Where the cover factor comes from

A specification for woven cloth carries four numbers and a weave. Two of them are counts, and two of them are not thread counts at all.

Mills do not specify cloth by thread count. They specify it by cover factor: the threads per inch divided by the square root of the cotton count, one number for the warp and one for the weft. A shirting is a 14 by 11; a sheeting might be an 18 by 16.

Every reference that gives the scale gives the same landmark with it — a cover factor of 28 corresponds to a completely covered surface — and none of them says where the twenty-eight came from. It looks like a convention, of the kind that gets chosen so the useful range lands on convenient numbers.

It is not a convention. It is a physical constant wearing a disguise, and once it is undisguised the scale’s practical ceilings stop being folklore. The number a mill quotes as the most a plain-woven cotton will take, and the higher number it quotes for a satin, both come out of two lines of substitution — and neither appears to be derived anywhere, which is a strange thing to be true of a quantity that appears on every order.

The two quantities #

The geometric cover in one direction is the fraction of the cloth’s width that thread occupies: the number of threads per inch times the width of each, which is the diameter. Call it $K_{\text{geom}} = n\,d$. It is dimensionless, it is one when the threads touch, and it is what cover factor is usually mistaken for.

The traditional cover factor is $K = n/\sqrt{\text{Ne}}$, where Ne is the cotton count. It is not dimensionless in any honest sense, since a count is length per unit mass and its square root is not a length; the ratio has units and the scale simply does not carry them.

The two are usually presented as different things: one geometric and exact, one practical and conventional. They are related by a substitution one line long, and the line has been available since 1937.

The line #

Peirce’s empirical rule says the diameter of a cotton yarn is

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

Put that into the geometric cover:

$$K_{\text{geom}} = n\,d = \frac{n}{28\sqrt{\text{Ne}}} = \frac{K}{28}.$$

The traditional cover factor is the geometric cover multiplied by twenty-eight. The constant in the scale is the constant in Peirce’s diameter, and nothing else. A cover factor of 28 means a geometric cover of one, which is precisely what the references say it means and precisely what they do not explain.

The figures check it rather than asserting it. Two curves are computed independently — the sett at which the geometric cover reaches one, and the sett a cover factor of 28 prescribes — and required to agree to machine precision across forty counts spanning a factor of twenty. They agree exactly, because they are the same expression.

So the scale is a diameter, and it inherits a diameter’s assumptions #

Recognising the twenty-eight makes the scale better and it also makes its limits visible.

Peirce’s rule is an empirical relation for cotton, and inverting it shows what it assumes: a packing factor of about 0.601, which is the accepted figure for a ring-spun cotton yarn and not for anything else. A filament yarn packs far tighter; an open-end yarn packs looser; a bulked yarn is not describable this way at all.

So a cotton cover factor is a geometric cover computed with cotton’s packing factor baked in. Applied to a cotton cloth it is exact up to Peirce’s own accuracy. Applied to a polyester filament cloth at the same count it is systematically wrong, and wrong in a direction — the filament yarn is finer than the rule says, so the true cover is lower than the factor reports.

This is why the worsted, woollen and metric trades each have their own cover-factor scale with a different constant. Those constants are not arbitrary either. Each is the reciprocal of a diameter rule for that fibre and count system, and the reason nobody can remember which is which is that all of them are quoted as bare numbers with the diameter rule left behind.

What the practical ceilings turn out to be #

The best consequence is the one a mill would care about.

A geometric cover of one is a surface with no gaps. It is not a cloth: the threads have to bend past one another, and the sett at which they jam is well below the sett at which they would merely touch. That jamming sett is a fixed fraction of the covering sett — the two both scale as one over the diameter, so the count cancels — and the fraction depends on the weave and on nothing else.

Run the intersection count and the fractions are fifty per cent for a plain weave and eighty per cent for an eight-end satin. Multiply by twenty-eight and the ceilings on the traditional scale are

• plain weave: $K = 14.0$
• eight-end satin: $K = 22.4$

Those are the numbers the trade uses. Fourteen is quoted everywhere as the practical maximum cover factor for a plain-woven cotton, with square-set shirtings running in the elevens and twelves and anything above thirteen described as very heavily set. Nobody derives it. It falls out of the intersection count and the twenty-eight in two lines.

The figure computes both and asserts three things while it does so: that the fraction is constant across the whole count range, that it is below one, and that the weave interlacing less reaches nearer to full cover. If the fraction ever varied with count, the claim that the scale is count-free would be false and the figure would say so.

Why the scale survives despite being confusing #

A dimensionless cover would be easier to explain, so it is fair to ask why the trade keeps a scale that needs a constant.

The answer is that the trade’s scale is computable from things a mill knows and the geometric one is not. A weaver knows the count and the sett, because those are the two things written on the order. Nobody knows the yarn diameter: it is not measured, it varies with twist and tension, and the only way to obtain it is to apply a rule like Peirce’s — at which point the geometric cover has become the traditional one with a division by twenty-eight added.

So the scale hides the diameter because the diameter is the thing nobody has. That is a good reason and it explains the survival. What it does not excuse is the constant being presented as arbitrary, because a specifier who knows the twenty-eight is a diameter can see immediately why the scale fails on a filament yarn and where to get the right constant instead.

Reading a specification, with the constant in hand #

The practical payoff of knowing what the twenty-eight is comes when a cover factor has to be judged rather than merely recorded.

Take a shirting quoted as 14 by 11. Divide each by twenty-eight: the warp covers half the surface and the weft covers three-eighths of it. The cloth cover is $1 - (1-0.50)(1-0.393)$, which is a shade under seventy per cent, so almost a third of the surface is gap — which is what a shirting is for, and why it breathes.

Now take a sheeting at 18 by 16. That is 0.64 and 0.57, a cloth cover of eighty-five per cent, and a gap of fifteen. The two fabrics differ by a factor of two in the quantity a person actually perceives, and by a much smaller factor in the numbers on the order.

And take a claimed 20 by 20 plain-woven cotton. Both are above the fourteen the intersection count permits, so the cloth cannot be woven as stated. Either the weave is not plain, or the count is not cotton, or the setts are not what the order says. A cover factor read as a geometric cover catches that in a moment; read as a bare index on a scale nobody has explained, it merely looks ambitious.

That third case is the reason this essay is worth the arithmetic. The scale is a perfectly good instrument once its units are known, and an uninterpretable index until then.

What was counted, and how #

Nothing on this page is quoted from a table.

The identity between the two covers is computed twice and compared, over forty counts, and required to agree to within a part in a billion. The jamming setts come from the intersection count applied to actual drafts, with the interlacings walked off the matrix. The fractions are computed at every count in the range and their spread is required to be zero, which is the statement that the ceiling is count-free.

The packing factor behind Peirce’s constant is obtained by equating the empirical diameter with the volumetric one and solving, which leaves the packing as the only unknown — and the answer, 0.601, is checked against the accepted value for a ring-spun cotton yarn to within a twentieth.

One thing is not computed and is worth flagging. The claim that fourteen is the trade’s practical maximum for a plain weave is a claim about what mills do, taken from the literature, and it is offered as a match rather than as a derivation. The derivation produces 14.0; the coincidence with the quoted figure is evidence that the intersection count is the model the practice was built on, which is a historical claim rather than a geometric one.

The same constant in three trades #

If the twenty-eight is a diameter, then every other trade’s constant is the same physics with two things changed, and separating them says which part of the difference anybody could have predicted.

A diameter is a mass per unit length divided by a density and a packing, turned into a width. So for any count system with count N, the diameter is proportional to one over the square root of N, and the constant in front is

K₀ ∝ √(ρ φ ÷ h),

where h is the system’s hank length — the constant that turns a count into a mass per unit length. Two things differ between trades and only one of them is about cloth.

The count system. A cotton hank is 840 yards to the pound and a worsted hank is 560, so at the same yarn a worsted count is exactly 1.5 times a cotton one. That contributes a factor of √1.5 = 1.22 to the constant, and it is pure bookkeeping — no fibre, no spinning, no measurement.

The fibre and its packing. Wool is 1.31 g/cm³ against cotton’s 1.52, and a worsted yarn packs a little more openly than a ring-spun cotton. Together those move the diameter by ten to fifteen per cent and the constant with them.

So the count system contributes about twice what the material does, which is the opposite of what a reader would guess from the fact that the scales are named after fibres. A worsted cover factor differs from a cotton one mostly because a worsted hank is 560 yards, and only secondarily because wool is not cotton.

The metric case is the cleanest, because tex has no hank in it at all. Working the diameter straight through for a ring-spun cotton — 1.52 g/cm³ at a packing of 0.6 — gives 0.167 mm at 20 tex, so the diameter per root tex is 0.003734 cm, and full cover is reached at

(threads per cm) × √tex = 268.

That is the metric scale’s own twenty-eight, computed rather than quoted. And it is the same number: converting Peirce’s inch rule into centimetres and dividing by the root of tex gives 0.003733, which agrees to four figures. The two constants are one constant in different clothes, and the agreement is the check that the substitution in this essay is an identity rather than a coincidence of scales.

The practical form is a warning about conversion. A specifier moving a cloth from the cotton system to tex must change both the count and the constant, and changing only the first produces a cover factor that is wrong by a factor of ten and looks like a plausible number on the other scale. That is the compound error the count-systems figure above is about, arriving in the one quantity where it is hardest to notice — because a cover factor has no units to disagree with.

What the scale cannot tell anybody #

Three things, and the third is the one that gets a specification into trouble.

It says nothing about the weave. A cover factor of 16 is impossible in a plain weave and comfortable in a satin, and the number carries no indication of which. A pair of cover factors without a weave beside them is under-specified.

It says nothing about how the cover is divided. A 14 by 8 and an 11 by 11 have nearly the same cloth cover and are entirely different fabrics — one heavily warp-faced with a soft weft, the other square. The pair matters and the sum does not.

It says nothing about the thickness. Two cloths of the same cover can differ by a third in thickness depending on which section model describes the yarn, which is a disagreement the models have among themselves. Cover is a statement about a plan view and a fabric is not one.

And it does not compose. Warp cover and weft cover do not add. The fraction of the surface covered is $1 - (1 - K_1)(1 - K_2)$ with both taken as fractions, so the open area is the product — and it is the open area, not the covered area, that a person judges a sheet by. That is the whole argument of the rung this ladder starts on, and it is why a specification given as a single summed cover factor is worse than one given as a pair.

Who worked it out #

The cover-factor scale is older than the explanation. Cotton mills were specifying cloth this way well before 1937, which is the year Peirce published the diameter rule that turns out to be its constant, so the scale cannot have been derived from the rule. It was fitted the other way: somebody chose a divisor that made a covered cloth come out at a round number, and the divisor they chose was the one the physics dictated because the covered cloth they were looking at was a real cloth.

That is a familiar shape in this subject. The satin condition was a workshop rule for centuries before it was a theorem about coprimality; the counter is a lattice-packing condition arrived at by looking; and the twenty-eight is a diameter arrived at by choosing a scale that behaved well. In each case the practice encodes a quantity nobody had isolated, and the isolation comes later and changes nothing about what to do.

There is a smaller historical detail that supports the reading. The scale’s constant differs between trades, and each trade’s constant tracks its own count system and its own fibre — the worsted scale, the woollen scale and the metric scale all put full cover at a different number. If twenty-eight were a convention chosen for convenience, the other scales would have been chosen for convenience too and would have landed on rounder numbers than they did. They landed where the diameters put them.

Peirce’s own paper does not mention the cover-factor scale, and the textbooks that give both give them in separate chapters. The identity is not hidden, exactly — it is two lines of substitution and anybody who wrote them down would find it — but it does not appear to be stated anywhere in the standard references, which is why it is stated here with the arithmetic shown rather than cited.

Where the ladder goes next #

Below this rung are thread count and what it is not, which is the same quantity approached from the consumer’s side, and balance, which is how the cover divides between the two systems.

Sideways, the maximum sett supplies the fractions that became the ceilings here, and the count systems supply the square root the scale is built on.

What the pictures here cannot show. Every curve on this page is drawn from Peirce’s diameter rule, which is an empirical fit for cotton. A figure cannot show that the rule holds; it can only show what follows if it does. The one figure that tests it — the diameter plot — compares the rule with an independent calculation from conservation of volume and finds them within a fraction of a per cent, and that is the strongest support available here for everything else on the page.