Width and height each carry half the counter's gain

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 black letter as the eye sees it: at the junctions, the inside angles where a bar meets a stem, and not at the stroke ends. A junction gathers by the width of its stem found the junctions’ share of the seen error ranking with the stem’s width in pixels, whatever the letter’s weight or size, and at equal stem widths the heavier weight gathering slightly more.

A closed counter adds a little to a junction asked what that remainder was. It held the stem at ten cells and moved the letters’ spacing from fourteen cells a unit to eight, setting each at the sizes that gave the stem the same width in pixels, and found the closed spacing’s junctions holding 0.043 more of their share than the open spacing’s, and its stroke ends 0.011 more. E and H, whose bars face each other across a counter, gained about twice what L and T did. The prediction had been the counter — the paper enclosed by the strokes, across which one arm’s halo reaches the next — and the direction held on every test, by an amount no larger than a sub-pixel position moves it.

But the spacing it moved was one number. Closing it narrowed the letter and lowered it at once: the H’s counter between its stems and the E’s counters between its bars closed together, and the whole letter shrank around its stem. That essay’s closing section proposed separating them the way a type family does. A condensed face narrows the letter at the same height and stem; an extended one widens it. The prediction was that the condensed letters’ junctions would gather four to five hundredths more than the extended ones’, as the closed spacing’s had, while the stroke ends stayed within a hundredth — because the counter across the letter had narrowed and the letter’s height had not.

Half the gain lies across the letter, and half down it #

At equal stem widths and one height, the condensed letters’ junctions hold on average 0.023 more of their share of the seen error than the extended letters’, and their stroke ends 0.004 more. That is half of what closing both spacings added. Shortening the letters at one width, which closes the counters that run down them, adds the other half — 0.023 at the junctions and 0.002 at the stroke ends — and the two halves together, 0.047, come within four thousandths of the 0.043 that closing both at once gave. Each direction gains most in the letter whose counter lies across it: condensing in the H, whose counter spans its width, and shortening in the E, whose counters stack down it. The T, with no counter across it, gains three thousandths when condensed. Neither half, on its own, is larger than one and a half standard errors of the scatter a sub-pixel position produces.

• The prediction is right that the stroke ends stay put, within half a hundredth in either direction.
• It is wrong about the size. The condensed letters carry half the gain, not all of it.
• The other half is the counters that run down the letter, closed by shortening it — which also shrinks the letter, and the T’s small gain there may be that.
• The split is arithmetic and not yet measurement. The halves add up, but neither is resolved from nought on its own.

The three bars are three ways of closing the same letters. The lowest is the earlier essay’s: both spacings closed together, +0.043. The upper two take one spacing each, and each comes to +0.023. The dashed line is their sum, +0.047, which sits on the third bar’s end within the width of its label. That additivity is the finding’s backbone, and it says the gain is a sum of what happens across the letter and what happens down it — not a property of one of them with the other along for the ride.

A condensed face at one height #

The letters are the collection’s sans capitals — E, H, L and T built from strokes on a grid, their stems ten cells thick — with the grid’s spacing split in two. Across, the spacing is eight cells a unit for the condensed face, ten for the regular and fourteen for the extended; down it stays at ten, so all three stand the same height. The regular face is the earlier census’s letter cell for cell, and with the two spacings made equal every letter here is the stem census’s, which is checked. A face set so its stem is eight pixels wide is then 19.2 point in all three widths: the stem, the height and the size are held, and only the width and the counters across the letter change. The H’s counter, the paper between its stems, is 22 cells wide condensed and 46 extended.

The other half of the census is the same letters with the across spacing held at ten and the down spacing at eight and fourteen: a short face and a tall one, the same width, the tall one sixty per cent taller. Shortening the letter closes the E’s counters between its bars and the H’s above and below its crossbar, and — because the stem is held in pixels — it also sets the short letter at a smaller size.

Each face is set at the size that gives its stem five to twelve pixels at 300 dots an inch, drawn in black on white at four sub-pixel positions, sharpened on its stored values and read through the eye at forty centimetres, exactly as the stem and counter censuses were. At each stem width the junctions’ share of the seen error is divided by their share of the outline, pooled over the four letters and four positions; a share over one means the junctions gather more than their length.

Why a junction faces its counter #

Sharpening on stored values rather than on light is an unsharp mask applied to numbers that are not proportional to light, and sharpened type errs on its dark side traced what that does at an edge: the dark overshoot the mask adds beside a black stroke is too dark in light, and the light overshoot on the paper side is too light by less. The error is a halo that hugs the stroke on the paper’s side. On a straight edge that halo lies in a thin strip; at a stroke end it spreads round the outside of a convex corner and thins out; at a junction it gathers into the inside of a concave corner, where two edges’ halos overlap. That overlap is why junctions carry more than their share and stroke ends less.

The counter enters because the paper beside a junction is not unlimited. Across a narrow counter the halo of the far stroke reaches back towards the junction, and the eye, which filters the page through its own contrast sensitivity before it compares anything — the step the eye keeps the lightness errors found reversing a whole ranking of resizes — adds the two halos where they meet. A wide counter lets each stroke’s halo fade into open paper; a narrow one keeps them close enough to sum. That is a mechanism with a direction in it: a counter narrowed across the letter should add to the junctions that face across it, and one narrowed down the letter to the junctions that face down it — which is the letter-by-letter pattern the census found, the H’s junctions gaining when the letter is condensed and the E’s when it is shortened.

The stroke ends face the other way. A stem’s end looks out over the paper beyond the letter, and a bar’s free end over the paper beside it; neither faces a counter in either direction, and in this census neither moves by more than half a hundredth. A hairline spills its error onto the paper found hairline tips holding less than their share at every size for the same reason: an end with nothing across from it has nowhere for a halo to gather.

Width by width, the halves are noisy #

Condensing gains at seven of the eight stem widths and shortening at six, and the spread at any one width is wide: condensing runs from −0.067 at eight pixels to +0.084 at eleven, shortening from −0.016 to +0.048. That is the same spread the earlier census found closing both spacings, and it has the same source. Where a letter’s edges land against the pixel grid changes which pixels carry each junction’s error, and the eye counts a corner’s error, not its peak found a junction’s seen error moving with that landing by more than its size moves it. Four sub-pixel positions average some of it away and not all.

Measured as a standard error over the thirty-two stem-width-and-position pairs, condensing’s mean gain is 0.023 ± 0.021 and shortening’s 0.023 ± 0.018. Neither is resolved from nought on its own. The earlier census’s both-at-once gain, 0.043 ± 0.030, was a little better resolved, and it is what the halves add to. So the honest reading is structural rather than statistical: the gain the counter census found divides between the two directions of the letter in the proportion the arithmetic gives, about evenly, and the census does not have the precision to say which half is larger.

Beneath the halves the stem’s width does what it did in every census before: the junctions’ share rises from about 1.05 at five pixels to 1.44 at twelve, and the three widths cross one another along that rise. Condensing is a small correction to a curve set by the stem, the same size as the scatter along it, and a proofing rule that knows a face’s stem width in pixels knows almost everything this census can tell it about where the error goes.

Each direction finds the counter across it #

The letter-by-letter gains are where the two halves show what they are made of. Condensing gains most in the H, 0.030, and least in the T, 0.003. The H’s counter is the paper between its two stems, and condensing narrows exactly that; its junctions, where the crossbar meets the stems, face straight into it. The T has no counter across it: its bar sits on top of its stem with open paper on either side, and narrowing the bar changes nothing its two junctions face. The E gains 0.027 — its arms shorten, and its junctions face the shortened counters between them — and the L, whose one counter opens to the right, 0.014.

Shortening gains most in the E, 0.029, whose counters stack down the letter between its three bars, and the H 0.022, whose counter is cut above and below by its crossbar. The L and T gain 0.016 each. The T’s gain here is the one number the counter cannot explain — it has no counter down it either — and it is about the size the letter’s own shrinking would contribute: the short T stands at a smaller size than the tall one to keep its stem at the same pixels, and at a fixed stem in pixels a smaller letter puts more of its own strokes within the eye’s reach of each junction — the eye’s contrast sensitivity at forty centimetres spreads each pixel’s error over a patch of fixed size on the page, how fine a colour edge can be measured how large, and a letter shrunk around that patch fills more of it. Read that way, perhaps a third of the height half is size and the rest the counters down the letter — a reading the per-letter scatter allows and does not require.

What this changes about proofing sharpened type #

Proof at the stem, and treat the counter as a correction. The stem’s width in pixels sets the junctions’ share to within the scatter a sub-pixel position gives, at every width and height here. What the counter adds is two to five hundredths of a share that runs from one to one and a half.

A condensed face is not a special case. The counter census could be read as saying narrow counters gather error; this one says a condensed face gathers about half the extra that a smaller, tighter face of the same stem does, because height matters as much as width. A proofing rule written for “condensed type” alone would catch the smaller half.

Look at the H and the E. The letters whose counters lie across the direction a face is narrowed or shortened are where the gain goes. A serif moves the error and adds little found serifs shifting error between parts of a letter without changing much in total; the counters do the same on a smaller scale, and the H and E are where to look first.

How the letters were set #

The strokes are the stem census’s: a stroke running from one unit to another across fills the cells from the first unit times the across spacing to the second unit times it plus a stem, and likewise down with the down spacing, on a margin of three cells. The stem is ten cells; the spacings are eight, ten and fourteen cells a unit across with ten down, and ten across with eight and fourteen down. A letter is six down-units and a stem tall, so one whose stem is set a given number of pixels wide stands that many pixels times six down-spacings plus a stem, over the stem — seven times the stem at the regular spacing — and is set at that height over the cap height of a point, so the three widths share a size at every stem width and the short and tall faces do not. Each letter is placed at four sub-pixel positions in a field of 128 pixels, sharpened on stored values and filtered by the eye’s contrast sensitivity at forty centimetres, and its seen error is assigned to the nearest outline sample; junctions and stroke ends are the outline’s parts as the stem census tagged them. The standard error pools the four positions at each of the eight stem widths.

What the census leaves out #

The letters are block capitals. Real condensed faces are not the regular face squeezed: their stems thin a little, their curves tighten and their counters change shape as well as width. A condensed grotesque’s H and E would differ from these by more than their spacing.

Four sub-pixel positions, along a diagonal. Each position moves the letter by a quarter pixel across and down at once. The scatter they leave is as large as the halves; more positions would shrink it, and nothing else in the census would change.

Height and size move together in the second half. Shortening a letter at a fixed stem width also sets it smaller, and the census does not separate the counters down the letter from the letter’s size. The T’s gain is the only hint at the size’s share.

Still open: whether sixteen positions resolve the two halves #

Each half is 0.023 and each has a standard error of about 0.02, set by where the letters land against the pixels. The standard error falls as the square root of the number of positions pooled, so four times as many positions would halve it.

The calculation is this census at sixteen sub-pixel positions — a quarter-pixel grid across and down, rather than four steps along the diagonal — for the condensed, extended, short and tall faces at every stem width, with the halves and their standard errors recomputed. The prediction is that both halves hold at about two hundredths, now each more than two standard errors from nought, and that the T’s height gain holds at between one and two hundredths, putting a number on the letter’s size as a third of the height half. If instead one half falls towards nought, the counter census’s gain was one direction’s all along, and four positions had hidden which.

Two halves of a noisy number #

The habit here is about splitting a small effect.

The counter census found a gain of four hundredths with a scatter of three, and the natural next step was to find which of its two causes carried it. The split worked as arithmetic: two halves, each a little over two hundredths, adding to the whole. As measurement it did not, because each half inherited the whole’s scatter and had half its size.

The failure mode is to split an effect into parts and read each part with the confidence the whole had. The test is to carry the error bar through the split: a decomposition that adds up is a statement about the whole, and each part is a statement only as far as its own error allows.