A random error hides and a periodic one shows

A warp of ordinary cotton varies in thickness by fifteen per cent from end to end — which is the population every number on this site was computed without — and the cloth it makes looks even. A reed with one dent set a tenth of a millimetre wide — an error of perhaps two per cent in one spacing out of two thousand — puts a stripe down the piece that a buyer will reject on sight.

Put like that it seems to need an explanation about the eye. It does not. The whole of the difference is in how the two errors are arranged, and the arithmetic that separates them has no free parameter in it at all.

The claim #

The same total error, arranged in a period, is √(2n/π) times as strong at its own frequency as a random arrangement is at any frequency — where n is the number of ends being looked at at once.

Over a band of two hundred and fifty-six ends that ratio is 12.8. Over a thousand it is 25.5. It grows as the square root of how much cloth is in view, so looking at more cloth makes the periodic fault worse and the random variation better, which is the exact opposite of what a careful inspector’s instinct suggests.

The argument #

Read the sequence of spacing errors along the warp as a signal and ask where its energy is.

Independent errors put their energy everywhere. A sequence of n independent deviations has the same total energy whatever their signs happen to be, and that energy is spread across all n/2 frequencies the band can hold. So the amplitude at any one frequency is about the error divided by the square root of n — smaller than the error itself, and smaller the more cloth is included.

A periodic error puts all of its energy at one frequency. One shaft set forward, one dent too wide, one guide bar out of place: each produces a deviation that repeats, and every repeat adds to the same frequency rather than to a new one. The amplitude there is the error over the square root of two, whatever n is.

Divide the second by the first and the ratio is √(n/2), and then one more factor arrives from the shape of the random case. The amplitude at a single frequency of a random sequence is Rayleigh distributed, and the mean of a Rayleigh variable is √(π/4) of its root-mean-square — so the background a periodic peak actually has to beat is a little smaller than the naive estimate, and the ratio is

$$\sqrt{\frac{2n}{\pi}}.$$

The π is not decorative. It is the mean of a Rayleigh amplitude, and it is the difference between a ratio of 11.3 and the 12.8 the simulation returns at two hundred and fifty-six ends.

What was counted, and how #

Two sequences are built for each trial: a cosine at the stated period, scaled so that its root-mean-square is exactly the amplitude asked for, and an independent sequence with the same root-mean-square. Both are transformed, the periodic one’s peak is taken at its own frequency, and the random one’s background is averaged over every frequency except the two adjacent to that peak — so the comparison is between one frequency and the frequencies beside it, which is what a beat in the cloth actually has to stand out from.

Two things are asserted while it runs and each catches a different mistake.

The periodic peak equals the closed form exactly. A cosine of a stated root-mean-square puts a known amplitude at its own frequency and nothing anywhere else, so if that number moved, the transform would be wrong rather than the argument.

The ratio matches √(2n/π) to within three per cent, across every width. Asserting the relation rather than the value at one width is the rule of this house, and it is what would fail if the background were being averaged over the wrong set of frequencies — a mistake that changes the constant and leaves the square-root growth intact.

Why the inspector stands back #

The practical consequence is a habit that every finishing room already has and nobody explains this way: faults are looked for from a distance.

Standing back increases n. It puts more ends into one view, which divides the random background by a larger square root while leaving the periodic peak exactly where it was. A stripe that is invisible at arm’s length is obvious across a room, and it is not because the eye is integrating more light — it is because the competing texture has averaged itself down and the stripe has not.

The same argument runs in reverse for the opposite job. A yarn’s evenness cannot be judged from a piece of cloth at a distance, because the distance is what removes it. Judging evenness needs the cloth close to, where n is small and the random component is at full strength relative to any periodicity.

And it explains a specific, well-known injustice. A mill can be entirely blameless in its yarn — even, well spun, correctly counted — and be ruined by one bent heddle, while another mill with visibly worse yarn ships cloth that passes. The first has a periodic error and the second has a random one, and at the widths a buyer inspects, the ratio between them is more than an order of magnitude.

What counts as periodic, in a loom #

The list of things that produce a periodic error is short, and every entry on it is a mechanism rather than a material.

• The reed. One dent wide or narrow repeats at the denting order — every second, third or fourth end — and that is the tightest period a cloth can have. It is also not the sett, which is why the mark it leaves survives the cloth’s own contraction.
• The healds. One shaft carrying its ends slightly forward puts an error at the period of the draft, which is where the strongest visible marks come from in a twill.
• The take-up and the let-off. A gear tooth or an eccentric roller puts an error into the pick spacing at the period of a revolution, which is a bar across the cloth rather than a stripe down it.
• The yarn’s own manufacture. A drafting roller of a given diameter leaves a thick place at its own circumference, which is a period in the thread rather than in the cloth — and what that does depends on how the two periods meet.

Everything else — the fibre, the twist, the ordinary variation of a spinning frame — is random, and randomness is what the square root hides.

The arithmetic of one bad dent #

It is worth putting the two errors on the same scale, because the sizes involved are so different from what the visible result suggests.

A reed for a cloth at twenty-four ends per centimetre has a dent every 417 µm at two ends per dent. Suppose one dent is 10 µm wide — 2.4 per cent of a dent, and the sort of error a reed acquires from a single knock. That error repeats every two ends, and its amplitude at its own frequency is 7 µm.

The yarn in the same cloth has a diameter of 167 µm at a coefficient of variation of fifteen per cent, so its ends deviate by 25 µm root-mean-square, and its gaps by more. Over a band of two hundred and fifty-six ends, that random deviation contributes about 1.6 µm at any one frequency.

The yarn’s total error is two and a half times the reed’s and its visible amplitude is a quarter of it. The bad dent contributes a twentieth as much error to the cloth and shows four times as strongly, and the entire difference is that one of them repeats.

That is the arithmetic behind a piece of trade wisdom that sounds like superstition: a reed is checked before a warp is put in, and a yarn is not checked at all beyond its certificate. It is the correct allocation of attention, and the ratio above says by how much.

Why a patterned cloth hides a streak #

The trade says a busy weave hides a fault, and the spectral argument says why — with a mechanism quite different from the one about float lengths that the same claim gets tested against elsewhere.

A periodic fault has to be visible against something, and what it competes with is whatever the cloth is already doing at nearby frequencies.

A plain weave has almost nothing to compete with. Its structure repeats every two ends, so all of its own energy sits at one very fine frequency, far from anything an eye reads as a stripe. Everywhere else in the spectrum the cloth is empty, and a fault at any coarser period stands alone in it.

A patterned cloth is not empty. A stripe pattern, a check, a figured ground: each puts energy at its own coarse periods, and a fault whose period is close to one of those is being asked to stand out from something of its own kind rather than from a blank field. The masking is strongest where the two periods are closest, which is also the case in which the fault and the pattern beat with one another — so what the fault produces there is not a stripe but a slow shading, which is exactly the sort of defect that survives inspection.

That gives a rule with teeth in it: a cloth is most vulnerable to a fault whose period the cloth does not have, and least vulnerable to one it shares. A plain weave is vulnerable to everything coarser than two ends. A four-shaft twill is vulnerable to everything except errors at four ends and its multiples. A cloth with a coarse stripe pattern is armoured against faults near the stripe period and just as bare as a plain weave everywhere else.

It also explains a specific and otherwise puzzling observation: that the worst faults in patterned cloths are the ones at simple ratios to the pattern. A fault every eight ends in a four-end twill does not hide inside the pattern — it doubles some of the twill’s own lines and not others, which is a bolder mark than a stripe on a plain ground.

The distance that is right is the one the fault chooses #

The inspector’s habit of standing back was explained above by n, and that explanation is incomplete in a way worth repairing, because taken alone it says an inspector should stand as far back as the room allows and no inspector does.

Standing back raises n and buries the random background. It also moves the fault’s period to a higher spatial frequency, and the eye’s sensitivity to a grating is not flat: it peaks at around four cycles per degree and falls away on both sides. So a fault whose period is already fine is pushed past the eye’s best band by the very act of stepping back.

The two effects pull opposite ways, and the second fixes a distance. A period p subtends its optimum when it covers a quarter of a degree, which is 4.36 milliradians, so

the best viewing distance is about 230 times the fault’s period.

Those are the distances a finisher’s inspection actually uses, and they are not a matter of comfort or of eyesight. A reed mark is found with the cloth held close; a heald mark at arm’s length; a bar across the room. An inspector sweeping a piece at one distance is inspecting for one band of periods and is nearly blind to the others, which is why the practice is to move — and why a fault found by one person and missed by another is so often a difference in how they stood.

And the ratio does become a threshold #

The essay’s own limit says a ratio is not a threshold, which is right and can be closed one step further with a crude optical model — crude enough that the number below should be read as an order rather than a value.

Take a cloth held against the light, so that what an eye reads is the open area. A sinusoidal displacement of amplitude A at period λ modulates the local spacing by A·2π/λ, and the cover by the same fraction of itself. For the essay’s own bad dent — 7 µm at a period of two ends, 834 µm — that is a 5.3 per cent modulation of the spacing, so a cover of 0.40 moves by 0.021 and an open area of 0.60 carries a contrast of about 1.8 per cent.

The eye’s threshold contrast for a grating at its best frequency is around a half of one per cent, and at the eight cycles per degree this fault sits at from 190 millimetres it is nearer one. So the bad dent is roughly twice its threshold and is seen.

Run the yarn’s random background through the same arithmetic and its 1.6 µm at one frequency gives a contrast of about 0.4 per cent — at or just under the threshold, which is exactly the observed situation: a cloth of fifteen per cent yarn looks even, with a faint texture that is sometimes visible and sometimes not.

Inverting gives the tolerance. The dent error that lands on the threshold is about 3 µm, which is seven tenths of one per cent of a dent. That is a brutal number for a mechanical part a metre and a half long, and it is why reeds are precision objects, why they are inspected before every warp, and why a reed that has been dropped is scrapped rather than straightened.

The whole calculation rests on contrast being proportional to open area, which is true of a scrim against a window and progressively less true of a dense cloth seen in reflection. What survives the crudeness is that the two amplitudes bracket the threshold from opposite sides — and that is the statement the essay needed and could not previously make.

Where the model stops #

The eye is modelled as a band of a stated width, and it is not one. The visual system responds to spatial frequency with a sensitivity that peaks and falls away at both ends, so a periodic error at a very fine period — every second end, say — is not seen as a stripe at all but as a change in texture. Nothing here has a contrast sensitivity function in it, and the essay claims a ratio of amplitudes rather than a threshold of visibility.

A ratio is not a threshold. Whether an error shows depends on how large it is as well as how it is arranged, and this argument says nothing about the absolute size at which either kind becomes visible. Two errors in the ratio of twelve to one may both be invisible or both be obvious.

Only one dimension is treated. A warp streak is a one-dimensional signal because it runs the length of the piece; a genuinely two-dimensional fault — a moiré, a patch — has its energy spread over a plane of frequencies and the arithmetic is different.

And the cloth is treated as a spacing, with the threads’ own thickness held constant. Real faults are usually both: a thin place is thinner and leaves a wider gap. The two contributions can reinforce or cancel, and which they do depends on whether the eye is reading the shadow between the threads or the light on them.

The generalisation #

Coherence is worth √n, and n is set by how much of the system is looked at at once. That is the whole of it, and it is why a small systematic error is more dangerous than a large random one in every measurement, every process and every signal — not because systematic errors are worse in principle, but because they add while random ones cancel.

The practical form is a question to ask of any disturbance: does it repeat? If it does, more data makes it more visible; if it does not, more data buries it. That single distinction decides whether the right response to a problem is to gather more of the thing and look again, or to stop looking and go and find the mechanism.

And it decides where to spend effort. Reducing a random variation by a tenth is worth a tenth. Removing a periodic error entirely is worth the whole of it, times the square root of however much cloth the customer will look at — which is why one bent heddle deserves more attention than a whole grade of yarn.

Who found it, and when #

That periodic faults are the visible ones is universal knowledge in the trade and is what every fault-classification scheme is built around: the categories are almost all named after mechanisms, because the mechanisms are what repeat.

The spectral argument is ordinary signal analysis, applied here rather than derived, and the same reasoning is what makes a lock-in amplifier work. What appears not to have been written for cloth is the size of the ratio and its growth with the width of the view — and the practical consequence that follows, which is that the distance an inspector stands at is a parameter of the inspection rather than a matter of comfort.

Where the ladder goes next #

The next step is the case where the period is in the thread rather than in the machine, and the cloth’s own pick spacing decides what becomes of it: a slub recurring along a weft yarn finds the width of the cloth, and what it draws there is a stripe or a diagonal depending on a remainder.

Sideways, the same coherent-versus-random distinction is what separates the three kinds of loom mistake from one another — a threading error, a lifting error and a tie-up error differ in exactly this way, and the largest of the three is the one that repeats in two directions at once.