphic arts "middle gray" is produced by an average reflectance of about 19%.

Note also that the average height of a reflectance curve is never 0%: all physical surfaces reflect some light. The blackest watercolors reflect about 10% of the light falling on them, and black acrylic paints or color samples reflect roughly 5%.  

The trilinear color specification — the relative proportion of L, M and S outputs produced by the reflected light — can be used to infer the surface color represented by a reflectance curve, and I provide two aids to help you do this. Each curve is overlaid with the log sensitivity curves for the L, M and S cones. To show how the curve is actually interpreted by the eye, most reflectance curves are accompanied by the matching cone response profile, the level of cone response created by the light mixture.
 

a simple method for interpreting spectral reflectance curves

 
The key landmarks are the crossover points where one cone sensitivity curve slips below another. These are conveniently visible in the spectrum as two narrow, distinct bands of color: the "cyan" boundary between "blue" and "green" wavelengths (at around 495 nm), and the "yellow" boundary between "green" and "red" (at around 575 nm). These crossovers divide the spectrum into three sections: blue, green, and red. Within each section, the S, M or L cone is the dominant receptor. As a rule of thumb, the proportion of reflected light in each section of the spectrum indicates the proportional contribution of the L, M or S cones to the color sensation.
 

luminous efficacies for
photopic and scotopic vision

curves show the number of lumens
produced by 1 watt of radiant power
at each wavelength between
400 nm and 700 nm

It is unwise to "read" the color appearance (lightness, hue or chroma) of a surface directly from its spectral reflectance curve: the curves show the percentage of light reflected at each wavelength, but not the importance or weight of each wavelength in color perception. For example, the reflectance for a scarlet paint (diagram, right) has peak reflectance in the "red" end of the spectrum, but what exactly is its dominant wavelength (hue)? The tail of "blue" and "green" reflectance can have a significant impact on the hue and chroma of the surface. It is also difficult to assess the lightness of the surface, as the "green" wavelengths contribute much more to lightness or brightness than do the "red" or "blue" wavelengths. Reflectance curves are most interpretable when one curve is compared to another — to indicate the relative reflectance difference between two paints or inks or papers — or to indicate the general color appearance — red versus green, or saturated red versus unsaturated red.
 

using reflectance curves to define a color mixture
equal parts ultramarine blue (PB29) and cadmium red deep (PR108)
 

Two reflectance curves can also be combined to model the color that would be produced by the mixture of two pigments, as shown above for a mixture of equal parts of ultramarine blue (PB29) and cadmium red deep (PR108).

The reflectance curve for watercolor paint mixtures (of paints having equal tinting strength, opacity and dilution) is approximately the geometric mean of their separate reflectances computed at each wavelength in the spectrum. (The geometric mean is the square root of the product.) For example, if ultramarine blue reflects 80% of a specific "blue" wavelength (say 480 nm), and cadmium red deep reflects only 8%, then their mixture will reflect roughly 25% of the 480 nm light (that is, 0.08 x 0.8 = 0.064, where the square root of 0.064 is 0.25). This averaging must be repeated for every wavelength, then the apparent color of the mixture is determined from the cone responses to the resulting average reflectance curve (white line).

For transmission filter mixtures, the simple product of the two transmission profiles gives the resulting light intensity: 0.80 x 0.08 = 0.064, or 6.4% for the 480 nm wavelength.

The geometrical mean gives more weight to the absorptance rather than the reflectance of the two paints: at every wavelength, the reflectance of the mixture is closer to the darker paint. However, to judge the approximate hue of the mixture or understand how the two paints will behave when mixed with each other or with light, the visual average — shifted somewhat toward the darker reflectance curve at each wavelength — may often work fine.
 

a spectral reflectance curve for a scarlet red paint

This illustrates that color perception is dominated by wavelengths emitted or reflected within the center of the spectrum, roughly between "cyan" and "red orange". Paints that mostly absorb the middle wavelengths and reflect the spectrum ends (such as deep red and blue violet) produce especially dark colors.

 
the spatial geometry of light
 
The conversion from a spectral power distribution to lumens is only the first step toward a useful measure of light intensity: it is also necessary to specify the spatial geometry that describes how the light is being measured or viewed. Different measurement situations yield very different descriptions of the light source.  

The Nine Photometric Elements. The nine elements necessary to define the spatial geometry of light are:

1. An imaginary point source to stand for the spatial location of the light source and to represent the radial property of the emitted light.

2. An imaginary measurement sphere that is centered on and completely encloses the point light source (the geometry of luminous flux).

3. An imaginary aperture or opening in the measurement sphere, made by cutting away a specific surface area (A) from the sphere, that reduces the total emitted light from the point source to the light that radiates through the aperture (the geometry of luminous intensity).

4. A straight line that defines the average direction of light emitted from the point source through the aperture.

5. The physical distance (D) measured along the direction of light from the point source to the surface of the measurement sphere, or to a surface that is equal in area to the aperture in the sphere (a key element in illuminance).

(The measurement aperture, direction of light and distance are combined as a single unit of measurement, the solid angle, which is the ratio between the measurement aperture emitting light, or the surface area receiving light, and the squared distance of the aperture or surface from the point source: A/D2. See solid angle geometry and inverse square law.)

6. The angle of incidence (θi) between the direction of light and a line perpendicular to the plane of a physical surface receiving the light (see cosine correction for surfaces).

7. The image area (S) of the physical surface of the light source, as viewed from a point on the surface that receives the light (the key element in luminance).

8. The angle of emittance (θe) between the direction of light and a line perpendicular to the surface of an extended light source (see cosine correction for light sources).

9. The pupil area admitting light to the eye (the key element in retinal illuminance).

In combination, these nine elements restrict the light to five different measurement geometries: each geometry creates a different unit of light measurement. The diagram below provides a summary and visual mnemonic for these five measurement units — luminous flux, luminous intensity, illuminance, luminance and retinal illuminance — the geometry that defines them, and the units of measurement based on them.
 

the relationships among the five photometric units

 
There is a parallel radiometric nomenclature (radiant flux, radiant intensity, irradiance and radiance, excluding the troland): these are based on identical measurement geometries, but do not apply a luminous efficiency function to translate the spectral power distribution into a light distribution.
 

Luminous Flux is the measure closest to the fundamental physics of light generation. It is measured in lumens (lm):

1/683 watt emitted = 1 lumen

As explained above, this fractional unit of power (the watt) was adopted to remain consistent with historical units of light measurement: one lumen is roughly equal to the total light emitted by a single wax candle.

Luminous flux is the generic term for the visible power a light source emits per second. Total luminous flux is specifically the light emitted by the light source in all directions. The spatial geometry of this concept is equivalent to enclosing the light source within a measurement sphere that captures on its inner surface all the light emitted (diagram, right).

The radius of the sphere does not change the total amount of light falling on its interior surface. So we can imagine that the sphere is so enormous that the light source is in proportion just a single point source at its center. Because all light radiates from the center of the sphere, all the light is perpendicular to the interior surface of the sphere. This describes the radial geometry of light.

In practical situations, luminous flux is calculated by measuring the output from a light source from many different angles at equal distances, then integrating these over a spherical area; or by measuring the reflected light at one point inside a diffusing sphere, and extrapolating that quantity to the total spherical surface. In fact, the quantity is usually obtained by measuring the light from a single direction and distance (illuminance or luminance), and then calculating backwards from that.

Luminous flux is a "source centric" definition of light: it describes the source without regard to the direction, distance or surface area of any surface, camera or eye that might receive the light. Conceptually it represents the light source independent of a physical point of view, and corresponds to the sense of elemental power we infer from the experience of outflowing light.
 

luminous flux defined by a measurement sphere

Luminous Intensity is the luminous flux emitted from a point source into a radial envelope called a solid angle (explained below).

The solid angle is essentially a "window" or aperture cut into the measurement sphere. To maintain the radial geometry of light, the average direction of light we want to measure must be centered within the area of this aperture, and the light must be both a point source and located at the center of the sphere when the aperture and direction are defined.

This aperture removes a spherical surface area from the measurement sphere, and this is the area of the solid angle. This aperture area can be any size or shape, but the standard or unit solid angle is the steradian, equivalent to a square aperture 57° on a side or a circular aperture 65° in diameter. The steradian defines luminous intensity as lumens per steradian or candelas (cd):

1/683 watt emitted into 1 steradian
= 1 lumen/steradian
= 1 candela

The steradian encloses an area equal to the square of the radius of the measurement sphere, or a square radian. It is the unit solid angle because it is defined on a unit sphere (of radius 1), which makes its area equal to 1. As a result, the steradian aperture area is equal to 1/4π (roughly 1/12th) of the total surface area of a sphere. Thus, assuming a point source that radiates equally in all directions:

Luminous intensity captures the notion of a light source as having a brightness or power in a specific direction: street lights illuminating the pavement underneath, ceiling lights illuminating an office work area, a spotlight turned toward a cabaret singer, the sun shining toward the earth.

However, luminous intensity is still a "source centric" or abstract measure of light, because we have not specified the distance to a viewer or illuminated surface, nor the size of a physical surface that receives the light. Luminous intensity is an abstract measure of the quantity of radiant power or flux density within a standard solid angle.  

The Viewing Geometry. In order to make light measurement "viewer centric", we must adapt the measurement of luminous flux to include aspects of the light geometry as a viewer defines it. These include, among other things, the area of a surface that is illuminated by the light, the distance of the surface from the light source, and the visual size of the light source to the viewer. The solid angle is a measurement unit that can combine all these aspects of the viewing geometry.
 

luminous intensity measured by steradian in a specific direction

the steradian is the area on the surface of a sphere equal to the square of the radius of the sphere

Solid Angle Geometry. The solid angle is simply a projected area:

• Numerically, a solid angle is a ratio between distance and area that applies at all distances, in the same way that the "golden rectangle" is a ratio between height and width (1:1.618) that applies to rectangles of all sizes. (Physicists therefore say it is a "dimensionless" number.)

• Geometrically, a solid angle is analogous to (but not the same as) a cone or pyramid: in applying the solid angle to the measurement of light, the idealized point light source generating the light we want to measure corresponds to the apex of a cone, the average direction of the light propagating through space corresponds to the central axis of a cone.

However the "base" of the cone or pyramid is defined as the area of the base as projected onto the surface of a sphere with the point source located at the center of the sphere.

Why is it necessary that the solid angle area is defined on a measurement sphere? Because this ensures that every part of the solid angle is measured at an equal distance from the origin of the light (all light is measured at equal intensity), and because the spherical surface defines the smallest possible projected area for a given distance D measured along the average direction of light.

For example, we can cut across the solid angle with a flat surface, creating a "base" for the cone, or we can place the cone over the surface of a sphere, such as the moon (diagram, right). The flat and the curved surfaces will define a greater area than the measurement sphere, because on both the curved and flat surfaces the outer edge of the solid angle (b in the diagram) will be farther from the point light source than the distance measured along the axis (a).  

It is simple to find the area of plane geometrical figures projected onto a sphere. For example, if we assume that the solid angle has a circular cross section (like a cone), then the area of the solid angle projected onto a measurement sphere is:

where theta (θ) is the angular subtense of the solid angle, equivalent to the interior angle of the circular cone measured at its apex. In other words, we use angular subtense to define the solid angle area, independent of any measure of distance from the point source.

First, we define the dimensions of the projected area in degrees of an arc, as measured at the point light source. Thus, for a circular projected area we measure the diameter, in degrees, of the circle; for a square projected area we measure the side, in degrees, of the square. This is the angular subtense of the projected area (ASdegrees).

Second, we convert the angular subtense into radians, which is the appropriate unit of distance on the surface of a sphere. The conversion formula is:

Third, we calculate the spherical solid angle area, using radians as the dimensional units in the usual formulas for the area of plane figures. This results in the solid angle area expressed in steradians. Thus, a circular measurement area defines a solid angle as:

and a square measurement area defines a solid angle as:

solid angle (steradians) = ASradians2.

Thus, from the earth, the moon appears as a circular disk with an average diameter (angular subtense) of 0.52° or 0.0091 radians. So the solid angle defined by the moon's circular area as viewed from the earth is π*(0.0087/2)2 or 0.000065 steradians.

Finally, a solid angle can be defined for any kind of projected area — an ellipse, a pentagon, a checkerboard of separate square areas, the profile of Abraham Lincoln. We first determine the angular subtense of all dimensions that are necessary to calculate the area of the projected figure, then convert those dimensions into radians, and finally apply the formula for the figure area on a plane surface to find the solid angle area in steradians.

A Simplified Solid Angle. In practical viewing situations it is usually sufficient and much easier to measure (or estimate) the distance of the light source than its angular subtense; and in situations where the width of the projected area is very small compared to the distance of the area from the light source, we can assume, with negligible loss in accuracy, that the receiving surface is flat rather than three dimensional. Then the area of a solid angle simplifies to:

In this formula the solid angle is still expressed in steradians when both the physical surface area and the distance between source and surface are expressed in the same units (feet, meters, kilometers). Thus, the moon's diameter is 3480 km, and its average distance from the earth's surface is 378,000 km, which defines a solid angle of 0.000067 steradians.
 

the solid angle and surfaces

the projected area is not the same
as an illuminated surface area

Although it is not a geometrical figure, the steradian is a useful perceptual proportion for visual estimates of brightness on a surface (diagram, right). A "distance wide" circle of surface area underneath a diffuse light appears more or less evenly illuminated by the light and anchors our judgment of whether the light's illuminance is adequate to its purpose. A reading lamp looks adequately bright or too dim according to the amount of light it casts on a 2 foot circle of desk underneath it, and a ceiling light according to the illumination on the 9 foot wide circle of floor below.  

Inverse Square Law. Although the steradian area of a solid angle is dimensionless (constant at all distances from the light source), we need to consider what happens to the actual projected surface area at different distances from the light source.

Because the sides of a solid angle and the beams of light it contains both radiate from the point source, the total luminous intensity radiating into a solid angle remains constant regardless of the distance from the point source where we may measure the light. However the cross sectional area becomes larger, so the energy is more spread out (diagram, below).
 

solid angle and standard surface area
the solid angle is used to measure luminous intensity, which does not change with distance; the surface area is used to measure illuminance, which does change with distance

 
If we consider a constant surface area (the same physical surface) at two different distances from the light source, then the surface may equal the cross section of the solid angle at one distance (Da), but cover only a small part of it farther away (Db). So the quantity of light falling on the surface at Db must be less than the quantity at Da — but by how much?

The inverse square law defines the relationship between the two distances and the difference in the quantity of light falling on each (Ia and Ib) as:

Ib = Ia * (Da/Db)2

Which means: if a quantity of light Ia is incident on a surface area at the initial distance Da from the light source, and the surface is then moved to a new distance Db from the light source, then the new quantity of light incident on the surface Ib is the initial quantity of light increased or decreased by the ratio of the two distances squared.

This proportion gets its name from the fact that we take the ratio of the new distance divided by the initial distance, invert the ratio, then square it. In the diagram (above), a surface area initially at 1 distance unit from the light source is moved to 3 distance units from the light source; there it receives only (1/3)2 or 1/9th the luminous intensity that was incident on it at 1 distance unit. If the light is first measured at 3 units, then moved to 2 units, the quantity of incident light is (3/2)2 or 2.25 times greater.
 

streetlight and steradian

Cosine Correction. Finally, we usually assume with little loss in accuracy that the physical surface area corresponding to the solid angle area is flat (a plane). But we also, more importantly, always assume that the surface is perpendicular to the direction of light. If the plane surface is tilted at an oblique angle to the direction of light, then physical surface area enclosed by the solid angle increases or the apparent size of the surface, as viewed from the point source, decreases (subtends a smaller visual angle), often by a large amount. This is called foreshortening.

Because foreshortening always makes the surface area as seen from the point source appear smaller, the surface always defines a smaller solid angle and therefore receives less light. The amount of this reduction in the incident light is determined with the cosine correction for foreshortening. We insert, into the formula for the solid angle, the cosine of the angle (θi) between the direction of incident light and a line perpendicular to the surface (diagram right):

If a surface is perpendicular to the direction of light then the angle of incidence θi = 0, the cosine = 1.0, and there is no change in the amount of light falling on the surface. If the angle of incidence θi = 45°, then the cosine = 0.707 and the quantity of light incident on the surface is reduced 71%.

With the simplified formula for the solid angle, an understanding of the inverse square law, and the cosine correction for foreshortening, we can develop the two most common and "viewer centric" measures of light: illuminance and luminance.  

Illuminance is the quantity of light incident on a surface from a light source or sources. No information about the distance or size of the light source(s) is implied. However, the standard measurement unit of illuminance is defined as the light energy incident on a surface area of one square meter at a distance of one meter from a point light source, in one second. This yields lumens per square meter (lm/m2) or lux (lx):

1/683 watt incident on 1 square meter
= 1 lumen per square meter
= 1 lux

There are several obsolete or nonmetric definitions of illuminance, including the foot candle (1 lumen incident on 1 square foot). As there are roughly 10 square feet in one square meter, the foot candle defines a larger unit of light than the lux:

1 foot candle = 10.76 lux.

The illuminance on a surface area 1 meter square has a simple relationship (via the inverse square law) to the two previously defined measures of output from a light source, luminous flux and luminous intensity:

Recall that the steradian is just the "radius wide" square area on a sphere, and therefore the steradian solid angle is equal to the radius (distance) squared. In formula [1], we derive luminous intensity by dividing the luminous flux, the total output of light in all directions, by the number of steradians in the surface area of a sphere ( or about 12.57); then in both formulas we use the inverse square law to adjust luminous intensity by dividing by the distance squared (in meters) of the light source from the light receiving surface. This gives us the illuminance, in lux, as the quantity of the total luminous flux that would be incident on one square meter at the given distance.
 

the basic geometry of illuminance
illuminance as the point luminous intensity times the distance squared to a standard receiving area of one square meter

 
Thus, a nonreflecting 60W incandescent bulb emits about 600 to 840 lumens; lumens denotes luminous flux, so by formula [1] its illuminance at 3 meters is about 5 to 7.5 lux. Recessed or reflector bulbs emit their luminous flux in one direction, as a diffuse beam or light cone. The solid angle of this light cone varies across different types of lighting, but a handy rule of thumb is that the cone fits within a solid angle of one steradian. Then the rated luminous intensity of the light in lumens is identical to its illuminance in lux (1 lumen equals 1 lux at 1 meter distance), and only the inverse square correction for different distances is necessary. For a 60W reflector or recessed light, rated at 650 lumens, the illuminance at 3 meters is about 72 lux.

If the light is obliquely rather than perpendicularly incident on the surface, then the illuminance is less, and the luminous intensity must be multiplied by the cosine correction for foreshortening:

Finally, there is a third definition of illuminance, based on the luminance and image size (solid angle) of the light source as it appears from the surface:

[3] illuminance = luminance * solid angle S

where luminance is measured in candelas per square meter, and illuminance is expressed in lux.

Because illuminance is commonly used in architectural or task specifications, it is natural to think of it in terms of how lighting appears in a given situation. But this can be misleading, and three key points will help clarify what illuminance actually measures.

The first and most important point is that illuminance is never directly visible as a quantity of light. We only see its reflected image as the luminance of physical surfaces. If the light source were behind us, and there were no surfaces in view (or the surfaces completely absorbed light), we would look into total darkness. If we turned to look directly at the source of illumination, we would perceive the luminance of the optical image of the light source on the surface of the retina, which depends not only on the quantity of illuminance but also on the visual size of the source that emits it. Our judgment that areas are brightly or dimly lit is actually based on our perception of the light reflected from familiar surfaces, such as white walls or the pages of a book, and covertly on our perception of visual adaptation cues, such as contrast and color.

Second, by itself illuminance does not specify light sources — it describes the light power incident on a specific surface at a specific location in space. From an illuminance measurement alone, we cannot infer the power, size or number of light sources. It is analogous to the "blind" skin sensation of heat on our hand that can be induced by the distant sun or by a nearby bare light bulb: the sensation of heat, by itself, tells you nothing about the source. This means, in practice, that illuminance measurements must be limited in one of two ways. If we want to describe (in lux units) the amount of light provided by a specific light source, then we must exclude all other light sources from the measurement; if we want to describe the light incident on a specific physical surface, then we must sum the illuminance of all the light sources illuminating it.

Third, distance has a huge impact on illuminance through the effect of the inverse square law. At one meter, a single candle (1 candela) yields an illuminance of 1 lux. At 10 meters, the candle produces an illuminance of 0.01 lux. To get back the illuminance of 1 lux from a distance of 10 meters, we would have to use 100 candles.

Finally, in real world situations illuminance is significantly affected by the geometry of obscuring objects in relation to light sources: it does not depend on light sources alone. For example, on a clear sunny day, a book held in the shadow of a traffic stop sign on a country road is illuminated by the entire visible sky; but if we stand in the shadow of a large building that blocks out half the sky, the illuminance is reduced by almost half. Similarly, if a single north facing window illuminates a room, drawing the blinds from a second window doubles the illuminance. In forests, canyons, alleys or overcast days, the reduction in natural daylight illuminance signals the amount of light obstruction, not changes in the light source.

This dependence on distance makes illuminance the key architectural measure of lighting, and the physical dimensions of an interior space, along with the activities that must be performed in the space, determine architectural lighting requirements. The same level of illuminance can be provided by many different lighting systems (number, power and physical size of light sources); illuminance only specifies the total light output that the system must deliver.

There are many illuminance standards for the amount of light desirable in different architectural settings or for different tasks. Office lighting standards require illuminances in the range of 300 to 500 lux at work surfaces; home lighting levels are typically lower. (See this section for a broad comparison of illuminance levels.) Many eyestrain problems are created by the effects of glare (reflected light) or excessive light contrast, and not by inadequate illuminance levels.

Lighting engineers and interior designers measure illuminance with an illuminance meter or light meter, which determines the amount of light through the electric current produced by light energy falling on a photosensitive surface. Lighting engineers often use a light meter that collects light arriving from a wide area, so that the measured illuminance corresponds to light sources of any size and shape in any direction; these all sum to the total light incident on the measurement device. Photographers similarly use a light meter to estimate the average quantity of light available to alter photographic film; however, they typically measure only the light reflected from an average gray card, or the average light reflected from only the surfaces included in the image.  

Luminance is the luminous intensity of a light source divided by the physical area of the source, or (equivalently) the illuminance received from a light source divided by the visual size of the light source.

Luminance is the most context specific definition of light, because it is based on a solid angle and a surface area, which gives candelas per square meter (cd/m2):

1/683 watt incident on
1 square meter at 1 meter distance
= 1 candela/meter2

Luminance assumes a light emitting area (such as the diffusing glass over an electric light bulb, or a reflecting sheet of white paper, or a window) that produces the light incident on a light receiving area (film, CMOS chip, retina), or alternately the light that passes through an aperture that controls the amount of light reaching the light receiving surface.

A surface one meter square at one meter distance is equivalent to the steradian solid angle. So why isn't luminance the same as luminous intensity? Because luminous intensity assumes a point light source, while the definition of luminance adds the surface area of the light source. So the unit definition of luminance includes the unit 1 meter square surface area receiving luminous flux from a point light source one meter distant from it (implicit in the steradian solid angle representing the candela of luminous intensity), standardized on (divided by) the 1 meter square representing the unit surface area of the light source.

As distance is involved, it is combined with one of the two surface areas to define a solid angle — either the visual size of the surface receiving the light as viewed from the light source, or the visual size of the light source as viewed from the light receiving surface.
 

the cosine correction for foreshortening

The diagram below illustrates the basic geometry of luminance in terms of a flat, diffuse light source radiating into a unit hemisphere.
 

the basic geometry of luminance
luminance is the solid angle of point luminous intensity, times the surface area of the light

 
This geometry yields five equivalent definitions of luminance based on different measurement units:

Formula [1] shows that luminance is defined by three geometrical quantities: the two surface areas (or a surface and an aperture area) and the common distance between them. Thus, the surface area of the light source (circular in the diagram) is defined by π*X2. (The area of a square light source would be (2X)2.) The circular area of the light receiving surface (or aperture) is π*Y2. This surface (or aperture) is at distance D from the source.

It is usually convenient to divide the distance squared into one or the other surface area to create a solid angle, as in formulas [2] and [3]. The light receiving aperture area defines the solid angle A, equal to π*Y2/D2 steradians; the light emitting source area defines the solid angle S equal to π*X2/D2 steradians.

As always, it is convenient to express X and Y as a visual angle in radians, as this excludes the surface geometry of both the source and the aperture area. If Y is the angular subtense (in radians) of the aperture as measured from a point s on the source and X is the angular subtense of the source measured from point d at the aperture, then their solid angles are π*Y2 steradians and π*X2 steradians.

The solid angle A defines the proportion of luminous flux passing through the aperture from any single "point source" s on the surface of the light source. The total number of points illuminating the aperture is equal to the physical area of the source (π*X2). So the solid angle A based on the spherical area of the aperture is multiplied by the area of the source, and this is divided into the luminous flux.

Note that distance is only included once in a luminance calculation, so two solid angles are never required. In formula [4], luminous intensity already contains one solid angle (in the "aperture" area of the steradian solid angle), so this is scaled by the source area only. In formula [5], illuminance only contains a surface area (the area of the surface/aperture that receives the light), so this must be scaled by the solid angle to the source (its visual size at the viewing distance).
 

As before, in any of the formulas above, the luminance quantity can be adjusted with the cosine correction (diagram, right), when either the aperture plane (the angle of incidence, θi) or the source plane (the angle of emittance, θe) or both (a foreshortened source shines onto a foreshortened surface) are not perpendicular to the average direction of light between them:

Formula [6], based on illuminance (easily measured with a photometer) and including the cosine corrections necessary for specific lighting problems, is probably the most commonly used calculation of luminance.

Now that I've introduced all the standard formulas, let's consider the perceptual implications of luminance.

First, luminance describes physically extended light sources, not imaginary point light sources. Image area is really the key concept here, since all physical lights or reflecting surfaces must have a measurable surface area, or a visual width and height (angular subtense), no matter how small or far away the source may be. (Note that the bottom limit on visual width is determined by the "pixel" structure of our retina or CMOS chip, or the "granular" structure of a film emulsion; these impose an angular subtense on the imaging of very tiny sources, such as stars.)

Second, luminance is the "area intensity" of the light source. It does not just depend on the quantity of light reaching the eye, but the degree to which the illuminance originates in a visually compact source.

As a simple example: you step into a business office and notice that the floor is brightly illuminated. Simply by looking at the floor and the material it is made of, you can infer the approximate level of illuminance, for example as compared to the illumination produced on the floor by direct sunlight. But you cannot see the luminance of the light source itself: to do that, you must look up. If the illuminance originates in several "spotlight" ceiling fixtures, each of which would appear very small from your vantage, they would probably be uncomfortably bright to look at — they would have high luminance. Or if the illuminance originates in an entire ceiling covered by diffusing light panels, which spread the light over a large area, the panels may appear comfortably dim — they would have low luminance. Lighting engineers apply this fact in the design of light fixtures that deliver the same illuminance while spreading the source luminance over a larger visual area.  

Third, the "area intensity" of luminance is constant across distance: the luminance of a light is the same, whether it is 1 meter or 100 meters away! Since the illuminance of a light source is proportional to the inverse square of its distance, the incident light decreases as a light moves farther away. But the visual size of the light also gets smaller from our viewpoint, and by the same inverse square proportion. As a result, the ratio between incident light and visual size remains constant. As lights recede from us, they become dimmer but also proportionally more concentrated in visual area, so we perceive the source as having a constant environmental brightness. The same is true for relative luminance (illuminance times surface reflectance): material surfaces appear to have the same lightness regardless of distance.

Fourth, luminance is the photometric unit that most closely approximates the perceived brightness of a light or the lightness of a surface as viewed by a camera or human eye. If illuminance is the measure of incident light, luminance is the most appropriate measure of perceptible light. Luminance is always visible, and is therefore complementary to illuminance, which is always invisible.

However, despite the fact that luminance is constant across changes in distance from the light source, luminance and the perception of brightness/lightness are only loosely related. Perceived brightness depends on the apparent distance of the light (lights appear fainter as they move farther away), and the lightness of surfaces depends on the local contrast with other surfaces; and both judgments are affected by the level of luminance adaptation.  

Luminance & Optics. The luminance of a light source as imaged in an optical system, such as a camera or the eye, introduces some specific issues.
 

the pinhole geometry of luminance
the solid angle of the image equals the solid angle of the light source surface; both are constant across distance

 
In the simplest case, a pinhole camera contains no lens, only a hole that is extremely small in relation to the distance D to the light source and the focal length F to the image plane. The pinhole causes each point s on the source to project a single point image of itself on the image plane. As the solid angles S and I are therefore equal, the total light passing through the pinhole is equal to the illuminance (not the luminance) of the source at the pinhole.
 

cosine corrections in luminance

The pinhole is in turn a point source inside the camera, creating the image at focal distance F. Since moving the image plane away from the pinhole makes the image area (I) larger by projection, but does not increase the illuminance into the image, increasing the focal length makes every part of the image dimmer. So the image luminance is determined by the pinhole illuminance divided by the focal length squared (F2):

luminance [image] = illuminance/F2

What happens if we increase the size of the pinhole aperture, to let in more light? This increases the area admitting light:

luminance [image] = aperture area * (illuminance/F2)

which produces a much brighter but optically blurred image. The blurring effect of increased aperture is overcome by a lens (or parabolic mirror).
 

the optical geometry of luminance
the solid angle of the image is proportional to the ratio of the distances D/F

 
The solid angles I and A on opposite sides of the lens are no longer equal — the lens refracts the light "rays" into a wider solid angle. However, the two solid angles have in common the surface area of the aperture or lens opening, therefore the ratio of the two solid angles equal to the ratio of the two distances, D2/F2. This is the power or light concentrating capability of the optical system.

Assuming D is very large relative to F (as for binoculars or a telescope), then the solid angle I becomes larger, and the power of the optical system increases, as the focal length F becomes shorter. When the lens produces a focused image, then the physical image area is proportional to the physical source area in the ratio F2/D2 — that is, the higher the optical power, the smaller the focused image.

Each ratio is a reciprocal of the other, so they neatly cancel each other out when the solid angle I is multiplied by the image area (or when solid angle A is multiplied by the source area). As a result, for a completely transparent lens:

luminance [image] = luminance [source].

This essentially restates the third luminance property of light sources — luminance remains constant across changes in distance — because the image of an object in any optical system grows larger or smaller in the same inverse square relation to the object distance.

The quirk here is that the image illuminance increases with optical power (shorter focal length), just as it does with increased aperture. This is because a shorter focal length images a larger solid angle (field of view), and this "wide angle" image captures more light at the same time that it reduces the image size of every image element, increasing its luminance. Both effects increase the incident light (illuminance) at any point within the image area and the luminance (brightness) of the image. For this reason, a wide angle lens exposes a photographic film more quickly than a long focal length telescopic lens of equal aperture, and a magnifying glass ignites a piece of paper in sunlight, because the sun's heat is condensed from the aperture area of the lens (its shadow area on the paper) while the short focal length concentrates this light into a tiny spot.

Luminance & Surfaces. The luminance of reflecting surfaces is potentially complex and depends on (1) the illuminance onto the surface, (2) the surface reflectance or albedo of the surface (the proportion of light falling on the surface that is reflected from it), (3) the angle of incidence of the light, (4) the angle of view to the surface, and (5) how much the surface diffuses or randomly scatters the light. These issues are explored on a later page, but a few points should be mentioned here.

There are several alternative luminance measures that attempt to equate the luminance of surface with the illuminance incident on it. The most common are the metric apostilb and millilambert (preferable to the inconveniently small lambert) and the USA foot lambert:

1 candela/meter2 (1 nit) = 3.1416 apostilbs
= 0.3142 millilambert
= 0.2919 foot lambert

These are all related to the general formula for the luminance of a surface that is viewed perpendicular to the surface and illuminated from all sides by a perfectly diffuse light:

The viewing geometry is reversible. Thus a perfectly matte surface (which does not produce the appearance of gloss or reflection) with a reflectance of 0.4 that is illuminated at 500 lux by a single light source perpendicular to the surface will have an average luminance from all viewing angles of about 200 apostilbs or 64 cd/m2.

Note that reflectance is not the lightness (L), which is a visual sensation, but the proportion of the total incident light reflected by the surface. Rescaling this quantity on pi is necessary because light arriving from a single direction onto a physical surface is scattered or diffusely reflected in all directions, dimming it from any single point of view.

The apparently straightforward concept of reflectance (the proportion of incident light that is reflected by a surface) has its own quirks. Reflectance is normally measured as the proportion of light reflected perpendicularly from a surface that is illuminated from all directions by an evenly diffusing light source (or, equivalently, the proportion of light observable from all angles of view from a surface illuminated perpendicularly). It is in effect the reflected light observed from a single viewpoint but averaged over all angles of illumination. However, it is sometimes more appropriate to evaluate colors using a limited angle of view and illumination — for example, in the graphic arts, to view the flat surface of an artwork perpendicularly while illuminating the surface from one side (at a 45° angle of incidence) to minimize the alterations in color appearance produced by surface gloss or reflections. In these situations reflectance is more commonly measured as the reflectance factor, defined as the light reflected from a surface as a proportion of the incident light that would be reflected, under identical lighting and viewing conditions, by a perfectly reflecting diffuser (an idealized physical surface that is both perfectly reflecting or "pure white", and a Lambert surface that reflects incident light equally in all directions). Reflectance is a constant for any surface, while the reflectance factor of a surface can change substantially depending on the angle of incident light and the angle of view.

In most situations surfaces reflect to the eye only a small portion of the light incident on them: a "middle gray" (with L = 50) reflects less than 20% of the incident light. In addition, surfaces are typically much larger in visual size than the light source, and the luminance of equally illuminating light sources varies with their size. This paradoxically causes smaller light sources to appear brighter at the same time that it causes the surfaces under them to appear dimmer in comparison.

As an explicit example, assume an evenly diffusing (matte) white surface 1 meter square, placed 5 meters below a perfectly diffusing light panel 1 meter square that provides 100 lux of illumination to the surface. Then the light source, viewed from 5 meters, will appear to have a luminance of 2500 cd/m2 while the surface will have a luminance of about 30 cd/m2: the light source appears more than 80 times brighter than the surface it illuminates. Replacing the diffusing panel with a single 6" spot lamp that again provides 100 lux of illuminance will increase the luminance of the source to over 137,000 cd/m2, while the surface luminance is unchanged; now the light appears almost 4600 times brighter than the surface. Thus, diffusing panels allow the same luminance output to reach across larger architectural spaces with the same surface luminance effect; for the same reason, the ground appears brighter than the sky on heavily overcast days. In general, all unfiltered light sources (the sun and concentrated incandescent lights such as a bare tungsten filament) always look far "too bright" or glaring compared to the illuminance they provide on surfaces. Alternately, sources that are comfortable to look at directly (such as the full moon, a candle or a small flashlight) typically produce a weak illuminance that is useful only in near darkness.

Paradoxically, glossy or shiny surfaces will usually appear darker than matte surfaces having the same reflectance, unless the light source is reflected directly to our eyes. This is why the highly reflective waters of a sea or lake appear dark in comparison to highly reflective beach sand: the sand diffuses light equally in all directions, whereas the water reflects light primarily in the direction where the sun's image is visible on its surface.  

Retinal Illuminance is a measure of the amount of light that actually enters the eye. It is measured in trolands and is derived as the luminance of the light source multiplied by the area of the observer's pupil in square millimeters. Because pupil sizes vary from one individual to the next and within the same individual across different light intensities, the troland has only an empirical or observed relationship to source luminance; but at typical indoor levels of illumination (around 300 lux):

1 candela/meter2 = ~10 trolands

Thus, the moon's retinal illuminance in trolands is greater at night than it is during the day, because at night our dark adapted pupils are larger.

The troland is primarily used in vision research, and to standardize measurement across different individuals, an artificial pupil is placed in front of the viewers' eyes. This is a small hole in an opaque screen that is of a fixed diameter smaller than the minimum pupil diameter found in normal eyes; the area of the artificial pupil is used to calculate the troland.

The troland does not take into account most of the perceptual changes involved in luminance adaptation between day and night, so (like luminance) the troland does not describe very accurately the subjective sensation of brightness. It simply allows more precision in the estimate of light actually incident on the retina.

James Calvert has posted a useful Illumination tutorial that explains the geometrical definitions and standard formulas for photometric units.

 
the colormaking attributes
 
The physical measurement of light must be joined with an accurate description the subjective color experience produced by the color stimulus. This description is based on three colormaking attributes: (1) brightness/lightness, (2) hue and (3) hue purity (chroma or saturation), first defined by Hermann Grassmann and Hermann von Helmholtz in the 1850's.

Exactly how the three colormaking attributes relate to the three L, M and S cone fundamentals, the physiology of color vision, is not a concern. The only goal is to provide an unambiguous way for individuals to describe a color experience as a number or quantity on three standardized and easily recognized attributes.

The appearance of a color is a judgment based on context — the setting in which the color is viewed and our luminance adaptation and chromatic adaptation to that setting. But at the same time we can make an absolute color judgment, a kind of color perception in an ideal color space independent of the viewing context or our visual adaptation. This allows us to compare and recognize colors across different situations, for example when we perceive that a white piece of paper is brighter under noon sunlight than midnight moonlight, or that the colors of sunset are redder than those of noon. The relativistic colormaking attributes, those influenced by our visual adaptation and the viewing context, refer to related colors, while the absolute color judgments are roughly equivalent to unrelated colors.  

The three context attributes most important to color perception are:

• the intensity and color of the illumination — by far the most important context element. Different terms apply to the two separate attributes of a light source and their combination:

– illuminance refers to the quantity or intensity of light incident on the color area, which (via the light reflected from surfaces) determines the level of luminance adaptation.

– illuminant refers to an abstract relative spectral power distribution that characterizes the chromaticity of an idealized light source independent of its brightness; the actual spectral power distribution of a light determines its color rendering properties and the chromatic adaptation imposed on the visual system

– illumination refers generically to the intensity and color of the light incident on the color area and surrounding surfaces.

• the relative luminance contrast between a color area and its surroundings, which determines its perception as an emitting light or a reflecting surface. Colors appear self luminous when their luminance is much greater than a recognizably "white" surface; they appear as object or surface colors when their luminances are less than "white". Colors that cannot be clearly identified as either lights or surfaces are called aperture colors. Additionally, the chromaticity contrast (the relative luminance contrasts within specific parts of the spectrum) between a color and the surrounding surfaces often alters color perception.

• the spatial interpretation of the scene, which determines the three dimensional relationship between different surfaces, and between all surfaces and the (usually multiple) light sources in the viewing context.

To make accurate color judgments, these contextual factors (and others, such as gloss, texture or specular reflections) either must be eliminated (as in unrelated colors) or explicitly standardized (as related colors viewed on a medium gray background under a diffuse, moderately bright, full spectrum "white" light).

Physical Stimulus, Perceptible Stimulus and Sensation. The colormaking attributes provide a flexible and unambiguous description of color sensations as experienced in lights or surfaces. But this entails that they do not describe the physical qualities of the color stimulus, and are not equivalent to any measured quantity of the stimulus.

For example, a subjective quantity of the visual sensation of brightness (for example, a light that is described as "painfully bright") is not consistently related to a specific physical quantity of light (say, 10 watts of radiant power) or a specific perceptible quantity of light (say, as 6800 lumens). The 10 watts might arrive as "green" or "red" light, which will alter its apparent brightness; the 6800 lumens might be viewed as a flash of light in high photopic adaptation or in complete scotopic adaptation, which will alter its visual impact. Many other qualifications or unique circumstances are possible.

Again, context matters to visual perception. It is important to keep distinct the three conceptions of the stimulus — as physical quantities, as perceptible quantities, and as sensations — because a generalization based on one conception may not apply to the others. The colormaking attributes literally describe color sensations, and nothing else.

Even so, provided the contextual issues are appropriately limited or standardized, and within a generous allowance for measurement error and individual differences in visual capabilities, correlates or equivalents to the colormaking attributes can be computed from the physical or perceptible quantities of a stimulus. These form part of the color specification in nearly all modern color models.  

Brightness/Lightness. The first and most important colormaking attribute is the light or dark of a color as it appears in emitted or reflected light. This is perceived in two distinct ways:

Brightness refers to the relative sensation of light as emitted or reflected from a color area, given the current level of luminance adaptation. This is a sensory definition; it is weakly correlated with the perceptible luminance of the color area, as explained below.

Lightness refers to the brightness of a colored surface as a proportion of the brightness of an area perceived as "white" under the same illumination and light adaptation. This is also a sensory definition; it is strongly correlated with the physical measurement of the relative reflectance (luminance factor or albedo) of surfaces in comparison to the reflectance of a perfectly diffusing ("bright white") surface, within the photopic to high scotopic range of illuminance and given a wide range of different reflectances in the field of view.

For objects or surfaces, extremes of lightness are usually described as dark or black up to light or white; for self luminous areas (lights) the terms are faint or dim up to bright. The example below shows variations in the lightness of a dull (low chroma) middle blue hue.
 

differences in lightness
hue and chroma held constant

 
Lightness is associated with reflectance or average luminance factor judged against the reflectance of a white standard (diagram, right), and this contextual "white" anchor makes lightness a related color attribute. An arbitrarily defined "white surface" is actually the benchmark that is used to compute correlates of lightness from the measured luminance of illuminance of a colored surface. Perceptually, a "white" standard somewhere in view is not essential in order to see lightness differences; we usually have a secure sense of the amount of light falling on surfaces, and our luminance adaptation to the light, because of the variety of surface reflectances within a scene.  

Brightness and lightness are correlated with the luminance of a surface or light source. This means that brightness and lightness usually go up or down as the color luminance goes up or down, but whether and by how much depends on the viewing context. Let's first review the relationships among context factors and then summarize how they affect brightness/lightness perception. (An expanded version appears in color in the world.)
 

context factors creating the perception of brightness/lightness

 
We perceive physical environments because of the light incident on material surfaces, which depends on the source intensity or illuminance of the light source. The illuminance is the luminous intensity of the light source reduced by its distance from the illuminated surfaces. Averaged across all directly lit surfaces, this is the scene illuminance.

Illuminance is separate from the source image or luminance of the light source, which depends on the source intensity, its distance, and its visual size from our viewpoint. An extended, diffuse light source, such as an overcast noon sky or ceiling light panel, can provide substantial illuminance but, as a source image, appear very dim. This is because, at equal illuminance, luminance increases as the visual size of the source image gets smaller: an incandescent filament has much greater luminance than a light panel.

Environmental surfaces reflect more or less light depending on their surface reflectances. The combination of scene illuminance, shadows and surface reflectances defines the surface luminance range — the variations in the brightness of the physical enviroment. This is the anchor of luminance adaptation for two reasons: the luminance of surfaces is constant across distance (as with lights); and, for diffusely reflecting surfaces, luminance is not significantly affected by the angle of view or the angle of incident light.

Exactly how luminance adaptation occurs is not clear, but it apparently requires three simultaneous adjustments in the visual response: (1) a receptor gain adaptation to the average scene luminance (the adaptation gray, Lg,equivalent to a reflectance of about 13%), (2) a cognitive lightness anchoring that links the highest surface reflectance (no more than 5 times the adaptation gray) to the perception of "pure white" (the adaptation white, Lw), and (3) a perceptual expansion or contraction of the lightness range so that a surface presenting a luminance of about 1/5 of the adaptation gray is perceived as black (the adaptation black). As a result, color areas with luminances within the range Lw to 1/20Lw are perceived as objects varying in lightness, or color with some gray content.

The lightness range orphans many specific color areas that have greater luminance than the adaptation white, including areas of spot illumination (sunlight falling on the floor through a window), gloss or specular reflections, and secondary or primary light sources. These appear as lights varying in brightness, or color with some brilliance content. A stimulus darker than the adaptation black is invisible unless silhouetted against a lighter background, where it appears as a void. Brightness and lightness are both necessary to perceive the total range of luminances, from voids to source image, that can appear in physical environments.

Brightness perceptions are powerfully affected by the level of luminance adaptation and the luminance of the surrounding area. Lightness perceptions are remarkably consistent and stable, provided all surfaces are under the same illuminance; but lightness differences can be powerfully affected by the perceived spatial geometry of material surfaces and light sources, especially when these define the perception of average luminance, spot light and shadow.  

Within this general context description, the brightness/lightness of a color area depends on:

• Luminance adaptation. The visual adaptation to light intensity sets the perceptual boundary between surface lightness and light emitting brightness, and sensitivity to luminance differences within each range.

In most environments, the anchor for light adaptation is the average quantity of light reflected from surfaces — the scene luminance range — which is actually the reflected image of the scene illuminance. For both surfaces or lights of constant luminance, brightness decreases as light adaptation increases, and conversely brightness increases as dark adaptation increases. The lights and colors of a film appear dim as we enter the theater but brighten after our eyes become adapted to the dark, though there is no objective change in the luminance levels of the film. Even "gray" sidewalks or building walls have a high brightness, and automobile colors appear more vivid, as we exit the movie matinee, but these effects are muted as our eyes adjust to the light. In the same way, emitting lights appear to grow dimmer, whites appear brighter, surface colors appear more chromatic, and the contrast among whites, grays and blacks is greater, as scene illuminance increases, although these effects partially disappear as we adapt to the new illuminance level.

Under mesopic or scotopic vision (dark adaptation) we also experience a sensory change in the appearance of lightness: lightness contrast declines and "white" surface appears perceptually to be gray, as compared to the memory color of a white surface under photopic illumination. Under scoptopic vision only light emitting sources (such as the moon) appear perceptually as a "pure white".  

• Luminance Contrast. Lightness and brightness are local contrast judgments, not direct perceptions of light acting on the retina. So the relationship between a color's luminance and its perceived brightness or lightness is strongly affected by the visual context.

The key factor is relative luminance contrast. The light emitting or "brilliance" quality of brightness is perceived in color areas with 2 or more times the luminance of a white surround, or roughly 40 times the luminance of a dark gray or black surround. Lights appear brighter in relative darkness because of the substantially reduced surround luminance and lower luminance adaptation. And brightness contrast is increased if the brighter color area is made visually smaller, even when the contrasting color areas are surfaces of constant reflectance.

At night a flashlight appears "bright", and "brighter" than a candle, because the contrast is with a dark surround a dark adaptation; under a noon sun, both the candle and flashlight are invisible, because they produce a negligible luminance increase in relation to the average surface luminance and the eye's light adaptation. "Bright" also describes specular reflections that are visually much smaller than the source image, and surfaces whose luminance exceeds the current adaptation white due to spot illumination.  

The reverse is true for lightness: lightness contrast increases with increasing illuminance (the Stevens effect). More gradations of lightness become visible, and the visual contrast between lightness intervals appears greater; whites appear brilliant and darks appear deep black. Lightness contrast is quite pronounced under noon sunlight and becomes softened or muted at twilight. As illuminance decreases, the visual contrast between lights and surfaces becomes more extreme, and even very dim lights acquire brightness. Hence the filmmaker's trick of day for night, which creates the illusion of night by shooting daylight scenes under reduced exposure, darkening the image luminance and reducing the image contrast.
 

lightness equated with the proportion of light reflected

in comparison to a white surface under the same illumination

So long as the pattern of lights and darks on a surface remains the same, then lightness appears constant across changes in illuminance (diagram, right). This is because lightness perception only depends on surface reflectances (surface luminances) relative to each other or as a proportion of white. The black print in a book reflects about 10% of the incident light, and the white paper about 90%, defining a ratio of 1:9; these proportions and ratio do not change if the quantity of incident light (the illuminance) is increased or decreased, so the perceived lightness is constant.

A restricted range of luminance contrasts usually creates the lightness scale of grays. It is sometimes claimed that we cannot see the color "gray" in lights, but this is belied by the grays in the diagram at right, which are generated by the pixel sized lights in your computer monitor. We cannot see gray in lights viewed in isolation or as recognizable sources of illumination; the lights appear veiled or dim instead.

Lightness and brightness are complementary regions on the luminance dimension: normally lightness masks direct perception of brightness, and vice versa. The "blacks" in a television picture have the same absolute luminance (brightness) as the "gray" monitor screen when the television set is turned off: they produce a black color in the video image through contrast with higher luminance pixels around them.

Lightness is substantially affected by the contrast between a color area and its surround. The lightness of a color area can change, sometimes radically, depending on the lightness of surfaces that are visually next to or behind it (simultaneous contrast). In particular, a dark background or surrounding color will make a color area appear lighter; a light valued surround will make the color appear darker. A classic example is the full moon in either the day or night sky, which appears white although it is actually very dark (its albedo, equivalent to its reflectance, is 7%).  

Finally, as the "radiance" visible on surfaces or from the source image of lights, brightness signals a change (contrast) in illuminance or luminance across space, time or context. Relative luminance differences are perceived as constant lightness patterns across changes in illuminance, but they are perceptually compounded of a fixed quantity (reflectance) and a variable quantity (illuminance). In particular, brightness is the sensory token (the conscious attribute) for (1) a luminance perceived to exceed the lightness range, or (2) a illuminance change or contrast that requires an up or down adjustment in the perceptual interpretation of the luminance quantities associated with whites, grays and blacks. If illuminance is everywhere constant and equal across surfaces, and we have adapted to the scene illuminance, then we only perceive surfaces of different lightness; the perceptual quality of brightness is absent.
 

context differences between brightness (left) and
lightness (right)

Brightness becomes more salient than lightness when:

• the scene illuminance changes by a large amount (the sun comes from behind a cloud; we exit a movie theater)

• the local illuminance changes (we move an object from shadow to light, we turn on a desk lamp)

• there is a brighter or darker spot illumination on a surface (cast shadows, volume shadows, a beam of light on the floor)

• we see a surface reflection (the moon on water, the sun in an automobile windshield)

• we see a luminous color area against a dark surround, or a void within light reflecting surfaces.

In all these cases we see light as a distinct attribute that is more or less separate from surface.

As a result, perceptions of brightness do not allow a luminance match between surfaces that differ both in lightness (grays) and in local illuminance. For example, it is difficult to adjust an indoor spotlight illuminating a light gray paper so that the brightness of the paper matches the brightness of a dark gray paper in sunlight (diagram, right). The local judgments of relative luminance under local illumination override the global comparisons of absolute brightness. (See also the tiled cube example, below right.) However, these brightness matches are quite easy to do if only the color areas, without any surrounding cues of the scene illuminances, are visible through small apertures.
 

brightness comparisons across different grays and illuminances
are unreliable

• Spatial Interpretation. Relative lightness differences are greatly affected by the spatial or three dimensional interpretation of an image. This is because the angle of surfaces in relation to each other, and to the light source, determine the illuminance incident on the surfaces (which is less for surfaces at a more oblique angle to the light) and in particular the contrast between light and shadow.

In general, we see illuminated darks as darker than their actual reflectance, and shadowed lights as lighter than their actual reflectance. In the example (at right, top), the central "dark gray" tile on the illuminated side of the cube is the same monitor luminance and measured lightness as the "white" tiles on the shadowed side, which is evident when all other tiles in the form are removed (diagram, right, bottom).

The spatial interpretation of illumination differences between light and shadow, and the apparent match in of surface patterns on all sides of the object, obliterate a direct comparison of the brightnesses: again, lightness masks the perception of brightness. The tiles on the shadowed side are perceived to have a higher lightness because the visual system compensates for the effects of the virtual shadow.  

• Spot Illuminance. Finally, the brightness or lightness of surfaces depends on the continuity of the scene illuminance.

In most cases of spot illuminance (a local area of increased illuminance), the visual system registers the absolute increase in luminance as an increase in surface brightness. (Refer to this discussion for the distinction between relative and absolute luminance changes.) Similarly, a local are of reduced illuminance is perceived as a shadow. We don't see patches of sunlight through leaves as white spots on the ground; we see them as brighter versions of the same lightness visible in the surrounding shadows.

A surface can appear mysteriously darker or lighter than it normally appears if we cannot perceive the relative illuminance difference in a visual comparison. If we are sitting in a dark room, and see the sunlit asphalt pavement outside through a narrow opening in a curtain, the pavement can appear white or light gray rather than black. In the same way, a black paper hung in complete darkness and illuminated by a narrow beam of intense light will appear quite white, so long as nothing else in the room is visible. Its appearance snaps to black if a gray or white surface is also placed in the beam of light.

It is even possible to contrive situations in which the spot illuminance cannot be perceptually separated from the spatial definition of surface patterns or surface edges, or attributed to visible light sources or cast shadows in the scene. For example, a beam of light or shadowing form can be arranged so that the edges of the light or shadow corresponded exactly to the edges of a single white or black tile in a checkerboard floor. In that situation the discrepant surface luminance will appear as an isolated gray tile, or — if the spot illuminance change is large enough — as a square light source embedded in the floor, or as a square hole.

Terminology. Artists usually talk about a painting without concern for the lighting of the situation where it is viewed, and they interpret landscape values into paint values that will appear under different kinds of illumination. For those reasons, the related color judgment of lightness (or the artists' term, tonal value) is the concept to use when discussing works of art; brightness should be used to describe the landscape, studio or gallery lighting.  

Hue. This is the most familiar color attribute, the one that answers the familiar question, what color is it?

Hue is identified by a categorical basic color name such as red, yellow, green or blue; or a compound of two basic names such as yellow green; or a secondary color name such as orange. The example below shows several different hues of equal lightness and hue purity.
 

differences in hue
chroma and lightness held constant (colors of equal nuance)

 

lightness judgments affected by the spatial interpretation of light

the gray tiles have identical lightness (L = 60) in both images

Hue is usually associated with the average or strongest wavelengths in the light spectrum, regardless of the total range of wavelengths present in the stimulus (diagram, right). As the language categories for hue are imprecise and inconsistent, hue is often described by matching it to the color of a monochromatic light, denoted by its wavelength. This is called the dominant wavelength of the color: the dominant wavelength of yellow is 575 nm.

However, monochromatic lights change hue slightly if their brightness or chroma changes (as discussed here), and can shift substantially in contrast to other hues around them (as discussed here). So the match between a specific hue and a spectral wavelength is relative to the viewing context. That is, any color stimulus can be described by a dominant wavelength, but the dominant wavelength does not define the appearance of the color stimulus in all situations.

If many wavelengths are involved, the hue is determined as the average or geometric mean of all the wavelengths on a chromaticity diagram, not on the linear spectrum. That is, the hue created by a mixture of "red" and "violet" light (at the ends of the spectrum) is not green (in the middle of the spectrum) but purple (outside the spectrum, but between red and blue on a hue circle). For these extraspectral hues, the dominant wavelength is the complementary color ("green" monochromatic light) that exactly neutralizes the color mixture, denoted by a "c" placed before the wavelength number. The dominant wavelength of magenta is c530 nm.  

As this matching procedure implies, hues are limited to spectral (prism) colors and the extraspectral mixtures of spectral "red" and "violet" light. In English these hue names are magenta, red, orange, yellow, green, cyan, blue and violet or purple; compounds of these names such as blue violet or yellow green; and specific names for saturated colors such as scarlet, orange, chartreuse or turquoise.

Hue specifies the location of the color around the circumference of a hue circle, not any color location toward the center. Names of dull or muted colors such as white, gray, black, brown, maroon, pink, tan, gold, russet, olive and so forth do not describe spectral colors, and this rules them out as hue names, even though they may be appropriate answers to the question, what color is it?

Even so, artists should learn the correct hue designations for dull colors. "Brown" for example is technically a near neutral, dark valued orange with a dominant wavelength around 610 nm; "olive" is an dull, mid valued yellow with a dominant wavelength around 570 nm. You will never be comfortable describing your coffee as dark orange and your martini olive as dark yellow, but that is what they are; and accurately recognizing the hue of any surface color will help you to mix that color using a color wheel and to understand how the color is likely to change appearance under different types of lighting or from light to shade.

Hue is an attribute of both unrelated and related colors. We can easily identify the hue of traffic lights at any time of the day or night, and we can judge the hue of any surface provided that we know whether the illumination is bright or dim, and "white" or tinted.  

In general, apparent hue remains constant across wide changes in daylight illumination. In particular, changes in the sun's light from morning to afternoon, or in cloud cover, don't significantly affect hue perception. However, the contrast between similar hues, and their saturation, does appear to increase as illumination increases, and under dark adaptation (at night) hue perception in surfaces disappears, though we can see hue in lights (such as the planet Mars and distant traffic lights).

Reliable hue recognition can go awry in several unusual or extreme situations: (1) the surface is viewed without surrounding colors and without an accurate idea of the intensity and color of the light source; (2) the viewer has been adapted to one colored light source, and the illumination changes to another color or to white; (3) the hue is viewed in contrast to adjacent color areas of strongly different color and brightness; (4) the illumination has an intense (pure) color; (5) the spectral power distribution consists of a broken spectrum that emits only a few wavelengths or many wavelengths at very different intensities; (6) colors are viewed at extremely high luminance levels that saturate or overwhelm the photoreceptor cells; or (7) colors are viewed through a positive or negative afterimage.

Most of us are familiar with the grossly distorted automobile colors that appeared under yellow sodium vapor lights, or the dulling effect of fluorescent lights on reds and yellows. Abrupt changes in lighting color, for example when we step from daylight into the red light of a photographic darkroom or bar, produce especially inaccurate hue judgments. Color distortion is obvious in surface colors around sunrise or sunset, but this effect is familiar enough, and sufficiently minimized by discounting the illuminant, that it has a trivial effect on hue recognition.

Terminology. Artists use both unrelated and related color judgments to determine the paint mixtures needed to match colors in the environment. Related color judgments refer to the "true" or local color as it would appear on a normally lit surface (which is how artwork is typically displayed), even when we see the surface under unusual lighting conditions. Monet's advice, that the artist simply match the hue of a retinal color patch, means a painter should ignore local color and instead match the hue as it appears under the influence of any contextual factors.  

Hue Purity. The third and last colormaking attribute is the clarity or intensity of hue, again where hue has the limited meaning of monochromatic spectral colors and extraspectral mixtures.

Hue purity ranges from intense or highly chromatic for pure hue sensations to neutral or achromatic for completely colorless (white, gray or black) sensations. However it is common to find a very chromatic color (such as a "blue" monochromatic light) described as saturated, pure, bright, brilliant, rich, vivid, luminous or glowing, and an achromatic or near neutral color as unsaturated, impure, dirty, dull, dead, veiled, dark, pale, whitish or subdued.

The substantial overlap in the adjectives that describe chroma and brightness (and between both of them and the adjectives that describe vitality and intelligence) signifies the sensory and "moral" similarity between the two. However it is a parallelism rather than a polarity: chroma has its null state (gray) and brightness its null state (darkness). They are otherwise polar opposites: luminance is a broadband quality while chroma is targeted to spectral subunits; the most saturated possible hues, spectral lights, appear black if viewed as surface colors; and a strong luminance contrast by itself can produce both a high chroma and a luminance color perception.

The example below shows variations in the chroma of scarlet at constant hue and lightness.
 

differences in chroma
hue and lightness held constant

 
Hue purity is the most fascinating and problematic colormaking attribute. Whereas hue is an unambiguous percept that can be associated with a precise physical property (wavelength), and brightness is a somewhat complex percept associated with a precise physical property (radiance or luminance), hue purity is at once the most striking color property and the property that is most difficult to define in terms of specific stimulus attributes. In fact, many definitions of hue purity are obtained as the residual dimension in a geometrical color model: hue purity is whatever stimulus contribution remains after luminance and hue are accounted for.

For these reasons, hue purity has gone by many names — Sättigung, colorfulness, chromaticness, chroma, saturation, excitation purity, colorimetric purity, chromatic content, brilliance — each defined in relation to a specific stimulus attribute or color viewing situation. For now I use hue purity to refer generically to the vibrancy or intensity of a hue, but give the term a specific definition at the end of this section.

Three related definitions of hue purity as a perceptual dimension of color are current in the color research literature (e.g., in Fairchild's Color Appearance Models):

1. Colorfulness is the attribute of a visual sensation according to which the perceived color of an area appears to be more or less chromatic.

2. Chroma is the colorfulness of an area judged as a proportion of the brightness of a similarly illuminated area that appears white.

3a. Saturation is the colorfulness of an area judged as a proportion of its own brightness.

These perceptual definitions of hue purity, despite the uncertainty the reader may have as to how exactly they compare with one other, highlight the sibling relationship between hue purity and brightness/lightness as color sensations. This is an issue I will address below, when we consider the context factors that affect perceptions of hue purity.  

Stimulus Definitions of Hue Purity. First, let's consider the physical side of color psychophysics: what is a good color stimulus definition of hue purity?
 

hue equated with the dominant
or "average" wavelengths of light

Start with lights. We know that the most saturated physical stimulus possible is single wavelength (monochromatic) light, and the least saturated physical stimulus possible ("white" light) is the approximately equal mixture of all wavelengths across the entire visible spectrum. So the logical first step is to define hue purity as the spectral breadth of the elevated part of an emittance curve: hue purity is the number of different wavelengths in the spectral power distribution of the light (diagram, right). By this principle, a low saturation color stimulus reflects or emits wavelengths across a large part of the spectrum. As wavelengths become concentrated within a single, narrow span of the spectrum, the color's hue purity increases; but as this happens, invariably, the brightness of the color area decreases.

The problem with the "wavelength purity" definition of hue purity is that the eye does not respond equally to the "breadth" of wavelength mixtures. The span of "red" wavelengths labeled "medium" in the diagram would appear just as saturated as a single wavelength of "orange" light, because the "red" wavelengths do not lose hue purity when mixed. In contrast, if the "medium" span of wavelengths were centered on the "yellow green" wavelengths, the resulting mixture would appear desaturated almost to "white". Emittance and reflectance curves do not represent these quirks of color perception, so hue purity cannot be reliably inferred from the wavelength purity of the color stimulus.  

A second solution: define hue purity in terms of a standard light mixture. Hermann von Helmholtz, the 19th century creator of colormaking yardsticks, proposed that Sättigung was the proportional mixture of "white" and pure monochromatic light of equal brightness. This confines all hue purity measures to a consistent wavelength standard, creates a constant hue purity "ruler" (a mixing line between 0% at "white" and 100% at the pure spectral hue), and defines a practical method to manipulate hue purity in a color vision experiment (mix "white" with a single wavelength light of equal brightness). Best of all, this method matches an observer's perception of colored lights, which always appear to whiten as they are desaturated.

This definition is also simple to apply to surface colors as the visual mixture of white, black and color, for example by means of the pattern developed by August Kirschmann (diagram, right) to produce equal changes in surface color chroma at constant lightness when the disk is spun as a color top.

But while the Helmholtzian approach is an elegant way to define the color stimulus, it has a serious flaw as a definition of the color perception: spectral hues do not create equal sensations of hue purity. "Yellow" monochromatic lights have a whitish, pale color and a weak tinting strength when mixed with "white" light; "violet" monochromatic lights are dark, extremely intense, and very potent at tinting "white" light. Some perceptual quality different from the stimulus quality of "spectral purity" is necessary for viewers to say that saturated "yellow" light, the most saturated possible yellow stimulus in lights, is still not very saturated.

This seems logically to implicate the judgment of hue purity as a comparison with something else used as a standard reference. That "something else" is achromatic luminance — the perceived brightness or lightness of white.
 

hue purity defined as the
breadth of an emittance profile

 

a kirschmann disc

(top) the fixed pattern of white, black and pure color, mixed in equal steps across concentric rings; (bottom) the color appearance as a color top

Colorimetric Definitions of Hue Purity. To clarify this comparative approach, let's start by looking at how chroma is implemented in the chromaticity plane of a uniform color space, for example CIELUV (diagram, right). A chromaticity plane is based on the proportion of stimulation produced by the color stimulus in each of the three types of cones: we are no longer considering the physical mixture of the stimulus but the response mixture in the retina.

First, compare this diagram with what we already know about spectral hue purity. The "yellow" to "red" spectral hues lie along a straight line, and therefore retain spectral hue purity when mixed; the "green" spectral hues are strongly bowed, which brings their mixture closer to the achromatic white point (WP); the distance from the white point to the spectrum locus of "yellow" (Y) is quite small, indicating that the light appears relatively pale, while the distance from the white point to spectral "blue violet" (B) is quite large, indicating that the light appears quite intense. (Recall that these variations in spectral color arise because of the overlap in cone fundamentals across the spectrum, which makes it difficult for a single type of cone to be stimulated independently of the other two.) The general perceptual facts are accurately represented.

Now we can define hue purity as the chromaticity distance from the white point to the color in a chromaticity diagram, in which by definition all colors have equal luminance to the white point or pure achromatic stimulus. In other words:

4. Chroma refers to the attribute of a visual sensation which permits a judgment of the amount of pure chromatic color present, regardless of the amount of achromatic color (CIE, 1982).

The wording here is infelicitous, as it implies we can simply add a huge amount of "white" pigment to a stimulus without affecting its apparent chroma. The meaning is that chroma remains constant across a change in the luminance of the achromatic standard, provided that an equal change in luminance is applied to the color stimulus as well. However chroma is an easily interpreted metric in terms of an established color model or color space, as it creates concentric circles of equal chroma centered on the achromatic point (diagram, right).  

Finally, when defined in terms of a color model, it is customary to refer to Helmholtz's Sättigung as a new color attribute, defined in terms of an achromatic and spectral hue mixture on the chromaticity plane:

5. Excitation purity refers to the chroma of a color area judged as a proportion of the chroma of the monochromatic hue of the same brightness and dominant wavelength.

That is, the chromaticity plane in a color model is used to define the chroma of the spectrum locus at a given luminance or lightness, and the chroma of the stimulus is divided by this hue specific quantity. As a result, excitation purity and chroma will yield very different estimates of hue purity, depending on the hue of the color (see diagram caption, above right).  

Context & Hue Purity. Unfortunately, a chromaticity diagram is a map of color sensations, a map that depends on the way we define receptor responses to light. We are still not talking directly about color perception, the psychological rather than sensory side of psychophysics.

To do that, we need to return to the perceptual definitions of colorfulness, chroma and saturation given above, and to think in terms of surface colors, where hue purity is affected by absolute luminance and by relative luminance contrast: it is not a fixed property of a colored material but a relative property of the material viewed in a specific context.

Fairchild illustrates the difference between colorfulness and chroma by means of illuminated cubes similar to the diagram (below).

 

luminance contrast effects on hue purity

 
In this context, the colors of the orange cube can be compared to the cube next to it, which in context we adopt as the "white" standard. Within the illuminance implied by this figure, the quantity of light falling on matching faces of the two cubes is the same, and we infer the cubes are each of a solid color on all sides. Then, just as we see the three sides of the white cube as having the same lightness, although they differ in luminance (brightness), we see the three sides of the orange cube as having the same chroma, although they differ in colorfulness.
 

chroma vs. excitation purity in the
CIE u'v' chromaticity diagram

colors C and c, or Y and y, have equal chroma but unequal excitation purity; colors Y, C and B have unequal chroma but equal (maximum) excitation purity

I find the diagram (right) makes this contrast between colorfulness and chroma easier to understand. Here orange, green and blue squares are shown under varying levels of illumination, which affects them all equally. At low illuminance (bottom), the orange color appears relatively dark and subdued; if we extract the color and display it in isolation, it appears quite brown. As we increase the luminance of the color — the incident illuminance, for surfaces; the luminous intensity, in lights — the colorfulness of the color increases (the square on the left becomes auburn, and finally orange), although, in context with the blue and green (on the right), we perceive a constant orange surface, brightening in tandem with the other colors due to changes in the illumination, which therefore has a constant chroma.  

What about saturation, or the comparison of a color's chromaticity to its own brightness? The natural context for saturation is the perception of lights, because we typically see lights as standing apart from surfaces, so that their inherent brightness is directly apparent. For surfaces we must eliminate a "white" context and instead surround the color with an achromatic background that matches the color's lightness. This yields a definition of saturation that is specific to surface colors:

3b. Saturation is the colorfulness of an area judged in relation to an achromatic (gray) area of equal lightness.

In other words, saturation is just colorimetric chroma divided by brightness/lightness (S =C/B or S = C/L): as lightness or brightness increases, the chroma must also increase by an equal proportion in order for the saturation to remain the same.

Shown below is the original red chroma series, now displayed against an achromatic background of matching lightness:

 

differences in saturation
hue and lightness held constant

 
The robustness of saturation as a perceptual judgment in both surface colors and lights is that any perceptible difference between the color area and a matching gray (or any departure from white in the color of a light) is, by definition, entirely due to the chromatic intensity. (Unfortunately, equating the brightness of monochromatic colors by means of heterochromatic brightness matching, or sorting high chroma Munsell color chips into a correct match with a gray scale, are perceptually difficult tasks, because hue purity appears as a kind of brightness. This is apparent in the red color series above: the intense red does not seem merely purer than the grayed colors, it seems brighter or glowing, as if hue contained a chromatic luminance.

Finally, here is the orange series presented above, now with matching gray surrounds.

 

saturation as elimination of lightness contrast

 
In this situation the difference between the background and the color sample appears constant, and therefore the saturation is constant, although the measured chroma of the color sample (as a digital color) increases to remain proportional to the increase in the color luminance (lightness). The difference between this diagram and the previous diagram (above right) is whether we impute the changes in lightness to a change in the surface color, or to a change in the illumination on the surfaces.

Now, if we are unclear about the source of the luminance change, or if we increase the relative luminance contrast between the color area and its surround, then increasing color luminance or decreasing surround luminance increases colorfulness, chroma and saturation. (This was studied by Ralph Evans under the rubric of brilliance.) The diagram below illustrates the basic effect.

 

luminance contrast effects on hue purity

 
In this example, which only hints at the actual impact of environmental luminance contrasts, the violet and magenta color areas each have a characteristic luminance (CIELAB lightness L*) at maximum chroma, as shown in relation to the adaptation white or black at left. Each color is shown (center) within a surround of matching lightness, and then (right) within a surround that is of a higher lightness (for the magenta) or of lower lightness (for the violet). Although the effect is small, you should see the violet in the darker surround as more saturated ("brighter"), and the magenta in the lighter surround as less saturated ("duller").

What's more, the contrast between a surface color and a brighter surround induces a quality of "blackness" in the color that does not appear in colors perceived as isolated lights or as surfaces brightly lit within a dark surround. Gray and some unsaturated "warm" colors (such as olive, brown or maroon) only live within the substantial luminance contrasts that produce the appearance of blackened surface colors. Dim lights viewed against a dark background do not appear gray or brown but instead as faint white or orange lights.

Provided the color luminance remains constant, minimizing surround luminance increases colorfulness. If the illumination on a color area is increased while surrounding color areas are kept at reduced illumination, the colorfulness of the color area increases. This is a gimmick widely exploited in "tourist trap" art galleries: by illuminating individual paintings with tightly vignetted spotlights, in a gallery space that is otherwise dimly lit, the colorfulness and lightness contrast in the paintings is artificially increased.  

Brightness, Whiteness & Hue Purity. Finally, we can examine the specific perceptual judgments necessary to define colorfulness, chroma and saturation.

Colorfulness is (like brightness) an unrelated color attribute, and like brightness it is related to the luminance of the color. To judge hue purity, a viewer must be able to perceive the colorfulness and the brightness of a color area as distinct attributes.
 

perceptions of brightness in hue purity judgments

 
In the "colored light" example (above, left), a viewer judges the saturation of a color viewed in isolation by comparing the brightness and colorfulness — two distinct and unambiguous qualities of the color sensation. In lights, the white content has a quality of "shine through" that makes the color content appear relatively thin or diluted. Saturated lights typically appear relatively dark, because no "white" light can shine through. The color can be made brighter by adding white light or by increasing the source luminous intensity, and each has a different perceptible effect on the colorfulness — the whiteness is perceived as diluting the colorfulness.

In the "colored surface" example (above, right), colorfulness and whiteness are again distinct qualities of the color sensation, but the origin of the luminance (brightness) is now problematic. What part of the chromatic luminance is due to the color's chromatic intensity, and what part to the illuminance or the absolute intensity of the incident light? To resolve this ambiguity, the viewer must judge the illuminance by viewing its undimmed reflection as the luminance of the "white" standard.

Implicitly, the white standard is only chosen after luminance adaptation to light source, so that the effect of adaptation on color appearance is taken into account. However, because viewers may quite readily adopt a gray (or even black!) as the "white" lightness standard when surfaces are viewed in isolation or differently illuminated, judgments of chroma can be highly dependent on context and can be radically changed if the structure of light in space is misperceived. This is explicit in the examples above: increases in the monitor chroma of the color in a digital file are perceived as the "same chroma" when embedded in the representation of an orange cube illuminated in space.

Chroma judgments depend on luminance contrasts (in lightness anchoring), on the spatial interpretation of the scene, and on memory color — so that, for example, one can adjust the brightness and contrast on a television image until it "looks right", regardless of whether the video image is of a sunny or cloudy day, in color or in black and white. We see beyond thousands of environmental variations in illuminance — as we walk from one room to another, look at objects close to or far from a window, perpendicular or slanted to the light, in light or in shadow — in order to judge colored surfaces.  

Optimal Color Stimuli. At this point we can address a new objective: whether it is possible to define a measure of hue purity that (1) applies specifically to surface colors rather than light mixtures; (2) is standardized on the maximum possible hue purity for any surface hue, luminance contrast or absolute illuminance level; and (3) has a verifiable perceptual validity, in the sense that colors determined to be at "maximum hue purity" really appear to be so. This can be achieved by using optimal colors as the perceptual standard of maximum hue purity.

Recall that the maximum hue purity for light colors is defined by a monochromatic light — a single wavelength from the visible spectrum, or the mixture of two wavelengths, "red" and "violet", for extraspectral hues. Spectral lights are the most saturated color stimuli possible, which means monochromatic stimuli can be used to benchmark the maximum excitation purity or Helmholtzian Sättigung of any colored light.

However, the hue purity for surface colors involves a different perceptual process, because a wide range of luminance contrasts between the color area and the background are involved. As surface colors become more and more saturated, or are illuminated more brightly against a constant background, they eventually reach a point where the chromatic luminance no longer appears to increase: instead the color appears to glow or fluoresce and then, transform into a light. The basic form of color appears to change. This zero grayness boundary provides a commonsensical benchmark for maximum hue purity.

This grayness boundary, where surface colors reach the maximum possible hue purity without a fluorescing or self luminous appearance, can be equated with the optimal color stimuli or MacAdam limits proposed by David MacAdam (who developed the concept in 1935 from ideas advanced earlier by Rösch and by Schrödinger). Optimal colors satisfy criterion 1, above.
 

context differences between colorfulness (left)
and chroma (right)

Optimal color stimuli are not actual surfaces or color samples, but theoretical reflectance profiles that meet two requirements:

• reflectance is either 0% or 100% at every wavelength

• the transition from 100% to 0% reflectance (or the reverse) occurs no more than two times across the entire spectrum.

These profiles are theoretical because no physical surface can reflect 100% or absorb 0% of light incident on them, and because the transitions from high to low reflectance are more gradual or rounded in physical colorants.

Optimal colors can take only three possible forms (diagram, right):

Type A: color created by an isolated spike or block of reflectance inside the spectrum (which requires no spectrum limits), which produce all colors of a spectral hue;

Type B: color created by a block of reflectance at either end of the spectrum (which requires one spectrum boundary in either the "red" or "violet" wavelengths), which cannot produce any color of a green hue;

Type C: color created by two blocks of reflectance at opposite ends of the spectrum (which requires two spectrum limits in both the "red" and "violet" wavelengths), which comprise all colors of an extraspectral hue.

In all cases, the width(s) w represents the reflected wavelengths that define the luminance and chromaticity of the color area.

Although optimal color stimuli cannot exist as physical surfaces, hue purity is dependent on luminance contrast; so it possible to simulate the appearance of a MacAdam color in a surface or object (for example by illuminating a patch of highly saturated paint with a vignetted spot of intense white or chromatic light), and to use this optimal simulator to represent the physical standard of chromatic intensity in a perceptual color matching task. The MacAdam limits represent theoretical or ideal colors, but the perceptual boundary they represent is quite real (criterion 3, above).  

In all the idealized profiles the width w is arbitrary. This width affects the luminance of the surface (and therefore its perceived lightness) and its chromaticity (as a location in a chromaticity diagram). By incrementally increasing the width w in the reflectance curves, and incrementally moving each profile across the entire spectrum, a complete inventory of optimal color stimuli can be generated. Assuming the visible spectrum extends from 380 nm to 750 nm, and calculations are made in integer wavelength intervals, there are 136,532 optimal colors, distributed roughly evenly across all levels of chroma and lightness.
 

optimal color boundaries for lights of different hue purity
in the CIE UCS diagram; numbers indicate the brightness of colors at each boundary; from Perales, Mora, Viqueira, de Fez, Gilabert & Martinez-Verdu (2005)

 
The diagram above shows optimal colors projected into the CIE u'v' uniform chromaticity scale diagram, where we have to interpret surface lightness levels as specific brightness levels across all hues (because the spectrum locus defines the chromaticity limits). These form optimal color boundaries as brightness increases from the spectrum locus.

The shrinking area of the boundaries shows that increasing the width w of optimal colors increases the light brightness but causes a proportional decrease in the maximum possible chroma. Note that hue alters the link between brightness and MacAdam limits. Hues in the circuit from green through blue and magenta contract toward the equal energy white point (EE) roughly in proportion to the width w: any of these hues at 50% brightness is limited to a chroma that is roughly halfway between the white point and the spectrum locus, and this contraction is perceptually most extreme in blue violet hues. But this is not true for hues from green yellow through red orange: yellow or orange can have maximum hue purity even when the spectral mixture (width w) is very broad. For lights from yellow green through orange red, hue purity is effectively uncoupled from spectral purity.
 

optimal color stimuli

the three possible profile forms;
in each case the widths "w" can be
any size up to the total spectrum
width, and for "B" the width can
be from either end of the spectrum

However, as we're talking about surface colors, it is more meaningful to show the optimal color boundaries within a color space designed for surface colors, for example the CIELAB a*b* plane (diagram, right). And the story for reflecting surfaces is very different from self luminous lights. Optimal surface colors, across all levels of lightness, mesh to form a wedge shaped "basket" whose width is widest in the middle values and narrowest at extreme light or dark values. The limits of this basket extend from a diagonal keel along extremely dark purple (lower right) upward into blue, green, red and yellow, then close at the top to form a narrow crest extending from green yellow to white. For comparison I've inserted the CIELAB location of common watercolor pigments and outlined, in white, the gamut of common watercolor pigments.  

Now a color's hue purity can be defined by a line drawn from the achromatic point through the location of the color on the chromatic plane and extended at the same hue angle to the optimal color boundary of equal lightness. This yields two chroma values: for the paint and for the optimal color of matching lightness and hue angle. This defines hue purity relative to the physically maximum possible chroma for a specific hue and lightness:

6. Hue purity refers to the chroma of a surface color as a proportion of the maximum possible chroma [MacAdam limit] for a surface color of matching lightness and hue angle.

Thus, for any surface color SHL of hue angle H and lightness L, there is a corresponding optimal color OCHL at the same hue angle and lightness. Then:

HP(SHL) = chroma(SHL) / chroma(OCHL)

As with excitation purity, this procedure standardizes the MacAdam limits so that they are equally far from the achromatic center at all levels of lightness and across all hues (that is, the wedge shaped optimal color "basket" is morphed into a cylinder whose axis is a gray scale). If we ignore differences in lightness as a separate (independent) dimension, the cylinder collapses along the gray scale to form a hue purity (chromaticity) plane. Then the hue purity of surface colors is their distance from the center of this hue purity circle, as shown below for 130 common watercolor paints.
 

CIELAB estimates of chroma and hue purity
hue purity standardized to unit radius; CIELAB chroma markers inserted for comparison, with chroma rescaled so that chroma and hue purity are equal for hansa yellow (PY97).

 
Obviously this procedure destroys the geometry of the CIELAB a*b* plane, with one exception: the hue angle remains intact. So the diagram superimposes the original CIELAB chroma markers over the hue purity values, scaled so that the chroma and hue purity of a saturated yellow (where lightness differences have the least effect on MacAdam limits) are the same, and connects the hue purity and chroma markers with a line. This allows comparison of the relative differences between the two measures around the hue circle.

The comparison suggests that yellow orange to violet red (magenta) paints, and blue violet to green blue (cyan) paints, should appear to viewers as closer to a pure hue than their CIELAB chroma values predict. Red oranges should also appear more saturated than yellow paints or any other paints with the exception of ultramarine blue. Indeed, many orange paints achieve a hue purity above 80%, just as a Munsell value of 9 achieves 80% of the possible reflectance for an achromatic surface. The blues also approach 70% or more of the maximum possible hue purity, and the disparity in chroma between reds and blues is altered almost to equality. In contrast, the hue purity of paints from blue green to green yellow is consistently below 50%: the most intense green pigments (the phthalo greens) have the same hue purity as burnt sienna or iron blue!  

When I developed this hue purity metric, it seemed quixotic in comparison to the CIELAB chroma scaling. So I was gratified to discover in 2006 that the CIECAM chroma placement of watercolor paints substantially matches their locations in these hue purity calculations (separate from the differing estimates of hue angle in CIELAB and CIECAM). To drive home this point, here is the hue purity distribution calculated for 174 watercolor pigments, using the CIECAM acbc chroma metric.
 

CIECAM estimates of chroma and hue purity
hue purity standardized to unit radius; CIECAM chroma markers inserted for comparison, with chroma rescaled so that chroma and hue purity are equal for hansa yellow (PY97). (For a table of CIECAM hue purity values, see this page.)

 

optimal color boundaries on
the CIELAB a*b* plane

watercolor gamut in white; hue purity is defined as HP = Cpaint/Cmax
where C is chroma on the a*b* hue plane

This diagram suggests that the CIECAM scaling in the b– and a– direction underestimates hue purity, but the overall match between chroma and hue purity is improved. However, separate from the disparity in hue angle estimates between CIELAB and CIECAM, the hue purity estimates (dispersion of dark blue markers) are remarkably similar between the two color systems. This occurs because the "basket" of optimal color boundaries is shaped by whatever idiosyncracies each system may produce in hue or chroma scaling, so that the peculiarities are expunged when chroma is standardized on the optimal colors.

Nevertheless, the chroma scaling in CIECAM is clearly better. This is because both the chroma and saturation metrics in CIECAM produce a roughly circular shape in the optimal color limits in contrast to the wedge shaped limits produced in CIELAB (diagrams, right). All these considerations are why I prefer CIECAM as the basis for an artist's color wheel. CIECAM shows the convergence between my hue purity metric and modern measurements of saturation.

Optimal colors introduce a departure from the traditional conceptions of saturation in a significant point: a viewer would have the perceptual sensation of a surface color that is pure but not colorful (a "pure pink", a "pure brown", even a "pure black"). And this perception would occur because optimal colors eliminate the grayness (whiteness or blackness) that consistently appears in material colors (at least, without manipulating the luminance contrast).

 

side view of optimal color boundaries in CIECAM
with the distribution of watercolor pigments at peak chroma

 
In fact, very light or dark valued colors, close to the media specific white (W) or black (K) lightness limits, will appear most "material" and least "optimal" because the material chroma limits are at a significant radius from the optimal color limits (dotted lines). This occurs because the reduced lightness range between the material white and black, the result of physical processes of absorptance and scattering that occur equally in all colors within the medium, causes all the material chroma limits to be gamut mapped toward a middle gray by approximately the same proportional amount. (The exceptions occur in deep red colors, where surface blackness causes a disproportiate darkening of the hue, and in the blue violet colors, where a pseudofluorescence from ultraviolet causes the chroma to appear especially bright.) Material colors seem least material and most "ideal" at the unique lightness where the ratio S/OC is at its maximum — which is around a mid value for red colors (a+, diagram above) but at a very light value for yellow colors.

All the definitions of chromatic intensity reviewed here, and evaluated in the next page, seem reducible to one of three concepts: (1) the tinting strength of a chromatic stimulus in additive mixture or in lightness contrast with another color, usually white (Evan's grayness); (2) as the chromaticity distance of a color from an achromatic reference point, either white or gray (saturation, CIE chroma); or (3) as a chromaticity ratio between the chroma of a color stimulus and the maximum chroma or saturation possible for that hue, where the maximum is either a physical stimulus (excitation purity, my hue purity) or an imagined or subjective ideal (colorfulness, NCS pure color).

Terminology. Artists normally work with color relationships in context, and attempt to match color intensities from the luminance range of a landscape onto the luminance range of a canvas without regard for the differences in luminance involved. For this reason saturation or chroma are the most appropriate terms. Colorfulness can be used to describe the relative color changes observed by changes in the illuminance level, for example, in the appearance of a painting viewed at higher lighting levels, or under lights of different color rendering properties.  

Are Three Attributes Enough? The final question to address: are more than three colormaking attributes required to describe surfaces? The answer is, yes and no.

Yes, because color attributes obviously change according to the overall level of luminance, and the luminance contrast between color and surround, as we see in our home in the contrast between early morning and noon light, or in the color change in surfaces illuminated by a shaft of direct light; and these changes also affect color purity. If we want to describe changes in color appearance from one illumination context to the next, or between surfaces and lights, or in an unfamiliar or confusing setting, then five attributes are required.

No, because if we describe lights viewed in darkness or surfaces viewed under daylight levels of natural illumination, or the color of familiar objects under dim or tinted illumination, then three attributes are usually sufficient to describe the color unambiguously. These perceptions are closely attached to our concept of the "real" colors of unchanging objects, and are closer to color ideas than color perceptions.

Surfaces can display many other properties such as gloss, crackelure, texture, grain, aggregation, opacity, depth, iridescence and pattern, but these are unambiguously surface or material qualities that are easy to distinguish from color as a global or average attribute. Three colormaking attributes are the norm because any more are usually superfluous.

There may be other color attributes reaching across categories of luminance, hue or color purity that remain unexplored in traditional color research but are worth further study because they have been important to artists. One of these is the warm/cool contrast, which I believe emerges from chromatic adaptation to the color changes of daylight. Manipulation of the warm/cool contrast is important to the representation of natural light in landscape painting, and judgments of warm or cool are a convenient and practical way to characterize comparative color differences in color design and color mixing.
 

optimal color boundaries on
the CIECAM ab plane

(top) chroma metric,
(bottom) saturation metric;
under EE illuminant, 20% background reflectance, white surface
luminance 318 cd/m2

We end up with these five (often three) colormaking attributes:
 

the colormaking attributes
unrelated colorsrelated colorscomments
brightnesslightnesslightness = value; independent of changes in illuminance, but affected by surround or background contrast; relative to light adaptation level and surround luminance contrast; white, gray and black are only visible as related colors
huedisappears at very low (scoptoic) or very high (blinding) luminance levels; constant across changes in illuminance; changes with change in luminance, with change in illumination color, with change in chroma, or with surround hue contrast; olive and brown only visible as related colors
colorfulnesscolor purity in relation to ideal pure hue; changes with change in luminance or illuminance, or any admixture with white and/or black; changes with surround chromatic or luminance contrast
saturationchromarelative hue purity in relation to own brightness (lights) or to white under equal illumination (surfaces); remains constant across changes in luminance (dim to bright) or illuminance (shadow to light)
excitation purityhue purityjudged by comparison to monochromatic lights for lights, optimal color stimuli for surfaces

 
These five colormaking attributes completely define any color sensation: lightness and brightness identify the proportion of the total illumination that is reflected by a surface; colorfulness and/or chroma describe the color purity as a proportion of the incident and reflected light; and hue identifies the average or dominant wavelengths of the light spectral power distribution within the closed span of a hue circle.

Once again: contextual factors can strongly affect the colormaking attributes. These three factors must be held constant in order to make accurate color judgments or color comparisons:

• intensity and color of the illumination

• relative luminance contrast with surrounding colors

• spatial interpretation.

All these phenomena reveal that the momentary, pixelated retinal image has been extensively adjusted and interpreted to produce perceptions that are consistent with a lifetime of visual experience with the real world.

 
painting saturation & value
 
This concluding section looks at the relationships among lightness, saturation and chroma in terms of traditional painting practice.  

How to Judge Chroma. To judge chroma or saturation accurately, the artist becomes familiar with it in qualitative terms. The trick is to use different perceptual tests for three levels of chromatic appearance.

• At one extreme — grayed, dull, near neutral or achromatic colors — the color is difficult to name as a hue. Unsaturated colors are mixtures of many different wavelengths of light, so the color does not have a narrow, precise location within the visible spectrum. So the question is whether the surface has a specific hue, and what that is. Usually the gray appears to be definitely either warm or cool before a specific hue can be assigned to it.

• At the opposite extreme, when the chroma is very intense, then the surface hue is quite specific and the best visual cue is how much the color appears to glow, in comparison to a gray of equal lightness. In paints, this is very easy to distinguish from the surface gloss. The color has an intensity that seems almost to rise off the paper. This glowing effect is not related to lightness: a good quality ultramarine blue applied full strength to white paper has a pronounced luster when wet, even though the color is quite dark and acquires a whitish overtone when dried.

• For colors at moderate saturation there is a third useful cue: as the chroma increases, the hue becomes more specific. When John Ruskin wrote that "you shall look at a hue in a good painter's work ten minutes before you know what to call it," he simply wanted to say that the good painter's colors are unsaturated. Burnt sienna appears to be a kind of red orange, but it is not clear whether the color is closer to orange or red.

The chroma or saturation of the color is determined by which test seems most relevant. If you are debating whether the color is warm or cool, it is very unsaturated; if you know it is warm but are unsure of the hue, it is moderately unsaturated; if you know the general hue but not the specific tint, then it is moderately saturated; if the color presents a specific hue but seems somewhat whitish, dark or veiled, then it is saturated; if the color appears to glow or shine, it is at maximum saturation.

Keep in mind that yellow green, the color of sunlight, is also the color with the smallest perceptual range of chroma, and is the surface color and light color closest to white. The chroma of yellow paints drops very quickly as they are mixed with any purple, blue or black; the lightness goes down as well. Saturated yellows, oranges and reds intermix very well with minimal impact on chroma; blues and greens tend to darken each other unless diluted or whitened, and blue will dull almost any other color except purple or magenta.  

Chroma, Lightness & Saturation. The next problem is to understand how chroma and saturation change with lightness or luminance contrasts. This means the artist know hows to translate surfaces under light into color changes.

The diagram shows an array of red violet color samples across the complete range of chroma and lightness. Lightness or tonal value increases vertically from black (bottom) to white (top), along equal steps of a gray scale or value scale. Chroma changes horizontally in equal perceptual steps from achromatic (gray) at left to maximum chroma at right. The colored rectangles in each row of the diagram are the same lightness; rectangles in each column are the same chroma.
 

lightness (L), chroma (C) and saturation (S) for
a red violet hue
reduced chroma and reduced light signal reduced illumination on the surface; this combined effect produces constant saturation

 
In this arrangement, lines of equal saturation are represented by a constant proportion or percentage of chroma in relation to lightness:

saturation = (chroma/lightness)*100

The line S = 100 passes diagonally through the squares that have equal lightness and chroma, from black (L =0, C = 0) through a pale purple (L = 7, C = 7). The line S = 200 locates colors where the chroma is twice (200%) the lightness; S = 50 the colors where chroma is half (50%) the lightness, and so on.

For a given hue, an illuminated surface has a specific lightness and chroma. As the surface falls into shadow, the lightness of the surface decreases down the same column. However, chroma decreases as luminance goes down under constant illuminance, so the color loses chroma as well, which means the color shifts leftward across a row in the figure. The combination of these two effects is a diagonal originating in a pure black value, which is a line of equal saturation for the color.

The diagram below represents in caricature the visual implications of a color change in lightness, chroma and hue. First look these examples over and consider which ones look the least and the most convincing as a representation of a shadow on one side of a dull, moderately light valued red orange sphere.
 

shadow contrasts in lightness, chroma and saturation
left column: lightness reduced 40%, constant chroma; middle column: lightness and chroma reduced 40%, constant saturation; right column: chroma reduced 40%, constant lightness; middle row: shadow hue = lighted hue; top row: shadow hue 15° greener; bottom row: shadow hue 15° redder

 
The column of spheres at right does not appear shadowed (reduced in lightness) but discolored. The spheres at left appear shadowed but discolored by the change in hue. The balanced reduction in lightness and chroma or constant saturation (middle column) is both the most color neutral and the shadow that shows the least bias from changes in hue.  

The Painters' "Broken Colors". The lightness of any paint can be altered in one of three ways without altering its fundamental hue. In the European painting tradition, any desaturating mixture of a pure pigment with another color, or with white and/or black, was called a broken color, and breaking a color was synonymous with dulling or whitening it.

In modern terminology these mixtures are called shades, tints or tones of the pure color, as shown in the figure below for a middle red.

 

shades, tones and tints of a middle red

 
The pure pigment color as raw paint from the tube (but properly diluted so that it does not bronze or blacken when applied to paper) represents the hue at its maximum chroma and optimal lightness. From that point, the paint color can be modified by:

• reducing both the chroma and lightness of the color (–L–C) by mixing with a black or dark near neutral paint, which can reduce the lightness to the darkest value possible in watercolors (around step 2 on a value scale), to produce shades of the hue

• reducing the chroma of the color without significantly changing the lightness (–C), by mixing with either (1) a gray mixture of black and white of approximately the same lightness as the pure color, or (2) another paint of similar lightness that is a mixing complementary color, which brings the mixture color closer to gray; these are tones of the hue

• reducing the chroma and increasing the lightness of the color up to the "white" lightness of the paper or canvas (+L–C), by mixing with pure water (in watercolors or acrylics) or a white paint or colorless medium (in acrylics and oils), to produce tints of the hue.

These changes in lightness and chroma within a constant hue are typically represented in modern color models as separate "pages" or sections through the color solid, for example in the Munsell Book of Color and the Swedish NCS. This presentation mirrors the practice of altering paints of a specific hue by mixing them with white and/or black paint, a technique advocated in painting treatises from Alberti to Chevreul. Here we will look at the color changes produced by these traditional methods of paint mixing.

Tints. Progressive dilution of a paint with water produces the tints or lightened, pastel versions of the hue. Watercolor paints can also be lightened by adding a white paint, such as zinc oxide or titanium oxide, which is especially effective with very dark or strongly tinting colors. (Note however that mixture with a white pigment can make some pigments, such as prussian blue [PB27] or dioxazine violet [PV23] significantly less lightfast.)

Lightening or dilution raises the lightness of the color. Because white has zero chroma, it follows that tints lose chroma as the lightness approaches white. In most paints, once the paint has been diluted to its maximum chroma on paper, any dilution of a paint reduces its saturation by an equal amount: a 50% increase in value toward white results in a 50% decrease in paint saturation and chroma.

Note that paints begin to shift toward tints only after they reach their maximum chroma on paper. For some paints, the dilution needed to produce the maximum chroma causes an increase in lightness somewhere between 20% to 50% of the raw paint lightness. This is often the case for very dark paints or moderately dull paints on the warm side of the color wheel, including: anthrapyramidine yellow (PY108), all the quinacridone pigments (quinacridone gold PO49, quinacridone orange PO48, quinacridone maroon PR206, quinacridone carmine PR N/A, quinacridone rose and quinacridone violet PV19), perylene maroon (PR179), perylene scarlet (PR149), all the iron oxide (earth) pigments (PBr7, PR101, PY42, and PY43), dioxazine violet (PV23), indanthrone blue (PB60), the darker shades of phthalocyanine blue (PB15 and PB16), and both shades of phthalocyanine green (PG7 and PG36). See the page on the secret of glowing color for details.

Shades. Paints also can be darkened by adding a dark neutralizing pigment such as neutral tint, payne's gray, carbon black or synthetic black. Darkening a colored paint produces shades of the hue, and expands the range of color values below the value of the pure paint.

A true black is a pure neutral, so shades lose chroma as the proportion of black in the mixture increases. They also lose lightness by an equal amount. Therefore the saturation of the paint should, ideally, remain the same, because lightness and chroma decrease in roughly equal proportion.

In fact, darkened paint mixtures also lose some saturation, because "black" paints are actually a dark gray, which pulls the color toward a point above the true black lightness. This causes the mixture to cross saturation lines as it approaches the neutral color. In perceptual terms, very dark paints typically seem to whiten slightly as they dry, and this whiteness acts as it does in tints, to reduce both the chroma and saturation.

If a tinted near neutral is used instead, the saturation can actually hold constant for paints of similar hue: blue, if payne's gray or indigo is used; orange or red, if sepia is used; purple, if neutral tint is used; green, if perylene black is used. In particular, paints named sepia should not be used to darken blue paints, which are the complementary color the sepia hue, as they will excessively dull the color; and similarly for the complements of the other near neturals just described.

Tones. Finally, the lightness of the paint can be held relatively constant, but shifted toward a neutral gray, by adding a mixing complementary color of higher lightness or a synthetic black diluted to the same value as the pure color. This produces tones of the hue.

As with shades and tints, tones lose chroma as the mixture approaches gray. Tones have approximately the same value as the pure paint, so the saturation of tones also decreases as increasing amounts of gray (complementary color) are added.

Mixing complements usually produce a neutral tone that is darker than the complementary paint, so once a neutral mixture is reached it must be diluted up to match the lightness of the pure color. Once this is done, tones can create a complete range of chroma or saturation within a constant hue and lightness.

Hue Shifts. It is common to find that the hue of the paint mixture changes slightly as it is altered across tones or tints, and especially across the lightness changes produced by tints. This is both a perceptual and material effect.

The shift is called the Abney effect in monochromatic (single wavelength) lights, which change hue as increasing amounts of "white" light are mixed with them. The fact that this effect occurs in light mixtures indicates that it arises in color perception: the y/b and r/g opponent functions exert different response compressions, which produce different relative changes in chroma as hue intensity increases or decreases. On an artist's color wheel the Abney shifts are clockwise for hues roughly between yellow green to blue violet, and counterclockwise for hues from yellow to magenta. The effect is largest for blue violet colors, which become violet or purple when strongly desaturated with "white" light.

In paints, a similar shift is caused by the differing tinting strength and hue of pigment particles. In general, smaller pigment particles will have a greater tinting strength and also a different hue (typically yellower) than the pigment powder as a whole. As paints are diluted with water or white paint, the smaller particles, with higher tinting strength, exert a greater effect on the color, shifting the hue. On the artists' color wheel, this shift is usually clockwise for paints roughly from red to yellow, and counterclockwise for paints from blue violet to green.

The size and direction of the tinting shift, which is reported as hue shift in the guide to watercolor pigments, depends on the range of particle sizes in the paint, the hue difference between small and large particles, and their relative tinting strengths. In general, the tinting shift overwhelms the Abney effect in paint mixtures: ultramarine blue and ultramarine violet becomes bluer, not redder, when diluted with water or white paint.

N E X T :   the geometry of color


 

Last revised 08.01.2005 • © 2005 Bruce MacEvoy