Textile Structure
how cloth is put together
Concept

Yarn count — where it appears

A yarn's fineness, quoted as a mass per length or a length per mass depending on the system. Every one of the seven in use converts to every other exactly, and the diameter this collection computes from a count is a conditioned mass rather than a dry one.

Named by 22 essays across 4 fields — each of them below, with the objects they name alongside it.

The same thread count, twice. Two cloths with identical thread counts and different yarn. The count is the same number in both; the fraction of the surface the threads actually occupy is not, and that fraction is what thread count is usually taken to mean.

Thread count is not quality

It counts threads. It says nothing about how much of the cloth they cover, it can be inflated by counting plies, and it ranks fabrics in nearly the wrong order. The quantity it is mistaken for is cover factor, and that one is computable.

setting · Cover Six ways of saying how fine a yarn is. The same four yarns written in six count systems, all placed on one axis of tex. A direct system states mass per unit length and rises as the yarn coarsens; an indirect one states length per unit mass and falls. Both kinds are in daily use and neither says which it is.

The yarn count systems, and why there are several

Half the ways of saying how fine a yarn is get bigger as it gets finer and half get smaller. That is not carelessness — and one of the constants buried in the oldest rule of thumb turns out to be a measurement nobody wrote down.

setting · Count Where the twenty-eight comes from. The setting a cotton cover factor of 28 prescribes, and the setting at which the threads would cover the surface geometrically, against yarn count. They are one curve: the trade's scale is Peirce's diameter with the units taken out. Below them is the sett at which each weave actually jams, which is a fixed fraction of it.

Where the cover factor comes from

A cotton cover factor is threads per inch over the root of the count, and the scale says twenty-eight means a covered surface. The twenty-eight is not a convention: it is the reciprocal of Peirce's yarn diameter, and the trade's practical ceilings of fourteen and twenty-two fall straight out of it.

setting · Cover Every cloth at 150 grams. The counts and setts that all weigh 150 g/m² at a crimp of 7 per cent and a balance of 1. Hollow marks are past the jam — arithmetic rather than cloth. Among the 10 that can be woven the cover factor runs from 0.28 to 0.77, a factor of 2.77, and every one of them is the fabric the specification asked for.

What a fabric weighs

Every fabric is sold by its weight in grams per square metre, and the number is a sum of four products in which no term appears alone. One equation, four unknowns: a hundred and fifty grams describes an open coarse cloth and a close fine one, and the cover factors differ by a factor of nearly three.

setting · Weight The evenness a cotton yarn cannot be better than. Lay staple fibres down at random and count how many cross a plane: the count is Poisson, its variance is its mean, and the coefficient of variation of the mass per unit length is therefore 1/√n with no material and no machine in it. The lower curve is that floor. At 20 tex there are 118 fibres in the section and the floor is 9.93 per cent; at 5 tex there are 29 and it is 19.86. The upper curve is what a yarn spun at an index of irregularity of 1.35 actually measures, which is the floor times a constant — so the whole shape belongs to the counting and none of it to the spinning. The exponent is exactly −½ and is asserted as such rather than fitted to the curve.

The spread was never free

This collection has taken a yarn's irregularity off a delivery note and used it as an input. It is not an input. Counting the fibres in a cross-section puts a floor under it that no spinner can beat, and the floor is one over the square root of the count.

cloth · Variation A V-fastened tuft. A cut pile bound into its ground by V fastening, drawn in section. The pile end passes beneath 3 of the 6 ground picks and wraps 1 half-turn around them in all. The integrity criterion says the tuft is attached; how hard it is held is a different question with a different model behind it.

A chenille is a yarn that is already a fabric

A chenille yarn is a woven gauze cut into strips. Every structural question this collection asks about cloth can be asked about it — does it hang together, how many systems has it, what holds the pile in — and the answers are the same answers one level down.

fancy · Fancy yarn 20 tex, counted. The cross-section of a 20 tex cotton yarn, with every fibre in it drawn. The count is a division and nothing else: a 20 tex yarn spun from 0.17 tex fibre has 117.6 fibres crossing any plane through it, and the yarn is 14.0 fibre diameters across because n fibres packed at 0.6 fill a circle √(n/φ) times as wide. The arrangement is drawn on a lattice and is not claimed: real fibres are not on one, they migrate between the core and the surface as they run, and everything this collection says about a yarn's strength turns on their doing so.

How many fibres make a thread

Every number in this collection began with a diameter, and a diameter is not a measurement — it is a count of fibres, divided. Once the division is written down, three quantities that had nothing to do with each other turn out to be the same number.

setting · Assembly Folding improves the evenness and not the yarn. Two independent singles of 15% give a fold of 10.61%, because independent errors add in quadrature: an improvement of exactly √2. The floor falls by exactly √2 as well, from 9.93 per cent to 7.02, because the fibre count is 2 times what it was. So the index of irregularity is unchanged — 1.511 before and 1.511 after, equal to twelve figures, not merely close. Folding does not make a better yarn; it makes a bigger one, and every part of the improvement is the part the count was going to give anyway. What folding does buy is elsewhere: the torque, the surface, and where the grip comes from.

A two-fold yarn is not twice a single

Folding halves nothing. It improves a yarn's evenness by exactly √2 and lowers the floor that evenness is measured against by exactly √2, so the index of irregularity comes out identical — folding does not make a better yarn, it makes a bigger one.

setting · Ply A designed thin place does not get worse and an accidental one does. The thinnest place a yarn reaches, against how many gauge lengths of it are tested, for a slub yarn and for a randomly uneven yarn of the same 36.3% coefficient of variation. The slub's floor is its base count — 89% of its mean — and it is a horizontal line, because a designed variation has a stated minimum and never goes below it. The random yarn's minimum is an order statistic and falls without limit: 56% over 10 lengths and 22% over 30000. So the advantage is not a number but a function of how much yarn is being asked about, running from 1.6× to 4.0×. What the curve cannot show is the break itself: a yarn's strength at a thin place is not proportional to its linear density there, and the conversion needs a fibre model.

A designed thin place is kinder than an accidental one

A slub yarn and a badly spun one can carry exactly the same coefficient of variation, and the number tells a mill nothing about which it has. The designed variation has a floor — its base count, and it never goes below it — while the accidental one has a tail that falls further the more yarn is tested. At 36% CV the slub bottoms at 89% of its mean and the random yarn reaches 24%, and the gap widens from 1.6 to 3.7 times as the test grows from ten gauge lengths to ten thousand.

fancy · Fancy yarn A bouclé at an overfeed of 0.80. A core of 20 tex running straight, an effect thread of 30 tex delivered 80 per cent faster than it, and a binder of 15 tex over the top. The surplus effect thread has nowhere to go but into loops, and their size is not a free choice: a semicircular loop of radius r consumes r(π − 2) of surplus, so at one loop every 3 mm the radius is 2.10 mm — 0.70 of the spacing. The yarn's resultant count is 89 tex and the sum of its three components is 65, because the effect enters multiplied by its own overfeed. What the drawing cannot show is the load path: the core and the binder are at the yarn's own length and the effect thread is longer than the yarn, so a pull on the yarn is carried by 39 per cent of its mass and the loops stay slack.

A fancy yarn has its crimp in the wrong thread

A bouclé is made by feeding an effect thread eighty per cent faster than the core it wraps, so the yarn contains more thread than it is long — which is crimp by this collection's own definition. A cloth's crimp is in the thread that carries the load, so removing it is the cloth's first extension. A bouclé's surplus is in a thread that carries nothing, so pulling the yarn stretches the core at once and the loops never straighten. Thirty-nine per cent of the yarn's mass is on the load path, and the other sixty-one is decoration.

fancy · Fancy yarn The depth belongs to the fibre and the density to the yarn. Over a 5-fold range of cotton yarn counts, the hair layer's decay length moves by 6.2% and its population moves by 2.37-fold against a square root of 2.24. Both follow from one cancellation. The shell's share of the section is 4d_f/D, the migration period is a fixed number of yarn diameters, and λ = ½(kD)(4d_f/D) — the yarn's diameter divides out and leaves λ = 2k·d_f, a length belonging to the fibre alone. The density has no such cancellation and goes as √(nφ)/L. The two small departures visible here are not two facts: they are the shell's second-order term, and they are the same number to the last bit of a double. The consequence for a spinner is that a coarse yarn is hairier and its hairs are no longer, so everything that depends on reach — pilling, prickle, a printed edge — is decided by the fibre and not by the count.

Hairiness goes as the root of the count

A coarse yarn is hairier than a fine one and everybody knows it. What nobody has said is that its hairs are no longer — the count and the length obey different laws, one rises as a square root and the other does not move at all, and the identity behind both was asserted on this site for an entirely unrelated reason.

setting · Hair layer A bouclé cloth jammed at its loops and at its count, overfeed 0.8. Sections across 36 mm of plain-woven cloth in a bouclé of 89 tex — a 20 tex core, a 30 tex effect overfed 80% with loops every 3 mm, and a 15 tex binder — drawn to scale. The loops stand 2.10 mm off the core, so the yarn's outline is 4.43 mm across while the diameter its count implies is 0.352 mm, 12.6 times smaller. Jammed at the outline the cloth takes 1.13 ends a centimetre and weighs 20 g/m²; jammed at the count, 14.2 and 252 g/m². What the drawing cannot show is where between the two a real cloth jams, which is how far its loops interleave with their neighbours'.

A bouclé is set by its loops and weighed by its count

A bouclé yarn has two diameters: the one its count implies, a third of a millimetre, and the one its loops occupy, over four millimetres. A cloth jams where its yarns touch, and a bouclé's yarns can touch at either — so its jamming sett is a bracket twelve and a half times wide, from 1.1 to 14 ends a centimetre. The weight follows the count wherever in that bracket the cloth is set, which puts the open end at a twelfth of the close end's weight with four per cent of its area covered by yarn.

fancy · Fancy yarn How much of a yarn has to hang before it stops snarling. The tension a 20 tex cotton needs to stay straight, at each twist level, expressed as the length of the yarn's own weight that would supply it. The two curves are the two ends of the stiffness bracket: the fibres free to slide, and the section coherent. At 800 turns a metre they are 1.9 metres and 616. Anybody who has let go of a twisted yarn knows which of those is right, which makes this one of very few places where the bracket can be closed from the everyday end. Both rise as the square of the twist, because the torque does and the criterion is quadratic in it.

How much yarn has to hang

The tension a thread needs to stay straight, converted into the only unit anybody has an intuition for: the length of the yarn's own weight. One bound says two metres and the other says six hundred, and everybody who has handled thread already knows which.

setting · Twist Three rules for how hard to fold a yarn, and which one the trade uses. For 2, 3, 4 folds of 20 tex cotton at 800 turns a metre: the ratio that makes the fold's surface helix angle equal its singles', the ratio that sets the fold's net moment to zero, and the ratio the trade actually folds at. The surface rule is 1/√n exactly, because a fold of n singles is √n times the diameter. The torque balance is C/(B+C), which for this fibre is 0.201 and is the same at every count, every twist and every number of folds. The trade's brackets contain the surface rule in all three rows and the torque balance in none of them.

A cabled yarn is a fold of folds

The rule that sets a fold's twist is one over the square root of the number of components. Apply it twice and a cabled yarn's three twist levels are fixed by two integers — which is a prediction with no free constants, about a class of yarn the trade quotes no rule for at all.

setting · Ply What each further binder takes off a snag. How far a snag drags thread through a bouclé, in binder points, against how many binders the yarn carries: 1, 7.43 points; 2, 3.72 points; 3, 2.48 points; 4, 1.86 points; 6, 1.24 points. The relation is exactly inverse, because the grip is the exponential of the wrap angle and the reach is a logarithm of it, so the second binder removes 3.72 binder points and the third a further 1.24.

The second binder buys half of everything

A bouclé's loops are held by their binder, and a snag drags thread from one loop to the next until the thread breaks. The essay before it left that reach at seven binder points on one binder and said a second would square the grip. It does — and because the grip is exponential in the wrap and the reach is a logarithm of it, the reach is exactly inverse in the binder count: 7.43 points, then 3.72, then 2.48. Half of everything any number of binders can buy is bought by the second one, which is how many the trade uses.

fancy · Fancy yarn The flattening a knitted fabric asks for, against its own tightness. Every point is a solved fabric — 18 of them, over five loop lengths, three relaxation states and three counts — and the vertical axis is the closest its adjacent courses come, in diameters. The values run from 0.707 to 0.839 and they fall on one curve against the tightness factor, which is the model's only dimensionless group. That is what makes this a structural requirement rather than an arithmetical accident of one example: a tighter fabric demands a flatter yarn, by an amount its own tightness decides, and nothing about the fibre or the count enters except through that group.

The count that decides how flat

A knitted fabric's demanded flattening is a function of its tightness factor, and a tightness factor is the square root of a count over a loop length. So a coarser yarn at the same loop is flatter — which is a prediction about a spinner's choice that nobody has framed as one.

setting · Count What two layers of one yarn buy. For each cloth in the table here, the thickness and the thermal resistance of a two-layer cloth carrying exactly the same yarn per unit area, as ratios to the single cloth. The thickness ratio is √2 everywhere, to three figures. The warmth ratio runs from 1.588 on the openest cloth to 1.717 on the closest, because dividing the yarn also divides the fibre fraction and the mixture conducts less.

Two layers are warmer than they are thick

Six essays have taken a double cloth apart from the draft's side. None has asked what the reader gets. Divide one cloth's yarn into two layers and the fabric is 41 per cent thicker — √2, which is the account's own law — and about 70 per cent warmer, because dividing the yarn also divides the fibre fraction and the mixture conducts less. The gap widens with every further layer and never closes, and a shaft loom stops at two.

fancy · Double cloth One dip, two yarns: the same depth of dye. The cross-section of two cotton yarns drawn to one scale after a 20-second dip in a bath whose dye the fibres take up 10 times as strongly as the liquor holds it, each fibre shaded by the dye that reached its distance from the surface. 20 tex: 118 fibres, 167.1 µm across, dyed to 19.7 µm — 24% of its radius and 42% of its section; 74 tex: 435 fibres, 321.4 µm across, dyed to 18.2 µm — 11% of its radius and 21% of its section. The depth in micrometres is the same for both because a dip dyes a depth; the share is not, because the coarse yarn has more section behind the same ring. The fibres are drawn on a lattice and the count is computed; the affinity is an assumed value and every depth moves with its square root.

A dip dyes a depth, not a share

A yarn dyed after spinning is a cylinder the dye has to diffuse into through the liquor between its fibres, and the fibres slow it by taking dye out of the liquor as it passes. A twenty-second dip therefore dyes to a depth, some tens of micrometres, and that depth is almost the same in a fine yarn and a coarse one. So the same dip that colours two fifths of a shirting yarn's section colours a fifth of a denim warp's; eight dips deepen the shade eightfold and leave the ring where it was; and a tighter yarn rings more thinly, because packing closes the pores faster than it narrows the yarn.

finishing · Dyeing Every count that can make 150 g/m² as plain jersey. Areal weight against count for plain jersey in the fully relaxed state: the shaded wedge is every knit between a tightness factor of 1.3 and 1.6, whose weights are kₛ times the tightness times the root of the count. The rule at 150 g/m² crosses it between 15.8 and 23.9 tex, a ratio of 1.51. A plain woven cloth of the same weight can be made from anything between 20 and at least 200 tex, a range of at least 10 to one against the knit's 1.51.

A knit's weight nearly names its yarn

A woven cloth's weight is one equation in four unknowns, and a hundred and fifty grams can be woven from anything between twenty tex and two hundred. A plain jersey's weight has the loop in it and nothing else to spare, and the loop is bounded by the yarn it is knitted from. Put the two together and the loop cancels: the weight is a constant times the tightness times the root of the count, so at one weight the count is fixed to within half again — and in each relaxed state it is fixed to a different half.

setting · Weight What a 2-fold yarn weighs against the sum of its singles. The count of a 2-fold cotton yarn against the sum of its singles' counts, as the folding twist runs from three tenths to nine tenths of the singles' twist, at twist factors of 3000, 3800, 4500 (turns a metre times the root of the tex). Below a ratio near two thirds the fold is lighter than the sum, above it heavier; the curves cross the sum at 0.661 at 3000, 0.658 at 3800, 0.654 at 4500, and in the small-angle limit at exactly 0.6667. The trade's folding band, 0.6 to 0.75, is shaded, and across it the fold is within −0.6% and +1.3% of its notation. What the curves cannot show is a real fibre's migration between the core and the surface of each single, which the ideal helix leaves out and which shortens the path both lengths are computed from.

Two thirds is where a fold weighs its singles

A folded yarn's count is written as its singles' count divided by the number folded, as though folding changed no length. It changes two. The fold's helix wraps each single round the yarn's axis, which puts more single into every metre, and the untwisting straightens each single's fibres, which puts less fibre into every metre of single. Each is three to seven per cent at a weaving twist. They cancel at a folding ratio that is exactly two thirds in the small-angle limit — the folder's own number — and six elevenths for three singles and two fifths for four.

setting · Count What a 2×2 cable weighs as its cable twist rises. A cotton cable of 2 folds, each of 2 singles, with the folds set at 0.707 of the singles' twist and the cable twisted the singles' way at ratios from nought to 0.60, at twist factors of 3000, 3800, 4500. A little cable twist makes the cable lighter, because it takes twist out of folds set above their own crossing; then the cable's own helix wins and the count climbs. It crosses the singles' sum at 0.260 at 3000, 0.250 at 3800, 0.239 at 4500. At the rule as quoted, a cable twist of 0.500, the cable is +2.21%, +3.54%, +4.95% heavy; at the rule read in the finished thread, 0.293, it is +0.16%, +0.32%, +0.54%. What the curves cannot show is a real cable's folds pressed out of round, which shortens the cable's helix and moves every crossing right.

A cable weighs its singles at the twists the finished thread keeps

Cabling untwists every fold and hands that twist back to the singles inside it, so beneath its own helix a cable is a fold folded at the difference of two twists — and the cable's helix is a length nothing below it can pay for. Twisted where the cabling rule is quoted, a two-by-two weighs three and a half per cent over its notation. Read in the finished thread, with every level's surface at one angle, the same rule sets the folding frame exactly where a plain fold's is and the cable at 0.293, and the cable comes back to within half a per cent.

setting · Count What a thread of twos weighs as its levels of twist are added. A two-fold, a cable of two two-folds, a cord of two such cables and a cord of two cords, all of 20 tex cotton singles, each twisted so that every level's surface sits at one angle in the finished thread, at twist factors of 3000, 3800, 4500. At 3,800 they weigh fold −1.35%, cable +0.32%, cord +2.02%, cord of cords +3.75% against the sum of their singles: a fold light, a cable on its notation, and every level after it one helix heavier than the last. Dashed, the same four twisted to the rule as quoted on the frames, +0.46%, +3.54%, +4.54%, +6.98%. What the chart cannot show is the folds' and cables' real shape, which is taken round.

The singles pay once and every level adds a helix

A cable twisted so that every level's surface sits at one angle weighs its notation, and it is tempting to read that as a law: equal angles, honest count. It is not one. With every surface at one angle the singles keep the same twist however many levels sit above them, so their saving is paid once, while every level adds a helix of the same length. A two-fold comes out light, a cable on its notation, and a cord of two such cables two per cent heavy — the cable's agreement is the walk passing through zero. For a cord to weigh its notation, everything above its singles has to sit at about half their angle.

setting · Count

Named alongside it

The objects these essays reach for when they reach for this one.

SpecificationPacking factorPlyFibre countTwist factorYarn diameterCoefficient of variationFancy yarnHelix angleSettCover factorFibre fineness

All concepts