A jacquard's harness has a depth after all

The harness has a depth is this collection’s answer to why a dobby carries sixteen or twenty-four shafts rather than two hundred. Each shaft stands further from the fell than the one in front of it; the warp sheet runs straight from fell to heddle, so a shed of a stated opening at the reed needs every shaft’s lift to be proportional to its own distance from the fell; the extension a lift costs an end goes as the square of that lift; and so a stated tolerance on warp strain is a stated distance from the fell, which is a whole number of shafts. One per cent buys thirteen.

A jacquard escapes it completely, the argument continued, because it has no shafts. Every end has its own hook and its own cord, every cord runs down to a mail, and every mail hangs at the same distance from the fell, so there is no back of the harness to work hardest.

That is true of a board with one row of holes. No jacquard has one.

A cord is thicker than an end spacing #

The comber board is a perforated board as wide as the cloth with one hole for every end in it. At an ordinary furnishing sett of sixty ends a centimetre the ends are a sixth of a millimetre apart.

A harness cord is not a sixth of a millimetre. It is a twisted linen or synthetic cord carrying a mail — a small metal eye the warp end passes through, of the kind a heddle is on a shaft loom — and the two together want something approaching a millimetre of room, with the mail the larger of them. So the holes cannot sit in a single line across the board at the cloth’s own spacing, and they never have: a comber board is ruled in rows, staggered fore and aft, with successive ends going through successive rows and the row count chosen so that the holes in any one row are far enough apart for the cords.

The arithmetic is one division. The rows a board needs is a cord’s own pitch over the end spacing, rounded up, and at sixty ends a centimetre with a cord wanting nine tenths of a millimetre that is six rows.

Rows are spaced along the warp — six millimetres apart is an ordinary figure — so six rows is a board thirty millimetres deep, and the ends passing through its back row hang thirty millimetres further from the fell than the ends passing through its front row.

The depth is real and it is a factor of eight #

The strain a shed puts on an end is computed exactly where the shaft-loom argument computes it: the warp runs straight from the fell to the mail and on to the back rest, so the sheet’s length is two hypotenuses, the extension is that length over the straight one less one, and the lift is whatever makes the opening at the reed the same from every row.

At sixty ends a centimetre the six rows strain their ends from 0.461 per cent at the front to 0.524 at the back, a spread of 0.064 per cent. A thirteen-shaft harness at sixteen millimetres a shaft is a hundred and ninety-two millimetres deep and spreads 0.499 per cent.

So the jacquard’s own depth spreads an eighth of what the dobby’s does. That is a real escape and it is not the release the earlier argument claimed: the quantity is smaller by a factor rather than absent, and a factor can be eaten into.

It grows with the cloth, not with the design #

The two depths are governed by completely different quantities, and this is the part that matters for anybody choosing a machine.

A shaft harness’s depth is a property of the design. A design needing eight shafts has a harness eight shafts deep; one needing twenty-four has a harness three times as deep. The strain spread is what the designer spends, and it is the reason a design is written to a shaft budget, and why the harness does not grow when the figure does.

A comber board’s depth is a property of the cloth, and the board is not the reed — the reed sets the spacing and the board only has to fit a cord through it. It is set by the end spacing and the cord’s thickness, and it has nothing to do with how complicated the figure is. A jacquard weaving a plain ground and one weaving a twenty-thousand-hook damask at the same sett have exactly the same board, exactly the same rows and exactly the same strain spread.

That is the escape the earlier argument was reaching for, stated properly. A jacquard does not turn a design’s complexity into warp strain. What it does instead is turn the cloth’s fineness into warp strain, which is a quantity a shaft loom does not pay for at all — a dobby weaving at a hundred ends a centimetre has the same thirteen shafts at the same pitch as one weaving at ten, whatever the sett its weave decides.

Which repairs the sideways fan, and dismisses it #

A jacquard harness needs three half-spans of height computed the other geometry above the board: the hooks stand in a machine a few tens of centimetres wide, the board is as wide as the cloth, and the cords fan out between them, so an edge cord leans and its mail rises by less than its hook does. That account stated its own exclusion plainly — a real machine’s hooks stand in a grid several rows deep, which leans the cords fore and aft as well as sideways, and the calculation assumed one row.

The board’s depth is the size of that exclusion, and it is now a number. An edge cord on a hundred-and-thirty-centimetre board under forty centimetres of hooks reaches four hundred and fifty millimetres sideways; the board’s own depth is thirty. The cord’s true reach is the hypotenuse of the two, which is 0.22 per cent longer than the sideways reach alone.

On the finest board here, forty-eight millimetres against the same four hundred and fifty, the correction is 0.57 per cent. Both are far below the five per cent tolerance the mount height was computed to, so the three-half-span rule survives untouched and now has its neglected term measured rather than named.

The one arrangement in which it would not survive is a narrow cloth. An edge cord on a sixty-centimetre board reaches only a hundred millimetres sideways, and a thirty-millimetre board depth against that is a 4.4 per cent correction — comparable to the whole tolerance. So the fore-and-aft lean is negligible on a wide cloth and is not on a narrow one, which is the opposite of the way the sideways lean behaves and is worth stating because it is the case a harness builder would not expect to be the awkward one.

The rows beat against the repeat, exactly as a reed’s dents do #

There is a second consequence of ruling the holes in rows, and it is not about strain at all.

The ends are assigned to rows in order and cyclically: end one to row one, end two to row two, and round again. So an end and the end r along are on the same row and at the same depth, and the depth pattern across the warp is periodic with the row count.

A ground weave is periodic too. If the row count shares a factor with the ground’s repeat, every repeat of the ground meets the rows in the same phase, and whatever small difference the depth makes lands in the same place in every repeat — which is a periodic variation lying exactly on the pattern. If it does not, the phase walks and closes only after a run of repeats.

That is the reed’s own arithmetic one level up. A reed groups the warp several ends to a dent and the grouping beats against the weave repeat, so a denting sharing a factor with the repeat treats every repeat identically and shows as a stripe, while one that does not is invisible. Here the grouping is by board row, the difference is a warp strain rather than a spacing, and the arithmetic is the same least common multiple.

The difference in size is enormous and worth saying so the parallel is not oversold. A reed’s denting moves ends by a fraction of their own spacing, which is a visible thing; a board’s rows differ in strain by six hundredths of a per cent, which is well below what any other source of unevenness contributes. The mechanism is identical and the magnitude is not, and the reason to state it is that a board is one more periodic structure laid over a warp, in a machine that already has a reed, a threading and a colour order beating against each other.

What the harness builder actually chooses #

Three of the four quantities in this arithmetic are the builder’s, and the arithmetic says which of them is worth spending on.

The cord’s pitch sets the row count directly and the depth through it. A cord and mail wanting half a millimetre rather than nine tenths takes a sixty-end board from six rows to three and halves its depth. That is the largest single lever and it is a matter of how fine a mail can be made.

The row pitch sets the depth for a given row count, in proportion, and the strain spread with it: three millimetres a row gives fifteen millimetres and 0.031 per cent where six gives thirty and 0.064. It costs nothing but the board’s own strength, which is why boards are made of laminated wood or a filled resin rather than of something that would need the rows spread out.

The board’s position — how far from the fell the whole thing hangs — sets the level of the strain rather than the spread, and the level is the larger number by a factor of seven: 0.46 per cent at the front row against a spread of 0.064 across the board. Moving the board back strains every end more and spreads no more.

And the sett is not the builder’s at all. It is the cloth’s, so the one quantity that drives the depth up is the one the machine has no say in — which is why a jacquard for very fine silk is a different mount from one for furnishing, and why the difference is in the board rather than in the machine above it.

What it does to a brocade’s binding warp #

One construction has a stake in all of this, and the earlier account named it without being able to price it.

A brocade’s pattern weft floats as far as the next figure unless an end is dropped under it, and on a jacquard the binding ends that do the dropping come through the same comber board as everything else. So the binding warp is on the same six rows, is strained by the same spread, and — because a binder is usually a finer thread than the ground warp — is the part of the cloth least able to spare it.

The arithmetic gives that a size. A binding end on the back row is strained 0.064 per cent more than one on the front row, which on a warp already at 0.46 is a fourteen per cent increase in the extension it carries. A fine binder run at the same tension as a coarse ground warp is already the weakest thing in the loom; putting an eighth of its neighbours on a back row adds a systematic difference that repeats with the row count.

And the binding ends are not distributed at random across the rows. A brocade’s binders sit at stated intervals in the warp — every eighth end, or every sixteenth — so their row assignment is the beat above with the binder’s own interval in place of the ground’s repeat. A binder interval sharing a factor with the row count puts every binding end on the same row, which is either the best or the worst arrangement available and is decided by an accident of two numbers neither of which anybody chose for this.

That is a small effect and it is the kind that is worth writing down anyway, because it is an interaction between two quantities chosen for unrelated reasons — a cord’s diameter and a brocade’s binding interval — with a periodic consequence in the cloth. This collection’s experience is that such pairs are where the unexplained faults live.

What was computed, and how #

The end spacing is ten over the sett in millimetres; the rows are the cord’s pitch over that, rounded up; the depth is one fewer than the rows times the row pitch. Each row’s strain is computed exactly as the shaft harness’s is — the warp sheet from the fell to the mail to the back rest as two hypotenuses, with the lift set so that the opening at the reed is the same from every row — and the comparison is against a thirteen-shaft harness of the same loom at its own shaft pitch. The row-against-repeat arithmetic is a greatest common divisor.

Four things are checked. A more finely set cloth needs at least as many rows, at every step of the sweep, and strictly more across the range a jacquard is woven at. Every row of a board strains its ends more than the row in front of it, at every sett, which is the check that the strain is being read along the warp and not across it. The whole board spreads less strain than a thirteen-shaft harness, at every sett in the sweep — the claim that the escape is real, stated as an inequality rather than a ratio, so that a sett fine enough to break it would fail the test rather than pass with a bad number. And rows dividing the repeat put every repeat in the same phase while rows that do not close after a computable run, checked on four rows against eight and on three against eight.

The cord’s pitch, the row pitch, the board’s distance from the fell and the loom’s own dimensions are inputs. The nine-tenths of a millimetre for a cord and mail is a plausible figure rather than a measurement of any particular harness, and every row count is a ceiling of it.

Where the model stops #

The cord’s pitch is assumed, and the row count is a ceiling of it. A real board is ruled to a standard — so many holes to the inch in so many rows — and the rows are chosen from a short list rather than computed. The arithmetic here gives the fewest rows that could work; a builder would use the next standard ruling above it.

The assignment of ends to rows is taken as cyclic and in order. A real harness is tied in one of several arrangements, and which end goes to which row is part of the tie. An arrangement that put whole blocks of ends on one row would change the beat entirely and would leave the strain spread where it is.

Nothing here is the board’s own sag. A board a metre and a half wide carrying several thousand cords is loaded along its whole length and bends; a sagging board is deeper in the middle than at its edges, which adds a variation across the width that this account does not have.

And the strain is computed for a straight warp sheet from a stationary fell. A shed is an extension computed at one moment; a loom’s fell moves, its let-off gives, and the strain a real end sees is a cycle rather than a value.

Still open: whether the board or the machine sets the finest cloth #

The rows rise with the sett and there is no bound in the arithmetic, so the obvious question is where it stops — and the answer is not in this calculation.

A board of r rows at the row pitch is deep, and a deep board is one whose back rows’ cords have to be longer, hang at an angle fore and aft, and pass through a board that is itself thick enough to guide them. At some fineness the board becomes the limit on what a jacquard can weave, and the limit would show as a cloth that cannot be set closer however fine its yarn is.

What the arithmetic does say is that the limit is not the machine. The hooks, the cards and the lifting are indifferent to the sett; the mount height is set by the cloth’s width; and the only quantity that runs away with fineness is the board. So if a finest-jacquard-cloth limit exists, it is a property of a perforated board and a mail, and it would be found by asking a harness builder how fine a mail can be made rather than by asking anything about weaving.

Who worked it out #

The comber board and its rows are as old as the drawloom and every harness builder rules them; the practice of staggering the holes fore and aft is universal and unremarked, because it is obvious to anyone who has tried to put two cords through one hole. What is added here is the row count computed from the sett, the strain spread it gives set against a shaft harness’s, and the observation that a jacquard’s depth is a property of the cloth where a dobby’s is a property of the design.