Silence lives in the lifting plan

A second wrong digit is silent only by cancelling the first walked every double threading error in the four-by-four catalogue and found 13,816 of 1,074,972 that leave the cloth exactly as it was meant — 1.29 per cent, five and a half times the single-error rate. It ended on the question the four-by-four walk cannot reach: at eight ends, with twenty-eight pairs of ends to exchange instead of six, does silence grow or shrink?

The answer this essay set out to give was “it shrinks”, and a first census said so. It was right about the drafts it sampled and wrong about the drafts a loom is actually pegged with, and the difference is the lifting plan. A threading error is silent when the cloth woven is another writing of the cloth meant, and whether such a writing is within reach of two wrong digits is decided almost entirely by how symmetric the lifting plan is. A plan of unrelated rows leaves a threading almost nowhere to hide; a twill’s plan, one row slid a pick at a time, leaves it room nine times in ten.

What a silent error is #

A threading error is an end entered on the wrong shaft. A lifting plan says nothing without a threading: the cloth woven is the lifting plan read through the threading, each end taking its shaft’s column, so a wrong digit replaces one column of the drawdown with another shaft’s. Two wrong digits replace two.

An error is caught if some end or pick stops interlacing, different if the drawdown is another cloth, and silent if it is a different matrix that is the same cloth — the same pattern slid along the ends or the picks, turned half round, or turned over, the writings how many cloths are there divides out. At eight ends a cloth has up to 256 such writings. The census builds all of them for each draft and asks whether the mis-threaded drawdown is one.

First, the drafts that cannot hide an error #

On a straight eight-shaft draft nothing is silent. Not one of 16,464 double errors on ten eight-end twills and the two eight-end satins, and not one of 672 single errors. The reason is short. Every end has its own shaft, so the drawdown’s eight columns are all different; a wrong digit that repeats a column leaves seven different columns, and no writing of a cloth with eight different columns has seven. The only errors that keep eight different columns are exchanges, and an exchange moves two columns where every writing of the cloth — a slide or a turn — moves six or eight.

Random drafts of four or more shafts behave the same way. A seeded sample of sixty drafts at each shaft count from two to eight, each with a random lifting plan and a random threading, found no silent error at all on four to eight shafts — 226,800 double errors, none silent — and silence only on two shafts (12.4 per cent) and three (0.21 per cent).

Read alone, that figure says the proof-reader’s job gets much easier as the repeat grows. The qualification in its caption is the whole of the next section.

A twill’s plan makes the shafts interchangeable #

A random lifting plan gives each shaft a column unrelated to the others. A real four-shaft plan does not. A 2/2 twill is pegged by lifting shafts one and two on the first pick, two and three on the second, three and four on the third, four and one on the fourth: each pick’s row is the last one slid along by a shaft. So each shaft’s column, read down the picks, is the previous shaft’s column slid by one pick.

And a slide along the picks is a writing of the same cloth. Weaving the whole cloth one pick later changes nothing about what cloth it is. Under a twill’s plan, then, “this end on shaft two” and “this end on shaft one, with the whole cloth slid a pick” are the same column — the shafts are interchangeable up to a symmetry the census already divides out. A threading error that moves ends to neighbouring shafts can be absorbed by that symmetry whenever the rest of the threading happens to line up, and on four shafts over eight ends it very often does.

The census walks every threading of eight ends that uses all four shafts — 40,824 of them — under each plan. Under the 2/2 twill’s plan, 36,320 of them, 89 per cent, can be mis-threaded silently by two digits; 0.97 per cent of their 10,287,648 double errors are silent. Under the 1/3 twill’s plan, which slides a single lift, 73 per cent and 0.55 per cent. Under a plan of eight unrelated rows — every pick and every shaft lifted and lowered, and no row a slide of another — 2.1 per cent of threadings and 0.008 per cent of double errors, every one of them an exchange, and none on a threading that uses every shaft twice.

A single draft shows the size of it. The point draft 1 2 3 4 3 2 1 2 — a diamond’s threading on four shafts over eight ends — under the 2/2 twill’s plan has 252 double errors, and 6 of them are silent, 2.4 per cent, three times the average. Entered on the straight draft 1 2 3 4 1 2 3 4 under the same plan, none are: the straight draft’s ends climb the shafts as the plan’s rows climb them, so any error breaks the one slide the cloth has, while the point draft turns back on itself and gives an error a second slide to hide in.

What the four-end catalogue was measuring #

The four-by-four census took every four-by-four matrix, and every four-shaft matrix at four ends is its own plan and threading at once: a draft that needs four shafts on four ends is a straight threading with its plan read off the matrix. Its 0.74 per cent on four shafts was an average over plans, and splitting it by plan says what the repeat does.

At four ends the 24 drafts whose plans are twills — each row the last slid by a shaft, the kind how many twills a repeat admits counts — are silent on 1.24 per cent of their double errors, and the 17,568 others on 0.74. Twills are more silent, but at four ends every plan is small enough to have symmetries by accident, and most of the silence in the catalogue comes from plans with no slide in them at all.

At eight ends the accident goes and the structure stays. A plan of unrelated rows falls to 0.008 per cent; the twill plans hold at 0.55 to 0.97. That is the answer to the question the earlier essay asked, put precisely: a longer repeat removes the silence that was coincidence and keeps the silence that was built into the plan.

That is also what connects this census to the other half of a draft. A mispick is one row in the wrong place: an error in the lifting plan rather than the threading. The argument here runs the same way with rows and ends exchanged — a lifting-plan error should be silent when the threading has a slide in it, as a straight draft does — and the transposition symmetry of the catalogue says the counts for mispicks under straight threadings should match the counts for threading errors under twill plans. That is a prediction from symmetry, not a count taken here.

Even one wrong digit #

Under both twill plans one wrong digit is silent 1,920 times in 979,776 — one error in five hundred. At four ends the single-error census found 576 silent errors, all on three-shaft drafts; the eight-end twill plans have them on four shafts, where the random sample and the straight drafts had none.

That changes the earlier essay’s sharpest finding. At four ends every silent pair was two audible errors cancelling, and no silent pair contained an error silent on its own. Under a twill’s plan a single digit can be silent by itself, because one end moved to the next shaft is one end’s column slid by a pick — and on a threading whose other ends already sit where a slide of the whole cloth would put them, that one change is the whole cloth slid.

The drafts a mill actually threads #

The threadings a four-shaft loom is most often entered with use every shaft twice in eight ends — the straight draft run twice, and the point and broken drafts that make diamonds, herringbones and broken twills. Of the 2,520 threadings with every shaft used exactly twice, 960 can be silently mis-threaded under a 2/2 twill’s plan and 704 under a 1/3’s — 38 and 28 per cent — and none under the plan of unrelated rows.

So the answer to the question as asked is that the repeat hardly matters. What decides whether a threading can lie is whether the lifting plan has a slide in it, and the plans a four-shaft loom is pegged with — twills, and every plan built by sliding a row — have exactly that. The four-by-four catalogue’s silence on four shafts, 0.74 per cent of double errors, and the eight-end twill plans’ 0.55 to 0.97 per cent are the same phenomenon at two sizes; the eight-end random sample’s zero was a sample of plans no loom uses.

Two shafts, and why they were always the exception #

On two shafts silence needs no help from the plan. A two-shaft threading is a pattern of two symbols, and a pattern of two symbols is very often one exchange away from a shifted copy of itself: 1 2 1 2 1 2 2 2 with its first and seventh ends exchanged is 2 2 1 2 1 2 1 2, which is the original started at its seventh end. That is why two shafts were silent 67 per cent of the time at four ends and 12 per cent at eight even with random plans. The repeat dilutes it; nothing removes it.

What it means for checking a threading #

The shortest notation has the longest mistakes found a threading digit the most dangerous symbol to get wrong, because one digit changes a whole column. The census here says when that danger is hidden. A mill checks a threading by weaving a sample and comparing it with the design, and that comparison misses exactly the silent errors: the sample looks right, and the cloth is the one designed, so in one sense nothing is wrong.

In another sense something is. Three mistakes and the shape each one leaves followed a fault into the cloth; a silent threading error leaves no fault in the cloth and a wrong threading on the loom, and the wrong threading matters the moment the lifting plan changes. Re-peg the same threading for a different weave and the silent error can become a loud one, because the new plan’s symmetries are not the old one’s. A mill that keeps a warp in the loom and changes its lifting plan — the ordinary way of weaving several cloths from one warp — is the mill a silent threading error catches, one cloth later.

So the instruction is not “check the simple drafts”. It is “check the threading against the draft, not against the cloth, whenever the plan is a twill”, because under a twill’s plan the cloth cannot say that the threading is wrong. How many shafts a draft needs counts the distinct columns; the twill’s plan makes distinct columns into slides of one another, and it is that relation, not the count, that decides whether a threading can be read back from its cloth.

How the census was taken #

Three populations, each stated. The twelve regular weaves on straight eight-shaft drafts — four two-run twills, six four-run twills and the two eight-end satins — walked completely. A seeded sample of sixty drafts at each shaft count from two to eight, each with a random lifting plan and a shuffled threading using every shaft, rejected unless the drawdown has exactly that many distinct columns and every end and pick interlaces. And every threading of eight ends that uses all four shafts, 40,824 of them, under each of three fixed four-shaft plans: the 2/2 twill, the 1/3 twill, and eight unrelated rows chosen so that every pick and every shaft is lifted and lowered.

For each draft every single error and every pair of errors is tried: each end moved to every other shaft the draft uses, singly and in pairs. A drawdown is silent if it lies in the set of the original’s writings — eight slides along the picks, eight along the ends, a half turn and a turning over, 256 combinations — built once per draft as eight-byte keys.

The four-end split re-walks the four-by-four census’s four-shaft drafts — 17,592 of them, each a straight threading of its own four columns — and classes a draft’s plan as a twill when every row is the row before it slid by one shaft in the same direction. Twenty-four drafts qualify; their double errors and the others’ are counted exactly as at eight ends, by comparing canonical forms.

What is required of it. No error on the regular weaves may be silent; no single error on the random-plan sample may be silent, and random-plan silence must lie only on two and three shafts. The twill-plan counts are observations, and the byte-key construction was checked against the first, string-keyed construction on every count before it replaced it.

What the census cannot say #

It covers four-shaft plans at eight ends. An eight-shaft twill plan slides its row too, and a threading that repeats shafts under it should behave as the four-shaft twills do; a straight eight-shaft threading cannot, for the reason given. Plans with five to seven shafts over eight ends cannot slide cleanly round a repeat they do not divide, and were not walked.

And a silent threading is only silent for one plan. Every count here is for a threading and a plan together; the practical danger is a threading that is silent under the plan it was checked with and loud under the next, and the census says how often the first happens, not which next plans expose it.

Who found which part #

Mills have always checked threadings by weaving, and the risk that a sample looks right with a wrong entry is folklore among hand-weavers, usually told about point drafts. The four-by-four census and its double-error count are this account’s.

What is new here is where the silence lives. The first census at eight ends drew its lifting plans at random and concluded that silence fades with the repeat; exhausting the four-shaft threadings under the plans a loom is actually pegged with found the opposite, and the difference is the one property a random plan never has — a slide.

Still open: which next plan exposes a silent threading #

A silent threading error is invisible under its plan and waits for the next one. The question a mill would ask is which plans expose it: given a threading silently wrong under a 2/2 twill, how many of the other plans a four-shaft loom can be pegged with — the 1/3 twill, the basket, the broken twills, the point twills — weave a visibly different cloth from it?

The census here has everything needed: the silent pairs under one plan, and a drawdown for any other. Crossing the two — every silent pair under each plan, re-woven under every other — would give the chance that a silent error survives a change of plan, which is the number that decides whether checking against the cloth is ever enough on a warp that will be re-pegged.