Goethe's Theories
J. W. Goethe developed a colour harmony theory on the basis of his hue circle. In this circle, colours are categorised into two sides, the positive and the negative. The former includes yellow, reddish yellow and yellowish red; the latter includes blue, reddish blue and bluish red. On the basis of these classifications, Goethe (1810) identified three types of visual effects:
Powerful – When component colours are all selected from the positive side of the hue circle, the colour combination generates "powerful" feelings, such as quick, lively and aspiring.
Soft – When component colours are all selected from the negative side, the colour combination generates "soft" feelings, such as restless, susceptible and anxious.
Splendid – When component colours are selected from both sides, the "splendid" effect is produced.
Goethe believed that colours are harmonious if they are located opposite to each other on both sides of his hue circle, and that these colours will generate the "splendid" effect. He wrote (Goethe, 1810, Note 810),
"Yellow demands red-blue;
Blue demands red-yellow;
Red demands green;
And contrariwise."
Note that colours selected from either side of the hue circle can produce specific emotional effects, i.e. powerful or soft, which no longer exist if some of the colours are replaced by those on the other side. This means that in Goethe's view, colour harmony is not intended to create specific emotional effects but is aimed at a balanced state.
Chevreul's Theories
M. E. Chevreul's theories on colour harmony are based on his colour circle of 64 hues derived from three primary hues: yellow, red and blue. He identified two types of colour harmony (Chevreul, 1839):
Harmony of Analogy, including those having neighbouring tones of the same hue and those having different hues of the same tone.
Harmony of Contrast, including those having the same hue but being far apart in tone and those having contrast hues.
Chevreul believed that complementary hue pairs can create greatest harmony. In addition, when two colours are seen as disharmonious together, they should be separated by either white or black; white should be used if the two colours are dark and black should be used if they are bright. Chevreul's theories have been regarded as the most important colour harmony principles, and have become part of colour education throughout the western world (Birren, 1987).
Ostwald's Theories
As a practical tool for colour harmony, Ostwald system has attracted considerable attention from colour scientists and artists since its first appearance in W. Ostwald's book "Die Farbenfibel" (Ostwald, 1916). This system is particularly favoured by artists and designers because of its superficial similarity of construction to the way artists mix their paints on the palette (Billmeyer, 1987).
Ostwald colour space can be illustrated as a number of triangles arranged
in a circle around a central axis, the grey scale. Central to this system
is the idea that every colour is a spinning-disk mixture of three elementary
sensations: full colour (the most highly chromatic colour available
at the time, labelled C), ideal white (W)
and ideal black (B), and 
(Foss
et al., 1944; Granville, 1994), as shown in the following.
The Ostwald hue circle is based on four major hues, red, yellow, green and blue, spaced 90º apart. Additional hues between these four were selected on the basis of equal spacing in each quadrant. Opposite hues, when combined in the proper proportions, must spin to a neutral grey using a spinning-disk technique. Ostwald lightness, the central axis, was derived as a logarithmic scale, based on the Weber and Fechner laws, providing approximately equal visual steps.
A major disadvantage of this system is that for each hue the full colour was selected from the most highly chromatic colour available at the time when a new edition of the Ostwald colour atlas, the "Color Harmony Manual", was released. The use of different full colours at different times, but with the same notation for those full colours, is a serious drawback of the Ostwald system.
W. Ostwald, the founder of this system, developed colour harmony principles based on this system. He believed that order is the most important factor of colour harmony (Ostwald, 1916). His principles of colour harmony can be summarised by:
Colours harmonise if they are located at the equal white and equal black
circle in the Ostwald system;
Colours harmonise if they have equal white content;
Colours harmonise if they have equal black content;
Colours harmonise if they have equal hue content.
These principles, also called ring star, can be illustrated in the above diagram. According to Ostwald, the reason why these principles can create harmonious colour combinations is the fact that these colours have properties "in certain simple relationships", which is what he meant by order.
Munsell's Theories
The
Munsell system, originated by artist A. H. Munsell in 1905, has been widely
regarded as an (approximate) uniform colour space (Hunt,
1998). Colours in this system are arranged such that the perceptual difference between
any two neighbouring colours is nearly constant in each of the three dimensions, Munsell
Hue, Munsell Value and Munsell Chroma. The Munsell
system can be illustrated in the diagram on the right (Munsell,
1921).
There are five elementary hues in the Munsell system: Red, Yellow, Green, Blue and Purple. These hues are spaced evenly around the vertical axis, the grey scale, so as to represent equality in perceived hue. According to the CIE definition, hue is "the attribute of a visual sensation according to which an area appears to be similar to one, or to proportions of two, of the perceived colours, red, yellow, green and blue" (Hunt 1998, Appendix 9). Note that the equality of visual spacing in hue, which is important to the Munsell system, is not included in this definition. Nevertheless, the equality feature in Munsell Hue makes it possible for this system to become a uniform colour space.
There are ten main steps in the grey scale, the Munsell Value, with white designated 10 and black 0. The greys have values from 1 to 9 as they become lighter. The difference in lightness between any neighbouring colours, say 2 and 3, is intended to be perceptually as great as between any other neighbouring pair, say 6 and 7. This means that the Munsell Value is also intended to represent uniform lightness scales.
The distances of colours from the central axis represent a uniform scale in colourfulness (Billmeyer, 1987) or, in other reviewers' opinion, in chroma (Fairchild, 1997; Hunt, 1998; Kuehni, 2000). This scale is called Munsell Chroma, which starts at the neutral axis and proceeds out at each Munsell Value and in the direction of each Munsell Hue to fill out the colour space. The maximum designation possible for Munsell Chroma depends on the limits of real colours. Therefore, the outer boundary of the Munsell colour space is not symmetrical, as shown in the above diagram.
Munsell saw balance as a key factor in determining colour harmony. He wrote in his "A Grammar of Color" (Munsell, 1921, p. 11),
"The sense of comfort is the outcome of balance that this approximate balance is desirable may be shown by reference to our behaviour, as to temperatures, quality of smoothness and roughness, degrees of light and dark, proportion of work and rest."
On the basis of the Munsell colour system, Munsell suggested two approaches to balance in a colour combination: a) the balance of "colour strength" and b) the balance of colour areas.
In the Munsell system, the colour strength is defined as the product of Munsell Value and Munsell Chroma. The higher the values of Munsell Value and Munsell Chroma, the "stronger" the colour appears. Munsell believed that a bright colour can balance a dark colour, and that a vivid colour can balance a greyish colour by organising the colour areas.
In the balance of colour areas, a "strong" colour should occupy a less amount of area to balance a "weak" colour. To do this, the areas should be inversely proportional to the product of Munsell Value and Munsell Chroma, namely
where 
are areas of colours 1 and 2;
are Munsell Value for colours 1 and 2; 
are Munsell Chroma for colours 1 and 2.
Munsell's idea of colour balance has widely influenced artists and designers in colour selection. However, his definition of "colour strength", the product of Munsell Value and Munsell Chroma, does not seem to be strictly true. For instance, when colours are presented on a white background, dark colours would appear to have more "strength" than light colours, providing that they are both at the same chroma level. This disagrees with the above equation, which indicates that dark colours always have less "strength" than bright colours. A different definition of "colour strength" was proposed by Moon and Spencer that "colour strength" was determined by the scalar moment of the colour in question about an adaptation point in a uniform colour space.
Similar to Ostwald's ring star, Munsell's practical principles of colour harmony are based on the idea that colours can harmonise only when they are located on a specific path in his colour space. These paths include:
a) The grey scale;
b) Colours of the same Munsell Hue and the same Munsell Chroma;
c) Complementary colours of the same Value and the same Chroma;
d) Colours of "diminishing sequences", in which each colour is
dropped down one step in Value as Chroma goes down one step, and so is
Hue;
e) Colours on an "elliptical path" in the Munsell colour space.
Moon and Spencer's Model
P. Moon and D. E. Spencer proposed a quantitative model of colour harmony, using predictors "colour interval" (i.e. colour difference), area factor and an aesthetic measure (Moon and Spencer, 1944a-1944c). Their work has attracted considerable attention, although the model was found to have poor predictive performance (Pope, 1944; Granger, 1953, 1955a-b; Sivik and Hård, 1994). Nevertheless, Moon and Spencer's effort was still important in the research of colour harmony. Granger (1955c) wrote, "in spite of the disappointing results obtained with the existing form of Moon and Spencer's aesthetic measure, the general quantitative approach which they adopt is to be welcomed in a field which for so long has been characterised by unchecked qualitative speculation."
Moon and Spencer believed that colours should harmonise when the colour difference between each component colour is unambiguous. Unambiguous colour difference is defined as intervals labelled identity, similarity and contrast, as illustrated in the diagram below. Between identity and similarity and between similarity and contrast are called "intervals of ambiguity".
Moon and Spencer believed that a harmonious balance among colour patches can be obtained if the scalar moments about an adaptation point in a uniform colour space are either equal or simple multiples of each other, i.e. the proportion of the scalar moments should be either 1, 2 or 3. The adaptation point is the point in a uniform colour space corresponding to the state of adaptation of the eye, normally a medium grey. The scalar moment of a colour is the product of the colour area and the distance in a uniform colour space between the adaptation point and the colour point. This distance represents the colour strength. The balanced colour areas have the following relationship (also called "scalar moment ratio"):
where 
are areas of colours 1 and 2; 
are colour difference values of colours 1
and 2 from the adaptation point (e.g. a medium grey), respectively.
Moon and Spencer's aesthetic measure of colour harmony is given below:
M = O / C
where C represents complexity and O represents order of a colour combination, as defined in the following:
where N is the number of colours; 
is
the number of colour pairs whose component colours have different values
in Munsell Hue; 
is the number of colour pairs whose component
colours have different values in Munsell Value; 
is the number of colour pairs whose component
colours have different values in Munsell Chroma.
where
= (No
of colour pairs whose component colours have a scalar moment ratio of 1) x 1.0 +
(No of colour pairs whose component
colours have a scalar moment ratio of 2) x 0.50 +
(No of colour pairs whose component
colours have a scalar moment ratio of 3) x 0.25
= (No
of colour pairs whose component colours are identical in Munsell Hue) x 1.5 +
(No
of colour pairs whose component colours are similar in Munsell Hue) x 1.1 +
(No of colour pairs whose component
colours are contrast in Munsell Hue) x 1.7
= (No
of colour pairs whose component colours are identical in Munsell Value) x (-1.3) +
(No of colour pairs whose
component colours are similar in Munsell Value) x 0.7 +
(No
of colour pairs whose component colours are contrast in Munsell Value) x 3.7
= (No
of colour pairs whose component colours are identical in Munsell Chroma) x 0.8 +
(No of colour pairs whose component colours are similar in Munsell Chroma) x 0.1 +
(No of colour pairs whose component
colours are contrast in Munsell Chroma) x 0.4
= (No
of colour pairs whose component colours are of 1st ambiguity
in Munsell Value) x (-1) +
(No of colour
pairs whose component colours are of 2nd ambiguity in Munsell
Hue) x 0.65 +
(No of colour pairs whose component colours are of 2nd ambiguity in
Munsell Value) x (-0.2)
Itten's Theories
The central idea behind J. Itten's colour harmony theories is that "two or more colours are mutually harmonious if their mixture yields a neutral grey." (1961) He developed a number of colour chords for creating colour harmony, which generate neutral greys when shown together on a spinning disk. These colour chords include dyads, triads, tetrads and hexads, for two-, three-, four- and six-colour combinations, respectively, as illustrated in the diagrams below (Itten, 1961):
Itten also studied the relationships between colour strength and harmony. He suggested that yellow, orange, red, purple, blue and green have equal "strength" if the size of each colour forms a proportion relationship 3 : 4 : 6 : 9 : 8 : 6. This relationship was also derived from his spinning-disk technique.
Coloroid System
The
Coloroid system was developed by A. Nemcsics (Nemcsics, 1980, 1985, 1987,
1993, 2003; Neumann et al., 2005). The aim of the system is to create aesthetic uniformity of a colour space.
According to Nemscics (1980), a scale is aesthetically
uniform if the whole scale from its starting point to its end, when viewed
simultaneously, appears to change evenly. This means that the Coloroid system is intended to provide
a global, rather than a local, visual uniformity of colour distribution.
The Coloroid system has three primary dimensions: Hue, Saturation and Lightness. Coloroid Lightness is represented by a central vertical grey scale, with perfect white at the top and perfect black at the bottom, from which the Hue planes radiate and the Saturation increases, as illustrated in the diagram on the right (Nemscics, 1980).
Coloroid Hue is specified by 48 evenly spaced hues to provide a hue circle with the definition of constant CIE dominant (or complementary) wavelength as constant hue. The hues are within the wavelength of 450 to 625 nm.
Coloroid Saturation is defined by analysing the test colour as an additive mixture of the saturated colour, perfect black and perfect white. The Saturation is the percentage of the saturated colour in that mixture.
Coloroid Lightness is defined as a square-root function of the CIE tristimulus Y, as the following:
where V is Coloroid Lightness; Y is the CIE tristimulus value and represents the luminance factor. For the perfect white (the perfect diffuser) Y = 100 and V = 100. For a perfect black, Y = 0 and V = 0. Thus, the V scale extends from 0 to 100.
On the basis of the Coloroid system, Nemcsics (1993) developed several colour harmony principles, considering colour contrast, "scalar relation", colour association and colour preference.
For colour contrast, Nemcsics claimed that colours can harmonise only when they have a contrast relationship in at least one of the three attributes, Hue, Saturation and Lightness. He agreed with Moon and Spencer's idea that colour harmony is determined by two types of colour interval, ambiguous and unambiguous (Moon and Spencer, 1944a). He believed that a colour pair would harmonise if their colour difference in Coloroid Hue angle is within 25º to 43º or within 121º to 139º. These ranges agree well with those proposed by Moon and Spencer.
By "scalar relation" Nemcsics meant the relationship between colours in a colour combination if their Coloroid Saturation and Lightness values constitute a uniformly increasing or decreasing series. This idea is similar to that proposed by Ostwald and Munsell in which orderly arrangement of colours are said to create colour harmony.
As for colour association, Nemcsics believed that if an emotional effect is unambiguously presented by a colour combination, then the combination could harmonise.
For colour preference, Nemcsics believed that the reason the Middle Ages had different colour harmonies from those in Late Baroque was that the two eras had different colour preferences (Nemcsics, 1993, p. 271). He claimed that colour harmony is dependent on colour preference within a given group of people at a given time.
Following are principles of colour harmony based on the Coloroid system (Nemcsics, 2003):
- Component colours have the same hue and saturation, but their lightness values constitute an arithmetical or geometrical sequence.
- Component colours have the same hue and lightness, but their saturation values constitute an arithmetical or geometrical sequence.
- Summing the above two special cases the colour characteristics are the same, but their lightness and saturation values change jointly on one straight line, where the distances of the points of division constitute an arithmetical or geometrical sequence. The (T , V ) pairs can be placed on more parallel straight lines as well, in each case according to the same arithmetical or geometrical sequence.
- The above rules can be equally related to one or more hues as well. Among the many hues the 3-hues or trichrome harmony is of prime importance. The SET of possible trichrome basic colours belonging to the A basic hue are: {A ± 1.0, A ± 4.6, A ± 6.6, Complementary hue (K) K ± 1.0, K ± 4.6, K ± 6.6} hues. From this set, the basic hue A and more two hues selected next to it, constitute a trichrome colour harmony.
- Any two hues selected from the above set constitute a dichrome hue harmony even if A basic hue is omitted
NCS
The NCS (Natural Colour System) was developed by T. Johansson and S. Hesselgren and, more recently, by A. Hård and L. Sivik (Hård and Sivik, 1981; Hård et al., 1996a, 1996b). In the NCS, colours are specified in terms of the relative amount of four elementary hues (red, green, yellow and blue) and of black and white. The structure of the NCS is shown in the diagram below.
The NCS Hue is defined in terms of the resemblance of the test colour to the two nearest elementary hues. For instance, orange may be described as bearing a 30% resemblance to red and a 70% resemblance to yellow. This definition agrees with that suggested by the CIE (Hunt 1998).
The second dimension of the NCS is Chromaticness, the resemblance of the test colour to the colour of the same hue having the maximum possible chromatic content. The third dimension is Blackness, the resemblance of the colour to the perfect black. Note that lightness is not among the dimensions of the NCS. This absence makes the NCS to be the only major colour order system that does not include lightness as one of its primary dimensions (Kuehni, 2000).
Hård and Sivik developed a variety of methods for colour selection for colour combinations on the basis of the NCS (Hård et al., 1996a, 1996b; Hård and Sivik, 2001; Sivik and Hård, 2004). These methods are not colour harmony theories, but classifications of colour combinations using the NCS specification system. For instance, they investigated the relations between colours using a measure called “colour interval”, which is determined by three attributes, the distinctness of border, interval kind and interval size.
Distinctness of border is defined as the sum of NCS Hue difference, Chromaticness difference and Blackness difference, each calculated with a weight of 0.3, 0.2 and 1.0, respectively.
Interval kind is a qualitative attribute that indicates which elementary hues are related to colours in question.
Interval size is determined by the distances of the related elementary hues in terms of the NCS Hue, Chromaticness and Blackness. On the basis of these attributes, Hård and Sivik selected a variety of colour combinations that are said to have similar characteristics in visual effect.
It should be noted that Hård and Sivik both doubt the idea that complementary colours can harmonise (Hård and Sivik, 2001).
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