What a tuck costs

A knitted fabric is sold by its weight, exactly as a woven one is. Two structures knitted from the same yarn on the same machine at the same setting weigh different amounts, and the difference is not small: a float jersey is nearly a fifth lighter than the plain jersey beside it.

Where that fifth goes is a count, and it is one of the few places on this site where a number about a knit can be produced without a stiffness, a friction or a fitted constant.

One constant, and it is not this site’s #

The whole calculation rests on a number from 1959 and it is worth naming before it is used.

Munden’s result is that a relaxed plain knit’s dimensions depend on one length only: courses per unit length are k_c over the loop length, wales per unit length are k_w over it, and the constants are properties of the relaxation state rather than of the yarn.

Read that the other way round. If a relaxed fabric has k_w wales per unit length for a loop length of one, then a loop is k_w wale spacings of yarn. For a fully relaxed fabric k_w is 4.3, so a loop is a little over four times the distance between two neighbouring wales.

That is a conversion, not a model. It says the same thing Munden’s constant says, with the division the other way up.

The other two actions #

A float across one needle is exactly one wale spacing. It is a straight line between the loop on one side and the loop on the other, and the distance between those is the wale spacing by definition. There is no approximation in that sentence and no constant in it.

So replacing a knit with a miss removes k_w − 1 wale spacings out of k_w, which is 76.7 per cent of a loop, every time, on every structure. The site asserts that as a relation rather than as a number: the saving per missed needle must come out the same fraction of a loop whatever the structure it is embedded in, because it is one subtraction repeated.

A tuck is a loop’s worth of yarn taken into the loop below. The needle takes as much yarn as it would for a knit — the cam draws it the same distance — but forms no new loop with it. Taking it as exactly a loop is a defensible first answer and it is not exact: the yarn is drawn a little further because it has to reach round the held loop as well, and the trade’s measurements put the tuck a little above a loop rather than at it. Every figure here states the factor it used and the default is 1.15.

That asymmetry is the finding. The float saves three quarters of a loop and the tuck spends a sixth of one, so the two actions are not opposite in size at all: a structure with a quarter of its positions floated is nineteen per cent lighter, and a structure with a quarter of them tucked is under four per cent heavier.

What was counted, and how #

Every named structure, counted over its own repeat:

Two things in that table are worth stopping on.

Single piqué and half-cardigan are the same number. So are single float and float jersey. That is not a coincidence and not an error: the arithmetic depends only on how many of each symbol the repeat contains, not on where they are, so two structures with the same composition cost the same yarn. Everything that distinguishes lacoste from half-cardigan — the appearance, the hand, the dimensional behaviour — is in the arrangement, and the yarn count cannot see arrangement at all.

The twill float halves the yarn. Two thirds of its positions are missed, so the fabric is nearly all float, and the array is still a fabric by both conditions: every wale knits once in three courses and every course knits somewhere. A fabric of long floats and few loops is exactly what it sounds like — snagging, dimensionally unstable, and half the weight — and the count says how much of the weight it saves without saying anything about whether it is any good.

The part that is exact, and the part that is not #

The temptation now is to turn yarn per position into grams per square metre, and it can be done — with one assumption that is false and whose falsity has a known direction.

Grams per square metre needs the area a repeat occupies. Munden’s constants give it for a plain knit: wales per centimetre, courses per centimetre, multiply. But a tuck fabric is wider and shorter than plain and a float fabric is narrower and longer, so using plain’s dimensions for either is wrong.

So the number is returned conditionally and its field is named for the condition: this is the weight the fabric would have if its dimensions were plain’s. At 20 tex on a 3.5 mm loop that is 135 g/m² for plain, 140 for half-cardigan and 109 for the float jersey.

The direction of the error is known even though its size is not. A tuck fabric is larger than the assumption, so the same yarn is spread over more area and its real weight per square metre is lower than the conditional figure. A float fabric is smaller, so its real weight is higher. Both errors run towards plain, so the conditional numbers overstate the spread.

That is an unsatisfying place to stop and it is the honest one. Predicting the dimension change needs the shape of a held loop and of a float under tension, which is a bending problem with the yarn’s own stiffness in it, and this site’s knitted geometry stops at the loop length.

Where the yarn goes at the machine, and where it goes afterwards #

There is a second way to read the same count and it is the one a knitter uses at the machine.

The cam setting fixes the loop length: how far the needle is drawn down decides how much yarn it takes, and it is the same for every knitting needle on the machine. So a structure does not change the loop length; it changes how many needles form loops at all, and the yarn per course follows.

That makes the arithmetic a production number as well as a fabric one. A course of a float jersey consumes 3.48 wale spacings per needle where plain consumes 4.30 — so at the same machine speed the yarn feed runs nineteen per cent slower, the package lasts nineteen per cent longer, and the fabric comes off the machine at the same courses per minute and a different weight. Nothing has been adjusted.

Run the same census with the tuck taken as exactly a loop rather than 1.15 of one — the one modelling decision in the whole calculation, made the other way — and cardigan’s excess falls from 7.5 per cent to nothing at all, because on that assumption a tuck and a knit consume identical yarn. The float column does not move by a hair, because there is no assumption in it. That is the essay in miniature. Where the arithmetic is a count, it is the same under either assumption; where it is a model, it moves by the whole of the effect. A specification quoting a tuck fabric’s weight to three figures is quoting the tuck factor to three figures without saying so.

Why the float saves so much more than the tuck spends #

The asymmetry — three quarters of a loop saved against a sixth of one spent — is the essay’s most useful number and it is worth saying where it comes from, because it is not a property of knitting.

A miss removes a loop and replaces it with a chord. The loop was 4.3 wale spacings and the chord is one, so the saving is the whole difference between a path that goes up, round and back and a path that goes straight across. That is a large fraction because a loop is a very inefficient way of covering a wale spacing — it spends four and a third units of yarn to advance one.

A tuck removes a loop and replaces it with nothing at all. The yarn the needle drew is still there; it has simply gone into the loop below instead of forming a new one. So the yarn count barely moves, and the only reason it moves at all is that reaching round a held loop takes a little more thread than reaching round an empty needle.

So the two actions are not opposites in the ledger even though they are opposites at the machine. A miss is a substitution of a short path for a long one; a tuck is a relocation of the same yarn. The first shows in a weight and the second does not.

That reading also says which of the two a knitter reaches for when the target is a weight and which when the target is a fabric. Floats are the weight control and tucks are the bulk control, and the arithmetic says so before any fabric is made: a structure that needs to be lighter has to miss needles, and a structure that needs to be fuller has to tuck them, and no amount of tucking makes a fabric lighter.

And it explains why the float’s saving is exact and the tuck’s cost is not. The float’s replacement path is a straight line between two known points, so its length is a definition; the tuck’s is a path round a body whose shape is what this site does not model. The one action that is a substitution has an exact answer and the one that is a relocation does not, which is the reverse of what the machine’s own descriptions would suggest.

There is a practical corollary about how far each can be pushed. A structure can miss most of its needles and remain a fabric — the twill float misses two in three — so the weight control has a long range, bounded by the conditions an array must satisfy rather than by the yarn. The bulk control has a much shorter one, because a needle can only hold so many courses before the latch will not close, and the machine’s hold limit binds long before the arithmetic does. One lever runs to half the weight and the other runs to a few per cent, which is the practical form of the same asymmetry.

Stitch density is not yarn density #

There is a second count in the table and the trade uses it interchangeably with the first, which it is not.

Stitch density is loops per unit area, and it is what a knitter measures with a counting glass. Yarn density is grams per unit area, which is what a buyer pays for. In a plain knit they are proportional, because every needle position holds a loop and every loop is the same length — which is the state the loop essay describes and the only state in which the two words mean one thing.

In a tuck or float structure they are not. Half-cardigan has three loops per four positions, so its stitch density is three quarters of plain’s while its yarn is three per cent above. Cardigan has half the loops and eight per cent more yarn. The twill float has a third of the loops and half the yarn.

A structure can have fewer stitches and more yarn in it, and the trade’s phrase “a denser fabric” does not say which is meant. The array settles it in one count each way.

The two columns move in opposite directions down the tuck rows and in the same direction down the float rows, which is the whole distinction in one table. A tuck removes a loop and keeps its yarn; a miss removes a loop and most of its yarn with it.

It is worth asking what the same question looks like on the other kind of cloth, because a woven fabric has no loop to spend yarn on and its weight moves for an entirely different reason.

The comparison with a woven cloth #

The woven version of this arithmetic is the areal weight identity, and putting the two side by side is instructive because they fail in opposite places.

A woven cloth’s weight is sett times count times one plus crimp, summed over the two systems. Every term is measurable and the crimp comes out of Peirce’s geometry, so the whole thing can be computed from a specification with no free parameter and no conditional.

A knit’s weight is loop length times loops per unit area times count — and the loops per unit area is the part that needs Munden, and Munden is a fitted constant from measured fabrics rather than a geometric derivation. The weave’s difficult quantity is the crimp and it is solved; the knit’s difficult quantity is the relaxed dimension and it is measured.

That difference is structural rather than historical. A woven cloth’s geometry is pinned by the reed and the take-up: the threads are where the machine put them. A knitted fabric has no such constraint — the loops settle wherever the yarn’s bending and the friction between loops leave them — so its dimensions are the outcome of a mechanical equilibrium rather than of a machine setting. Munden’s achievement was to find that the outcome depends on one length, which is a much stronger statement than it looks.

Where the model stops #

The tuck factor is a stated calibration. Nothing here derives it, and every number involving a tuck moves with it. A tuck taken as exactly a loop lowers half-cardigan’s excess from 3.7 per cent to 2.5.

The float is one wale spacing per needle passed, which assumes it is straight and that it spans an integer number of wale spacings. A float in a relaxed fabric is not straight — it is slack, because it was laid at the needle pitch and the fabric contracted afterwards — so the real float is longer than the arithmetic and the saving is overstated.

The whole calculation is single jersey. A rib or an interlock has two beds and the loops on the two are not the same length, so the conversion from loop length to wale spacing is a different one and k_w is a different constant.

And there is no fibre in any of it. Two structures of the same yarn are compared, so the count cancels; comparing a wool and a cotton at the same structure needs a density and a packing factor and is the woven weight arithmetic again.

Who found it, and when #

Munden’s 1959 paper is the whole foundation and it is worth being clear about what it did and did not claim. It measured relaxed plain-knitted fabrics of several yarns at several loop lengths and found that courses per unit length and wales per unit length each go as one over the loop length, with constants that depend on the relaxation state and on nothing else — not on the yarn, not on the fibre, not on the machine gauge.

He did not extend it to tuck and float structures, and the extension has never been made cleanly. The knitting literature has tables of dimensional constants for named structures, obtained the same way Munden obtained his — by knitting fabrics and measuring them — and the tables do not compose: knowing the constants for half-cardigan and for single float says nothing about a structure using both.

That is exactly the gap an array-based account would close if it could, and it is why this essay is careful to separate the count from the dimensions. The count composes and the constants do not. Any structure’s yarn can be computed from its array; no structure’s size can be computed from anything but a measurement of that structure or one very like it.

Where the ladder goes next #

The float has appeared twice now — as the thing that saves the yarn here, and as the thing that pulls a fabric in. It is also the object this site has spent five rungs on in a woven cloth. The next essay puts the two side by side and asks how far the analogy holds, which turns out to be further than expected in the optics and not at all in the mechanics.