33. Pitch classes against the colour wheel

Hunt #75 (hunts/chroma_hue/). Verdict: PRETTY BUT TRIVIAL for the note-to-hue question; the reformulation in section 9 is the one worth keeping. The whole correspondence reduces to one statement: a note-to-hue bijection can carry exactly one piece of structure, the choice of a character of Z_12, and Z_12 has two injective characters up to symmetry, chromatic and fifths. Everything interesting that follows is a fact about Z_12 and about hearing, with the colour wheel serving as a set of twelve labels. The colour side, measured in a perceptual space, is not a 12-gon and has no 12-fold structure to match.

Nothing here is evidence about RH or about anything in docs/08. This document keeps the operator's requested separation: definitions, derivable facts, empirical results, candidate patterns, failed ideas, speculation.

1. Definitions

• Pitch classes. Z_12 with transposition T_c(x) = x + c and inversion I(x) = -x; together the dihedral group D_12 of order 24. Interval class ic(d) = min(d mod 12, -d mod 12), six values.
• Hue 12-gon. Twelve equally spaced points on the circle, with the cyclic metric d(i, j) = min(|i - j|, 12 - |i - j|) and its own D_12 of rotations and reflections.
• Wheel. A bijection f : Z_12 -> hue 12-gon. Two wheels are equivalent if they differ by a hue rotation or reflection. The operator's question asks which wheels preserve something.
• Affine wheels. f(x) = u x + c with u a unit mod 12, u in {1, 5, 7, 11}. u = 1 is chromatic order, u = 7 (equivalently 5) is the circle of fifths, u = 11 is inversion. 48 maps; two equivalence classes.
• Colour of a chord. Under wheel f, the colour of a pitch-class set S is the centroid (1/|S|) sum_{x in S} e^{i theta(f(x))} in the chroma plane. This is the additive mixture of the notes' hues at equal chroma, which is what the eye does to superposed lights and what the ear does not do to superposed tones.
• Dissonance measures on interval classes, kept separate and never averaged: Sethares' roughness curve for harmonic complex tones (six partials, 0.88 decay, computed), Tenney height log2(pq) of the textbook just ratio, and the plain ordinal ranking P5 < M3 < m3 < M2 < m2 < tritone. Their consonance orders agree except on the tritone (Sethares places it between the thirds and the whole tone).
• Perceptual hue. The OKLCH hue angle (Ottosson 2020), with CIEDE2000 (Sharma, Wu, Dalal 2005) as the difference formula. Both are implemented in hunts/chroma_hue/colorspace.py and pinned: all 34 published CIEDE2000 test pairs reproduce to 5e-4.

2. Derivable facts

Each of these is an elementary argument, and each is re-derived by enumeration in tests/test_chroma_hue.py so the document cannot drift.

F1. A wheel makes hue distance a function of interval class if and only if it is affine. If d(f(x), f(y)) depends only on x - y, the adjacency graph of the hue 12-gon pulled back to Z_12 is a translation-invariant 2-regular graph, i.e. a Cayley graph Cay(Z_12, {g, -g}), and it is a single 12-cycle only when gcd(g, 12) = 1. Then f maps consecutive multiples of g to consecutive hues, so f(x) = ±g^{-1} x + c. Backtracking over all 12! bijections finds exactly 48, and they are the affine set. So there are exactly two wheels up to symmetry with any transposition-invariant meaning: chromatic and fifths. Every other bijection assigns different hue distances to the same interval in different keys.

F2. Every transposition-invariant circular embedding is a Fourier mode. A dissimilarity on Z_12 that depends only on interval class is a circulant matrix; classical MDS doubly centres it and the result is still circulant, so its eigenvectors are the DFT modes. The best two-dimensional embedding is therefore the wheel x -> kx for the mode k with the largest eigenvalue, and it is a regular 12-gon if k is a unit and a collapsed polygon otherwise. This is why "rotate the wheel until structure appears" cannot work: the only knob is k.

F3. The colour of a chord under wheel k is its k-th Fourier coefficient. Centroid of {e^{2 pi i k x / 12} : x in S} = f_k(S) / |S| with f_k the DFT of the indicator of S. Chroma of the mixed colour = |f_k| / |S|; hue of the mixed colour = Fourier phase. The whole DFT of a pitch-class set is the list of its colours under the six wheels k = 1..6, of which only k = 1 and k = 5 are injective. Under the chromatic wheel, "grey" chords (f_1 = 0) are the balanced sets; under the fifths wheel they are the sets with f_5 = 0. Counted by enumeration: 98 subsets are grey under each injective wheel, 922 under k = 6 (the whole-tone wheel, two hues).

F4. The tritone is complementary under every affine wheel. 6 is the unique element of order 2, fixed by every automorphism, so u·6 = 6 and the tritone lands opposite on every wheel. Checked for all 48.

F5. Injective wheels for n-TET number phi(n)/2 up to symmetry. n = 3, 4, 6 have one; n = 5, 8, 10, 12 have two; every n > 12 has at least three. So 12 is the largest equal temperament whose circle admits exactly one non-chromatic arrangement. This is a fact about phi(n) = 4, not about music or colour.

F6. Maximally even sets are the most saturated chord colours (Clough and Douthett 1991; Amiot 2007, 2016). Among 7-note sets the diatonic scale is the unique maximiser of |f_5|; among 5-note sets the pentatonic; among 7-note sets the chromatic cluster uniquely maximises |f_1|. Reproduced by exhaustive enumeration of all subsets. Under the fifths wheel the diatonic scale is the most vivid seven-note colour and a whole-tone scale is grey; under the chromatic wheel the cluster is the most vivid and the diatonic scale is dull.

3. Empirical results

E1. Exhaustive consonance search (probe_consonance.py). Objective: Spearman correlation over the 66 unordered pairs between hue distance d(f(x), f(y)) and dissonance of ic(x - y). All bijections with f(0) = 0 and f(1) <= 6 (21,772,800 maps, every equivalence class under hue rotation and reflection) were scored exactly, against 200,000 uniformly random bijections.

The circle of fifths is the unique argmax under all three measures, 4.5 to 5.2 null standard deviations above the mean, and no map of the other 21,772,799 reaches it. The 48 affine maps take exactly two scores. The chromatic wheel is the worst structured choice: adjacent hues are the most dissonant intervals. MDS (F2) says the same in the continuous setting: the mode-5 eigenvalue dominates in every measure and the mode-1 eigenvalue is negative for two of three and below 3 percent of mode 5 for the third.

E2. The colour side is not a metric 12-gon (probe_perceptual.py).

The wheel a colour picker draws (HSL) is far from any regular polygon in a perceptual space. A wheel built to be regular in OKLCH is regular only because it was built so, and even then its twelve steps are not perceptually equal. The sRGB primaries and secondaries sit at OKLCH hues 29, 110, 143, 195, 264, 328 degrees: gaps of 81, 33, 52, 69, 64, 61. There is no 12-fold, 6-fold or even 4-fold symmetry in perceptual hue; the four unique hues of opponent theory are irregularly spaced and Munsell's perceptually spaced circle is 5-fold and 10-fold, not 12.

E3. The wave version (probe_spectrum.py). Visible light, 380 to 750 nm, spans 0.98 octaves of frequency, the coincidence Newton used. Twelve equal-tempered steps of light frequency mapped to monochromatic hue (CIE 1931 colour matching functions via the Wyman, Sloan, Shirley 2013 analytic fit, validated on the equal-energy white, the luminous-efficiency peak and the sRGB dominant wavelengths 611, 549, 463 nm) give, across the 7.8 semitones from 660 to 420 nm where the fit is trustworthy and hue is monotone: hue steps per semitone of 1.4, 38.6, 71.8, 21.5, 27.4, 55.5, 54.1 degrees (ratio 50), a total hue span of about 275 degrees, and a non-spectral purple arc of about 85 degrees that no wavelength produces. So the physical octave of light is not a circle (it is an interval, closed only by the purple line that the eye invents), and equal steps in log-frequency are nowhere near equal steps in hue. The ear keeps the spectrum; the eye projects it onto three broad cones and discards the rest. That projection is the reason F3 holds: colour mixing is linear and sound is not mixed at all.

E4. Other temperaments (probe_fourier.py). For n = 5..31 the dominant MDS mode of the Sethares dissonance curve was computed. The chromatic mode is the worst embedding for every n >= 9 and its eigenvalue grows more negative with n. The dominant mode is injective and generated by the best fifth for n in {5, 12, 13, 16, 17, 19, 23, 26}; for n in {6, 8, 10, 14, 20, 21, 28} the best circular embedding of consonance collapses pitch classes (in 24-TET the fifth is 14 steps and gcd(14, 24) = 2, so no bijective wheel puts fifths adjacent). 12 is in the good set; so are 19 and 5. This is one objective and one roughness model, and it says 12 is unremarkable within that set.

4. Candidate patterns, and what killed them

• "Rotate the colour wheel against the chromatic scale until structure appears." There is nothing to rotate into: by F1, the hue side sees only the choice of character, and rotation is a symmetry of the objective. Killed by F1 and by the 48 affine maps taking exactly two scores.
• "The circle of fifths fits colour better than the chromatic scale." It fits dissonance better, uniquely and decisively (E1), and this is a statement about the harmonic series (3/2 is the simplest non-octave ratio) with no colour in it. Replace the hue 12-gon by any twelve labels on a circle and the result is unchanged.
• "Complementary colours are the tritone, triadic harmony the augmented triad, analogous colours the cluster." True under every wheel (F4 and the k = 4 wheel), and empty: regular polygons on a circle are regular polygons on a circle. Itten's colour-harmony polygons were borrowed from music in the first place, and the empirical colour-harmony literature (Ou and Luo; Schloss and Palmer) finds preference driven by hue similarity and lightness contrast, not by polygons.
• "Equal hue steps match equal semitones because both are waves." Killed by E3: hue per semitone of light varies fiftyfold and the spectrum covers three quarters of the wheel.
• "Maybe a non-affine bijection encodes consonance better than fifths." Searched exhaustively; none does (E1). The best non-affine map trails by 0.04 to 0.06 and is not invariant under transposition, so it would assign different colours to the same interval in different keys.

5. The strongest reason it is trivial

Both objects are circles, and that is the entire overlap. The music side is a group: Z_12 is homogeneous, every pitch class looks like every other, and its representation theory (six Fourier modes, two of them faithful) is the complete list of ways to draw it on a circle. The colour side is not a group: perceptual hue has four irregularly spaced unique hues, a 2.6-fold variation in attainable chroma, an 85-degree arc of non-spectral purples, and no rotation symmetry at all. A bijection between them can only transport the group structure onto the labels. Any pattern you find is a pattern of Z_12, and all of those were already in the DFT literature (Lewin 1959, Quinn 2006, Amiot 2016, Tymoczko and Yust 2019 on phase as pitch-class sum).

6. What survived adversarial testing

Two things, both about hearing rather than colour, and both rediscoveries in substance if not in this exact form:

1. The circle of fifths is the unique bijection of Z_12 onto a 12-gon that maximises rank agreement between cyclic distance and dissonance, over all 12! maps and under three unrelated dissonance measures. Original computation; the continuous version (F2, mode 5 dominates) is a two-line consequence of circulant structure and is well known to the DFT school in the form "the fifth Fourier coefficient measures diatonicity".
2. The colour of a chord is its Fourier coefficient (F3). This is the one formulation in which colour adds something real: the eye's linear mixing is exactly the operation that turns a pitch-class set into a Fourier coefficient, and the twelve-bin wheel is exactly a character. It makes Amiot's maximal-evenness theorem a statement about saturation: the diatonic scale is the most vivid seven-note colour under the fifths wheel. Good for teaching the DFT of pitch-class sets; not new mathematics.

7. Speculation and follow-up hunts

• Fourier-phase colouring of real music (Tymoczko and Yust 2019 note that the k-th phase of a transposition orbit is the pitch-class sum). Hue = phase under the fifths wheel, chroma = |f_5|, lightness = |f_1| or cardinality gives a three-channel colour for any chord or windowed passage, with key changes as hue rotations and chromaticism as desaturation. That is a visualisation, not a theorem, and it should be built as one.
• n-TET and the injective-generator set {5, 12, 13, 16, 17, 19, 23, 26} from E4 is one roughness model away from meaning anything. A hunt with Plomp-Levelt, Vassilakis and Hutchinson-Knopoff curves and a stated objective could say whether that set is stable. Cheap, probably empty.
• Perceptual colour harmony as a dissimilarity on the hue circle could be run through the same circulant test: if a measured hue-harmony matrix is far from circulant (E2 says the distance matrices are), there is no "circle of fifths for colour", which would close the analogy from the colour side as E1 closes it from the music side.

9. Round two: the cyclotomic ontology, and the paths first skipped

The operator pushed back on the first verdict and asked for bolder reformulation. Four skipped paths were run (probe_round2.py, probe_cyclotomic.py); one of them changed the language.

R1. Metamerism. If the eye is a linear projector, chords with equal Fourier coefficient are metamers. Over all 4096 subsets, the chromatic wheel distinguishes 1763 (cardinality, colour) pairs, the fifths wheel 1763, the six-hue wheel k = 2 347, k = 3 175, k = 4 125, k = 6 49, and the full DFT all

1. The two injective wheels together also distinguish exactly 1763.

The fifths wheel carries no information the chromatic wheel lacks. The largest metamer class is the 24 grey hexachords (three tritone pairs, or two augmented triads).

R2. Why: the two wheels are Galois conjugates. Write a chord as the cyclotomic integer a(S) = sum_{x in S} zeta^x in Z[zeta_12], zeta = e^{2 pi i/12}. Then f_k(S) is the image of a(S) under zeta -> zeta^k, and for k a unit mod 12 that is an element of Gal(Q(zeta_12)/Q) = (Z/12)^x. So a colour wheel is a complex place of the cyclotomic field Q(zeta_12), and chromatic versus fifths is the Galois group modulo complex conjugation. Q(zeta_12) has degree phi(12) = 4, hence two complex places, hence two wheels; the largest n with [Q(zeta_n):Q] = 4 is 12 (F5); the collapsed wheels k = 2, 4, 3, 6 are the places of the subfields Q(omega), Q(omega), Q(i), Q; the tritone is complementary because zeta^6 = -1 in every place (F4); transposition is multiplication by the unit zeta^c; inversion is complex conjugation. Checked: f_5(S) = sigma_5(f_1(S)) for all 4096 subsets. Every result of sections 2 and 3 is a sentence in this language.

R3. The prediction it makes: chromatic saturation times fifths saturation is a rational integer. |f_1(S)|^2 |f_5(S)|^2 = N(a(S)), the field norm from Q(zeta_12) to Q of an algebraic integer, so it is a non-negative integer for every chord. In interval-vector terms, with iv_d the number of pairs at interval class d,

|f_1|^2 = a + b sqrt(3), |f_5|^2 = a - b sqrt(3), a = |S| + iv_2 - iv_4 - 2 iv_6, b = iv_1 - iv_5, N = a^2 - 3 b^2.

Checked for all 4096 subsets. Consequences, enumerated over the 224 set classes under D_12: only ten norms occur, {0, 1, 4, 9, 13, 16, 25, 36, 37, 64}; 18 classes are grey (N = 0, the sets with f_1 = 0); 84 classes are units (N = 1), among them the major and minor triads, the diminished triad, the pentatonic and the diatonic scale, the semitone and the major third. Units are the classes with (a, b) in {(1, 0), (2, 1), (7, 4)} up to sign, so a unit chord either has iv_1 = iv_5 and a = 1, or one more semitone than fifth (or the reverse) and a = 2. The invariant is coarse: the interval vector takes 200 values on the 224 classes, the norm ten, and nine norm values are shared by different interval vectors. The pair (|f_1|^2, |f_5|^2) lies on the hyperbola xy = N (figures/chroma_hue_norm.png): a chord vivid under one wheel is dull under the other unless its norm is large, and the pentatonic (0.07 against 13.9) is the extreme case with N = 1.

So "the major triad is a unit of Z[zeta_12]" is true and not special; 37 percent of set classes are. The honest grade: a derivable fact, checked by enumeration, whose knownness was searched once (section 8) without finding the norm stated as a chord invariant; Amiot's book places the coefficients in cyclotomic fields and Z-related sets share the norm trivially (same interval vector), so the framing is probably folklore to the DFT school.

R4. Orbifold radius against chroma. Tymoczko's voice-leading distance from a chord to the nearest perfectly even chord of its cardinality, against |f_1| (chroma under the chromatic wheel): Spearman 0.955 for trichords, 0.913 for tetrachords, 0.917 for hexachords, and in no cardinality is |f_1| a function of the radius (two to six distinct values per radius). Tymoczko 2008 ("Set-class similarity, voice leading, and the Fourier transform") already makes this connection; this is a rediscovery with numbers.

R5. The torus: two collapsed wheels at once. Z_12 = Z_3 x Z_4 by the Chinese remainder theorem, so the k = 4 wheel (three values, the augmented cosets) and the k = 3 wheel (four hues, the diminished-seventh cosets) are jointly injective. That embeds Z_12 in a torus, which is the neo-Riemannian Tonnetz rather than a circle, and it is the one note-to-colour code in which the four opponent unique hues get a job (hue = dim-7 coset, lightness = augmented coset). The 24 major and minor triads get 24 distinct codes. Pretty; no theorem.

R6. Real perceptual geometry. Replacing the cyclic metric by the measured CIEDE2000 matrix of a perceptual wheel (OKLCH fixed chroma, and HSL) and searching bijections again (hill climb with 40 restarts, 100,000-map null): fifths remains best on five of six (wheel, measure) pairs; on HSL with Sethares a non-affine map gains 0.018 (0.506 against 0.488), below the null's standard deviation of 0.134. And a small theorem fell out: for any affine wheel and any distance matrix on the hues, every transposition- invariant objective is invariant under hue rotation, because each hue pair is paired with a function of its hue difference. Rotating the wheel does nothing even when the wheel is irregular.

Verdict after round two. Unchanged for the question as posed. The better question is R2: a colour wheel is a complex place of Q(zeta_12), which turns the note-to-colour story into the arithmetic of Z[zeta_12] and produces one integer invariant (R3) that is true, cheap, and coarse. Not a pursuit; a paragraph worth keeping.

8. What was searched

The knownness check is partial. Consulted from memory and confirmed by one web search on 2026-08-22: Amiot, Music through Fourier Space (2016); Quinn, "General equal-tempered harmony" (2006-07); Clough and Douthett, "Maximally even sets" (1991); Tymoczko and Yust, "Fourier phase and pitch-class sum" (2019); Yust's review of Amiot (MTO 23.3). None of these discusses hue; the search found no published bridge between Fourier phase of pitch-class sets and a perceptual hue circle, which is weak evidence of absence and is reported as such. The colour-side references (Ottosson 2020 for OKLab; Sharma, Wu and Dalal 2005 for CIEDE2000; Wyman, Sloan and Shirley 2013 for the CMF fit) are used as published and pinned by tests.

Reproduce

.venv/bin/python hunts/chroma_hue/probe_consonance.py   # ~4 min, exhaustive
.venv/bin/python hunts/chroma_hue/probe_perceptual.py
.venv/bin/python hunts/chroma_hue/probe_spectrum.py
.venv/bin/python hunts/chroma_hue/probe_fourier.py
.venv/bin/python hunts/chroma_hue/probe_round2.py      # ~3 min
.venv/bin/python hunts/chroma_hue/probe_cyclotomic.py
.venv/bin/python hunts/chroma_hue/make_figures.py
.venv/bin/python hunts/chroma_hue/make_figure_norm.py
.venv/bin/python -m pytest -q tests/test_chroma_hue.py

Figures: figures/chroma_hue_wheels.png (Z_12 painted by its six characters), chroma_hue_consonance.png (the exhaustive score distribution with fifths and chromatic marked), chroma_hue_perceptual.png (the HSL hues in the OKLab plane, and the spectral octave), chroma_hue_spectrum.png (MDS eigenvalue by Fourier mode, three measures), chroma_hue_norm.png (the 224 set classes on ten hyperbolas xy = N).