The harness does not grow

Open almost any account of weaving and somewhere near the description of a dobby will be a sentence of the form a sixteen-shaft loom can weave a repeat of sixteen ends. It is said in passing, as background rather than as a claim, and it is wrong by an unbounded factor.

It is true of one threading, and the threading it is true of is the one every weave in the foundation and the satin essays uses. On a straight draw — ends threaded 1, 2, 3, 4, 1, 2, 3, 4 and so on — the repeat of the cloth is the repeat of the threading, so the shaft count is the repeat and the sentence holds exactly. Every weave in the foundation of this site is a straight draw, which is probably why the sentence survives: it is true of everything a beginner meets.

Where the economy actually is #

Reverse a twill and the threading turns round. Instead of 1, 2, 3, 4, 1, 2, 3, 4 the ends go 1, 2, 3, 4, 1, 3, 2, 1 — out and back — and the cloth acquires a chevron with a repeat of eight ends. The harness has not changed. It cannot change: reversing the order in which ends are assigned to frames does not invent a new frame.

Keep going. Run out over twelve ends and back over twelve and the repeat is twenty-four; over twenty-four and back and it is forty-eight. The threading is a walk on four shafts and a walk can be as long as anybody likes.

So the two quantities are independent, and the one a designer cares about is not the one the loom is sold by:

• The harness is the number of distinct behaviours an end can have. It is bought once and cannot be exceeded.
• The repeat is the number of ends before the threading pattern comes round again. It is drawn once when the loom is dressed and costs nothing but time.

A dobby is worth owning because of the first, and the patterns it makes are wide because of the second, and conflating the two makes the machine look far more limited than it is.

What the reversal costs, which is not shafts #

The harness is free. Three other things are not, and the essay on reversed twills worked two of them out at foundation:

A point reversal duplicates exactly two ends per repeat, at every width. Mirror the column index about the last end before the turn and the two ends either side come out identical — two adjacent ends doing the same thing, which reads in the cloth as one thick thread and is what a weaver calls a cracked herringbone.

The float lengthens. A two-and-two twill has a float of two everywhere; reversing it on a point lengthens the longest float to four, and reversing cleanly to three. The turn is where two floats meet end to end, and they join.

The tightest reversals stop being cloth. A point reversal every two ends falls into three separable layers. That is the integrity criterion doing its job on a construction that looks entirely reasonable on paper.

None of those is a shaft. All three are properties of the cloth, and all three are exactly the kind of thing this site’s machinery decides — which is the point worth drawing out. The harness constraint and the cloth constraints are different constraints with different currencies, and a design is limited by whichever binds first.

The same argument runs the other way #

Everything above is about the warp, because the harness is a warp device. The weft has an exactly parallel story and it is worth stating, because the symmetry is the thing that makes the point general rather than a trick of threading.

A lift is a set of shafts raised together, and a pick calls for one. Two picks that call for the same set are the same lift. So the pattern length — how many picks before the treadling comes round — is bounded by the number of distinct lifts a draft needs, in precisely the way the pattern width is bounded by the number of distinct columns, and the mechanism that stores lifts is a different budget from the harness that performs them.

Reverse the treadling of a two-and-two twill and the same thing happens: the picks run 1, 2, 3, 4, 1, 3, 2, 1 over eight picks, calling for the same four lifts. A cloth reversed in both directions — the diamond twill of the herringbone essay — has a repeat of eight by eight, four shafts and four lifts, and a diamond in it — which is the entire content of most of what a four-shaft weaver ever makes.

The two reversals are independent, so the repeat area grows as the product while both budgets stay flat. Thirty-two ends by thirty-two picks is a repeat of 1,024 intersections carried by four frames and four lifts, and the whole of that growth is bookkeeping about order rather than about behaviour.

How far a point draw actually goes #

The harness is free, so something else has to bind, and it is worth doing the arithmetic to see what and when.

A shaft carries heddles, and a heddle is a physical object about a millimetre and a half thick standing on a frame perhaps a metre wide. Six or seven hundred heddles on one frame is an ordinary maximum, and a thousand is a crowded frame that opens badly.

Take a shirting warp: sixty ends per inch across forty-five inches is 2,700 ends. On a straight draw over four shafts that is 675 heddles per frame — comfortable. Now thread a point draw with a repeat of forty-eight ends. The repeat contains twelve ends per shaft, the warp contains 2,700 ÷ 48 ≈ 56 repeats, and each frame therefore carries 56 × 12 = 675 heddles. Exactly the same number.

That is not a coincidence and it is the useful form of the result. Every end has to hang somewhere, so the total heddle count is the warp’s end count regardless of the threading, and widening the repeat redistributes nothing. The threading is free in heddles as well as in frames.

What a point draw does cost is evenness. A straight draw puts an equal number of ends on each frame by construction; a reversal need not, and an asymmetric threading can load one frame with twice the ends of another. On four shafts and a symmetric reversal the imbalance is small. On sixteen shafts with a threading designed around a motif it can be severe, and it is the first thing a weaver checks after the shaft count, which the previous rung computes.

The rule stated properly #

The correct version of the textbook sentence takes one more clause:

A loom with n shafts can weave any draft whose matrix has at most n distinct columns.

That is a statement about the matrix, not about its width, and it has a pleasant consequence: it is decidable in one pass and it is exactly the count of the previous rung. A designer with a sixteen-shaft dobby does not ask whether a repeat is under sixteen ends; they ask whether the draft, however wide, calls for more than sixteen behaviours.

The two questions agree on a straight draw and diverge immediately on anything else, and the divergence is the entire commercial reason a dobby exists. If the shaft count really were the repeat, a sixteen-shaft loom would be a machine for weaving sixteen-end patterns, which nobody would buy.

It is worth being concrete about how much the correct version buys. A sixteen-shaft dobby weaving a straight draw reaches repeats of sixteen ends. The same loom threaded on a point draw reaches a repeat of thirty ends with the same sixteen behaviours; threaded on a compound draw with several blocks it reaches whatever the designer can lay out, bounded by the warp’s width. The difference between the two readings is not a refinement. It is the difference between a machine for weaving small geometric repeats and a machine for weaving a damask border, and the sentence that conflates them is describing the wrong machine.

The satins are the counter-example that keeps the rule honest, and they are the reason the confusion is forgivable. An eight-end satin needs eight shafts for eight ends and there is no threading that reduces it, because every column of a satin genuinely differs from every other. So the weaves where shaft count and repeat are equal are not only the beginner’s straight draws — they include the most prized cloths in the trade, and the satins are exactly where a weaver most wants more shafts.

What was counted, and how #

For each width in the sweep, three reversals of the two-and-two twill are generated from their rules — point, clean and broken — and each is threaded. The threading is the walk over distinct columns described in the previous rung; the shaft count is its size; the reduction is the repeat divided by it.

Three things are asserted while the census runs.

Every reversal at every width weaves on the base twill’s shaft count. This is the finding, and it is asserted rather than reported because a failure would mean the threading code had found a distinct column where the construction cannot produce one.

The reduction is the repeat divided by a harness that never grew. Stated separately, because the first assertion could be satisfied by a bug that returned a constant.

A straight draw saves nothing. The control, and the reason the textbook sentence exists.

An earlier version of the third assertion demanded a reduction of at least twelvefold, which is a property of the default list of widths rather than of the finding — and asking the same census about narrower widths made a true statement fail. That is a small thing and worth recording, because it is the commonest way an assertion goes bad: it is written against the numbers in front of it rather than against the claim, and it then rejects perfectly good input the first time the generator is asked something new.

How much the free threading is actually worth #

The essay says the textbook sentence is wrong “by an unbounded factor” and demonstrates it by reversals. The size of the space that opens up can be counted exactly, and the count is worth having because it says what the real constraint on a shaft loom is.

A threading of w ends on n shafts, with every shaft used and with the shafts themselves interchangeable, is a partition of the ends into n non-empty groups — so the number of distinct threadings is the Stirling number of the second kind, S(w, n).

On four shafts:

A four-shaft loom threaded over sixteen ends has a hundred and seventy million distinct threadings available to it, against exactly one for the straight draw the textbook sentence describes. The point draw and the broken draw this essay uses are two of them.

The lifting side multiplies. Four shafts admit fourteen useful lifts — every subset but all up and all down — so an eight-pick repeat admits 14⁸, which is a billion and a half lifting plans. Multiplying the two, a four-shaft loom weaving a sixteen-by-eight repeat has of the order of 10¹⁷ distinct drafts available to it.

Against 22,874, which is the whole of this collection’s four-by-four census.

Which relocates the constraint entirely #

A number that large says something the reversals cannot, and it is not about the loom.

A designer’s binding constraint is search, not shafts. The space of threadings a modest harness reaches is astronomically larger than anything anybody can enumerate, inspect or hold in mind, so the shaft count stopped being the limit a very long way back and the limit that replaced it is the human one.

The trade’s response is visible in its vocabulary and now has a reason. Point, broken, block, skip, undulating, herringbone, diamond: every named threading is a rule — a small structured family inside the space, generated by a construction rather than chosen from a list. The rules exist because the space is not searchable, and a rule that produces a hundred usable threadings from one line of description is worth more than a catalogue.

That also explains a shape this collection keeps finding. Its own censuses — every cloth at four by four, the plane groups, the separable drafts — are all at a repeat small enough to exhaust, and exhaustion stops at about four by four. Everything wider is explored by rule and by taste, and the reason is not that nobody has tried but that 10¹⁷ is not a number anybody enumerates.

And it sets a bound on what a census can ever say #

One consequence for reading this collection’s own results. The four-by-four census covers 65,536 matrices, of which 22,874 are cloths — thirty-five per cent. If a similar fraction of the sixteen-by-eight space is weavable, the usable design space on four frames is still of the order of 10¹⁷.

So no statement of the form most drafts do such-and-such can ever be checked outside the smallest repeats, and every one this collection makes should be read as a statement about four by four. Where a result generalises, it generalises because it was derived — the coprimality condition, the separation of the maximum, the alternating-column count — and not because a larger census confirmed it.

A census is a way of finding a rule and never a way of proving one, and at these sizes it is not even a way of checking one.

Where the model stops #

The loom’s width is a real bound and is not in any of this. A repeat of forty-eight ends is only useful if the warp is wider than forty-eight ends, and a threading of ninety-six ends on four shafts puts twenty-four ends on each frame per repeat — which at a fine sett across a wide loom is a great many heddles on one shaft. Heddle capacity is what actually limits an ambitious point draw, and it is a property of the hardware that no matrix contains.

The shed suffers. Ends threaded on the back shafts travel further than ends on the front, so a threading that puts long runs on the back frames opens less cleanly. Weavers reorder frames for exactly this reason, and the reordering is invisible in the cloth and in the count.

A wide repeat is not automatically a good pattern. Nothing here says the reversals are worth weaving; it says they are affordable. Which of them is worth weaving is the question the float limit answers, and the answer has no shafts in it.

And the count is minimal rather than obligatory. Four shafts is the fewest a reversed twill can be threaded on; a weaver with sixteen is free to spread the same design over more frames, and often does, because more frames means fewer heddles apiece and a cleaner shed. So the number computed here is a floor on the cost and not a description of what anybody actually threads.

Who found it, and when #

The point draw is old — it is in the earliest European drawloom and pattern-loom descriptions, and versions of it are older still — and no one appears to have thought it needed an argument, because a weaver who has threaded one can see that the frames did not multiply.

What did need an argument, and got one from the mid-eighteenth century onward, was the machinery for changing the lifting plan quickly enough for a wide repeat to be worth threading. That is the dobby’s actual invention, and it is the subject of the next rung: the threading buys the pattern’s width, and the chain buys its length.

The word dobby is generally taken to be a worn-down form of draw boy — the child who stood at a drawloom and pulled the cords that raised the pattern, one lift at a time, for as many picks as the design ran to. That etymology is the whole history of the machine in one word: the threading had always been able to carry a wide repeat, and what was missing was a way to sequence the lifts without a second person. A drawloom could weave enormous repeats and did, at the cost of an extra pair of hands per loom.

The mechanical answer arrived in pieces through the nineteenth century, and the shape that stuck is the pattern chain: pegged lags, or later punched bars, cycling past a set of feelers that decide which shafts rise. The important thing about it is what it stores. It stores lifts, not picks, and it does not know or care how wide the threading is — which is why the two budgets in this essay stayed separate all the way through the machine’s development, and why nobody who built one would have said the shafts limited the repeat.

The confusion this essay is about probably comes from teaching order rather than from the trade. A beginner threads a straight draw because it is the simplest thing to do, meets the shaft count and the repeat as the same number, and is never afterwards told that they came apart. By the time a point draw appears, the identity has been used often enough to feel like a definition — and a false identity that is true of every example so far is exactly the kind that survives.

Where the ladder goes next #

The threading has been shown to be free of the harness. The other half of the loom’s budget is what the pattern chain stores, which is lifts rather than picks and is a smaller number than almost anybody expects. Past that is the machine with no shafts at all, where every end is its own shaft and the threading stops being a constraint entirely — at which point what the design costs is something else again.