Textile Structure
how cloth is put together
Series

Notation — the series

9 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. What four notations can say. Four ways of writing a weave down, with how many of the 426 four-by-four cloths each can express and whether expressing one identifies it: point paper, 426, identifies; a four-part draft on 3 shafts and 4 treadles, 110, identifies; a fraction name, 5, does not identify; a longest-float specification, 426, does not identify. A longest-float specification names every cloth and separates them into only 3 classes.

    Four ways to write a weave down

    A weave's name in the trade is a pair of numbers — 2/2, 3/1, five-end satin — and the notation is so universal that it is easy to forget it is a notation. It has an image and the image is computable: of the 426 cloths at four by four, five have a name of that kind, and there are four names for the five.

    part 1 · cloth
  2. What a profile draft can reach at 4 by 4. The share of the 22,874 interlacing 4-by-4 drafts that a profile draft can express, at two block sizes. A profile is a grid of blocks each carrying a figure weave or a ground weave, so its image is every draft reachable by any choice of the two weaves and any assignment — which is enumerated here rather than argued: 4,096 combinations at the larger block, and the distinct results counted. With two-by-two blocks it reaches 306 drafts, which is 1.34 per cent. With one-by-one blocks the profile is the draft and it reaches all of them, which is the control. What the bars cannot show is that the reachable drafts are the useful ones: every figured cloth ever woven is in the small set, and the notation is narrow because designs are.

    A profile draft is a notation whose alphabet is weaves

    The rung below measured four notations for a single weave and left open the notations for something larger. A profile draft is the first of them: a grid of blocks, each carrying a figure weave or a ground weave. Its image is enumerable and it is tiny — every pair of two-by-two weaves against every assignment of two-by-two blocks reaches 306 of the 22,874 interlacing four-by-four drafts, which is 1.34 per cent. And the 306 are the ones anybody weaves.

    part 2 · cloth
  3. The draft for 2/2 twill, as a loom holds it. The 2/2 twill written the way a weaver writes it: the threading above, saying which shaft each end hangs on; the lifting plan to the right, saying which shafts rise on each pick; and the cloth below, which is not copied from the weave but produced by running those two against one another and then checked against it.

    A lifting plan says nothing without a threading

    The second of the notations for something larger than a weave is a pair, not a notation: a threading and a lifting plan, and neither alone expresses anything. The pair's image is exactly the drafts with no more distinct columns than there are shafts — 98 at two shafts, 5,282 at three, all 22,874 at four — and it is not nested with the profile draft's in either direction. The profile reaches 192 drafts that need all four shafts, and misses 64 of the 98 a two-shaft loom weaves.

    part 3 · cloth
  4. Which specifications a 30 tex yarn can be woven into. Warp sett across against weft sett down, in threads a centimetre, for a 30 tex cotton yarn. Each cell is one specification. 220 are left empty because no cloth in the catalogue can be woven at those two setts; 16 are filled and outlined in the warp's colour because every one of the 426 can; and 20 are filled and outlined in the float's colour because some can and some cannot. The admissible region is a rectangle because the warp sett and the weft sett are bounded by two different counts.

    A specification can name a cloth that is not there

    The three notations measured so far write drafts, and every draft is a cloth somebody could weave. A mill works from none of them: it works from a count, two setts and a weave quoted together, and that is the first notation whose image has holes in it. Of 2,560 specifications across the trade's own working range, 2,097 name no cloth at all — and in the 240 where the weave field decides anything, both numbers the trade quotes to decide it rank the catalogue wrongly.

    part 4 · cloth
  5. What one wrong symbol reaches. Five kinds of symbol, with how many of them a document holds and how many intersections of a 1800-by-1800 piece one of them decides. A threading digit reaches 810,000; a jacquard card's hole reaches one. The product of the two columns is 3,240,000 in every row, because each set of symbols decides every intersection of the piece exactly once.

    The shortest notation has the longest mistakes

    Four essays of this account have asked what each notation can express. None has asked what happens when one is written down wrong, and the answer is an identity rather than a tendency: a notation's symbols partition the cloth, so the count of symbols times the reach of one is the piece, in every notation, with nothing to trade. One threading digit decides 810,000 intersections of a square metre and one of a jacquard's holes decides one — and only the threading can be wrong while the cloth is right.

    part 5 · cloth
  6. One wrong threading digit against two. What becomes of a threading when one digit is wrong and when two are, over every draft in the four-by-four sweep: one digit wrong, 252,968 cases, 45.0 per cent refused by the drawdown, 54.8 per cent a different cloth, 0.228 per cent the same cloth; two digits wrong, 1,074,972 cases, 59.1 per cent refused by the drawdown, 39.6 per cent a different cloth, 1.285 per cent the same cloth.

    A second wrong digit is silent only by cancelling the first

    One wrong threading digit leaves the cloth exactly as it was 576 times in 252,968, all on three-shaft drafts. The obvious guess about two was that silence would become the rule, since two changed digits can swap two shafts and swapping shafts is the harness's own freedom. Counted over all 1,074,972 pairs, two wrong digits are silent 1.29 per cent of the time — five and a half times as often, on fifteen times as many cloths — and never by adding one silent slip to another. Every silent pair is two audible errors cancelling, and at two errors point paper and lifting plans, which can never be wrong silently by one symbol, are silent almost as often.

    part 6 · cloth
  7. Silent threading errors under three lifting plans. For every four-shaft threading of eight ends, under three lifting plans: 2/2 twill, 36,320 of 40,824 threadings can be mis-threaded silently by two digits, 0.97% of double errors and 0.20% of single errors are silent; 1/3 twill, 29,728 of 40,824 threadings can be mis-threaded silently by two digits, 0.55% of double errors and 0.20% of single errors are silent; unrelated rows, 864 of 40,824 threadings can be mis-threaded silently by two digits, 0.0084% of double errors and 0.0000% of single errors are silent. A twill's plan slides one row a pick at a time, so its shafts are interchangeable by a slide along the picks, which is a writing of the same cloth. What the chart cannot show is a plan with more shafts.

    Silence lives in the lifting plan

    Two wrong threading digits leave a four-end cloth exactly as it was 1.3 times in a hundred, and the question left was whether a longer repeat makes that rarer or commoner. At eight ends the answer depends on something the question did not name. With a lifting plan of unrelated rows, silence all but vanishes; with a twill's plan — one row slid a pick at a time — nine four-shaft threadings in ten can be silently mis-threaded, and even single errors are silent one time in five hundred. The repeat hardly matters. The plan's symmetry does.

    part 7 · cloth
  8. Which lifting plan exposes a threading error another plan hides. For every four-shaft threading of eight ends and every double threading error silent under the row's lifting plan, the share that the column's plan exposes — weaving a cloth that is caught or visibly different — when the same threading is re-pegged. Silent under the 2/2 twill (99,872 pairs): 2/2 other way 0%, 1/3 twill 44%, 3/1 twill 44%, broken 70%, point 34%, plain 0%, hopsack 34%, unrelated 99%. Silent under the 1/3 twill (56,352 pairs): 2/2 twill 0%, 2/2 other way 0%, 3/1 twill 0%, broken 76%, point 50%, plain 0%, hopsack 47%, unrelated 98%. Silent under the broken twill (29,888 pairs): 2/2 twill 0%, 2/2 other way 0%, 1/3 twill 55%, 3/1 twill 55%, point 0%, plain 0%, hopsack 53%, unrelated 97%. Silent under the point twill (66,208 pairs): 2/2 twill 0%, 2/2 other way 0%, 1/3 twill 57%, 3/1 twill 57%, broken 55%, plain 0%, hopsack 47%, unrelated 99%. A zero means the column's plan hides every error the row's does; the plan of unrelated rows exposes all but the 864 pairs that are the threading itself started elsewhere. What the table cannot show is which plans a given mill pegs one warp with.

    A proof plan has no slide in it

    A threading error that a 2/2 twill hides is invisible in the cloth and still wrong on the loom, and it shows only when the loom is re-pegged. Re-weaving every silent error under eight other plans says which do the showing. The twill run the other way and plain weave never expose one. A 1/3 twill exposes 44 per cent, a broken twill 70, and a plan of eight unrelated rows all but 864 — and those 864 are not errors at all, because each is the right threading started at another end. So the plans nest: whatever a broken or a point twill hides, the 2/2 hides too, and a threading proved under a 2/2 twill has passed the weakest test there is.

    part 8 · cloth
  9. Twelve eight-shaft lifting plans' silent errors, nested. The double threading errors silent under twelve eight-shaft lifting plans, over every threading that puts one end on each of eight shafts: 1/7 twill, 2/6 twill, 3/5 twill, 1/2/3/2 twill, satin, move 3, satin, move 5: 1,536; 4/4 twill: 1,888; 3/1/1/3 twill: 448; 2/1/1/4 twill: 256; broken 4/4 twill: 384; point 4/4 twill: 576; unrelated rows: 0. Plans in one box hide exactly the same errors. An arrow means every error silent under the first is silent under the second: 1/7 twill into 4/4 twill, 3/1/1/3 twill into 4/4 twill, 2/1/1/4 twill into 1/7 twill, 2/1/1/4 twill into 3/1/1/3 twill, broken 4/4 twill into point 4/4 twill, point 4/4 twill into 4/4 twill. The order is not total — the 3/1/1/3 twill and the satins each hide errors the other does not — but it has a top, the 4/4 twill, which hides every error any of the others hides. The plan of unrelated rows hides none.

    An eight-shaft error hides in the row's mirror

    At four shafts the lifting plans nest, with the 2/2 twill hiding every threading error the others hide. An eight-shaft loom has satins as well as twills, and their slides are different slides, so the order was expected to break. It does not. Over every threading that puts one end on each of eight shafts, the 4/4 twill hides every double error any of eleven other plans hides, and a satin hides exactly what a 1/7 twill hides, error for error. What decides it is not the slide. Every silent error is an exchange that turns the threading into a copy of itself, the slide accounts for one in six of them, and the rest hide in the row's mirror.

    part 9 · cloth

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