The other half of the twist curve

Everybody who has handled yarn knows the shape. Add twist to a bundle of fibres and it gets stronger; keep adding and at some point it starts getting weaker again. There is a best twist, and a spinner’s whole trade is knowing where it is.

This collection has drawn that curve once, in the essay that gave the helix angle a page of its own, and drew it honestly incomplete. The falling half is obliquity and is trigonometry. The rising half — the grip that twist supplies — was declared to need fibre length, fineness, friction and migration, none of which the collection had, so it was drawn as a saturating model with a constant in it, and the caption said in as many words that the optimum belonged to the constant rather than to the finding.

The constant is no longer needed. The grip has a derivation now, and what it leaves behind is one measured number that scales the answer and cannot change its shape.

The claim #

The optimum twist is a property of the twist factor and not of the count — the same twist factor at 6 tex, at 20 and at 60 — and it moves in the two directions the fitted number cannot touch.

• The maximum for an ordinary cotton falls at a twist factor of 3,100 on the lower bound of the obliquity bracket and 4,340 on the upper. The trade twists cotton wefts at about 3,200 and warps at about 4,200, so the whole of ordinary practice sits inside the bracket.
• A longer staple wants less twist. 2,930 at a 38 mm staple against 3,440 at 22 mm, and the ordering holds across a tenfold range of the fitted number.
• A coarser fibre wants more twist at the same staple, for the same reason with the sign reversed.

The first of those is the one worth dwelling on, because it explains a convention rather than a measurement.

Why the answer is a twist factor #

A spinner does not specify turns per metre. They specify a twist factor: turns per metre times the square root of the count, or one of the several equivalent quantities in the older systems.

This collection has already shown why that combination is the natural one: a yarn’s diameter goes as the square root of its count, the tangent of the surface angle is π times the diameter times the turns, so the angle depends on turns × √tex and on nothing else. Two yarns at the same twist factor have the same surface angle, to twelve decimal places.

That was a statement about geometry. This is the statement that makes it useful: every quantity in the strength curve is a function of the angle alone. The obliquity is cos²α. The critical length is d_f over 4μη times a function of α. The cohesion factor is the critical length over the staple. Not one of them contains the yarn’s count.

So the optimum comes out at the same twist factor at every count, and the collection can check that rather than assert it: computing the curve at 6 tex, 20 tex and 60 tex gives maxima at 1,266, 693 and 400 turns per metre respectively — three quite different numbers — and a twist factor of 3,101 in all three cases.

The trade’s unit is the variable the problem is actually a function of. That is a stronger statement than “the unit is convenient”, and it is the reason a spinner can carry one number in their head across the whole count range.

The two halves, and what each is worth #

At the ordinary warp twist of 20° the two factors are 0.883 and 0.873, and their product is 0.771. Neither half dominates, which is what makes a maximum rather than a shoulder.

The shape of the loss on either side of the maximum is asymmetric and the asymmetry is practical. Going ten degrees below the optimum costs 34 per cent; going ten degrees above costs nine. A spinner in doubt should over-twist, and the trade does: the ordinary practice sits a little above the lower bound’s optimum rather than below it.

There is a second reason for that and it is not in this curve. A warp is abraded thousands of times in the loom by the heddles and the reed, and abrasion resistance keeps rising with twist well past the point at which strength stops. So a warp is twisted harder than its strength optimum on purpose, and a weft — which is thrown rather than dragged — is not.

That gap is the honest headline. The obliquity bracket was introduced as a statement about strength; here it turns into a statement about what twist to use, which is a decision rather than a number, and the two bounds disagree about it by more than the difference between a weft and a warp.

The orderings, which the fitted number cannot move #

The contact efficiency scales the critical length inversely, so raising it shortens the gripped length, raises the cohesion curve, and moves the optimum down in angle: 26° at η = 0.02, 20° at 0.05, 14° at 0.2, 9° at 1. That is a wide spread and it is the reason no absolute optimum is claimed here as a result.

What survives is the comparison between two yarns computed at the same η.

The staple ordering is checked at η = 0.02, 0.05 and 0.2 and holds at all three. So does the fineness ordering, in the other direction: at the same staple, a coarser fibre has a longer critical length, spends more of itself being gripped, and needs more twist to compensate.

The fineness ordering has an awkward feature worth stating, because it is the kind of thing that makes a prediction untestable if it is not noticed. In natural fibres, coarse and long go together. A wool is three times coarser than a cotton and its staple is nearly three times longer, and the two effects nearly cancel — which is why comparing cotton with wool tests nothing. The prediction is only sharp within a fibre: two cottons of the same staple and different fineness, or two of the same fineness and different staple.

Why the loss is so much steeper below the maximum than above #

The asymmetry in the table — a third of the strength lost ten degrees below the optimum against a twelfth ten degrees above — is not a feature of the numbers. It is what the two factors are each doing at that point, and it can be read off their shapes.

Below the optimum the cohesion is still collapsing and the obliquity has barely started. The critical length goes as the reciprocal of sin²α, so the cohesion is in the steepest part of its rise, changing by tenths per degree; the obliquity is cos²α, which near twenty degrees is flat to within a few thousandths per degree. One term is falling fast and the other is not yet defending anything.

Above the optimum the roles have swapped, but not symmetrically. The cohesion has saturated — it is 0.873 at twenty degrees and 0.950 at thirty-five, so there is very little left for it to give — while the obliquity has begun to fall, at a rate that is still modest because cosine is flat at its own maximum and only steepens later. So above the optimum one term has stopped paying and the other has not yet started charging much.

A maximum where one factor saturates and the other declines gently is a maximum with a shoulder on its high side and a cliff on its low side, and that is the whole explanation of the practical rule. It also says where the rule expires: the obliquity’s fall does eventually steepen, and by thirty-five degrees the product is down to 0.638. Over-twisting is cheap for about fifteen degrees past the optimum and stops being cheap after that.

The count-independence has an exception, and the exception is not in the arithmetic #

The optimum is the same twist factor at every count because nothing in either half of the curve contains the count. That is exact and it is checked. It is also, as a prediction about what a mill does, not quite what happens — and the reason is worth having, because it is a case of a variable hiding inside a constant.

A spinner cannot make a very fine yarn out of any cotton. The count sets a floor on how few fibres may lie in the yarn’s cross-section, and below about a hundred the yarn’s evenness collapses, which is a separate and unforgiving constraint. So a 6 tex yarn has to be spun from a fine long-staple cotton and a 60 tex yarn need not be.

Both of the model’s orderings then apply. The fine yarn’s cotton is finer, which wants less twist, and longer, which also wants less. So the twist factors a mill actually uses drift downwards as the yarn gets finer — and the drift is real, and the model predicts it, and it is not a failure of the count-independence.

What the argument says is that the drift belongs to the fibre and not to the count. Two yarns of different counts spun from the same cotton want the same twist factor; two yarns of the same count spun from different cottons do not. A table of twist factors against count, which is what the trade’s tables are, is reading a fibre-quality gradient through a variable that is not causing it — which is exactly the kind of confusion a count-independent model is good for detecting and no amount of tabulation would.

What a spinner is actually choosing #

The curve is a strength curve and a spinner is not choosing strength alone. It is worth listing what else moves when the twist factor moves, because every one of the other quantities is one this collection computes and every one of them points a different way.

The diameter. Packing rises with twist, so a harder-twisted yarn is thinner at the same count — and a thinner yarn changes the cover factor, the jam and the size of every hole in the cloth. That is a separate essay’s worth of consequence and it is the reason a crepe cloth and a poplin of the same construction are not the same cloth.

The stiffness. A harder-twisted yarn presses its own fibres together harder, which is what decides whether they can slide when the yarn is bent.

The liveliness. A twisted yarn stores a moment, and a fabric made from a lively yarn tries to untwist it. In a woven cloth the crossings resist and almost nothing happens; in a knit the loop is free to rotate and a jersey leans.

The lustre and the hand. Fibres lying at a steeper angle catch light differently, which is what the twill-line rule is about, and a harder yarn feels harder.

And the retraction, which is a cost in fibre: a yarn twisted to 40° is a seventh shorter than the strand that went into it, so the spinner buys a seventh more fibre for the same length of yarn.

So the strength maximum is one constraint among six, and the reason it deserves a curve of its own is that it is the only one of the six that has a maximum rather than a monotone trend. Everything else is a trade that gets steadily worse in one direction; strength is the one that turns round.

What was closed, and what it closes #

This settles a refusal this collection made deliberately and recorded at the time. The caption on the earlier curve said that the rising half depended on quantities the site did not have and that quoting an optimum without quoting the cohesion assumption was quoting a fitted parameter as a discovery.

That was right about the state of the machinery and wrong about the reach of the argument. The quantities were: fibre length, fineness, crimp, surface friction and migration. Three of them — length, fineness, friction — now enter through a single derived group. Migration enters as a bracket rather than a number. Crimp does not enter at all, and its absence is a genuine gap.

What has not changed is the epistemic status of the optimum’s position. It is still true that moving the fitted contact efficiency moves the optimum, and it is still true that quoting “3,100” as a result would be quoting a fit. The improvement is that the fit is now one physical quantity used consistently, and that the claims made from the curve are differences rather than values.

What was counted, and how #

The count-independence of the optimum is computed rather than argued. The curve is built at three counts spanning a factor of ten; the maxima are at three different turns-per-metre and one twist factor.

Both orderings are asserted across a tenfold range of the contact efficiency, and asserted as strict inequalities between computed optima rather than as a visual reading of two curves.

The cohesion factor’s two branches are checked to meet at a half, which is where a yarn none of whose fibres is fully gripped still realises half of them.

And the machinery refuses a filament. Asking for this curve for polyester raises an error naming the reason: the whole rising half is about fibre ends, and a filament has none between the ends of the package. A generator that returned a plausible curve there would be the most dangerous kind of wrong.

Where the model stops #

The contact efficiency is fitted and it moves the optimum by 40 per cent over its honest range. Nothing here narrows it. What can be said is that the value which puts the optimum where the trade twists is 0.05, and that this is a calibration rather than a prediction — one datum spent, with everything else then following.

Migration is a bracket and the bracket is wide. The two bounds put the optimum 40 per cent apart in twist factor. A measurement of migration in real yarns would close this and this collection has none.

Fibre crimp is absent. A cotton fibre is not straight; it has a natural crimp and a permanent twist of its own, and both change how it packs and how it grips. Nothing here has a term for either.

The cohesion model averages over uniformly distributed fibre-end positions and gives every fibre the nominal staple length. A real staple is a distribution, and the short-fibre fraction is what the model is most sensitive to.

Abrasion is not in the curve at all, and it is half of why a warp is twisted the way it is. This curve is about breaking a yarn once, and a warp end is rubbed a hundred thousand times before it is ever pulled hard.

And nothing here is about a folded yarn, whose singles sit at their residual twist rather than their spun twist and are pressed on from outside as well as from within.

Where the ladder goes next #

Into the folded yarns, where the same two halves appear with one important addition: the ply’s own helix supplies pressure of its own, so a single inside a folded yarn is gripped by something a single on its own does not have. The consequence is that the folding twist costs almost nothing in strength, which is what allows a spinner to choose it for balance instead.

And back down into the assembly, where the same radial pressure decides something quite different: whether a yarn’s fibres can slide when it is bent rather than pulled. That turns out to settle which end of the stiffness bracket a thread in cloth is at, and the answer is not the middle.