Hairiness goes as the root of the count

Every survey of yarn hairiness ever published shows the same thing: coarser yarns are hairier. The relation is usually drawn as a scatter with a line through it and quoted as a rule of thumb, and it is one of those regularities that is so obviously true — a fatter yarn has more fibre in it, and more fibre means more ends — that nobody has asked what the exponent should be.

The cloth #

The rule of thumb answers the wrong question, because hairiness is not one quantity. A hair population has a count and a length, and a spinner asking whether a yarn will pill, prickle or print sharply is asking about the second while every instrument reports something dominated by the first.

So the question worth asking is not how hairiness scales but how each of its two factors scales, and whether they scale the same way. They do not.

The claim #

The number of hairs per unit length of yarn goes as the square root of the count. Their mean protruding length does not depend on the count at all. So a coarse yarn has more hairs and not longer ones, and the total protruding length goes as the square root of the count because one factor moves and the other does not.

And the sharper half, which is a statement about the fibre rather than the yarn:

The decay length is fifty-six fibre diameters, whatever the yarn. A wool yarn’s hairs are nearly twice as long as a cotton yarn’s, and no count either was spun to changes it.

The identity both laws come out of #

This site asserted, for an entirely different reason, that

d_yarn / d_fibre = √(n / φ)

to twelve figures — n fibres of diameter d_f packed at φ into a circle of diameter d_y, which is conservation of volume used twice. The assertion exists because two routes through the same conversion could have drifted apart. It turns out to decide the whole of this essay.

The density. Hairs come from fibre ends in the outermost shell one fibre thick. The number of ends is 2n/L, exactly. The shell’s share of the section’s area is 4d_f/D to first order, which by the identity is 4√(φ/n). So

N₀ = (2n/L)·4√(φ/n)·e = (8e/L)√(nφ)

— the root of the fibre count, and since n is proportional to the yarn’s count at fixed fibre, the root of the count.

The depth. A hair’s length is the residual of a surface excursion, so it is half the migration period times the shell’s share. The migration period is a fixed number of yarn diameters, k, because migration is a journey across the section and the section is its scale. So

λ = ½ (kD)(4d_f/D) = 2k·d_f

and the diameter divides out. λ is a length belonging to the fibre. For cotton at k = 28 that is 620 micrometres; for wool, whose fibre is 22 micrometres rather than 12, it is 1,100.

Neither law has an adjustable exponent in it. Both fall out of the shell’s area share, which is geometry, and the migration period’s units, which is a modelling decision stated in one sentence.

The check that is not a tolerance #

The two laws are approximations, because the shell’s exact area share is 1 − (1 − 2d_f/D)² and not 4d_f/D. Over a fivefold count range the decay length moves by six per cent and the density’s ratio departs from the square root by six per cent.

Those are the same six per cent, to the last bit of a double. The assertion in the file demands equality to 10⁻¹², not agreement to within a tolerance, and it demands it for every fibre in the table.

The reason is worth writing out because it is what makes the check strong. λ is proportional to D·s and N₀ is proportional to n·s, and D is proportional to √n. So the ratio between two counts is √(tex ratio)·(s₂/s₁) in the first case and, after dividing by the root, (s₂/s₁) in the second. Both departures are s₂/s₁ − 1. A model with an error anywhere in the shell, the migration period or the count would break that equality long before it broke a six-per-cent tolerance, and it would break it in a way no plot would show.

What the two laws do to a cloth rather than a yarn #

A cloth is not a yarn, and the conversion between them — the sett a construction is set to — undoes some of the count dependence in a way worth following, because it explains why the whole subject has stayed invisible.

Per unit area a fabric presents the sum of its two setts of yarn — ends per centimetre plus picks per centimetre, one centimetre of each per square centimetre. So the hairs per unit area are that sum times N₀. And a cloth made of a coarser yarn is set more openly, because a coarse yarn jams sooner: the sett falls roughly as the inverse of the diameter, which is the inverse root of the count.

The two dependences very nearly cancel. N₀ rises as √tex and the sett falls as 1/√tex, so the hairs per square millimetre of cloth are almost the same whatever the cloth is made of. Across this site’s whole table of constructions — from a twelve tex voile to a sixty tex duck, a fivefold range of counts and a twofold range of setts — the hair density spans less than a factor of two.

So a weaver cannot make a cloth hairier or smoother by construction, and that is why nothing in the weaving literature connects hairiness to anything. The two levers that do move it by useful factors are the fibre, which sets the length, and the finishing, which multiplies or truncates the population. Both are somebody else’s department, which is a reasonable explanation for why the quantity has sat in a gap between two trades.

What it means for a spinner #

Reach is a fibre decision and quantity is a count decision, and they are made by different people.

A mill choosing a coarser count for a heavier cloth is choosing more hairs. It is not choosing longer ones, so it is not choosing a worse print, a worse pill or a prickly garment — those need reach, and reach did not move. Conversely a mill choosing a coarser fibre — a stronger wool, a different cotton grade — is choosing longer hairs at every count it spins, and every reach-dependent property moves at once.

That separation is not in the trade’s vocabulary, which has one word for hairiness and one instrument reading for it. It follows directly from the two laws and it is the practical content of the essay.

Where the model and the trade disagree, and it is not smoothed over #

Hold the count and change the fibre’s fineness instead. A finer fibre means more fibres in the section, so more ends; and it means a thinner shell, so a smaller share of them near the surface. The first wins:

N₀ ∝ √(n) ∝ 1/√(tex_f)

so the geometry says a finer cotton spins a hairier yarn, with more hairs each proportionately shorter and a total protruding length that barely moves. The trade says the opposite, firmly and with a century of practice behind it: a finer cotton spins a smoother yarn, which is one of the reasons a finer fibre commands a price.

The disagreement lands entirely in the escape fraction, which is the model’s one measured constant and the one thing the geometry does not supply. A finer fibre is more flexible, so the twist bends it back into the body more readily; and there are more neighbours in the shell to catch it. Both effects lower the escape fraction and neither is computed here.

That is not a repair. It is a statement that the model’s fitted constant is not a constant, and that the place it varies is exactly the place the model’s prediction is wrong. An essay that quietly introduced a fineness dependence into the escape fraction to make the curve go the other way would have produced a model that agreed with everything and predicted nothing.

What a total protruding length actually reports #

The two laws settle what a hairiness instrument is measuring, and the answer is that the commonest reading is the product of a quantity that moves and one that does not.

Total protruding length per unit of yarn is the population’s count times its mean reach, so it goes as the root of the count times a constant — the root of the count, again. The instrument’s headline number is therefore a measurement of the hair count wearing the units of a length, and every conclusion drawn from it is a conclusion about how many hairs there are.

That has a consequence for how two yarns are compared. Two cottons of the same count from the same fibre have the same predicted reach and can differ in total protruding length only through the escape fraction, which is what the spinning did — so a comparison at one count is a clean comparison of spinning systems. Two cottons of different counts differ by a square root before any spinning enters, and comparing their raw hairiness figures compares mostly the counts.

The trade knows this and handles it by normalising, usually by dividing the hairiness by the square root of the count or by comparing only within a count. Both of those are the right move and neither is usually justified; the first is exactly the law above, arrived at empirically and used as a correction.

The interesting case is the one normalisation cannot reach. A wool yarn and a cotton yarn at the same count and the same escape fraction have the same hair count and reaches nearly twice as far apart, so their total protruding lengths differ by a factor of two from the fibre alone. No count normalisation removes that, because the difference is in the factor the normalisation does not touch — and it is exactly the factor that decides pilling, prickle and print definition.

That is a stronger caution than it sounds, because the two fibres a mill is most likely to want compared are cotton and wool.

Which is a second reason, quite separate from the escape fraction, that hairiness figures do not travel between fibres. A normalised reading compares two spinning systems within one fibre and compares nothing at all across two.

What was counted, and how #

Four fibres and a fivefold count range, with three assertions each: that the depth moves by under a seventh, that the density follows the root to the same figure, and that the two departures are one departure. Then, separately, that the depth agrees with 2k·d_f to within an eighth for every fibre in the table — which is the claim that the cancellation is real and not merely that the answer is flat.

The fineness sweep is asserted in three parts: that the count rises, that the length falls, and that the product moves by less than a fifth across a fourfold change in fineness. The third is the one that would break first if either of the first two had the wrong exponent.

Nothing here is compared with a measured hairiness, deliberately. The escape fraction was set once, on a different comparison, and the scalings are what the model says with it held fixed. Fitting it per count would have made the square root unfalsifiable.

The one measurement that would settle it #

Both laws are falsifiable and one of them is falsifiable cheaply.

A hair-counting instrument reports counts at one, two and three millimetres, which is the tail rather than the total — the distinction the two instruments turn on. The ratios between those three counts are a measurement of the decay length and of nothing else — they do not involve the population’s size, the escape fraction, or anything else the model has fitted. So spinning one cotton to four counts and reading the ratios settles the depth claim in an afternoon, and the prediction is that the ratios do not move.

The density claim is harder, because it needs the counts themselves and those carry the fitted constant. But the ratio of the counts between two yarns of the same cotton at different tex is again free of it, and the prediction is a square root.

Neither test needs a hairiness value, an absolute calibration, or a comparison between laboratories — which is fortunate, because comparisons between hairiness laboratories are famously unrepeatable, for reasons the next rung but one is about. Both tests are ratios within one instrument on one day.

Where the model stops #

The migration period in yarn diameters is one constant carrying the entire depth result. If it is not a fixed number of diameters — if it is partly a fixed length, as it might be if the driver is the drafting zone rather than the section — the depth acquires a count dependence and this essay’s headline weakens to a statement about which factor moves more.

The staple length is taken as a single number. It is a distribution, and a short-fibre tail contributes ends out of proportion to its mass, so a real yarn has more ends per millimetre than 2n/L with the mean staple in it. That biases the density up and does not touch the depth.

Twist does not appear at all. The shell’s area share has no twist in it, though the packing factor a diameter comes from does, and the migration period certainly does — a harder-twisted yarn migrates over a shorter length. So the depth should fall with twist and this essay does not say by how much.

And the second-order term is being trusted twice. The six per cent is used as evidence that the two laws are one law, which it is; it is not evidence that six per cent is the right size of correction, because both sides of the equality come from the same expression.

The generalisation #

When a quantity is a product of two factors that scale differently, the quantity’s own scaling is the least useful thing about it.

The transferable form is to resist reporting the product. Hairiness is a count times a length; a fabric’s cover is a sett times a diameter; a strength is a number of fibres times a tenacity. In each case somebody has an instrument that reports the product, and in each case the two factors are set by different decisions and predict different things.

The diagnostic is cheap: change one input and see whether the two factors move together. If they move apart — or, as here, if one moves and the other does not — the product is hiding the mechanism rather than summarising it.

Who found it, and when #

The observation that hairiness rises with yarn count is as old as the instruments, and Barella’s surveys from the 1960s onward are where the numbers live. The observation that the decay length barely moves is implicit in every published set of hair counts at one, two and three millimetres — the ratios between them are strikingly similar across counts — and has not, as far as this collection can tell, been remarked on.

Fibre migration is Morton and Yen’s, 1952. Expressing its period in yarn diameters is this essay’s step and is the one to check.

Where the ladder goes next #

To the other input a spinner controls. The spinning triangle decides the hair takes the escape fraction — the one number the geometry does not supply — and asks what a spinning system does to it, which turns out to be the cleanest explanation available for why compact spinning changes one hairiness instrument’s reading and not the other’s.

Then, up one rung, to why there are two instruments at all: two hairiness meters read two moments.