A yarn has a diameter for every instrument

Every diameter on this site comes from one division. A count is a mass per unit length; a fibre has a density; a yarn is a cylinder packed to a stated fraction; and the diameter follows by conservation of volume with nothing else in it. That number is exact given its inputs, it is the number every cover factor and every jammed sett on this site is built from, and it has one property that has never been examined: nothing whatever measures it.

The cloth #

A spinner measuring a yarn has several routes and they do not agree. A projected image on a screen gives one width. A yarn wound side by side on a black board under a glass gives another. A permeameter, which infers a diameter from how much air the yarn obstructs, gives a third. The width at which two neighbouring ends stop moving closer under a reed gives a fourth, and it is the one that decides a jammed sett. The trade’s response to the spread has always been to name a preferred method and get on with it.

The spread is not scatter. The hair population is a distribution with a decay length of six hundred micrometres on a yarn a hundred and sixty-seven micrometres thick, and every one of those instruments is asking a different question of the same distribution.

The claim #

A spun yarn does not have a contact diameter. It has a coverage profile, and each instrument reports the height at which that profile crosses whatever level will trigger it. The heights are hundreds of micrometres apart, so the diameters are a factor of three apart, and no experimental care can narrow them because they are not measurements of the same quantity.

The corollary is the useful half. The one number this site has been using — the mass diameter plus twice twenty-five micrometres — is a description of where most of the material is and not of where the yarn ends. It is right for the questions it was introduced to answer and wrong for every question about reach.

Coverage, which is the right quantity to threshold #

The population gives hairs per millimetre of yarn standing at least a given height off it. What an instrument meets is not a count but an obstruction, so the quantity to threshold is how much of the space beside the yarn a hair occupies. Per millimetre of yarn there are N(h) hairs at height h, each a fibre’s diameter across, so the fraction of a band beside the yarn that is fibre is

c(h) = N(h) · d_f

which is dimensionless, and which is a coverage rather than a solid fraction — it counts shadows, not volume. At the yarn’s own surface the long population covers about one per cent of the space beside it. Grossed up by the measured split between the long and short populations, the whole protruding fibre covers something under a tenth.

That single figure settles the shape of every answer below. A layer that is nine parts in ten gap cannot stop anything by bulk. What it can do is stop things that are triggered by a little material, and different things need different amounts.

The three contours #

A threshold that needs half the space filled is never met at all. Nothing about a hair layer is dense enough to carry a neighbouring thread’s own load by itself, at any height including zero. So the diameter a reed presses two ends to — the jamming diameter that decides a sett — is very nearly the mass diameter, and this site’s whole apparatus of cover factors and jammed setts is not disturbed by the hair layer at all. That is a relief and it is also a result: the arithmetic that was built without hairs is the arithmetic that did not need them.

A threshold that needs a twentieth of the space is met a third of a millimetre out. That is roughly what an optical width measurement stops at, because a projected image reads as edge where about that much of the background is obscured. Such an instrument reports a yarn nearly nine hundred micrometres across against a mass diameter of a hundred and sixty-seven.

And a hair counter triggers on one hair. It stops where the count falls below its own detection rate, which on the exponential is well past two millimetres.

Three thresholds, three answers, one profile, and the spread between the loosest and the mass diameter is a factor of nearly thirty. No amount of care with any one instrument moves any of them.

The three are not equally useful, and the ranking is the opposite of the one a reader might expect. The tightest threshold is the one that reproduces the site’s existing arithmetic, so it is the one that changes nothing. The loosest is the one that predicts something new — because reach is what a pill, a printed edge and a prickling fibre all need, and reach lives entirely in the tail nobody has been measuring.

What this does to a cover factor #

The cover factor is a sett times a diameter, so it inherits the question. This site computes it from the mass diameter, which is the right choice and which now needs saying rather than assuming: the cover factor is used for jamming, for air permeability and for the hole between four threads, and every one of those is a question about where the material is.

But the cover a cloth looks to have is a different contour and a much looser one, and that is not a small effect. Opacity is not cover argued the point from a different direction: a cloth blocks more light than its geometric cover predicts. The hair layer is part of why, and the arithmetic is now available rather than asserted — the coverage integrated over the whole layer adds several points to the optical cover of an ordinary cloth while adding nothing measurable to what a permeameter sees, because air takes the largest channel and a nine-tenths-gap layer is not a channel.

Two quantities that a single cover factor was standing in for, moving in opposite directions. A finish that removes hairs makes a cloth measurably more transparent and not measurably more permeable.

The narrowness of that spread is itself the answer to a question a spinner would ask next. If a cloth’s optical cover is being lifted by hairs, can it be lifted deliberately by construction? Almost not at all: doubling the sett and halving the count leaves the hair coverage where it was. Everything that moves this quantity by a useful factor is a finishing operation, which is why the two operations that matter to it — singeing and raising — are finishing operations and always have been.

What was counted, and how #

The check the argument needs is an ordering and a failure, and both are asserted.

The ordering is that the looser the threshold, the wider the yarn is found to be, over three thresholds that differ by more than two decades in level. It would be broken by a sign error anywhere in the profile.

The failure is the interesting half: the load-carrying threshold is required not to be met, at any height. A model that quietly returned a jamming diameter larger than the mass diameter would be claiming that a hair layer holds threads apart, which would move every jammed sett on this site — and it is the kind of claim that looks like a refinement and is a wrecking ball.

And the whole spread is required to exceed a factor of three, because an essay whose point is that a quantity is not a length has to show that the alternatives are not within rounding of each other.

The measurement this site should have been suspicious of #

There is a place in this collection’s own history where the question already bit and was answered by a different route.

The diameter was quoted at one twist took Peirce’s 1937 rule that a cotton yarn is 1/(28√Ne) inches across and inverted it against conservation of volume, recovering a packing factor of 0.601 — the accepted figure for a ring-spun cotton, which nobody chose for the convenience of the arithmetic. That agreement is the reason this site trusts its diameters, and it stands.

What it now also says is which contour Peirce was measuring. He got the mass diameter to two figures, from a black-board measurement of yarns wound side by side, on hairy 1930s cotton. The reason he could is that winding yarns side by side under tension is a jamming measurement — the threads are pushed together until they stop — and the jamming contour is the one contour the hair layer does not move. Had he measured by projection he would have got a number half as large again and the packing factor would have come out near 0.27, which is not a packing factor for anything.

A ninety-year-old constant survives because the method that produced it happened to pick the contour with no hairs in it. That is luck rather than judgement, and it is worth recording where the luck was.

It also decides which of this site’s own numbers are exposed. Anything derived from a diameter that was got by jamming is safe: the jammed setts, the closure condition, the crimp, the thickness, the whole Peirce apparatus. Anything derived from a diameter that a reader would obtain by looking is not — and there is exactly one such quantity in the collection, which is the impression a cloth gives of being closed.

The contour is a logarithm, and that decides how reproducible it is #

The three contours were located one at a time. They can be located in one expression, and the expression says something about instruments that no single contour does.

Coverage falls exponentially, because the population does: c(h) = c₀e^(−h/λ), with c₀ the coverage at the yarn’s own surface and λ the decay length. Setting that equal to an instrument’s trigger level c* and solving,

h = λ ln(c₀ ÷ c*),

so the width reported is the mass diameter plus twice that. With c₀ just under a tenth and λ six hundred micrometres, a trigger at a twentieth of the space filled puts the contour 353 µm out and the reported width at 873 µm against a mass diameter of 167 — which is the number this essay reached the long way round, and reaching it twice is worth something.

The form is the finding. A contour is logarithmic in the threshold, which sounds reassuring and is not.

Logarithmic sensitivity means an instrument’s trigger level can be wrong by a factor and the contour moves by only λ times the log of that factor. But the contour is multiplied by two and added to a yarn a sixth of a millimetre thick, so a modest error in the trigger is enormous compared with the thing being measured. A factor of two between two machines’ sensitivities moves the contour by λ ln 2 — and the reported diameters differ by 830 micrometres, which is five times the yarn.

That is the calibration statement, and it is severe:

two optical width instruments must have their trigger levels matched to within about four per cent to report one yarn’s diameter to within five per cent.

Nobody matches optical thresholds to four per cent, and nobody needs to for the purpose such instruments are usually put to, which is comparing yarns on one machine on one day. It does explain why between-laboratory agreement on yarn diameter has always been poor while within-laboratory repeatability is excellent — the repeatability is real and the agreement was never available.

Which contour needs no calibration at all #

Set against that, the jamming contour has a property the others do not, and it is the reason this collection’s arithmetic has been safe.

The load-carrying threshold is never met. There is no height at which the hair layer fills half the space beside the yarn, so a jamming measurement is not sitting on a contour of the coverage curve — it runs off the end of the curve and stops at the yarn’s own material. It has no threshold in it, so it has nothing to calibrate, and two instruments doing it agree because there is nothing for them to disagree about.

The one diameter that is reproducible is the one with no threshold in it. Everything else on the list is a level somebody chose, reported as though it were a length.

Why an optical width will not convert into a count #

One practical consequence, and it is the kind of thing a mill discovers empirically and attributes to the yarn.

The mass diameter goes as the square root of the count, which is conservation of volume and is exact. The optical diameter is that plus 2λ ln(c₀/c*) — and c₀ is itself a function of the count, because the hair density rises roughly as the root of the count while the decay length does not move at all.

So an optical width is a square root plus a logarithm, and no single factor converts one into the other. A calibration fitted over a narrow range of counts will look excellent, because a root and a log are both slowly varying and their sum is very nearly a straight line over any short interval. Carried two or three counts away it drifts, and the drift is systematic rather than noisy.

A conversion from optical width to count is therefore a local fit and must be refitted for each range, which is exactly the practice such instruments carry and is usually presented as a peculiarity of the machine. It is a property of the yarn’s surface: a quantity with a logarithm in it does not scale.

Where the model stops #

The short population is doing most of the covering and is not modelled. Every coverage above is the long population grossed up by one measured ratio. The shape of the profile below about a tenth of a millimetre is therefore assumed similar to the shape above it, which it certainly is not — the short population is denser and steeper. Every contour below a tenth of a millimetre should be read as an order of magnitude and not as a number.

No instrument’s actual threshold is known. The three levels used here are plausible for the three kinds of instrument and are not specifications. What the argument needs is that they are far apart, which survives any reasonable choice; what it cannot supply is the diameter a particular machine will report.

And a hair is taken as vertical for the purpose of coverage. A hair lying nearly along the yarn contributes its whole length to obscuring the background and almost nothing to reach, and the two are not distinguished here.

Nothing bends. A projected image is taken of an unloaded yarn and a jamming measurement is taken of a loaded one, and a hair under load is a cantilever with a very small second moment. That difference is not in the coverage arithmetic at all; it is treated separately, and it is what the next rung but one is about.

The one place the ambiguity is not a nuisance #

There is a use for a quantity with three values, and this site already has the machinery for it.

A cover factor computed at the mass diameter and a cover factor computed at an optical contour are two numbers whose difference is a measurement of the hair layer, taken with instruments a mill already owns. Neither on its own says anything about hairiness; the gap between them is proportional to the sett times the coverage, so dividing it by the sett recovers the coverage directly.

That is not a proposal for an instrument. It is the observation that the ambiguity carries information, and that the information is thrown away every time somebody picks a preferred method and reports one number. A pair of covers is a hairiness measurement hiding inside a discrepancy, in the same way that the gap between a cloth’s two thicknesses is one hiding inside a tolerance.

The generalisation #

When two careful measurements of one quantity disagree by a factor, the first thing to suspect is that the quantity is a contour.

The transferable form is that any graded boundary — a surface with a nap, an interface with a diffusion layer, an edge with a gradient — has as many positions as there are things that might stop at it, and asking which is right is asking the wrong question. What is right is to report the profile and to say which contour a given use needs.

The practical version, for anything on this site that quotes a yarn diameter: say what the diameter is for. A jamming question wants the mass diameter. An optical question wants a contour hundreds of micrometres out. A question about reach — a pill, a print, a prickle — wants the tail and not a diameter at all.

Who found it, and when #

The disagreement is not a discovery. Yarn diameter has been known to be method-dependent for as long as anybody has measured it, and the standards bodies dealt with it by specifying the method rather than by explaining the spread. What is new is the arithmetic that makes the spread a prediction: the profile, its decay length, and thresholds that produce the three answers rather than being fitted to them.

The recovery of Peirce’s packing factor is this site’s own, from the essay that inverted his rule; what is added here is why the number he got was the mass diameter and not something a third larger.

Where the ladder goes next #

Up the ladder, to the two instruments that measure hairiness itself rather than diameter: two hairiness meters read two moments, and the reason they have never correlated is the same reason three diameters do not agree.

Sideways, into what happens when the threshold is a force rather than a level. A light touch never reaches the crowns asks what a plate meets at a stated pressure and finds the crossover is a tenth of the pressure a thickness gauge presses at — so the contours in this essay reappear as pressures, and a fabric’s thickness becomes a property of the instrument in exactly the way its diameter is.