What a dobby stores

A loom has two budgets and the previous rung spent only one of them. The harness decides how many distinct things an end may do; the pattern mechanism decides how many distinct things a pick may ask for. They are separate purchases, they are exceeded at different moments, and a design is limited by whichever runs out first.

The second is the less familiar, partly because the object that holds it has changed several times — pegged wooden lags, punched paper, a bar of steel pins, a file — and partly because the word pick gets used for both the thing and its instruction. A pick is a single passage of weft across the cloth. The instruction that produced it is a set of shafts, and a fabric of ten thousand picks may contain four instructions.

That gap is the subject here, and it is worth saying at the outset how large it usually is. For nearly every draft anybody weaves the instruction count is a small integer and the pick count is not, so the ratio between them runs into the thousands and the whole of the design’s length is carried by order rather than by content. The pattern mechanism is a device for repeating a very short list very accurately, which is a much humbler description than “the machine that weaves the figure” and a considerably more accurate one.

A lift is a set, not a pick #

On any pick, some shafts are up and the rest are down. That set is the lift, and it is all the mechanism needs to know: it does not need to know which pick it is, only what to raise.

So two picks calling for the same set are the same instruction, and a pattern chain that stored one entry per pick would be storing duplicates. Every dobby ever built stores the distinct sets and cycles through them in the order the design requires — which means the cost of a draft in chain is the number of distinct rows in its matrix.

That is the exact transpose of the shaft count, which is the number of distinct columns, and the symmetry is not a coincidence: a draft is a matrix and the loom is a machine for factoring it. The threading factors the columns, the chain factors the rows, and the cloth is the product.

Where the two budgets differ #

For the weaves this site has drawn so far the two numbers are usually equal, and it takes a moment to see why that is an accident of small repeats rather than a rule.

A twill on four ends has four distinct columns and four distinct rows. A satin on eight has eight and eight. A plain weave has two and two. Every one of those matrices is circulant — each row is the previous row shifted — and a circulant matrix has exactly as many distinct rows as distinct columns when the shift generates the whole cycle. So the whole family of twills and satins spends the two budgets in step, and a weaver working only in that family never sees them separate.

They separate the moment the design stops being a shift.

A reversal spends chain and not harness. Reverse the threading — as the previous rung does at every width — and the harness stays at four while the repeat widens; reverse the treadling and the lifts stay at four while the pattern lengthens. Both reversals are re-orderings of instructions the loom already has.

A block design spends harness and not chain. Divide a cloth into blocks each woven in a different weave and the ends in different blocks behave differently — new columns, new shafts — while the lifts may be very few, because the same handful of sets serve every block.

A satin spends both, in step, and there is no way round it. Which is why the shaft count of a satin is quoted as its order and nobody argues.

What the census says about lifts #

The four-by-four census gives the distribution directly, and it is the same distribution as the shaft count — necessarily, because transposing a draft is a symmetry of the census. Ninety-eight drafts need two lifts, 5,184 need three, and the rest need four.

That equality of distributions is worth separating from equality of counts. Transposing turns a draft with two shafts and four lifts into one with four shafts and two lifts, so the two histograms match while individual drafts are perfectly free to be lopsided. Asking how many drafts have equal counts is a different question from asking how the counts are distributed, and running the two together is an easy mistake to make with a symmetric table.

A worked case: the block draft #

The clearest place to watch the two budgets come apart is a block design, because it is the one construction where a designer is deliberately buying one and not the other.

Take a cloth divided across its width into two blocks. In the first, the ends weave a warp-faced satin; in the second, the same ends weave the weft-faced complement. This is the ordinary way of making a damask on a shaft loom, and it is called a block damask precisely because the figure is built from rectangles rather than drawn freely.

Count the columns. Every end in the first block does one thing and every end in the second does another, and within a block the ends are the satin’s own distinct columns. So the shaft count is the sum over blocks: two blocks of an eight-end satin need sixteen shafts, three blocks need twenty-four, and the harness grows linearly with the number of blocks. This is the reason a block damask loom is a large loom.

Now count the rows. Each pick raises a set that is some satin lift in the first block and some satin lift in the second, and there are only eight of each — so at most sixty-four distinct lifts exist, and the design typically uses eight of them, because the two blocks are usually driven from the same pick sequence. The chain barely notices.

So a block damask is expensive in harness and cheap in chain, and it is exactly the case that makes the jacquard worth its price: the jacquard’s cost scales with the ends rather than with the blocks, so it wins the moment a designer wants a figure that is not made of rectangles.

The tie-up, which is a third thing #

Between the shafts and the chain, on a treadle loom, sits an object that belongs to neither and is regularly confused with both: the tie-up. It is the set of cords connecting treadles to shafts, and it says which shafts a given treadle raises.

A treadle is a lift with a foot attached. So the tie-up is the store of distinct lifts, the treadling sequence is the order they are called in, and the number of treadles is the chain capacity of a loom that has no chain. A four-shaft loom with six treadles holds six lifts; a draft needing eight is not weavable on it without retying, however few shafts it wants.

This matters here for one reason. The count in this essay — distinct rows — is what a treadle loom needs treadles for, and a weaver meets the constraint as a shortage of feet long before meeting it as an abstraction about matrices. The mechanical dobby’s whole contribution was to make that number large and cheap, and the fact that a hand weaver still counts treadles is why the two budgets are more visible on a hand loom than on a power loom.

There is a fourth quantity lurking in the same place and it is worth naming so as not to trip over it. Some looms tie treadles to raise shafts and some to lower them, and a rising shed and a sinking shed reading the same tie-up produce complementary cloth — the face of one is the back of the other. Nothing in the matrix distinguishes them, in exactly the way nothing in the matrix distinguishes a Z twill from an S twill, and the site’s rule about saying which model applies here too.

Why the chain was the hard part #

The threading has been able to carry a wide repeat since somebody first threaded a point draw, and a drawloom could carry an enormous one. What a drawloom could not do without a second pair of hands was sequence the lifts, and a boy pulling cords is the original pattern store.

So the whole nineteenth-century mechanisation effort went into the second budget rather than the first. Pegged lags on a rotating cylinder, then punched cards, then steel bars and electronic files: each is a way of holding more instructions and changing between them faster, and none of them touches the harness at all.

That history explains a small oddity in how looms are described. A dobby is advertised by its shaft count and almost never by its chain capacity — sixteen-shaft, twenty-four-shaft, thirty-two-shaft — even though the chain was the innovation. The reason is probably that chain capacity stopped binding first: once a mechanism could hold a few hundred lags, most designs fitted, and the number that kept mattering was the one that had never been mechanised away.

What was counted, and how #

The lifting plan is read off the threading: shaft s is up on pick i when the ends assigned to it are up, and the ends on a shaft are identical by construction so there is nothing to choose. The distinct rows of that plan are keyed and numbered in order of first use, exactly as the columns were for the threading, and the count of them is the number reported as lags.

Every figure in this ladder then weaves its own drawdown from the two panels and compares it against the draft they were derived from, cell by cell. This is worth doing rather than assuming because the two factorisations are computed by nearly identical code operating on the matrix in two directions, and nearly identical code is exactly where a transposed index survives a reading.

Over the census the lift counts are accumulated alongside the shaft counts. The assertion that both lie between one and four is weak and would catch a gross error; the assertion that the totals sum to the census size is the one that would catch a filter applied twice.

Where the model stops #

A lift is not free to be anything. Some shed mechanisms cannot raise arbitrary subsets — a counterbalance loom ties shafts in opposition, so lifting a set implies lowering its complement and unbalanced lifts fight the tie-up. The matrix does not know this, and a draft perfectly affordable in both budgets can still be unweavable on a particular shed.

Order costs something even when instructions do not. A chain holds lags in sequence, and a design calling for four lifts in a sequence four hundred picks long needs four hundred lags carrying four distinct patterns. The instruction count and the sequence length are yet another pair of quantities that are easy to conflate, and the second is what a mechanism physically holds.

Nothing here is about time. A dobby changes lifts at loom speed and that is the end of it; a drawloom did not, which is why a damask took what it took. Cost in instructions and cost in labour are different currencies and this essay only counts the first.

And the count is a floor rather than a plan. The distinct rows are the fewest instructions that could produce the cloth. A weaver may well hold more — a chain written straight through the repeat, one lag per pick, is easier to check and easier to correct than one written as a table of lifts with a sequence beside it, and on a hand loom the easier thing wins. So the number here is what the design costs and not what anybody’s chain contains.

What the two budgets are worth to buy #

A loom’s two budgets are bought at different prices and by different means, and setting the prices beside the counts says which one a design should be economical about.

A shaft is a frame, a set of heddles and a place in the harness, so it costs metal, weight and a slot. It also costs speed: a deeper harness is heavier to move and demands a bigger shed, and both slow the loom. So shafts are expensive, they are bought in units of one, and a loom’s count is fixed when it is built.

A lift is an entry in a store, and it has been cheap since the store stopped being a boy. A pegged lag is a strip of wood; a punched card is a card; a modern dobby’s store is a file. So chain capacity is bought in hundreds, it costs nothing per unit, and it is the budget that has been mechanised away.

That asymmetry has a consequence for how a design should be arranged, and it is the reverse of what the counts suggest. Two designs of equal cost in the matrix are not of equal cost on a loom, because one of the two factors is expensive and the other is nearly free. A design that can be re-expressed to spend fewer columns and more rows is cheaper, and the operation that does that is a transposition — weaving the cloth sideways.

And weaving sideways is a real operation. A cloth’s warp and weft can be exchanged at the design stage: what was a threading becomes a treadling and what needed shafts now needs lifts. Nothing about the finished cloth changes except which system is which, and the two systems are not interchangeable in the cloth — but for a design whose two directions are structurally alike, the transposition is free and buys the whole difference between an expensive budget and a cheap one.

That is presumably why so many block designs are woven with their long direction along the picks. The trade’s explanation is about how the pattern reads and how the cloth is cut; the arithmetic adds that the design is being laid out along the axis whose budget the machine gives away.

Reading a draft sideways #

There is a habit worth acquiring from all of this, and it is cheap: look at every draft twice, once down the columns and once along the rows.

The two readings answer different questions with the same machinery. Down the columns is the harness, and it asks how many kinds of end there are. Along the rows is the chain, and it asks how many kinds of pick. A draft that looks expensive one way is very often cheap the other, and which of the two matters depends entirely on what loom is in the room.

It also catches errors. The two counts are related by transposition, so a draft whose column count and row count are wildly different is telling something true about the design — and a draft whose counts are wildly different from what the designer expected is usually telling something true about a mistake. A stray end that differs from its neighbours in one place adds a shaft and no lift; a stray pick adds a lift and no shaft. The two failures look identical on point paper and are immediately distinguishable in the counts.

Who found it, and when #

Nobody, again, in the sense of a result with a name on it. That a pattern chain stores distinct lifts is how the machine was built, not a discovery about it.

What is worth recording is the shape the two budgets make together, because it is a genuinely modern way of looking at the object: a loom is a device that reconstructs a matrix from a factorisation, the harness holds one factor, the chain holds the other, and the cloth is what comes out when they multiply. Stated that way the jacquard’s innovation is obvious in advance — it is what happens when the first factor is abandoned entirely, and the machine holds the matrix itself.

Where the ladder goes next #

Both budgets have now been counted and both have been shown to be properties of the machine rather than of the cloth. The next rung is the machine that abandons one of them: a jacquard gives every end its own shaft, which removes the threading constraint completely and replaces it with an entirely different one. After that, a damask is the cloth that motivated the whole enterprise, and its arithmetic turns out to be gentler than its reputation.