The autistic savant paradigm
When autistic savants graciously demonstrate their outstanding skills for us, be they musical, mathematical or artistic in nature, we find their talents spectacular. We do not know how they can retrieve information or solve problems with such ease and rapidity. Nor do they know. Take for example the autistic twins John and Michael, who have I.Q.s well below the average, and are unable to understand basic mathematical operations such as multiplication or division. In spite of these and other limitations, they are adept at making lightning calendar calculations, and are able to simply see prime numbers of a surprising order without even using pencil and paper.
Dr. Oliver Sacks has had the opportunity to work with John and Michael, and in that joyous blend of inquisitiveness, wit and charm, The Man Who Mistook his Wife for a Hat, he shares some of his clinical observations regarding their ability to mentally calculate very large prime numbers. He tells how the twins quickly reduce any number thrown at them to shambles, factoring it and sifting out the decomposable elements. The prime number kernels that remain seem to be favoured as perceptual events that have a higher reality value for them: they find them as remarkable and awe inspiring as we do the process itself.
Unconscious algorithms
What process permits such intuitive feats? Dr. Sacks suggests that these calculations are arrived at with the use of an unconscious algorithm. He says that the twins "must have 'sense' in their numbers-in the same way, perhaps, as a musician must have harmony", and he quotes Leibniz who said that "the pleasure we obtain from music comes from counting, but counting unconsciously." It is generally accepted, especially amongst musicians, that music is closely related to mathematics, but what of color now? Could the pleasure that we derive from juxtaposing "harmonious" colors also be satisfying our love for counting? When I read Sacks' remark that these savants have a 'Pythagorean' sensibility and that "what is odd is not its existence, but that it is apparently so rare", I thought that this kind of sensibility might actually be commonplace but unnoticed. I have long suspected that colors might be perceived according to the Pythagorean laws that explain our understanding of scales and harmony in music, and DR. Sacks' comment reminded me of some numbers that I had played with while studying color in television broadcasting.
Rational colors
To understand how harmonics may be applied to color vision, consider the analogy of music. If you pluck a guitar string and then place your finger lightly at its midpoint, you will hear the same pitch sounded one octave higher. All natural harmonics occur at whole integer fractions of their fundamental. The smaller the denominator of the fraction, the more harmonious the harmonic sounds to our ear. In fact, the first harmonics to appear in the series above a given fundamental give a pure major triad. Getting back to the octave for a moment, recall that a frequency that is exactly the double of the other will be recognized as having the same abstract pitch. Doubling the frequency or halving the wavelength amounts to the same thing. If we consider the visual field as ranging from 750nm at the red end of the visible spectrum to 380nm at the violet end, it will be noted that the octave of red (requiring half the wavelength of the deepest red, or 375nm) is not quite reached. Nevertheless, we feel that violet tends towards red, and we have even invented the color wheel to express this subjectively felt continuity.
In spite of the fact that the visible spectrum falls just short of encompassing a whole octave, I believe that our brain has learned to tackle this wave phenomenon with the same software (perhaps located in the thalamus) that it uses to make sense of sound, a phenomenon that actually encompasses several octaves. It is not so farfetched to imagine that by recognizing patterns of relationships between visual data--patterns that it has already seen in auditory phenomena--the brain would be capable of organizing such data in a similar fashion. To verify that these harmonic relationships do exist, we will be justified in using a virtual fundamental lying outside of the visual field to appreciate the unconscious algorithms that the brain utilises to determine color hue and to appreciate visual harmony.
More specifically, I have found that a scale or rainbow of colours may be generated mathematically by considering the visible range of colours as a series of harmonics beginning at the 16th partial above a given fundamental red. For the purposes of conscious (and more laborious) calculations, a particular red wave length may be thought of as the brain's virtual color yardstick. This might vary amongst individuals but would always be exactly four octaves below a visible red perceived as a discrete hue by the individual. By using scalar ratios to calculate the harmonics, it is not necessary to worry about octaves, however.
Scalar ratios
The scalar ratios lie between 1 and 2, including 1 but not reaching 2 since dividing a wavelength by 2 would again bring us to the same abstract pitch. Scalar ratios may be used to quickly calculate the wavelength of a given partial. Now the 16th partial, being one of the octaves of the fundamental, will use 1 as its ratio because it has the same pitch as the fundamental when we disregard octaves. To build a color scale out of harmonics that would have a 652nm wavelength red as its 16th partial, we simply calculate the 17th partial by dividing 652nm by 17/16 (or 1.0625), which is about 614nm and is perceived as orange. The 18th partial then very neatly falls on 580nm, which we perceive as yellow. The next color in this scale is a yellow/green, followed by green, aqua, light blue, blue, indigo, and so on.
Primary colors
If we could see the entire range of colors in the lower octaves that serve as a virtual scaffolding for our single octave vision, we could note that someone using 652nm as a visual yardstick would first encounter an octave of the same RED as the 2nd partial; the 3rd partial (equivalent to the 5th degree in a musical scale with just intonation) would be BLUE; the 4th partial would be another octave of the fundamental (the same red); the next new color to appear (the 5th partial, equivalent to the 3rd degree of a musical scale) would be GREEN. I find it fascinating that these colors, equivalent to the major triad in music, happen to be the three colors that are used as primary colors in visual applications that use additive color, such as the television. An individual using a larger wavelength as the preferred yardstick might have a different subjective feeling for these colors. If one uses a (scalar) fundamental red of 725nm as the yardstick, then the first new colors to appear in this harmonic series are RED, BLUE and YELLOW, exactly the colors that are preferred by children and that are used as primary colors in the subtractive approach to color. In all cases, the resulting scale rests on a fundamental from the red end of the spectrum, and this is why I say that we perceive our world in the key of red.
When autistic savants graciously demonstrate their outstanding skills for us, be they musical, mathematical or artistic in nature, we find their talents spectacular. We do not know how they can retrieve information or solve problems with such ease and rapidity. Nor do they know. Take for example the autistic twins John and Michael, who have I.Q.s well below the average, and are unable to understand basic mathematical operations such as multiplication or division. In spite of these and other limitations, they are adept at making lightning calendar calculations, and are able to simply see prime numbers of a surprising order without even using pencil and paper.
Dr. Oliver Sacks has had the opportunity to work with John and Michael, and in that joyous blend of inquisitiveness, wit and charm, The Man Who Mistook his Wife for a Hat, he shares some of his clinical observations regarding their ability to mentally calculate very large prime numbers. He tells how the twins quickly reduce any number thrown at them to shambles, factoring it and sifting out the decomposable elements. The prime number kernels that remain seem to be favoured as perceptual events that have a higher reality value for them: they find them as remarkable and awe inspiring as we do the process itself.
Unconscious algorithms
What process permits such intuitive feats? Dr. Sacks suggests that these calculations are arrived at with the use of an unconscious algorithm. He says that the twins "must have 'sense' in their numbers-in the same way, perhaps, as a musician must have harmony", and he quotes Leibniz who said that "the pleasure we obtain from music comes from counting, but counting unconsciously." It is generally accepted, especially amongst musicians, that music is closely related to mathematics, but what of color now? Could the pleasure that we derive from juxtaposing "harmonious" colors also be satisfying our love for counting? When I read Sacks' remark that these savants have a 'Pythagorean' sensibility and that "what is odd is not its existence, but that it is apparently so rare", I thought that this kind of sensibility might actually be commonplace but unnoticed. I have long suspected that colors might be perceived according to the Pythagorean laws that explain our understanding of scales and harmony in music, and DR. Sacks' comment reminded me of some numbers that I had played with while studying color in television broadcasting.
Rational colors
To understand how harmonics may be applied to color vision, consider the analogy of music. If you pluck a guitar string and then place your finger lightly at its midpoint, you will hear the same pitch sounded one octave higher. All natural harmonics occur at whole integer fractions of their fundamental. The smaller the denominator of the fraction, the more harmonious the harmonic sounds to our ear. In fact, the first harmonics to appear in the series above a given fundamental give a pure major triad. Getting back to the octave for a moment, recall that a frequency that is exactly the double of the other will be recognized as having the same abstract pitch. Doubling the frequency or halving the wavelength amounts to the same thing. If we consider the visual field as ranging from 750nm at the red end of the visible spectrum to 380nm at the violet end, it will be noted that the octave of red (requiring half the wavelength of the deepest red, or 375nm) is not quite reached. Nevertheless, we feel that violet tends towards red, and we have even invented the color wheel to express this subjectively felt continuity.
In spite of the fact that the visible spectrum falls just short of encompassing a whole octave, I believe that our brain has learned to tackle this wave phenomenon with the same software (perhaps located in the thalamus) that it uses to make sense of sound, a phenomenon that actually encompasses several octaves. It is not so farfetched to imagine that by recognizing patterns of relationships between visual data--patterns that it has already seen in auditory phenomena--the brain would be capable of organizing such data in a similar fashion. To verify that these harmonic relationships do exist, we will be justified in using a virtual fundamental lying outside of the visual field to appreciate the unconscious algorithms that the brain utilises to determine color hue and to appreciate visual harmony.
More specifically, I have found that a scale or rainbow of colours may be generated mathematically by considering the visible range of colours as a series of harmonics beginning at the 16th partial above a given fundamental red. For the purposes of conscious (and more laborious) calculations, a particular red wave length may be thought of as the brain's virtual color yardstick. This might vary amongst individuals but would always be exactly four octaves below a visible red perceived as a discrete hue by the individual. By using scalar ratios to calculate the harmonics, it is not necessary to worry about octaves, however.
Scalar ratios
The scalar ratios lie between 1 and 2, including 1 but not reaching 2 since dividing a wavelength by 2 would again bring us to the same abstract pitch. Scalar ratios may be used to quickly calculate the wavelength of a given partial. Now the 16th partial, being one of the octaves of the fundamental, will use 1 as its ratio because it has the same pitch as the fundamental when we disregard octaves. To build a color scale out of harmonics that would have a 652nm wavelength red as its 16th partial, we simply calculate the 17th partial by dividing 652nm by 17/16 (or 1.0625), which is about 614nm and is perceived as orange. The 18th partial then very neatly falls on 580nm, which we perceive as yellow. The next color in this scale is a yellow/green, followed by green, aqua, light blue, blue, indigo, and so on.
Primary colors
If we could see the entire range of colors in the lower octaves that serve as a virtual scaffolding for our single octave vision, we could note that someone using 652nm as a visual yardstick would first encounter an octave of the same RED as the 2nd partial; the 3rd partial (equivalent to the 5th degree in a musical scale with just intonation) would be BLUE; the 4th partial would be another octave of the fundamental (the same red); the next new color to appear (the 5th partial, equivalent to the 3rd degree of a musical scale) would be GREEN. I find it fascinating that these colors, equivalent to the major triad in music, happen to be the three colors that are used as primary colors in visual applications that use additive color, such as the television. An individual using a larger wavelength as the preferred yardstick might have a different subjective feeling for these colors. If one uses a (scalar) fundamental red of 725nm as the yardstick, then the first new colors to appear in this harmonic series are RED, BLUE and YELLOW, exactly the colors that are preferred by children and that are used as primary colors in the subtractive approach to color. In all cases, the resulting scale rests on a fundamental from the red end of the spectrum, and this is why I say that we perceive our world in the key of red.
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