The reed is not the sett

A reed is a comb. Its pitch is fixed by manufacture, the warp is drawn through it a stated number of ends per dent, and the product of those two numbers is the warp sett — ends per centimetre — held between the reed’s wires at the moment the pick is beaten up.

It is not the sett of the cloth, and the difference is not a tolerance. It is exactly the weft’s crimp.

The reason is one sentence. The weft’s thread length in a pick is set at the reed, because that is where the shuttle crossed. Once the cloth has left the reed the weft takes up its crimp — it goes over and under the warp rather than straight across — and a thread of fixed length that undulates spans less than one that does not. The cloth is therefore narrower than the reed by exactly the fraction the crimp accounts for, and the warp sett rises by exactly the same factor.

The arithmetic, which has no free parameter in it #

Write c₂ for the weft crimp — the fraction by which the weft thread is longer than the cloth it crosses. Then a length ℓ of weft thread spans ℓ/(1 + c₂) of cloth.

At the reed the thread spans the reed’s width, so the cloth’s width is that divided by (1 + c₂), and the contraction — how much narrower the cloth is, as a fraction of the reed width — is

contraction = c₂ / (1 + c₂)

The sett moves the other way by the same factor. A reed at n ends per centimetre produces cloth at n(1 + c₂), so a weaver working backwards from a specified cloth sett divides by one plus the weft crimp.

Every term in that is already on this site. The crimp comes from the Peirce solution at the cloth’s own construction, which is the same solve that produces the crimp in the interchange essays, the thickness in the section drawings and the extension available in the tensile ladder. Nothing is fitted here and nothing is new; what is new is noticing that the same number answers a question about the machine.

Why the spread is so wide #

The contraction follows the weft crimp, and the weft crimp follows the cover.

An open scrim has room: the warp ends are far apart, the weft passes between them with very little undulation, and its crimp is under two per cent. A close balanced sheeting has no room: the warp is set near its jam, so the weft has to climb over and under a nearly continuous surface, and its crimp reaches fifteen per cent.

So the contraction is largest exactly where the fabric is most tightly specified. A sheeting is bought to a width and a sett; a scrim is bought by the metre. The construction whose customer cares most about the number is the one where guessing it costs most.

The warp does the same thing along the piece #

Turn the argument through a right angle and it says something about length rather than width, and it is the loom’s own way of measuring cloth.

A warp end is longer than the cloth it lies in by its own crimp, so a beam of stated length yields a shorter piece: 1,000 metres of warp gives 872.5 metres of sheeting, 926.8 of muslin, 956.1 of a voile. That difference is the take-up, and it is what the loom’s take-up motion is named after.

The loom does not know how much cloth it has made. It knows how far it has turned the take-up roller and how much warp it has let off, and the two differ by a quantity that is a property of the yarn and the weave rather than of the machine. A weaver ordering a warp for a stated length of cloth is doing this arithmetic, usually from a table.

The width and the length are the same fact applied to the two thread systems, and it is worth being explicit that they are not independent: the two crimps are tied by the interchange, so a cloth beaten harder takes up more in the warp and less in the weft, and the reed allowance and the beam allowance move in opposite directions together.

What was counted, and how #

The crimps are not quoted. Each cloth in the table is solved at its own stated construction — warp and weft counts in tex, warp and weft setts in threads per centimetre — by the same Peirce geometry the rest of this site’s setting work uses, with diameters from the counts by conservation of volume at a stated packing factor.

The solve is checked before it is used: the geometry has an internal consistency condition and a cloth that solves to a negative crimp is refused outright, because a thread shorter than the cloth it crosses cannot exist and every downstream number would be arithmetic about a fabric that is not there.

Three assertions carry the table. Every cloth must come out of the reed narrower than it went in, which is a statement about the sign and would fail if the contraction had been written the wrong way up. The spread across the table must be more than a factor of two, which is the claim that a single allowance will not do. And the beam yield must be shorter than the beam, for the same reason as the first.

The table’s own numbers are worth stating plainly, since they are the essay’s content: sheeting 12.75 per cent, duck 10.02, filter cloth 10.59, poplin 8.23, muslin 7.32, batiste 6.66, voile 4.39, cheesecloth 1.87.

What a weaver actually does with it #

The arithmetic runs in both directions and the two uses are quite different jobs.

Forwards, from a reed to a cloth: a mill with a reed already in the loom asks what sett it will get, multiplies by one plus the weft crimp, and has the answer. That is the easy direction and it is the one a table serves well, because the reed is a fixed object with a stamped count on it.

Backwards, from a specification to a reed, is the direction that bites. A customer specifies the cloth’s sett, and what has to be chosen is a reed count and a number of ends per dent whose product, multiplied by one plus a crimp that is not known until the cloth has been woven, lands on the specified value. The crimp depends on the sett, the sett depends on the crimp, and the loop closes.

In practice it is solved by one iteration and a swatch, which is the right answer and is worth saying plainly rather than dressing up: the geometry here gives a first estimate good to a per cent or so, and the second estimate comes off the loom. What the geometry does buy is a starting point that is right for the construction rather than right for the last cloth somebody wove, and a sanity check afterwards — a measured contraction two points away from the computed one is evidence that something else is different, most often the tension the warp was held at.

The width follows the same loop and is the number a finishing plant cares about, because a piece that comes out narrow cannot be widened without stretching it, and stretching it moves the crimp back the other way — which is the shrinkage field’s whole subject seen from the other end.

The two allowances multiply, and the third number is the weight #

Width and length are the same fact at right angles, so a mill setting a beam and a reed is applying two corrections at once and what it actually gets is their product.

A warp of stated length at stated reed width encloses an area. The cloth that comes out of it encloses that area divided by (1 + c₁)(1 + c₂), because the length is spent on the warp’s crimp and the width on the weft’s. For the balanced sheeting, whose two crimps are both 14.61 per cent under the equal-share convention, that product is 1.313: a thousand metres of warp at a metre of reed width yields 761 square metres of cloth, and the missing quarter is entirely inside the threads.

The same product appears somewhere the site has been using it for other reasons, and noticing that it is the same product is the point of this section. A cloth’s areal weight is the threads’ weight per square metre multiplied by exactly those two factors — each system’s contribution is its sett times its count times one plus its own crimp — so the sheeting weighs 31 per cent more per square metre than the same threads would if they were laid flat and straight in the same numbers.

That is one arithmetic answering three questions. How wide the cloth comes out, how long it comes out, and how much it weighs are all the two crimps, used once each, twice, and once each again. A weaver who has computed a reed allowance has already computed the beam allowance’s partner and the weight’s, and a table that lists the three separately is a table with one number in it three times.

Beating harder makes the cloth wider and shorter #

The two crimps are tied by the interchange, so they cannot be adjusted separately, and the consequence for the loom is concrete enough to be worth stating as an instruction.

Beat the pick up harder and the warp is forced to bend further round it: the warp’s crimp rises and the weft’s falls, since the two share one thickness between them. A higher warp crimp means more take-up, so the beam yields less cloth. A lower weft crimp means less contraction, so the cloth comes out wider. A change in beat-up moves the two allowances in opposite directions at once, and a mill correcting a narrow piece by beating harder has just shortened its own output.

What is much steadier than either is the product. The interchange holds the thread lengths fixed and moves the crimp between the systems, so (1 + c₁)(1 + c₂) changes by far less than either factor does — which means the area a beam yields, and the cloth’s weight per square metre, are both much less sensitive to how the loom is running than the width and the length are separately.

That is why a mill measuring output by weight sees a steadier process than one measuring it by width.

The same insensitivity is what makes weight a poor acceptance test for a construction, and it is the reason a specification carries a sett as well as a weight: two cloths of identical areal weight can have arrived there with the crimp divided quite differently between their systems, and everything the crimp decides — the extension, the thickness, the cover, the width — will differ between them. It is also a warning about the reverse: a piece whose weight is exactly to specification can still be out of tolerance on width, because the quantity that was controlled is the one that barely moves.

Where the model stops #

Peirce’s geometry is a plain-weave geometry, and every row of the table is a plain weave. A twill’s crimp is smaller at the same sett because it interlaces less often, so its contraction is smaller too, and the arithmetic is unchanged with a different number in it. What this essay does not do is compute that number for other weaves, because the model that would is the one the site already says stops at plain weave.

The crimp ratio is a free parameter and it is set to one. Peirce’s solve leaves how the two systems share the thickness undetermined; taking it as an equal share is a convention, and a cloth off the loom does not have an equal share because the warp has been held under tension throughout. That is a real effect in the direction that matters — it moves the weft crimp up and the warp crimp down — and it is a processing fact the geometry cannot supply.

The reed’s own denting is absent. A reed does not space a warp evenly; it groups it, several ends to a dent, and the grouping is a second grid with its own consequences. The sett used here is the mean, which is the right quantity for a width calculation and the wrong one for a mark.

And nothing here is about tension. The width a cloth reaches when it leaves the reed is not the width it reaches when it is finally relaxed; that is a further contraction with a different mechanism, and this site has a whole field about it. The number in this essay is the loom’s, not the finisher’s.

The generalisation #

The shape of this is a conserved quantity crossing a boundary between two states, and reading the conservation off gives the correction exactly.

The thread’s length is what is conserved. The reed fixes it and the cloth spends it, and the difference between how the two states spend a fixed length is the whole of the correction. No property of the machine enters, no dynamics, no tension, no beat-up force — only the geometry of the two states and the fact that the same thread is in both.

That is why the answer is exact rather than empirical, and it is why a table of reed allowances measured in a mill agrees with it. The mill’s table is a table of weft crimps with a different heading.

The general form is worth carrying: whenever a fixed quantity is laid down in one geometry and consumed in another, the ratio between the two is a property of the geometries alone. A great deal of textile arithmetic is that observation applied to thread length, and the two rungs of this ladder are two instances of it at right angles.

Who found it, and when #

Reed calculations are as old as the reed, and every weaving manual has a table or a rule of thumb. The trade’s usual form is a percentage allowance — “allow eight per cent for take-up in width” — quoted for a class of cloth, which is right in the middle of the range and wrong at both ends by a factor of two.

What the manuals do not usually say is that the allowance is the weft crimp, exactly, and therefore that it is the same number the fabric’s crimp interchange, its extension, its thickness and its cover factor are all computed from. Stating it that way turns a look-up into a solve, and it means a construction nobody has tabulated is answerable from the same geometry as one that has been.

Peirce’s model is 1937 and it was written to answer questions about cloth geometry rather than about looms. The application here is direct and, as far as this site knows, uncontroversial — the interest is not in the model but in noticing that a machine setting is one of its outputs.

Where the ladder goes next #

The next rung asks a different question about the same two numbers: not what they are but what they can be. The warp sett is a reed and a whole number of ends per dent, so it takes the values a catalogue reaches and nothing between; the pick density is a pair of change wheels, so it takes thousands of values. Over the range ordinary cloth is woven in, the loom can choose one of the two setts ninety-nine times more finely than the other — and the matrix that describes the cloth does not distinguish them at all.

Sideways, the state this essay ends at is what comes off the loom, which is the finishing field’s starting point; the crimp it is all computed from is the interchange; and the reed’s other consequence, which is a mark rather than a width, is the beat it makes against the weave.