Watered silk is a beat

Moiré taffeta — watered silk — has a figure on it like the surface of a river: broad wavering bands several centimetres across, in a cloth whose finest visible feature is a weft rib a half-millimetre wide. Nothing in the fabric has a period of centimetres. The pattern is a hundred times the scale of anything woven into the cloth, and it is made by folding the cloth on itself and pressing.

This is the one pattern on the site that is not in a draft, not in a colour order and not in a finish applied to a surface. It is a difference, and the arithmetic that produces it is one subtraction.

One subtraction #

A grating of pitch p is a periodic thing, and the useful way to write a periodic thing is as a wave with a wavevector: a vector pointing across the bars, of length 2π/p. A fine grating has a long wavevector and a coarse one a short one.

Lay two gratings over each other and the light that gets through is the product of what each one lets through — a place is open only if both are open. A product of two waves carries their sum and their difference, which is the identity behind every beat in every part of physics. The sum has a wavevector longer than either, so it is finer than the cloth and disappears into it. The difference has a wavevector that is the vector difference of the two.

And the moiré is that difference. Its pitch is 2π divided by the length of the difference vector, and its fringes run perpendicular to it. Everything else in this essay is that statement read in two special cases.

Parallel grids of different pitch. The two wavevectors point the same way, so the difference has length 2π/p₁ − 2π/p₂ and the beat pitch is p₁p₂/|p₁ − p₂|. Two grids differing by four per cent beat at twenty-five times their pitch; by one per cent, at a hundred times.

Identical grids at an angle. The two wavevectors have the same length and differ by an angle θ, so the difference has length 2(2π/p) sin ½θ, and the beat pitch is p/(2 sin ½θ). At one degree that is fifty-seven times the pitch.

Those two formulas appear in the textbooks as two results. They are one result read twice, and the site computes both from the vector difference and asserts each against its own closed form — which is worth more than either formula alone, because a sign error in a vector difference produces perfectly plausible numbers.

What was counted, and how #

A closed form nobody tested is a guess with confidence attached, so the period is also measured.

Two square-wave grids — bars of a stated width on a stated pitch, which is what a rib actually is — are sampled along the direction the moiré varies in, and the shortest shift that reproduces the profile is searched for. The search is told nothing about the predicted answer: it walks outward from the finest structure in the profile and stops at the first genuine recurrence.

The averaging step is exact rather than approximate and it is worth a sentence, because it is the part that took two attempts. The difference vector is by definition perpendicular to the fringes, so along a fringe both gratings advance in phase at the same rate — the difference of their wavevectors has no component in that direction, which is what “perpendicular” means. Averaging over exactly one period of that common rate therefore removes the bars entirely and leaves nothing but the beat.

The first version of the search skipped that reasoning, smoothed over a fixed window instead, and returned a twentieth of the right answer for the parallel case while looking exactly like a measurement.

The measurement also draws its own boundary, and the boundary is a finding rather than a limitation. It refuses when the beat is less than about twenty times the grid pitch, because below that there is no separate pattern to find — the profile is the bars, modulated. That is very nearly where an eye stops seeing a figure and starts seeing a coarser texture, which is not a coincidence: both are asking whether the slow variation can be separated from the fast one.

Why the figure is so large, and so unrepeatable #

The angle formula is the one the finish uses, and the whole practical character of the process is in its shape.

A figure has to be large enough to read as a figure and small enough to fit on the cloth. Call that one millimetre to sixty. For a half-millimetre rib that puts the usable angle between 0.48° and 29° — but the two ends of that range are nothing alike, because the period goes as one over the sine of half the angle.

At the wide-figure end, a hundredth of a degree moves the figure by 1.26 millimetres. A ply of cloth laid on another with a hundredth of a degree of error — which is a millimetre of skew over five metres — produces a visibly different fabric.

So the wandering, irregular, no-two-alike character of moiré taffeta is not a defect of the process and not an artistic decision. It is the angle varying by hundredths of a degree across the width of the cloth, which is what happens when two plies of a woven fabric are laid together by hand. A perfectly registered moiré would have a perfectly regular figure and would look like a print.

The rib is the grid, so the weave decides the pitch #

The cloth is not incidental. A moiré finish needs a surface with a strong periodic relief, and in the trade that means a weft rib — a taffeta or a faille, where a coarse weft is covered by a fine dense warp so that each pick stands as a ridge.

That gives the pitch directly: the rib pitch is the pick spacing, so a cloth at twenty picks per centimetre has a half-millimetre grid. Nothing about the moiré cares what the ribs are made of; it cares that they are periodic and that they can flatten where they cross.

The flattening is the calendering half. Pressing two plies together at heat and pressure flattens the yarn where a rib meets a rib and leaves it standing where a rib meets a hollow — and a flattened yarn reflects light specularly where a round one scatters it, which is the same mechanism the site’s calendering essay computes for a single ply. So a moiré is a calendering effect applied through a mask, and the mask is the second ply.

Two consequences follow that a reader can check on a real fabric. The figure is on both plies, mirror-imaged, because both were pressed against each other; a finisher gets two matched pieces from one operation. And the figure is a lustre effect, not a relief one, so it disappears at the angle where the specular reflection does — which is why moiré looks flat in a photograph taken with the light behind the camera.

The cloth it is done to, and why it is that cloth #

A moiré finish is applied to one family of fabrics and the reasons are all structural.

The cloth must be unbalanced, and heavily. A taffeta is a plain weave with many more ends than picks — an unbalanced cloth in the site’s own sense — so the warp covers the surface completely and the weft is buried, and each pick shows as a ridge because it is thicker than the ends lying over it. A balanced plain weave has no ridges and nothing to superimpose.

It must have a fine, dense warp, because the ridge is only as sharp as the covering yarn is fine. That is the same argument the site makes about thread count from the other side: a high warp sett of a fine yarn is bought here for a surface property rather than for a hand.

And it must take a calender. The figure is a difference in lustre between flattened and unflattened yarn, so the fibre has to flatten and stay flattened — which is why the classic fabric is silk, why acetate and polyester took it over, and why cotton moiré needs a resin. Mercerised cotton will hold a calender better than raw, for the same packing reason the site computes there, and it is still the wrong fibre for the job.

There is one more requirement and it is easy to miss: the rib must be regular. The beat period is a difference of two nearly equal quantities, so it magnifies any irregularity in either grid by the same factor it magnifies the angle — thirty or fifty times. A cloth whose pick spacing wanders by two per cent produces a figure whose local period wanders by everything. That is a defect in a printed moiré and it is the whole of the effect in a woven one, which is a distinction the trade makes by eye and has never had a number for.

The angle a fold can actually deliver #

The usable band runs from about half a degree to twenty-nine, and a fold does not sample that band uniformly. Working out what a fold can give explains why the classic figure looks the way it does.

Fold a piece of cloth back on itself along a line and the two plies are, geometrically, at the same angle: a fold reflects, and a reflection maps the weft direction onto itself when the fold line is square to the selvedge. A perfectly square fold gives an angle of exactly zero, which by the parallel formula gives an infinite beat and therefore no figure at all — one uniform tone across the whole cloth.

So the classic operation is not a fold at an angle; it is a fold that is very nearly square, with whatever departure the cloth and the folder between them produce. That departure is what supplies the angle, it is a fraction of a degree, and it is where the sensitivity lives.

Which is why the figure wanders rather than repeating. A cloth folded by hand is not square to a hundredth of a degree anywhere and is a different fraction of a degree out at every point across its width, so the local beat period changes continuously along the piece — long where the plies happen to be nearly parallel, short where they are not. The wandering bands are a map of the folder’s own error, magnified by a factor that runs into the thousands at the near-square end.

There is a second contribution that has nothing to do with the folder and everything to do with the cloth. The pick spacing is not perfectly uniform, so the two plies are not two identical gratings even when they are exactly parallel — and that is the parallel formula rather than the angle one, working on a difference of a fraction of a per cent. The two mechanisms are present together in every real fold, they magnify by comparable factors, and no examination of the finished cloth separates them.

That is the honest reason the finish cannot be specified. A watered figure is the superposition of two magnified errors, one in the folding and one in the weaving, neither of which anybody measures and both of which are the product.

The same arithmetic, in the places nobody wanted it #

A beat appears whenever two periodic things are laid over each other, and a mill is full of periodic things.

Two plies of net curtain — or of a mock leno, whose holes are a grid of exactly this kind — at a small angle show it immediately, and everyone has seen it. A printed screen against a woven ground beats when the screen’s mesh and the fabric’s sett are close, which is why a rotary screen’s mesh count is chosen against the cloth rather than against the design. A digital photograph of a fine shirting beats against the sensor’s pixel pitch, which is the reason fabric photography uses a fine cloth at a large magnification or a coarse one at a small one and never the middle.

And the reed against the weave repeat beats too, in whole threads rather than in millimetres. That case is different enough to have its own rung: the two grids there are counted rather than measured, so the arithmetic is a least common multiple rather than a subtraction, and the remedy is a divisor rather than an angle.

The common thread is worth stating plainly. A moiré is not a property of either grid, and looking at either one will never predict one. It is a property of the pair, which is why every one of these appears at an integration stage — folding, printing, photographing — and never at the point where the periodic thing was made.

Where the model stops #

Both grids are treated as flat and infinite. A real fold has a curvature at the crease and the two plies are not in contact everywhere, so the figure fades where they part — which is a real feature of real moiré and is not in this arithmetic at all.

The transmission is a product and the real surface is not. Two plies of an opaque cloth do not multiply their transmissions; they press on each other, and what varies is how much yarn is flattened. The product model gets the geometry right — the period and the fringe direction are properties of the two lattices and nothing else — and says nothing about contrast.

Nothing here has a sett in it. The rib pitch is taken as given, and the relation between it and the cloth’s structure is one line above rather than a model. A cloth whose ribs are irregular has an irregular figure and the arithmetic does not describe it.

And a real moiré is not periodic. The whole finding of the angular band is that the angle varies across the piece, so the local period varies, so the figure is not a beat at one scale but a slowly changing one. Every number here is a local statement.

Who found it, and when #

The word is the older of the two things. Moiré is a French participle applied to watered fabrics well before anybody analysed the effect, and the finish is documented in silk-weaving from at least the seventeenth century: fold, press, and the water appears.

The optics came much later. Lord Rayleigh used the superposition of two gratings in 1874 as a way of testing gratings — the beat magnifies a small error in the ruling by a factor of a hundred, which makes moiré a metrology technique rather than a curiosity — and that use is still the main one. Moiré deflectometry, strain measurement by grid superposition, and alignment marks on semiconductor masks are all Rayleigh’s observation applied.

There is a nice reversal in that history and it is the one worth carrying. The silk finishers were using an effect whose sensitivity is its whole character — a hundredth of a degree matters — and treating that sensitivity as the product. The physicists two centuries later used the same sensitivity in the other direction, to measure the thing the finishers could not control. Neither had a use for the effect’s regularity; both were interested in exactly how sharply it responds.

Where the ladder goes next #

The other grid a weaver cannot avoid is the reed, and it beats against the weave. The next rung computes that beat, which is counted in whole ends rather than in millimetres, and arrives at a rule a weaver can act on: dent so that the ends per dent share no factor with the repeat.