A junction gathers by the width of its stem

A hairline spills its error onto the paper followed the error of sharpening scanned type on its stored values, rather than on light, from text sizes to display sizes. It found that black type’s serifs hold less than their share of the seen error at every size, whatever their weight, and that the corners which do gather are the junctions — the inside angles where a bar meets a stem. Pooled over four sub-pixel positions, the junctions of four unserifed capitals climbed from ×1.05 of their share of the outline at 12 point to ×1.64 at 48.

The explanation it gave was the one sharpened type errs on its dark side had given for the same climb over a shorter range. The difference the eye keeps lies on the darker side of an edge. An inside angle has dark on three sides, so it gathers in proportion to how much dark falls inside the eye’s pool around it. At text sizes a stem is a few pixels wide and the pool reaches past it onto paper; at display sizes the stem is wider than the pool and the angle is full of ink.

If that is the whole mechanism, then the point size is standing in for something simpler: the width of the stem in pixels. Size is only one of the two things that set that width; the other is the letter’s weight. A bold capital at 18 point and a regular one at 24 have stems of nearly the same width, and the mechanism says their junctions should hold nearly the same share. The prediction the earlier essay wrote down was that letters of three weights fall on one curve against stem width, rising from about one at a stem of three pixels to about ×1.7 at twenty.

A later finding corrects the light weight. The light letters’ H was drawn with its crossbar starting one spacing in, which on a stem six cells thick left the crossbar floating four cells clear of both stems, with no junctions of its own. A closed counter adds a little to a junction found it and joined the crossbar. The regular and bold letters are unchanged. The light letter’s junctions now hold ×1.01, ×0.98, ×1.16, ×1.30 and ×1.41 of their share at 12, 18, 24, 30 and 48 point, where this essay says ×0.99, ×0.98, ×1.19, ×1.30 and ×1.39, and its stroke ends move by up to three and a half hundredths. The weights at matched stem widths agree within an eighth rather than a tenth — the light letter at 24 point against the bold at 12 now differ by 0.12 — the stem’s width now ranks the junctions at 0.98 and the point size at 0.88, and the heavier weight still gathers more in five pairs of six. The numbers below are the floating crossbar’s.

The stem’s width orders the junctions #

Over a light, a regular and a bold weight at five sizes each, the junctions’ share of the seen error ranks with the stem’s width in pixels at 0.97 and with the point size at 0.85. Wherever two weights have stems within six tenths of a pixel of each other, their junctions’ shares differ by under a tenth. The curve runs from ×0.99 at a stem of 3.2 pixels to ×1.80 at 26.5, close to the prediction. What the width leaves over is small and has a sign: at the same stem width, the heavier letter’s junctions hold the larger share in five pairs of six.

• The prediction holds. Placed by stem width, fifteen letters of three weights make one rising band; placed by size, they make three curves.
• Below about five pixels a junction holds no more than its share. The light weight at 12 and 18 point sits at ×0.99 and ×0.98, the regular weight at 12 point at ×1.05.
• Above it the share climbs steadily, by about five hundredths a pixel up to thirteen pixels and more slowly after.
• The stroke ends fold onto one curve too, falling as the junctions rise, so the width governs both kinds of corner the unserifed letters have.
• The residual belongs to the weight, and the likeliest place it comes from is the paper between the strokes, which a heavier letter has less of.

One letter at three weights #

The hairline census drew its capitals — E, H, L and T, the letters a serif moves the error and adds little began with — on a grid with ten cells to a stem, one stem to a unit of spacing, four units across and six down. Weight is varied here by keeping the spacing and changing the stem: a stroke still starts at the same unit, but it is six cells thick for the light letter, ten for the regular and fourteen for the bold. So the bars and stems of the three letters stand in the same places, and what changes between them is how much of the letter is ink.

The regular weight is the earlier census’s sans letters cell for cell, and the check is run before anything else is read: at 24 point its junctions hold ×1.3597 of their share, which is the earlier number to the last digit. That matters because the whole comparison rests on the regular weight standing for the curve the earlier essay drew, so that the two new weights are read as departures from something already measured rather than as a second, unconnected census.

A design with fixed spacing and a changing stem is not how a type family is drawn. A real bold widens its letters as it thickens them, so that its counters — the paper enclosed by the strokes — do not close up. Here the counters do close: the gap between the E’s arms is twenty-four cells in the light letter, twenty in the regular and sixteen in the bold. That is a limitation for reading these letters as a family, and it turns out to be useful for reading the residual, because it means the weights differ in their counters as well as their stems, and the residual can be asked which of the two it follows.

Three curves against size #

Against the point size, the three weights draw three curves, stacked in order of weight. At 12 point the light letter’s junctions hold ×0.99 of their share, the regular letter’s ×1.05 and the bold letter’s ×1.28. At 24 point the three are ×1.19, ×1.36 and ×1.48, and at 48 point ×1.39, ×1.64 and ×1.80. Every curve rises at every step, which is the earlier finding at each weight separately: a larger letter’s junctions gather more.

What the stacking says is that a size is not a property a junction can gather by on its own. A bold letter at 12 point already holds more than the light letter does at 24, and the bold at 24 holds more than the light at 48. If the reader of a proof wanted to know where a sharpening on stored values would show, the size of the type would not answer it without the weight.

The same numbers can also be read down a column. At a fixed size, the heavier weight always gathers more, and the step from light to bold is large: three tenths to four tenths of the outline share, depending on size. That is the same size of effect as doubling the point size. So a weight change is not a small correction to a size effect; the two are the same kind of change.

Where the stems are the same width #

The stem’s width in pixels is its share of the cap height times the cap height. The cap height is seven tenths of the point size, which at 300 dots an inch is about 35 pixels at 12 point and 140 at 48, and the stem’s share of it is six, ten or fourteen parts in sixty-six, seventy or seventy-four. So the three weights’ stem widths overlap across sizes: the light letter at 30 point has a stem of 7.9 pixels and the regular at 18 point one of 7.5; the regular at 24 point has 10.0 and the bold at 18 point 9.9; the light at 48 point, the regular at 30 and the bold at 24 have stems of 12.7, 12.5 and 13.2.

Those overlaps are the test. Six pairs of letters in the census have stems within six tenths of a pixel of each other while differing in weight, and every one of them differs in size as well, by between a factor of 1.3 and a factor of two. If size itself mattered, over and above the width it gives the stem, those pairs would disagree by roughly what the stacking in the previous figure showed — three to four tenths. If only the width mattered, they would agree.

One band against width #

Placed by the stem’s width, the three curves fold onto one band. The six pairs agree to within a tenth: the light letter at 18 point and the regular at 12, with stems of 4.8 and 5.0 pixels, hold ×0.98 and ×1.05; the light at 24 and the bold at 12, 6.4 and 6.6 pixels, ×1.19 and ×1.28; the light at 30 and the regular at 18, 7.9 and 7.5 pixels, ×1.30 and ×1.27; the regular at 24 and the bold at 18, 10.0 and 9.9 pixels, ×1.36 and ×1.39; the light at 48 and the regular at 30, 12.7 and 12.5 pixels, ×1.39 and ×1.44; and the light at 48 and the bold at 24, 12.7 and 13.2 pixels, ×1.39 and ×1.48. Against the three to four tenths that separated the weights at a fixed size, that is agreement.

Ranked over all fifteen letters, the junctions’ share follows the stem’s width with a Spearman correlation of 0.97 and the point size with 0.85. The second number is not small, because the stem’s width grows with size within each weight; what matters is that the width does better, and that it does so by folding the weights together rather than by any trick of the ranking.

The shape of the band is the prediction’s. Below about five pixels the junctions hold their share and no more: the light letter at 12 point, with a stem of 3.2 pixels, sits at ×0.99. The share then rises by about five hundredths a pixel, to ×1.4 near thirteen pixels, and flattens a little after, to ×1.64 at twenty pixels and ×1.80 at 26.5. The prediction was about one at three pixels and about ×1.7 at twenty. It was drawn from the regular weight alone, and the other two weights land on it.

The flattening is where the mechanism says it should be. The eye’s pool at forty centimetres, as the eye counts a corner’s error, not its peak set it out, is a few pixels across at 300 dots an inch, set by where contrast sensitivity falls off for the lightness channel. Once a stem is several pools wide, a wider stem adds dark to the angle only at the pool’s edge, where the pool weights little. The first few pixels of stem width buy the angle most of its ink; the next ten buy much less.

The stroke ends fall on one curve as well #

An unserifed letter has two kinds of corner, and the stroke ends are the other one. Where a bar or a stem simply stops, the corner is outer: ink on one side of it, paper on three. In black type the dark side of that corner is the stroke itself, and the halo the sharpening leaves on the paper side is where the eye’s lightness scale is shallow — the slope the straight piece under the cube root set out — so a stroke end holds less than its share. The earlier census found them at ×0.76 at 12 point and ×0.52 at 48, falling as the junctions rose.

Against the stem’s width, the three weights’ stroke ends fold onto one curve that falls. The light letter at 12 point, with a stem of 3.2 pixels, gives its stroke ends ×0.95 of their share — nearly a fair share, because a stroke three pixels wide is hardly wider than the pool, and the halo on its paper side and the error on its ink side are counted together. By five pixels the share is ×0.74 to ×0.76, by ten pixels ×0.60 to ×0.63, and past twenty pixels ×0.51 to ×0.52. At the matched widths the weights agree even more closely than the junctions do: the light letter at 30 point and the regular at 18 both hold ×0.66, and the light at 48, the regular at 30 and the bold at 24 hold ×0.55, ×0.56 and ×0.58.

The two curves are one mechanism read from two sides. A junction is an inside angle whose dark side grows with the stem; a stroke end is an outer corner whose dark side is the stem’s own end, and as the stem widens the end becomes a long edge with two short corners, each holding less of the error the eye pools around it. What moves a letter’s error from its ends to its junctions is the one width, and neither kind of corner shows the stacking by weight the size plot showed.

The stroke ends carry a residual of their own, and it has the same sign. At the matched widths the heavier weight’s ends hold the same share or up to three hundredths more, never less, which is the direction the junctions lean too. The effect is smaller at the ends than at the junctions, which is what would be expected if what the heavier letter adds is error near the paper between its strokes, since the junctions face into that paper and the ends mostly face away from it.

What the width leaves over #

The agreement between weights is within a tenth, and it is not random. In five of the six pairs the heavier weight’s junctions hold the larger share, by three to nine hundredths. The one exception is the light letter at 30 point against the regular at 18, where the light letter is ahead by three hundredths. In that pair the light letter’s stem is the wider by 0.45 of a pixel, the largest such gap among the six, and at five hundredths a pixel the band itself would put the light letter about two hundredths ahead; the exception is the band’s own slope showing through. The other two pairs in which the lighter letter has the wider stem — by 0.07 of a pixel and by 0.23 — still have the heavier letter ahead, by three and five hundredths.

So there is a small effect of weight over and above width, and it has a sign. The two things that differ between weights at the same stem width are the size of the letter around the stem and the paper inside it. At the same stem width a light letter is larger, so its arms are longer and farther apart in pixels; a bold letter is smaller, with shorter arms and a narrower counter.

The census as drawn cannot say which of these the residual follows, because in these letters the two go together. It can say how large the residual is: under a tenth of the outline share, against a spread of eight tenths across the whole band. For a proofing decision, that is below the difference between sub-pixel positions the earlier essay found for a single rendering, and a sharpening test that uses stem width alone to predict where the error shows will be right to within it.

What this means for proofing a sharpening #

A test pattern for sharpening on stored values can be specified by stem width rather than by size and weight. A row of unserifed capitals whose stems are five, eight, twelve and twenty pixels wide at 300 dots an inch covers the range from no gathering at the junctions to about ×1.65, whatever face the job is set in. A heavy face at text size and a regular face at display size are the same test at the same width.

The band is stated for one resolution and one distance. At 300 dots an inch and forty centimetres, a stem’s width in pixels and its width on the page are the same statement, and the band can be read in either. At another resolution they part: the eye’s pool is fixed on the page by the viewing distance, while the unsharp mask’s radius is fixed in pixels, so a 600-dot device halves the halo against the pool as well as doubling the stem’s pixel count. Which of the two widths the band follows there is not settled by a census run at one resolution, and a test pattern for another device should be measured, not scaled.

The cure is unchanged by any of this. Sharpening in light removes the difference, as it did for every letter in the earlier censuses, and a halftone is a luminance object is why the lightness error is the one that shows. What the stem width buys is a way to say in advance where a sharpening on stored values will be seen before the job is proofed, from one number the type already has.

How the letters were set #

Each letter is a union of cells on a grid with ten cells a unit of spacing and a three-cell margin on each side. A stroke from unit a to unit b across fills cells from ten a to ten b plus the stem, and likewise down, so a one-unit stroke is exactly one stem thick and the regular weight, with a stem of ten, reproduces the earlier letters. The light and bold weights use stems of six and fourteen. Each letter is rendered at a cap height of seven tenths of the point size on a 300-dot page by exact area coverage, at offsets of 0, 0.25, 0.5 and 0.75 of a pixel along both axes, encoded as sRGB, and sharpened by an unsharp mask of radius 1.5 pixels and amount one, once on the stored values and once on light. The difference between the two is filtered by the three spatial channels of the model eye at forty centimetres, as in a pool with an edge, and read as CIEDE2000 per pixel. Each pixel’s error goes to the nearest point of the outline, each junction’s share of the error within two pixels of its corners is divided by those corners’ share of the outline’s length, and the four letters and four offsets are pooled by adding lengths and errors before dividing. The stem’s width in pixels is the stem’s share of the letter’s height times the cap height.

What this leaves out #

The weights are not a type family. A drawn bold opens its letters as it thickens their strokes; these letters keep their spacing, so the bold’s counters close. That is why the residual cannot be assigned, and a family with proportional counters might show a smaller residual, or none.

The letters are rectilinear and black on white. Rounds carry no junctions, and reversed type gathers at its outer corners instead, where the ground is the dark side; the stem’s width governs neither. The hairline census found reversed junctions holding a third of their share, and nothing here changes that.

The eye is a model fitted at threshold, with the caveats the eye keeps the lightness errors records. The band’s shape depends on the pool’s size; a different sensitivity curve would move where the band flattens without changing the fact that width, not weight or size, sets it.

And the pixel grid is pooled over four positions a quarter of a pixel apart, which the hairline census showed is necessary: one position moved a hairline tip’s share by a factor of 3.7. A single rendering of any one letter can sit well off the band for that reason alone, by more than the residual; the residual is read only because it keeps one sign across six pairs, pooled.

Still open: whether the counter or the letter’s size carries the residual #

The residual can be taken apart by varying what these letters tie together. Hold the stem at ten cells and change the spacing — eight, ten and fourteen cells a unit — so that the counters open and close while the stems stay the same, and set each at the sizes that bring the stem to the same widths in pixels. If the junctions’ share then moves with the counter, the residual is the counter: the halo of one arm reaching across narrow paper to the next, adding to an angle’s error what the stem alone does not. If it moves with the letter’s height instead, the residual is the pool seeing more of the rest of a small letter around each angle.

The prediction is the counter. The stroke ends, which face away from the paper between the strokes, carry a smaller residual than the junctions, which face into it; if the counter is the cause, a letter whose counters close while its stem stays fixed should show its junctions gathering more than the stem’s width allows, and its stroke ends barely moving. If both kinds of corner move together, the cause is the letter’s size.

A variable a result is stated in is not always the variable it depends on #

The earlier finding was correct as stated: the junctions of black type gather more as the letter grows. It was stated in point sizes because the census varied the size. It would have stayed stated that way, and been applied that way, if the mechanism given for it had not named a quantity the size only carried.

The failure mode is to report a trend against the parameter that was stepped, and to let the report outlive the reason. Size and stem width moved together in every letter the earlier census drew, so nothing in that census could tell them apart, and a proofing rule written from it — look at the junctions above 30 point — would have been wrong for every bold face at text size. The test that tells them apart is cheap once the mechanism is written down: find a second way to reach the same value of the quantity the mechanism names, and see whether the result follows the quantity or the parameter. Here it followed the quantity, and what it left over was small enough to measure only because the agreement was otherwise so close.