Folding is untwisting

A great many of the yarns in ordinary use are not single. A sewing thread is three singles twisted together, a shirting warp is often two, a hosiery yarn is two, and every carpet yarn is folded. The trade has a rule for how hard to fold: about two thirds of the singles’ twist, in the opposite direction, and the yarn then hangs straight instead of curling up on itself.

The rule is quoted everywhere and derived nowhere, and the reason is that deriving it needs a torsional rigidity that nobody can supply. What can be derived is the thing the rule is a rule about, and it turns out to be a subtraction.

The claim #

Folding removes the folding twist from the singles, exactly. A single spun at T_s and folded at T_f is left with T_s − T_f turns per metre of its own.

Three consequences, and the third is the one nobody states:

• The folding twist cannot reach the singles’ twist. At T_f = T_s the singles have no twist left, no grip on their own fibres and no strength. So the balance ratio is below one for a reason that has nothing to do with torque, and any account of folding that begins with the torque has skipped the constraint that binds first.
• The surface fibres lie nearer the finished yarn’s axis than they did in the single. The single’s helix and the ply’s helix are opposite hands, so the two subtract, and at a folding ratio of exactly one half they are equal and cancel.
• A folded yarn hands the arithmetic two diameters rather than one, forty-one per cent apart for a two-fold, and every cover factor and every jam in this collection is computed from a diameter.

Why folding untwists #

Take one single and hold both of its ends. Now wind it once round its neighbour and bring it back to where it started, still holding the ends.

The single has gone round the ply’s axis once. It has also — and this is the part that is easy to miss — turned once about its own axis, because holding the ends means it cannot swivel. A single being wound is a single being rotated, and the rotation is one turn per turn of the fold.

If the fold runs the other way from the spinning, that rotation removes twist. One turn of S fold takes one turn of Z spin out of each single, and over a metre:

$$T_{\text{residual}} = T_s - T_f.$$

There is no model in that. It follows from the singles being wound rather than swivelled, which is what a folding frame does — the packages are held and the yarn is taken up, and nothing in the machine rotates a single about its own axis independently.

The consequence is large. At the trade’s ratio a single that was spun to a 22.8° surface angle is sitting in the finished yarn at 7.8°, which is a soft-twist yarn by any standard. The singles inside a folded yarn are much softer than the singles that were sold, and any measurement made on the singles before folding is a measurement of a different object.

The exact cancellation at one half #

The single’s surface fibres lie at an angle to the single’s own axis. The single’s axis lies at an angle to the ply’s axis. The two helices are opposite hands, so to first order a fibre’s inclination to the finished yarn is the difference between them.

Write both out. The single’s residual angle has

$$\tan\alpha_r = \pi d_s \frac{T_s - T_f}{1000},$$

and the ply’s helix, whose radius is half the single’s diameter for a two-fold, has

$$\tan\beta = 2\pi\frac{d_s}{2}\frac{T_f}{1000} = \pi d_s \frac{T_f}{1000}.$$

The same π d_s appears in both. So the two angles are equal exactly when T_s − T_f = T_f, which is

$$\frac{T_f}{T_s} = \tfrac{1}{2},$$

with no dependence on the count, the fibre, the packing or the twist level. At a folding ratio of exactly one half, the singles’ residual helix and the ply’s helix are equal and opposite, and a surface fibre of a single runs — to first order — along the finished yarn’s axis.

That is the geometrically special point of the whole operation, and the trade does not fold there. It folds at 0.6 to 0.75, past it, which means the ply’s helix has overtaken the singles’ residual and the surface fibres have crossed the axis and started leaning the other way.

Why the trade folds past it #

Two reasons, and only one of them is in this collection’s reach.

The one that is. At a ratio of one half the singles retain half their twist, which is a soft-twist yarn — its grip on its own fibres is correspondingly weak, and what grips the end of a fibre falls quadratically as the angle falls. Folding further does not fix that; it makes it worse. What it does supply is pressure from outside, and that is a computation of its own with a genuinely surprising answer.

The one that is not. A folded yarn is folded to be balanced: to have no residual moment, so that a loop of it lies flat instead of kinking back on itself. That is a torque condition, and computing it needs the torsional rigidity of a fibre assembly under lateral pressure — a quantity this collection has a bracket for and not a value, one level along in bending rather than in torsion.

So the two thirds is carried here as a measurement. It is labelled as one everywhere it is used, it is carried as a range rather than a value — 0.6 to 0.75 for a two-fold, lower for a three-fold — and every result computed from it is computed at both ends.

This is the same refusal the collection makes about where a yarn sits inside its bending bracket, and it is made for the same reason: the quantity that would settle it is a friction problem inside an assembly, and assembling a plausible answer out of assumptions would produce a number nobody could check.

The two diameters #

A folded yarn’s diameter is not one number and the disagreement is not small.

Read the other way round, the gap is a statement about packing. A folded yarn sitting at its envelope diameter is a yarn packed at 0.30 rather than 0.60, so how tightly is it folded and which diameter are the same question. A hard-folded yarn is nearer its mass diameter; a soft-folded one is nearer its envelope.

What folding is for, in order of how much it matters #

Set out plainly, because the trade’s reasons and this collection’s arithmetic do not agree about the ranking.

Balance. A single yarn is lively: a loop of it kinks. That is a nuisance in knitting, where the loop is free to rotate and the fabric leans, and it is a nuisance in sewing, where a lively thread snarls at the needle. Folding in the opposite direction cancels the moment, and this is the reason most often given.

Roundness and smoothness. Two singles wound together present a surface with fewer protruding ends than one single of twice the mass: the folding twist traps the outer fibres of each single against its neighbour. That is a real effect and it is why a folded yarn feels smoother, and it belongs to the hair layer rather than to anything in this essay.

Strength. Usually given as the main reason and, on this collection’s arithmetic, the least of the three: the singles inside the yarn are much softer than they were, and what they gain from the ply’s own grip they very nearly give back.

Evenness. Never given as a reason and worth exactly what the doubled count is worth, which is the subject of the next rung.

The ordering is worth having because it says what folding cannot do. It cannot make a badly spun yarn well spun; it cannot lower the index; and it does not deliver strength in proportion to the twist put into it. What it delivers is a thread that lies flat and feels smooth, which is a great deal if the yarn is going into a knitting machine or a needle.

Notation, briefly #

Folding notation is one of the places where a convention carries information and is routinely mangled. A two-fold 20 tex yarn is written R 40 tex/2 in the resultant system — the finished count, then the number of singles — and 20 tex × 2 in the singles system, and the two mean the same yarn while looking like a factor-of-two disagreement.

The twist directions are written as a pair: Z/S for a Z-spun single folded S, which is the ordinary case. Z/Z exists and is not a mistake: folding in the same direction adds twist to the singles rather than removing it, giving a very hard, lively, compact yarn used for crepe and for some sewing threads. Everything in this essay applies to it with the sign reversed, and the residual twist becomes T_s + T_f, which is why a Z/Z yarn reaches surface angles no single could be spun to directly.

The cancellation point for any number of folds #

The one-half is a two-fold’s number and the general one is worth writing, because it is one line and it explains where the trade’s several balance bands sit.

For n singles arranged in a ring, their centres lie at a radius of half a single’s diameter divided by the sine of π over n. Setting the ply’s helix angle equal to the singles’ residual angle and cancelling the π d_s that appears in both gives

T_f / T_s = 1 / (1 + 1/sin(π/n))

which is 0.500 for two singles, 0.464 for three, and 0.414 for four. The cancellation point falls as the fold count rises, because more singles sit further from the axis and their helix is therefore steeper at the same folding twist.

Set the trade’s balance bands against those. A two-fold is folded at 0.6 to 0.75, which is 0.1 to 0.25 past its cancellation; a three-fold at 0.5 to 0.65, which is 0.036 to 0.19 past its own. Both are folded past the geometric cancellation and by a broadly similar margin, which is a small piece of evidence that the balance condition tracks the same geometry rather than being an unrelated empirical band that happens to sit nearby. It is not proof of anything — the torque calculation is still declined — and it is the sort of consistency that would be worth noticing if it failed.

The same-hand fold, which does the opposite of everything above #

A Z-spun single folded Z is not an error, and running the arithmetic on it reverses every statement in this essay.

The rotation is still one turn of the single per turn of the fold. What has changed is the sign: the fold now turns each single in the direction it was already twisted, so the residual is T_s + T_f rather than the difference. A single spun at 800 and folded Z at 540 is sitting at 1,340 turns a metre inside the finished yarn.

Four consequences follow, and each is the mirror of one above.

The singles inside are harder than the singles that were sold, not softer. Their surface angle rises rather than falls, their grip on their own fibres rises with it, and the failure this essay’s title is about does not arise.

There is no cancellation at any ratio. Both helices are the same hand, so they add wherever they are compared, and the surface fibres lie further from the finished yarn’s axis than they did in either component. That is the least lustrous arrangement available, which is exactly what a crepe wants.

There is no upper bound from the untwisting side. Nothing runs out, so the folding twist may be taken as far as the machine and the yarn’s own strength allow — and a Z/Z yarn reaches surface angles a single could not usefully be spun to, because a single at that twist would be far past its own strength optimum while the yarn as a whole is not.

And it is maximally lively rather than balanced. The two moments add, so a loop of Z/Z yarn kinks harder than a loop of either component, and that liveliness is the property being bought: it is what makes a crepe yarn contract and pucker its cloth in the finishing.

Three folds and more #

Everything above was written for a two-fold and generalises with one change: the ply’s helix radius is no longer half the singles’ diameter.

For three singles arranged in a triangle the centres sit at d_s/(2 sin 60°) from the axis, which is 1.155 times the two-fold’s offset, so the ply’s helix is steeper at the same folding twist. That moves the cancellation point below one half, and it is why the trade’s balance band for a three-fold — 0.5 to 0.65 — sits lower than the two-fold’s.

The mass diameter goes as √folds: 1.414 for two, 1.732 for three, 2 for four. The enclosing circle goes as 2, 2.155 and 2.414. The two converge, which is the packing statement above, and the practical consequence is that a three-fold yarn is a better-behaved object for this collection’s arithmetic than a two-fold: there is less room for the diameter to be wrong in.

Beyond four nobody folds, and the reason is that a cabled construction — folding folded yarns — becomes the better arrangement. That is a third level of the same argument and everything in this essay applies to it once more.

What was counted, and how #

The untwisting is asserted against the helix rather than restated. Nine folded yarns — three singles twists by three ratios — with the residual angle computed by the fold’s own arithmetic and again by asking the helix directly about T_s − T_f, and the two required to agree to twelve decimal places. Two routes to one angle is exactly the shape that drifts.

And a folding twist at or above the singles’ twist is refused rather than returned. A yarn with no twist left is not a folded yarn and the machinery says so, naming the two numbers.

The two-fold diameter ratio is asserted to be exactly √2, which is the check that the mass route and the enclosing-circle route are the two different things they are claimed to be.

Where the model stops #

The composition of the two helices is first order. Saying that a fibre’s inclination to the ply axis is the difference of two angles is right for a fibre at the outside of the ply and only approximately right elsewhere; the exact path is a helix wound on a helix and its inclination depends on where round the single the fibre is. The equality of the two angles at a ratio of one half is exact; the statement that the fibre then lies along the axis is not.

The balance ratio is a measurement and is the largest thing declined here. Nothing in this essay derives it, and the reason is stated rather than glossed.

The singles are supposed identical and identically tensioned during folding. They are not: one package runs out before the other, tensions differ, and a real folded yarn has one single doing more of the work — which is one of the mechanisms behind a folded yarn’s own irregularity.

Nothing here has any migration in it. The obliquity of a fibre in a folded yarn is a bracket in exactly the way a single’s is, and the bracket is wider because there are two helices to migrate along.

Where the ladder goes next #

Into the evenness, where folding does something exactly and disappointingly: it improves a yarn’s coefficient of variation by √2 and lowers the evenness floor by √2, so the index of irregularity is unchanged and nothing about the spinning has improved at all.

And into the strength, where the surprise is. The folding twist takes grip away from inside the singles and puts pressure round the outside of them, and the two very nearly cancel: a folded yarn’s strength barely notices the folding twist, which is precisely what leaves a spinner free to choose the ratio for balance.