The Shape of a Palette

Colour theory names a handful of arrangements that are supposed to look right together: complementary, split-complementary, analogous, triadic, tetradic, monochromatic. They are usually taught as diagrams on a wheel, and illustrated with paint chips.

This study asks a narrower and more answerable question. Take 7,677 measured shots from six films — Delicatessen, The City of Lost Children, Amélie, A Very Long Engagement, Sisu, and Zatoichi — and fit every one of those templates to every shot. Which frame in the corpus comes closest to each shape?

The answer is eight frames, each shown the same way: the picture, where its colour actually sits on the wheel with the fitted template drawn over it, and the same frame as a vectorscope. Two of the eight are here to show that the template they are named for does not occur.

The purest fit is not always the most persuasive picture, and it is worth being honest about that. So where the corpus has one, each shape is also given a louder frame: the most forceful legible example of the same template, with the arithmetic for why it came second. Purity is where the colour sits. Force is how much of it there is. The study ranks on the first and the eye reacts to the second.

Two stages, because the two instruments cost differently. Each shot already carries a five-colour palette in CIELAB — an area-weighted summary of the frame. That is cheap to fit and coarse: five clusters, one of them usually near-neutral, cannot fairly judge a four-leg template. So stage one uses it only to shortlist 250 candidates.

Stage two decodes those stills and scores them again on a continuous saturation-weighted hue distribution measured from the active picture at 512 pixels, in 72 bins of 5°. Every winner comes from stage two.

Fitting a template means rotating it to every angle and keeping the best. A fit has two parts: coverage, the share of chromatic weight the template's windows capture, and balance, how evenly that weight is spread across its legs. Balance is what stops a triadic template scoring well on a frame with one empty leg — three anchors with nothing at the third is not a triad.

A frame must also win its own template. Templates nest: anything monochromatic also sits inside analogous, and a square tetradic contains two complementary pairs. Every exemplar is reported with its nearest rival reading, and two of them lose to it.

3,637 of 7,677 shots were eligible — the rest were too dark or too colourless to have a harmony at all. One further frame was set aside by hand, and the package records which and why.

One hue family, varied only in lightness and chroma. One anchor: every chromatic pixel sits in a single hue family.

The best-fitting frame in 7,677 shots is Delicatessen shot 0017, at 01:02:14:10, held for 5.8 seconds. It fits: 1.000 against the template, and the nearest rival reading (analogous) reaches only 0.002.

Its hue spread is 2°, its mean chroma C* 41.9, its mean key L* 34.4, and 57% of its pixels carry enough colour to vote on hue at all. There is no louder alternative for this one: no other frame in the corpus is both a legible fit for this template and a more forceful picture than the exemplar.

Neighbouring hues, no opposition. 3 anchors, 32°, 32° apart, each with a ±22° window.

The best-fitting frame in 7,677 shots is Zatoichi shot 0069, at 01:07:12:11, held for 5.5 seconds. It fits: 0.931 against the template, and the nearest rival reading (monochromatic) reaches only 0.594.

Its hue spread is 27°, its mean chroma C* 12.3, its mean key L* 25.4, and 17% of its pixels carry enough colour to vote on hue at all.

The louder frame is the stronger picture. 51% of it carries enough colour to have a hue at all, against 17% of the exemplar, and that colour sits 3.5× further from grey — mean chroma C* 43.3 against 12.3. That is what makes it read as its harmony from across a room.

The quieter frame is the better fit. It scores 0.842 against the template where the exemplar scores 0.931: less of its chromatic weight lands inside the template's windows (92% against 98%) and what lands is shared less evenly between the legs (balance 0.92 against 0.95).

Purity and force are different questions, and this pair is the clearest place to see it: the template describes where a frame's colour sits, not how much of it there is. A frame can obey its template almost perfectly and still whisper.

Two hues directly opposite. 2 anchors, 180° apart, each with a ±24° window.

The best-fitting frame in 7,677 shots is Sisu shot 0138, at 01:14:10:10, held for 6.9 seconds. It fits: 0.960 against the template, and the nearest rival reading (monochromatic) reaches only 0.518.

Its hue spread is 109°, its mean chroma C* 13.1, its mean key L* 51.0, and 17% of its pixels carry enough colour to vote on hue at all.

The louder frame is the stronger picture. 32% of it carries enough colour to have a hue at all, against 17% of the exemplar, and that colour sits 2.1× further from grey — mean chroma C* 27.3 against 13.1. That is what makes it read as its harmony from across a room.

The quieter frame is the better fit. It scores 0.641 against the template where the exemplar scores 0.960: less of its chromatic weight lands inside the template's windows (80% against 99%) and what lands is shared less evenly between the legs (balance 0.80 against 0.97).

Purity and force are different questions, and this pair is the clearest place to see it: the template describes where a frame's colour sits, not how much of it there is. A frame can obey its template almost perfectly and still whisper.

One hue against the two neighbours of its opposite. 3 anchors, 150°, 60° apart, each with a ±22° window.

The best-fitting frame in 7,677 shots is Zatoichi shot 0254, at 01:35:03:08, held for 20.0 seconds. It fits: 0.797 against the template, and the nearest rival reading (complementary) reaches only 0.520.

Its hue spread is 78°, its mean chroma C* 13.5, its mean key L* 28.4, and 16% of its pixels carry enough colour to vote on hue at all.

The louder frame is the stronger picture. 29% of it carries enough colour to have a hue at all, against 16% of the exemplar, and that colour sits 1.2× further from grey — mean chroma C* 16.8 against 13.5. That is what makes it read as its harmony from across a room.

The quieter frame is the better fit. It scores 0.652 against the template where the exemplar scores 0.797: less of its chromatic weight lands inside the template's windows (81% against 93%) and what lands is shared less evenly between the legs (balance 0.81 against 0.85).

Purity and force are different questions, and this pair is the clearest place to see it: the template describes where a frame's colour sits, not how much of it there is. A frame can obey its template almost perfectly and still whisper.

Three hues evenly spaced around the wheel. 3 anchors, 120°, 120° apart, each with a ±24° window.

The best-fitting frame in 7,677 shots is Zatoichi shot 0643, at 02:20:49:16, held for 26.9 seconds. It fits: 0.581 against the template, and the nearest rival reading (split-complementary) reaches only 0.349.

Its hue spread is 85°, its mean chroma C* 12.1, its mean key L* 25.8, and 12% of its pixels carry enough colour to vote on hue at all.

The louder frame is the stronger picture. 16% of it carries enough colour to have a hue at all, against 12% of the exemplar, and that colour sits 1.1× further from grey — mean chroma C* 13.8 against 12.1. That is what makes it read as its harmony from across a room.

The quieter frame is the better fit. It scores 0.444 against the template where the exemplar scores 0.581: what lands is shared less evenly between the legs (balance 0.56 against 0.75). On this frame the template loses outright to a split-complementary reading at 0.674.

Purity and force are different questions, and this pair is the clearest place to see it: the template describes where a frame's colour sits, not how much of it there is. A frame can obey its template almost perfectly and still whisper.

Four hues at right angles. 4 anchors, 90°, 90°, 90° apart, each with a ±22° window.

The best-fitting frame in 7,677 shots is Zatoichi shot 0937, at 02:52:22:02, held for 2.1 seconds. It does not fit. This is the closest the corpus comes, and even here the template scores 0.168 while a complementary reading scores 0.508. The frame is better explained as something simpler.

Its hue spread is 91°, its mean chroma C* 9.2, its mean key L* 18.0, and 11% of its pixels carry enough colour to vote on hue at all. There is no louder alternative for this one: no other frame in the corpus is both a legible fit for this template and a more forceful picture than the exemplar.

Two complementary pairs, unevenly spaced. 4 anchors, 60°, 120°, 60° apart, each with a ±22° window.

The best-fitting frame in 7,677 shots is Zatoichi shot 0207, at 01:30:30:05, held for 6.9 seconds. It does not fit. This is the closest the corpus comes, and even here the template scores 0.629 while a split-complementary reading scores 0.758. The frame is better explained as something simpler.

Its hue spread is 95°, its mean chroma C* 13.5, its mean key L* 46.6, and 6.6% of its pixels carry enough colour to vote on hue at all. There is no louder alternative for this one: no other frame in the corpus is both a legible fit for this template and a more forceful picture than the exemplar.

No hue survives the chromatic threshold. Nothing here clears the chromatic floor, so the frame has no hue to be harmonious with.

The best-fitting frame in 7,677 shots is Zatoichi shot 0484, at 02:02:17:08, held for 49.0 seconds. There is no template to fit. Only 0.25% of the frame carries enough colour to be placed on the wheel at all, which is what makes it the corpus's clearest achromatic frame.

Its hue spread is 72°, its mean chroma C* 1.8, its mean key L* 21.4, and 0.25% of its pixels carry enough colour to vote on hue at all.

The louder frame is the stronger picture. 1.1% of it carries enough colour to have a hue at all, against 0.25% of the exemplar, and that colour sits 1.8× further from grey — mean chroma C* 3.2 against 1.8. That is what makes it read as its harmony from across a room.

The quieter frame is the better fit. There is no template to lose: what an achromatic exemplar must not have is colour, and this frame has 4.6× as much of it — 1.1% against 0.25%. It is the more dramatic picture and the less colourless one.

Both are worth looking at. Only one of them is the measurement.

Two templates have no example in this corpus, and that is the most interesting result here.

Neither square tetradic nor rectangular tetradic appears in 7,677 shots. The closest candidates are in the panels above, and both are beaten on their own frames by a simpler reading. A four-leg template needs chromatic weight at four places on the wheel at once, roughly evenly — and a photographed frame almost never does that. Light has a colour, and everything it falls on inherits some of it; a costume department can put four hues in shot, but the grade then pulls them toward one another.

This is worth holding onto when reading colour-theory writing about film. The templates that survive contact with measurement are the ones with one or two hue families and an opposition — which is most of what a wheel diagram is for, and considerably less than what it promises.

Zatoichi supplies 6 of the eight exemplars, and the reason is instructive rather than flattering.

A frame can only demonstrate a hue relationship if it has several hue families with enough chroma to be located on the wheel and enough separation to be told apart. Amélie is saturated but overwhelmingly warm, so almost every frame collapses to one lobe. The City of Lost Children is the darkest of the six, and darkness costs hue reliability before it costs anything else. Sisu is cool and low-chroma, which produces excellent complementaries and little else.

A film with moderate saturation and several distinct colour families is the one that produces textbook diagrams. That is a fact about what this measurement can see, not a ranking of the films.

It is also why the louder frames matter. Zatoichi is graded low in saturation, so its exemplars are unarguable in the numbers and quiet on the screen — the shape is there, and you have to look for it. The second frame in each pair is the same shape at volume, usually from Amélie, which is the opposite kind of film: enormous chroma, narrow hue, almost no separation. Between the two of them the pair says the whole thing. One shows you where the colour sits. The other shows you why anyone cares.