What a balanced yarn is balanced about

A yarn specification will say that a yarn is balanced. Nobody asks about what.

There are two conditions in circulation, both called balance, and they are not close. One asks the fold’s net moment to be nought; the other asks the fold’s surface helix angle to equal its singles’. For a cotton the first is at a fifth of the singles twist and the second at seven tenths, and a yarn satisfying one badly fails the other.

The two conditions, stated #

Torque balance. The fold has no net moment about its own axis, so a loop of it hangs as a loop rather than kinking. Two moments cancel: what each single’s residual twist supplies, and what bending each single onto its helix costs. The condition is that their sum is nought, and the closed form is the ratio of the two stiffnesses over their sum — for a cotton, 0.200.

Surface balance. The fold’s surface helix angle equals its singles’. Since a fold of n singles is √n times the diameter, the condition is that its twist is 1/√n of theirs — 0.707 for two folds, whatever the fibre.

The two conditions have nothing in common except a word.

How differently they behave #

The two differ in every respect that matters for a specification, and listing the differences is the fastest way to see that they are not two estimates of one thing.

Fibre. Torque balance varies by a factor of three across this collection’s fibre table, from 0.143 for a polyester to 0.444 for a wool, following the stiffness ratio exactly. Surface balance does not depend on the fibre at all.

Fold count. Surface balance falls as one over the square root of the fold count. Torque balance does not depend on the fold count at all — the radius on which the singles sit multiplies one term and divides the other, and cancels.

Twist level. Neither depends on it, which is the one thing they agree about.

Count. Neither depends on it either.

So the two are distinguished by exactly the two variables a specification is most likely to change: the fibre and the number of folds.

Which one folding achieves #

The trade folds at the surface condition. That was established by putting three candidate rules against the trade’s own three brackets: the surface rule is inside all three and the torque balance is inside none, and the two are separated by their dependence on the fold count as well as by their magnitude.

So a folded yarn as delivered is surface-balanced and torsionally live, and everybody who handles one knows it: folded yarns are steamed, folded yarns left in a loop take up a small twist, and a fold that has been over-twisted is lively in the opposite sense — which is why a slack yarn snarls at all.

If folding balanced the torque, none of that would happen.

Which one a specification usually means #

The uncomfortable answer is that a specification usually means the torque condition and gets the surface one, because the sentence in the textbook is about torque and the number in the mill is the surface ratio.

That is not a disaster in practice, because the torque left over is small and steaming removes it. It is a problem when anybody reasons from the specification, and the two places it bites are predictable.

A new fibre, and this is not hypothetical: the collection’s own tables put folding at the heart of what a plied yarn is and would have been consulted by anybody working from them.

A new fibre. Somebody folding an aramid, whose stiffness ratio is 0.044, and reasoning from torque balance would fold at 0.042 of the singles twist and produce something that is not a yarn. The right answer is 0.707, the same as everything else.

An unusual fold count. Somebody folding a six-fold cord and reasoning from torque balance would use the same ratio as for a two-fold, because the balance does not depend on the count. The right answer is 0.408.

Both of those are errors a reasonable person makes by following a stated rule correctly.

The word to use instead #

The remedy is not to abandon the word but to qualify it, and the qualification is short.

Torque-balanced means the yarn has no residual moment. It is testable directly: hang a loop and see whether it kinks, or use the trade’s own liveliness test and count the turns taken up.

Angle-matched means the fold’s surface twist angle equals its singles’. It is testable directly too: measure both helix angles, or measure both twist factors, which is the same statement in the units the trade already uses.

Those two tests are different measurements with different equipment and they will disagree on any real folded yarn. A specification that names which one it wants is checkable; one that says “balanced” is not.

The third meaning, which is a state rather than a construction #

There is a third use of the word and it is worth separating too, because it is the one that resolves the practical tension.

A yarn is sometimes called balanced when it has been set — steamed, conditioned or left long enough that its residual torque has relaxed away. That is a statement about the yarn’s history rather than about its geometry, and it produces something that satisfies the torque condition without having been constructed to.

So the trade’s actual practice is: construct to the surface condition, then set to the torque condition. Two conditions, two mechanisms, and the specification collapses them into one word.

That is a sensible way to make yarn and a bad way to write a specification, because the setting is undone by conditions that do not undo the geometry — wetting a set cotton brings some of its liveliness back, and a set that was done at the wrong moisture regain does not hold — which is the same fragility a set loop has.

What each condition is worth to a user #

A reader who is buying yarn rather than making it wants to know which condition to care about, and the answer depends on what the yarn is for.

For sewing thread, the torque condition matters most. A live thread snarls as it comes off the spool and into the needle, and a snarl in a sewing machine is a stoppage. That is why sewing threads are folded and then set, and why sewing thread specifications talk about liveliness explicitly rather than about balance.

For a knitting yarn, the torque condition matters differently. A live yarn knits into a fabric that leans, and the lean does not come out in washing. So the specification a knitter needs is about residual torque, and a yarn that is angle-matched but unset will produce a spiralled fabric however carefully it was folded.

For a weaving yarn, the surface condition matters most, because the warp is under tension throughout and cannot snarl, and what the cloth looks like is decided by the yarn’s surface. A weaver can use a livelier yarn than a knitter can.

So the two conditions map onto the two ways yarn is used, and the trade’s practice of constructing to one and setting to the other serves both. What it does not do is tell anybody which they are getting.

A cheap test that distinguishes them #

Both conditions are testable and the tests are unusually cheap, which is worth setting out because a specification that cannot be checked is not a specification.

For torque: cut a length of about half a metre, hold the two ends together, and let the middle hang free. A torque-balanced yarn hangs as an open loop. A live one takes up turns, and counting them is the trade’s own liveliness measurement.

For angle: measure the fold’s surface helix angle and one of its singles’, which needs a microscope with a protractor eyepiece and takes a minute. Or, without a microscope, compute both twist factors from the twist tester’s own reading and the count — they should be equal.

The second test is the one nobody runs, and it is the one that would show that folding is doing what this ladder says it is doing.

What was counted, and how #

Both conditions are computed rather than quoted, from the same arithmetic, on the same fibres.

The torque condition is a bisection on the fold twist between nothing and twice the singles twist, on the exact moments rather than the small-angle forms, checked against its own closed form and against the requirement that the departure grows with the helix angle rather than scattering.

The surface condition is an identity: matching the two angles divides the twist by exactly the square root of the fold count, checked to twelve figures across the fold counts the trade quotes.

The trade’s brackets are the collection’s own table, copied from practice several ladders ago and unchanged.

How much torque is left over, in numbers #

The practical question behind all of this is how live a surface-balanced yarn actually is, and the arithmetic answers it.

At the surface condition a two-fold cotton has had about seventy per cent of its singles twist taken out, against the eighty per cent that would balance it. So a residual of about a tenth of the original twist remains, in the singles’ own direction.

What that is worth can be read off the snarling threshold, which goes as the square of the residual twist. A single at eight hundred turns a metre needs about two metres of its own weight hanging below it to stay straight. A fold carrying a tenth of that residual needs about a hundredth of the tension — two centimetres of its own weight — which is nothing at all.

So the practical answer is that surface balance leaves a yarn almost torque-balanced, in the sense that matters to a sewing machine. The factor of three between the two conditions sounds alarming and produces a residual liveliness two orders of magnitude below a single’s, because the threshold is quadratic.

That is why the confusion has cost nobody anything, and it is also why it went unnoticed: the wrong explanation predicts a yarn that behaves almost exactly like the right one.

Why it still matters that the explanation is wrong #

Given that, it is fair to ask whether any of this matters, and the answer is that an explanation is machinery for reasoning about cases nobody has met.

A wrong explanation that happens to give nearly the right answer in the familiar case gives arbitrarily wrong answers outside it. Fold an aramid by torque balance and the yarn falls apart. Fold a six-fold cord by torque balance and it is under-twisted by forty per cent. Choose a fold ratio for a new fibre by looking up its shear modulus and the shear modulus is irrelevant.

Every one of those is a reasonable person following a stated rule correctly, and every one produces a bad yarn. That is the cost of a wrong explanation and it is paid entirely by whoever is doing something new.

This collection’s whole method is aimed at that cost. A number quoted from practice is worth having; a number quoted from practice with the wrong derivation attached is worth less than nothing, because it travels.

Where the model stops #

Neither condition is what a folder optimises. A folder is choosing for strength, evenness, appearance, hand and cost, and the fold ratio affects all five. The claim here is that the number they arrive at coincides with the surface condition, not that they compute it.

The torque condition assumes an elastic yarn. A set yarn has no residual torque at any fold ratio, so the condition is vacuous for the thing most folded yarn actually is. That is the largest caveat on this rung and it is the reason the practical tension is smaller than the conceptual one.

And “surface angle” is a single number for a distribution. A yarn’s fibres run at every angle from nothing at the axis to the surface angle at the outside, and matching the surface does not match the distribution. Two yarns with matched surfaces and different internal structures behave differently under load.

The generalisation #

The habit is one this collection has now needed three times and it deserves a name.

A word that names a condition must name which quantity is being conditioned. Balance is the third instance. The first was a dimension quoted without its state, where the word “width” named three different numbers depending on how the fabric had been relaxed. The second was set, which is both a process a yarn has been through and a fraction of its natural curvature.

In every case the failure mode is identical: the word is used correctly by everybody, the quantity meant is obvious from context to somebody in the trade, and the ambiguity only bites when a number is carried from one context to another — which is exactly what a collection like this one does all the time.

The defence is cheap. Whenever a condition is named, write the equation it is a condition on. A sentence with an equals sign in it cannot be ambiguous about what is equal to what.

What a specification would have to say #

Putting the rung to use means writing down what a yarn specification would need in order to be checkable, and it is three lines rather than one word.

The twist factor of the singles and of the fold, which are the same number if the yarn is angle-matched and are the quantities a twist tester reads directly. That settles the geometry, and it settles it in the units the trade already uses.

Whether the yarn has been set, and how. Steamed, conditioned, autoclaved, or not at all; and at what regain. That settles the torque, because setting is what removes it.

And the liveliness as measured, by the hanging-loop test, in turns taken up per unit length at a stated weight. That is a direct measurement of what is left, and it does not require anybody to agree about mechanisms.

Three lines, all of them measurable with equipment every mill has, and none of them containing the word balanced. A specification written that way says everything the current one gestures at and nothing it cannot support.

Why this is the third time #

It is worth counting, because the pattern is now clear enough to act on.

Three times in this collection a single word has been carrying two quantities: state, which is both a relaxation procedure and a set of dimensions; set, which is both a process and a fraction; and now balance, which is both a moment condition and an angle condition.

In every case the two quantities are used correctly by everybody in the trade, because context disambiguates them for anybody who is there. In every case the ambiguity became a problem the moment a number was carried out of its context — which is what a collection like this one does constantly, and is the whole point of building one.

So the general rule is not about vocabulary hygiene for its own sake. It is that a collection that moves numbers between contexts has to disambiguate words that context was doing the work for, and the cheapest way to do that is to write the equation.

Who found it, and when #

Both conditions are old. The surface-angle relation is Gégauff’s, from 1907. The torque balance of a plied yarn has been analysed repeatedly in the textile literature since the 1950s.

What is not in the literature, as far as this collection can find, is the observation that the trade’s folding ratios satisfy one and not the other, and that the word used for the practice names the condition it does not satisfy.

Where the ladder goes next #

The arithmetic goes up a level. A cabled yarn is a fold of folds, with three twist levels, two surface conditions and a great deal of scope for confusion — and the surface rule applies at each level unchanged, which is a cabled yarn is a fold of folds.

And the torsion ladder then leaves the yarn for the cloth, where a structure exists that does to warp ends what folding does to singles: a leno twists what a weave only crosses.