The yarn count systems, and why there are several

A yarn has a fineness, and there are at least six ways of writing it down. Some of them get larger as the yarn gets finer. Some get smaller. Two of them differ by a factor of nine for no reason connected to anything physical, and one of them is defined in terms of a length of eight hundred and forty yards.

This looks like a mess and mostly is one, but it is not an arbitrary mess, and the shape of it says something about how the trade measures things.

Two ways to weigh a thread #

Everything follows from one fork in the road, taken independently by spinners and by merchants.

A direct system fixes a length and weighs it. Tex is grams per thousand metres; denier is grams per nine thousand; decitex is grams per ten thousand. A bigger number means a heavier thread of the same length, so a bigger number means a coarser yarn.

An indirect system fixes a weight and measures the length. The metric count Nm is metres per gram. The English cotton count Ne is the number of eight-hundred-and-forty-yard hanks in a pound. The worsted count is the same idea with five-hundred-and-sixty-yard hanks; the linen lea uses three hundred. A bigger number means more length for the same weight, so a bigger number means a finer yarn.

Both are natural, and which is natural depends on what is in front of the person doing the measuring. A spinner watching yarn come off a machine measures a length and weighs it: direct. A merchant with a bale to sell knows the weight and wants to know how far it will go: indirect. Neither convention is wrong and neither displaced the other, because both survived in the trades that invented them.

The consequence is a permanent hazard. A quoted count without its system is not merely ambiguous about magnitude, it is ambiguous about direction. Told that a yarn is a 40, a reader cannot say whether it is fine or coarse until they know which 40.

Reducing everything to one number #

The way out is a canonical unit and a conversion, and the canonical unit chosen here is tex.

Tex has one advantage over the others that is worth the switch: it is additive. Ply two yarns of 20 tex together and the result is 40 tex, because mass per length adds. Ply two Ne 30 yarns together and the result is Ne 15, because reciprocals do not add. The direct systems are linear in the thing being combined and the indirect ones are not, which makes tex the sane place to do arithmetic even for people who quote in something else.

The conversions are then arithmetic on definitions rather than remembered constants. A cotton hank is 840 yards, a yard is exactly 0.9144 metres, a pound is exactly 453.59237 grams — so

$$\text{Ne} \times \text{tex} = \frac{1000 \times 453.59237}{840 \times 0.9144} = 590.54$$

and every other indirect system differs only in the hank length. Nothing here is typed in from a table; the constants are computed from the definitions each time, which is the difference between a conversion that can be wrong by a factor of nine and one that cannot.

The check is worth stating because it is easy to skip. Converting a value out to every system and back must return it exactly, and a wrong hank length used twice cancels only if it is used the same way twice — which conversion out and conversion back deliberately do not do. A conversion table that is wrong by a constant factor produces numbers of exactly the right order, and nothing downstream ever complains.

From count to diameter #

Fineness is what the trade quotes. What the geometry of a cloth actually needs is a diameter, and getting from one to the other requires an assumption that is easy not to notice.

A yarn is not solid. It is fibres with air between them, and the fraction of its cross-section that is fibre is its packing factor. Given that, the diameter follows from arithmetic alone: mass per length divided by fibre density gives an area weighted by the packing, and area gives a diameter.

$$d = \sqrt{\frac{4\,\text{tex}}{\pi \times 10^5 \times \rho\,\phi}}$$

with $\rho$ the fibre density in grams per cubic centimetre and $\phi$ the packing factor. A 20 tex cotton yarn at a packing factor of 0.6 comes out at 0.167 mm.

The packing factor is the assumption. It is not a constant of nature; it depends on how the yarn was spun, on the twist level, and on how hard it has since been squeezed. A ring-spun cotton yarn is usually taken at about 0.6, an open-end yarn rather less, a filament yarn much more. Quoting a diameter without it is quoting an answer without one of its inputs.

What the oldest rule assumes #

There is a much older way to get a diameter, and it is still the one the trade uses. Peirce, in 1937, gave the diameter of a cotton yarn as

$$d = \frac{1}{28\sqrt{\text{Ne}}} \text{ inches}$$

and offered no derivation. It is a rule of thumb, and it works.

The interesting thing to do with a rule of thumb is not to use it but to ask it what it assumes. Both expressions scale as one over the square root of the count, so their ratio is a pure number independent of which yarn is chosen. Set them equal and the only unknown left is the packing factor — which means Peirce’s constant of 28 can be inverted.

The answer is 0.601.

That is not a round number and it is not a number anyone chose for the convenience of the arithmetic. It is very nearly exactly the packing factor a ring-spun cotton yarn is independently measured to have. Peirce’s 28 is a packing factor in disguise, and the rule works on cotton because cotton yarns are packed that way.

Two things follow, and the second is the useful one.

The rule is not a general result about yarn. Carry it to a polyester filament yarn, whose fibre density is 1.38 rather than 1.52 and whose packing is much higher, and the implied packing factor comes out at 0.66 — which is to say the rule is now assuming something about the yarn that is not true of it. The 28 encodes cotton twice over, in density and in packing, and neither is announced.

And the agreement is a genuine cross-check on the arithmetic here. Two derivations from different premises — one geometric, one empirical from 1937 — landing within a fraction of a per cent of each other over a five-fold range of counts is the kind of coincidence that does not happen when one of them has a mistake in it.

Twist, and the thing counts do not say #

A count says how much material there is per unit length. It says nothing whatever about how that material is arranged, and the arrangement is set by twist.

Twist is quoted as turns per unit length, and the number that matters is not the turns but the twist factor — turns per unit length divided by the square root of the count in an indirect system, or multiplied by it in a direct one. That grouping is not a convention; it is what makes two yarns of different counts have the same surface helix angle, which is the thing that actually governs how they behave.

Twist decides three things a count cannot, and none of them is visible in the matrix that decides the structure. It decides the strength, which rises with twist to a maximum and then falls as the fibres become too oblique to carry load. It decides the packing factor, and therefore the diameter — so a count and a diameter are related through twist, and the relationship in the previous section quietly assumed a normal one. And it decides the direction of the surface helix, which is the property that interacts with the direction of a twill line to make the same weave read soft or crisp.

So a yarn is specified by a count and a twist, and a great deal of writing about fabric quotes only the first.

Where this bites: the count in the cloth #

The reason any of this belongs on a site about structure is that count is one of the two inputs to every geometric statement about a fabric.

The cover factor — the fraction of the cloth’s area the threads occupy — is thread spacing times thread diameter, and the diameter comes from the count. That is why cover factor in the trade is quoted as thread count divided by the square root of the yarn count: it is the geometric cover with the square-root relationship folded in and the constants dropped.

The maximum sett is likewise a statement about diameters, so a jamming calculation done in threads per centimetre is a calculation done in yarn counts wearing a disguise. And the crimp follows from the same place, since how much longer a thread is than the cloth it crosses depends on how far it has to travel round the threads it meets — which is a statement about diameters and nothing else.

Where eight hundred and forty yards came from #

The hank lengths look arbitrary and are not quite. They are products of the units the trades measured in.

A cotton hank is seven leas of 120 yards, and a lea is 80 threads round a reel of 54 inches — which is a yard and a half. Multiply it out: $80 \times 1.5 = 120$ yards to the lea, seven leas to the hank, 840 yards. Every number in that chain is a piece of equipment: the reel’s circumference, the count of threads a worker wound before making a tie, the number of ties before the skein was taken off.

The worsted hank of 560 yards is the same reel with a different tie count, and the linen lea of 300 yards is a different reel entirely. Each system is a fossil of a workshop.

The denier is stranger and older. It began as the weight in French deniers of nine thousand metres of silk, and nine thousand metres is itself a metric restatement of an earlier length — so denier is a metric unit wrapped around a pre-metric practice, which is why it sits so awkwardly beside tex. The factor of nine between them has no physical meaning whatever.

Knowing this does not make the systems easier to use. It does explain why they are not going to be rationalised away by argument: each of them is the natural unit of a particular workshop, and the workshops came first.

What the diameter is for #

The point of getting a diameter out of a count is that the diameter is what every geometric model of a cloth actually consumes — and the models then disagree about what to do with it.

That comparison is the subject of the next rung, but the shape of it belongs here because it is a warning about what a count buys. A count gives an area. An area gives a diameter only if the section is circular, and a yarn in a cloth is squashed flat by the threads crossing it — the more so the more often it crosses them, which ties the question back to the interlacing count.

So the honest statement is that a count fixes an area, and everything past that is a model. This essay’s arithmetic — count to tex to diameter — is the first half of a calculation whose second half has choices in it.

The systems that will not die #

A last word on why the mess persists, because the answer is not simply inertia.

Tex was standardised in 1960 with the explicit aim of replacing all of it, and sixty-five years later the cotton trade still quotes Ne, the worsted trade still quotes Nw, filament yarn is still sold in denier and decitex, and linen is still occasionally sold in leas.

Part of that is ordinary conservatism. But part of it is that the indirect systems encode something real about how their trades work. A cotton spinner thinks in hanks because a hank is a unit of production. A merchant thinks in length per weight because length per weight is what a bale is worth. The unit is a piece of the trade’s own logic, and replacing it with a physically cleaner one loses information that the people using it find convenient.

The workable position, and the one taken here, is to convert everything to tex for arithmetic and quote in whatever the reader expects. What is not workable is quoting a bare number.

Where the additivity of tex stops #

Tex is recommended above because it adds: ply two 20 tex yarns and the result is 40 tex, where two Ne 30 yarns give Ne 15 and reciprocals do not add. That is true of the mass and it is not quite true of the count, and the gap is worth naming because it is systematic and it reaches every diameter computed from a folded yarn’s nominal count.

Mass per unit length adds only if the length is unchanged. Folding twist shortens the assembly: the singles take a helical path round the folded yarn’s axis, so a metre of folded yarn contains rather more than a metre of each single. The contraction is a few per cent at ordinary folding twists — the same helix that makes a twist an angle — so

a two-fold yarn is coarser than twice its singles, by the folding take-up.

Two 20 tex singles folded at a three per cent contraction give 41.2 tex, not 40. The nominal resultant count printed on the package is the sum; the true one is the sum divided by one minus the contraction.

Three consequences, and the second is the one that reaches this collection’s own arithmetic.

The nominal count is always fine and the real yarn is always coarser. The error has one sign, because a contraction cannot be negative, so it does not average out across a specification.

Every diameter computed from a nominal folded count is low by half the contraction, since a diameter goes as the square root — one and a half per cent at a three per cent take-up. And every cover factor, every jam and every hole computed from it carries the same error. That is small and it is a real systematic in a collection that computes covers to three figures.

And the error is larger for a hard-folded yarn. The contraction grows with the folding twist, so a sewing thread — folded hard, precisely so that its singles are locked — carries the largest discrepancy of anything in the trade. A 40 tex sewing thread quoted as two-fold 20 may be 42 or 43, which is worth knowing when the same figure is used to choose a needle.

The remedy is the one this essay recommends throughout and it needs nothing new: quote the resultant count as measured rather than as summed. A folded yarn’s tex is a mass per unit length like any other and can be weighed, and a package that carries the weighed figure is carrying the number every downstream calculation actually wants.

That the summed figure survives is for the same reason the indirect systems survive. It is what the spinner knows at the moment of folding, before the yarn exists to be weighed, and it is right to within a few per cent — which is exactly the accuracy at which the difference stops being visible and starts being carried.

What none of this measures #

Three limits, stated plainly.

A count is a mass, not a size. Two yarns of the same tex can have quite different diameters if they are packed differently, and the conversion in this essay is only as good as the packing factor it is given.

A count says nothing about uniformity. Real yarn varies along its length, and the variation is what makes cloth streaky. Every number here is a mean.

And nothing in this essay is about fibre. Two yarns of the same count in cotton and in polyester have different diameters because the densities differ, and they behave differently for a hundred reasons that are outside any of it. The count system is a way of writing down one number, and the number is a linear density.

Where the ladder goes next #

The count is one input to the geometry; the other is the shape of the section, and that is where the models start disagreeing. Peirce against the racetrack runs two of them side by side on the same yarn and finds they agree closely on one question and not at all on another.

The reason any of it was wanted is how close threads can be set, and the reason it matters commercially is that thread count is not quality — a claim which, restated in this essay’s vocabulary, is that a count of ends per inch means nothing without the count of the yarn in them.