A specification can name a cloth that is not there

Three notations for something larger than a weave have now been measured, and all three write the same kind of object. A point-paper draft, a profile draft and a threading with a lifting plan each denote a pattern that tiles the plane, and the three questions asked of them — what can each express, how much does each express twice, and does either image contain the other — are questions about sets of drafts.

A mill works from none of them. The document on the bench carries a yarn count, a warp sett, a weft sett and a weave, quoted together, and it is what a cloth is ordered against, costed against and rejected against. It is a notation by any reasonable use of the word: a finite string that picks out a fabric.

It is also the first one in this account whose image has holes in it.

A draft cannot fail to exist and a specification can #

The difference is worth stating precisely, because it is the whole of what makes the fourth notation a different kind of object.

A four-by-four matrix of ones and zeros written down is a draft. If every end and every pick reaches both faces, it is one of the 22,874 in this account’s sweep, and there is a cloth it describes. Nothing about the notation can produce a string that denotes nothing. The same is true of a profile, whose cells are weaves, and of a threading, whose cells are shaft numbers: every string in the language names something.

A specification’s five fields are not independent, and the document does not say so. A yarn has a diameter; threads cannot be set closer than their own diameters plus the room their crossings take; and the room the crossings take depends on the weave. So the count, the two setts and the weave constrain one another, and a specification that violates the constraint is a string in the language that denotes nothing.

That is not an exotic failure. It is most of the language.

How much of the grid is empty #

The grid is the trade’s own working range rather than a sweep of everything: ten counts from 10 to 100 tex, which runs from a fine voile to a heavy canvas, and sixteen setts from 10 to 70 threads a centimetre in each direction, which runs from a scrim to a poplin. Each specification also names a weave, and the weave is drawn from the 426 cloths at four by four.

Taking the setts and the count alone gives 2,560 specifications, and asking of each how many of the 426 cloths can carry it gives three answers rather than two:

Four specifications in five name nothing. They are not unusual documents — they are an ordinary count with a sett that its own yarn will not reach, which is the commonest mistake in specifying a fabric and is invisible on the page because no field of the document is out of range on its own.

The second row is the surprise. In 223 of the grid’s cells the weave field carries no information whatever, because every cloth in the catalogue satisfies the other four numbers. A document quoting a plain weave and a document quoting a five-shaft sateen describe the same admissible set there, and the weave is a statement about the fabric’s face rather than about whether it exists.

So the weave field decides something in 240 cells, nine and a half per cent of the grid. Everything that follows is about those.

The empty region belongs to the coarse yarns, and so does the silence #

Splitting the grid by count says which specifications the notation is most dangerous for, and it is not the one intuition offers.

At ten tex, 44 per cent of the grid names no cloth. At a hundred, 98 per cent does. The diameter goes as the root of the count, so a tenfold coarser yarn is 3.16 times thicker and its ceilings are 3.16 times lower, and the sixteen setts on the grid run off the top of the admissible range one after another.

The band behaves the same way and it matters more. At ten tex the weave field decides in 63 cells and at a hundred tex in 3. A fine cloth is specified in a region where the choice of weave is a real constraint on the setts; a coarse one is specified in a region where either every weave works or none does, and the weave field is then a statement about appearance with no weavability content at all.

Which inverts the usual advice. A specifier who has learnt to check the weave against the sett has learnt a habit that is most useful on fine yarns and nearly vacuous on coarse ones — and the coarse specifications are exactly the ones where the empty region is almost the whole grid, so the check that matters there is the one nobody is watching for.

The band is an L, and that is a fact about the constraint #

The pale and the solid regions in both figures meet along two edges rather than along a curve, and the shape is the constraint itself rather than an artefact of the grid.

The jamming argument says a repeat of E ends must occupy E thread diameters plus one more diameter for every weft crossing the picks make across it, because a crossing thread has to pass through. So

$$\text{ends per unit width} \le \frac{E}{d\,(E + c_\text{pick})}$$

where $c_\text{pick}$ is the number of times an average pick changes face across the repeat. The weft sett is bounded by the same expression with the two systems exchanged, and there $c_\text{end}$ appears — the number of times an average end changes face down the repeat.

Those are two different counts. The warp sett’s ceiling is set by what the picks do and the weft sett’s ceiling by what the ends do, and a weave can be generous in one and mean in the other. The admissible region is therefore a rectangle: a ceiling on the warp sett and an independent ceiling on the weft sett, and no interaction between them at all.

Which means the catalogue’s position in this argument is a point in a plane, not a value on a line.

The number the trade quotes is the wrong shape #

Firmness is the interlacing count per intersection: the mean of the two crossing counts, up to the repeat size. It is the number every account of setting uses, this account’s included, and it is what decides how densely a cloth can be set.

Over the grid taken as a whole it is exactly right.

Eight firmness values, eight shares, monotone, a clean factor of two from end to end. If the question is how much of the space of specifications does a weave admit, firmness answers it and answers it in one number.

The question a specification actually asks is different. It is not how much a weave admits; it is whether this weave admits this specification. And there the mean of two constraints is the wrong object, because the two constraints bind in different directions and a mean cannot tell which.

The catalogue’s nineteen positions collapse onto eight firmness values, so eleven distinctions are thrown away before the question is asked. The longest float — the other number a specification usually carries — does no better: the 426 cloths have nine distinct pairs of warp and weft float maxima, and the single longest float is a maximum over both systems, which discards the asymmetry in exactly the same way.

Both of them get the band wrong, and one gets all of it wrong #

The check is mechanical. In each of the 240 decisive cells, take every ordered pair of cloths in the catalogue and ask whether the ranking puts an unweavable cloth below a weavable one — which is a ranking saying, of two cloths, that the one that cannot be woven is the safer choice.

The longest float mis-ranks a pair in all 240 cells. That is not a marginal failure; a float specification is simply not about weavability, and reading it as though it were is reading a statement about the face as a statement about the construction.

Firmness mis-ranks in 225 of the 240, which is 93.8 per cent. The 15 it gets right are cells in which the admissible set happens to be a firmness class, and nothing in the specification says which cells those are.

The split by balance says where the failure comes from. Of the 240 decisive cells, 20 have equal setts and 220 do not. Firmness is wrong in 9 of the 20 balanced cells — 45 per cent — and in 216 of the 220 unbalanced ones, which is 98.2 per cent.

So the mechanism is exactly what the plane suggested. A balanced specification asks one question of a mean and gets a defensible answer about half the time; an unbalanced one asks two questions and gets one answer, and is wrong almost always. And nearly every fabric anybody specifies is unbalanced, because a warp is set by a reed and a weft by a gear and the loom cannot choose the two at the same granularity.

Two cloths that tie on everything the document carries #

The failure is not a matter of close calls. The catalogue contains pairs of cloths whose crossing counts are transposes of one another, and a transpose forces every scalar the trade quotes to agree exactly.

Both cloths are three-shaft. Both have a longest float of three. Both have a firmness of exactly 0.5625, which is not a rounding — the transpose makes the sum identical term by term. A specification naming either of them carries the same numbers in every field a mill would check.

And they are separated by setts eight per cent apart. The least unbalanced specification in the whole grid that tells them apart is 10 tex at 54 ends and 50 picks a centimetre: the left cloth’s ceilings are 56.4 and 52.1 threads a centimetre and the right cloth’s are 52.1 and 56.4, so the warp sett of 54 sits inside one and outside the other.

Eight per cent is inside the tolerance most specifications are written with. The two cloths are not exotic and the setts are not extreme; the document simply does not carry the field that would decide.

Why the fourth notation fails differently from the first three #

The three notations for a repeat failed by being small. A fraction name writes few cloths; a profile writes 306 of 22,874; a three-shaft harness writes 5,282. Each has an image, the image is a proper subset, and the failure mode is that a wanted cloth is outside it — which a designer discovers immediately, because the notation will not accept what is being written.

A specification fails by being too large. Its language admits strings that denote nothing, and it admits them silently: every field is in range, the arithmetic of the other fields is not performed, and the document reads exactly like a document for a cloth that exists. There is no analogue of this in the first three, and the reason is structural rather than a matter of care.

A notation for a repeat is a notation for a combinatorial object, and combinatorial objects either are or are not in the language. A specification names a physical object, and a physical object has to satisfy an inequality that no amount of reading the notation will evaluate. The matrix cannot say what is not in it, and a specification’s fields cannot say what their own conjunction requires.

This is the same shape as the observation that a thread count is not a quality — a number in a document standing in for a property it does not determine — but it is one level worse. A thread count is a true statement about a cloth that exists; an over-set specification is a true-looking statement about a cloth that does not.

What a document would have to carry to close it #

The repair is not a new instrument and it is not more decimal places. The specification needs the two crossing counts rather than one firmness, and both are computable from the weave it already names.

$c_\text{pick}$ and $c_\text{end}$ are two small rational numbers — over the whole four-by-four catalogue they take values between 2 and 4 — and the check is two inequalities that a pocket calculator settles. No mill would find them burdensome; nobody quotes them because nobody has had a reason to separate them, and the reason is that the mean is the number that appears in every setting rule ever published.

Which is the honest summary of the ranking result: firmness is the right number for the question setting rules ask and the wrong number for the question a specification asks, and the two questions look identical until the grid is drawn.

What was counted, and how #

The catalogue is this account’s own, unchanged: the 426 cloths at four by four, taken as equivalence classes of the 22,874 interlacing drafts under the operations that leave a cloth the same cloth.

The yarn diameter is Peirce’s, computed from the count and a packing factor of 0.6 for cotton — the same function the cover factor is built on, so the ceilings here and the covers there are one model rather than two.

The ceilings are this account’s maximum-sett rule, which is the jamming argument stated as a width per repeat rather than as a spacing per thread. It is the same inequality; the per-repeat form is the one that makes the two crossing counts appear separately, and the per-thread form is the one that hides them in a mean.

The rankings are checked over every ordered pair, 426 × 426 in each decisive cell, rather than over a sample or over the named weaves. A ranking that is wrong about one pair in one cell is wrong, and sampling would have found the firmness failure and missed the float failure, which is total.

The crossing pair’s zero is required rather than reported. It has to be zero, because the pair is the inequality; a single violation would mean the inequality had been transcribed wrongly, and the check exists to catch that rather than to establish a result.

What the grid cannot show #

The catalogue is four by four and a mill’s weaves are not. A five-end sateen has a longest float of four and crossing counts below anything here, so the real range of $c_\text{pick}$ and $c_\text{end}$ is wider than 2 to 4 and the empty region at a given sett is correspondingly smaller. The shape of the argument does not depend on the repeat size — the two counts are defined for any weave — but every proportion on this page is a proportion of this catalogue.

The jamming limit is geometric and a real cloth can be beaten past it. Threads flatten, and a flattened yarn covers more at the same sett because it is wider, which moves the ceiling. So the pale region is not a region of impossible fabrics; it is a region of fabrics that are not possible in the round-section model the specification’s own arithmetic uses. That is the right claim for a document to be measured against, and it is a narrower one than “cannot be woven”.

And the grid is uniform where the trade is not. Weighting each cell equally is what makes 81.9 per cent a statement about the language rather than about the market; mills specify inside the admissible region most of the time because they specify by precedent. The measurement is of what the notation permits, not of what is ordered.

Who found it, and when #

That a sett is bounded by the yarn and the weave together is old and is in every manual, usually as a table of maximum setts by weave with the arithmetic left out. Ashenhurst’s setting rules of the 1880s are the canonical form, and Brierley’s of the 1930s are the one that puts a weave-dependent exponent on it; both are rules for a balanced cloth, and both produce a single number per weave.

The separation of the two crossing counts is this account’s, and it is not a discovery so much as a refusal to average: the per-repeat form of the jamming width has the two counts in it, and every published form of the rule collapses them before the question is asked. What is new here is measuring what the collapse costs, which needs a complete catalogue to measure over.

The result that both quoted numbers mis-rank is the kind that only appears when the band is isolated. Over the grid as a whole firmness is monotone and looks unimprovable; the 9.4 per cent of cells where the weave field decides anything are the only place the failure is visible, and they are invisible to anyone who has not drawn the grid.

Still open: what a fourth notation’s redundancy is #

Each of the first three notations had a redundancy that turned out to mean something — the harness’s was the weaver’s freedom to balance the shafts, the profile’s was where each weave starts.

A specification’s redundancy has not been measured here at all, and it is the harder half. Two specifications that name different counts and different setts can describe cloths a customer would not distinguish, because a weight fixes the fibre and not the drape and several of a specification’s fields trade against one another at constant handle. The equivalence that matters is therefore not “denotes the same fabric” but “denotes a fabric that would pass the same inspection”, and that relation has no arithmetic in this account yet.

What can be said is which direction it runs. The image has holes and the redundancy is large, so the fourth notation is both partial and coarse — it names less than it appears to and distinguishes less than it appears to, and the two failures are independent.