A net over a voile beats through a harmonic

Two identical sheers a few centimetres apart make a moiré with no angle between them and no difference in their threads, because a grid further from the eye looks finer. Every result there took the two layers to be the same grid, and the commonest double window is not. A net hung in front of a voile, or a lace panel over a lining, is a coarse grid and a fine one, five or ten threads of one to a mesh of the other.

The obvious prediction is that two grids so different make no moiré at all. They do. Their fundamentals are far apart and beat at a spacing finer than the voile’s own threads, but a grid is not only its fundamental: a coarse net carries harmonics, and one of them is almost exactly the voile. The net beats with the voile through that harmonic, and the whole arithmetic of two identical sheers comes back with one number changed.

That also settles what the pattern does as a viewer walks towards the window. It was expected to jump between families of fringes at particular distances. It changes family at particular distances, and it does not jump there — it dissolves into a texture twice the net’s pitch and re-forms from the next family — while between those distances it swells to nothing wherever the apparent ratio of the two pitches is a whole number.

Two grids this different have no beat between their fundamentals #

Take a voile of about thirty-three ends to the centimetre, a pitch of 0.3 millimetres, and a net of 1.55-millimetre mesh whose bars cover a tenth of its pitch, hung five centimetres apart.

Two gratings beat at the difference of their frequencies, and here the difference is between 3.33 lines a millimetre and 0.65. The beat between the two fundamentals repeats every 0.37 millimetres, which is barely coarser than the voile it is made from. That is a texture at the scale of the threads, not a pattern across a curtain.

So the moiré a double window shows cannot come from the fundamentals. It has to come from something in one grid whose pitch is close to the other’s.

A net is a stack of finer grids #

A grating of thin bars is a sum of sinusoids: its fundamental at the grating’s own pitch, a second harmonic at half the pitch, a third at a third, and so on. Each is a perfectly good grid in its own right, and each can beat.

The net’s fifth harmonic has a pitch of 1.55 over 5, which is 0.31 millimetres. Against the voile’s 0.3 that is a grid 3.3 per cent coarser. The net therefore contains, at the fifth harmonic, a copy of the voile stretched by a third of a tenth — and a pair of sheers one of which is slightly stretched is exactly the case the identical-sheers arithmetic already solved.

The general rule has one line. A coarse grid’s k-th harmonic beats with a fine grid as a grid of the coarse pitch divided by k would, and the family that shows is the k nearest the ratio of the two pitches as the eye sees them. Here the ratio is 5.17, and the family is the fifth.

The fifth harmonic is a stretched voile #

With the net behind the voile, the stretched copy is the far layer. A far layer coarser by a fraction ε looks finer by perspective and coarser by the stretch, and the two cancel at one distance, the gap over ε: five centimetres over 0.033 is 1.5 metres.

From 1.5 metres the net’s fifth harmonic, seen through the extra five centimetres, has a pitch of exactly 0.3 millimetres on the voile, and it lies on the voile thread for thread. There is no beat because there is no difference. Step in to a metre and the fringes are 18.6 millimetres apart; back out to two metres, 37.2; to three, 18.6 again; to ten, 10.9. Far away they settle at 9.3 millimetres, the spacing of the stretch alone.

Every one of those numbers is the identical-sheers arithmetic applied to a voile and a copy of it 3.3 per cent coarser. The net does not need to resemble the voile. It needs a harmonic that does.

Walking up to the window, the family changes #

The identical sheers had one family of fringes at every distance. A net and a voile have as many as the net has useful harmonics, and which one shows depends on where the eye is.

Seen through the gap, the net behind looks finer than it is, by the distance over the distance plus the gap. So the ratio of the net’s apparent pitch to the voile’s is 5.17 far away and falls as the eye comes closer: 5.08 at three metres, exactly 5 at 1.5, 4.77 at 60 centimetres. The family is the whole number nearest that ratio, so it is the fifth everywhere beyond 34 centimetres, where the ratio passes four and a half, and the fourth inside it.

Each family has its own null where the ratio is its own whole number. The fifth vanishes at 1.5 metres and the fourth at 17 centimetres; the change from fourth to third comes at 10.5 centimetres, closer than anybody stands to a curtain. Walking up to the window the pattern swells to nothing, narrows, changes family, and swells to nothing again, and the distances at which it does so crowd towards the glass.

At a change of family the fringe is at its finest #

The distance at which the family changes is where the lead expected a jump, and the arithmetic says something quieter happens there.

A family’s fringes are spaced at the coarse pitch divided by how far the apparent ratio is from that family’s whole number. The nearest whole number is never more than a half away, so the fringe spacing never falls below twice the net’s pitch as the eye sees it — 3.1 millimetres for the net’s real pitch, less when the net hangs behind and looks finer — and it reaches that floor exactly where the ratio is a half, which is where two families are equally near.

At 34 centimetres the net behind looks 1.35 millimetres in pitch, both the fourth and the fifth harmonic are half a cycle out of step with the voile, both beat at 2.7 millimetres, and neither is a pattern: a spacing under three millimetres seen from a third of a metre is a fine stripe in the cloth. Step closer and the fourth family’s fringes open out from that texture; step back and the fifth family’s do. The pattern does not jump from one family to the next. It dissolves into the net’s own scale and re-forms, at a distance set by the ratio passing a half.

A thin net’s harmonics are nearly as strong as its first #

A harmonic can only make a fringe as strong as the harmonic is. For a grating whose bars cover a fraction c of its pitch, the k-th harmonic’s amplitude goes as the sine of π·k·c over k.

For thin bars that sine is nearly π·k·c itself, so the first several harmonics are almost equal. A net of fine threads has a flat spectrum: with bars covering a tenth of the mesh, the fifth harmonic is 65 per cent as strong as the first, the fourth 77 and the sixth 51. The fifth-harmonic fringes are not a faint echo of a moiré. They are most of one.

The sine has a second consequence and it is the sharper one. Wherever k·c is a whole number, the family is missing. A net whose bars cover a fifth of its mesh has no fifth harmonic at all, and over this voile the fifth is the only family whose fringes are more than two millimetres across at any ordinary distance — the fourth and sixth, far away, beat at 1.33 and 1.86 millimetres. So the same mesh with bars twice as thick shows no moiré across a room, only a fine texture, from a pair of layers that would otherwise show fringes a centimetre wide.

A net in front has no null across a room #

Put the net in front of the voile instead and the stretched copy is the near layer. A coarser near layer adds to perspective rather than cancelling it, and it has no null at any distance.

Seen through the gap it is now the voile that looks finer, so the apparent ratio of the pitches is 5.17 far away and rises as the eye comes closer. It passes five and a half at 78 centimetres, six at 31 and seven at 14. Beyond 78 centimetres the fifth family shows and never vanishes; its fringes are 3.6 millimetres apart from a metre, 6.1 from three and 8.1 from ten, closing on the same 9.3 from below.

So the order of the two layers is not a detail. A net behind a voile makes fringes that swell to nothing a metre and a half from the window; the same net in front makes fine fringes that never do, and from across a room the first pair’s fringes are two to four times the size of the second’s.

The fringes walk as a stretched pair’s do #

A viewer stepping sideways in front of identical sheers sees the fringes move with them one for one, which is the parallax of the horizon. A stretched pair’s fringes move at the gap over the gap less the distance times the stretch, and since the net’s harmonic is a stretched voile, that is what these do.

With the net behind, a fringe moves three times as far as the eye does from a metre, and in the same direction. At the null the rate is unbounded, and beyond it the fringes run the other way: three times as fast against the viewer from two metres, exactly as fast against them from three, a third as fast from six. With the net in front there is no null to pass, and the fringes lag the viewer: 0.61 of each step from a metre, 0.34 from three.

That makes the sideways step the vernier essay’s instrument once more. A pair whose fringes run against a viewer has its coarser layer — here, the net’s harmonic — at the back, and the rate says by how much the pitch ratio misses a whole number.

A whole-number net behaves like a second voile #

The stretch that did all of this is how far the ratio of the real pitches is from a whole number, and nothing forces that to be anything in particular. A net of 1.5-millimetre mesh over the same voile has a ratio of exactly five.

Then the net’s fifth harmonic is the voile itself, unstretched, and the pair behaves in every respect like two identical sheers: the fringes are the voile pitch times the distance over the gap, six millimetres from a metre and eighteen from three, and they cover the same third of a degree from every distance. Only inside 45 centimetres, where the ratio passes four and a half, does the net show that it is a net.

So the double curtains that look most like a single moving pattern are not the pairs whose threads match. They are the pairs whose pitches stand in a whole-number ratio, whatever that number is.

Pressed together, gratings in a whole-number ratio have met this field twice already. The reed’s mark is a grouping of ends laid over a weave repeat, and a colour order beats the weave it is threaded on; in both the arithmetic is a least common multiple, and a denting or colour order that shares a factor with the repeat is the whole-number case. What hanging the two grids apart adds is that perspective moves the ratio, so the same pair passes through whole numbers as a viewer walks.

A piano tuner listens for the same beat #

The trade that has always worked with harmonic beats is not a textile one. A piano tuner setting an octave cannot hear a beat between the two strings’ fundamentals, which are a whole octave apart; what is heard is the lower string’s second partial beating against the upper string’s first, and the octave is set by slowing that beat to nothing. A fifth is set by the lower string’s third partial against the upper’s second.

A net and a voile are tuned the same way, by perspective instead of by a tuning lever. The net’s fifth partial beats against the voile’s first, and walking towards or away from the window moves the net’s apparent pitch the way turning a pin moves a string’s. The null at 1.5 metres is the distance at which the curtain is in tune, and the change of family at 34 centimetres is where the nearest interval stops being the fifth partial and becomes the fourth.

The analogy is exact in its arithmetic and in one further respect. A tuner listens to partials because two strings whose fundamentals are far apart share no beat at their fundamentals, which is precisely why the net and the voile share none either.

A harmonic magnifies the net’s own irregularity #

A moiré is a vernier, and it magnifies the error too: its fringes stay countable only while a grid’s spacing irregularity times the gain stays under 0.84. A harmonic sharpens that ceiling, and the reason is plain even though it has not been simulated here.

A thread of the net displaced by a fraction of the net’s pitch displaces the net’s fifth harmonic by five times that fraction of the harmonic’s own pitch, because the harmonic’s pitch is a fifth as long. Seen through its k-th harmonic, a net is k times as irregular as it is. A net laid as evenly as a voile carries its fifth-harmonic fringes as far as a voile five times less even would carry identical-sheer fringes — so a net’s moiré breaks up into wandering bands closer to the window than a pair of voiles’ does, at the same fringe size. It is the rule that a random error hides and a periodic one shows turned against the pattern: the beat is periodic and shows, and the irregularity it magnifies is random and, multiplied by the harmonic, stops hiding.

That is an inference from the vernier result, not a run of it: the ceiling there was found for two identical grids, and a harmonic beat’s ceiling has not been measured.

Counting one layer five times over #

The closed forms were checked by looking, in the same ray-traced scan that confirmed the identical sheers. A point on the near layer is joined to the eye, the ray is carried on to the far layer, and the two grids’ positions are counted in their own pitches — with the net’s count multiplied by the family’s harmonic, so that the scan finds the places where the net’s fifth or fourth harmonic, rather than its fundamental, comes into register with the voile. Nothing about the period is told to the scan.

For two equal grids the first family was confirmed to be the identical-sheers period exactly. For the 1.55-millimetre net in front and behind, at one, three and six metres, the family’s period was confirmed to equal the identical-sheers period of a grid of the net’s pitch over k, and the scanned fringes to agree with it to half a per cent. The spacing was confirmed never to fall below twice the net’s apparent pitch across distances from 15 centimetres to 20 metres; the null behind at 1.5 metres, and the absence of any beyond 32 centimetres in front, were confirmed from the closed form; each null’s family was confirmed to have no frequency there; and the family strengths were checked to vanish at one over the cover.

What a pair of gratings leaves out #

Both layers are one-dimensional bar gratings. A voile has warp and weft, and a curtain net is usually a hexagonal mesh with knots or bonded crossings, not a grating of straight bars. Each direction of threads beats with its counterpart as here, but a hexagonal net has three directions against the voile’s two, and the fringe families at sixty degrees are not computed.

Strength is an amplitude, not a darkness. How dark a fringe looks depends on the voile’s cover and on what two layers pass together at each registration, and a sheer’s threads let light through; none of that is in the family strengths, which only rank one family against another.

The eye is a point and looks square-on, as it did for the identical sheers, and the layers hang flat. A folded net changes its apparent pitch across every fold, and since a fold’s foreshortening is a stretch of several per cent it moves the family’s distances as much as the pitch ratio does.

And nothing is said about what is seen. A 3.1-millimetre texture from a third of a metre, or a 9-millimetre fringe from ten metres, is at the edge of what reads as a pattern, and where that edge lies is a question about vision.

Still open: a hexagonal net over a square voile #

The commonest curtain net is not a square grid. A tulle’s mesh has threads running three ways at sixty degrees, and a voile’s run two ways at ninety, so a net over a voile pairs harmonics that are not parallel. Each of the net’s three directions has its own nearest harmonic against whichever of the voile’s two directions it lies closest to, and those beats are fringes at an angle as well as at a pitch — the watered figure’s arithmetic, where a small angle made an enormous fringe, arriving by a third route. Whether a hexagonal net over a square voile shows one family of fringes, three, or a lattice of them, and at what distance each vanishes, needs the two-dimensional harmonics followed through the same perspective, and has not been done here.