A two-fold yarn is not twice a single

“A folded yarn is more even than a single” is one of the things everybody in the trade knows, and it is true. Fold two singles of fifteen per cent irregularity together and the resulting yarn measures 10.6, which is a large and easily demonstrated improvement.

It is also worth nothing at all, and the reason is that the improvement is exactly the improvement that doubling the count was going to give anyway.

The claim #

Folding improves a yarn’s coefficient of variation by exactly √folds and lowers its evenness floor by exactly √folds, so the index of irregularity is unchanged.

That single sentence disposes of the usual argument for folding on evenness grounds, and it leaves three genuine differences standing:

• A folded yarn is balanced, or can be, which a single cannot be.
• A folded yarn is smoother, because the folding twist traps the outer fibres of each single against its neighbour.
• A folded yarn’s grip on its own fibres comes from a different place — from outside as much as from within — which is an argument with a surprise in it.

None of the three is an improvement in spinning. The index is what measures that, and folding leaves it exactly where it found it.

The two √2s #

The first is elementary and is a statement about adding independent errors.

Two singles are drawn from the same process and their thick and thin places are uncorrelated, so the mass per unit length of the pair is a sum of two independent variables. Variances add; the mean doubles; the coefficient of variation therefore falls by the square root of the number of singles:

$$\mathrm{CV}_{\text{folded}} = \frac{\mathrm{CV}_{\text{single}}}{\sqrt{n_{\text{folds}}}}.$$

The second is the one that cancels it. A yarn’s evenness floor is 100/√n per cent, where n is the number of fibres in a cross-section. A two-fold yarn has twice the mass per unit length of one of its singles and therefore twice the fibres in its section: n doubles, and the floor falls by √2 as well.

$$\mathrm{CV}_{\text{lim, folded}} = \frac{\mathrm{CV}_{\text{lim, single}}}{\sqrt{n_{\text{folds}}}}.$$

Divide. The √folds cancels and the index is identical.

The two statements are the same statement. Independent errors adding in quadrature is the Poisson argument, seen at the level of singles rather than at the level of fibres; the floor’s √n and the folding’s √folds are one square root counted twice. The check is worth making anyway, because two derivations that reach the same answer by different routes is the shape that goes wrong.

What a folded yarn is, for the arithmetic #

This collection computes cloth properties from a count and a diameter. Both of them, for a folded yarn, are the folded values.

A 2/20 tex yarn has a resultant count of 40 tex. Its mass diameter is √2 times a single’s, which is exactly the diameter a 40 tex single would have at the same packing. Its cover factor at a given sett is the 40 tex cover factor. Its areal contribution to a cloth’s weight is the 40 tex contribution. Its evenness floor is the 40 tex floor.

So for four of this collection’s own quantities, a folded yarn and a single of the resultant count are the same object, and no calculation here can tell them apart.

That is the honest summary of the difference and it is a loosening rather than a difference: a folded yarn is a less definite object than a single of the same count, and the collection’s arithmetic is correspondingly less certain about it.

Why the identity is worth more than the improvement #

An identity that says a familiar improvement is worth nothing invites the response that the improvement is real and the identity is a definitional trick. It is worth answering that directly, because the answer is what the index is for.

The improvement is real and it is a property of the count. A 2/20 tex yarn genuinely is more even than a 20 tex single, by √2, and a cloth woven from it genuinely will look more even than a cloth woven from the singles. Nothing in this essay withdraws that.

What the identity establishes is that the improvement was available without folding. Spin a 40 tex single to the same index and it has the same coefficient of variation, because the floor and the process are the only two inputs and both are unchanged. So the improvement is what doubling the count buys, and folding is one of two ways of doubling a count.

That reframes the question a specification should be asking. Not is folding better than not folding — which compares two different counts and is therefore not a question — but is a folded yarn better than a single of the resultant count, which is a comparison at fixed count and is the one the index makes possible.

The answer to that question is: not on evenness, at all, ever. And yes on three other things — the balance, the surface, and the route to a fine appearance at a coarse mass. Every one of those is a reason to fold and none of them is the reason usually given.

That is the general use of a normalised measure. A raw coefficient of variation is a comparison between a process and a count muddled together; the index separates them, and the separation is what makes a claim about the process testable. A trade that had only the raw figure would have no way to notice that its commonest argument for folding was an argument about arithmetic.

What happens at four folds and beyond #

The identity holds at every fold count, and it is worth asking why anybody stops, because the arithmetic offers no reason to.

The evenness gain per fold falls as a square root, so the second single buys 29 per cent, the third buys 13 and the fourth buys 8. Against that, every fold costs a doubling in machinery — a doubling frame runs at a fraction of a spinning frame’s speed, and a four-fold yarn has been through two folding operations if it is cabled or one very slow one if it is not.

And the index does not move, so none of the gain is a gain in quality. A four-fold yarn is a coarse yarn with a fine surface, four times over, and the surface is the only thing that has improved beyond what the count already gave.

So the fold count in practice is decided by the surface and the balance rather than by the evenness: two for a warp that wants clarity, three for a sewing thread that wants roundness, and beyond that a cabled construction, which is a different object with its own arithmetic. The place the trade stops is exactly where the reasons that are not evenness stop paying, which is a small piece of evidence that the trade has never really been folding for the evenness — whatever its specifications say.

Where folding does buy something #

Balance, which is the reason it is usually done. A single yarn stores a moment and a loop of it kinks; a folded yarn can be made not to, and this matters most where the yarn is free to rotate — in a knitting machine, where a lively yarn makes the fabric lean, and in a needle, where a snarling thread stops the machine.

Smoothness, which shows up as a lower hair count. Each single’s protruding fibres are pressed against its neighbour by the folding twist, and the hairs are what a neighbouring thread meets — so a folded yarn has a contact diameter closer to its mass diameter than a single does, which is the one place the folding tightens the arithmetic rather than loosening it.

A different grip. The singles inside a folded yarn are at their residual twist and are soft, but the ply’s own helix presses on them from outside. The two effects nearly cancel, which is why the trade can pick the folding ratio for balance and ignore the strength.

And a route to a fine even yarn. This is the one that follows from the floor and is not usually stated in these terms. A spinner asked for a fine, even yarn is stuck: fineness raises the floor and nothing about the process can lower it. Folding is the only move available, because it produces a thread of the required fineness of appearance — two fine singles, each with its own surface — while the mass per unit length, and therefore the floor, is that of a coarser yarn. That is why the finest shirtings are woven from two-fold yarns and the coarse ones are not.

What the cloth sees #

Take the split above into the weaving shed and it decides several things at once.

The sett. How closely threads may be set is computed from a diameter, and a folded yarn arrives with two. In practice a folded warp sets closer than its envelope diameter predicts and further apart than its mass diameter does, and the discrepancy is the largest single source of error in setting a folded cloth from first principles.

The cover. Cover factor is sett times diameter and inherits the same bracket. A folded cloth designed to a cover factor computed from the mass diameter comes out more open than intended.

The weight. Areal weight is the one quantity that is exactly right for a folded yarn with no bracket at all, because it is a count times a sett and never passes through a diameter. It is the only number in a folded cloth’s specification that can be trusted to three figures.

And the appearance. A cloth woven from folded yarn shows two singles’ worth of surface per thread. At a distance the thread reads as one; close to, the two are visible as a fine cord, which is the characteristic look of a two-fold poplin and the reason it is specified.

The claim that does not survive #

There is a version of the folding argument that this arithmetic refutes rather than qualifies, and it appears in specifications.

“A 2/20 tex yarn is a better yarn than a 40 tex single because it is more even.” It is not more even. It has exactly the same coefficient of variation as a 40 tex single spun to the same index, and the comparison that was actually being made — folded 2/20 against single 20 — is a comparison between two different counts, which is the thing the index exists to prevent.

The version that does survive is narrower and true: a 2/20 tex yarn is more even than a 20 tex single, which matters if what the cloth needs is two threads of 20 tex appearance rather than one of 40. That is a real design decision and it is about the surface rather than the arithmetic.

What a specification should carry #

The practical form of all this is a rule about how to write a yarn down, and the current conventions get it half right.

A folded yarn is quoted as R 40 tex/2, which carries the resultant count and the number of singles — enough to reconstruct both the singles’ count and the fibre count, and therefore enough to compute the floor. So far so good.

What is usually also quoted is a coefficient of variation, and it is quoted bare. Bare, it is uninterpretable: 10.6 per cent is an excellent single 20 tex, an ordinary 40 tex, and a poor 80. The number that would be interpretable is the index, which is 1.51 in all three cases and says the same thing about the process each time.

So the rule is: divide by the floor of the count the yarn actually is. For a folded yarn that is the resultant count, not the singles’, and a specification that quotes a singles count beside a folded yarn’s measured evenness is quoting two numbers that do not belong together.

Folding as a way of buying a count #

There is one more way to read the identity and it is the most useful of the readings.

A spinner has two independent quantities to sell: a count, which the customer specifies, and an index, which is the whole of what the process contributes. The floor sits between them and is nobody’s to move.

Folding changes the count and leaves the index alone. So folding is, exactly and only, a way of buying a different count while keeping the appearance of the singles — and that is a genuine thing to want. A 2/60 tex worsted is a 120 tex yarn for every purpose the floor and the cloth arithmetic care about, and a 60 tex yarn for every purpose the surface cares about: the fibres a finger meets belong to one single, not to the pair.

That split — the mass behaves as the resultant and the surface behaves as the singles — is the cleanest statement of what a folded yarn is, and it is why folded yarns dominate exactly where surface and mass are wanted in different proportions. A worsted suiting wants a fine, clear surface and enough mass to hang; a sewing thread wants a fine surface and enough strength to pull; a carpet yarn wants a coarse mass and a surface fine enough to look like more tufts than there are.

It also predicts where folding is pointless. A cloth whose surface nobody looks at closely and whose mass is the whole requirement — a sacking, a duck, a backing cloth — gains nothing from folding except the balance, and those cloths are woven from singles.

What was counted, and how #

The identity is asserted across folds and counts rather than at one point. Nine cases — three counts by three folds — with the improvement required to equal √folds to twelve decimal places and the index ratio required to be one to the same tolerance. An identity that holds at one point is arithmetic; one that holds across a grid is a derivation.

The floor used in the folded case is computed from the folded count through the same function the single used, rather than divided by √folds directly. That is deliberate: dividing would be assuming the thing being checked, and the check is that two independent routes agree.

And the fold is required to be at least two. A folded yarn of one single is not a folded yarn, and the machinery refuses it rather than returning the single.

Where the model stops #

The singles are supposed independent and they are not quite. Two singles spun on the same frame from the same sliver share some of their periodic faults, so the folded yarn’s variance is a little above the quadrature sum. The direction is known and the size is not.

Nothing here is about the folded yarn’s own irregularity of construction. A folding frame can deliver one single at higher tension than the other, which produces a yarn with a straight core and a wrapped surface rather than two equal helices — a real defect, with its own name in the trade, and one this arithmetic has no term for.

The mass diameter is used throughout and it is a bound. Every statement above that a folded yarn is “the same object” as a single of the resultant count is a statement about the mass diameter, and the envelope diameter says otherwise by 41 per cent.

And the index is a rating of the singles’ spinning, so saying that folding leaves it unchanged is saying that folding is not a spinning operation. It is not; it is an assembly operation, and this essay is the statement that the two are measured differently.

Where the ladder goes next #

Into the strength, which is the one property of a folded yarn that this arithmetic cannot get by doubling anything. The singles inside the yarn are soft and the ply presses on them from outside, and the two nearly cancel across the whole practical range of folding twists — which is a result about what a spinner is free to choose rather than about what a yarn is.

And outward into the cloth, where a folded yarn’s two diameters have to be reconciled with a construction. How closely threads may be set is computed from a diameter, and a folded yarn arrives with a bracket instead — so the sett a folded warp will take is less predictable than a single’s by exactly the width of that bracket.