Twist and the twill line

Open any weaving manual at the chapter on twills and there is a sentence like this one: for a prominent twill, use a warp whose twist runs opposite to the twill; for a subdued one, use a warp whose twist runs the same way. It is given as a rule, without a reason and without a number, and it is correct — which is the usual condition of a trade constant on this site, and the usual invitation to ask what it assumes.

Both halves of it are geometry, and the site has already computed each half for another purpose. Putting them on the same square of cloth takes one subtraction.

The twill’s angle is a sett ratio #

A twill’s diagonal is not a thread. No yarn in a woven cloth runs diagonally; the line is a sequence of interlacings, one step across for each step down, and its angle on the cloth depends on how far a step is in each direction.

One pick down is one over the weft sett; one end across is one over the warp sett. So the line’s angle from the weft direction is the arctangent of the warp sett over the weft sett, and its angle from the warp is the complement — which is the form this essay needs, because a yarn’s twist is measured from the yarn’s own axis and a warp end’s axis runs down the cloth.

A square-set cloth gives 45°. A warp-faced cloth gives less: at forty ends and twenty picks the line stands 26.6° from the warp, which is why denim’s twill looks steep. A weft-faced one gives more.

This is why “a 45° twill” means a balanced cloth and not a weave. The step is one end per pick in all of the above, and the angle moved by nineteen degrees without the draft changing at all.

The fibres’ angle is a twist factor #

The previous rung reduced twist to a single number: the surface fibres of a yarn lie at α to its own axis, tan α is πd**T, and substituting the diameter arithmetic makes α a function of the twist factor and of nothing else.

A warp end’s axis runs down the cloth. So its surface fibres run at α from the warp direction — up to the right for a Z-twist yarn, up to the left for an S-twist one.

At 20 tex and 900 turns per metre that is 25.3°, on either side depending on the handedness.

The subtraction, and what it means for the eye #

Two directions on a surface, both measured from the warp, both signed by a handedness. The angle between them is a subtraction:

When the twill and the twist run the same way the fibres lie along the line. When they run opposite, the fibres cross it.

The optics do the rest. A bundle of parallel fibres reflects light specularly along its own direction — the same argument that makes a satin shine, applied at the scale of a fibre rather than of a float. So fibres running along a twill line make a continuous streak of reflection down the line, which fills it in and blurs the boundary between the line and the ground: the twill reads as subdued. Fibres running across it break the reflection at every fibre width, so the line stands out against a ground that scatters differently: the twill reads as bold.

The rule is exactly right and it is upside down from the first guess. Making the fibres agree with the line does not emphasise the line; it camouflages it.

What was counted, and how #

Quoting the four rows of that table would be quoting an example. The rule is asserted instead, in the form that can fail.

Thirty-two settings — four sett ratios, four twist levels, two twill directions — each giving a pair of cases that differ only in the twist’s handedness. For every setting, the agreeing case must put the fibres closer to the line than the opposing case, and the assertion carries the setting and both angles in its message so that a failure says which case broke it.

The census is not decorative. The rule as usually stated has no sett in it, and a sett ratio moves the twill line by twenty degrees; the assertion is what says the rule survives that.

The one place the rule breaks, and it is a sett #

On a square-set cloth the four cases fall into two symmetric pairs and the rule is clean. On an unbalanced cloth they do not, and one of them does something the rule does not describe.

At forty ends and twenty picks the twill line is 26.6° from the warp. A Z-twist warp at 900 turns per metre puts its fibres at 25.3°. The two are 1.3° apart — the fibres are lying almost exactly along the line, far closer than the 19.7° of the balanced case.

That is the extreme of the subdued end, and it has a name in the trade that is never connected to twist: a warp-faced steep twill of hard-twisted warp is the classic “cracked” or “clear-finished” construction whose line is deliberately faint. The arithmetic says why, and it says that the effect depends on a coincidence between two quantities the weaver sets independently.

And there is a twist level at which they coincide exactly. Inverting tan α = πd**T for the twill’s own angle gives the twist that puts the fibres along the line: on this 40 × 20 cloth it is 952 turns per metre, a factor of 4,258, which is an ordinary warp twist. On a square-set cloth it is 1,905 turns per metre — a crepe twist, hard to spin and harder to weave.

The refusal at the other end is a real bound rather than an inconvenience. A weft-faced cloth — sixteen ends and forty picks — puts its twill at 68.2° from the warp, and matching that needs 4,762 turns per metre in a 20 tex yarn. No spinner produces that outside a crepe, so on a weft-faced twill the fibres can never lie along the line, and the subdued option is not available at any twist. The arithmetic reports it as a refusal with the required twist in the message rather than returning a number nobody could use.

What else moves when the setts move #

The essay so far treats the sett ratio as a knob that turns the twill line. It is not a free knob: it is the same ratio that decides four other things the site has already computed, and a weaver adjusting it for the twill’s appearance moves all of them.

The balance. A cloth with forty ends and twenty picks is unbalanced, and the site’s own measure of that is the same ratio. A steep twill is a warp-faced cloth by construction.

The crimp division. Peirce’s geometry does not decide how the crimp splits between warp and weft, and the sett ratio is most of what does in practice: the closely set system is the straighter one. So the steep-twill cloth has a low-crimp warp and a highly crimped weft, and the exchange between them is correspondingly lopsided.

The cover. Cover factor is sett times diameter per system, so raising the warp sett to steepen the line raises the warp cover towards its jam. There is a limit past which the line cannot be steepened at all because the warp cannot be set closer.

And the weight. Areal weight is sett times count times one plus crimp, summed — so a cloth resett to change the twill’s angle is a different weight and a different price.

So “choose the twist to suit the twill” is the cheap adjustment and “choose the sett to suit the twill” is the expensive one, which is the practical reason the rule is stated in terms of twist. A spinner’s twist direction costs nothing to specify; a mill’s sett is the fabric.

Where the model stops #

Only the warp is counted. A twill’s surface is warp on the warp floats and weft on the weft floats, and the weft’s own twist has its own angle to the same line — measured from the weft direction, so its relation to the twill is the complement of the warp’s. On a warp-faced twill the warp dominates and the essay’s arithmetic is most of the story; on a balanced one both systems show and the surface carries two fibre directions at once.

Nothing here is a reflectance model. “A streak along the fibres” is a geometric statement about specular reflection from parallel cylinders, and how strong the effect is depends on the fibre’s own lustre, on the finish and on the light. The site can say which combination puts the fibres nearer the line; it cannot say by how much the twill’s contrast changes.

The fibres are taken as lying at the surface angle. A yarn’s surface is not all at α — the fibres visible on a cloth’s face include ones that have migrated inward and outward along their length, and a hairy yarn’s protruding ends lie at every angle. The clean-finished, hard-twisted, low-hairiness yarns the rule is usually applied to are the ones the model fits best, which is not a coincidence: those are the yarns whose surface is a bundle of parallel helices.

And the twill’s angle assumes a step of one. A twill of step two lays its line at a different angle, and the site’s own twill-angle arithmetic carries the step. Every figure here uses a step of one because that is what a manual’s rule assumes.

The rule has a third case the trade does not name #

The four rows of the table are two pairs, and the pairing hides that the rule’s two options are not equally reachable. Reading the arithmetic as a continuum rather than as a choice turns up a case with no name.

The agreeing case can be made arbitrarily close and the opposing case cannot be made arbitrarily far. The fibres lie between nought and about forty degrees from the warp; the twill line lies between nought and ninety. So the agreeing case can reach zero degrees of separation — the coincidence above — while the opposing case’s separation is the sum of two angles, both bounded, and reaches at most about a hundred and thirty degrees, which is seventy the other way round.

That asymmetry means the rule’s two ends are not symmetric about anything, and the subdued end has a limit at exactly zero while the bold end has a limit somewhere in the middle of the range. A weaver wanting the boldest possible line does not want the fibres as far from it as possible; they want them at ninety degrees, which is a specific pair of sett and twist rather than an extreme of either.

That is the case with no name. A twill whose warp fibres cross it at a right angle is the maximum-contrast construction, and reaching it needs the twill’s angle from the warp and the fibres’ angle to sum to ninety. On a square-set cloth that means fibres at forty-five degrees, which is a twist factor of about 7,000 — a crepe twist. On a steep warp-faced twill at 26.6° it means fibres at 63.4°, which is further still.

So the bold end of the rule is unreachable and the subdued end is reachable exactly. Every real “bold” twill is somewhere on the way towards a maximum nobody can weave, and every real “subdued” one can be tuned to vanish. That asymmetry is invisible in a rule stated as two options and is the whole shape of the arithmetic when it is stated as one subtraction.

It also says which of the two ends a mill can actually specify to. A subdued twill can be aimed at exactly, because the target is a coincidence and both quantities are settable; a bold one can only be aimed towards, and how bold it comes out depends on how far from the unreachable maximum the construction happens to land. That is a fair description of the trade’s own confidence in the two: a clear-finished worsted is a specification and a bold cheviot is a hope.

The same subtraction on the other surfaces #

The argument is about a twill only because a twill is the case with an obvious line. It applies wherever a cloth has a direction on its surface, and there are three others on this site.

A satin has no line and its fibres still have a direction. A satin shines because its floats are long and parallel to the warp; the fibres on those floats lie at α to the warp, so a satin’s specular streak is not along the warp but along the fibres, tilted by the twist angle. A hard-twisted satin warp reflects at a noticeably different angle from a soft-twisted one, which is a real difference in how a fabric “throws” its light and is not usually connected to twist.

A herringbone reverses the twill and not the twist. So one half of a herringbone is the agreeing case and the other half is the opposing one, in the same piece of cloth, from the same yarn — which is exactly why a herringbone’s two halves differ in apparent tone and why the effect is stronger in some fabrics than others. It is the sharpest available demonstration of the rule and nobody uses it as one.

And a crepe has no direction to agree with. A crepe is designed to have no offset at which its surface correlates with itself, so there is no line for the fibres to lie along or across, and the twist direction has no appearance consequence at all. Crepe fabrics do use very hard twist — but for liveliness and retraction rather than for lustre, which is the previous rung’s business rather than this one’s.

The herringbone case is the one worth carrying, because it is a controlled experiment sitting in every wardrobe. One draft, one yarn, one finish, two twill directions, and the two halves of the cloth are visibly not the same tone.

Who found it, and when #

The rule is in the nineteenth-century pattern books and is older than they are. Weavers had two twist directions available and two twill directions, and the four combinations gave four fabrics from one draft; a mill that wanted a bold cheviot and a mill that wanted a clear-finished worsted would specify opposite ways round, and the reason given was that the twill “showed up” or “did not”.

The optics were understood in outline much later, and the connection to the yarn’s helix angle appears in twentieth-century textile-physics texts as an aside rather than as a result. Nobody, as far as this site can find, has written the subtraction down — because doing so requires the twill’s angle to be treated as a quantity set by the setts rather than as a property of the weave, and the trade’s habit is to speak of “a 45° twill” as though the number belonged to the draft.

That habit is the reason the rule has no sett in it. Put the sett in, and the rule stops being a choice between two options and becomes a continuum with a coincidence in it and a bound at one end — which is what the arithmetic above says and what the four-row table hides.

Where the ladder goes next #

Both rungs of this anchor have been about a yarn’s surface. What neither has is the yarn’s interior: how the fibres are arranged through the section, how they migrate between core and surface, and how much of a yarn’s strength that migration is worth. That is where a spun yarn stops being a helix and starts being a structure, and it is recorded here as not done.