Textile Structure
how cloth is put together
Setting and geometry

The sett owns the pitch and the yarn owns the height

Set a cloth twice as close and its surface does not get smoother. The crowns come twice as often, because that is what a sett is, and they stand at very nearly the same height, because the closure condition pins the amplitude to the yarn — so a fine cloth is finer-grained rather than flatter, and the two are confused in every description of handle.
16 min read 6 figures A mechanism, not a materialStructure before fibreMeasured, not claimed

Worth reading first: A cloth has an outside · How close can threads be set · The weave decides the sett, and two models disagree about it.

The trade’s language about fabric handle runs everything together. A close-set cloth is smooth, a coarse one is rough, and a fine yarn at a high thread count is smoother still. Behind those words there are two independent geometric quantities, they answer to different things, and one of them barely moves.

The sett sets the pitch of the relief and not its height. A 2/2 twill in sheeting set from 14 to 29 ends per centimetre. The spacing of the crowns falls from 714 µm to 345 µm — in exact proportion to the sett, because it is the sett — while the height the surface swings through moves from 381 µm to 381 µm, which is not at all. The reason is the closure condition: the two crimp heights must add to the sum of the two diameters whatever the spacing, so the amplitude of the surface is pinned by the yarn and only its wavelength is free. The third curve is the root-mean-square roughness measured off the sampled surface, which wanders by a few per cent because it depends on where the sample grid falls relative to the crowns — it is drawn to show that it has no trend, not to be read off. A closer sett therefore makes a finer-grained cloth and not a smoother one, and the two are confused in every description of fabric handle. At 32 ends per centimetre the geometry refuses altogether: the cloth is close enough that its crimp can no longer divide equally, which is the jam arriving as a loss of symmetry rather than as a loss of room.
Fig. 1 A two-and-two twill in sheeting set from fourteen to twenty-nine ends per centimetre, with the spacing of its crowns, the height its surface swings through, and the sampled roughness on one axis. The spacing halves, in exact proportion to the sett, because it is the sett. The height is 381 micrometres at every step, because the closure condition says the two crimps add to the two diameters whatever the spacing. The measured roughness wanders a few per cent and has no trend worth reading.

The claim

A woven surface has a wavelength and an amplitude, the sett decides the first exactly and has almost no effect on the second, and the reason the second is pinned is the closure condition.

Two consequences, and both contradict something that is said routinely.

A closer sett makes a finer-grained cloth, not a flatter one. Twice the threads per centimetre means twice as many crowns per centimetre and crowns of the same height.

And the yarn decides the amplitude, through a condition that has nothing to do with how densely the cloth is woven. Peirce’s closure says h₁ + h₂ = d₁ + d₂: the two crimp heights add to the sum of the two diameters, whatever the spacings are. So the total excursion of the surface is fixed by the yarn count before any sett has been chosen.

The argument

The closure condition is the whole of it and it is worth restating in the form that makes the result obvious.

A cloth is two thread systems, one passing above the other at every crossing, and between them they fill the thickness of the fabric. If the warp’s centre line swings through h₁ and the weft’s through h₂, then a warp end at the top of its swing and a weft pick at the bottom of its own are separated by exactly the sum of the two radii plus the two half-swings — which is to say the cloth is d₁ + d₂ thick, and h₁ + h₂ = d₁ + d₂ follows.

Nothing in that sentence mentions a spacing. Peirce’s other two equations do, and they decide how the total divides between the systems and what weave angle each takes to get there, but the sum is settled by the two diameters alone.

So the height of a crown above the mid-plane, h/2 + d/2, has a fixed sum across the two systems whatever the sett, and the peak-to-valley range of the surface is very nearly d₁ + d₂ at every sett a cloth can be woven at.

The surface of a 2/2 twill, in plan. One repeat of a 2/2 twill in sheeting at 16 ends/cm, drawn 2 × 2 times, with every point painted in the colour of whichever thread owns the outside of the cloth there and at an opacity set by how high it is. The range is 212.3 µm from the highest point to the lowest, the root-mean-square roughness is 75.1 µm, and 53% of the plan is hole rather than surface. The bright ribbons are the float plateaux, where a thread lies straight across the threads it passes over and its outside is a horizontal line rather than a point — which is the whole of what follows. The warp crowns at 189.9 µm and the weft at 183.7 µm, a step of 6.2 µm, so the cloth touches the world on its warp alone until anything pressing on it has sunk that far.
Fig. 2 The same twill set openly, at sixteen ends per centimetre. The crowns are far apart, the holes between them are large, and the surface swings through nearly the full sum of the two diameters.
The surface of a 2/2 twill, in plan. One repeat of a 2/2 twill in sheeting at 28 ends/cm, drawn 2 × 2 times, with every point painted in the colour of whichever thread owns the outside of the cloth there and at an opacity set by how high it is. The range is 221.0 µm from the highest point to the lowest, the root-mean-square roughness is 74.9 µm, and 30% of the plan is hole rather than surface. The bright ribbons are the float plateaux, where a thread lies straight across the threads it passes over and its outside is a horizontal line rather than a point — which is the whole of what follows. The warp crowns at 190.8 µm and the weft at 182.8 µm, a step of 8.0 µm, so the cloth touches the world on its warp alone until anything pressing on it has sunk that far.
Fig. 3 And at twenty-eight. There are more than three times as many crowns in the same area of cloth and each of them stands at the same height as before. Nothing has been smoothed; the same relief has been drawn at a smaller scale.

What was counted, and how

The sweep runs one weave through a series of setts, solving the cloth’s geometry at each and sampling its surface.

Three numbers are read at each step: the thread spacing, which is arithmetic; the root-mean-square deviation of the surface from its own mean plane, measured over the covered area only because the holes are not surface; and the peak-to-valley range.

Across fourteen to twenty-nine ends per centimetre the spacing falls by a factor of 2.07, exactly as the setts demand. The peak-to-valley height is 381 micrometres at both ends of the range and at every step between them, because it is the sum of the two diameters and Peirce’s closure condition says so. The sampled roughness wanders between seventy-five and eighty micrometres with no monotone trend inside the range, the wobble coming from where the sample grid happens to fall relative to the crowns rather than from the cloth — which is why the exact quantity is the one to quote and the measured one is drawn only to show it has no slope.

The ratio of the two is what changes. At the open end a crown stands about one part in nine of its own spacing; at the close end one part in four and a half. A cloth set closer is not smoother in any absolute sense — it is steeper, its relief occupying a larger fraction of its own wavelength, which is exactly the opposite of what the word smooth suggests.

The surface of a 2/2 twill, in plan. One repeat of a 2/2 twill in sheeting at 22 ends/cm, drawn 2 × 2 times, with every point painted in the colour of whichever thread owns the outside of the cloth there and at an opacity set by how high it is. The range is 258.8 µm from the highest point to the lowest, the root-mean-square roughness is 77.0 µm, and 30% of the plan is hole rather than surface. The bright ribbons are the float plateaux, where a thread lies straight across the threads it passes over and its outside is a horizontal line rather than a point — which is the whole of what follows. The warp crowns at 191.6 µm and the weft at 182.1 µm, a step of 9.5 µm, so the cloth touches the world on its warp alone until anything pressing on it has sunk that far.
Fig. 4 A third setting, between the two. The pitch of the relief has moved again and the height has not: the same yarn makes the same hills whatever spacing they are put at, which is the claim in its most direct form. Two levers, two outputs, and neither crosses over into the other.

What the sett does to the contact, which is a third thing again

If the amplitude does not move and the wavelength halves, what happens to the area a cloth touches with?

It rises, and not because the surface has flattened. The crown line of a float is a horizontal segment whose length is the number of crossings the float spans less one, times the spacing of those crossings — so at a closer sett each individual segment is shorter, and there are proportionally more of them. Those two effects cancel exactly in the total length of crown line per unit area, which is therefore independent of the sett.

What does not cancel is the width of the strip that crown line presents. That width is set by the thread’s radius, which has not changed, so the bearing area at a given depth is very nearly the same at every sett.

The sett therefore moves the contact hardly at all, which is the third quantity in this essay and the least expected. Making a cloth denser does not make it touch more. It makes it touch in more places, each of them smaller, with the same total.

What a 2/2 twill touches with, at a depth of 7.5 µm. A flat plate brought down onto a 2/2 twill in sheeting at 16 ends/cm until it has sunk 7.5 µm below the highest point of the cloth. Everything it is touching is picked out; everything else is the cloth beneath. That is 6.64% of the plan, against 4.07% from the closed form, which counts the same area without sampling anything. The shape is the point: this weave carries 2.67 mm of crown line per repeat, so the contact is a set of ribbons and it widens as the square root of the depth. Nothing here is pressed: this is the geometry of the surface, and what it takes to reach this depth is a separate question with a fitted stiffness in it.
Fig. 5 The openly set twill with everything within two per cent of its thickness of the top marked. Compare the same picture at a closer sett: the ribbons are longer here and there are fewer of them, and the total area they cover is nearly the same. What the sett has changed is the size of each contact patch, which is what decides whether a load is carried at many small points or at fewer large ones — a distinction that matters for wear and for pressure and not at all for total area.

That result has a consequence for the cover factor, which is the quantity the trade uses for density. Cover rises steadily with the sett towards one; the bearing area does not rise at all. So the two most natural measures of how much cloth is present — how much of the plan is covered, and how much of it can be touched — come apart completely, and they come apart because one is a projection and the other is an extreme.

Where the sweep stops, and why the stop is informative

The sweep does not continue indefinitely, and the way it fails is the interesting part.

A sheeting of twenty-five tex yarn set at thirty-two ends per centimetre has, at that spacing, no state at all in which the two systems divide their crimp equally. The geometry admits a narrow band of divisions near 1.28 and nothing else: the warp must take more than its share of the thickness because the weft, at that spacing, has nowhere to put a bend.

That is the jam arriving, seen from an unfamiliar direction. The usual statement of it is that a cloth cannot be set closer than its threads are thick; the surface’s version is that as the jam is approached the division of the crimp stops being free, and the crown heights are forced apart before the sett itself becomes impossible.

So the last thing to go, as a cloth is crowded, is the symmetry of its two faces. A cloth near its jam has one system standing well above the other whether anybody wanted that or not, which is a fact about what such a cloth will touch and wear on, and it arrives before any of the more familiar signs of over-setting.

Why the confusion is worth clearing up

Two properties of a fabric are routinely attributed to sett and only one of them belongs to it.

Grain, which does. How far apart the crowns are is what decides whether a surface reads as a texture or as a smooth field to a finger or to an eye at a stated distance, and it is proportional to the reciprocal of the sett with no other term in it. A cloth at forty ends per centimetre has crowns two hundred and fifty micrometres apart; at sixteen they are six hundred and twenty-five apart.

Depth, which does not. How far the surface swings between its high and low points is the sum of the two diameters, and the only way to change it is to change the yarn — or to flatten the yarn after weaving, which is what a calender does and is the only other route there is.

The two are conflated because they usually move together in practice: a finer yarn is set closer, so a fine cloth has both a shorter wavelength and a smaller amplitude, and the eye reports one impression. Separating them says which lever is doing the work. To halve the depth of a cloth’s relief, halve the yarn diameter — that is, quarter the tex — and the sett will follow because it must. Raising the sett at constant count buys grain and buys cover and buys nothing at all in depth.

What a 2/2 twill touches with, at a depth of 7.5 µm. A flat plate brought down onto a 2/2 twill in sheeting at 28 ends/cm until it has sunk 7.5 µm below the highest point of the cloth. Everything it is touching is picked out; everything else is the cloth beneath. That is 7.42% of the plan, against 5.13% from the closed form, which counts the same area without sampling anything. The shape is the point: this weave carries 1.54 mm of crown line per repeat, so the contact is a set of ribbons and it widens as the square root of the depth. Nothing here is pressed: this is the geometry of the surface, and what it takes to reach this depth is a separate question with a fitted stiffness in it.
Fig. 6 What the close setting touches with, at the same depth as the open one. More crowns come into contact and each of them is the same crown — so the bearing area rises with the sett and the shape of an individual contact does not. That is the pitch and the height, read off a contact map instead of a relief.

The two quantities, in one table

Set side by side, the arithmetic of an ordinary shirting cloth taken from fourteen to twenty-eight ends per centimetre at constant count says this.

The crown spacing goes from 714 micrometres to 345, a factor of 2.07, because it is ten divided by the sett and nothing else. The height the surface swings through goes from 381 micrometres to 381 — unchanged to three figures, because it is the sum of the two diameters both times. The height of a warp crown above the mid-plane goes from 190.4 micrometres to 190.7, which is a change of two parts in a thousand and is the arithmetic of the closure condition arriving twice. The steepness — the sampled roughness divided by the spacing — goes from 0.106 to 0.217, a factor of 2.05, and it is the only one of the four that has moved for a reason.

Three of those numbers are constant and one is a reciprocal. That is the whole of the finding, and it is worth having in that form because the trade’s own vocabulary offers no word that distinguishes them: fine is used for all four.

Where the model stops

The sweep changes the sett and holds the count. A real close-set cloth is usually a fine-yarn cloth as well, and the two changes have opposite effects on the amplitude: raising the sett does nothing to it and lowering the count reduces it. Nothing here says which dominates in any particular pair of fabrics, because that depends on which of the two a mill actually changed.

The crimp ratio is held at one throughout, until the geometry refuses. How the crimp actually divides between the systems is not a measurement but a convention, and a different convention would move both crown heights while leaving their sum where it is. The amplitude result survives that untouched — it depends on the sum — and the step between the crowns does not.

Relaxation is not in the sweep. A cloth off the loom contracts and its crimp rises, so its finished sett is not the sett the reed set, and the sweep here is a sweep over finished setts rather than over loom states. The closure condition holds at every state, so the conclusion is unaffected; the numbers on the axis are.

And nothing here is a handle measurement. Whether a finger reports a surface as smooth depends on a great deal more than its geometry — friction, compressibility, thermal conduction and the hair layer among them — and this essay claims only that two geometric quantities usually confused are independent, not that either of them is what a hand feels.

What a finger can and cannot resolve

The essay is careful to say it is not a handle measurement, and that is right. It is worth taking one step towards one anyway, because the two quantities it separates land on opposite sides of a threshold the hand actually has, and that decides which of them a person can possibly be reporting.

A fingertip’s static spatial resolution is about a millimetre. Gratings finer than that are not resolved as patterns; they are detected, but as roughness rather than as spacing.

Every crown spacing in this essay’s sweep is finer than that. Fourteen ends per centimetre gives 714 micrometres and twenty-eight gives 345, so the whole of the ordinary weaving range is below what a stationary finger can resolve. A cloth’s sett is not something a hand can feel as a pattern at any construction anybody weaves.

Which sharpens the essay’s claim into a prediction. The amplitude is pinned by the yarn and the wavelength is below acuity, so at constant yarn count, changing the sett should change nothing a stationary finger can detect at all — and every reported difference between a close-set and an open cloth of the same yarn must be coming from somewhere else: the cover, the bending stiffness, the air, or the hair layer.

That is falsifiable in an afternoon with two cloths and a blindfold, and a positive result would say the geometry here is missing something.

The sett is felt as a pitch

There is a second channel, and it puts the sett back into the answer in a form nobody would guess from the word smooth.

A finger dragged across a fabric is a mechanical oscillator being driven at the frequency at which crowns pass under it: the scanning speed divided by the crown spacing. At an ordinary exploratory speed of a hundred millimetres a second,

sett spacing frequency
14/cm 714 µm 140 Hz
20/cm 500 µm 200 Hz
25/cm 400 µm 250 Hz
28/cm 345 µm 290 Hz
40/cm 250 µm 400 Hz

The receptors that carry vibration in the fingertip peak in sensitivity at around 250 hertz, which on this table is a sett of about twenty-five ends per centimetre — the middle of the shirting range.

So the prediction inverts the vocabulary completely. A more closely set cloth is not felt as smoother; it is felt as more strongly textured, up to about twenty-five ends per centimetre, because its relief is driving the hand nearer the frequency the hand is best at. Past that the frequency runs above the peak and the sensation falls away again, which would put the least detectable constructions at both ends of the weavable range and the most detectable in the middle.

That is a strange shape and it is the shape a resonance gives. It also offers an explanation for something otherwise arbitrary: the setts the trade calls fabric-like are the setts that ring the hand hardest, and cloths at the open and close extremes both read as less characteristically textile — a scrim as a net, a densely set cotton as paper or plastic.

Two caveats, both real. The scanning speed is a free parameter and a person exploring a fabric varies it, which smears the resonance across a band rather than a point; and the amplitude of the driving is what the hair layer transmits rather than what the crowns present, which is a different surface entirely below a fifth of a kilopascal.

What survives both is the direction. The sett enters handle as a frequency and not as a smoothness, and a frequency has a best value rather than a monotone one.

The generalisation

A periodic surface has two independent parameters and a lattice decides only one of them.

The transferable statement is that the spacing of a repeating structure and the amplitude of that structure answer to different constraints, and it is very common for the spacing to be free and the amplitude to be pinned by a conservation condition. Here it is the thickness of the cloth being shared between two threads. In another system it might be a volume, or a length of material, or an area.

Whenever an amplitude is pinned that way, tightening the lattice steepens the structure rather than smoothing it — and every intuition that says finer means gentler has the sign wrong.

What this says about the fourth-power rule

This collection already has an essay on the rule that a close cloth’s properties go as the fourth power of its cover, and the surface adds a boundary to it.

Rules of that shape are about what happens between the threads: the size of the aperture, the resistance to air, the light that gets through. Every one of them collapses violently as the sett rises, because the gap between two threads goes to zero much faster than the threads themselves approach one another.

Nothing about the surface behaves that way. The crowns are on top of the threads rather than between them, so no quantity in this essay has a high power of anything in it. The amplitude is constant, the wavelength is a first power, and the bearing area is very nearly flat.

So a cloth has two quite different regimes of behaviour as it is crowded, and which one a property belongs to is decided by whether it is a property of the holes or of the threads. Air, light, water and dust are hole properties and are violently sensitive to sett. Thickness, relief, contact, wear and shine are thread properties and are almost indifferent to it. That division is worth carrying, because it explains why a specification that pins down thread count says a great deal about how a cloth breathes and almost nothing about how it feels.

Who found it, and when

The closure condition is Peirce’s, from 1937, and is stated in his paper as the equation that closes an otherwise underdetermined system. Its use here is a corollary rather than a discovery: he wrote it to make the geometry solvable and it happens to say something firm about the surface as well.

The observation that a woven relief has a wavelength set by the sett and an amplitude set by the yarn is, as far as this collection can tell, nowhere in the literature as a statement, though every sectional drawing ever published contains it. What is new is only that the two can now be computed and swept, so the claim is a measurement rather than an impression.

Where the ladder goes next

Into the catalogue, where the other half of the surface lives: two drafts of twenty-two thousand holds the cloth fixed and varies the draft instead, and finds that the range of surfaces one construction can produce is much wider than the range the sett can reach.

Sideways, the amplitude result is what makes the next question answerable at all. Because the swing of a cloth’s surface is pinned by the yarn, the area in contact at a given depth is decided by the draft alone — which is how much of a cloth is touching, and it turns out to be a fraction of one per cent.

Where this goes next

The surface of a 2/2 twill, in plan. One repeat of a 2/2 twill in sheeting, drawn 2 × 2 times, with every point painted in the colour of whichever thread owns the outside of the cloth there and at an opacity set by how high it is. The range is 221.0 µm from the highest point to the lowest, the root-mean-square roughness is 74.9 µm, and 30% of the plan is hole rather than surface. The bright ribbons are the float plateaux, where a thread lies straight across the threads it passes over and its outside is a horizontal line rather than a point — which is the whole of what follows. The warp crowns at 190.8 µm and the weft at 182.8 µm, a step of 8.0 µm, so the cloth touches the world on its warp alone until anything pressing on it has sunk that far. A cloth has an outside Every quantity in this collection is a property of the inside of a fabric — a crimp, a cover, a hole, a fibre count. None of them says where the cloth stops. The outside is a height field the draft computes, its crowns stand at two different levels, and which of the two is higher decides what the cloth touches the world with. What an interlacing costs. Each weave's interlacing count beside the closest it can be set in the same yarn. The two run opposite ways, because a thread that changes face often has to bend often and a bend takes room. How close can threads be set There is a maximum. Push more threads into a cloth than the geometry allows and they simply will not go, and the limit depends on the weave as much as on the yarn. Where Poiseuille's rule starts being true of a cloth. Every account of a fabric's air permeability quotes the fourth power: flow down a channel goes as the fourth power of its width, so a small change of sett is a large change of flow. That is Poiseuille's result and it is about the viscous drop. A channel through a cloth is about as long as it is wide and the air in it is moving fast, so most of the pressure goes on giving that air its kinetic energy instead — and the inertial term does not depend on the hole's size at all. Plotted against cover, the viscous share of the drop runs from 1.0 per cent in an open muslin to 61.6 in a close one, passing half at a cover of 0.571. Quoting the fourth power alone below that is wrong by a factor rather than by a correction: at the open end it overstates the flow 98-fold. The fourth power is a close cloth's rule Every account of a fabric's air permeability quotes the same thing: flow down a channel goes as the fourth power of its width, so a small change of sett is a large change of flow. That is Poiseuille's result, it is about the viscous drop, and in an open cloth the viscous drop is two per cent of the pressure. The rule becomes true as the cloth closes, and where it starts being true is a number.

What links here

Computed from the collection rather than written here: the essays that point at this one.

Shares its objects with

Essays naming at least two of the same things, that neither author linked.

Named objects

A flat tag is an object no other essay names yet.

Closure conditionCover factorCrimp heightCrown heightJammingSettSurface heightSurface roughness