A finer yarn is a worse yarn

Fineness is what a yarn is sold on. A finer count makes a lighter cloth, a smoother surface and a higher thread count, and the price of a cotton rises steeply with the count it will spin to. Every part of the trade treats fineness as quality.

The arithmetic disagrees, and it disagrees by a fixed exponent.

A yarn’s irregularity cannot go below one over the square root of the number of fibres in its section. The count is the fibre count. So halving the count halves the fibres, raises the floor by forty-one per cent, and raises everything downstream of the floor by whatever power that quantity carries. Fineness is bought with irregularity, and the exchange rate is not negotiable.

The claim #

Every quantity this collection corrects for a population is corrected by more when the yarn is finer, and the amount is a fixed power of the count.

• A linear quantity is unaffected at any count. A cover factor stays exact.
• A quantity with a curvature is biased by half the variance times the second derivative, and the variance goes as 1/n — so a bias goes as the reciprocal of the count. A bending rigidity, which is a fourth power, is biased 2.7 per cent high at 80 tex and 41.8 per cent high at 6.
• An extreme is worse than either, because its excess grows with both the spread and the amount of cloth sampled, and only one of the two is under anyone’s control.

The consequence for this collection is uncomfortable and worth stating plainly: its own predictions are most accurate for the coarse, cheap yarns and least accurate for the fine, expensive ones.

The arithmetic, which is one substitution #

The population argument gave a closed form for what a spread does to a power. If a diameter is lognormal with coefficient of variation CV, then the population average of the k-th power exceeds the k-th power of the average by

$$\frac{\mathbb{E}[X^k]}{\mathbb{E}[X]^k} = (1 + \mathrm{CV}^2)^{k(k-1)/2},$$

which is where the 2.25 per cent on a mass and the 14.3 on a rigidity came from at a fifteen per cent spread.

The substitution is to stop treating CV as a number. It is

$$\mathrm{CV} = I \times \sqrt{\frac{1 + \mathrm{CV}_f^2}{n}},$$

the index of irregularity times the floor, and n is the count divided by the fibre’s fineness. Put that into the moment ratio and the excess becomes a function of the yarn’s specification and its spinning system, with no measurement left in it.

The price, quantity by quantity #

The table this collection computes for the excesses is the same table it always was; what has changed is its input, which is now a construction rather than a measurement somebody took off a delivery note.

Reading the two together gives the exchange rate. Going from 80 tex to 6 tex is a factor of thirteen in count, 3.7 in the floor, and — for the fourth-power quantity — fifteen in the bias. The exponent is what makes the trade so unfavourable: a bias built on a variance built on a reciprocal count multiplies the count’s exponent by two and then by the quantity’s own power.

A cover factor is the exception and it is exact. Cover is linear in the diameter, so the population average is the average thread’s value at any spread whatever — at 6 tex as at 100. That is the control on the whole method and it is why a cover factor remains the most trustworthy number in this collection.

The rest of the table moves by its own exponent and the ordering never changes. A mass and an areal weight are squares and move as CV²; a cloth’s thickness and its cover move as the first power of the diameter and do not move at all; a hole is a difference and is amplified by the ratio of diameter to spacing before any exponent touches it; a permeability carries an exponent between two and four depending on how close the cloth is, so the same yarn substitution costs more in a sheeting than in a muslin. Every one of those is the same substitution done once, and none of them needs a new measurement.

The one escape, and it is the fibre #

The floor is 1/√n and n is the yarn’s count over the fibre’s count. There are therefore two ways to raise n, and only one of them is available to somebody who has been told what yarn to spin.

Spinning a coarser yarn raises n and is not an option: the count is the specification.

Spinning from a finer fibre raises n at the same yarn count, and it is the only lever left. A 20 tex cotton at 1.7 decitex has a hundred and eighteen fibres and a floor of 9.9 per cent. The same 20 tex in a 1.5 decitex polyester has a hundred and thirty-three and a floor of 8.7 — and in a 0.5 decitex microfibre it has four hundred and a floor of 5.0, which is half the cotton’s. A cut man-made fibre also has almost no fineness variation of its own, so it escapes the eight per cent correction that a natural fibre pays as well.

That is what a microfibre is for, stated in one line: it is a way of buying a lower evenness floor at a fixed yarn count. The smoothness and the drape usually given as the reason are downstream of it, and so is the fact that a microfibre cloth can be set closer without looking coarse.

The same argument says why the fine cottons are the long, fine ones and why they cost what they do. A grower cannot make cotton finer at will; a spinneret can be changed in an afternoon. Every advantage a manufactured staple has over a natural one in evenness is this one advantage, and it is a count.

The extremes, which do not have an exchange rate #

A bias can be corrected. An extreme cannot, because it is the answer to a different question — over how much cloth? — and no factor converts one sample size into another.

That is the practical version of the argument and it is the one a weaver meets. A warp jams where its threads are thickest, and the excess of that extreme over the mean grows with the spread. So the same move to a finer count that made the bias worse by an exponent makes the jam worse by a tail — and the two failures show up in different places, the first in a prediction that is out and the second in a warp that will not weave.

There is a second reason the extremes are the harder half, and it is about what a specification can carry. A bias is a property of the yarn and travels with it: a certificate quoting a count and an index determines every one of the curvature corrections above, for anybody, anywhere. An extreme is a property of the yarn and the amount of it being asked about, so the same yarn has a different answer on a narrow loom and a wide one, in a short seam and a long one, under a small presser foot and a large one. Nothing on a delivery note can state it, and no amount of measurement by the spinner will produce it.

The same is true at the other end. A yarn breaks at its thinnest place, a bundle is weaker than its threads, and both are minima over a sample. Fineness makes each of them worse and by more than it makes the means worse.

The exchange rate, stated as a single exponent #

The essay’s claim is that fineness is bought at a fixed rate, and it is worth writing the rate down as one number for each quantity, because then a designer can read off what a count substitution costs without recomputing anything.

The chain is short. The floor goes as the inverse root of the count. A variance is a square, so it goes as the reciprocal of the count. A curvature bias is half the variance times a second derivative, so it goes as the reciprocal too — and the quantity’s own exponent multiplies it once more, as k(k − 1)/2.

So the bias in a quantity of exponent k goes as 1/tex, with a coefficient of k(k − 1)/2 times the index squared times the fibre’s fineness. Reading that off:

Every row is one over the count, times a number that depends on nothing but the quantity’s own power. So halving the count doubles every bias in the table, exactly, and the ordering of the rows never changes.

That makes the exchange rate quotable in one sentence. Halving a yarn’s count doubles every population correction in this collection, and leaves the cover factor at nought. Nothing about the spinning, the fibre or the construction enters that statement; it is the reciprocal in the floor, squared into a variance, and it is the same reciprocal for every quantity.

And it says where the arithmetic stops being worth doing. A bias of half a per cent is not worth a correction; a bias of forty-two is not worth a prediction. Somewhere between the two is a count at which this collection’s numbers stop being usable without the correction and start being usable with it — which is around 20 tex for a fourth-power quantity and around 5 for a square. Below that the corrections themselves are large enough for the quadratic approximation to fail, and the honest answer is the full integral or nothing.

And the ordering of the two thresholds is the awkward part. A fourth-power quantity needs the correction at a count where a square-law one does not, so a single cutoff cannot be stated for the collection as a whole — the right question is always about a particular quantity, and a designer who applies the corrections at one count and not at another is making a decision per number rather than per yarn.

What this says about thread count #

There is a claim in the trade that a high thread count means a good cloth, and this collection has taken it apart once already on the grounds that a thread count says nothing without a yarn count beside it. The floor adds a second and sharper objection.

A high thread count requires a fine yarn, because threads can only be set so close. A fine yarn has a high floor. So a cloth advertised on its thread count is a cloth made of the most irregular yarn its construction admits — and the irregularity is visible, because a cloth cannot be more even than its yarn except by averaging, and the averaging is what a close sett does.

The two effects genuinely oppose each other, which is why the practice is not simply wrong. A close cloth of fine irregular yarn averages more threads into each patch the eye resolves; a coarse cloth of even yarn averages fewer. Which wins is a computation rather than a slogan, and it is the subject of its own rung.

What to do about it, which is not obvious #

A cloth designer reading the above has three moves and they are worth separating, because two of them are commonly confused.

Accept the bias and correct for it. The curvature corrections are exact functions of the count and the index, both of which are on the yarn’s own specification. A rigidity computed from the mean thread and then multiplied by (1 + CV²)⁶ is right; a rigidity computed from the mean thread and left alone is 41 per cent low in a fine yarn and three per cent low in a coarse one. This costs nothing and is the whole of what the arithmetic asks.

Buy the index rather than the count. Between two yarns of the same count, the one with the lower index is genuinely more even, and the difference is the only part of the measurement that a spinner earned. Between two yarns of different counts, comparing raw coefficients of variation is comparing floors and tells nobody anything.

Fold, if fineness is the requirement. Folding is a doubling of the count for evenness purposes and lowers the floor by exactly √2 — although, as that rung shows, it lowers the measured irregularity by exactly the same √2, so the index is untouched and nothing about the spinning has improved. What has improved is the thread, which is what the cloth is made of.

What does not work is the move the trade language invites: paying more for a finer yarn and expecting a more even one. Those are opposite directions on the same axis.

What was counted, and how #

The floor’s exponent is asserted at exactly −½ over the whole range of counts, rather than fitted to the curve. A relation that merely falls would be satisfied by any decreasing function and would not support the exchange-rate claim, which is entirely about the exponent.

The moment ratio is checked against quadrature to twelve digits at every power used, and the quadratic approximation is quoted beside the exact value in the excess table so that a reader can see where it stops working. At the spreads a coarse yarn has it is good to a part in a thousand; at a 6 tex yarn’s spread it is out by several per cent, and the direction of the error is not constant.

The two spreads used in the tables above are computed, not chosen. They are the floor at the stated count times an index of 1.35, and the index is the mid-band ring value. Substituting the rotor band’s 1.8 raises both tables and leaves every ordering in them alone.

Where the model stops #

The index is assumed constant across the count range and it is not. A coarse yarn is easier to draft evenly than a fine one, so the index itself tends to rise as the count falls. That makes the conclusion stronger rather than weaker, and it is left out because nothing here can predict by how much.

Only the diameter is given a distribution. Twist, strength and stretch vary too, and not independently of it. The exchange rate priced here is the exchange rate for one of the yarn’s several irregularities.

The comparison holds the fibre fixed. In practice a finer count is spun from a finer and longer cotton, which raises the fibre count at the same mass and partly offsets the loss. That is a real effect and it is the reason the trade’s fine counts exist at all; it is also bounded, because the fibre’s own fineness sets the finest count and a spinner who has bought the finer cotton has spent the offset getting there.

Nothing here is about a fault. A thin place bad enough to break is not a large deviation from a mean but a different object, and it is priced by a different argument.

Where the ladder goes next #

Into the two things the floor reaches through a route other than a bias.

The first is strength, which is a minimum rather than a mean: a yarn breaks at its thinnest section, and the thinnest section over a test length is an order statistic of the same distribution the floor bounds. That gives the strange result that evenness and strength are one measurement taken two ways.

The second is the cloth’s appearance, where the effect runs the other way: weaving averages many threads into every patch the eye resolves, so a cloth is more even than its yarn by the square root of the number of threads in view. Whether the fine irregular yarn or the coarse even one gives the smoother-looking cloth is then a computation, and it has an answer.