The Shape of a Palette #2

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 is the second run of a question the first edition asked of 7,677 shots from six films: fit every one of those templates to every measured shot, and see which frame comes closest to each shape. The corpus has since grown to 18,235 shots from 11 films — Delicatessen, The City of Lost Children, Amélie, A Very Long Engagement, Sisu, Zatoichi, Skyfall, Sicario, Blade Runner 2049, Oppenheimer, and Bugonia — and the whole search was run again over all of them.

5 of the 8 exemplars changed: a frame the first edition had never seen now fits those templates better than the one it chose. That is the reason to run it twice. "The best-fitting frame in the corpus" is a claim about a corpus, and this one is nearly three times the size it was.

What did not change is the finding. The two four-leg templates still have no example, in a population this much larger, and both are still beaten on their own best candidates by a simpler reading. An absence that survives a corpus tripling is a much stronger result than an absence found once.

Each shape is 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. Then, where the corpus has them, two more frames — a louder one, the most forceful legible example of the same template, and a third, the best legible example from a film neither of the first two used. Purity is where the colour sits. Force is how much of it there is. The third frame asks whether the shape survives a change of film, which is a question six films could not answer and eleven can.

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 643 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.

7,495 of 18,235 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. Two things about this run in particular. The shortlist carried into stage two is 120 candidates per template rather than the 45 the first edition used: holding that number over a corpus this size would quietly have handed the decision to the coarse instrument. One analysis run is deliberately outside this corpus: 1917 is sampled on a clock rather than cut into shots — it has no shot durations, and the observational unit of this study is the shot.

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 18,235 shots is Delicatessen shot 0164, at 01:17:45:21, held for 5.2 seconds. It fits: 1.000 against the template, and the nearest rival reading (analogous) reaches only 0.000.

Its hue spread is 3°, its mean chroma C* 36.6, its mean key L* 35.8, and 47% of its pixels carry enough colour to vote on hue at all.

The louder frame is the stronger picture. 50% of it carries enough colour to have a hue at all, against 47% of the exemplar, and that colour sits 0.9× further from grey — mean chroma C* 34.1 against 36.6. That is what makes it read as its harmony from across a room.

The quieter frame is the better fit. It scores 0.778 against the template where the exemplar scores 1.000: less of its chromatic weight lands inside the template's windows (78% against 100%).

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.

A third film, on purpose. Delicatessen and Amélie supplied the frames above, and two frames from one or two films cannot tell you whether a shape belongs to the template or to a grade. Bugonia (2025) is here because it is the best legible example of this template in a film neither of them used.

It fits the template at 0.975, against 1.000 for the exemplar — enough to be a legible example of the shape, which is the bar a companion frame has to clear. Its own photography is a different proposition: mean chroma C* 22.6 against 36.6, mean key L* 17.7 against 35.8, and 39% of the frame carrying colour against 47%.

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

The best-fitting frame in 18,235 shots is Bugonia shot 0783, at 02:16:32:12, held for 5.8 seconds. It fits: 0.860 against the template, and the nearest rival reading (monochromatic) reaches only 0.493.

Its hue spread is 34°, its mean chroma C* 14.1, its mean key L* 20.3, and 18% 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 18% of the exemplar, and that colour sits 3.1× further from grey — mean chroma C* 43.3 against 14.1. 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 0.860.

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.

A third film, on purpose. Bugonia and Amélie supplied the frames above, and two frames from one or two films cannot tell you whether a shape belongs to the template or to a grade. A Very Long Engagement (2004) is here because it is the best legible example of this template in a film neither of them used.

It fits the template at 0.853, against 0.860 for the exemplar — enough to be a legible example of the shape, which is the bar a companion frame has to clear. Its own photography is a different proposition: mean chroma C* 21.1 against 14.1, mean key L* 38.8 against 20.3, and 24% of the frame carrying colour against 18%.

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

The best-fitting frame in 18,235 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. 65% 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* 46.4 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 0.570 against the template where the exemplar scores 0.960: what lands is shared less evenly between the legs (balance 0.58 against 0.97). On this frame the template loses outright to a monochromatic reading at 0.818.

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.

A third film, on purpose. Sisu and Blade Runner 2049 supplied the frames above, and two frames from one or two films cannot tell you whether a shape belongs to the template or to a grade. Bugonia (2025) is here because it is the best legible example of this template in a film neither of them used.

It fits the template at 0.700, against 0.960 for the exemplar — enough to be a legible example of the shape, which is the bar a companion frame has to clear. Its own photography is a different proposition: mean chroma C* 24.3 against 13.1, mean key L* 31.6 against 51.0, and 34% of the frame carrying colour against 17%.

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

The best-fitting frame in 18,235 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. 35% of it carries enough colour to have a hue at all, against 16% of the exemplar, and that colour sits 1.5× further from grey — mean chroma C* 19.7 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 0.536 against the template where the exemplar scores 0.797: less of its chromatic weight lands inside the template's windows (77% against 93%) and what lands is shared less evenly between the legs (balance 0.69 against 0.85). On this frame the template loses outright to a monochromatic reading at 0.635.

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.

A third film, on purpose. Zatoichi and Bugonia supplied the frames above, and two frames from one or two films cannot tell you whether a shape belongs to the template or to a grade. Oppenheimer (2023) is here because it is the best legible example of this template in a film neither of them used.

It fits the template at 0.556, against 0.797 for the exemplar — enough to be a legible example of the shape, which is the bar a companion frame has to clear. Its own photography is a different proposition: mean chroma C* 19.1 against 13.5, mean key L* 32.0 against 28.4, and 23% of the frame carrying colour against 16%.

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

The best-fitting frame in 18,235 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 18,235 shots is Bugonia shot 0814, at 02:19:40:01, held for 5.6 seconds. It does not fit. This is the closest the corpus comes, and even here the template scores 0.036 while a analogous reading scores 0.354. The frame is better explained as something simpler.

Its hue spread is 59°, its mean chroma C* 11.1, its mean key L* 15.4, and 16% 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 18,235 shots is Skyfall shot 0931, at 01:48:27:19, held for 2.9 seconds. It does not fit. This is the closest the corpus comes, and even here the template scores 0.508 while a monochromatic reading scores 0.553. The frame is better explained as something simpler.

Its hue spread is 88°, its mean chroma C* 16.3, its mean key L* 19.9, and 19% 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 19% of the exemplar, and that colour sits 1.0× further from grey — mean chroma C* 16.8 against 16.3. That is what makes it read as its harmony from across a room.

The quieter frame is the better fit. It scores 0.474 against the template where the exemplar scores 0.508: less of its chromatic weight lands inside the template's windows (75% against 78%) and what lands is shared less evenly between the legs (balance 0.64 against 0.66). On this frame the template loses outright to a split-complementary reading at 0.652.

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.

A third film, on purpose. Skyfall and Amélie supplied the frames above, and two frames from one or two films cannot tell you whether a shape belongs to the template or to a grade. Sicario (2015) is here because it is the best legible example of this template in a film neither of them used.

It fits the template at 0.444, against 0.508 for the exemplar — enough to be a legible example of the shape, which is the bar a companion frame has to clear. Its own photography is a different proposition: mean chroma C* 14.9 against 16.3, mean key L* 14.3 against 19.9, and 26% of the frame carrying colour against 19%.

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 18,235 shots is Oppenheimer shot 3042, at 03:31:59:16, held for 0.0 seconds. There is no template to fit. Only 0.00% 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 70°, its mean chroma C* 3.4, its mean key L* 53.1, and 0.00% 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 0.00% of the exemplar, and that colour sits 0.9× further from grey — mean chroma C* 3.2 against 3.4. 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 Inf× as much of it — 1.1% against 0.00%. It is the more dramatic picture and the less colourless one.

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

A third film, on purpose. Oppenheimer and Sisu supplied the frames above, and two frames from one or two films cannot tell you whether a shape belongs to the template or to a grade. Zatoichi (2003) is here because it is the best legible example of this template in a film neither of them used.

Only 0.77% of it carries any hue at all. Its own photography is a different proposition: mean chroma C* 1.3 against 3.4, mean key L* 28.7 against 53.1, and 0.77% of the frame carrying colour against 0.00%.

Two templates have no example in this corpus either, and that is again the most interesting result here — more so than the first time, because the corpus that failed to produce one is now nearly three times the size.

Neither square tetradic nor rectangular tetradic appears in 18,235 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.

The first edition could say that a four-leg harmony did not appear in 7,677 shots. This one can say it did not appear in 18,235, across films shot on different stocks, in different decades, by different hands. That is no longer a quirk of a small corpus.

The exemplars are drawn from a wider spread of films than the first edition managed, and that is the concrete gain. When six films were searched, one low-saturation film supplied most of the winners and the report had to spend a section explaining why. With 11 in the pool the shapes are found in several different kinds of photography, and the third frame for each template is chosen specifically to come from a film the other two did not use.

That matters for what the study can claim. A shape found repeatedly in one film's grade is a fact about that grade. The same shape found in the purest example from one film, the most forceful from another and a legible one from a third is a fact about the template — and the reader can see all three and disagree.

One exemplar deserves a note the arithmetic cannot give it. The clearest achromatic frame in the corpus comes from a film that alternates a colour strand with a photographed black-and-white one, so it is achromatic because of how it was shot, not because a grade drained it. The measurement cannot tell those apart — it reports that almost nothing in the frame carries a hue — and this is the one place in the study where the reason happens to be known from outside it.

What has not moved is the shape of the answer. The templates that survive contact with measurement are still the ones with one or two hue families and an opposition. Growing the corpus did not produce a square tetradic frame, and there is no reason left to expect that growing it again would.