Textile Structure
how cloth is put together
Weaves

The other crepe is in the yarn

A crepe weave puts the texture in the matrix. A crepe yarn puts it nowhere the matrix can see: the cloth is a plain weave, and the surface comes from a thread twisted so hard that it shortens by a seventh and spends the rest of its life trying to untwist.
15 min read 6 figures A mechanism, not a materialStructure before fibre

Worth reading first: A crepe cannot be structureless · Twist is one angle · The other half of the twist curve.

This collection has an essay about crepe already, and it is about a weave: a crepe cloth’s draft is a matrix arranged so that no repeat is visible, which turns out to be a harder problem than it sounds and to have an exact statement.

There is a second crepe and its draft is a plain weave.

Take an ordinary plain-weave construction and put in it a weft twisted to two thousand turns a metre — three or four times the twist a weaving yarn is normally given. Weave it, wet it, and let it relax. The cloth comes out with a fine, irregular, pebbled surface, twenty per cent narrower than it went in, and nothing whatever in its matrix to account for the texture.

Z twist at 1600 turns per metreA 20 tex cotton yarn, 0.167 mm across, with its surface fibres drawn as the helices they are. The angle between a surface fibre and the yarn's axis is 40.0°, and it is the only quantity in this family: the yarn is 13.8% shorter than the fibre in it and carries 59% of the strength the same fibre would give lying straight.surface angle 40.0°twist factor 7155 — the same angle at any count40.0°20 tex cotton0.167 mm diameterone turn every 0.63 mmretraction 13.8%obliquity keeps 59%tan α = πdT, with d from the count and the packing factorZ 1600/m
Fig. 1 A crepe yarn: 1,600 turns a metre at 20 tex, a twist factor of 7,155, a surface helix angle of 40°. The fibres are lying nearly halfway between the yarn’s axis and its circumference. Everything the finished cloth does — its shrinkage, its texture, its weakness and its dull surface — is a consequence of that angle, and none of it is in the weave.

The claim

A crepe yarn’s texture is the retraction and the torque of a very high twist, released in the cloth — and both are computable from the helix angle that produces them.

  • At 40° the yarn is 13.8 per cent shorter than the straight fibre in it. Relax the cloth and that shortening is imposed on the fabric, which has nowhere to put it except out of the plane.
  • The yarn’s residual torque is enormous and it has nothing to push against but the crossings. Where a crossing lets go, the thread kinks — and a kink is a snarl of a few millimetres, which is the pebble.
  • The yarn realises 0.562 of its fibres against 0.771 at its own strength optimum, so a crepe cloth pays twenty-seven per cent of its strength for the surface.

The last number is the honest one and it explains why crepe constructions are not used where strength matters.

The retraction, which is where the shrinkage comes from

A fibre at the surface of a twisted yarn follows a helix. Its length per unit of yarn is the secant of its own angle, so the yarn is shorter than the fibres it contains — and the shortening, averaged over the section, is what this collection computes as retraction.

At ordinary weaving twists it is a footnote: 4.1 per cent at 800 turns a metre. At crepe twists it is not.

twist, turns/m twist factor surface angle retraction
800 3,580 22.8° 4.1%
1,200 5,370 32.2° 8.5%
1,600 7,155 40.0° 13.8%
2,000 8,944 46.4° 19.3%

Now think about what happens in the cloth. The yarn was woven at a length; the twist in it is held by the friction of the crossings and by the tension on the loom. Wet the cloth and the fibres swell and slide, the friction falls, and the yarn does what a twisted elastic body does when it is released: it partly untwists, and as it untwists it lengthens.

The cloth cannot lengthen. The threads are locked into a weave, so the extra length has to go somewhere, and the only direction available is out of the plane. The thread buckles between its crossings.

That is the crepe surface: a plain weave whose threads are each a little too long for the space they are in, buckling in a pattern that is irregular because the untwisting is irregular.

The one number, pulling two ways. The fibre count decides two quite different things about a yarn and it decides them in opposite directions. The stiffness bracket — the ratio between a yarn whose fibres slide freely and one that bends as a solid rod — is n/φ², so it widens as the yarn gets coarser: 98 at 35 fibres and 1307 at 471. The evenness floor is 100/√n, so it narrows: 18.1 per cent down to 5.0. There is no count at which both are favourable, and the trade-off is not a matter of degree: the two exponents have opposite signs. Both curves are drawn on their own scale because they are in different units; what the figure claims is the crossing, not the values.
Fig. 2 Where the shrinkage comes from: the one number pulling two ways. A twist that retracts the yarn is the same twist that carries the torque, so the two effects a crepe wants cannot be ordered separately — asking for more pebble asks for more retraction, at a rate the yarn decides.

The torque, which is where the pebble comes from

Retraction alone would give a uniformly puckered cloth. The characteristic irregularity of a crepe — the reason it reads as pebble rather than as ribbing — comes from the torque.

A hard-twisted yarn is lively: it stores a moment, and a free loop of it snarls back on itself. In a cloth the moment has to be resisted by the crossings, and a plain weave has more crossings per unit length than any other weave — which is why crepe cloths are woven plain rather than in a float weave, and it is a design decision with a reason.

But the resistance is not uniform. Every crossing resists by friction, and friction varies from crossing to crossing with the local pressure, the local diameter and the local crimp. Where the resistance is least, the thread turns a little; where the turning is enough, it snarls.

The pebble is a set of local instabilities, each one a length of thread that has kinked, distributed at random because the friction that held them was distributed at random. Nothing in the weave places them, which is exactly what makes the surface look unrepeating.

The trade’s practice follows from the mechanism in two ways. Crepe wefts are used in S and Z alternately, two picks of each, so that the torques of neighbouring picks oppose and the cloth does not skew — a whole-cloth version of the imbalance count that decides whether a knit leans. And the cloth is woven with a hard-twisted weft and an ordinary warp, because a warp under tension for hundreds of metres would snarl at every stop.

What the twist costs

Everything above is a description of what the twist buys. The bill is on the strength curve.

Both halves of the twist curve, at 28 mm staple. The falling curve is obliquity and is exact — the affine end of the bracket, computed from the helix and nothing else. The rising curve is cohesion and is the half this collection had declined: a fibre end is gripped by friction under the twist's own radial pressure, the fibre's strength and the yarn's load cancel out of the comparison, and what is left is a critical length that depends on the twist through a pure function of the angle. Their product has a maximum at a twist factor of 3101 — 693 turns per metre at 20 tex, a surface angle of 20° — which is inside the range spinners use. The scale of the rising curve is fitted, through a contact efficiency of 0.05, and moving it moves the optimum; what it cannot move is the ordering between two staples or two fibres, which is what the two claims made from this figure are about.
Fig. 3 The strength–twist curve with a crepe twist marked far off to the right of it. The maximum is at a twist factor of 3,101; a crepe yarn is at 7,155, more than twice as far along, where the obliquity has fallen to 0.587 and the realisation to 0.562. A crepe yarn gives up twenty-seven per cent of the strength the same fibre would have delivered at its best twist, and the cohesion half of the curve — which is still rising — has nothing left to offer.

There is a second cost that is subtler and is about predictability rather than strength.

The strength bracket of a twisted yarn. Two exactly computable models of the same yarn, and no real yarn is outside them. The lower curve is affine: each fibre stays at its own radius, is strained cos²θ of the yarn's strain, and the core reaches breaking first — the classical cos²α. The upper curve is equal tension: every fibre migrates between the core and the surface, has the same mean strain, and they break together — 2cos α/(1 + cos α), which comes out of the same integral with the tension held constant instead of the strain. The gap between them is what migration is worth, and it is 2/(cos α(1 + cos α)) exactly: nothing at no twist, 1.158 at a shirting warp's 25°, and 1.478 at a crepe's 40°. A spun yarn is made of fibres that wander, and this is the price of their not doing so.
Fig. 4 The obliquity bracket, which is the gap between a yarn whose fibres keep their radius and one whose fibres migrate. At an ordinary warp twist it is 16 per cent; at a crepe’s 40° it is 48. So a crepe yarn is not only weaker than a normally twisted one — it is the twist at which this collection is least able to say how strong it is, because the whole of the uncertainty about migration is multiplied by the angle.

And a third, which is the one the cloth is actually bought for: a crepe yarn is dull. The fibres lie at 40° to the yarn’s axis, so a surface covered in them has no direction along which parallel cylinders reflect together. The same geometry that gives a satin its lustre — a long uninterrupted length of thread lying one way — is here disrupted at the fibre scale rather than at the float scale. A crepe is matt by construction, and that is a large part of why it is used.

The construction, which follows from all of it

Take the three costs together and the way a crepe cloth is actually built stops looking like tradition.

Plain weave, because the torque has to be resisted at as many crossings as possible and plain has the most. A crepe woven on a float weave skews and snarls.

Hard-twisted weft, ordinary warp, because a warp is under tension for hundreds of metres through the loom and a lively yarn snarls at every stop mark. The weft is thrown, held for a moment, and beaten up.

Alternating S and Z, in pairs of picks, so the torques of neighbouring picks oppose and the cloth does not run off square.

Open sett, because the buckling needs somewhere to go: a thread that is too long for its space cannot buckle if the space is full. This is the same constraint a honeycomb works under and it is why crepe cloths are set noticeably more openly than plain cloths of the same yarn.

And a wet relaxation as the last step, without which the cloth is a rather weak plain weave with nothing on its surface at all.

Every one of the five is a consequence of a number in this essay, and none of them is a consequence of the weave matrix — which is the sense in which a crepe cloth’s construction is not in its draft.

Two crepes, one fabric

Set the two mechanisms side by side, because they produce fabrics that are sold under one name and are structurally unrelated.

The crepe weave puts the irregularity in the matrix: a draft arranged so that no repeat is visible, with floats of assorted lengths scattered so that the surface has no line in it. It can be woven from ordinary yarn, it has ordinary strength, and its texture is exactly as regular as its repeat — which is to say, regular at a scale large enough not to be seen.

The crepe yarn puts the irregularity nowhere the matrix can reach. The draft is plain. The texture is the buckling of threads that are too long for their spaces, at positions decided by friction.

The pair is a good example of a general point this collection keeps meeting: a fabric’s appearance underdetermines its structure. A honeycomb gets its cells in the wash and a seersucker gets its stripes at the loom, and both look like a cloth with a raised pattern. Deciding which mechanism produced a given cloth requires unweaving it, and in the crepe case the test is simple: unpick a thread and see whether it snarls.

Which end of the bracket a thread in cloth is at. A yarn bends as a solid rod while its fibres cannot slide and as a loose bundle once they can, and the crossover is a curvature: the coherent state demands an axial force in the outer fibre that has to be built up by friction under the twist's own radial pressure. The curves are the crossover radius against twist at four contact efficiencies, the top one being 1.0 — the claim that fibres touch along their whole length, which nobody makes. The rule at the bottom is the radius a thread is bent to by its own crimp in a cloth, about 0.25 mm. Every curve is above the rule by at least 19-fold, so a thread in cloth is at the free end of its bracket at every twist and every efficiency, and the collection's habit of using the lower bound is a result rather than a convention.
Fig. 5 Which end of the bracket a thread in a crepe is at. The retraction is quadratic in the twist angle where the strength is not, so the two curves cross somewhere in the usable range — and a crepe yarn is set past the crossing, which is why it is weak and why nobody sets one from first principles.

The retraction is quadratic where the strength is not

Two quantities in this essay depend on the twist and they depend on it in ways that pull apart, which is why a crepe twist is where it is rather than anywhere else.

What a packing factor decides. Every diameter on this site comes from a count through a packing factor of 0.6, and that number was obtained by inverting a rule published for cotton yarns at one particular twist. This is what moves if it is wrong by the width of the range real yarns occupy — 0.45 to 0.75, which is the whole of it. An areal weight does not move at all, because it is a count times a sett and never passed through a diameter; a cover factor moves by 15%; a bending rigidity moves by 78%, because it goes as the fourth power. The exponents are exact and are asserted, not read off the bars.
Fig. 6 What the packing does, which is the third quantity moving with the twist. The retraction is quadratic in the twist angle, the strength is not, and the packing rises monotonically — so a crepe yarn is denser as well as shorter and weaker, and the density is the part a weaver notices first.

Read the retraction table as a shape and it accelerates: 4.1 per cent at 800 turns a metre, 8.5 at 1,200, 13.8 at 1,600, 19.3 at 2,000. Each step of four hundred turns costs more retraction than the one before it, because the retraction is one minus the mean cosine of an angle whose tangent is rising linearly — so it starts flat and steepens, and it has no limit short of the yarn tying itself in knots.

The strength does the opposite. Past the maximum the realisation falls, and it falls gently at first and then faster: from 0.771 at the optimum to 0.562 at a crepe twist is twenty-seven per cent lost, spread over more than a doubling of the twist factor.

So the thing being bought accelerates and the thing being paid accelerates too, and the crepe twist is where the first has become large enough to be worth the second. That is not an optimum in any computable sense — nothing here can price a pebbled surface against a newton — but it does explain why the crepe range is narrow and why it sits so far out. Below about 1,200 turns a metre the retraction is too small to buckle a cloth and the strength is barely touched; the yarn is simply a hard-twisted yarn. Above about 2,000 the retraction is enormous and so is the loss, and the yarn is difficult to handle at every stage before the loom.

Between those two the retraction has grown by a factor of two and a quarter and the realisation has fallen by a fifth. That ratio is the whole of what a crepe twist buys, and it is favourable only because the retraction curve is the steeper of the two over exactly that stretch.

It also says what would change the answer. A fibre with a flatter strength curve — one whose obliquity loss is smaller, which is to say one whose fibres migrate — would tolerate a higher crepe twist for the same strength cost, and the retraction would be larger still. That is a prediction the migration bracket cannot settle, and it is the one place in this essay where the bracket’s width has a consequence a mill would notice.

It also puts a floor under the fibre choice. A short-stapled or coarse fibre needs more twist to be gripped at all, so it arrives at the crepe range having already spent part of its budget on cohesion rather than on retraction — which is a second reason, quite separate from the finish, that crepe cloths are made from the finer and longer cottons and from silk. The fibre is not chosen for its handle; it is chosen because it reaches a forty-degree surface angle with something left over.

Why nobody sets a crepe from first principles

There is a practical note worth making, because this essay has priced several things and predicted none of them.

A cloth designer working with ordinary yarn can compute a great deal before weaving: the cover, the crimp, the weight, the sett a warp will take, the air a cloth will pass. Every one of those computations goes through a diameter and a construction, and both are known.

A crepe cloth defeats all of it at once. Its diameter is a hard-twisted yarn’s, so the packing factor is far from the one this collection uses; its finished sett is a relaxation away from its woven sett and the relaxation is not computed here; its surface is a buckling nobody predicts; and its strength sits inside a bracket half again as wide as an ordinary yarn’s.

So crepe is the fabric this collection is least able to describe, and the reason is not that crepes are exotic. It is that every quantity the arithmetic needs is evaluated at the edge of the range the arithmetic was built for. That is worth saying plainly rather than leaving as an impression: an essay that computes six numbers about a fabric and cannot predict its most obvious property has located a boundary, and locating boundaries is most of what a collection like this one can honestly do at its edges.

What was counted, and how

The retraction is the integral this collection already had, checked against its own closed form to a part in a million at two thousand quadrature points, and quoted here at four twist levels rather than at one so that the shape is visible.

The realisation figures come from the strength curve with its stated contact efficiency, and the comparison made is between two points on one curve — a crepe twist against the same yarn’s optimum — which is a ratio the fitted number very nearly cancels out of.

The bracket at 40° is a closed form checked against its integral, and it is quoted because the uncertainty is the point rather than the value.

Where the model stops

Nothing here computes the pebble. The buckling of a thread that is too long for its space is a post-buckling problem in a system with friction at every crossing, and this collection has a criterion for whether a cloth can carry a push and nothing that predicts the wavelength or the amplitude of what happens after it cannot. The mechanism is argued; the surface is not computed.

How much the yarn untwists is not computed either. It depends on the torsional rigidity of the yarn, on the friction at the crossings and on how wet the cloth is, and the first of those is a quantity this collection declines to supply in the folded-yarn case for the same reason.

The retraction is quoted for a fully relaxed yarn. In the cloth only part of it is released, because the crossings hold and because the yarn does not fully untwist, so the shrinkage a crepe cloth actually shows is a fraction of the table above rather than the table itself.

The lustre claim is geometric and is not a reflectance model, in the same way and for the same reason as every other optical statement in this collection.

And the twist is treated as uniform along the yarn. It is not: twist runs from thin places to thick ones, so a crepe yarn’s liveliness varies along its own length with its own irregularity — which is very likely part of why the pebble is as irregular as it is, and is not something this collection can quantify.

One more consequence follows and it is a commercial one. Because so little of a crepe can be computed, crepe constructions are inherited rather than designed: a mill runs the setts and twists that worked, and a new crepe is developed by weaving samples. That is a reasonable response to the arithmetic above, and it is worth distinguishing from the cases where the trade proceeds by trial because nobody has done the calculation.

Where the ladder goes next

Back into the finishing, where the untwisting is actually released. A crepe cloth is a fabric whose finishing route is its construction, in the way that a honeycomb’s is — and the sequence, the temperature and the tension decide the surface as much as the yarn does.

And sideways into the other place a very high twist appears: the voile and the organdie, where the same hard twist is used not for texture but to make a fine open cloth stiff enough to hold its shape. The mechanism is the retraction again, resisted rather than released, and the yarn is chosen at the same angle for the opposite reason.

Where this goes next

How much a draft agrees with itself. On the left the draft; on the right its correlation at every offset, one cell per offset, with the offset of nothing at the top left. Warp-up counts as plus one and weft-up as minus one, so the number in each cell is agreements minus disagreements out of 64. The correlations away from the origin sum to exactly -64, whatever the draft — structure can be moved about and not removed. A crepe cannot be structureless A crepe weave is designed to have no line in it anywhere. The correlations of a draft with itself sum to a number fixed by the repeat alone, so structure can be spread and never removed — and on eight ends the floor turns out to be half the repeat, set by a fact about binary words with nothing textile in it. Z twist at 800 turns per metre. A 20 tex cotton yarn, 0.167 mm across, with its surface fibres drawn as the helices they are. The angle between a surface fibre and the yarn's axis is 22.8°, and it is the only quantity in this family: the yarn is 4.1% shorter than the fibre in it and carries 85% of the strength the same fibre would give lying straight. Twist is one angle Every model on this site treats a yarn as a cylinder with a diameter. It is a bundle of loose fibres, and what makes it behave like a cylinder is twist — which is a helix, so the whole subject is one angle, and the trade's twist factor turns out to be the only combination of count and turns that decides anything. A seersucker in section. A seersucker in section across four stripes, at a feed ratio of 1.30 — the slack warp let off 30 per cent faster than the tight one — over a 6.0 mm stripe. The surplus has nowhere to go in the plane, so it buckles, and the standard small-amplitude result gives 2.09 mm of rise, which is 4.2 times the cloth's own thickness of 0.500 mm. Nothing has to relax for this to appear: unlike a honeycomb, a seersucker comes off the loom already puckered, and washing deepens it rather than creating it. What the drawing cannot show is that the buckle's shape is an assumption — a sinusoid pinned at the stripe's edges — while its amplitude follows from the surplus and the half-wavelength alone. A seersucker is made at the loom Every other relief weave in this collection gets its shape after the loom, from a difference of crimp between two regions of a few per cent. A seersucker's surplus is thirty per cent and is put in as the cloth is woven — an order of magnitude more, which the square root turns into a factor of four in depth and no more than that.

What links here

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

Reads more easily once this is understood

Essays that name this one as worth reading first.

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.

Fibre migrationHelix angleLustreObliquityRelaxationShrinkageTwist factorWeave matrix