Mathematical Process Summary:
The concept of a torsor within the context of hues and the spectral octave can be illustrated through an example.
While color science typically employs wavelength measurements (nm) within the electromagnetic spectrum, music utilizes audio frequencies (Hz). These quantities are inversely related. Analogous to musical frequency ratio calculation from string lengths (or wavelengths), where the specific frequency value is less important than the ratio itself (assuming constant string tension), the precise terahertz values or photon energy are not directly employed here. Wavelength units (nm) are sufficient for determining proportional frequencies, calculated as inverses of the wavelengths. For example, the frequency ratio from red (700 nm) to cyan (495 nm) is calculated as follows:
Red (base): 700/700 = 1
Cyan frequency ratio: 1 × (700/495) ≈ 1.414
In the generated scales, ratios are calculated relative to red. However, given the cyclical nature of the system, the choice of base color is arbitrary; the proportional intervals remain invariant regardless of which color is chosen as the root or unison. This invariance exemplifies the torsor nature of hues.
The position, wavelength, and corresponding musical note assigned to magenta are derived from the observed complementary relationships. Specifically, the frequency ratio assigned to magenta is the frequency ratio of green multiplied by √2. This methodology accounts for individual variations in the spectral octave range (e.g., 370–740 nm, 405–810 nm), which are dependent on the location of the green attractor. While the graphics presented here utilize a constant 375–750 nm range for illustrative purposes, this choice reflects the torsor nature of hues.
Examples of Unique Hue-Based Scales:
- Modern Trichromat Research: This scale utilizes median unique hue data from contemporary color vision studies on normal trichromats.
- Tetrachromat Data: This scale is derived from studies on individuals with genetic predispositions to a fourth photopigment.
Auditory Examples:
The following auditory examples demonstrate the translation of unique hues into musical scales, revealing perceptual and structural parallels between light and sound.
Musical Properties of Hue-Derived Scales and the Role of Uniform Distribution
If strikingly unusual or exotic microtonal sonorities are anticipated from these hue-derived scales, their relative conventionality may be surprising. While subtle microtonal inflections may be perceptible to trained listeners, the overall impression is often surprisingly consonant with established musical practice. As previously mentioned, not only the tritone is frequently approximated by frequency ratios derived from hue data, but also other stable musical intervals, such as the major third and perfect fifth, emerge from various color combinations. The resulting scales exhibit major and minor chords, and each scale features varying degrees of consonance with other traditional intervallic relationships, corresponding to intervals such as sixths and sevenths. However, a single diatonic scale is not derived from a single root; multiple intervals are present, but their non-uniform distribution prevents direct transposition of chords derived from one color to another. The fact that these scales exhibit musical usability with common timbres, as demonstrated by the piano example in Audio:Trichromats01, is notable.
This observation raises the question of whether this musical usability is merely coincidental. To address this, the implications of randomness in tuning systems are considered. Prior research ([link]) explored the musical properties of randomly generated tunings, examining various interpretations of randomness, order, and predictability. A key finding was that uniform distribution of pitches within the octave space—even with some allowance for clustering—facilitates conventional musical usage, including tonicization and consonance on standard instruments. This arises from the inherent tendency of random subdivisions of the octave to approximate low-integer rational values, regardless of timbre (within certain tolerances). Constructing a scale with ten unusable pitches proves more challenging than constructing a usable one.
"Octave space" is defined here as any pitch range of the form (a, 2a], where a ∈ ℝ⁺. Uniformity of pitch distribution is considered within this space, ensuring that any octave-equivalent range within the audible spectrum contains a reasonable density of pitches (approximately 5 to 20). This definition excludes trivial cases such as uniformly distributed pitches concentrated within a narrow frequency range or sparsely distributed across the audible spectrum without regard for octave equivalence.
This conclusion is further supported by analysis of the Scala archive, a database of over 5,000 world tunings. Interval matrix analysis revealed that approximately 80% of the database exhibits congruence, indicating that many scales share the same intervallic content but with different starting points (modal transpositions/cyclic permutations, thus exhibiting the torsor property). Furthermore, randomly generated numbers, even from pseudo-random number generators (PRNGs), often approximate existing scales within a tolerance of approximately 5 cents. This suggests that tunings resembling established, structurally organized systems can emerge from seemingly random values. This observation led to the development of an "Average Tuning System," a 14-note system capable of approximating at least five notes from any of the 5,000 tunings in the archive within a 10-cent tolerance.
As demonstrated in the aforementioned study, music created with numbers derived from diverse sources, including planetary sizes, temperatures, mountain heights, and subatomic particle energies, consistently exhibits musical usability due to the emergence of stable, familiar intervallic relationships. This reinforces the principle that uniform distribution within the octave is a primary factor in creating musically usable scales.
Therefore, the relative conventionality of the hue-derived scales is not entirely unexpected. The color attractors themselves are well-distributed across the "color octave," naturally facilitating traditional tonal and modal usage.
However, this statistical predictability does not diminish the significance of these findings. While the musical usability of these scales may be statistically probable, their origin in physical reality and human perception imbues them with additional meaning. These are not merely arbitrary numerical values; they are rooted in the fundamental properties of light and its perception.
If the visible spectrum spanned a significantly different range—either much smaller (e.g., 400–430 nm) or spanning multiple "spectral octaves" (e.g., 400–3500 nm)—the relationship between color and chroma would become less compelling. The fact that colors exist within a single spectral octave strengthens the perceptual analogy.
This limited range also addresses the question of whether sufficient color distinctions exist to represent functional harmonies. The answer is affirmative. The fine distinctions made in color perception are analogous to the subtle distinctions made in musical intervals. Just as musicians may debate whether an interval is a "super major second" or a "sub minor third," distinctions are made between colors such as "yellowish orange" and "orangish yellow." This shared phenomenon highlights the fine granularity of both auditory and visual perception.
Color Wheel Construction: Addressing Color Space Transformations and Limitations
The construction of the color wheels presented in this analysis requires careful consideration of color space transformations and the inherent limitations of representing the visible spectrum within the RGB color space. Converting a specific wavelength to RGB values involves several factors that can influence the final color representation:
- CIE XYZ Model Version: Different versions of the CIE XYZ color space (e.g., 1931, 1964, 2012) have slightly different color matching functions, leading to variations in the resulting XYZ coordinates for a given wavelength.
- Illuminant: The choice of standard illuminant (e.g., D65, A, C) affects the white point of the color space and, consequently, the mapping of wavelengths to XYZ coordinates.
- Gamma Correction: Gamma correction is a non-linear transformation applied to RGB values to account for the non-linear response of display devices. Different gamma values will result in different RGB representations for the same XYZ coordinates.
Consequently, obtaining a specific RGB value like (0, 255, 255) for cyan from a wavelength requires careful selection of the CIE XYZ model, illuminant, and gamma. Furthermore, achieving fully saturated RGB values for all spectral hues is often impossible. If a median render of the spectrum with equal power distribution is used, for example, the perceived saturation of red tends to decrease at longer wavelengths, making it difficult to accurately represent individual "best red" values at wavelengths like 710 nm.
It is crucial to emphasize that the color wheels presented here are primarily concerned with the hue/chroma dimension of color, not with precise representations of luminance or gamma. The goal is to accurately represent the relative positions of hues within the spectrum and their complementary relationships, rather than to create a photometrically accurate rendering of the spectrum.
Therefore the final color attractor representations in the wheel are ultimately based on standard RGB values, chosen to represent the perceived hue as accurately as possible within the limitations of the RGB color space. The choice of RGB values for the attractors is done with a focus on maximizing saturation and perceptual distinctiveness, with the understanding that this might not perfectly align with a strict radiometric conversion. (a pseudo-color)
This approach acknowledges the inherent limitations of representing the full spectrum in RGB screens while prioritizing the accurate representation of hue relationships, which are central to the analysis presented.
Ultimately, it is important to note that this color wheel, constructed through a combination of standard color matching functions (CIE XYZ 2006) and individual perceptual data (color attractors and bands), possesses significant predictive power. (Stereo color mixing, color constancy-afterimages)
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Prediction Mechanism and Perceptual Parallels
Beyond the general correspondence implied by the "spectral octave," several other parallels exist between the auditory and visual sensory domains. These parallels manifest in two distinct ways: (1) through shared complementary structures, as revealed by the octave-based color model; and (2) through shared descriptive characteristics or related conceptual frameworks.
For example, within the context of sensory conflict, binocular rivalry is often compared to the phenomenon of binaural beats. While this comparison highlights similarities in sensory conflict resolution, it does not typically propose a shared model based on chroma or color. Binocular rivalry is analyzed in the context of stereoscopic color vision, where the octave and chroma models are essential for predicting color mixing outcomes.
The auditory analog of afterimages, the "aftersound" effect, is briefly mentioned as a further example of sensory adaptation. While not directly related through chroma or octave models, the afterimage phenomenon is analyzed to demonstrate its adherence to the same complementary pairs predicted by the octave chroma model.
Color constancy is also addressed, drawing a parallel with the auditory phenomenon of tonal constancy. These phenomena are not directly linked by the octave model but share similar resolution mechanisms for interpreting "neutral" or ambiguous sensory information to maintain a coherent perceptual experience, particularly in the context of musical scales and color perception. Subsequently, color constancy is analyzed using the octave model, demonstrating that the same complementary color pairs emerge when the brain interprets a physically achromatic object under colored illumination. The perceived hue of the object under colored light corresponds to the complementary of the illuminating light's hue.
Image Presentation and Viewing Instructions
The images presented in this study serve as carefully constructed visual demonstrations of various color perception phenomena. Due to the precision required for accurate depiction of these effects, image compression, re-encoding, or resizing can introduce significant noise and artifacts that compromise the integrity of the demonstrations. Standard image formats, such as JPEG, even at high quality settings, employ compression algorithms that introduce RGB artifacts and unwanted perceptual filtering. These artifacts are particularly problematic for demonstrations requiring precise control of color manipulation, such as those illustrating color constancy.
For optimal viewing and accurate reproduction of the intended effects, it is strongly recommended to access the original, uncompressed image files. These are provided in PNG (16-bit) format for most examples, or as uncompressed bitmaps. While the thumbnail images included in the article are functional for illustrative purposes, they have been resampled and interpolated, potentially introducing subtle but significant alterations that may affect the perception of the intended effects. This is especially critical for the color constancy demonstrations, where the objective neutrality of gray areas is paramount.
For instance, in the color constancy demonstrations focusing on complementary color pairs, the "constancied color" (the color perceived in a region objectively defined as grayscale RGB 127, 127, 127) can be altered by the color filtering inherent in JPEG compression. This can introduce unintended color casts into the nominally gray areas. While the general color constancy effect may still be discernible, the crucial demonstration of a color emerging solely from a region of objective gray due to contextual interpretation (scene, object, and light/material reflection) is compromised. A detailed explanation of this effect and its nuances is provided in the corresponding section.
The stereoscopic images presented in this study employ two techniques: true 3D rendering with depth information and dichoptic presentation of duplicated images to demonstrate the interaction of complementary colors with afterimages and color constancy. All stereoscopic images are designed for cross-eye viewing. While all images can be viewed using VR headsets with appropriate scaling and settings, certain effects and induced artifacts are best observed using the cross-eye technique. Specifically, the cross-eye method is essential for demonstrating the interaction of afterimages in stereoscopic vision, allowing for analytical observation of the residual images along the mixed percept. Similarly, the cross-eye technique is valuable for directly demonstrating the nuances of color conflict resolution and its interaction with other regions of the visual field. For cross-eye viewing, the left image is intended for the right eye, and vice versa.
All images adhere to standardized RGB values for the color attractors (see Table01 for specific values). Even with a perfectly calibrated display, minor individual variations in color perception are expected. These variations may arise from factors such as lens pigmentation, macular pigment density, cone sensitivity shifts, and individual perceptual and subjective differences. Consequently, some observers may perceive slight variations in the precise point of achromatic cancellation in stereoscopic mixing experiments, or subtle differences in afterimage and color constancy complementary mappings. Such individual differences are inherent in color perception research.
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| (Image.07) - Stereo vision demo. Orange/Yellow snake |
Stereoscopic Color Mixing and the Integration of Binocular Information
The integration of binocular information in stereoscopic vision raises important questions about the role of trichromacy and opponent processing at various stages of visual processing, from the initial encoding in the retina and early visual pathways (retinal and post-retinal opponency) to the formation of a unified perceptual experience. The fact that color information undergoes at least one further transformation during stereoscopic processing before reaching conscious awareness suggests that opponent mechanisms may operate at later stages of visual processing. This also highlights the binary nature of color opponency in achieving achromatization (the perception of gray or white).
While opponent processing is well-established in the retina and early visual areas of the brain, the phenomenon of stereoscopic color mixing suggests the possibility of a further stage of opponent processing specifically dedicated to integrating color information from the two eyes. This proposed "final" opponent process could be responsible for the observed cancellation of complementary colors when presented to the two eyes in a stereoscopic configuration. This hypothesis aligns with known mechanisms involved in binocular rivalry and stereoscopic depth perception, both of which require the integration and resolution of potentially conflicting signals from the two eyes. Further research is necessary to fully elucidate the neural basis of this proposed "final" opponent process.
The propagation of color information in the brain, originating from discrete photoreceptors and culminating in continuous image perception, necessitates interpolation of the discrete signals. This interpolation, evident in the filling-in of the blind spot and the perceived continuity of peripheral vision despite decreasing resolution, represents a point where the limits of qualia become apparent, merging with a lack of conscious experience. This interpolation may occur concurrently with or prior to stereoscopic color mixing, which exhibits complementary relationships resembling subtractive models.
This suggests that color mixing occurs prior to the formation of unified qualia but interacts with other phenomena, such as color constancy, in complex and not entirely predictable ways, as will be discussed.
Binocular Rivalry and Resolution
Binocular rivalry occurs when two different images are presented to each eye. The brain is unable to fuse these disparate images into a single coherent percept, resulting in an alternating perception of the two images, with each image intermittently dominating conscious awareness.
This rivalry can be directly observed with colored stimuli. When viewing stereoscopically merged blue and yellow squares, for example, rivalry ensues. However, color alone does not fully account for this conflict. Introducing contextual cues, such as the outline of an object, facilitates fusion and resolves the rivalry. In the landscape example, where one image is tinted blue and the other yellow, stereoscopic viewing successfully merges the images, and the colors are no longer perceived as conflicting, but rather mix, exposing the non-complementary nature of blue and yellow, mixing into green. (The blue-yellow "problem" will be addressed later.)
Demonstrating the Influence of Luminance on Stereoscopic Color Mixing:
The following image pair (Image.01-Conflict and Image.02-Resolution) is designed to isolate the influence of luminance variations on stereoscopic color mixing, specifically addressing the non-complementary mixing of RGB yellow (255, 255, 0) and blue (0, 0, 255). Previous examples, containing more complex luminance information, demonstrated that these colors combine to produce green, rather than the expected achromatic (gray) percept observed with true complementary pairs. This deviation from expected achromatic mixing can be attributed, at least in part, to the influence of varying luminance cues present in those images.
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| Image-01-Conflict |
To directly examine the interaction of saturated yellow and blue patches in the absence of confounding luminance variations, the subsequent image (Image.02-Resolution) is constructed with minimal luminance differences. Shadows and luminance variations are removed, leaving only subtle object edges to facilitate binocular fusion. This contrasts with the first image (Image.01-Conflict), which presents the same saturated yellow and blue patches without any object cues, in which rivalry is more likely to prevent color fusion.
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| Image-02-Resolution |
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| Image-02b-Stereo Mix Result |
Results and Interpretation:
In Image.01-Conflict, the absence of visual cues prevents (in most subjects) binocular fusion, resulting in binocular rivalry – the alternating perception of the yellow and blue patches. However, in Image.02-Resolution, the addition of minimal outline details enables binocular fusion. Critically, despite fusion, the combined percept is green, not gray. This confirms that RGB yellow and blue do not behave as true complementary colors in stereoscopic mixing. Unlike true complementary pairs, such as RGB red (255, 0, 0) and cyan (0, 255, 255), which do achromatize (mix to gray) in stereoscopic vision, yellow requires violet for achromatic mixing, and blue requires orange.
This observed deviation from expected complementary mixing demonstrates a discrepancy between the standard RGB color space and the brain's internalized, neurally represented color space. The brain's internal color space appears to be organized around a more uniform distribution of color attractors and canonical complementaries, which do not perfectly align with the RGB primaries. The use of near-saturated and spectrally extreme stimuli (yellow and blue) highlights this divergence. The results suggest that luminance information plays a significant role in how the brain resolves color conflicts in stereoscopic vision, and that the brain's internal representation of color may be more consistent with a model of truly complementary opponent processes.
(It's important to note that monocular rivalry also exists, where the alternating perception occurs even when only one eye is stimulated with two different images presented in rapid succession. This further emphasizes the brain's role in resolving sensory conflict, which isn't inherent of stereo inputs)
The following stereoscopic images are designed to demonstrate two key aspects: (1) binocular complementaries, defined as those opposed in the logarithmic wheel mapping; and (2) color mixtures exhibiting subtractive-like characteristics. For clarity, these demonstrations focus on pairs of color attractors.
Visual Processing Hierarchy in Stereoscopic Vision
To investigate the precedence, order, and interactions of various visual processes and effects, several stereoscopic images were designed to elucidate the conditions necessary for a unified perceptual experience and to isolate specific perceptual conflicts. The goal was to create stereoscopic image pairs that fuse naturally while introducing controlled conflicts in specific visual attributes. The analysis of these experiments suggests a hierarchical organization of visual processing, where depth information derived from binocular disparity exerts a dominant influence, often resolving conflicts arising from color and luminance information.These findings also enable the creation of images where perceptual conflicts can be induced in otherwise harmonious stereoscopic image pairs.
Summary of Observations:
1. Scene Influence on Color Mixing:
- Image (a): Colored Background, Colored Ball: This image depicts a soccer ball positioned against a uniform background. The left eye's view has a blue (0, 0, 255) filter applied to the entire image, while the right eye's view has a yellow (255, 255, 0) filter. The soccer ball is rendered in orange (255, 127, 0) in the right eye's view. When these images are fused stereoscopically, the observer perceives a black and white (achromatic) soccer ball against a green background. The disparity information from the ball, combined with the luminance cues, facilitates stable binocular fusion. The disparate color information from the backgrounds is integrated through stereoscopic color mixing, resulting in the perception of green.
- Image (b): Colored Ball, Monochromatic Background: This image uses the same yellow and blue filters applied only to the soccer ball in each eye's view (yellow for the left eye, blue for the right). The background is rendered as monochromatic gray in both views. When these images are fused stereoscopically, the blue-yellow conflict present in the ball is not resolved into green. Despite the ball being the primary focus of attention and providing depth cues, the consistent achromatic information from the background and the matching depth information facilitate binocular fusion. The color mixing "instruction" is likely interpreted as "deliver the color information to qualia as is," preventing the typical blue-yellow mixing seen in other contexts. Depth information continues to dominate the perceptual strategy. To further highlight the binocular color conflict, small blue and yellow patches are introduced in the respective images, positioned so as not to overlap with the ball (top-right). These patches are perceived as floating, distinct colored regions within the 3D scene, demonstrating clear binocular rivalry. In contrast to Image (a), where the same color information was integrated into a unified green percept, these patches remain distinct due to the lack of a global color mixing instruction.
- Image (c): Color Inversions: This image pair explores the effects of inverting complementary color filters between the two eyes. The right eye's view features an orange background and a blue ball. The left eye's view inverts these filters, presenting a blue (0, 0, 255) background and an orange (255, 127, 0) ball. When these images are fused stereoscopically, the global color conflict created by the complementary backgrounds is resolved towards a near-achromatic (gray) percept, driven by the depth and luminance information. However, both the ball and the small colored patches (also using the same orange-blue color pair) exhibit pronounced binocular rivalry. This setup demonstrates that the color mixing strategy is determined globally. Despite using the same colors, the intended balance of the global mixing scheme influences the entire visual field. The right eye's view exerts a "push" towards blue, while the left eye's view exerts a "push" towards orange, resulting in the gray background. Crucially, this global influence extends to the local color information as well. The colors of the ball and patches are pushed along in the same direction as their respective backgrounds, amplifying their chromatic contrast and resulting in a more intense perceptual conflict. This amplification manifests as a "brighter" or more saturated rivalry. Close observation of the ball's edges reveals the conflicting colors "bleeding" into the nearby achromatic (gray) grass. This observation directly demonstrates that color interpolation is processed independently of the luminance channel, which retains sharp detail without interference from the color conflict.
Stereoscopic Color Mixing and the Subspace Mixing Strategy
The following images demonstrate a crucial aspect of stereoscopic color mixing: the concept of a subspace mixing strategy/instruction This refers to the phenomenon where, once the visual system identifies a region of hue interaction and determines a resolution for color conflict (using disparity or luminance information), it applies this resolution to the entire subspace. This occurs even if internal inconsistencies remain within the region and regardless of whether some colors are, in fact, identical. (This principle was previously illustrated in the soccer ball examples).
Critically, the hue shifts induced by stereoscopic mixing do not propagate outside each defined region.
Image.07 illustrates this principle. It uses red and cyan as stereoscopic complementaries, applied as filters to an image of a snake. Specific areas within the snake image are designed for additional color mixing demonstrations. When the image is successfully fused via cross-eye viewing, the snake is perceived primarily in grayscale with yellow/orange details. Simultaneously, the original red and cyan images are still visible at the periphery of the fused image. Importantly, the hue shifts necessary for achieving the achromatic (gray) state in the fused region do not extend beyond this region. The observer's perception of the surrounding environment (the room, the computer screen background) remains unaffected.
However, within the defined region of stereoscopic mixing, the mixing instruction does apply globally. Objects or details within the red or cyan filtered areas are "dragged" along by the forces that are uniting the parent colors into the achromatic state.
In this specific example, two additional color mixtures are observable:
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Orange: Created by the combination of red areas in the red-filtered image and yellow details in the cyan-filtered image.
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Yellow-Green: Created by the combination of orange areas in the red-filtered image and the cyan areas in the cyan-filtered image.
These resulting color mixtures are consistent with predictions based on the spectral octave and the principle of the subspace binocular mixing strategy. The sharp edges of the images within the computer screen window likely define the boundaries of the region to which the mixing instruction is applied. The brain may interpret this as viewing the scene through a window, effectively isolating the stereoscopic mixing effects to the defined area. This subspace is analyzed in depth later.
Out-of-Gamut Color Shifts in Stereoscopic Mixing
This image demonstrates how the subspace mixing strategy in stereoscopic vision can "drag" colors out of the standard color gamut. A stereoscopic image depicting a landscape with true depth information is employed. The right-eye image is filtered with yellow, containing some orange areas, while the left-eye image is filtered with blue-violet. When fused stereoscopically, the landscape is perceived with green trees and near-gray areas (while the rest of the observer's visual field remains unaffected).
Crucially, two red patches, identical in both the left- and right-eye views, are included in the image. These patches are slightly displaced vertically (as opposed to horizontally, which would be interpreted as a depth cue). This vertical displacement ensures that the patches are perceived as separate entities superimposed on the fused, near-gray background.
Upon stereoscopic fusion, these red patches undergo a dramatic transformation. One patch is perceived as a dark, purplish hue, while the other appears as a lighter, orange hue. Neither of these perceived colors corresponds to the original red of the patches. Furthermore, the overlapping regions of the patches create a highly saturated orange, often perceived as being out of gamut. This demonstrates how the mixing strategy can not only shift colors but also push them beyond the boundaries of typical color representation. (The question of the afterimage of this complex, out-of-gamut hue is explored in a later section.)
Stereoscopic color mixing and its interaction with other effects such as color constancy and afterimages provide more details about the nature of color phenomena. Each next section introduces Color Constancy and Afterimages and integrates with binocular color mixing.
Contextual Perception: Tonal and Color Constancy
In music, tonal constancy describes a phenomenon analogous to color constancy in vision. While these phenomena are not necessarily mediated by identical adaptive mechanisms, familiarity may contribute to their enhancement. In music, tonal constancy, as analyzed subsequently, refers to the brain's ability to interpret musical scales with nominally equal step sizes and neutral intervals (from a diatonic perspective) as exhibiting non-equal steps and resolved intervals when required by the musical context. This can be illustrated with more extreme examples than the stretched diatonic scales discussed earlier. Given any melody in any tuning system, each note can be subjected to a degree of pitch variation without losing its tonal meaning. A specific interval that functions as a minor third in one chord or cadence may be perceived as a major third in a different context or melodic trajectory. Similarly, other intervals can exhibit a superposition of functional roles. A sharpened second, for instance, may function as a melodic minor third but, when transposed an octave higher and combined with a suitable fifth or seventh, can function as a ninth. Tonal constancy is further elucidated with auditory examples later in this analysis.
Color constancy is related to chroma through the octave color wheel and complementary colors. This visual phenomenon, intimately linked to afterimages, refers to the brain's capacity to interpret the color of objects under varying illumination conditions. The visual system adapts to changes in illumination and takes into account both illumination and material properties to discriminate colors.
As a consequence of color constancy, when an object is illuminated with light of its complementary color, it is perceived as achromatic (gray or white). Conversely, objectively achromatic objects are perceived as tinted with the complementary color of the illuminating light. This effect can be readily demonstrated on computer screens, further confirming the objective nature of gray and its susceptibility to perceptual adaptation.
As previously mentioned, familiarity plays a role in shaping these effects. Research has shown that color constancy is more pronounced when the shape and actual color of the object are known; in the absence of such prior knowledge, the perceived hue is less salient.
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| (Image.08) The "orange" guitar. |
Another factor indicating the active participation of the brain in this effect is the difficulty in simply simulating it. For color constancy to occur, sufficient contextual cues must be present for the brain to interpret a scene, rather than merely an image. This is analogous to binocular rivalry, where contextual cues resolve perceptual conflict. For example, a pure blue image with a small gray square at its center is typically perceived as a blue background with a gray square; the color constancy effect is not elicited by simple color-gray contrast alone. However, a more realistic scene (Image.08) generates a vivid effect, even with a less saturated "simulated" blue light. The image depicts an "orange" classical guitar, which is objectively gray, with the rest of the scene rendered using a pure RGB blue filter (0, 0, 255).
Given the influence of familiarity on the strength of color constancy, the subsequent images employ Rubik's Cubes within a scene. Rubik's Cubes are commonly used in color constancy demonstrations because, while viewers may associate them with color, they do not typically associate them with a single, fixed color. This object provides sufficient cues to establish a "natural" scene and elicit the color constancy effect across various complementary color settings.
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| (Image.09) Color Constancy and Complementary Colors. This image demonstrates eight configurations of illuminating light and the corresponding perceived complementary color on the gray cube. The effect is sufficiently pronounced that discerning the objectively gray regions of the cubes may require careful observation. Some viewers may even be inclined to download the images to verify that the target areas are indeed gray (RGB 85, 85, 85). |
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| (Image.10) |
Image Pair (image.cc-10-11) Demonstrating Color Constancy and the Role of Familiarity
The following image pair (image.cc-10-11) is designed to demonstrate several key aspects of color constancy, including the influence of object recognition and familiarity. While previous research has convincingly shown that color constancy can be enhanced by object recognition and memory, the primary goal of these images is to establish a baseline for subsequent demonstrations. Specifically, these images illustrate that carefully controlled illumination and filtering can produce perceptually compelling color experiences that are solely attributable to color constancy mechanisms, independent of object familiarity. In all these images, tiger's colored fur is objectively the same monochromatic RBG values.
Image cc-10 presents four copies of a tiger art pencil drawing, each at a different luminance level. This image demonstrates how the "constancied" orange hue remains relatively consistent across varying intensities of blue illumination. Critically, the regions perceived as orange in this image are objectively grayscale (RGB 127, 127, 127). While object familiarity (the knowledge that tigers typically have orange fur) might contribute to the perception of orange in this image, it is essential to establish that the effect can be achieved with objectively neutral gray areas.
Image cc-11 explores the limits of familiarity's influence. It presents four variations of the same scene, each with different illumination settings. In these variations, the tigers are perceived as orange, red, yellow, and green, respectively. Remarkably, the corresponding fur areas in each tiger image are also objectively grayscale (RGB 127, 127, 127). The perception of these diverse hues arises because the brain interprets the varying illumination as realistic, and color constancy mechanisms then generate the corresponding complementary colors. In essence, we "trust" our interpretation of the light source more than our prior knowledge that tigers are not typically green, yellow, or red. More concretely, the opponent processing cells likely respond to the blue light across the entire region, leading to the emergence of the complementary orange, red, magenta, yellow or green sensation within the grayscale areas.
Constancy and Stereoscopic Mixing
The color constancy effect can be predictably manipulated and mixed stereoscopically. This complex scenario is illustrated in images [cc09-10-11].
Two identical grayscale tiger images are used, each filtered with a different color: cyan for one and violet for the other. The cyan-filtered image elicits a reddish percept of the tiger due to color constancy mechanisms operating on the objectively gray areas. Conversely, the violet-filtered image elicits a yellowish percept. These induced colors are perceived as "normal" due to the brain's compensation for the filtering.
Upon binocular fusion (using the cross-eye technique), the background, now perceived as a subtle blue light, appears nearly achromatic (gray) compared to the saturated orange percept of the tiger. Remarkably, in this arrangement, where perceived colors are effectively "rebuilt" through constancy and stereoscopic vision, the only areas lacking direct monochromatic information in both eyes (the objectively gray areas) are the only areas exhibiting color after binocular fusion (the orange percept). A diagram [cc11] below the images illustrates this color mixing process.
Diagram [cc11] Labels:
M = Monocular: Indicates information presented to a single eye.
B = Binocular: Indicates information resulting from binocular fusion.
L = Left: Refers to information presented to the left eye.
R = Right: Refers to information presented to the right eye.
SDCI = Subspace Dominant Chromatic Information: Represents the chromatic information presented to each eye before constancy effects.
CDCI = Constancy Driven Chromatic Information: Represents the chromatic information resulting from the color constancy mechanism (the "constancied" colors).
The strength of the color constancy effect in this demonstration is notable. All tiger images are objectively grayscale. The first image [cc09] is designed for cross-eye viewing, and image [cc10] illustrates the resulting percept. The subtle saturation difference in the blue background is sufficient to elicit a strong orange percept in the tiger, demonstrating the robustness of color constancy.
The CDCI (Constancy Driven Chromatic Information) determined for each SDCI (Subspace Dominant Chromatic Information) can be mixed stereoscopically, validating its function as genuine chromatic information that influences perception.
The binocular chromatic fusion strategy is determined for each SDCI. The visual field can contain multiple SDCI(object detection of stereoscopic images on the screen). When fusing images using the cross-eye technique, colors mix predictably, while the surrounding visual field (level 0) remains unaffected. Each object's size and salience influence the fusion strategy within its respective SDCI. This means that nested images inherit, rather than create, their fusion strategies from their superspace. Consequently, while multiple simultaneous stereo images can be mixed independently, color conflicts within nested images are not resolved independently and are "dragged" by the fusion strategy of their superspace. Even with identical images that don't inherently require "mixing," color conflicts can still occur, as demonstrated in subsequent examples.
Image Series (image.cc-[12-19]): Color Constancy Demonstrations with an Electric Guitar
This series of eight images utilizes an electric guitar to demonstrate color constancy under varying color attractor illumination settings. These demonstrations highlight the complementary relationships between illumination and perceived color, as well as the varying salience of different "constancied" hues. The images also illustrate how the perception of these hues, derived from objectively grayscale regions, can be influenced by the surrounding color context.
The images were carefully adjusted to equalize the average salience of the perceived colors. It is observed that some hues are more readily elucidated (perceived as saturated) than others. For example, "constancied" cyan is typically perceived as more salient than "constancied" red. This difference in salience mirrors the ranking of afterimage hues, where red afterimages are often the least vivid. One possible explanation for this phenomenon is the larger size of the neural "blobs" representing red in V1 compared to other canonical hues. If afterimage and color constancy are active, feedback-driven processes involving V4 and subsequent visual areas looping back to V1, the weaker signal reaching the initial loop stage for red (due to the larger V1 blob size) might explain its lower salience. However, it is likely that multiple factors contribute to this difference. It is important to note that afterimages produced with natural pigments and daylight exhibit higher saturation for red, suggesting that the lower saturation observed with RGB displays might be a limitation of the display technology or the additive color mixing process.
Each guitar image features two grayscale areas, one slightly darker than the other, with RGB values near (100, 100, 100) and (150, 150, 150), respectively. These areas contain subtle variations in gray details and shadows. The rest of the guitar is "illuminated" with different colored lights, inducing the perception of the corresponding "constancied" color within the grayscale regions.
Adjacent to the guitar is a circular arrangement of eight guitar picks, each representing a canonical color attractor: red, orange, yellow, green, cyan, blue, violet, and magenta. These picks serve two purposes. First, they demonstrate the interaction of the colored illumination with other hues beyond the grayscale areas. Second, and more importantly, one of the picks in the arrangement is also gray, matching the grayscale areas on the guitar. This gray pick, along with the spectrally ordered arrangement of the other picks, resolves potential ambiguities in hue perception.
For example, an isolated guitar under blue (0, 0, 255) illumination might be perceived as either orange or yellow. However, the presence of the gray pick and the surrounding picks (yellow and red) clarifies that the guitar's "constancied" hue falls between yellow and red, confirming it as orange. Similarly, a guitar under yellow light might be reported as either violet or blue; the presence of the gray pick and the surrounding picks (blue and magenta) helps the observer correctly categorize the "constancied" hue as violet/purple. Constructing these images is analogous to performing a spectral ordering task with "imaginary" hues, a conceptually challenging but revealing process.
Tonal Constancy
This auditory phenomenon shares conceptual similarities with color constancy, although the underlying mechanisms and models differ. Analogously, if one considers a set of pitches (musical chromas), such as the diatonic scale, as analogous to a set of colors, these pitches can be substantially shifted and retuned without losing their tonal meaning, just as a set of colors can remain identifiable under varying illumination conditions. The stretching or alteration of notes can be considered analogous to different "illuminations" of the set of musical chromas, which nonetheless retain their identifiable relationships.
As discussed previously, a key difference exists between musical and visual chromas. The color of light can be perceived based on a single frequency or wavelength. In music, however, chroma is relative; a single isolated note does not possess an inherent "color" but acquires a contextual chroma within a chord or melody. In this sense, the color spectrum functions as a torsor relative to sound. Once a note is incorporated into a harmonic or melodic context, its role and degree are defined by its relative chroma. Each note, therefore, possesses multiple chromas relative to the other notes within the musical context. Consequently, different intervallic configurations of the diatonic scale not only remain functionally viable but can also generate additional chromas through transposition. There is no single "yellow" in this analogy; there are multiple "roots," each with its own set of relative chromas.
In general, tonal constancy refers to the brain's tendency to interpret musical intervals and progressions within a tonal context, even when the actual intervals deviate from standard tunings. A clear demonstration can be provided using 7-tone equal division of the octave (7-EDO), a tuning system in which no interval perfectly corresponds to those of 12-tone equal temperament.
Audio Examples
The following audio examples present melodies in both 7-EDO and their 12-tone equivalents. Despite the equal step sizes in 7-EDO, the melodies evoke a sense of familiar tonal functions. For example, the second step in 7-EDO, a neutral third at 342 cents (approximately halfway between a major and minor third), is often perceived as having either a "major" or "minor" quality depending on the surrounding musical context, such as the implied harmony or the melodic contour. This effect, which persists even with pure sine waves (thereby eliminating harmonic artifacts), demonstrates tonal constancy: the listener's brain interprets the neutral intervals within a tonal framework, resolving them into functionally familiar pitches. When the same progression is rendered in 12-tone equal temperament, the listener/performer naturally resolves each step into the "correct" functional pitch to satisfy the implied cadence.
Ξ Example A - 7edo
(Further examples exploring this phenomenon in other tuning systems, including more complex modulations, can be found on my YouTube channel.)
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(Image.11) This geometric visualization compares 7-EDO with the diatonic scale in 12-tone equal temperament on a logarithmic scale. Transposition of the 7-EDO structure yields identical intervallic relationships, whereas transposition of the diatonic scale reveals the seven familiar modes of 12-tone music.
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This phenomenon raises questions regarding the limits of tonal functions. How much can these intervals be shifted or stretched before they lose their tonal meaning? This is a complex question involving individual perceptual variations and the continuous nature of pitch space. Color, chroma, and hue play a central role in exploring this question within the visual domain.
While the examples demonstrate how a single interval can serve different functions depending on context—particularly within melodic sequences or trajectories—two additional factors warrant consideration. First, familiarity, while difficult to define and quantify precisely, intuitively influences perception and reinforces tonal constancy. Second, the non-Euclidean nature of pitch space, where the cumulative perception of small intervals can lead to an overestimation of the total perceived distance, contributes to the effect. These factors, combined with the influence of surrounding notes on the perception of otherwise neutral intervals, provide a comprehensive explanation for tonal constancy.
Sense Adaptation and Aftereffects:
Both vision and audition exhibit phenomena related to sensory adaptation, termed aftereffects. In vision, this is known as the afterimage, while in audition, it is the aftersound. Although both involve sensory adaptation, their underlying mechanisms differ significantly.
Auditory Aftereffects (Aftersounds):
Auditory aftereffects manifest as the perception of a residual pitch following exposure to broadband noise with a rejected frequency band. The perceived pitch corresponds to the logarithmic center of the rejected band. This phenomenon is distinct from visual afterimages and will not be discussed further in this section, which focuses exclusively on visual afterimages.
Visual Aftereffects (Afterimages):
Visual afterimages are perceptual phenomena in which a residual color sensation persists after the removal of an initial stimulus (the inducer). While the initial trigger is attributed to temporal adaptation of photoreceptor cells (cone fatigue), the phenomenon is more complex than a simple depletion of cone sensitivity.
Complexity of Afterimage Perception:
Several observations highlight the complexity of afterimage perception:
- Temporal Integration: The perceived afterimage hue is determined by the integrated exposure time to the inducer, even if the inducer color changes rapidly. For instance, a stimulus alternating rapidly between dark red and yellow will produce a similar afterimage to a spatially mixed red-yellow stimulus, provided the total exposure times are equal. This demonstrates a temporal integration process in adaptation.
- Edge and Object Influence: Afterimages can be perceived in regions where the inducing color was not directly present. For example, if a sharp-edged shape (e.g., a star) is presented with only its corners tinted red, the afterimage will encompass the entire shape, including the gray central area. This suggests that post-retinal processing, including edge and object detection, influences the perceived afterimage. This influence occurs pre-stereoscopically, as evidenced by the persistence of individual afterimages in each eye during stereoscopically fused images (as demonstrated in previous sections).
These observations indicate that while cone fatigue initiates afterimage formation, subsequent visual processing stages modulate their final appearance.
Types of Negative Afterimages:
The common conception of afterimages as simple RGB complementary colors is an oversimplification. Modern research has revealed more nuanced relationships between inducer and afterimage hues. It is crucial to distinguish between two types of negative afterimages:
- Instant Afterimages: These appear immediately upon removal of the inducer.
- Delayed/Conflict Afterimages: These require longer exposure times and are best observed in a dark environment. They are often described as "negative images" that exhibit oscillation (with a frequency similar to binocular rivalry) and gradual fading. When multiple inducer colors are used sequentially, delayed afterimages can appear sequentially as well.
Early studies primarily focused on instant afterimages, attempting to explain them solely through cone fatigue and a simple subtraction of the inducer color from the background. These explanations failed to account for delayed afterimages, which are a common experience (e.g., the afterimage seen after staring at a bright light and then closing the eyes). Delayed afterimages are more readily elicited with natural pigments and daylight and are optimally observed in complete darkness (by covering the eyes).
The Octave Hue Wheel and Afterimage Prediction:
While empirical data on inducer-afterimage hue relationships exhibit some variation across studies (due to individual differences and methodological variations), a consistent pattern emerges when compared to the logarithmic octave hue wheel proposed in this study. This wheel demonstrates remarkable predictive power for afterimage hues.
Specific Complementary Pairs:
- Red and Cyan: Red and cyan are conventionally considered complementary and exhibit reciprocal afterimage relationships. Empirical studies consistently report afterimages within the red and cyan bands, corresponding to their directly opposing positions on both the sRGB color wheel and the logarithmic octave hue wheel.
- Green and Magenta: Green and magenta also exhibit reciprocal afterimage relationships, with afterimages consistently falling within their respective color bands. These colors also occupy opposing positions on both color representations.
- Blue and Yellow: Blue and yellow present a crucial deviation from simple reciprocity. Blue induces an orange afterimage, while yellow induces a purple afterimage. Conversely, orange induces a blue afterimage, and purple induces a yellow afterimage. This non-reciprocal relationship is accurately predicted by the logarithmic octave hue wheel, where blue is opposite orange and yellow is opposite purple, unlike the sRGB wheel, where blue and yellow are directly opposed.
This accurate prediction of afterimage relationships provides strong support for the validity of the logarithmic octave hue wheel and its underlying logarithmic representation of the visible spectrum.
GIF Animations (Gif.01-02): Edge Detection and Afterimage Perception
The following GIF animations (Gif.01-02) demonstrate the influence of edge and object detection on afterimage perception. Each animation, consisting of three frames played in a continuous loop, comprises a 2-second inducer presentation followed by 1-second presentations of two different edge guides. These animations illustrate how afterimages are selectively perceived only where edges are present, with colors disappearing in areas lacking defined boundaries. This observation strongly supports the active construction of afterimage perception, suggesting that while cone fatigue may play a role in triggering the effect, higher-level visual processing, including edge detection, is crucial in shaping the afterimage's appearance. Gif.01 utilizes the classic example of two superimposed four-point stars, while Gif.02 employs differently colored circles.
Analysis of Afterimage Research Data:
Analysis of existing afterimage research data reveals evidence of a color organization consistent with the octave hue model proposed in this study.
Reciprocity and Stereoscopic Confirmation:
Numerous studies report a statistical reciprocity between inducer and afterimage hues, often modeled by iterative functions that map inducer hues to their perceptual inverses. This reciprocity is also qualitatively evident in stereoscopic color vision. For instance, when two complementary colors (e.g., red and cyan, precisely determined for a given individual using minimal intensity adjustments to achieve perceptual achromatization) are presented dichoptically (one color per eye), the resulting afterimages are also complementary. The fused, binocular afterimage appears achromatic, demonstrating mutual cancellation of the opponent afterimage hues.
Conversely, when colors are fused stereoscopically to create a chromatic mixture (rather than complementary cancellation), the afterimage reflects both the fused color and the individual afterimage components. For example, if blue and yellow are fused to create a perceived green, the subsequent afterimage will be magenta/purple. This resulting color percept is conceptually "doubly justified": it arises both as the afterimage of the fused green percept and as a stereoscopic combination of the individual afterimages induced by each eye (blue inducing orange, yellow inducing violet). The stereoscopic fusion of these individual afterimages (orange and violet) results in the perceived magenta afterimage. The following image is designed to illustrate these phenomena. (Viewing instructions are provided at the beginning of the stereoscopic vision section.)
Deviation from RGB Complementarity and Hue Distribution:
However, a more significant pattern emerges from the distribution of afterimage hues, specifically a clear deviation from simple RGB complementarity. Studies employing 24 uniformly spaced inducer samples from an RGB color space, including pure RGB and CMY primaries, reveal a non-uniform distribution of afterimage hues.
These studies identify three distinct clusters of inducers, each mapping to a disproportionately large region of afterimage hues compared to their corresponding region on the RGB color wheel. For example, a relatively small 20° region of green in the inducer set can map to nearly 100° of red and magenta hues in the afterimage set. Other clusters, while smaller in angular extent, exhibit the same "stretching" or non-linear mapping.
Connection to the Octave Hue Model:
This non-linear mapping demonstrates that the RGB color wheel does not accurately represent the relationship between inducer and afterimage hues. If the afterimage mapping is "unstretched" to achieve a more uniform distribution of hues, the resulting arrangement closely resembles the spectrum generated by the octave hue method proposed in this study. Specifically, this "unstretching" reveals the key non-reciprocal relationships predicted by the octave model, such as blue mapping to orange and yellow mapping to violet/purple, which are not directly opposite on the RGB wheel.
Afterimages and Stereoscopic Mixing: Evidence for a Shared Mechanism
These experiments explore the interaction between afterimages and stereoscopic mixing, revealing a direct relationship between these two phenomena and providing further support for the proposed model of color perception. The observation that afterimage hues interact stereoscopically in the same manner as colors in stereoscopic mixing experiments strongly suggests a shared underlying mechanism.
Reciprocity of Afterimages and Dichoptic Complementaries
The reciprocity of afterimages and dichoptic complementaries, where an inducer hue triggers its complementary afterimage and vice versa, has been previously discussed and is well-supported by modern research. While minor discrepancies in hue matching may arise due to sampling methods or the use of median sample values, the general reciprocal relationship is clear. Furthermore, even when perfect reciprocity is not observed, the afterimage hue typically falls within the expected color category, representing a minor variation rather than a fundamental deviation. Stereoscopic mixing experiments provide even more direct evidence for this reciprocity.
Stereoscopic Mixing of Afterimages
The following series of images is designed to minimize luminance and depth cues to isolate the interaction between afterimages and stereoscopic mixing. Minimizing these cues helps to avoid binocular rivalry and allows for a clearer observation of the color mixing effects.
Image [] employs red and cyan filters for the right and left eyes, respectively. Upon stereoscopic fusion, the image is perceived as gray, with the original red and cyan images still visible at the periphery. After adaptation (inducing afterimages) and subsequent replacement of the red/cyan images with grayscale or outlined versions (while maintaining cross-eye viewing), the central fused image remains gray. This suggests that the afterimages, like the original colors, are also subject to stereoscopic cancellation.
This is confirmed by the following observation: Slight head movements while fixating on the fused image, or covering one eye, reveals that each eye has its own corresponding afterimage (cyan for the red-adapted eye, red for the cyan-adapted eye). These afterimages, generated before the point of binocular combination, are then mixed stereoscopically, just as real chromatic information would be.
Stereoscopic Mixing of Afterimages: Complex Interactions
The interaction becomes more complex when considering other color mixtures. The selection method for color conflict resolution can influence the outcome, potentially "dragging" in other colors that would normally be expected to cancel out or produce a predictable mix. However, this is not always the case, revealing a clear hierarchy in the operation of these effects.
The green percept created by the stereoscopic mix of blue (0, 0, 255) and yellow (255, 255, 0) was previously demonstrated with the inclusion of orange areas to provide an achromatic reference. In the following image, the minimal luminance information simplifies the interaction.
This blue-yellow stereoscopic pair produces a cyan-green percept. After adaptation (inducing afterimages), the blue stimulus generates an orange afterimage, and the yellow stimulus generates a violet afterimage. The stereoscopic mix of these afterimages results in a red-magenta percept. This doubly justified color (red-magenta) is also the afterimage of the cyan-green percept, demonstrating an algebraic-like nature of hue interaction. (A more detailed exploration of this concept is presented later in this study.) As an initial analogy, consider a unit group where each element has an inverse. Multiplying an element by its inverse results in the identity. Multiplying other pairs results in a different element, whose inverse is the result of multiplying the inverses of the initial pair. Hue interactions in the brain appear to follow a similar cyclic multiplication scheme.
Complementary Colors and the Central Role of Green:
As previously discussed, the symmetrical arrangement of color attractors on the logarithmic octave hue wheel allows for a degree of predictability. Knowing the positions of red, orange, and yellow for a given individual allows for the prediction of cyan, blue, and violet by locating the points directly opposite them. This corresponds to a wavelength ratio of approximately 1:√2 (or √2:1 when mapping back into the visible range).
However, this raises a crucial question: can any color attractor be predicted independently, without relying on the positions of other hues? Green emerges as a strong candidate.
Green as the Logarithmic Center:
Green's position on the octave hue wheel is defined by the logarithmic midpoint of the visible spectrum. Because green lies opposite the non-spectral color magenta, its location is effectively determined by the extreme edges of the visible spectrum. In other words, the perceived limits of the visible spectrum, where red and violet are no longer perceived, define the central point of green. This makes green's position independent of other hue positions within the spectrum.
Empirical Support for Green's Central Role:
This independent predictability of green is supported by empirical observations. The perceived transitions between green and its adjacent hues (yellowish-green and bluish-green) often correspond closely to the perceived limits of the individual's visible spectrum—the points where red and violet disappear.
Predicting Spectral Limits from Green:
This relationship can be demonstrated using approximate wavelength ranges for Newton's "principal" hues (which, as discussed earlier, are modern interpretations of his qualitative descriptions). Taking the approximate range for green as 500-570 nm, red band limit as 700nm and violet band limit as 400nm:
- Multiplying the short wavelength edge of green (500 nm) by √2 (approximately 1.414) yields approximately 707 nm, close to the perceived limit of red.
- Dividing the long wavelength edge of green (570 nm) by √2 (or multiplying by 1/√2, approximately 0.707) yields approximately 403 nm, very close to the perceived limit of violet.
This demonstrates that the perceived edges of the green band can effectively predict the perceived limits of the visible spectrum, reinforcing its central and independently defined position on the hue wheel.
(draft)
The Logical Foundation of the Color Continuum
The structure of distinct color appearances can be logically deduced from fundamental axioms defining color as a phenomenon, specifically cyclicity-continuity and achromatism-symmetry. This deduction is independent of the physical, biological, or perceptual specifics of light and vision, focusing on the abstract essence of color differentiation. Color is envisioned as a foundational, primal space, manifesting across various realms of reality—from abstract color spaces to physical, biological, neurological, subjective, and perceptual layers. These layers represent transformational points in color information processing, forming a continuum where boundaries inevitably blur. This abstract definition allows for color to exist even without strictly defined, discrete layers of manifestation.
The model focuses on the hue dimension of color, addressing not the number of components but the limits of their distinctiveness. It analyzes discrete hue categories within the continuum and argues against the notion of "new colors," which is often associated with tetrachromacy—a point that will be elaborated upon further...
Within this continuous phenomenon, we naturally categorize regions, assigning discrete names as change along the continuum becomes perceptually significant. This categorization is influenced by perceptual thresholds and evolutionary pressures, highlighting certain "attractor" hues as reference points, like red, green, and blue. We name other appearances as mixtures, such as yellowish-orange. This leads to the intuitive, yet ultimately misleading, concept of "unique" or "primary" colors. In reality, colors are not truly irreducible; they arise from transformations and mixtures within the continuous underlying space.
To clarify, we must refine the notion of "unique hues," distinguishing it from the idea of color attractors. The common definition of a unique hue as "without any tint of another" is subjective and scientifically weak.Instead, a relational definition of co-unique hues is proposed.
Definition of Co-Unique Hues: A pair of hues (A and A⁻¹) are co-unique if and only if neither hue contributes to the perceptual experience of the other, representing opposing tendencies relative to the neutral point of achromatism. This definition is independent of qualia, physical properties, spectral location, and naming conventions, focusing on their fundamental relational opposition.
Fundamental Axioms of Color
- Continuous and Cylic: Hues transition smoothly and cyclically into one another, forming a continuous spectrum without discontinuities or gaps. This cyclical nature is essential to the concept of hue.
- Achromatism (Complementarity): Colors exist in at least complementary pairs, co-unique hues (A and A⁻¹) that, when combined, produce a neutral achromatic sensation (grey). This process is termed achromatization and is fundamental to color perception.
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The following mathematical definitions, while implying algebraic characteristics and a group-like structure, are intended to be concise and focus on the essential properties of the hue dimension.
Concepts & Definitions
- \( H \): The set of hues (points in hue space)
\( H \cong S^1 \), a compact, cyclic continuum isomorphic to the unit circle.
- \( N \): The achromatic state
- \( \ast \): Abstract binary mixing operation
The 'mix' operation represents how hues perceptually combine or interact, not necessarily a direct physical or computational process. It encompasses additive, subtractive, and, fundamentally, stereoscopic color mixing, emphasizing the binary nature of hue interaction.
For hues \( h_1, h_2 \in H \):
\[
\ast(h_1, h_2) =
\begin{cases}
N, & \text{if } \Delta \theta = \pi \text{ (complementary)} \\
h_1 + \frac{\Delta \theta}{2} \mod 2\pi, & \text{otherwise}
\end{cases}
\]
where \( \Delta \theta = \min(|h_2 - h_1|, 2\pi - |h_2 - h_1|) \).
- \( d \): A perceptual distance metric \( d: H \times H \to \mathbb{R} \)
- Core Axioms
1. Cyclicity:
\( \forall h_1, h_2 \in H, \, h_1 \ast h_2 \in H \cup \{N\} \)
Mixing is closed and cyclic.
2. Continuity:
\( \forall h_1, h_2 \in H, \, \exists f_1, f_2: [0, 1] \to H \text{ such that } f_1(0) = f_2(0) = h_1, f_1(1) = f_2(1) = h_2 \)
Two disctinct continuous path connects any two hues.
3. Co-Uniqueness:
\( \forall A \in H, \, \exists A^{-1} \in H : d(A, A^{-1}) = \max \)
Every hue has a maximally distant complement.
4. Achromatism:
\( \forall A \in H, \, A \ast A^{-1} = N \iff \Delta \theta = \pi \)
Mixing complements yields achromaticity.
5. Symmetry:
\( \forall A, B \in H, \, d(A, B) = d(B, A) = \min(\Delta \theta, 2\pi - \Delta \theta) \)
Perceptual distance is symmetric and respects path equivalence.
Derivation of the Four-Hue Structure (Theorem):
\[
\exists A, B \in H : A \neq B \land A^{-1} \neq B^{-1} \land A^{-1} \neq B
\]
At least four fundamental hues are required to satisfy continuity and achromatism.
Proof Sketch:
1. Existence of a Co-Unique Pair: By Axiom 3, let \( A \) and \( A^{-1} \) exist.
2. Achromatic Discontinuity: \( A \ast A^{-1} = N \), creating a perceptual gap.
3. Bridging the Gap: Continuity (Axiom 2) necessitates intermediate hues \( B \) and \( B^{-1} \).
4. Symmetry & Closure: Placement of \( B \) and \( B^{-1} \) equidistant from \( A/A^{-1} \) restores cyclicity.
Result:
A closed, continuous cycle \( A \rightarrow B \rightarrow A^{-1} \rightarrow B^{-1} \rightarrow A \).
Emergent Properties
- Transitional Hues: Midpoints between fundamentals (e.g., \( AB \)) arise from mixtures.
- Perceptual Categorization: Evolutionary and perceptual thresholds create "attractor" hues (e.g., red, green).
- Illusion of Primary Colors: "Unique hues" are relational, not absolute, emerging from symmetry.
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Distinctiveness Decay Theorem
In a continuous, cyclic hue space \( H \cong S^1 \), recursive subdivision of the hue continuum yields hues with exponentially diminishing perceptual distinctiveness. Specifically, after \( n \) subdivisions, the distinctiveness \( D(n) \) of new hues relative to their neighbors is given by:
\[
D(n) = \frac{D_{\text{max}}}{2^n}
\]
where \( D_{\text{max}} = \pi \) is the maximal distinctiveness in \( H \).
Proof Sketch
1. Base Case: For \( n = 0 \), the co-unique pair \( A \) and \( A^{-1} \) has \( D(0) = D_{\text{max}} = \pi \).
2. Inductive Hypothesis: Assume that after \( k \) subdivisions, the distinctiveness of new hues is \( D(k) = \frac{D_{\text{max}}}{2^k} \).
3.Inductive Step: For \( k + 1 \) subdivisions, each new hue is inserted at the midpoint of an interval with distinctiveness \( D(k) \). By definition, the new distinctiveness is:
\[
D(k+1) = \frac{D(k)}{2} = \frac{D_{\text{max}}}{2^{k+1}}
\]
4. Conclusion: By induction, \( D(n) = \frac{D_{\text{max}}}{2^n} \) holds for all \( n \geq 0 \).
Base Case
- Initial Pair: Let \( A \) and \( A^{-1} \) be a co-unique pair of hues, with \( d(A, A^{-1}) = D_{\text{max}} = \pi \).
- Distinctiveness: \( A \) and \( A^{-1} \) are maximally distinct (100% different).
Inductive Step
- First Subdivision: Insert hue \( B \) at the midpoint between \( A \) and \( A^{-1} \).
- By definition, \( d(A, B) = d(B, A^{-1}) = \frac{D_{\text{max}}}{2} = \frac{\pi}{2} \).
- Distinctiveness: \( B \) is 50% distinct from \( A \) and \( A^{-1} \).
- Second Subdivision: Insert hues \( C \) and \( D \) at the midpoints between \( A \) and \( B \), and \( B \) and \( A^{-1} \), respectively.
- \( d(A, C) = d(C, B) = d(B, D) = d(D, A^{-1}) = \frac{D_{\text{max}}}{4} = \frac{\pi}{4} \).
- Distinctiveness: \( C \) and \( D \) are 25% distinct from their neighbors.
- General Case: After \( n \) subdivisions, the distinctiveness of new hues is:
\[
D(n) = \frac{D_{\text{max}}}{2^n} = \frac{\pi}{2^n}
\]
- Distinctiveness Decay: Each subdivision halves the distinctiveness of new hues relative to their neighbors.
Implications
- Limits of Distinctiveness: As \( n \to \infty \), \( D(n) \to 0 \). New hues become indistinguishable from their neighbors.
- No New Categories: Subdivision refines resolution but does not create fundamentally new distinct hues.
- Tetrachromacy: Even with additional photoreceptors, the theorem predicts diminishing returns in perceptual distinctiveness.
Interpretation
Necessity of a Four-Hue Foundation:
Four-Component Basis: Due to the inherent symmetry and continuity of hue space \(H\), it can be logically decomposed into four quadrants. This decomposition is naturally defined by the four fundamental hues \( {A, A^{-1}, B, B^{-1}} \), strategically positioned to ensure continuity and avoid any discontinuity, especially at the neutral point \(N\)
The principle of achromatism necessitates at least one co-unique pair (A and A⁻¹). A rudimentary two-hue spectrum (A – N – A⁻¹) emerges, but this structure violates continuity due to the achromatic point creating a discontinuity between A and A⁻¹. To restore continuity, at least one additional co-unique pair (B and B⁻¹) is required, positioned to bridge the achromatic gap and, crucially, create a balanced and symmetrical structure within the hue circle. This logically necessitates a minimum four-hue structure: A – B – A⁻¹ – B⁻¹ – A, forming a closed, continuous cycle.
For continuity, B and B⁻¹ cannot be placed arbitrarily. They must "bridge" the gap between A and A⁻¹. The most symmetrical and continuous way to do this is to place B and B⁻¹ such that they are also maximally distant from each other and equidistant from A and A⁻¹. This creates a balanced and symmetrical structure.
Emergence of Transitional Hues:
The transitions between these four fundamental hues are continuous, generating further hue categories. Points of equal contribution from adjacent fundamental hues (midpoints in the continuum) represent distinct transitional hues (e.g., between A and B). These transitional hues are doubly justified: they arise from the mixture of two adjacent fundamental hues (e.g., AB) and as complements of the opposite mixture (e.g., A⁻¹B⁻¹), reinforcing the symmetrical and relational nature of color space.
Logical Limit to "Most Distinct Hues":
Starting with any co-unique pair and applying continuity and achromatism inevitably leads to this four-fundamental-hue structure, subsequently populated by transitional hues to fill the continuum. Further subdivision within this continuum yields finer variations of these categories, not fundamentally "new" distinct hues. Imagine a cycle where co-unique pairs represent maximal perceptual difference (100%). The maximum difference from one hue to another within this cycle is limited. The second co-unique pair (B/B⁻¹) can only be maximally 50% distinct from the first (A/A⁻¹). Recursive subdivision creates hues of diminishing distinctiveness (25%, 12.5%, etc.), highlighting that the continuum is filled by variations, not an endless proliferation of fundamentally new categories.
Logical Implications and the Four-Component Model
This framework logically predicts the existence of four fundamentally different unique hues as a necessary condition for a cyclic and continuous spectrum of appearances. This prediction is independent of specific color spaces or physical implementations and provides a fundamental, axiomatic explanation for the discrete nature of our core color categories. Starting with any arbitrary co-unique pair will inevitably generate the same fundamental four-hue structure to satisfy achromatism and continuity. These relationally defined hues form the basis of our conventional color categories, one iteration or subdivision from the Distinctiveness Decay Theorem renders, roughly, as color attractor names: red, orange, yellow, green, cyan, blue, violet and magenta.
This model aims to explain why we perceive the hues we do, not just how we represent them. The model logically derives the structure of the hue cycle from first principles, therefore, the notion of independently arising "new hues" becomes unnecessary within this framework. The perceived color space is thus not an arbitrary construct, but a logically constrained interdependent system.
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Practical Implications and Limitations
It is important to consider that the presented model, particularly the S¹ hue wheel representation and the logical framework derived from it, describes an idealized case. This ideal assumes a perfectly uniform perceptual color space and a hypothetical observer with maximally fine just-noticeable differences (JNDs). Even under such idealized conditions, when presented with a cyclical continuum of uniformly distributed, yet initially unknown, color components, the model predicts a maximum perceptual distance between co-unique hues and the fundamental logical structure remains consistent.
Consider a scenario where the physical spectrum is drastically non-uniform, for example, heavily skewed towards yellow, with only a minimal representation of other colors and transitions (e.g., 99% yellow and 1% encompassing all other spectral appearances). While a linear mapping of such a spectrum to the hue wheel would distort the simple angular and symmetry relationships, the core principle of complementarity remains valid perceptually. Every hue point, regardless of its physical prevalence in the spectrum, will still possess a complementary, or 'neutralizer,' point that, when appropriately combined, leads to an achromatic sensation. This fundamental symmetry, the existence of complementary pairs, does not necessitate a physically uniform spectrum; the maximal perceptual distance between co-unique hues is a perceptual, not a strictly physical, attribute.
Furthermore, when considering real-world color mixing, such as with paints, the idealized hue dimension must be understood in the context of saturation and lightness. Imagine mixing spectrally narrow, 'pure' colors like a highly saturated red and a highly saturated green paint. While the idealized S¹ hue wheel might predict an orange or yellowish hue as the midpoint, in practice, paint mixing often yields a darker, desaturated orange-brown. This deviation arises because real-world paints introduce changes in saturation and lightness, dimensions not explicitly represented in the simplified S¹ hue wheel. However, the hue component of the mixture is still directionally consistent with the model's prediction: the mixture is indeed perceptually intermediate between red and green in hue, even if modified by other color dimensions.
Even in cases where visual intuition might be misleading, such as mixing a slightly orange-leaning red and a slightly green-leaning cyan paint, the S¹ model remains analytically valuable. While a cursory visual assessment of the paint mixture might suggest a near-achromatic gray, spectral analysis could reveal a subtle, yet measurable, spectral distribution peak corresponding to the yellowish hue predicted by the S¹ model for the idealized mix. This highlights that while real-world color mixing is complex and multi-dimensional, the hue dimension, as structured by the S¹ model and the logical framework, provides a consistent and accurate underlying principle for understanding color relationships, even when those relationships are not immediately apparent in everyday color experiences.
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| Co-Uniques(Complementaries) as generating sets |
Deconstructing the "New Color" Misconception in Color Vision Research
The notion of "new colors" or "unique hues" frequently emerges in discussions about tetrachromacy and animal color vision, often lacking precise definitions and rigorous justification. This ambiguity fosters misinterpretations, particularly regarding the perceptual experiences of organisms with more than three cone types, leading to overstated claims. This section argues that the concept of discrete, standalone "new hues" is fundamentally incompatible with a logically consistent understanding of color perception, which is grounded in continuity, complementarity (achromatism), and relational structure.
The Ambiguity of "Unique Hues" and "New Colors"
A primary issue is the absence of a clear, universally accepted scientific definition for "unique hue." The traditional subjective definition – a color "without any tint of another" – lacks scientific rigor and contributes to overinterpretations, especially in studies of animal and human tetrachromatic vision. A more robust, relational approach defines "co-unique hues" based on mutual cancellation: two hues are co-unique if neither contributes to the perceptual experience of the other, representing opposing tendencies towards achromatic neutrality. The very idea of a "new color" or "new hue" is problematic as it often implies entirely discrete, qualitatively different sensations outside of human experience. Without a precise definition of "hue" anchored to physical, neural, and abstract color models, claims of "new colors" are difficult to validate and can be misleading.
The Flawed Analogy of Additional Cones and the Dichromat-Trichromat Transition
A common, yet flawed, analogy attempts to explain tetrachromacy by comparing it to the transition from dichromacy to trichromacy, suggesting that each additional cone simply adds a new dimension to color perception. This analogy often proceeds without a clear definition of "color," assuming a direct, quantifiable link between cone activation and perceptual experience. While it's true that some animals transition from dichromacy to trichromacy with a third cone mutation, this analogy overlooks a crucial distinction: dichromatic vision inherently includes a neutral gray point formed by mixing their two hues, creating a discontinuity in their perceptual space. Trichromatic vision, conversely, is continuous, lacking a spectral gray (except through complementary color mixing). This fundamental difference is key: dichromats require this gray point to differentiate their two hues because two hues alone cannot reconstitute white light. Trichromats, however, can reconstitute white light from any complementary pair, making their spectrum continuous. A trichromat can conceptually approximate dichromatic vision by replacing parts of the spectrum with gray, but a dichromat cannot conceive of the trichromat's continuous spectral experience. The addition of a cone, therefore, fundamentally alters the structure of color space by making it continuous, not just expanding its dimensionality in a simple additive manner.
Misinterpretations of Tetrachromacy and the Overemphasis on Photoreceptors
Claims of tetrachromats perceiving "millions of unimaginable colors" are often based on flawed reasoning and an overemphasis on the number of photoreceptors. Studies claiming tetrachromacy often fail to investigate "complementary behavior"—a crucial aspect of color perception. If a genuinely "new hue" exists, there must be a corresponding complementary hue that, when mixed, produces a neutral sensation. This has not been demonstrated in tetrachromacy research. Furthermore, these claims often misinterpret Just Noticeable Difference (JND) data. Enhanced JNDs, indicating finer discrimination between similar wavelengths, are frequently mistaken for the perception of new hues. However, increased JNDs simply suggest finer discrimination within existing color categories, not the emergence of new qualia. Phrases like "satisfies the criteria for behavioral tetrachromacy" are often used vaguely, contributing to confusion. The ability of a tetrachromat to distinguish between similar oranges, for example, can be explained by variations within the normal trichromatic range of cone sensitivities and doesn't necessitate the perception of a novel color sensation. Many such studies also lack crucial qualitative data – interviews and descriptions of perceptual experiences – and comparative JND tests against trichromats.
Similar misconceptions extend to claims about mantis shrimp vision. The assertion that mantis shrimp, with their 15 photoreceptors, experience colors "beyond human imagination" is equally flawed. Having more photoreceptors does not inherently equate to perceiving more distinct hues or finer spectral distinctions. Mantis shrimp vision, like human trichromatic vision, operates on the principle of relative receptor responses, creating a continuous mapping between wavelength and receptor stimulation. Their "colors" are determined by the relative responses across their photoreceptor array, not by discrete, otherworldly hues. The analogy of spiders having more legs but not moving in more spatial dimensions aptly illustrates the flawed logic of equating photoreceptor number directly with the dimensionality of perceptual experience.
While the mantis shrimp's 15 photoreceptors are often cited as an example of "super color vision," an alternative hypothesis suggests that their proliferation might be better explained by the principle of evolutionary efficiency. Rather than evolving a single, broadly tuned photoreceptor, it might have been evolutionarily simpler to duplicate existing photoreceptor types and subtly shift their spectral sensitivities. This "instantiation" of similar receptors at different wavelengths would effectively cover the visible spectrum without requiring drastic alterations to the underlying biochemistry of the photoreceptors themselves. This hypothesis emphasizes the fundamental need for broad spectral coverage in any visual system and offers a plausible explanation for the mantis shrimp's unique visual apparatus, even if it doesn't necessarily imply an extraordinary capacity for color discrimination.
Logical Constraints and the Structure of Color Space
The prevailing simplistic view of color dimensionality, directly translating cone types to independent color channels, overlooks the crucial role of post-receptoral processing, particularly stereoscopic color mixing. Color perception's dimensionality is fundamentally dual, not simply determined by the number of cone types. The third cone acts as a reference point for the other two; the intensity response from two cones is meaningless without this reference. Retinal neurons encode proportional responses, not pre-formed color qualia. It's illogical for fully formed color percepts to be encoded at the retinal level when subsequent processes like stereoscopic color mixing operate on distinct principles, with binary input.
Critique of Current Color Vision Models and Cone-Level Qualia
This analysis challenges current color vision models that overemphasize photoreceptors and the concept of unique hues. The assumption that each cone type has a pre-assigned, fundamental color at the cellular level is illogical, implying fully formed color percepts at the retinal level, prior to conscious awareness. This inappropriately relocates the objectivity of color from wavelength or the mind to the cones themselves. The widespread dissemination of information about tetrachromacy, often with exaggerated claims of "millions" of additional colors, highlights a significant misunderstanding. Studies presented as evidence for novel color experiences in tetrachromats, often relying solely on metameric failure and lacking qualitative data, are insufficient to establish the perception of new color sensations. The conventional understanding of color dimensionality, based on a simplistic photoreceptor-centric view, needs revision.
Conclusion
In conclusion, claims of "new colors" in tetrachromacy and other forms of non-standard color vision often overemphasize the role of photoreceptors while neglecting the crucial roles of post-receptoral processing and the inherent logical constraints of continuity, complementarity, and relational structure within color space. The color space, as we understand it, leaves no room for discrete, standalone "new hues" without fundamentally disrupting its organization, symmetry, and balance. Color is an interdependent system. The dimensionality of color perception is not simply dictated by the number of cone types but by the complex interplay of receptor signals and their integration by the brain, particularly within the context of stereoscopic vision. Therefore, a critical re-evaluation of the "new color" concept and a more nuanced understanding of color perception beyond a simplistic photoreceptor-centric view are necessary.
Applying the Axioms to Visual Experience: A Logically Constrained Color SpaceThis abstract model, grounded in the principles of co-unique pairs, achromatism, and continuity, provides a powerful framework for understanding the organization of human color experience. It demonstrates how the existence and approximate arrangement of primary unique hues can be logically deduced, starting from fundamental principles.
Green occupies a unique and independently predictable position. Located at the approximate logarithmic center of the visible spectrum (defined by its approximate boundaries of 375 nm and 750 nm), green serves as a foundational anchor point. This central placement, determined by the physical limits of human vision, makes green a natural starting point for the organization of color space.
The axiom of continuity dictates that hues must transition smoothly into one another. Therefore, hues must exist on either side of green. At the extreme edges of the visible spectrum, we perceive red and violet. These two hues, to maintain continuity, must blend, creating the non-spectral hue magenta. By the axiom of complementarity, magenta is the complement of green.
Applying the principle of complementarity to red and violet, we deduce the existence of their respective complements: cyan and yellow. These hues naturally emerge adjacent to green on the hue circle, positioned to be complementary to red and violet.
It's important to note that, unlike the purely abstract deduction, we don't begin with a pre-defined co-unique pair. Instead, we start with the empirically observed extremes of the visible spectrum (red and violet), whose mixture defines magenta, the complement of green. From these three anchor points (green, red/magenta, and violet/magenta), the requirement of complementarity alone yields cyan and yellow as the necessary transitions between green and red/violet. This process generates six distinct hues: red, yellow, green, cyan, violet, and magenta.
Orange and blue emerge as a direct consequence of both complementarity and the continuity axiom. Orange smoothly fills the perceptual transition between red and yellow, while blue fills the gap between cyan and violet. These transitions, while maintaining the complementary relationships and following the same distinctiveness decay observed elsewhere in the hue circle, do not lead to achromatization because each hue arises logically in its correct order with its opponent.
This demonstrates how, starting with the independently defined boundaries of the visible spectrum, the axioms of complementarity and continuity necessitate the existence and approximate positions of all primary unique hues. No further unique hues can be added without violating these fundamental principles.
This abstract color model, based on the logical necessity of co-unique pairs and a continuous spectrum, demonstrates a remarkable alignment with physical reality, revealing a consistent mathematical structure rooted in logarithmic perception and octave-like cycles. This strongly suggests that color, at a fundamental level, possesses an objective basis, even though its ultimate experience is subjective.
Connecting to Cone Function and Dichromacy:
The model predicts four relative primaries (two co-unique pairs) from which all other hues can be derived. However, any single point on the color continuum is composed of a mixture of at most three, not all four, of these relative primaries. This follows directly from how these primaries were logically derived: each transition involves only two adjacent hues.
This observation leads to a hypothesis about the function of the three cone photoreceptor types in the human retina. Rather than directly encoding opponent color channels (red-green, blue-yellow), each cone type primarily records the intensity of light it absorbs across a broad range of wavelengths. Opponent processing, crucial for color perception, occurs at a later stage, where neural circuits calculate the proportions of stimulation between the different cone types. This proportional encoding allows the visual system to "retrieve" or infer the approximate wavelength or spectral characteristic of the incoming light, which is then processed to create a specific visual experience.
In dichromats, the absence of one cone type limits the system to just two dimensions of color information. They experience only two unique hues and their mixtures, which blend into gray. In terms of our model, they are essentially trapped in a two-hue cycle, unable to access the transitions that give rise to the full color continuum. This is the paradox of breaking the continuity axiom: they have no way to distinguish the different hues and their transitions.
The role of the third cone is thus not merely to add a third color dimension but to provide the necessary reference point for calculating proportions. Just as the proportion between two mountain heights changes drastically if the sea level changes, the relative stimulation of two cone types is meaningless without a third reference point. Each cone pair needs a third cone as a "sea level" to determine the relative proportions and, therefore, the perceived color. This proportional encoding mechanism elegantly explains how the brain extracts rich color information from the limited and overlapping spectral sensitivities of the cones.
The Octave Principle: A Neural and Physical Basis for Color Perception
Our perception of color may be optimized by a principle analogous to the musical octave, wherein the visible spectrum spans a specific, brain-preferred range—from a base frequency \( \lambda \) to its double \( 2\lambda \). This range appears to be a “sweet spot” for information processing, providing both efficiency and clarity in how color is decoded and represented by our visual system.
The Spectral Octave and Information Processing
The hypothesis begins with the observation that the visible spectrum covers roughly one octave in electromagnetic frequency. This is significant because, much like an octave in music where the first and second harmonics resonate in a clearly defined relationship, the visible range might be naturally selected for its role in efficient information processing. The analogy extends to phenomena such as the double-slit experiment, where only certain wavelengths—those falling within a particular harmonic range—interfere constructively to produce clear, visible patterns(see next section). In a similar vein, wavelengths within the \( (\lambda, 2\lambda) \) range can be processed by the visual system without the ambiguity or interference that might arise if a broader spectrum were used.
If light were to contain wavelengths far beyond this octave (for example, spanning from 200 nm to 3000 nm), the resulting temporal integration could become exceedingly complex. Different wavelengths might interfere with each other, creating overlapping “orders” or patterns that confound the neural decoding of color. Thus, by confining the effective range to a single octave, the visual system minimizes potential interference, ensuring that color signals are extracted in a clear and unambiguous manner.
Neural Representation: The Hue Wheel
Neurophysiological studies support this octave-based view through the organization of color-responsive cells in the visual cortex—specifically in areas such as V1, V2, and V4. These cells are arranged in a circular fashion, forming what is often referred to as a neural hue wheel. Unlike a linear spectrum, this circular mapping accommodates the cyclical nature of color perception, providing an elegant explanation for phenomena like the existence of magenta—a hue that does not correspond to any single wavelength.
Magenta arises not as a direct sensory input but as an emergent property of the brain’s interpolation between the spectral extremes of red and violet. In other words, the brain completes the color circle by generating magenta from the overlapping neural representations of these ends of the spectrum. This internal construction aligns with the octave principle by highlighting how the visual system organizes information: by using harmonic relationships to create a stable, redundant, and efficient representation of color.
The Octave Principle in Neural Encoding
The concept of an octave is not limited to the physical properties of light; it also reflects the brain’s strategies for encoding and processing sensory information. By confining color representation to a limited octave, the brain can prevent unwanted frequency synchronization among neural populations. For instance, if two colors were encoded by firing rates that differed by a factor of two (spanning an octave), the risk is that these signals might synchronize excessively, leading to redundancy or even mutual cancellation. Keeping the representation within a controlled octave range avoids such pitfalls, ensuring that each hue maintains a distinct and non-interfering identity.
This confined range also establishes a crucial “stability point”—the achromatic state. When colors are logarithmically encoded within a range (from \( f \) to \( 2f \)), the brain can reliably compute gray (achromatism) by balancing complementary colors. Since complementary hues within this octave naturally oppose each other, their balanced neural representations effectively cancel out, resulting in a stable perception of gray. This mechanism is particularly important for stereoscopic vision, where the brain must integrate slightly different inputs from each eye into a cohesive, stable percept.
Mapping the Spectral Octave to Neural Frequencies
One intriguing aspect of the octave principle is its potential linkage to the brain’s frequency-based processing. Neural circuits communicate via oscillatory signals, often operating in distinct frequency bands to minimize interference. This frequency-based organization suggests that the visual system might map the electromagnetic frequency of light directly onto a corresponding range of neural frequencies. In doing so, the hues we perceive would be represented by separate, non-overlapping channels within the brain’s processing architecture.
Such a mapping offers several advantages. First, it directly connects the physical properties of light with the internal, frequency-specific operations of the brain. Second, it aligns with well-established principles of neural processing—such as resonance and frequency-specific communication—thereby offering a plausible mechanism for the emergence of the octave-like structure in hue perception. Moreover, if the final color percept is determined by the ratio of neural firing frequencies from each eye, then fixed ratios (such as the square root of 2) might be linked to the perception of achromatic colors. This idea could help explain why diverse color combinations can result in the same gray, highlighting the brain’s reliance on stable frequency ratios for accurate color decoding.
Implications and Open Questions
While the octave principle provides a compelling framework for understanding color perception, several challenges and open questions remain. For instance:
- Mapping Specific Hues to Neural Frequencies: Is the relationship between wavelengths and neural frequencies strictly linear, or does it involve a more complex mapping?
- Mechanisms of Frequency Comparison: What precise neural mechanisms detect and compare these frequency ratios, particularly across inputs from both eyes?
- Integration with Other Explanations: How does this frequency-based hypothesis coexist with other factors—such as evolutionary adaptations or developmental influences—that might shape our perception of color?
These questions underscore that, although the octave principle is a promising and conceptually elegant model, further empirical research is needed. Neurophysiological experiments and computational modeling will be crucial to test whether the brain indeed uses an octave-based strategy to optimize color representation.
In summary, the octave principle offers a unified explanation for several aspects of color perception—from the spectral limits of the visible range to the neural organization of color information in the brain. By constraining the effective range of light to an octave, the visual system minimizes interference, prevents unwanted frequency synchronization, and achieves a stable point for achromatic cancellation. Moreover, the circular arrangement of color-responsive cells and the emergent properties of hues like magenta lend further support to this model.
While the mapping between electromagnetic frequencies and neural processing remains an open area of inquiry, the hypothesis that our perception of color is optimized by an octave-like structure is both compelling and consistent with known principles of neural communication. As further research unfolds, it may reveal that the octave is not merely a characteristic of musical harmony but a fundamental organizing principle of our sensory and perceptual systems.
Probabilistic Color and the Spectral Octave
The limitations of the human visual spectrum, confined to approximately an octave (a 1:2 frequency ratio), are not arbitrary but are rooted in the fundamental physics of light and wave interference, as demonstrated by the double-slit experiment.
In the double-slit experiment, even single photons passing through two narrow slits behave as waves, interfering with themselves to create a pattern of bright and dark fringes on a screen. When white light (a mixture of wavelengths) is used, each wavelength creates its own interference pattern. Because the position of the bright fringes depends on the wavelength – a phenomenon called dispersion – the white light is separated into its constituent colors.
The position of these bright fringes (maxima) is described by the diffraction grating equation (also applicable to the double-slit experiment):
d sin(θ) = nλ
where:
`d` is the distance between the slits.
`θ` is the angle of the bright fringe.
`n` is the order of the maximum (0, 1, 2...).
`λ` is the wavelength.
This equation shows that shorter wavelengths (blue) have fringes closer to the central maximum (smaller θ), while longer wavelengths (red) have fringes farther out. The central maximum (n=0) is white because all wavelengths interfere constructively there.
The crucial point is that the fringes for different wavelengths and different orders (n) can overlap. Let 'a' be the shortest visible wavelength. Its first-order maximum (n=1) is at angle θ₁:
d sin(θ₁) = a
A wavelength 2a (double the shortest) has its first-order maximum (n=1) at θ₂:
d sin(θ₂) = 2a
But the second-order maximum (n=2) for the shortest wavelength 'a' is also at θ₂:
d sin(θ₃) = 2a => θ₂ = θ₃
This means the first-order maximum of 2a overlaps with the second-order maximum of 'a'. Wavelengths longer than 2a would overlap with even higher-order maxima of shorter wavelengths.
This physical overlap leads to perceptual mixing. Our visual system, with its broadly sensitive and overlapping cone types (S, M, L), cannot disentangle these overlapping wavelengths. The ratios of cone responses, which determine perceived hue, become less distinct, resulting in desaturated, whitish, or ambiguous color sensations.
Therefore, the "spectral octave" is a physical constraint, ensuring that the interference patterns of different wavelengths are sufficiently separated to produce distinct cone response ratios and, consequently, distinct hues. It's also biologically efficient, avoiding the need to process wavelengths that wouldn't contribute to useful color information.
From Deterministic Waves to Probabilistic Waves:
The example implicitly uses the classical wave description of light to derive the diffraction grating equation. While it mentions single photons, it should be highlighted that this doesn't fully embrace the probabilistic interpretation of the wave function.
In quantum mechanics, the wave associated with a photon (or any particle) is not a wave of physical displacement like a water wave. It's a probability wave. The square of the wave's amplitude at a given point represents the probability density of finding the photon at that point.
Shifting to this probabilistic view of the light wave makes the analogy more fundamental. It's not just about waves interfering; it's about probability distributions interfering.
Cone Response as Probabilistic Absorption:
For simplicity the study interpreted cone response as a relatively straightforward function of wavelength. While it mentions "overlapping sensitivities," doesn't explicitly frame it as a probability.
The absorption of a photon by a photopigment molecule in a cone cell is a quantum event. It's not guaranteed. Each cone type has a probability of absorbing a photon of a given wavelength. This probability is described by the cone's spectral sensitivity curve. This curve is not just a measure of "how much" light is absorbed; it's a measure of the probability of absorption.
Framing cone response as probabilistic absorption aligns it perfectly with the probabilistic nature of light detection. Both the arrival of a photon at a location and its absorption by a cone are governed by probabilities.
Time Integration of Probabilities:
The study already discussed "temporal integration" in the context of afterimages, but it must also be explicitly connected to the probabilistic nature of light and cone response.
Our perception of a continuous, stable color arises from the time integration of countless probabilistic events. We don't see individual photon arrivals or individual cone absorptions. Our visual system integrates these events over time, averaging out the probabilistic fluctuations.
Each cone is essentially acting as a probabilistic "photon counter." The rate of photon absorption (which is proportional to the probability of absorption multiplied by the intensity of the light) determines the cone's output signal. The visual system then compares the rates of absorption from the different cone types.
A Unified Probabilistic Framework:
Light arrives at the retina as a spatial distribution of photon arrival probabilities.
Each cone type has a wavelength-dependent probability of absorbing a photon.
The visual system integrates these probabilistic events over time, effectively "sampling" from the probability distributions.
The perceived color is a result of comparing the time-averaged rates of photon absorption by the different cone types.
This framework places both the physics of light and the initial stage of visual processing on the same fundamental footing: probability. It's not just an analogy between waves; it's an analogy between probabilistic processes.
Conclusion: Parallels and Open Questions in the Chroma of Color and MusicThis study, focusing on the relationship between visual and musical chroma, has revealed numerous parallels while also raising profound questions about the fundamental nature of both perceptual domains.
Continuity, Cyclicity, and Discretization:
Both visual and musical chroma are continuous and cyclic. However, the discretization of these cycles into named categories (colors, notes) is primarily a product of cultural and individual experience. This discretization is limited by the inherent ambiguity that arises at the boundaries between categories.
The Octave as a Framework:
The octave, as a frequency range that doubles, provides a suitable, though not strict, framework for containing the discretized components of both pitch and light. While physically distinct (sound waves vs. electromagnetic radiation), both can be described in wave terms. Ultimately, however, the brain processes both as neural activity – firing neurons, not air rarefactions or photon absorption.
Complementarity and Achromatism:
Within both cycles, each point has an inverse, an "achromatic" counterpart. Just as gray can be created by numerous complementary color pairs, in music, the tritone acts as a kind of "achromatic" interval. Stacking tritones can effectively mask the fundamental pitch, making it difficult to discern individual notes within the complex sound. (It's important to note that the term "complementary notes" in music has a different meaning than in color. Musical "complementary" often refers to the additive inverse of pitch classes, a definition that varies across musical traditions. The tritone, however, exhibits a more direct analogy to color complementarity, as it tends to obscure or "cancel" the fundamental.)
The Subjectivity of Harmony:
Throughout this study, the concept of "color harmony" has been deliberately avoided. While inspiring and thought-provoking, much of what is written about color harmony is ultimately subjective. In music, while harmony also involves subjective elements, there are more objective, measurable aspects. This difference made it difficult to offer meaningful insights into color harmony beyond subjective experience.
Musical Scales: Coincidence or Inevitability?
The musical scales derived from hue-wavelength matchings are musically valid and practically useful. However, it's likely that their correspondence to recognizable musical ratios is more a coincidence arising from the distribution of hues within the spectral octave than a strict metaphysical principle.
Musical scales, including the pentatonic, do not have a purely scientific origin. They are likely a complex product of cultural development and learned preferences for specific timbral combinations that resonate with the brain's processing mechanisms. There are no "unique notes" in any fundamental sense. Instead, there's a differentiable subset of the pitch cycle that allows for increasingly complex musical creation and performance. This subset can be altered substantially without loss of tonal meaning, just as colors don't need to be spectrally pure to be identifiable.
The Missing Fundamental and Perceptual Preferences:
The missing fundamental effect highlights the brain's tendency to seek out and respond to harmonic timbral relationships. This preference may be related to the aesthetic pleasure we derive from music, which could share a common neural mechanism with color perception. When we play or listen to music, our brains might be engaging similar processes to those involved in combining and experiencing colors.
Universality in Music:
The strongest evidence for some degree of universality in music lies in musical form. Studies have shown that people from isolated cultures can often recognize the function of music from other cultures (lullabies, dance music, ritual music, love songs).
The diatonic scale, while seemingly ubiquitous, is another area of debate. While some theorists downplay its significance, attributing it to cultural borrowing, the diatonic scale appears in numerous cultures independently. It's a naturally occurring, manageable, and uniformly distributed subset of the pitch cycle. The idea that it was "stolen" from one culture to another likely oversimplifies a complex, multi-faceted development. Many musical concepts, like instruments themselves, likely have multiple origins, not a single linear path of invention.
Pitch Attractors and Evolutionary Influences:
While seemingly universal, the prevalence of the diatonic scale raises the question: why? Are there "pitch attractors" analogous to color attractors? While pitch attractors are less clear than color attractors, they might exist. It's possible that they arise from speech patterns or, perhaps even more likely, from the sounds of animals (birdsong, etc.), which could have unconsciously influenced our musical development over evolutionary time.
The Interconnectedness of Color and Music Perception:
Neither discrete colors nor discrete pitches are fundamental in any deep perceptual sense. This shared characteristic is what links them most profoundly.
The Octave and Color Space:
The logarithmic color representation within the octave, while requiring further research with larger sample sizes, offers a compelling explanation for deviations in RGB complementaries, afterimage effects, color constancy, and stereoscopic vision. However, the question remains: why the octave? Why the √2 ratio for true complementaries?
The Dimensionality of Color Space:
The dimensionality of color space is not adequately described by the number of photoreceptor types. Equating photoreceptors with dimensions ignores the complex neural transformations, including dichoptic (two-eye) color mixing. This simplified view undermines claims about tetrachromacy and "four-dimensional" color space, especially given the difficulties in even defining "unique hue."
The Logical Color Model and Pitch Space:
The logical color model, based on simple, accepted axioms (cyclicity-continuity, achromatism-symmetry), can be applied with similar success to pitch space. Both domains exhibit a limited number of fundamentally different sensations, with further subdivisions contributing to ambiguity and blurred boundaries.
Conclusion:
Just as there are no truly "new colors," there are no truly "new notes." The cycles are complete. That's all there is.
Practical Applications:
One practical application of this study is the development of more accurate color coding for musical notation, avoiding the distortions inherent in RGB-based systems. The canonical subtractive model presented here offers a more perceptually consistent alternative.
This study, while providing answers to some questions, has opened up many more avenues for future exploration. The journey into the fascinating intersection of color and music perception continues.
(draft)
Methodological Framework for Investigating Color Perception (Integrated with Web Application)Research into color perception is fraught with conceptual ambiguities, particularly concerning terms like "unique hues," "color bands," and "richer color experience." These terms often lack precise definitions, leading to potential misinterpretations and methodological challenges. A rigorous approach is crucial to disentangle genuine perceptual phenomena from terminological confusion and methodological artifacts. This framework outlines a structured methodology, integrating insights from both discrete and continuous spectral presentations, to investigate color perception and address claims of expanded chromatic experiences, such as those associated with tetrachromacy. A valuable tool for implementing these methodologies is a web application designed for exploring perceptual color thresholds within an RGB color space.
Clarifying Key Terminology:
Before outlining the experimental procedures, it is essential to define key terms that are central to color perception research and often used ambiguously:
- Color Attractors (Unique Hues): These are perceptually "primary" color sensations associated with discrete names (e.g., red, green, blue, yellow). They represent focal points in color space around which other hues are perceived to cluster.
- Color Bands: Discrete, distinguishable regions within a continuous spectrum, analogous to perceptual "steps". The term is often used to describe segmented portions of a spectrum perceived as different hues. It's crucial to note the ambiguity of this term, as it can be confused with "unique hues" or simply describe distinguishable spectral appearances.
- Just-Noticeable Difference (JND): The smallest detectable change in a sensory stimulus (in this context, hue). JNDs are a quantitative measure of perceptual discrimination.
- Richer Color Experience: This is a polysemous term encompassing several potential aspects of vision, including:
- Increased Precision/Resolution: Enhanced ability to distinguish fine details and resolve spatial information.
- Increased Number of Hues: Perception of a wider range of distinct hues, representing qualitatively new color sensations.
- Increased Saturation/Chroma: Perception of more intense and pure colors.
It is critical to differentiate "finer discrimination" (quantified by JNDs) from a genuinely "richer color experience," especially when investigating claims of novel color perception.
It is conceivable that, in some studies, the conclusions regarding a "richer" color experience could be rephrased in terms of a "poorer" experience without altering the consistency of the reported observations, highlighting the ambiguity of the terminology, and lack of robust data.
Continuous vs. Discrete Spectra:
The method of spectral presentation significantly influences perceptual discrimination:
- Discrete Presentation: When the spectrum is presented as distinct, separate bands, observers can typically distinguish over 60 regions due to the clear boundaries between them (corresponding to an average JND of 1–2% of the visible range).
- Continuous Presentation: In a smooth, continuous gradient, the number of distinguishable regions decreases significantly, averaging around 11, as the transitions between hues become less distinct.
Experimental Stages for Investigating Color Perception:A progressive, multi-stage approach is recommended to rigorously assess color perception, starting with foundational validity checks and moving towards complex investigations of hue discrimination and categorization. A web application, as described below, can facilitate many of these stages.
Web Application for Perceptual Color Threshold Exploration:This web application provides a robust and versatile platform for exploring perceptual color thresholds, particularly Just Noticeable Differences (JNDs), within a metameric RGB color space. While focused on RGB metamers, the fine-grained control over various visual parameters allows for comparison with similar experimental setups. Its design directly addresses the fundamental differences between continuous and discrete spectral representations, offering insights into the perception of color bands and color attractors, aligning perfectly with the methodologies outlined in this framework.
Key Features and Functionality of the Web Application:- CIE XYZ Color Model Selection: Users can select from various CIE XYZ color models (1931, 1964, 2006) and convert wavelengths to RGB, ensuring accurate colorimetric calculations.
- Adjustable White Point and Gamma: Flexibility in setting white point and gamma allows for approximating real-world lighting conditions or using purely perceptual gradients.
- Four-Row Interface: The application's interface is structured into four rows, designed to systematically investigate different aspects of color perception:
- Continuous Spectrum: Displays a continuous (pixel-sized) spectral gradient defined by a user-selected wavelength range.
- Discretized Spectrum: Presents a discretized version of the same spectral range, with the number of discrete samples controlled by a "step" parameter.
- Adjusted Discretized Spectrum: Renders the same discrete samples with user-adjustable noise and blur parameters to simulate individual visual noise and acuity thresholds.
- Separated Discrete Samples: Displays the same adjusted discrete samples but with spacing between color patches to minimize contrast effects and isolate hue perception.
Stage 1: Validation of Screen-Based Spectral Representation:
Before utilizing screen-based displays, it is crucial to assess their validity. While the web application itself doesn't directly validate displays, its accurate RGB conversion and CIE XYZ model integration are essential for ensuring that screen-based experiments are meaningful and minimize metameric failures. The flexibility to adjust white point and gamma allows for better control over the displayed colors.
- Procedure: Present participants with both:
- A real spectrum generated by a prism or diffraction grating.
- A simulated spectrum displayed on an RGB screen, carefully calibrated and accounting for gamut limitations.
- Expected Outcomes:
- Identical Perception (within saturation limits): If participants perceive the real and simulated spectra as identical (apart from potential saturation differences due to RGB gamut limitations), it suggests that standard RGB-based color matching is applicable for these individuals.
- Perceived Differences Beyond Saturation: If participants report differences that cannot be attributed to saturation, it indicates regions of metameric failure for that observer with the RGB display. For individuals claiming tetrachromacy, if no difference is perceived, it challenges the notion that their purported "new colors" are outside the standard tristimulus framework.
Stage 2: Assessing Perceived Color Bands in a Continuous Spectrum (Utilizing Web Application):
This stage investigates color band perception using the "Continuous Spectrum" row of the web application.
- Procedure (Web Application Integration):
- Use the "Continuous Spectrum" row of the web application to display a continuous spectrum across a chosen wavelength range (e.g., 400-700nm for the visible spectrum, or a narrower range like 400-500nm).
- Instruct participants to mark perceived color band transitions directly on the displayed continuous spectrum within the application interface. The application can be designed to allow users to click or draw on the spectrum to indicate boundaries.
- Expected Results (Web Application Context):
- Trichromatic Norm: Using the application, expect most individuals to identify approximately six to seven color bands, corresponding to the major unique hues, consistent with typical trichromatic vision.
- Atypical Color Perception: The application allows for easy recording of the number and location of boundaries marked by participants, facilitating the identification of individuals with potentially atypical color perception (reporting more or fewer bands).
- Color Band Identification Task (Web Application Feature): The web application directly facilitates the "Color Band Identification" experimental paradigm. Users can be asked to identify and mark "color band transitions" directly on the continuous spectrum displayed by the application. For example, with a 400-500nm range, users might identify regions like "violet fade," "violet peak," "blue/indigo," "blue/cyan," and "cyan" within the application.
Stage 3: Discrete Color Step Grouping Test (Utilizing Web Application):
This stage leverages the "Discretized Spectrum" and "Separated Discrete Samples" rows to investigate color grouping in discrete presentations, mitigating ambiguities of continuous spectra.
- Procedure (Web Application Integration):
- Utilize the "Discretized Spectrum" row and the "Separated Discrete Samples" row of the web application.
- Display a spectral range (e.g., 400-500nm) in both rows simultaneously, varying the "step" parameter in the "Discretized Spectrum" row to present different numbers of discrete color chips. The "Separated Discrete Samples" row can be used to minimize contrast effects.
- Instruct participants to group the color chips in the "Discretized Spectrum" (and "Separated Discrete Samples") rows into sets based on perceived color similarity, potentially using drag-and-drop features or labeling functionalities within the application interface.
- Expected Outcomes (Web Application Context):
- Convergence of Categorization: The application allows for direct comparison of results from the "Discretized Spectrum" and "Separated Discrete Samples" conditions. Participants should ideally converge to a similar number of color groups in both conditions as they did for color bands in the continuous spectrum (Stage 2), reinforcing the consistency of categorical color perception even when using discrete stimuli and minimizing contrast effects.
- Color Grouping by Similarity Task (Web Application Feature): The application is designed to directly implement the "Color Grouping by Similarity" paradigm. By presenting a discretized spectrum, the question of color bands is reframed as grouping discrete chips by perceived similarity, which the application facilitates through its visual presentation and potential interactive grouping features.
Stage 4: Stepwise Color Differentiation Task (Discrete JND Testing and Spectral Ordering) (Utilizing Web Application):
This stage utilizes the "Discretized Spectrum" row and the "step" parameter of the web application to systematically increase the number of discrete color steps and investigate the perception of "all distinguishable colors."
- Procedure (Web Application Integration):
- Use the "Discretized Spectrum" row of the web application.
- Set the wavelength range to span a broad portion of the visible spectrum (e.g., 375-750nm).
- Start with a small "step" value (e.g., 2) and progressively increase the "step" parameter, displaying a new sequence of evenly distributed color chips at each step. The application automatically generates and displays these sequences based on the chosen spectral range and step count.
- Instruct participants to indicate when they believe the "Discretized Spectrum" row displays "all the colors" they can distinguish within the chosen spectral range, mirroring the "Identifying 'All the Colors'" paradigm.
- Record the "step" value at which the participant indicates they perceive "all the colors." This step count serves as a proxy for the number of distinguishable hues.
- Expected Outcomes (Web Application Context):
- Trichromatic Threshold: Using the application, typical trichromatic observers are expected to stop at a "step" value around 6 or 7, consistent with the canonical color categories (red, orange, yellow, green, blue, violet), even as the specific colors presented at each step change. The application directly facilitates the "Identifying 'All the Colors'" paradigm and allows for easy recording of this "trichromatic threshold."
- Tetrachromacy Assessment and "New Color" Emergence: Individuals claiming tetrachromacy might stop at a higher "step" value in the application, potentially suggesting finer discrimination. The application allows for recording this "stepwise color differentiation" data, which can be analyzed to compare thresholds between groups and investigate claims of expanded color perception. While the application itself doesn't directly assess "new color" sensations, the systematically varied color sequences it presents, coupled with participant feedback on "all colors" being represented, provides valuable data for analyzing such claims in conjunction with qualitative reports.
Stage 5: Evaluating Claims of "New Colors" and Color Attractors:
Data gathered from the web application in Stages 2, 3, and 4, particularly the step counts, color band markings, and grouping data, can be analyzed to evaluate claims of expanded color perception and investigate the underlying perceptual mechanisms. The application facilitates the collection of quantitative data (step counts, band boundaries) and qualitative observations (participant reports) necessary for this evaluation.
This final stage focuses on interpreting the data from previous stages, particularly in the context of claims regarding "new colors" and expanded color perception, such as in tetrachromacy. It also addresses the identification of color attractors and their "best exemplars."
- Conceptual Framework for "New Colors":
- A "new color" must be perceptually distinct from existing known colors to a degree comparable to the difference between established unique hues (e.g., red vs. orange).
- The existence of a truly novel hue would challenge the established isomorphism of hue space with a simple circular topology.
- Increased discrimination capacity (finer JNDs) does not automatically equate to the perception of genuinely new colors.
- If "new colors" are claimed, it is crucial to investigate if these colors can be synthesized or matched using standard RGB primaries. If so, it challenges the notion that they represent truly novel perceptual dimensions beyond the tristimulus framework.
- Procedure and Analysis:
- Analyze data from Stage 4 Stepwise Color Differentiation: Examine the step counts at which participants indicate all colors are represented. Compare step counts between groups (e.g., trichromats vs. potential tetrachromats).
- Analyze reported "new color" sensations: If participants report "new colors," meticulously analyze the specific colors presented in the sequence at the step where these sensations emerged. Investigate if these colors can be characterized as mixtures of existing hues or if they represent something genuinely novel.
- Color Attractor Investigation (following JND establishment): After establishing JND thresholds (especially in discrete presentations), investigate color attractors and their "best exemplars." This stage should be separated from initial discrimination tasks to minimize the influence of learned color associations and terminology on perceptual judgments. Variations in color choices for "best exemplars" (e.g., different shades of "blue") are more likely to reflect individual preferences and learned associations than fundamental differences in spectral perception itself.
Open-Source Availability and Further Application:
The web application's open-source nature is a significant advantage. It can be freely used and adapted for research purposes, allowing for standardized and easily reproducible experiments across different labs. Its simple interface and direct data recording capabilities make it a valuable tool for routine use and for collecting comparative data alongside more complex experimental setups. The application can be further extended to incorporate features for JND measurements using adaptive procedures, and for more sophisticated analysis of color grouping and categorization data.
Conclusion:
By integrating this web application into our methodological framework, we provide a practical and powerful tool for researchers to investigate color perception rigorously. The application directly supports key experimental paradigms for studying color bands, discrete color discrimination, and stepwise differentiation, addressing the critical methodological challenges and conceptual ambiguities in the field. Its features are specifically designed to facilitate the implementation of the proposed multi-stage approach, enabling a more systematic and data-driven exploration of color perception, including claims of expanded chromatic experiences.
This integrated framework, incorporating the web application, now offers both a conceptual roadmap and a tangible toolset for advancing research in color perception
Physically-Based, Octave-Modeled, Logarithmic Hue Wheel
The following sequence of graphics illustrates the construction of this hue wheel, providing a more intuitive visualization than the abstract group mappings presented in the mathematical section. It also clearly demonstrates how and which region is assigned to non-spectral magenta, which, interestingly, has ample "space" to close the wheel smoothly.
While electromagnetic waves are often described in terms of wavelengths in color science, music theory typically focuses on frequency ratios. To bridge these perspectives, the graphics and accompanying explanations incorporate both metrics, referencing each value as appropriate for clarity and context.
Key Graphics and Steps:
1. Linear Spectrum (200–1600 THz): A linear representation of the electromagnetic spectrum, highlighting the visible range (~400–750 THz) and including black regions beyond visible light for reference.
2. Octave Doubling: Frequencies are repeated at \(2^k\) generating three "rainbows" separated by black gaps. Non-spectral magenta, which does not exist in the physical spectrum, will be placed within these gaps.
3. Logarithmic Scale: The three rainbows are equalized in size by compressing the scale logarithmically.
4. Magenta Addition: A normal distribution curve fills the gaps, smoothly blending red and blue without altering their intensity.
5. Hue Wheel: A single rainbow from magenta to magenta forms a continuous logarithmic hue wheel.
