Four ways to write a weave down

This site has one essay about what a weave matrix decides and one about what it cannot say. Both take point paper as the object and ask what it is good for.

There are other notations, and they are the ones actually used. A mill order says “2/2 twill”. A weaving draft is four blocks on a page — threading, tie-up, treadling, drawdown. A fabric specification says the longest float is three. Each of those is a way of writing a weave down, each has an image — the set of cloths it can express — and the images are computable against a complete catalogue.

The comparison turns out to separate them on a property that is not size. Three of the four are lossy, in three completely different ways, and only one is lossy in the way people assume notations are lossy.

A fraction name is a strong condition #

“2/2 twill” says: read up any end and there are two over and two under, and moving one end along slides the same reading by one pick.

Both halves are load-bearing, and the second is not in the name. The composition — the ordered list of run lengths summing to the repeat — is the fraction. The step is supplied by the reader from the words “twill” or “satin”, and the notation has no room for it.

So the image of the fraction notation is exactly the drafts of the form W[i][j] = c[i − s·j] for some column c and some step s: every column a rotation of one column. That is a strong condition. Plain, every twill, every satin and every sateen satisfy it by construction, because the rule is that the column slides. Nothing else does.

Counted over the four-by-four catalogue: five cloths of 426, which is 1.2 per cent. And there are only four names for the five, because three of the four names cover two cloths apiece.

The ambiguity is not an obscure one. The two cloths under “2/2” are the S and the Z twill, and this site has a whole essay on how the two differ: every quantity the matrix produces is identical for both, and what separates them is the yarn’s own handedness against the twill line. So the fraction notation fails to identify a cloth in exactly the place the trade already knows it fails, and has to add a letter to the name to patch.

A four-part draft is bounded rather than lossy #

The weaving draft proper is four blocks: which shaft each end is threaded on, which shafts each treadle lifts, which treadle is pressed on each pick, and the drawdown that results. It is a factorisation of the matrix — end j is up on pick i exactly when the shaft it is on is up on that pick — and multiplying it back out reproduces the draft exactly.

So it is not lossy at all. Given a four-part draft the cloth is determined, and given the cloth the draft can be constructed. What it has instead is a bound: a loom with s shafts can only realise a cloth with at most s distinct columns, and one with t treadles at most t distinct rows.

At four by four that bound is not a restriction, and this is worth stating because it is not what a reader expects. Every four-by-four matrix has at most four distinct columns and at most four distinct rows, so four shafts and four treadles reach the whole catalogue — all 426. Drop to three shafts and the reach falls to 110; drop to two and it falls to nine.

That symmetry is worth a sentence of its own. The catalogue does not distinguish warp from weft: transposing a draft gives a draft, and the two budgets therefore buy exactly the same amount. The machine that weaves it is not symmetric in any other respect — one sett is a reed and the other is a gear — and this is the one place where the two directions cost the same.

A float specification is a partition #

The thing a fabric is actually bought against is not a weave at all. It is a set of properties, and the structural one is usually a float limit: nothing longer than three, nothing longer than four.

That notation writes every cloth, in the sense that every cloth has a longest float. It identifies nothing, in the sense that it sorts the whole catalogue into three classes: one cloth with a longest float of one, eight with two, and 417 with three.

Four hundred and seventeen cloths under one heading is not a description; it is a filter. And it is the filter the trade uses, because the float is the number that decides lustre, snagging, abrasion, tear strength and how densely the cloth can be set — so a specification that names the float has named the thing that matters and has said nothing whatever about the construction.

The four failures are four different failures #

Setting them side by side is what the comparison is for.

Point paper writes every cloth and identifies it. It is complete and faithful, and it is why every argument on this site is a statement about a matrix.

A four-part draft is faithful and bounded. It writes what a machine of stated size can make, which is a set that grows with the machine and is the whole catalogue at this repeat.

A fraction name is class-restricted and ambiguous. It writes a strong-condition subset — 1.2 per cent — and does not identify its members.

A float specification is complete and partitioning. It writes everything and separates it into three classes, the largest holding 98 per cent of the catalogue.

Those are four different relations between a notation and its objects, and only the third is what “lossy notation” usually means. The interesting one is the fourth, because it is the notation with the most commercial weight: the number a fabric is bought against is the one that says least about what the fabric is.

The same loom counted a third way makes the comparison between the notations concrete rather than a matter of taste.

What was counted, and how #

Each notation’s image is computed over the same catalogue, and the catalogue is computed twice by independent routes and required to agree.

The fraction notation’s membership test is a search over steps rather than a pattern match: for each candidate step the whole matrix is checked against a rotation of its own first column, and a weave passes only if some step works for every column. The composition is then read round the column from its first change, so that the name does not depend on where the writing started — otherwise the same cloth would acquire different names from different drawings of it.

Names are counted over cloths rather than over drafts, because a name is a name for a cloth. That matters: at draft level the same fraction appears many times over, once for every place the writing could have started, and counting those would inflate the image by a factor of sixteen.

The loom’s image is computed by counting the distinct columns and distinct rows of every cloth in the catalogue and comparing both against the budgets, which is the same computation this site uses for a single draft’s shaft count.

Three assertions guard it, and each could fail. Most cloths must have no fraction name, which is the essay’s headline; a fraction name must fail to identify its cloth, which is the ambiguity; and a loom below full size must not reach the whole catalogue, which is what makes the bound a bound. That last one is the reason the table is drawn at three shafts rather than four: at four the row is 426 and the assertion would be vacuous.

What a float specification actually carries #

The float notation’s failure can be given a size rather than a description, and the size is startling enough to be worth the arithmetic.

Naming one cloth out of 426 takes a little under nine bits. A float specification’s three classes hold one cloth, eight cloths and 417 cloths, so the answer is almost always “three” — and a message that almost always says the same thing carries very little. Working the entropy of that partition out gives about a sixth of a bit.

So a float limit conveys under two per cent of what it would take to say which cloth a fabric is. That is not a criticism of the specification, which is not trying to identify a cloth; it is a measurement of how completely the trade’s structural specification and the trade’s structural vocabulary have come apart. The number a fabric is bought against is nearly independent of the construction it is bought as.

It also puts the fraction notation in perspective. A fraction name reaches 1.2 per cent of the catalogue and, within that reach, very nearly identifies its member — one name in four is ambiguous, which costs a fraction of a bit. The name that reaches almost nothing says almost everything about what it reaches; the specification that reaches everything says almost nothing about anything. Those are the two ways a notation can be nearly useless for identification, and the trade uses both, for different purposes, without either being a mistake.

Point paper is faithful because it is redundant #

The one complete and injective notation is also the wasteful one, and the waste is not incidental.

A four-by-four matrix is sixteen bits and there are 426 cloths, which need nine. So point paper spends nearly twice what the catalogue requires, and the surplus is spent on a specific thing: a cloth has many drawings, one for every place the writing could have started, and point paper keeps all of them.

That redundancy could be removed. Take each cloth’s lexicographically least writing over all translations and the result is complete, injective and shorter than what point paper uses — a canonical form, in the ordinary sense. Nobody uses one, and the reason is a third property the essay’s two axes do not measure.

A reader has to be able to look at the notation and see the cloth. Point paper works because the drawing is a picture of the interlacing at the scale the interlacing happens: an end is a column, a pick is a row, over is filled. A canonical form is a picture of a cloth that has been rotated to a position nobody wove it in, and a weaver reading it would have to undo the rotation before recognising anything.

So the trade’s notations are optimising something none of the counts above can see. The fraction name is short and speakable. The four-part draft is a set of instructions for a machine and reads in the order the work is done. Point paper is a scale drawing. Every one of them is redundant or restricted in exchange for being legible, and the notation that would be optimal on the two measured axes is the one nobody would be able to read.

Why a notation’s image is worth measuring at all #

A notation is usually judged on whether it is convenient, and convenience is a real property that nothing here measures.

What the image measures is something else: whether a question can be asked in the notation. A mill that describes its cloths by fraction names has a catalogue of five cloths at this repeat and cannot record a sixth, not because the sixth is hard to write but because there is nothing to write. That is a property of the vocabulary rather than of the mill, and it is invisible from inside — the cloths that cannot be named are also the cloths that never come up, because they never come up in a conversation conducted in names.

Where the model stops #

Everything here is at one repeat size. The fraction notation’s image at eight by eight is larger absolutely and much smaller as a fraction, because the shift condition is a strong one and the catalogue grows fast. The loom’s bound goes the other way and becomes a real restriction immediately: at eight by eight a four-shaft loom reaches a small corner.

The four-part draft is treated as a factorisation and not as a document. A real weaving draft carries a threading order — straight, pointed, broken — which is information about how to thread rather than about the cloth, and two drafts with the same image can be very different to work from. None of that is here.

A float specification is not the whole of a specification. A real one carries weight, sett, count, finish and a dozen test results, and the essay’s claim is only about its structural content — that the number naming the construction names one class of 417.

And “identifies” is used in one exact sense: two cloths with the same expression in a notation are indistinguishable to it. Whether a reader with trade knowledge could tell them apart is a different question, and in the S-and-Z case they obviously can, because the trade added a letter.

The generalisation #

The useful move is to stop asking whether a notation is good and start asking two separate questions: what is its image, and is it injective on that image?

Those are independent, and the four rows here demonstrate all the combinations that matter. Complete and injective is point paper. Complete and non-injective is the float specification. Restricted and injective would be a notation that writes only twills and writes each one exactly once — which is what the fraction notation would be if the step were in the name. Restricted and non-injective is what it actually is.

The shape worth carrying is that the two failures feel the same from inside a practice and are completely different in consequence. A restricted notation makes some objects unsayable, which is discovered the first time somebody wants one. A non-injective notation makes some distinctions unsayable, which is never discovered at all — until a fabric comes back from a mill with the twill running the wrong way.

Who found it, and when #

The fraction notation is old and universal, and the S and Z letters were added to it precisely because it was found to be ambiguous. The four-part draft is the standard European weaving notation and is at least four hundred years old in something like its present form; its bound has always been understood, because the bound is the loom.

What is not usually done is measuring the images against a complete catalogue, because until there is a catalogue there is nothing to measure against. The 426 is this site’s own, and every number here is a count over it.

The one number worth pausing on is the four names. A notation with an image of five and a vocabulary of four is not economising; it is at the point where its own coarseness has caught up with it, and the patch — a letter for the handedness — is the notation admitting that the step it left out was load-bearing.

Where the ladder goes next #

This opens a notation ladder in the cloth field, and what it leaves open is the notations that are not about a single repeat: the profile draft that carries a block design, the lift plan a dobby stores, the card a jacquard reads. Each of those is a notation for something larger than a weave, with its own image and its own faithfulness, and the interesting question is which of them can express a cloth that the others cannot.

Sideways, the object all four notations are notations for is the matrix, and what that encoding itself decides, hides and cannot express is set out in one place. The catalogue every count here is taken over is every cloth at four by four, and the reason the fraction notation reaches so little is the same shift condition that makes the manuals’ derivations reach nine.