Complementary Colors Analysis

 Why This Analysis Exists

This analysis emerged indirectly from work on Tonal Constancy, a developing framework that extends probabilistic models of tonal cognition into pitch inference, categorization, and microtonal structure.

The core analogy is drawn from color constancy in vision: just as a palette of colors remains categorically stable under different illuminants, pitch sets can remain functionally recognizable across distortions, tunings, and deformations. The theory builds on work by David Temperley, William A. Sethares, Carol Krumhansl, Diana Deutsch, and Roger Shepard, while remaining exploratory and model-driven.

Across multiple stages of development, color analogies repeatedly reappeared—not metaphorically, but structurally.

Tonal Experiments and Perceptual Structure

Several experiments used tuning systems generated from arbitrary numerical sets unrelated to 12-tone equal temperament. Music composed within these tunings still exhibited recognizable diatonic function.

This cannot be explained by simple categorical tolerance windows (e.g., ±20 cents). Instead, tonal stability appears to depend on: hierarchical pitch relationships, trajectory-based weighting, cadence formation, expectation accumulation...

These factors redistribute perceptual tolerance dynamically. The result is the emergence of tonal illusions, functional multivalence, and invariance effects that resemble perceptual phenomena typically described as "illusions." In reality, these are structured perceptual outcomes arising from probabilistic inference under constraint.

One of the pitch datasets used in these experiments came from color vision research: spectral locations associated with “unique hues” in trichromats and tetrachromats. Some symmetries in this dataset motivated the present analysis.

Parallel Work in Color Systems

Independently of the tonal research, I had been working on color constancy, afterimages, and binocular color fusion for design and visual applications. These studies revealed consistent deviations from RGB-based complementarity, especially in: afterimage pairing, constancy across lighting conditions, stereoscopic fusion.

A simple internal color model developed for these purposes was later used in the interval-matrix application (a microtonal theory tool). It is not a radiometric mapping of the spectrum; rather, it is a pseudo-wheel constructed from proportional relationships that systematically deviate from RGB complementarity.

The spectral dataset analyzed here intersects with that earlier model.

Dataset and Methodological Position

The dataset used consists of individual reports (15 trichromats, 23 tetrachromats) in which participants marked spectral locations of color attractors (red, orange, yellow, green, cyan, blue, violet) without visual stimuli.

This is significant because: responses are individual, not averaged, methods are consistent across participants, no immediate visual cue constrained selection, responses reflect learned perceptual associations.

Much skepticism in color science toward wavelength–hue proportionality arises from aggregated datasets compiled across heterogeneous methods. Averaging across experiments obscures internal structure. Even within controlled studies, substantial inter-individual variation is well documented (one individual’s "best red" may overlap with another’s "best orange").

The present analysis does not claim definitive evidence. The dataset is small. However, it is one of the few available that preserves individual structure without methodological averaging.

Why So Many Complementary Models?

Color science contains numerous models of complementary wavelengths, especially for non-spectral hues like magenta. Many of these combine heterogeneous datasets and methods, producing inconsistent results. The question is whether internal proportional symmetries exist at the individual level that are obscured in standard linear representations.

Logarithmic Representation and Spectral Octave

The key transformation in this analysis was representing each individual’s visible range as a spectral octave: \((\lambda,2\lambda]\)

Using a logarithmic wheel within that range reveals a symmetry that is not visually apparent in linear tables or conventional 400–700 nm circular mappings.

When plotted logarithmically within each observer’s range (commonly approximated here as 375–750 nm for consistency of visualization), complementary pairs frequently approach symmetry:

red – cyan, orange – blue, yellow – violet, green – magenta (derived position)

The symmetry point corresponds to: \( \sqrt{\frac{\text{red end}}{\text{violet end}}} \)

For most observers, this approximates √2.

Thus: \( \frac{W_X}{W_{X′}} \approx  \sqrt{2}\) for complementary pairs.

More precisely, the ratio is individual: \( \sqrt{\frac{W_{\text{red end}}}{W_{\text{violet end}}}} \)

In most typical ranges (e.g., 800/400 nm), this approaches √2.

Magenta closes the wheel opposite green under the same proportional logic.

(Image.06) Color attractor locations (red, orange, yellow, green, cyan, blue, violet; magenta is artificially mirrored across green) for trichromats (left) and tetrachromats (right), plotted on a logarithmic scale within the spectral octave of 375–750 nm. (Illustrative RGB values).

Why This Is Not Obvious in Linear Plots

(Image.05) Color Attractor ("Unique-Hues") 380-780nm, linear scale.

Standard linear tabular plots conceal proportional symmetry. Even circular models often rely on perceptually uniform step sizes derived from separate assumptions, further obscuring internal structure.

The logarithmic octave representation exposes relationships of the form: \(W^2_G = W_X \times W_{X′} \)

Which can also be written as symmetry around "green": \( \frac{W_G}{W_X} = \frac{W_X′}{W_G} \)

These relations are visually apparent in logarithmic wheel form but not in tabular data.

Violet–Blue Ambiguity and Logarithmic Sampling

Greater inter-individual overlap appears in violet/blue than in red/orange regions. This pattern is consistent with sampling a logarithmic phenomenon linearly. Just noticeable differences (JNDs) for hue are known to follow approximately logarithmic scaling.

Thus, greater clustering at short wavelengths is expected under linear representation.

Internal Structure Despite Individual Variation

Individual differences in color perception are real: cone sensitivities, lens pigmentation, neural weighting, semantic learning, etc...

However, the internal relational structure between color categories appears preserved. If one observer’s red shifts, the entire attractor structure shifts proportionally. Regression analysis shows strong multicollinearity: hues "move as a whole" rather than independently.

This resembles scale stretching in music: endpoints may vary, internal ratios remain coherent.

(Image.07) Individual Color Attractors Symmetries - The image presents the original tabular visualization of color attractor data for each observer. Adjacent to this table, data from selected subjects are represented within a logarithmic octave wheel visualization. This visualization reveals that the near-perfect symmetrical patterns observed in some individuals are not readily apparent in the original tabular format.

Trichromats vs Tetrachromats

Preliminary statistical observations:

Both groups show symmetry around green.
Complementary ratios cluster near √2 individually.
Trichromats show strong inter-hue correlations, especially between neighboring hues.
Tetrachromats show similar patterns, except green exhibits weaker correlation with others.
Principal component structure differs: cyan dominates variance in trichromats (~60%), yellow in tetrachromats (~60%).
No single hue explains overall structure; at least three components are required to explain >90% variance.

These results are exploratory and not definitive.

Scope and Limitations

This dataset is small. The findings are preliminary. Larger and more diverse samples are required.

The analysis does not claim that hue is reducible to wavelength proportions. Rather, it suggests that when individual perceptual ranges are treated proportionally (logarithmically), a consistent internal symmetry emerges.

This symmetry aligns with afterimage pairing, predicts deviations from RGB complementarity, aligns with binocular color fusion results, mirrors proportional relationships found in tonal systems, Whether this reflects deep perceptual constraints or learned structural organization remains open.

The present work isolates the proportional pattern. Explanation comes later.