The spread was never free

A yarn’s coefficient of variation is the most measured quantity in the whole industry. It is on every delivery note, it is the number a spinner is judged by, and this collection has used it as an input: fifteen per cent, taken as given, pushed through a cover factor and a rigidity and a jam, and reported as a set of biases and extremes.

Fifteen per cent of what, though, and why fifteen?

The question sounds rhetorical and it has an exact answer. A yarn is an assembly of a countable number of fibres, and once the count is written down the irregularity stops being a free parameter. There is a value it cannot go below, that value is a property of the count and of nothing else, and it is not a bound in the loose sense of something a good process approaches — it is a floor that the arithmetic of counting puts there before any machine has touched the fibres.

The claim #

A yarn made of fibres laid down at random has a coefficient of variation of mass per unit length of exactly 1/√n, where n is the mean number of fibres in a cross-section, and no spinning process can produce a lower one.

Two things follow at once and both are more useful than the floor itself.

• A measured irregularity means nothing until it is divided by its own floor. The ratio is the index of irregularity, it is dimensionless, its unbeatable value is one, and it is the only quantity that compares a fine yarn with a coarse one. Fifteen per cent is a good ring-spun yarn at 10 tex and an unspinnably bad one at 80.
• The fifteen per cent this collection has been using is an index of 1.51 at the count it was used at, which is a rotor yarn or an indifferent ring-spun one. Nothing said so, because nothing had a floor to say it against.

The argument, which is counting and nothing else #

Take a very long yarn and cut it across at a random place. Count the fibres that cross the plane.

If the fibres are laid down independently — each one starting at a position uncorrelated with the others — then the number crossing any given plane is a Poisson variable. That is not a modelling choice made for convenience: it is what “independently placed” means. And a Poisson variable has the property that makes the whole argument one line: its variance equals its mean.

So if the mean number in a section is n, the variance of the count is n, the standard deviation is √n, and

$$\mathrm{CV} = \frac{\sqrt{n}}{n} = \frac{1}{\sqrt{n}}.$$

The mass per unit length of the yarn is the number of fibres in the section times one fibre’s mass per unit length, so the coefficient of variation of the mass is the coefficient of variation of the count, and it is 1/√n.

There is no material in that. No modulus, no friction, no twist, no machine, no fitted constant. Two yarns of the same fibre count made from wool and from polyester on a ring frame and on a rotor have the same floor, and the floor is the same today as it was in 1830.

The one correction worth making #

Real fibres are not all alike. A cotton’s fineness varies from fibre to fibre by about forty per cent, and a section that happens to catch forty thick fibres weighs more than one that catches forty thin ones. The count is Poisson; the mass now has a second source of variance.

Weighting by mass rather than by number gives

$$\mathrm{CV}_{\text{lim}}^2 = \frac{1 + \mathrm{CV}_f^2}{n},$$

with CV_f the coefficient of variation of the fibre’s own linear density. For cotton at 0.4 that raises the floor by eight per cent; for a cut man-made staple, whose fineness is a spinneret setting and varies by a few per cent, it leaves it alone.

That correction is worth carrying for one reason: it is the only place where the fibre enters the floor at all, and it says that a natural fibre is at a permanent disadvantage to a manufactured one of the same fineness. The disadvantage is eight per cent, which is small, and it is a real one — a floor is a floor.

What the index measures, and what it does not #

Divide the measured irregularity by the floor and the result is the index of irregularity: one if the fibres are laid down at random, more than one if the process has added something.

Everything a spinning process does is in that number. Drafting waves, roller eccentricities, the periodic thick places a badly set frame produces, the fibre hooks a card leaves — all of it raises the measured CV above the floor and none of it changes the floor. And nothing lowers the index below one, because randomness is what a floor of independence means: to beat it a machine would have to place each fibre knowing where the others had gone.

The trade bands are worth stating because they are what a reader can check the whole argument against. A ring-spun cotton runs at an index of about 1.15 to 1.6; rotor spinning, whose back-doubling averages the feed but whose fibre arrangement is worse, at 1.5 to 2.1; air-jet between them. Those are measurements and they are bands rather than values, and every one of them is above one, which is the check on the floor being a floor.

What this does to the earlier arithmetic #

The essay that put the population back computed what a fifteen per cent spread does to seven of this collection’s own quantities: nothing at all to a cover factor, 2.25 per cent to a mass, 14.3 per cent to a bending rigidity, and an unbounded amount to a jam or a thickness reading. Every one of those numbers was computed from a CV that arrived from outside.

It does not arrive from outside any more. It arrives from the count, times the index — and the count is the yarn’s own specification.

The practical difference is that the biases can now be predicted for a yarn nobody has measured. Give a count, a fibre and a spinning system, and the floor comes from the first two and the index from the third; multiply, and the CV follows; push it through the curvature table and the excesses follow. That was not possible before, and it is the only reason a floor is worth having.

It also settles a question the earlier arithmetic raised and could not answer. The corrections it computed all get worse as the yarn gets finer, and it was not obvious whether that was a property of yarns or of the particular fifteen per cent. It is a property of yarns: the floor goes as n to the minus a half, so a finer yarn is a worse yarn by an exponent rather than by custom.

Where the floor meets the other floor #

There is a second floor in this subject and the two meet in a way that is worth drawing.

A spun yarn needs a minimum number of fibres in its section for twist to hold them — about thirty-five on a ring frame — and that sets the finest yarn a fibre can make. For cotton it is about 6 tex, or Ne 99, which is where the finest commercial counts sit.

Put the two floors together. A yarn at the spinning limit has thirty-five fibres in its section, so its evenness floor is 100√(1.16/35.3), which is 18.1 per cent — and at an ordinary ring-spun index of 1.35 it will measure about 24.5. The very finest yarn a cotton spinner can make is, of necessity, an extremely irregular one, and no improvement in machinery can change that: the two floors are the same count read twice, and they close on each other from opposite sides.

That is the sharpest form of the argument in this essay. The two most desirable properties of a fine yarn — fineness and evenness — are in direct competition through a single count, and the competition is settled by the fibre before the spinner starts.

It also explains a piece of practice. Fine yarns are spun folded far more often than coarse ones are, and the usual explanation is strength. Strength is part of it, but folding is also a doubling of the count, and a doubled count is a floor lowered by √2 — which is the only route to a fine, even yarn that this arithmetic leaves open.

What was counted, and how #

The floor is checked by counting rather than by algebra. An identity verified algebraically proves that the algebra was copied twice. So the claim is tested by simulation: draw Poisson counts at three means, measure the coefficient of variation of the sample, and require it to match 1/√n within three per cent at twelve thousand samples — which is about twice the sampling error of the estimate itself, tight enough that a wrong power of n could not survive and loose enough that the check does not fail on a bad afternoon. At n = 20 the count gives 22.29 per cent against a claimed 22.36; at 60, 12.85 against 12.91; at 200, 7.13 against 7.07.

And the floor is required to move. A check that a quantity matches a formula is satisfied by a constant if the formula happens to be constant over the range tested, so the sweep additionally asserts that the floor at the coarsest count is less than half the floor at the finest.

The exponent is asserted at exactly −½ rather than fitted. Over the range 5 to 100 tex the measured log-slope of the floor against the count must equal −0.5 to twelve decimal places. It is not a curve that happens to look like an inverse root.

Three factors, and none of them interacts with the others #

Written out, a yarn’s coefficient of variation is a product of three things and nothing else:

CV = index × √(1 + CV_f²) × √(fibre tex / yarn tex).

The first belongs to the machine, the second to the fibre’s own uniformity, the third to the count and the fibre’s fineness. They multiply, which means their logarithms add, which means an improvement in one is worth the same fractional amount whatever the others are. That is a stronger statement than it looks, and it is the reason the index is a usable rating at all: a process improvement of ten per cent is ten per cent at 6 tex and at 80, in cotton and in wool, on a ring frame and on a rotor.

It also makes the levers comparable, which the raw numbers do not.

Spinning better. Moving the index from 1.35 to 1.15 is a fifteen per cent improvement, and it is what combing buys — at the cost of the short fibre it removes, which is between twelve and twenty per cent of the bale.

Spinning coarser. Doubling the count improves the floor by 29 per cent, since the square root of two is 1.41 and the improvement is its reciprocal. That is the largest single lever in the list and it is almost never available, because the count is the specification.

Choosing a finer fibre. The floor goes as the square root of the fibre’s linear density, so a fifteen per cent improvement needs a fibre 28 per cent finer — 1.7 decitex cotton down to 1.2, which is a different and much more expensive cotton, or 1.7 decitex polyester down to 1.2, which is a spinneret setting and costs almost nothing. That asymmetry is most of why fine man-made staple exists.

Choosing a more uniform fibre. The (1 + CV_f²) term is at 1.077 for cotton at a forty per cent fineness spread and at 1.001 for a cut synthetic, so the whole lever is worth 7.2 per cent and is exhausted the moment a manufactured fibre is used. It is the smallest term in the product and the only one with a hard ceiling.

Reading the four together gives the order a spinner should work in, and it is not the order the trade’s attention follows. The count dominates, the fibre’s fineness comes second, the process third and the fibre’s own uniformity last — and yet the index is the number that gets argued about, because it is the only one of the four that is anybody’s fault.

There is one honest defence of that attention. The other three are decided before the yarn is ordered, by a buyer specifying a count and a merchant selecting a bale. The index is the only term still open once the fibre is in the machine, so it is the only term a spinner can be judged on — which is a statement about responsibility rather than about arithmetic, and it is worth keeping the two apart.

Where the model stops #

The fibres are assumed independently placed, and in a real yarn they are not. A carding engine, a drawframe and a ring frame each impose their own periodicities, and the correlated part of the variation is exactly what the index measures. So the floor is a floor for an idealised random assembly, and the claim that no process beats it rests on the argument that correlation can only add variance — which is true for correlations of either sign only if the process cannot place a thick fibre deliberately next to a thin one. Nothing in the ordinary spinning line can.

The correction for fibre fineness assumes fineness is independent of position. It is not quite: a card and a comb both preferentially remove short and fine fibres, so the population in the yarn is not the population in the bale.

It is a floor on mass per unit length and on nothing else. A yarn also varies in twist, in strength, in stretch and in hairiness, and none of those is bounded by this argument. In particular there is no floor here on strength variation, although where a yarn breaks is closely tied to where its mass is thin.

The measuring length is not in the floor and it is in the measurement. An evenness tester reads mass over a short length of yarn — a centimetre or less — and the coefficient of variation of a longer average is smaller. Quoting a CV without its measuring length is half a number, and the floor derived here is the floor at the length over which one fibre’s presence or absence is a whole event.

The index is a rating of a process, not a prediction of one. Nothing here says what index a given machine will reach; the bands are reported measurements. What the argument supplies is the denominator, and it supplies it exactly.

Who found it, and when #

The limit irregularity is Martindale’s, published in 1945, and it belongs to the same decade as the industrial evenness tester — which is not a coincidence, because a measurement that could not be compared between counts was a measurement asking for a denominator. The index of irregularity has been the standard way of quoting yarn evenness ever since.

What this collection adds is the connection to its own arithmetic. The floor is 1/√n; the stiffness bracket is n/φ²; the diameter is √(n/φ) fibres across. Three quantities in three different fields, all of them the same count, and until the count was written down none of them knew about the others.

Where the ladder goes next #

Directly into the consequence, which is that fineness costs evenness by an exponent: every correction the population arithmetic computes gets worse as the count falls, and the cost can be priced for each of this collection’s own quantities.

And into the two places where a floor on the mass reaches something else. A thin place is where a yarn breaks, so the floor reaches the strength through an extreme rather than a mean; and a cloth woven from an irregular yarn averages that irregularity over the threads in view, so the floor reaches what the cloth looks like through a spatial average. The first makes the variation worse and the second makes it better, and the two are the same number entering by different doors.