Where a seam is allowed to go

The stations are also a staircase ended on a suspicion that nobody had stated as a rule. A join in a several-station landscape stays invisible only while nothing of the depicted world spans it — no river running out of the near band into the middle one, no road, no line of trees — because any such object is drawn by two rules at once and shows the discontinuity the mist is there to hide. If that is true, a painter choosing where to put a seam is not exercising judgement about composition. They are choosing among the depths at which the landscape happens to be empty.

There is a version of this convention that has no such problem, and naming it sharpens what the problem is. A map along, and a picture across describes a handscroll, whose eye moves continuously along the roll, so there is no join anywhere and nothing to span: every depth is drawn by the rule that belongs to it. The several-station landscape buys the same gain — room for the far distance — in three lumps rather than continuously, and a lump has edges. Everything below is about the price of the edges.

Two things have to be measured before that is a constraint rather than a remark. The first is whether emptiness is really the only defence, or whether a seam can simply be put somewhere the symptoms are small. The second is how much emptiness a landscape actually has — and the answer to that turns out to depend entirely on what it is counted in.

Distance buys off the wrong symptom #

A seam has two symptoms and they behave differently as the seam is moved.

The offset — how far a crossing object’s foot lands from where the other band’s rule would put it — is the riser’s own image, the focal length times 2.4 metres over the join’s depth. It is 210 px if the seam is put at eight metres, 64.6 px at the twenty-six metres the convention uses, and 24.0 px at seventy. Moving the seam back is a real cure for it.

The turn is not touched. A road of width $w$ crossing a join between eyes at $H_1$ and $H_2$ has its edges drawn at $\arctan(2H/w)$ under each rule, and the difference of those two angles contains the two eye heights and the road’s width and nothing else. At every seam depth there is, a road eight metres wide turns 23.20 degrees.

That settles the first question, and it settles it against the painter. A seam breaks direction, not size found that the offset is exactly what butting two bands gives a painter freedom to absorb, and the turn is exactly what it does not. So the symptom a painter can already hide is the one that moving the seam reduces, and the symptom that convicts is immovable. Distance is no defence at all.

Nor is smallness. The thing being hidden is not a subtlety: read as one station over stepped ground, the step at the first join is as tall as the entire band that precedes it. Whatever else is true about where a seam may be placed, it may not be placed anywhere a reader can watch something cross it.

What a landscape’s emptiness is counted in #

So the constraint is emptiness, and emptiness has an obvious measure: a join is one depth, so an object of depth-extent e blocks an interval of length e, and what is left is the free share. Scatter six objects of three to twenty metres through a landscape running from eight metres to two hundred and about three quarters of its depth is still available.

That number is almost useless, and the reason is that a painter does not choose a depth. A painter chooses a place on the page.

Within a band the row of a ground point is the station’s height times the focal length over the depth, so a depth interval is worth fH(1/a − 1/b) pixels and the front of a band is worth many times the back of it. The same twelve-metre object blocks 6.3 per cent of the range wherever it stands and between 1.0 and 35.6 per cent of the page depending on where that is — a factor of thirty-six between standing at the front of the near band and standing at the far end of the picture.

A single tree at the front of the picture takes a third of the painter’s available page. The same tree two hundred metres off takes a hundredth. Counting in metres cannot see the difference and reports both as the same object.

So the room a painter has is a matter of luck #

Once the page is the currency, the free share stops being a property of how crowded a landscape is.

Counted in metres the answer is 76.4 per cent, give or take 2.3 points, and it hardly moves: six objects of stated sizes block about fifty metres of a hundred and ninety-two however they are arranged. Counted in the picture the average is 80.6 per cent — slightly more, because objects scattered evenly through depth mostly fall where the page cares least — and the standard deviation is 9.5 points, four times as large, with the four hundred scatterings running from 46.8 per cent to 95.4.

That is the finding, and it is not the one the question invited. How much room a painter has for a seam is set by where the landscape is crowded, not by how much. The same six objects leave nearly the whole picture free if they happen to sit in the distance and take half of it away if two of them stand near the front. A metre count reports the same landscape both times.

It also explains why a painter’s practice would look like judgement from outside. A painter who reports choosing a seam depth for compositional reasons, and a painter who takes the only gap the foreground leaves, are indistinguishable on one painting. They differ in how often the seam falls in the same kind of place across many paintings, and that is a measurement on a corpus rather than on a picture.

And it goes away quickly #

The last thing to measure is how much crowding it takes before there is no choice at all.

At a quarter of the convention’s own furnishing, 93.5 per cent of the range is free; at the convention’s own, 77.1; at double, 55.1; at quadruple, 30.5. The page curve falls more slowly — 94.4, 77.7, 60.8, 43.0 — because the objects added are being added where they already are, which is mostly in the distance.

Two readings of that. A landscape sparse enough to look like the genre’s own — a few trees, a village, a spur of rock — leaves a painter most of the depth to work in, and the constraint is real but not binding. A landscape crowded enough to be a scene rather than a prospect takes it away, and does so before the picture looks full. The convention is for the first kind of landscape, which may be the whole reason it is for that kind of landscape.

That also sharpens what a shadow decides which landscape it is found. A shadow crossing a join settles which of the two readings a picture is, and a crossing shadow needs a tall enough object near enough to the seam. In a sparse landscape those are exactly the objects a painter has most freedom to place elsewhere, and in a crowded one they are exactly the objects the painter has no room to avoid.

The convention’s own two joins cost the same #

The landscape this field has been measuring has two joins, and they look like very different objects. The first is at twenty-six metres between stations 1.6 m and 4.0 m apart in height; the second is at seventy metres between stations 4.0 m and 11.0 m apart. One is close with a small riser and the other is far with a riser nearly three times as large.

They cost almost exactly the same. The offset is 64.6 px at the first join and 70.0 px at the second — eight per cent apart — and a road eight metres wide turns 23.20 degrees at the first and 25.02 at the second. The near join’s advantage of a small riser is spent paying for its closeness, and the far join’s advantage of distance is spent paying for its riser, and the two cancel to within a tenth.

That is not a coincidence the convention stumbled into; it is what choosing the station heights for a reason produces. Three distances in one landscape found the stations chosen so that each band is given back the picture the depth-squared law takes from it, and a station that rises in proportion to the depth it is drawing is a station whose riser rises in proportion too. The riser’s image is the riser over the depth, and a ratio held constant is a constant.

So a painter of this convention has no cheap seam and no expensive one. Both joins are equally loud, which means both need the same treatment, which is what the mist across both of them looks like.

A tenth of the depth is two fifths of the picture #

The same arithmetic says something blunter about where a landscape of this kind actually lives.

The near band runs from eight metres to twenty-six, which is 9.4 per cent of the depth range and 41.0 per cent of the page. The far band runs from seventy metres to two hundred, which is 67.7 per cent of the range and 30.3 of the page. The middle band is 22.9 and 28.7. Two thirds of the depicted distance is drawn in under a third of the picture, and a tenth of it takes two fifths.

That is why the page count and the metre count disagree so violently about a single object, and it is worth having as a number rather than as a direction. An object at eleven metres is standing in the part of the landscape that holds two fifths of the drawing; an object at a hundred and thirty is standing in the part that holds a tenth of it. A painter searching for a gap is searching a range whose front eighteen metres are more of the picture than its last hundred and thirty.

The blunt corollary is how few things it takes. Two objects, one spanning the near band and one the middle, block 69.7 per cent of the picture between them and leave a painter the far band alone — and the far band is where a seam is least useful, because its whole purpose is to separate distances the near rule is already failing. Six scattered objects leave four fifths of the page on average because they mostly fall where the page is cheap; two placed where a landscape actually puts its incident leave under a third.

That is the sense in which the metre count misleads rather than merely being coarse. It is not that it gives a number that is too large. It is that it gives a number that is stable, when the quantity it stands for is not — and a stable wrong number invites the conclusion that the constraint can be ignored.

It is also the reason the convention exists. A landscape that changes its rule halfway up recovered each band’s own station and found no common ground across a join; the reason a painter pays that price is that one camera would give the far band even less than the 30.3 per cent it gets here. The seam is bought with the same currency it is measured in, which is why a constraint on where it may go is a constraint on the whole convention rather than a detail of it.

What this does not settle #

The scattering is uniform in depth and nothing says a landscape is. Objects were placed with equal probability per metre, which puts most of them far away because most of the range is far away. A real landscape has its incident in the middle distance and its foreground deliberately clear, and both of those choices move the page count more than they move the metre count — which is the whole burden of this essay applied to its own model.

Blocking is treated as all or nothing. An object spans a depth or it does not, and the measurement charges the full extent either way. A seam breaks direction, not size showed the symptom is graded — a road’s turn is worst at a middling width and falls away for very narrow and very wide ones — so a real count would weight each blocked depth by how loudly the thing spanning it would complain. That weighting would raise the free share and would not change its variability, which is the finding.

Nothing here reads a painting. Every number above comes from the convention as this field models it — three stated bands, three stated stations, objects scattered by a rule nobody in the fifteenth century agreed to. What the measurement produces is a prediction about where seams would have to fall, not an observation that they do. Testing it means locating seams in real paintings of this kind, which the stations are also a staircase already found is the hard part, since the convention’s mist is laid exactly where the evidence would be.

And a seam need not be one depth. The convention butts two bands along a horizontal, and a painter who ran the join along a river bank or a ridge would be choosing a curve through the picture rather than a depth, which has far more room in it and is a different measurement entirely. Nothing here says the seams in real paintings of this kind are straight.

Emptiness, and what it is worth #

Moving a seam further off cuts the offset it forces from 210 pixels at eight metres to 24 at seventy, and leaves the turn at 23.20 degrees exactly, at every depth. Since the offset is the symptom a painter can absorb by butting the bands and the turn is the one that cannot be absorbed, distance buys off only what was already free. Emptiness is the only defence a seam has.

Emptiness counted in metres says a landscape of six objects leaves three quarters of its depth available, give or take two points. Counted in the picture it says four fifths, give or take nine and a half — and the scatterings run from 46.8 per cent to 95.4. One twelve-metre object blocks 6.3 per cent of the range from anywhere and 35.6 per cent of the page from the front of the near band, thirty-six times what it blocks from the back of the picture.

So the constraint the stations are also a staircase suspected is real, and it is not the constraint it looked like. A painter is not choosing among the depths where the landscape is empty in any evenly-weighted sense. A painter is hostage to the foreground: one thing standing near the front costs more room than everything standing in the distance put together.

Still open: whether a seam has to be a depth at all #

Everything above takes a join to be one depth, because that is what butting two bands along a horizontal makes it. That is a fact about the convention as this field has modelled it rather than about anything a painter is obliged to do.

A join laid along a river bank, a ridge or a line of trees is a curve across the picture, and the two bands meet at whatever depth that curve happens to be at in each column. It would not have to avoid an object; it would have to go round one, which is a far weaker requirement — and it would put the seam exactly where a reader’s eye already expects a discontinuity, which is what a river bank is.

The measurement that settles it is a geometric one before it is a historical one. A curved join means the riser’s height varies along the seam, so a crossing object meets a different offset at each of its own columns and its turn is no longer one number. The question with a number in it asks what that costs: whether a curved seam’s symptoms are smaller than a straight one’s at the same crowding, how much curvature a seam can take before the bands stop tiling the picture without a gap or an overlap, and whether the free share a curve enjoys is large enough to explain a painter never appearing to search for a gap at all.