A closed counter adds a little to a junction

A scanned page sharpened on its stored values, rather than on light, darkens the dark side of every edge more than it lightens the light side, and sharpened type errs on its dark side found where that error lands on a letter as the eye sees it. In black type it gathers at the junctions — the inside angles where a bar meets a stem — and avoids the stroke ends. A hairline spills its error onto the paper followed it from text sizes to display sizes and found the junctions’ share climbing with size. A junction gathers by the width of its stem found the simpler variable underneath: at three weights and five sizes, the junctions’ share of the seen error ranks with the stem’s width in pixels, and letters of different weights whose stems are the same width gather alike.

Alike, but not the same. At equal stem widths the heavier letter’s junctions gathered slightly more in five pairs of six, by three to nine hundredths. The heavier letter differs from the lighter in two ways at the same stem width: its counters — the paper enclosed or half-enclosed by its strokes — are narrower, and it is smaller around its stem, so the eye’s pool sees more of the rest of the letter from each junction. That essay’s closing section proposed separating the two by holding the stem and moving the spacing. The prediction was the counter: a junction faces into the paper between strokes, where one arm’s halo reaches across to the next; a stroke end faces away from it. If the counter carries the residual, closing the counters should raise the junctions’ share and leave the stroke ends alone. If the letter’s size carries it, both should move together.

Closing the counter raises the junctions and not the stroke ends #

At equal stem widths, the closed spacing’s junctions hold on average 0.043 more of their share of the seen error than the open spacing’s, and its stroke ends 0.011 more — a quarter as much. The junctions gain at six of eight stem widths, and E and H, whose bars face each other across a counter, gain 0.05 where L and T, whose counters open to the side, gain 0.02 to 0.03. The prediction’s direction holds on every test. Its size is small: width by width the junctions’ gain runs from −0.05 to +0.12, the spread a sub-pixel position alone produces, and the stem’s width in pixels still orders all three spacings together. Drawing the letters apart also found a fault in the earlier census: its light H had a crossbar that did not reach its stems.

• The residual is the counter, by the lead’s own test: the junctions move and the stroke ends barely do.
• It is a small effect. Four hundredths of a share against a curve that climbs from ×1.0 to ×1.4 over the same widths.
• The letters with the enclosed counters show it most, which is the test the lead did not name and the counter passes.
• The earlier census’s light letter is corrected, and its conclusions survive with a wider tolerance.

The upper bar in each row is the junctions’ gain and the lower the stroke ends’. The lower bars are short and nearly all positive; the upper bars are longer, mostly positive and uneven — tall at six and eleven pixels, below zero at seven and eight.

Holding the stem and moving the spacing #

The letters are the stem census’s four sans capitals — E, H, L and T — drawn on a grid of cells, each stroke from one unit of spacing to another and as thick as the stem. There the spacing was held at ten cells and the stem varied from six to fourteen. Here the stem is held at ten cells and the spacing varied: eight cells a unit, closer than the stem census’s regular letter; ten, which is that letter exactly; and fourteen, wider. Closing the spacing narrows every counter — the gap between the E’s arms goes from thirty-two cells on the open spacing to fourteen on the closed — and shortens the letter around its stem from ninety-four cells to fifty-eight.

Each spacing is set at the sizes that give its stem the same widths in pixels: five to twelve, in steps of one. At a stem of eight pixels, the closed letter is set at 15.9 point and the open one at 25.8; the stem is the same width on the page and the letters around it differ by more than half. At every stem width the closed letter’s counters are less than half the open letter’s in pixels and its height five eighths. The two things the lead wanted separated are still tied together by the spacing, but the stroke ends are the test that tells them apart: the letter’s size reaches every corner, and the counter only the corners that face into it.

With the spacing at ten cells, the letters set at stems of five and ten pixels are the stem census’s regular weight at 12 and 24 point exactly, junction and stroke end alike, which is checked.

The stem’s width still orders everything #

The three spacings climb one curve. The junctions hold ×1.0 of their share at a stem of five pixels and ×1.4 at twelve on every spacing, and the stem’s width ranks all twenty-four cells at 0.94 — lower than the stem census’s 0.98 only because the three spacings tie at every width. The closed spacing’s line sits a little above the open one’s more often than not, and falls just below it at seven and eight pixels. What the figure shows is the stem census’s finding, repeated on letters of a different shape: the variable is the stem.

The stroke ends fall along one curve too, from about ×0.75 at five pixels to ×0.56 at twelve. No stem width separates the closed and open spacings’ stroke ends by more than 0.022; the regular spacing wanders a little further from both, by up to 0.044 at nine pixels, which is the size of the scatter a sub-pixel position leaves in a pooled share. The stroke ends are the corners that face away from the paper between strokes, and closing that paper leaves them where they were.

The enclosed letters gain most #

The lead named the stroke ends as the test. The letters give a second one it did not name. E and H have bars that face each other across a counter — the E’s three arms enclose two, the H’s stems enclose two either side of the crossbar — and their junctions open onto that paper. L and T have one junction or two, opening onto paper that runs off the letter’s edge. If closing the counter is what raises a junction’s share, it should raise E’s and H’s more.

It does. Averaged over the eight widths, closing the spacing adds 0.051 to E’s junctions and 0.049 to H’s, and 0.028 to L’s and 0.021 to T’s. And the stroke ends go the other way: L and T, whose stroke ends are longer and more exposed, gain 0.016 and 0.017 there; E and H 0.006 and 0.009. A change in the letter’s size would not sort the letters this way. A change in the paper between its strokes does.

As large as a sub-pixel position #

The effect is real in every way it can be tested and small in every way it can be measured. Width by width, the junctions’ gain runs from −0.046 at eight pixels to +0.115 at eleven. A hairline spills its error onto the paper found that where a scan lands an outline against the pixels moves a hairline tip’s share by a factor of three, and every census since has pooled over four sub-pixel positions to keep that out; the pooled share still moves by about a tenth from one stem width to the next, as the stem curve’s own wobble shows. Four hundredths is below that. It shows up as a lean — six widths of eight, the enclosed letters twice the open ones, the stroke ends a quarter as much — rather than as a difference any one width would reveal.

Why a narrow counter should add anything is the mechanism the stem census proposed and did not test. An unsharp mask’s error on stored values is darker than the truth on each edge’s dark side and lighter on its paper side, and in a counter narrower than a few pixels the paper-side halos of two facing strokes overlap. The corner of a resized patch is lighter than its edges found the same arithmetic at a patch’s corner, where two edges’ lobes add; in a narrow counter the two walls’ halos add along the whole gap, and the junction at the gap’s inner end sits in the brightest part of it. The eye’s pool, a few pixels across at forty centimetres, takes that doubled halo into the junction’s account. On the open spacing the counter is wider than the pool and the halos never meet. That is a mechanism the numbers fit rather than one they prove — the census does not separate the halo’s contribution from the dark side’s — but it predicts exactly the pattern of letters seen: E and H, whose junctions open onto a gap between facing bars, gain; L and T, whose junctions open onto paper with no facing stroke, gain less.

That is also the size of the stem census’s residual. Its heavier letters gathered three to nine hundredths more than lighter ones at the same stem width, and a heavier letter on a fixed spacing is a closer-spaced one: its counters narrower by the difference in stems. The residual it saw is this one.

A floating crossbar in the earlier census #

The letters are drawn by the stem census’s own rule, and on a spacing wider than the stem that rule drew an H that is not an H. Its crossbar was written to run from one unit of spacing in to three units plus a stem, which meets the stems only when the stem is as thick as the spacing. On the regular letter, stem and spacing both ten cells, it did. On the light letter, a six-cell stem on a ten-cell spacing, the crossbar started four cells short of the left stem and stopped four short of the right: a bar floating between two posts, with stroke ends where its junctions should have been.

The H is now drawn with its crossbar spanning the letter, which covers the same cells wherever the old one reached the stems — the regular and bold letters, and this essay’s closed spacing, are unchanged cell for cell — and joins the light letter’s crossbar to its stems. Re-read with the joined H, the light letter’s junctions hold ×1.01, ×0.98, ×1.16, ×1.30 and ×1.41 of their share at its five sizes, where the stem census reported ×0.99, ×0.98, ×1.19, ×1.30 and ×1.39. The stem’s width now ranks the junctions at 0.98 rather than 0.97 and the point size at 0.88 rather than 0.85; the heavier letter still gathers more in five pairs of six; and the pairs of weights at matched widths agree within an eighth rather than a tenth, the light letter at 24 point now 0.12 below the bold at 12. The stem census carries a note saying so.

A floating crossbar did not break the census because the census pooled four letters and one of them lost its junctions: the other three carried the light weight’s reading. It would have broken a census of H alone, and it was invisible in every figure the census drew, which showed the E.

What this means for proofing a sharpening #

Read the stem, and then the counter. The stem’s width in pixels says where a sharpening on stored values will be seen; a letter with tight counters — a condensed face, a heavy weight on a fixed width — gathers a few hundredths more at its junctions than its stem alone predicts. For a proof that is a nuance rather than a rule: the stem moves the junctions’ share by four tenths over the range here, the counter by one twentieth.

The cure is unchanged. Sharpening in light removes the difference altogether, as it did for every letter before; a halftone is a luminance object is why the lightness error is the one a reader sees, and a pool with an edge is why the eye’s pooling decides which corners show it. The eye counts a corner’s error, not its peak found the same pooling reordering corners on resized patches; a junction facing a narrow counter is a corner whose pool holds more of its neighbour’s halo.

How the letters were set #

Each letter is a union of cells: a three-cell margin either side, strokes from one unit of spacing to another, each as thick as the stem, on spacings of eight, ten and fourteen cells a unit with a ten-cell stem. Each is set at the point size that gives the stem five to twelve whole pixels at 300 dots an inch — the stem’s share of the letter’s height times the cap height, seven tenths of the point size — rendered by exact area coverage at offsets of 0, 0.25, 0.5 and 0.75 of a pixel, encoded as sRGB, and sharpened by an unsharp mask of radius 1.5 pixels and amount one, on the stored values and on light. The difference is filtered by the model eye’s three spatial channels at forty centimetres and read as CIEDE2000 per pixel; each pixel’s error goes to the nearest point of the outline, and each part’s share of the error within two pixels of its corners is divided by those corners’ share of the outline’s length. The four letters and four offsets are pooled by adding lengths and errors before dividing, exactly as a serif moves the error and adds little pooled them.

What this leaves out #

Counter and size still move together. The spacing narrows the counters and shortens the letter at once; the stroke ends and the letters’ shapes tell the two apart, but a design that widened the counters while keeping the letter’s height — a condensed and an extended face of one weight — would separate them directly.

The pixel phase is pooled over four positions. Eight or sixteen would shrink the width-by-width scatter and might let a single width show the counter’s effect; four is what every earlier census used, and changing it here would have changed what the comparison is against.

Four letters. E, H, L and T are all straight-sided. Round letters — O, C, the bowl of a P — have counters without junctions, and letters like A and V have junctions at acute angles that no census here has read.

Still open: whether a condensed face gathers more at the same height #

The spacing here moved the counter and the letter’s size together. A type family separates them: its condensed width narrows the counters at the same height and stem, and its extended width opens them.

The calculation is these four letters at three widths — the horizontal spacing eight, ten and fourteen cells with the vertical held at ten — set at the sizes that give the same stem width and the same cap height in pixels. The prediction is that the junctions of the condensed letter gather four to five hundredths more than the extended, as here, while the stroke ends stay within a hundredth — because the counter has narrowed and the letter’s size has not. If the junctions stay level instead, the effect here belonged to the letter’s height after all, and the enclosed letters’ larger gain has another cause.

A rule drawn outside its range #

The habit is about what a function’s own users have tested.

The stem census’s rule for drawing a letter was written for letters whose stem equalled their spacing, and there it drew them exactly. It was then applied, as its terms allowed, to a stem narrower than the spacing, where one of its four letters came apart — and the census pooled that letter with three sound ones and reported a smooth result. The failure mode is to trust a rule across the whole range its terms admit because it was right where it was written. The test is to draw what it draws at the ends of that range and look.