A slub finds the width of the cloth

A spinning frame leaves periodicities in its yarn, and a yarn’s ordinary variation is not one of them. A drafting roller of a given diameter puts a thick place in every turn of its circumference; an eccentric bearing does the same at its own period; a worn gear does it at the period of a tooth. The result is a yarn with a thick place every few centimetres, at a spacing set by a machine part rather than by anything about the fibre.

What that yarn does when it is woven as weft is not obvious, and the obvious guess is wrong.

The guess is that a fault every so many millimetres along the weft will make a bar every so many picks — a horizontal marking, at a spacing set by the fault’s period against the pick spacing. It will not, and the reason is a piece of arithmetic that nobody has to compute to see, but which nobody sees without computing.

A pick does not consume one pick spacing of weft. It consumes a whole width of it.

The claim #

A periodic weft fault produces a set of diagonals, one every fault period across the width, each stepping sideways by the remainder of the cloth’s width divided by that period. When the remainder is zero it produces warp-way stripes.

A fault every 42.86 mm in a cloth 1,500 mm wide gives thirty-five thick places per pick and a sideways step of nothing at all: thirty-five permanent stripes down the piece, produced by a defect that is entirely in the weft. A fault every 43.2 mm in the same cloth gives thirty-four marks per pick and a step of 31 mm a pick, which is a shallow drift that reads as a texture.

The step has two readings and the eye takes the nearer one. A mark that lands 31 mm to the right of its predecessor lands 12 mm to the left of the next mark along, since the marks are 43 mm apart — so what a reader follows is a drift of 12 mm a pick leftwards rather than 31 rightwards. The two describe the same set of marks; the figures draw the nearer, because that is the line anybody looking at the cloth joins up.

So the appearance of the fault is decided by the width of the cloth, and the same yarn on two looms of different widths produces two entirely different defects — one of them serious and one of them invisible.

The argument #

Number the thick places along the weft yarn: they sit at 0, p, 2p, and so on. Pick n uses the yarn from nW to (n+1)W, where W is the width. So the position of a thick place within pick n, measured from the selvedge, is a multiple of p less nW — which means that from one pick to the next, every mark shifts by W mod p.

Three regimes follow, and they are separated by fractions of a millimetre in p.

• Locked. W is a whole number of periods. The shift is zero, the marks stack, and the cloth has stripes down it that no amount of looking at the weft will explain.
• Diagonal. A small shift produces a steep diagonal, walking across the cloth and returning to its starting column after p divided by the shift picks. This is the classic weft bar, and it is diagonal rather than horizontal — which is why it is so often described as a “shadowy” or “streaky” fault rather than as a bar.
• Scattered. A shift of a good fraction of a period puts consecutive marks nowhere near one another, and the fault contributes texture rather than pattern. The drift is still there and is still exact; it is simply too fast across the cloth for an eye to follow a line.

What was counted, and how #

The marks are placed rather than described: the position of every thick place is computed modulo the width, pick by pick, and drawn where it falls. The sweep then takes the remainder at six hundred periods across a six-millimetre range and reports where it vanishes.

The lockings are checked against what they have to be. A locking is a period that divides the width a whole number of times, so each one found by the sweep is required to equal the width over an integer, to within a twentieth of a millimetre. That check catches the mistake this arithmetic is prone to — a remainder taken the wrong way round, which produces spikes in the right number and the wrong places.

And the detection tolerance is derived rather than chosen. A locking is never hit exactly on a grid, and the residual scales with the grid’s own step multiplied by how many periods fit the width. The first version of this sweep used a fixed tolerance and found no lockings at all at one resolution and every point at another, which is the characteristic failure of a threshold that does not know the scale of what it is thresholding.

Why this is worse than it looks #

Three things make this particular fault mechanism nastier than most.

It is invisible in the yarn. A thick place a fifth over the mean diameter, every four centimetres, is well inside what a yarn’s ordinary variation covers. Nothing about the yarn looks wrong; its coefficient of variation is unremarkable, and the periodicity is a small part of its total variation. It is only when the cloth’s width picks that period out that anything is visible at all.

It is invisible in a sample. A short length of yarn, or a small woven sample, may contain only a few periods, and the pattern needs many picks to establish itself. A ten-centimetre sample of a cloth that will show stripes at full width shows nothing.

And it survives everything. A stripe caused by a locked period is in the structure of the cloth. It cannot be finished out, dyed over, or brushed away, because it is a real periodic variation in how much yarn is in each column.

The one consolation is the mirror of the last point: since the appearance depends on the width, a fault of this kind can be destroyed by changing the width. Weaving the same yarn a few centimetres wider moves the remainder away from zero and turns the stripes into a diagonal so shallow that nobody sees it. That is a strange and unsatisfying remedy — it changes nothing about the cause — and it is arithmetically exact.

The same argument in the warp #

Everything above is about weft, and the warp version is different in a way worth stating because it is the easier case.

A warp end is continuous down the length of the piece, so a periodic thick place along one end lands at the same position across the cloth every time — it is already locked, by construction, and it shows as a series of thick places down one end. But that is a fault in one end out of two thousand, so its visible amplitude is one two-thousandth of the cloth’s width, and it is essentially invisible unless it is large.

The asymmetry is worth naming because it inverts the usual one. A warp fault normally condemns far more cloth than a weft fault, because it runs the length of the piece. For periodic faults the ordering reverses: a weft periodicity is spread over the whole width and can lock into stripes, while a warp periodicity stays in its own column and does nothing.

The numbers a mill already has #

Everything this argument needs is measured somewhere in an ordinary mill, and none of it travels to where the division would be done.

The fault’s period comes from a yarn spectrogram, which is a routine test: it plots how much of a yarn’s variation sits at each wavelength, and a peak at a machine’s own dimension is exactly what it exists to find. A peak at 43 mm is reported as a mechanical fault in the spinning, traced to a roller of that circumference, and acted on there.

The cloth’s width comes from the loom, and it is the least ambiguous number in the mill.

The division of the second by the first takes a moment and answers a question the spectrogram cannot: will this yarn make a visible mark in this cloth? A peak at 43.2 mm in a 1,500 mm cloth is nothing. The same peak at 42.86 mm is thirty-five stripes. The spectrogram reports both identically, because to a spectrogram they are the same fault at the same amplitude, differing by less than its own resolution.

And the same table read the other way is a purchasing decision. A yarn with a known periodicity is not simply better or worse than one without; it is unusable in some widths and harmless in others, and which is which is decided by arithmetic rather than by grade.

Why the marks stack rather than blur #

One detail of the locked case deserves its own paragraph, because it is what makes the fault permanent rather than gradual.

When the remainder is zero, the thick places do not merely tend towards the same columns — they land in exactly the same columns, at every pick, for as long as the yarn’s period holds. A stripe made that way is not a statistical accumulation that a longer piece would average out. It is a systematic excess of yarn in thirty-five columns and a corresponding deficit everywhere else, and it grows in visibility with the length of the piece rather than diminishing.

That is the same distinction the coherent-versus-random argument makes about spatial frequency, arriving here through a different door: the locked case is coherent in two directions at once — periodic across the width and identical down the length — while the diagonal case is coherent across and drifting down, which is a much weaker thing.

How exactly locked is locked #

The locked case is stated above as a remainder of zero, and a remainder of exactly zero is not a thing a real machine part produces. It is worth asking how close is close enough, because the answer is much sharper than the argument so far suggests and it changes what a stripe is.

A mark stays in its column for as long as its accumulated drift is less than its own width. A thick place is perhaps ten millimetres of yarn, so the condition is that the step per pick, multiplied by the number of picks in view, comes to under ten millimetres.

Near a locking the step is the number of periods per width — thirty-five here — times the error in the period. So:

the period must be known to within the mark’s width, times the period, divided by the width and by the picks in view.

Put a metre of cloth in view, at twenty-two picks a centimetre, and that is 2,200 picks and a permitted error of 0.13 microns. Ask for the same stripes down a fifty-metre piece and it is 2.6 nanometres.

Those are not tolerances any roller has. Thirty-five periods in fifteen hundred millimetres is three parts in a million for the metre, and steel expands by twelve parts in a million per degree — so a one-degree change in the spinning room moves the period by four times the tolerance a single metre of stripes requires.

Which says what a locked fault actually is. Not a permanent register down the piece: a diagonal so shallow that it does not leave its column within the length anybody looks at, and which therefore wanders over a full piece rather than running true. The stripes are real, they are visible, and they are locked to the inspector’s field of view rather than to the cloth.

So the three regimes are not three cases but one continuum with a viewing length in it, and the locked band’s width in period space shrinks in exact proportion to how much cloth is being asked to agree.

Where the model stops #

Every pick is treated as using exactly one width of yarn. It does not: there is the crimp, which adds a few per cent, and there is the selvedge and whatever is wasted at it. Both change the effective width the remainder is taken against, and the sensitivity is such that a crimp of eight per cent moves the lockings by eight per cent — so the numbers here name the arithmetic rather than a real loom’s periods.

The insertion is treated as unidirectional. A shuttle loom lays alternate picks in opposite directions, which reflects every second pick and turns a diagonal into a zigzag. The lockings survive that unchanged, because a reflected column is still the same column, and everything between them becomes a chevron rather than a line.

A thick place is treated as a point. It has a length along the yarn — a slub is a centimetre or two — so what is drawn as a mark is a short segment, and neighbouring marks merge when the step is small. That makes the locked case even stronger and the diagonal case blurrier than the figures show.

And nothing here says whether the fault is visible. The marks are placed; how much thicker a thick place must be before an eye finds a pattern in it is a question about contrast, and this collection has no instrument for one.

What it does to a specification #

There is a general lesson at the end of this essay, but a narrow one belongs here first, because it is the kind of thing a specification could actually carry.

A yarn is bought against a count, a twist, a strength and an evenness. None of those four says anything about periodicity: a yarn with a clean spectrogram and one with a strong peak at a machine’s wavelength can have identical coefficients of variation, because a periodicity at one wavelength is a small part of a yarn’s total variation.

So the number that decides whether this fault appears is not among the numbers that change hands. The information exists, in the spinner’s own test data, and stops at the spinner’s door.

What would carry it is one line: the wavelength and amplitude of the largest peak in the yarn’s spectrogram. A weaver holding that line, and the width of the cloth to be made, can do the division before the warp is drawn in rather than after the piece is condemned.

The generalisation #

When a periodic thing is laid down in fixed-length runs, what appears is decided by the remainder of the run against the period — not by either of them alone. That is the transferable shape, and it turns up wherever a repeating supply meets a repeating consumer: a printing cylinder against a sheet length, a coating pass against a repeat, a data record against a page.

The practical form is a warning about smoothness. The remainder is a sawtooth, so the output is a discontinuous function of a continuous input, and everything about the usual approach to tolerances fails: there is no sense in which a fault period of 42.9 mm is “close to” one of 42.86 mm, because the first is harmless and the second makes stripes. A specification that bounds a period to within a millimetre has bounded nothing at all.

And the second lesson is that a system’s dimensions can be a remedy. If a defect depends on a remainder, changing the divisor removes it without touching the cause. That is not a repair and it should not be mistaken for one — but it is available, immediate, and exact, and it is worth knowing that it exists before spending a month looking for a bearing.

Who found it, and when #

Periodic yarn faults and the marks they leave are the whole subject of evenness testing, and the standard instrument for finding them has plotted a yarn’s variation as a spectrum since the 1950s — precisely because a periodicity is what matters and a total variation is not. The wavelengths a spectrogram flags are the ones that correspond to machine parts, and the practice of matching a peak to a roller diameter is routine.

What appears not to be routine is the second half: matching the wavelength to the cloth’s width rather than to the machine that produced it, and predicting from the remainder whether the fault will show as a stripe, a diagonal, or nothing. That step costs one division and it decides whether the yarn is usable in a given cloth.

Where the ladder goes next #

The other kind of periodic fault is the one whose period comes from the loom rather than from the yarn, and there the arithmetic is about where in the machine the mistake was made: a threading error, a lifting error and a tie-up error leave a line, a bar and a lattice respectively, and differ in size by four orders of magnitude.

Sideways, this whole ladder rests on the distinction that makes a periodic fault matter at all — a random error of the same size hides and a periodic one shows — and on the population argument beneath it, in which a cloth is not one thread but many.