Algebras have elements, such as integers. Algebras have operations on elements, such as addition. Integer additions result in integers. Integer addition commutes: 13 + 27 = 40 = 27 + 13. Integer addition is associative: (13 + 27) + 5 = 45 = 13 + (27 + 5). Integer identity element adds to integers to make the same integer: 13 + 0 = 13, and 0 + 0 = 0. Integer inverse elements add to integers to make zero: 13 + -13 = 0, and 0 + 0 = 0. Finite or infinite tables can show operation results for all element pairs.
If elements are colors and operation is additive color mixing, adding two colors makes color, by wavelength-space vector addition, following Grassmann's laws {algebra and color}. Order of adding two colors does not matter, so color addition is commutative. Sequence of adding three colors does not matter, so color addition is associative. Colors have complementary additive-inverse colors, and adding both colors makes white, so color addition has inverses. Adding black, white, or gray to color does not change color hue but does change saturation, so black, white, or gray are like identity elements. Unlike integer addition, adding color to itself makes same color.
distributive property
Identity, inverse, commutation, and association work whether colors come from light sources or reflect from pigments. Colors from light sources and colors from pigment reflections can mix. If reflected color mixes with mixture of two source colors, or if reflected color mixes with each of two source colors and then mixtures combine, same color results, like the distributive property.
Tone and color frequencies and wavelengths have harmonic ratios {harmonic ratios, color}.
harmonics
Harmonic ratios have small integers in numerator and denominator. In increasing order of denominator, harmonic ratios are 1:1, 2:1, 3:2, 4:3, 5:3, 5:4, and so on.
color wavelengths
The purest red color is at light wavelength 683 nm, with orange at 608 nm, yellow at 583 nm, green at 543 nm, cyan at 500 nm, blue at 463 nm, and violet at 408 nm. Magenta can be at 380 nm or 760 nm.
color wavelength ratios
Color wavelength ratio for red/yellow, 683/583 = 1.17, and green/blue, 543/463 = 1.17, is 7/6 = 1.17 or 6/5 = 1.20. Color wavelength ratio for red/green, 683/543 = 1.26, and yellow/blue, 583/463 = 1.26, is 5/4 = 1.25. Color wavelength ratio for red/blue, 683/463 = 1.48, is 3/2 = 1.5. Color wavelength ratio for yellow/green, 583/543 = 1.07, is 13/12 = 1.085. Color wavelength ratio for red/violet, 683/408 = 1.67, and magenta/indigo, 725/435 = 1.67, is 5/3 = 1.67. See Figure 1.
color frequency ratios
Color frequency ratio for yellow/red, 518/436 = 1.19, and blue/green, 652/556 = 1.17, is 7/6 or 6/5. Color frequency ratio for green/red, 556/436 = 1.28, and blue/yellow, 652/518 = 1.26, is 5/4. Color frequency ratio for blue/red, 652/436 = 1.50, is 3/2. Color frequency ratio for green/yellow, 556/518 = 1.07, is 13/12. Color frequency ratio for violet/red, 740/436 = 1.70, and indigo/magenta, 694/420 = 1.66, is 5/3.
additive complementary colors
Additive complementary color pairs have same wavelength ratio, 4/3 = 1.33. Red/cyan is 683/500 = 1.37 to 650/500 = 1.30. Yellow/blue is 583/463 = 1.26 to 583/435 = 1.34. Chartreuse/indigo is 560/435 = 1.29 to 560/408 = 1.37. Magenta/green is 722/543 = 1.33.
Additive complementary-color triples have three color-pairs, whose average wavelength ratio is also 4/3. For three additive complementary colors, ratios are red/blue, 683/463 = 1.48, red/green, 683/543 = 1.26, and green/blue, 543/463 = 1.17. Arithmetic average is (1.5 + 1.25 + 1.2)/3 = 1.32. Geometric average is (1.5 * 1.25 * 1.2)^0.333 = 1.32. For three subtractive complementary colors, ratios are magenta/cyan, 722/500 = 1.45, magenta/yellow, 722/583 = 1.24, and yellow/cyan, 583/500 = 1.17. Average wavelength ratio is 4/3.
Three complementary colors have same relative values: red = 1.5, green = 1.2, and blue = 1, or magenta = 1.5, yellow = 1.2, and cyan = 1.
subtractive complementary colors
Because mixing darkens and blues colors, subtractive complementary color pairs have increasing wavelength ratios. Red/green is 683/543 = 1.26. Orange/blue is 608/463 = 1.31. Yellow/indigo is 583/435 = 1.34. Chartreuse/violet is 560/408 = 1.37.
color wavelength ratios starting at red
Starting with red at 1/1 = 683/683, orange is 8/7 = ~683/608, yellow is 7/6 = ~683/583, green is 5/4 = ~683/543, cyan is 4/3 = ~683/500, blue is 3/2 = ~683/463, violet is 5/3 = ~683/408, and magenta is 7/4 = ~683/380.
color wavelength ratios starting at green
Magenta is 2/3 = ~380/543. Violet is 3/4 = ~408/543. Blue is 5/6 = ~463/543. Cyan is 8/9 = ~500/543. Green is 1/1 = 543/543. Yellow is 17/16 = ~583/543. Orange is 9/8 = ~608/543. Red is 5/4 = ~683/543. Magenta is 4/3 = 720/543.
color wavelength ratios starting at red
Starting with red at 1/1 = 683/683, orange is 8/7 = ~683/608, yellow is 7/6 = ~683/583, green is 5/4 = ~683/543, cyan is 4/3 = ~683/500, blue is 3/2 = ~683/463, violet is 5/3 = ~683/408, and magenta is 7/4 = ~683/380.
color wavelength ratios starting and ending at magenta
On color circles, complementary colors are opposites. Complementary-color pairs have same wavelength ratio, so cyan/red = blue/yellow = magenta/green. Colors separated by same angle have same wavelength ratio, so yellow/red = green/yellow = cyan/green = blue/cyan = magenta/blue = red/magenta. Example color circle has red = 32, yellow = 16, green = 8, cyan = 4, blue = 2, and magenta = 1 and 64. Put into octave as exponentials, red = 2^0.83, yellow = 2^0.67, green = 2^0.5, cyan = 2^0.33, blue = 2^0.17, and magenta = 2^0 and 2^1. Put into octave, magenta = 2/1, red = 9/5, yellow = 8/5, green = 7/5, cyan = 5/4, blue = 9/8, and magenta = 1/1. Complementary colors have ratio 1.412 = ~7/5. Neighboring colors have ratio 1.125 = 9/8. Example wavelengths with these ratios are magenta = 750 nm, red = 668 nm, yellow = 595 nm, green = 531 nm, cyan = 473 nm, blue = 421 nm, and magenta = 375 nm, close to actual color wavelengths.
color harmonic ratios
Color frequency categories are at harmonic ratios: 48 Hz for red, 60 Hz for green, and 72 Hz for blue. 60/48 = 1.25 = 5/4. 72/48 = 1.5 = 3/2. 72/60 = 1.2 = 6/5. See Figure 2. Color-pair wavelength ratios have harmonic relations. Red/magenta = 7/4. Red/violet and magenta/indigo = 5/3. Red/blue = 3/2. Complementary colors red/cyan, yellow/blue, chartreuse/indigo, and magenta/green = 4/3. Red/green and yellow/blue = 5/4. Red/yellow and green/blue = 6/5 or 7/6. Red/orange = 8/7. Green/cyan = 9/8. Yellow/green = 13/12. See Figure 3.
Red, green, and blue add to make white. Magenta, cyan, and yellow add to make black. For red, green, and blue, and for magenta, cyan, and yellow, average of the three color-pair wavelength ratios is 4/3.
Looking at only primary colors red, green, and blue, color-pair wavelength ratios are red/blue 3/2, red/green 6/5, and green/blue 5/4. Red:green:blue relations have 6:5:4 ratios.
Looking at wavelength differences rather than wavelength ratios, magenta, red, orange, yellow, green, cyan, blue, and violet have approximately equal wavelength differences between adjacent colors. See Figure 2. Setting wavelength difference equal to one, color wavelengths form series 8, 7, 6, 5, 4, 3, 2, and 1. See Figure 3.
Assuming colors are like tones, colors can fit into one octave. Primary colors red, green, and blue, and complementary colors cyan, magenta, and yellow, respectively, are equally spaced in octave from 2^0 to 2^1. Magenta, red, yellow, green, cyan, blue, and magenta form series 6, 5, 4, 3, 2, 1, and 0. Magenta = 2^1, red = 2^0.83, yellow = 2^0.67, green = 2^0.5, cyan = 2^0.33, blue = 2^0.17, and magenta = 2^0. Adjacent colors have ratio 2^0.17 = 1.125 = 9/8. All complementary colors have the same ratio, 2^0.5. All complementary-color triples, such as red/green/blue, average 2^0.5. White, gray, and black have average color-pair wavelength ratio 2^0.5. In this arrangement, color-pair ratios are red/magenta ~ 9/5, yellow/magenta ~ 8/5, green/magenta ~ 7/5, cyan/magenta ~ 5/4, and blue/magenta ~ 9/8. See Figure 3. In this arrangement, whites, grays, and blacks are farthest from being octaves and so have dissonance. Other colors have smaller integer ratios and so more consonance. Color categories are at harmonic ratios.
multiple harmonics
One pair has two or three categories, like tone intervals or red/green or red/green/blue. Two pairs make six or seven categories, like octave whole tones or main spectrum colors. Three pairs make 12 categories, like octave half tones or major spectrum colors. Four pairs make 24 categories, like octave quarter tones or major and minor spectrum colors.
summary
Using physical-color wavelengths, wavelength ratios are red/magenta = 7/4, red/violet = magenta/indigo = 5/3, red/blue = 3/2, red/cyan = yellow/blue= chartreuse/indigo = magenta/green = 4/3, red/green = yellow/blue = 5/4, red/yellow = green/blue = 6/5 or 7/6, red/orange = 8/7, green/cyan = 9/8, and yellow/green = 13/12.
Additive complementary-color pairs, such as red/cyan, yellow/blue, chartreuse/indigo, and magenta/green, have same 4/3 wavelength-ratio.
For red, green, and blue additive complementary colors, average of the three wavelength ratios, red/blue, red/green, and green/blue, is 4/3. For magenta, cyan, and yellow subtractive complementary colors, average of the three wavelength ratios, magenta/cyan, magenta/yellow, and yellow/cyan, is 4/3.
These intervals are harmonic musical tones in an octave: C, E, and G in the key of C. Blue and red make a major fifth interval. Blue and green make a minor third interval. Green and red make a major third interval.
Mathematical groups have elements, such as triangles. Operations map group elements to the same or another element. For example, if element is equilateral triangle, rotations around center by 120 degrees result in same element. Finite or infinite tables can show operation results for all elements.
If elements are colors and operation is additive color mixing, adding two colors makes color, by wavelength-space vector addition, following Grassmann's laws {mathematical group and color}. Adding black, white, or gray to color does not change color hue but changes color saturation, so color addition is not a single operation.
Because they cannot be negative but can complement each other, color qualities are vectors {vectors and colors}. Color vectors have three components: hue, saturation, and brightness, or red, green, and blue.
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Date Modified: 2022.0225