The spinning triangle decides the hair

The hair population is built from four numbers and three of them are arithmetic. The fibre count comes from a division; the ends per millimetre come from a staple length; the shell’s share of the section comes from an identity this site asserted for another argument altogether. The fourth is a measurement, it is called the escape fraction, and it is the only fitted quantity in the whole construction.

It is also the only place a spinner appears. Nothing about the count, the fibre or the sett is a decision about how the yarn was made, and the escape fraction is entirely a decision about how the yarn was made.

The cloth #

A ring frame draws a ribbon of fibres out of a pair of rollers and twists it into a yarn. The ribbon leaves the nip flat and a few millimetres wide; the yarn is a fifth of a millimetre across. Between the two there is a region — the spinning triangle — in which the ribbon is converging and the twist has not yet reached the fibres at its edges.

A fibre entering at the middle of the ribbon is surrounded on both sides from the moment it leaves the nip. A fibre entering at an edge is not: it has neighbours on one side only, it must travel further to reach the yarn’s axis, and it spends the whole length of the triangle unbound. If its end arrives during that journey, nothing catches it.

That is the whole mechanism, and every spinning system that claims to reduce hairiness is an intervention in it. Compact spinning condenses the ribbon pneumatically before the twist, so the triangle nearly disappears. Rotor spinning has no triangle at all and has wrapper fibres instead. Combing removes the short fibres, which are the ones most likely to be entirely inside the triangle when their ends arrive — and a staple length is a distribution rather than a number.

The claim #

The escape fraction is a property of the spinning triangle, and an intervention in the triangle removes hairs from one of the two populations and not from the other. So it moves the two hairiness instruments by different amounts, and the ratio between those amounts is predictable without knowing what the intervention was.

Stated as arithmetic: an integrating hairiness meter falls by a few per cent while a long-hair counter falls by two thirds, and the ratio between the two drops is the reciprocal of the long population’s share of the total protruding length.

Two populations, and why an intervention can only reach one #

The hair model records, and does not model, a second population. Setting the modelled long population’s total protruding length against an integrating instrument’s reading on the same yarn leaves a factor of about eight unaccounted for. There is a dense cloud of very short protrusions — loops, slack fibre lying on the surface, ends that stand off by a fibre diameter or two — which contributes most of the length and almost none of the reach.

The two populations have different origins and that is the point of this essay.

A long hair is an end that was never caught. It got free during the triangle, its free length is the residual of a migration excursion, and it reaches hundreds of micrometres.

A short protrusion is an end that was caught and is not quite flush. It is bound into the yarn body a fibre diameter or two below where it emerges, and it stands off by that much because a fibre is a cylinder and not a thread of paint.

An intervention in the triangle acts on the first and cannot act on the second. Condensing the ribbon stops ends escaping; it does not press the surface flat. So compaction removes a large fraction of the long population and essentially none of the short one.

The asymmetry, computed #

Let the long share be s and let compaction multiply the long population by f. Then the total protruding length goes from L to L(sf + 1 − s), so it falls by s(1 − f); and the long-hair count goes from N to fN, so it falls by (1 − f).

The ratio of the two drops is 1/s, and s is about an eighth. So a compaction that takes two thirds off the long hairs takes under a tenth off the total, and the two instruments disagree by a factor of eight about how much good it did.

That is what is measured. Compact yarns test better on a long-hair count by sixty to eighty per cent and better on an integrating meter by twenty or thirty, and the discrepancy has been reported so often that it is treated as an instrument problem. It is not an instrument problem. It is two instruments correctly reporting two different populations, one of which was the target.

Why the model under-states one of the two #

The prediction above puts the fall in total hairiness at about eight per cent and the measurements put it at twenty or thirty. The model is wrong in one direction and says so.

The reason is the honest one: compaction certainly does something to the short population as well, and nothing here models the short population. A condensed ribbon presents a smoother surface to the twist, so fewer ends stand proud by a fibre diameter, and that is a reduction the arithmetic cannot see.

What survives is the ordering and the sign of the asymmetry, which is what the essay claims. What does not survive is the magnitude of the smaller of the two drops. The direction of the error is stated because it is the conservative one: the model under-predicts the benefit and over-predicts the disagreement between the instruments.

What the asymmetry is worth to somebody buying yarn #

The arithmetic has a commercial edge on it, and it is sharp enough to be worth stating plainly.

A yarn bought on an integrating hairiness specification is bought on a number that an eighth of the population controls, in the way a mean diameter is a number a spread controls. Two yarns meeting the same specification can differ by a factor of thirty in the quantity that decides whether the cloth pills, whether a print holds its edge, and whether a garment prickles — because all three of those need reach, and reach is the part of the distribution the specification barely weighs.

That is not a hypothetical. It is the direct reading of the curve above: at a fixed total hairiness, the long-hair count runs over more than a decade across the range of decay lengths real yarns have. And the decay length is a fibre property, so a blend change, a grade change or a change of supplier moves it without moving anything the specification names.

The specification is not wrong; it is answering a different question. An integrating reading is an excellent predictor of how a yarn will behave in the shed — how much it will shed lint, how much it will cling to its neighbours, how much size it will need — because all of those are about total fibre in the way. It is a poor predictor of anything that happens to the finished cloth’s surface.

Twist, which acts on the same fraction from the other side #

Twist enters the escape fraction rather than the geometry, and it enters it twice with opposite signs.

More twist means the fibres at the ribbon’s edge are caught sooner, because the twist runs further back up the triangle: the surface helix angle rises with twist, and a steeper helix propagates the binding further toward the nip. That lowers the escape fraction.

More twist also means a smaller yarn diameter, and therefore a different cover factor, because a harder-twisted yarn packs denser — the packing factor rises with twist, which this site has recorded as a caveat on every diameter it computes. A smaller diameter is a larger shell share, so more ends are near the surface to begin with. That raises the density.

The two effects are opposed and neither is computed here, which is why this essay does not claim a twist law. What it can say is that both act, that they act on different factors, and that a measured hairiness-against-twist curve with a minimum in it — which is what such curves have — is the signature of two opposed mechanisms rather than of one with a subtlety.

What was counted, and how #

The assertion is the asymmetry, in two parts: that the long-hair count falls further than the total, and that it falls further by a factor of at least three. The second is the one that matters, because an essay claiming a qualitative difference between two instruments has to show that the difference is not a shading.

The escape fractions themselves are a table and are measurements. What is not asserted anywhere is that any particular system has any particular escape fraction — the table’s values are ordinary trade figures and the file says so. The arithmetic they feed is the ratio, and the ratio is insensitive to all four of them: it is 1/s and s comes from a different comparison entirely.

Nothing in this essay was fitted to a compaction measurement. The long share was set once, from the model’s total against a hairiness reading on a conventional ring yarn, before any spinning system other than ring was considered.

Where the model stops #

The triangle’s geometry is not computed. A ribbon width, a triangle length and a twist propagation distance would give the escape fraction from first principles, and this essay has none of the three. What it has is a mechanism that says which population an intervention reaches.

The short population is still not modelled, and it now carries a known error rather than an unknown one.

Rotor spinning is in the table and is not really described by the mechanism. A rotor yarn has no spinning triangle, so the escape route this essay is about does not exist; its hairs come from the fibres deposited on the rotor groove and from wrapper fibres, which are a different object with a different length distribution. Its entry in the table is a number without an argument behind it and should be read that way.

And a compacting device is treated as a single multiplier. Real ones differ — a perforated drum, a magnetic condenser, a suction slot — and there is no reason to think they all act on the same part of the distribution.

The staple distribution is not in it either. The mechanism says an end escapes if it arrives during the triangle, so the fibres most likely to escape entirely are the ones shorter than the triangle is long — which is exactly what combing removes, and which is why the combed entry in the table sits where it does. A model with a staple distribution in it would predict the combing benefit rather than tabulating it, and that is the clearest next piece of work on this rung.

What a staple distribution would predict for combing #

The limits section names the missing piece — a model with a staple distribution in it, which would predict the combing benefit rather than tabulating it — and one part of that model needs no triangle geometry at all.

An escaping hair is a fibre end, so the density of candidates is the number of fibre ends per unit length of yarn, and for a given mass of fibre that is one over the number-average staple length. Short fibres contribute ends out of all proportion to their mass, because a given weight of short fibre has more of them.

Put a carded cotton through it. Take eighteen per cent by weight at eight millimetres and the rest at twenty-eight: the number fractions go as weight over length, which is 0.0225 against 0.0293, so the number-average length is 19.3 millimetres — well below either component and dominated by the short one.

Comb it, removing the short fraction, and the number-average rises to 28. Ends per unit mass fall by the ratio, which is

thirty-one per cent fewer candidates, from removing eighteen per cent of the fibre.

That is a prediction rather than a table entry, it needs no escape fraction, and it is of the size the trade reports for combed against carded hairiness. The combing benefit is mostly an arithmetic consequence of which fibres carry the ends, and only the remainder — whatever the triangle does differently to a short fibre — needs the geometry the model does not have.

Which makes staple length the stronger of the two levers #

The same accounting says how the staple length compares with the other quantity a spinner controls, and they do not enter at the same power.

Ends per unit length of yarn go as the fibre count times one over the number-average length. Only the shell’s fibres can escape, and the shell’s share falls as one over the root of the fibre count, so the escaping candidates go as

√tex ÷ L̄ —

a square root in the count and a first power in the staple length.

So the staple length is twice the lever the count is. Raising the number-average staple by fifteen per cent removes as many candidates as making the yarn thirty per cent finer, and a spinner choosing between a longer cotton and a finer count is choosing between a linear term and a root.

It also says which comparison is being made when a mill reports that its finer yarns are less hairy. They are, per unit length — but a finer yarn is normally spun from a better cotton as well, and the two effects are not separable in any comparison of commercial yarns. The count’s contribution is the smaller half of what such a comparison measures, and the fibre’s is the larger.

The caveat is that neither term is the escape fraction itself, which is what the triangle decides and which multiplies both. A long, well-prepared cotton spun through a badly set triangle will be hairier than a short one spun through a good one, and nothing in the candidate count says otherwise. What the arithmetic gives is the supply of candidates; the triangle decides what fraction of them get away, and only the second is a spinning decision rather than a purchasing one.

The one intervention that is not in the triangle at all #

There is a fifth way to change the escape fraction and it happens after the yarn exists, which is why it sits outside every table of spinning systems.

A warp is sized before it is woven — a film of starch or of synthetic polymer laid on the yarn to survive the loom — and size glues the hairs down. The hairs in the loom is where this site first recorded it: a sized warp is a warp with its escape fraction temporarily reduced to nearly nothing, because the hairs are stuck to the body rather than absent from it.

The word temporarily is the whole of it. Desizing removes the film and returns the hairs, and the population that comes back is the one spinning left, because nothing was removed. A sizing is a mask on the escape fraction and a compaction is a change to it, and the two are indistinguishable in the loom and completely different in the finished cloth.

That distinction matters for the same reason the asymmetry does: a weaver measuring hairiness on a sized warp is measuring a property of the size. The measurement that predicts anything about the cloth has to be taken on the yarn before sizing or on the fabric after desizing, and those two agree with each other and not with the one in between.

The generalisation #

An intervention that targets one part of a mixture is judged by an instrument that measures the whole of it, and the judgement is wrong by the mixture’s own proportions.

The shape is general and it is not about textiles. A treatment that removes the tail of a distribution moves the mean a little and the tail a lot; a measurement of the mean therefore reports a small effect and a measurement of the tail reports a large one, and both are correct. The trap is that the two get compared and one is declared unreliable.

The diagnostic is to ask what fraction of the reported quantity the intervention could possibly have reached. Here it is an eighth, and an eighth is the whole explanation for a thirty-year argument about hairiness instruments.

Who found it, and when #

The spinning triangle is as old as ring spinning and the observation that edge fibres become hairs is standard. Compact spinning is Fehrer’s and Rieter’s, from the 1990s, and its hairiness advantage was its selling point.

The disagreement between hairiness instruments is thoroughly documented and is usually attributed to their different optical principles — one measures obscuration, the other counts pulses — which is true and is not the cause. What is new here is that the two principles happen to sample two physically distinct populations with different origins, and that a triangle intervention can only reach one of them.

Where the ladder goes next #

To the instruments themselves. Two hairiness meters read two moments shows that the disagreement is arithmetic rather than optics: a first moment and a tail probability are not the same statistic and cannot be made to correlate.

Sideways, to the operation that does what a spinning triangle cannot. Singeing truncates the population rather than scaling it, so it removes the reach entirely and the length hardly at all — which is the same asymmetry as this essay’s, pointed the other way and arriving from a flame rather than from a suction slot.