What a fabric weighs

Ask for a fabric and the first thing quoted is its weight. A hundred and twenty grams per square metre for a shirting, three hundred and forty for a denim, sixty for a lining. It is what a buyer specifies, what a mill costs against, and what is printed on every swatch card in the trade.

It is also a sum of four products in which no term appears alone.

mass per square metre = Σ ( sett × count × (1 + crimp) ) ÷ 10

summed over warp and weft, with setts in threads per centimetre and counts in tex. Everything in it is a length of yarn: how many threads there are per centimetre, how heavy a metre of one is, and how much longer than the cloth each one has to be because it goes over and under.

Two things follow, and the second is why the number is so much less informative than it looks.

The crimp term is not a correction #

A weight computed from setts and counts alone is the weight of a cloth whose threads run straight, and a cloth whose threads run straight is not a cloth.

This site has computed crimp from Peirce’s geometry since its foundation, so the term can be filled in rather than estimated. The route is the site’s usual one: the two diameters come from the counts by conservation of volume, the two spacings from the setts, and the weave enters through the float — a thread that floats over three picks bends once in three, so its bending pitch is three spacings.

That last step makes the weave a term in the weight, which is not how anybody quotes it.

Same yarns, same setts, five weaves, 12.8 grams a square metre between the ends of the column — eleven per cent, from nothing but how often the threads bend. A specification that names a weight and not a weave has left out a tenth of the answer.

The assertion behind that table is written against the mechanism rather than the numbers, which is a rule this site has had to learn three times over. It says that a weave which interlaces more often weighs more, checked over every pair whose firmness actually differs — and it deliberately exempts the 2/2 and 3/1 twills, which interlace exactly as often and differ only in where. Those two weigh very nearly the same, and which of them comes out heavier is decided by how the floats divide between the systems rather than by the mechanism the assertion is about. Asserting a strict order over a tie is how a true statement gets a failing test.

One equation, four unknowns #

The identity cannot be inverted, and that is the practical half of the essay.

A weight of a hundred and fifty grams is consistent with an eight tex yarn at eighty-eight ends per centimetre, with a hundred tex at seven, and with everything between. After the balance is fixed there is still a one-parameter family, and the members of it have nothing in common.

Four of those six rows can be woven, and across the whole sweep ten of fifteen counts can, spanning a cover factor from 0.28 to 0.77 — a factor of 2.8. One of those cloths is a close fine shirting and another is a loose coarse sacking, and both weigh a hundred and fifty grams a square metre. A specification that names only the weight has named almost nothing, and it is what most of them name.

The check on that family is worth stating because it is the kind that is easy to skip: every row is fed back through the weight identity and required to come out at the weight asked for, to a part in a billion. A sett solved from an equation and not checked against it is a place where a factor of ten hides.

The optimum is at the edge, and that is the result #

Inside the family there is a best cloth by cover, and where it sits is exact and slightly surprising.

Cover goes as sett times diameter. Diameter goes as the square root of the count. Weight goes as sett times count. So halving the count and doubling the sett holds the weight exactly — and multiplies the cover by exactly √2, because the diameter fell by √2 and the sett doubled.

There is no interior optimum. Every step towards a finer yarn improves the cover and holds the weight, so the improvement continues until something stops it, and the only thing that stops it is the jam: the sett at which neighbouring threads touch and no loom can insert a weft.

So the most covered cloth at a fixed weight is the finest yarn its jam allows, and the site asserts exactly that — that the best row is the finest weavable one, rather than checking a value.

That is the arithmetic behind a piece of trade knowledge everybody has and nobody derives: a fine-count cloth at a given weight feels denser, covers better and drapes differently from a coarse-count cloth of the same weight, and the fineness is the whole of the difference. It is also why fine counts are expensive — they are the only way to buy cover without buying weight.

Where the family runs out #

At the heavy end the arithmetic refuses, and the refusal says something rather than reporting a failure.

At three hundred and forty grams — a denim weight — only three counts in the sweep are weavable, and all three are coarse. Every finer count needs a sett past its own jam: the arithmetic returns a number, the loom cannot. And at nine hundred grams nothing is weavable at all, which is the arithmetic saying that a fabric of that weight is not a single cloth of one yarn. It is a double cloth, a pile, a felt, or a coating — a change of construction rather than of specification, which is the same shape of answer the applied field keeps arriving at.

The refusal quotes both ends of the sweep and the jam each would need, so the message says which direction the specification is impossible in.

What a second number buys, and which second number #

Naming the weight leaves three degrees of freedom and the trade always names a second thing. Which one it names is not arbitrary, and setting the candidates against each other says how much each is worth.

The count is the most informative. Weight and count together fix the sett at a stated balance, and the sett fixes the cover — so two numbers determine the cloth’s density, its openness, its jam margin and very nearly its handle. That is why a mill’s own record is a weight and a count, and why the pair is what a technician asks for first.

The thread count is the least. Ends plus picks per centimetre, with the weight, fixes the product of sett and count and therefore says only what the weight already said in another arrangement. It does eliminate one degree of freedom, but it eliminates the one that was already implied — which is a formal way of saying what the thread-count essay says by argument: the number is close to redundant beside a weight and is quoted as though it were independent.

The cover factor would be ideal and is never quoted. Cover and weight together fix the count directly, because cover goes as sett times the root of the count and weight as sett times the count, so their ratio is the root of the count and nothing else. Two numbers, one division, and the whole family collapses to a point. Nothing in the trade’s vocabulary is better suited to specifying a fabric in two numbers, and it is a quantity a mill computes and a buyer never sees.

That is a small, exact and slightly damning result. The two numbers universally quoted are weight and thread count, which between them leave two degrees of freedom; the two numbers that would leave none are weight and cover; and the difference between the two pairs is that one of them is what a merchant can measure by holding a cloth up and counting.

The weave remains outside all of it. Every pair above fixes the geometry and none of them fixes the crimp, which is worth up to eleven per cent of the weight and is decided by the interlacing rate. So even the ideal pair is a pair plus a weave, which is three — and a specification carrying weight, cover and weave would be a complete description of a cloth in three numbers a mill already has.

What the weight does say #

Having spent a section on what the number does not carry, it is worth being clear about what it does — because the trade quotes it first for a reason and the reason is not laziness.

It is the cost. Yarn is bought by mass and a fabric is mostly yarn, so grams per square metre is very nearly the raw-material cost per square metre. Nothing else on a specification is — not the thread count, which the trade has spent decades misusing for the purpose, and not the weave.

It is conserved through finishing in the only way that matters. A cloth shrinks and gains weight per unit area without gaining any mass, so a metre of loom-state cloth and the finished piece it becomes carry the same yarn — which makes weight the one quantity that can be tracked from the spinner to the shelf.

And it bounds the others. A fabric of a hundred grams cannot be a close cloth of coarse yarn, because the arithmetic will not allow it; a fabric of five hundred cannot be a fine open one. The weight does not determine the construction and it does exclude most constructions, which is why a buyer who knows the weight and one other number knows nearly everything.

That last is the practical shape of the whole essay. One equation with four unknowns leaves three degrees of freedom; naming the count leaves two; naming the balance leaves one; and naming the weave closes it. Four numbers, and the trade routinely quotes two.

The balance is the fourth number, and it moves everything #

Every figure above is balanced — the same sett and the same count both ways. Real cloths are not, and the identity has the balance in it twice.

Raise the warp sett and hold the weight: the weft sett falls to compensate, so the cloth becomes warp-faced, the warp cover rises towards its jam and the weft cover falls away from it. The total cover — the fraction of the surface hidden by either system — moves surprisingly little, because the two systems overlap: a surface hidden twice is still hidden once.

That is the arithmetic reason a specification can move a long way from balance without changing the weight or the apparent density, and it is why so many real fabrics are unbalanced. The firmness of the construction sets how far the two setts can diverge before the closer system jams. It is also why the weight identity underdetermines a cloth by more than the count alone suggests: the family is one-parameter at a fixed balance and two-parameter without one.

What was counted, and how #

Three computations, each with the check that would catch it being wrong.

The weights. The identity is arithmetic and needs no check; the crimp in it does. Peirce’s geometry is solved for each weave at its own bending pitch and the solution is re-run forwards through the equations it came from, so a residual larger than a part in a billion stops the build. That check has been in place since its earliest essays and it is the reason a crimp on this site can be quoted rather than hedged.

The family. Fifteen counts, the sett solved from the weight for each, and every one fed back through the identity and required to reproduce the weight to a part in a billion. Then each is compared against its own jam, computed from the section model, and the ones past it are marked rather than dropped — because a specification that can only be met past the jam is a fact worth showing.

The scaling. Halving the count and doubling the sett is asserted to hold the weight exactly and to multiply the cover by exactly √2, to twelve decimal places, at a stated count and sett. That is one line of algebra and it is the line the whole “no interior optimum” argument rests on, so it is run rather than believed.

Where the model stops #

The crimp division is stated, not derived. Peirce’s equations leave one degree of freedom — they do not decide how the crimp splits between warp and weft — so the crimp ratio is an input here as it is everywhere else on this site. A cloth whose warp was held harder on the loom has a low-crimp warp and a highly crimped weft at the same total, and the weight is unchanged only if the two counts are equal.

The float-to-crimp step is a model. Handing Peirce’s geometry a longer spacing for a floated thread assumes the floated crossings leave the thread straight, which is what a float is, and ignores the crossing threads pressing on it in passing, which a real cloth does not.

Nothing here is finished cloth. A fabric contracts in finishing and gains weight per unit area without a gram being added — five to ten per cent is ordinary — so a loom-state weight and a finished weight are different numbers and the trade quotes the second — and how far apart they are is the crimp coming back. The identity is the same and the setts in it are the finished ones.

And the jam is a single-yarn jam. The bound used is the round-section closure condition. A flattened yarn jams at a different sett and reaches a different cover, so the optimum’s position moves with the section model — which is one more reason the site runs Peirce’s circle and Kemp’s racetrack side by side rather than blending them.

Who found it, and when #

The identity is not anybody’s. It is a bookkeeping statement that has been in mill practice for as long as yarn has had a count, and it appears in every fabric-analysis text in the same form: weight equals ends times count plus picks times count, with a crimp allowance.

What varies between texts is that allowance, and the variation is the interesting part. Some give a flat percentage — five, eight, ten — some give a table by weave, and some omit it and quote the straight-thread weight. The site’s own arithmetic says the correct figure runs from half a per cent to twelve depending on the weave and the sett, so a flat allowance is right for one construction and wrong for the rest.

The inversion problem is not discussed anywhere this site can find, which is the more surprising absence. It is immediate — one equation, four unknowns — and it undermines the way fabrics are actually specified in a way that would be worth a paragraph in any of those texts. The nearest the trade comes is the working knowledge that a weight is quoted alongside a count and a construction, which is exactly the admission that the weight alone is insufficient, made as a habit rather than as a statement.

Where the ladder goes next #

The weight identity has the sett, the count and the crimp in it, and the site has essays on all three. The one thing it does not have is a thickness: two cloths of the same areal weight can differ by a factor of two in how thick they are, and thickness is what decides warmth, bulk and how a garment sits. That needs the section model rather than the count model, and it is the next rung this anchor wants.