A moiré is a vernier, and it magnifies the error too

The first rung of this ladder computed why a watered figure is so large: two identical grids at a small angle beat at p/(2 sin ½θ), which for a tenth of a degree is six hundred times the rib pitch. The figure appears because the arithmetic makes it appear, and the essay stopped there.

A magnification is also an instrument. If a pattern a reader can see across a room is the difference of two pitches half a millimetre apart, then reading the pattern is reading the difference — and reading it far more finely than any eye reads half a millimetre. That is exactly what a vernier scale is, and a moiré is one.

What the gain is #

Two gratings of pitches p and p(1 + d) beat at a period of p/d, so a difference of one part in a hundred gives fringes a hundred pitches apart. Call that ratio the gain: it is how many threads there are inside one fringe, and it is how much larger the pattern is than the thing making it.

For two identical gratings crossed at an angle the gain is 1/(2 sin ½θ), which for small angles is simply 1/θ in radians. At half a degree it is 115; at three degrees it is 19.

The instrument follows immediately. A displacement of one thread’s position moves the fringe by that displacement times the gain, so a fringe pattern reports thread positions at a magnification the observer chooses by choosing the angle. Watch a fringe move as a reference grating is slid across a cloth and the fringe’s motion is the pitch mismatch, amplified.

That is a real technique and it is not this collection’s: optical moiré metrology is a whole field, strain gauges are built on it, and the reason it works is exactly the reason a moiré taffeta shows a figure.

And what it magnifies, which is not only the signal #

An instrument that multiplies a displacement by a hundred multiplies every displacement by a hundred, including the ones nobody asked about. A cloth’s threads are not where they should be — a spacing is a physical quantity with a distribution, exactly as a yarn’s diameter is — and the moiré magnifies that too.

The obvious way to think about it is cumulative. A thread’s position is the sum of every spacing before it, so a spacing that varies with coefficient of variation cv gives a position that random-walks: across the gain threads inside one fringe the accumulated position error has a standard deviation of cv·p·√gain, and the fringe’s own spacing therefore varies by cv·√gain relative to itself.

That model is right about what it describes, and it is not what ends the pattern.

The measurement, and the model it refuted #

Simulating it is cheap and worth doing rather than arguing about. A grating whose spacings are drawn from this collection’s own seeded lognormal — the same population every other irregularity result here is built from, rather than a second sampler written for the occasion — is laid against a perfect one, the fringe centres are found where the two gratings’ phase difference passes a multiple of 2π, and two things are measured: how much the fringe spacing varies, and how many fringes there are.

The second is the one with a right answer. The ideal beat says exactly how many fringes fit in the span, so a count that departs from it is a fact rather than a threshold on a statistic.

The count departs, and the gain at which it departs was recorded for four irregularities. What comes out constant is not cv·√gain. It is cv × gain, at 0.84, to within three per cent across a factor of eight in irregularity.

So the ceiling is inversely proportional to the irregularity and not inverse-square, and the difference is not academic: at a two per cent irregularity the two models give 42 and 1,764.

Why it is local rather than cumulative #

The mechanism the measurement points at is a different one, and once seen it is obvious.

Two gratings of pitches p and p(1 + d) beat because the phase difference advances by 2πd at every thread. A thread whose own spacing is wrong by more than p·d advances the phase difference backwards instead, and where the phase difference is not monotonic a fringe splits into two.

That is a single thread against a single number. It does not accumulate, it does not care how far along the grating it happens, and it happens when cv·p is comparable with p·d — which is cv × gain ≈ 1, because gain is 1/d.

A beat built on a difference of one part in fifty is destroyed by a spacing error of one part in fifty. The accumulated wander is real, the closed form for it is right, and it is simply not the thing that arrives first.

The 0.84 rather than 1 is what a distribution does where an argument uses a threshold: the count departs a little before the mean crossing, because the spacings that matter are the tail ones.

What that says about watered silk #

The first rung worked out the band of angles a watered finish can be made at: a figure has to be large enough to read and small enough to fit on the cloth, and for a half-millimetre rib that puts the crossing angle between about half a degree and twenty-nine.

Half a degree is a gain of 120. Twenty-nine degrees is a gain of two.

Now put the ceiling against it. A gain of 120 needs a spacing irregularity below 0.84/120, which is 0.7 per cent. A gain of 40 — a two-centimetre figure, which is an ordinary watered taffeta — needs 2.1 per cent.

And this collection’s standing figure for a yarn’s coefficient of variation is fifteen per cent, which allows a gain of 5.6 and a figure under three millimetres. No such cloth exists; watered silk shows figures of centimetres and always has.

So one of two things is true, and the arithmetic says which. Either the ceiling is wrong, or the pitch a watered finish beats on is far more even than the yarn that makes it — by a factor of seven to twenty.

That is not a difficulty, it is the answer, and it is a statement about the loom. The rib pitch of a taffeta is the pick spacing, and a pick’s position is set by the take-up motion and the beat-up, not by the yarn’s diameter. A yarn that varies fifteen per cent in thickness can be laid at spacings that vary one per cent, because the thing setting the spacing is a ratchet and a reed rather than the thread.

A moiré finish is therefore a measurement of the loom, and the fact that the finish exists at all is the measurement. A cloth whose picks were spaced as unevenly as its yarn is thick could not be watered, and every piece of watered silk ever made is evidence that they are not.

Which explains the thing the first rung left as an observation #

That essay ended on the practical fact about moiré taffeta: no two pieces match, because no two are folded at the same angle, and at the large-figure end a hundredth of a degree moves the figure by more than a millimetre.

The ceiling adds the other half, and it is about one piece rather than about two. A watered figure is not regular within itself either, and the reason is now computable: at a gain of 40 and a one per cent pick irregularity the fringe spacing varies by cv·√gain, which is six per cent, so the fringes wander visibly across the width even before any of them splits.

That is what the finish actually looks like. A moiré taffeta does not carry a ruled pattern of parallel bands; it carries bands that wander, widen, close up and occasionally fork — and a reader who has been told the effect is a beat between two regular grids has been told something that would predict ruled bands.

The wandering is the cloth’s own unevenness, at forty times. It is the most direct picture of a fabric’s irregularity anybody has ever made, and it was made for four hundred years by people who thought they were decorating silk.

Reading a sett with it, and how well #

The instrument half of this is worth making concrete, because it is a thing anybody with a printed transparency can do and because its precision is computable.

Lay a grating of known pitch p over a cloth and count the fringes across a measured length L. The number of fringes is L times the difference of the two spatial frequencies, so the cloth’s own pitch comes straight out of the count — and the resolution of the count is one fringe, which is a pitch difference of p/L.

At a half-millimetre reference over a hundred millimetres of cloth that is one part in two hundred: a sett read to 0.5 per cent by counting a dozen bands, with no magnifier and no thread counter. A pick glass counts threads over a centimetre and the count is an integer, so it reads a sett to about one part in twenty-five; the moiré is an order of magnitude better and costs nothing.

The ceiling is what bounds it, and it bounds it in a way that is easy to state wrongly. Making L longer improves the resolution without limit and does not raise the gain, because the gain is fixed by the two pitches; so the limit is not on the resolution at all. It is on whether the fringes can be counted, and that is cv × gain — which for a reading near the cloth’s own pitch is a small gain and a safe one.

So the technique is at its best where the moiré finish is at its worst. A watered figure wants the largest gain the cloth will carry, right at the ceiling; a measurement wants a modest gain and a long baseline, nowhere near it. The two uses of the same effect pull in opposite directions, and only one of them is limited by the cloth’s evenness.

What this does not give is the thing a mill actually quotes, which is threads per centimetre in a finished cloth. A cloth’s sett moves when it is wetted and moves again when it is pressed, so a reading taken at one moment is a reading of that moment — which is true of a pick glass as well and is worth saying once.

The same ceiling stands over the rest of this ladder #

Every rung of this anchor computes a period, and every one of them has now acquired a condition it was written without.

The reed’s mark is a beat between the denting and the weave repeat, with a period in whole ends. Its gain is that period divided by the dent pitch, and the same argument applies: a reed mark at a period of fifty-six ends is only visible if the ends are spaced evenly enough for the fifty-sixth to still be in phase. Since the reed itself is doing the spacing, that condition is comfortably met — which is precisely why a reed mark is such a persistent fault. The grating causing the beat is also the thing enforcing the regularity.

A slub’s diagonals are the opposite case. The period there is a fault spacing along a weft yarn against the cloth’s width, and nothing enforces it: a slub-drawing mechanism has its own tolerance, and the diagonals wash out over a distance set by exactly the arithmetic above. That is why a slub pattern reads as a texture rather than as a line, and why making the slub distribution deliberately irregular destroys it completely rather than merely blurring it.

One ceiling, three phenomena, and it explains why they behave so differently: what matters is whether the thing making the grating is also the thing holding it in place.

What was counted, and how #

The gain is a closed form and the ceiling is a measurement, and they are separate functions so that neither can quietly borrow the other’s assumptions. moireGain does the arithmetic from two pitches and an angle; fringeMeasure builds a jittered grating and finds fringes in it.

The spacings come from spread.js’s population, which is the seeded lognormal this collection uses for every irregularity result. Writing a second sampler here would have been a second distribution to keep true, and the whole value of the comparison with fifteen per cent depends on the two numbers meaning the same thing.

The fringes are found from the phase difference and not from the transmission. Sampling the product of two gratings and looking for maxima works and is what the rung below does; here the quantity wanted is the fringe centre to better than a pitch, so the phase difference is evaluated on a grid of an eighth of a pitch and its crossings of 2π are interpolated.

The ceiling is found on the count, not the spread. A pattern that has started splitting has more fringes than the ideal beat allows, which is a departure from a known number. The spread has no known value to depart from, and using it would have meant choosing a threshold — which is exactly how a measurement turns into an assumption.

Both models are kept and both are asserted. The claim is that cv × gain is one number across the irregularities and that cv × √gain is not, and the second half is what distinguishes the two laws rather than merely fitting one. A version of this that asserted only the surviving model would have passed on data that supported either.

And the limit the file carries is checked against what the measurement returns rather than typed in beside it, so a change to the simulation that moved the breakdown would fail rather than silently disagree with the constant.

Where the model stops #

The spacing irregularity is not measured, anywhere on this site. Every number in the second half of this essay is a prediction with an unmeasured input: the ceiling curve is solid, the cloth’s position on it is a guess bounded by the existence of watered silk. What is needed is a coefficient of variation for pick spacing in a real fabric, and this collection does not have one — the fifteen per cent it uses everywhere is a yarn diameter figure and is being used here only to show that it is the wrong quantity.

The simulation is one-dimensional. Real fringes are two-dimensional curves and a real cloth’s threads wander sideways as well as being unevenly spaced. Both make the pattern worse and neither is modelled, so the ceiling computed here is an upper bound on the usable gain rather than an estimate of it.

The spacings are drawn independently. A loom’s pick spacing is almost certainly correlated — a slack warp affects several picks together, and a shedding fault repeats — and a correlated error accumulates faster than an independent one over short distances and more slowly over long ones. Which way that moves the ceiling depends on the correlation length against the fringe spacing, and is not computed.

And “countable” is not “legible”. The ceiling is where the fringe count departs from the ideal, which is a sharp and checkable event; whether a reader would call the pattern spoiled somewhat before or somewhat after that is a question about vision. What can be said is that the two are within a factor of about two of each other, because the wander at the ceiling is a tenth to a fifth of a fringe spacing and that is already visible.

Who found it, and when #

Moiré metrology is old and well developed — the amplification of a small displacement by a fringe pattern is in every optics text, and moiré strain gauges have been standard laboratory equipment since the middle of the twentieth century. None of the first half of this essay is new.

What does not appear in that literature is the textile case, and the reason is that an optical grating is ruled on glass and its lines are where they should be. The question “what irregularity in the grating destroys the pattern” is not an interesting question about a ruled scale and is the only interesting question about a cloth.

The measured law — that the ceiling goes inversely with the irregularity because the failure is one thread against the pitch difference, rather than inverse-square because the failure is accumulated wander — is this collection’s, and it arrived as a correction. The accumulated-wander model was written first, predicted a ceiling of 1,764 at a two per cent irregularity, and was refuted by the simulation written to confirm it.

The inference back to the loom is this collection’s too, and it is the part worth carrying: an ornament that has existed for centuries is a measurement of the machine that made it, if anybody works out what it could not have been made from.

Where the ladder goes next #

Five rungs of this anchor have all been about two gratings that are nearly the same. The other regime is two gratings that are nothing like each other — a cloth photographed by a sensor whose pixel pitch is ten times its sett, a print screen against a weave, a knitted fabric on a display — where the beat is between a pitch and one of its own harmonics, and where the pattern that appears is not the difference of the two fundamentals at all.