A cloth cannot be more even than its yarn

An ordinary shirting yarn varies in mass by thirteen per cent along its own length. The cloth woven from it does not look thirteen per cent uneven; it looks, to an ordinary eye at an ordinary distance, entirely uniform.

Nothing has been fixed. The threads in that cloth are the same threads, with the same thick and thin places in the same order. What has happened is that a great many of them are being looked at together.

The claim #

A patch of cloth is more even than its yarn by the square root of the number of independent contributions in it — and at a fixed cloth weight, that reduction cancels the yarn’s own irregularity exactly.

The second half is the finding. It settles a question this collection raised two rungs back and left open: a fine yarn is more irregular, but a cloth of fine yarn has more threads in every patch, so which of the two effects wins?

Neither. They are the same exponent with opposite signs.

• The yarn’s coefficient of variation goes as the inverse square root of the count.
• The number of threads in a patch, at a fixed cloth weight, goes as the count.
• The averaging takes a square root of the second, so the two cancel — and a 130 g/m² cloth has the same patch unevenness whether it is woven from 10 tex at 61 ends per centimetre or from 40 tex at 15.

The averaging, which is two square roots #

A patch of cloth of side L contains threads two ways.

Across. Each system contributes sett × L threads, each running through the patch and contributing its own mass. Different threads are independent — they came off different parts of the spinning frame — so their errors add in quadrature.

Along. A single thread crossing the patch contributes its mass averaged over the length L, and that average is over L/λ independent stretches where λ is the correlation length. The correlation length of a yarn’s mass is its staple length, because two planes closer together than one fibre share most of their fibres. For a patch smaller than a staple — which is every patch the eye is judging — there is one such stretch and this term does nothing.

So for the patches that matter,

$$\mathrm{CV}_{\text{patch}} \approx \frac{\mathrm{CV}_{\text{yarn}}}{\sqrt{s\,L}},$$

with s the total of the two setts and L the patch. Nothing else is in it: no weave, no crimp, no cover. The weave decides where the threads are and this argument does not care.

Where the eye is, and why three millimetres #

The patch size that matters is not the smallest one a viewer can resolve. It is the one at which the eye is most sensitive to contrast, and human contrast sensitivity peaks at a few cycles per degree — which at reading distance is a period of two to five millimetres.

That is why the figures above are drawn at three. A finer variation than that is present in the cloth and is not what anybody complains about; a coarser one is a barré or a weft bar and is a different defect with a different cause.

The trade’s own word for the quantity is cloudiness, and it is a spatial-frequency judgement. A cloth whose mass varies at the millimetre scale looks slightly grainy; the same total variation arranged at the centimetre scale looks like cloud, and is much more objectionable. The averaging above is the reason: the eye’s own aperture is doing the same square root, and moving the variation to a larger scale moves it out from under the averaging.

This is the same argument as the one that separates a random error from a periodic one, read in space rather than in frequency. A random variation is spread over every scale and is beaten down by the averaging at the scale the eye uses; a periodic one puts all of itself at one scale, and if that scale is the eye’s it is not beaten down at all.

The cancellation #

Now put the floor into the averaging and the count disappears.

The yarn’s coefficient of variation is its index times its floor, and the floor is 100√((1 + CV_f²)/n) with n = tex ÷ fibre tex. So CV_yarn ∝ tex^(−½).

A cloth’s areal weight is its setts times its count: W ∝ s × tex. At a fixed weight, s ∝ tex^(−1).

Substitute into the averaging:

$$\mathrm{CV}_{\text{patch}} \propto \frac{\text{tex}^{-1/2}}{\sqrt{s\,L}} \propto \frac{\text{tex}^{-1/2}}{\sqrt{W\,L/\text{tex}}} = \frac{1}{\sqrt{W L}}.$$

The count cancels exactly. A cloth’s patch unevenness depends on its areal weight, on the size of the patch, and on the spinning index — and not at all on the count it was woven from.

A worked pair makes it concrete. A 20 tex yarn at 24 × 24 per centimetre and a 6 tex yarn at 80 × 80 are both 103 g/m². The first is woven from a 13.4 per cent yarn and the second from a 24.5 per cent one. Both cloths come out at 3.53 per cent over a three-millimetre patch.

What this does and does not say about thread count #

It disposes of one argument and leaves another standing.

Disposed of: “a higher thread count is a smoother cloth.” At the same cloth weight it is not — it is exactly as smooth, because the finer yarn required to reach the higher count is worse by precisely the amount the extra threads recover. This collection has already objected to thread count on other grounds; this is a sharper objection, because it is an exact cancellation rather than a missing variable.

Left standing: the cloth’s weight is what decides its evenness. A heavier cloth is smoother, and it is smoother as the square root of its weight. That is a real and useful design rule and it is the one the arithmetic supports.

And left standing: everything that is not mass. The cancellation is about mass per unit area. A coarse yarn set open has visible individual threads where a fine yarn set close does not, and that is a different perception with a different cause — it is the cloth’s structure being resolved, not its mass varying. The two cloths above are equally even and do not look the same.

The weight rule, and what it costs #

If a cloth’s evenness goes as the square root of its weight and nothing else, then the only way to make a smoother cloth is to make a heavier one — and that is a real constraint rather than a slogan, because it applies at every count.

Doubling a cloth’s weight improves its patch evenness by 1.41. Halving it makes the cloth 1.41 times cloudier. So a lightweight cloth is necessarily cloudier than a heavy one of the same yarn quality, and every complaint about the appearance of very light shirtings and voiles is, in part, this exponent.

The three ways out are worth listing because each is used.

Raise the index. The spinning index divides straight through, so a yarn spun at 1.15 rather than 1.35 gives a cloth 15 per cent more even at every weight. This is the honest route and it is what a combed yarn buys.

Move the variation to a scale the eye averages better. Nothing changes the total variation, but a cloth whose faults are at the millimetre scale reads better than one whose faults are at the centimetre scale. Mixing many packages, alternating ends from two sources, and running a warp from a large number of small packages rather than a small number of large ones are all ways of pushing correlated variation down to a finer scale.

Or hide it. A patterned, printed or coloured-woven cloth has structure at the scale the eye is most sensitive to, and that structure masks the mass variation underneath it. That is why a plain white lawn is the hardest cloth in the trade to make acceptably and why a printed one of the same construction is not.

Two regimes, and the eye is in the wrong one #

The averaging above was written for patches smaller than a staple length, which is every patch the eye judges. It is worth following it past that size, because the law changes there and the change explains why the argument is so much less help than it ought to be.

Below the staple, a thread crossing the patch contributes one sample of its own mass, whatever the patch’s size. So the only thing growing with L is the number of threads, which grows linearly, and the reduction is √L.

Above the staple, each thread also contributes L/λ independent stretches along its own length. Now both counts grow linearly, the total number of independent contributions grows as L², and the reduction is L rather than its square root.

So a cloth’s cloudiness falls as 1/√L up to about thirty millimetres and as 1/L thereafter. The corner is at the staple length, and it is a corner in a log–log plot rather than a discontinuity — the along-thread count passes through one smoothly.

The consequence is the uncomfortable half. The eye’s peak sensitivity sits at two to five millimetres, which is well inside the shallow regime, so the eye is judging the cloth in exactly the range where extra area buys the least averaging. Doubling the scale of the patch a viewer attends to improves the evenness by 1.41 there, against 2.0 in the far field.

That is why standing back works, and works only so far. A cloth held at arm’s length and a cloth seen across a room are not the same object to a viewer: the second is being averaged in the steep regime, and its mass variation is genuinely invisible. The first is in the shallow regime, and a fifteen per cent yarn will show. Every buyer who inspects cloth by holding it up and then walking away from it is sampling both regimes without knowing there are two.

It also says something about the sampling in the trade’s own procedures. A cloth is graded on a table under a lamp, at a viewing distance of half a metre, and the defect scale a grader is trained on is calibrated at that distance — so it is a scale calibrated inside the shallow regime, and it will report a cloudiness that a wearer at conversational distance will never see. That is not an argument for grading less carefully. It is an argument for stating the distance, which the four-point systems do and the eye does not.

And it makes the staple length a cloth-appearance quantity, which nothing in the cloth’s specification suggests. A long-stapled cotton makes a cloth whose corner sits at 38 mm rather than at 22, so its mass variation is averaged in the shallow regime over nearly twice as wide a range of scales. The usual reasons given for preferring a long staple are strength and evenness and fineness, all of which are true and all of which act through the count. This one is separate, acts through the correlation length, and is invisible to every measurement made on the yarn.

Where the averaging fails #

Three cases, and all three are cloths where somebody complains.

When the variation is not independent between threads. Two adjacent ends from the same package, or a whole warp from one badly running frame, share their faults — so the quadrature sum is wrong and the reduction is smaller than √N. This is the ordinary mechanism behind a barré: a mixing fault that puts correlated yarn side by side.

When the variation is periodic. A periodic thick place in the weft repeats across the width at a spacing set by the fault’s own period, and the appearance is decided by the cloth’s width rather than by any averaging. The averaging beats down the random part and leaves the periodic part untouched, which is why a small periodic fault is far more visible than a large random one.

When the eye is looking at something other than mass. A shade variation, a lustre variation from twist, a difference in how two batches take dye — none of these is mass and none is averaged by this argument. They are averaged by the same √N, but their own coefficients of variation are not the yarn’s.

What a specification could say and does not #

Every quantity in the averaging is on a cloth’s own specification already, which makes the omission worth pointing at.

A cloth is specified by its two setts, its two counts, its weave and its finished weight. From the counts and the fibre comes the fibre count; from that the evenness floor; from the spinning index the yarn’s own coefficient of variation; and from the setts and a patch size the cloth’s. The whole chain is arithmetic, and nothing in it is a measurement of the cloth.

So a cloth’s expected cloudiness is computable before it is woven, from numbers a buyer already has. What is not on the specification is the index, which is the one quantity in the chain that measures how the yarn was spun — and it is the one a buyer is most often quoted a raw coefficient of variation for instead, which as the floor argument shows is uninterpretable without the count beside it.

The practical rule is short. Ask for the index, not the CV; compute the cloth from the weight, not from the thread count. Both halves follow from the two cancellations in this essay and neither is current practice.

What was counted, and how #

The cancellation is asserted at three counts over a fourfold range, to a part in 10¹⁰, after first asserting that the three cloths are the same weight — because a cancellation demonstrated on cloths of different weights would demonstrate nothing. It additionally asserts that the yarns differ by half again, so that a change which made all three yarns identical would fail rather than pass.

The patch sweep is asserted monotone at every step. A larger patch must be more even than a smaller one, at each adjacent pair rather than at the ends.

And the two independence terms are kept separate. The across-threads count and the along-thread count are computed separately and multiplied, so the regime change at the staple length is visible in the curve rather than smoothed into a single fitted exponent.

Where the model stops #

Independence between threads is assumed and is the assumption that fails first. Real warps have correlated ends; real wefts have correlated picks, because consecutive picks come from the same package. The reduction quoted here is an upper bound on the averaging and therefore a lower bound on the cloudiness.

The eye’s aperture is quoted from contrast-sensitivity measurements and is not derived. Three millimetres is a defensible middle of a range that runs from about one to about ten, and the figures are drawn across the range so the choice is visible.

Mass is taken as the thing being seen, and what is seen is light. A patch of cloth with more mass in it is darker or more lustrous, but the mapping from mass to appearance is a reflectance question this collection has declined and continues to decline.

Crimp is held constant across the constructions being compared. A cloth of fine yarn set close crimps differently from a coarse one set open, so the areal weights are not quite matched in the way the cancellation assumes. The correction is a few per cent and it does not have an obvious sign.

And nothing here is about a fault. A missing end, a wrong pick or a slub are not large deviations of a mass distribution; they are different objects with their own arithmetic.

Where the ladder goes next #

Into the fabric that cannot do the averaging. A knit is made of one thread, so a thick place in the yarn does not appear once among many independent neighbours — it runs along a whole course and rings the fabric. The same irregularity displays completely differently in a knit, and the difference is a statement about how many threads a fabric has rather than about how good the yarn is.

And sideways into the weight rule, which is now the one that survives: a cloth’s evenness goes as the square root of its areal weight, and every construction decision that changes the weight changes the appearance by that exponent — while every decision that trades count against sett at a fixed weight changes nothing at all.