Two knits with one tightness factor are one knit

Count the free quantities in the loop model. A loop length, a yarn diameter, a wale spacing, a course spacing. Three of those four are proportional to the first — the spacings by Munden’s constants — so the whole geometry is the loop length times a shape, and the shape depends on one ratio: the yarn’s diameter over the loop length.

Two knits that share that ratio therefore have the same loop, in units of their own loop length, whatever they are made of and whatever count they are spun to. A 12 tex cotton and a 30 tex wool matched on it give solved shapes agreeing to fifteen figures.

Why the ratio is the tightness factor #

A yarn’s diameter goes as the square root of its count, so the ratio d/ℓ is √tex over the loop length times a constant made of the fibre’s density and its packing factor.

And √tex over the loop length in centimetres is K, the knitter’s tightness factor, quoted at thirteen to seventeen for an ordinary cotton jersey. It is a trade index, arrived at empirically, correlated with handle and cover, and not derived from anything.

It is the model’s only variable. Every result on this ladder is a function of K and one constant per fibre, and there is nothing else to vary.

What that explains #

Why an empirical index works as well as it does, which is a question nobody has needed to ask because it works.

An index that correlates with several unrelated properties is usually a proxy for something, and the something here is the loop’s own shape. Cover correlates with K because the occupancy is K times a constant. Handle correlates because the bending rigidity does. Extensibility, run resistance, pilling and dimensional stability correlate for the same reason, each through its own function of the same ratio.

So the trade found the right variable by finding what predicted things, which is what an empirical index is for. The model’s contribution is to say why that variable and not another, and to say what it does not cover.

What it does not cover #

The forces, in absolute terms. The tightness factor is a pure number and a force is not, so no ratio of lengths can produce one.

Two knits at the same tightness factor have identical shapes and forces differing by the ratio of their yarn’s bending rigidities over the squares of their loop lengths. That is the exact statement and it is asserted rather than described: the computed force ratio and the predicted one agree to fifteen figures across a cotton–wool pair and a cotton–polyester pair.

A cotton and a wool knit at the same K look the same, measure the same and press on themselves quite differently. Munden’s constants are a statement about a knit’s dimensions and were never a statement about what it feels like.

The constant per fibre #

The bridge between d/ℓ and K carries the fibre’s density and its packing factor, and it is worth writing out because it is where a fibre change enters.

A yarn’s diameter is the square root of four times its linear density over π times its fibre’s density and its packing factor. So d/ℓ is √tex/ℓ times √(4/(π ρ φ)), which for cotton at a packing of nought point six comes to a definite number and for wool to a different one.

Two fabrics at the same K in different fibres are therefore at slightly different d/ℓ, in proportion to the square root of the density ratio. Cotton at one point five two against wool at one point three one is a four per cent difference in the ratio — small, real, and the reason a tightness factor is quoted per fibre in careful specifications.

A defect found in this collection’s own arithmetic #

Testing the collapse turned up a mistake here rather than in the trade, and it is worth recording plainly.

The two shape constants used by this site’s occupancy calculation were written as a literal pair — one over four point one and one over five point five — and the two came from different rows of Munden’s table. The wale constant was the wet-relaxed one and the course constant the fully relaxed one.

That is precisely the defect this collection’s own essay on quoting a dimension without its state is about, committed in its own arithmetic, and it moved every occupancy by five per cent. The constants are read from a named state now. Nothing turned on the five per cent — the occupancy was over one before and is further over one after — but a number that is right for no fabric is worth correcting whatever it decides.

What the corrected numbers are #

A 20 tex cotton at a three-and-a-half millimetre loop fills 1.129 of its own cell rather than 1.077, and wetting takes it to 1.355 rather than 1.292. The proportionality constant between occupancy and tightness factor is 0.0884 rather than 0.0843, and the tightness at which a loop is exactly full is 11.3 rather than 11.9.

Every one of those is a five per cent move in the same direction, because they are five readings of one quantity.

Why it went unnoticed #

Because both numbers are plausible and the assertion checking them was checking the wrong thing. The site asserts that the occupancy is proportional to the tightness factor to twelve figures, and that assertion passes for any pair of constants — proportionality is a statement about the loop length and the count, and the shape constants only set the constant of proportionality.

That is a general lesson about assertions rather than about knitting. A check that verifies a relationship cannot verify the inputs to it, and an input that is wrong by five per cent will sail through a check that is exact to twelve figures.

What each quantity does with it #

Since everything is a function of one variable, the useful form is a table of exponents rather than a set of separate results.

The occupancy goes as K directly, linearly, with the constant computed above. The slack barely moves, changing by five points across the whole band. The peak curvature goes as K nearly linearly, crossing one yarn wrap at about thirteen. The contact force rises by two and a half over the band. The extensional modulus goes as the inverse cube of the loop length at fixed count, so it rises steeply. The geometric extension ceiling moves by two per cent and is effectively flat.

That last pair is the useful contrast for anybody specifying: tightness is an enormous lever on how hard a knit is to stretch and almost none at all on how far it goes.

The collapse, tested #

The claim that two knits at one tightness factor are one knit is asserted rather than asserted-and-hoped, and the test is arranged so that it could fail.

Two yarns are chosen, their diameters computed, and the loop lengths set so that d/ℓ matches. The shapes are then solved independently and their coefficients compared: agreement to fifteen figures. And the deliberate mismatch is fed in as well — two loops of the same yarn at different loop lengths, which are at different tightness factors — with the assertion that they agree, and that assertion has to fail. It does.

The same collapse elsewhere in this collection #

A one-variable geometry is not unique to knitting and it is worth naming its woven counterpart, because the two are structurally different in an instructive way.

That is the whole difference. A woven cloth has two systems and therefore two covers, two crimps, two setts and a closure condition tying them; a plain knit has one thread doing one thing and therefore one number. The woven side of this collection solves equations and the knitted side quotes a table for exactly that reason, and the table is short because there is only one variable to tabulate against.

What a knitter can do with it #

Two things, and the second is the one that changes a decision.

A specification collapses. A fabric asked for at a stated tightness factor is a fabric whose whole geometry is fixed, so quoting the count and the loop length separately says no more than quoting K. That is already the trade’s practice.

A fibre substitution does not. Swapping cotton for wool at the same K keeps the cover, the dimensions and the extension available, and changes the contact force, the modulus, the bending rigidity and everything that depends on friction. A substitution made on the geometry alone will be a fabric that measures identically and behaves differently, and this is the arithmetic that says by how much.

What a fibre substitution actually changes #

The collapse says two knits at one tightness factor have one shape and different forces, and the forces differ by the ratio of the yarns’ bending rigidities. That ratio is worth writing out in terms a buyer has, because it does not go the way anybody would guess.

At the free bound a spun yarn’s rigidity is the sum of its fibres’, so it is the fibre count times one fibre’s — and one fibre’s goes as its modulus times the fourth power of its diameter, while its diameter goes as the square root of its own linear density over its density. Multiplying through, the yarn count cancels once and what is left is

B ∝ (yarn tex) × E × (fibre tex) ÷ ρ².

A coarser fibre gives a stiffer yarn at the same yarn count, linearly, because a few fat fibres resist bending far better than many thin ones. That is the term nobody expects to dominate and it usually does.

A wool jersey at the same tightness factor as a cotton one presses about half again as hard, despite wool’s modulus being under half of cotton’s, because a wool fibre is three times coarser and the fineness enters linearly while the modulus does too. The two fabrics measure identically, cover identically and extend to the same place; the wool one is stiffer to get there.

Three consequences for a substitution made on geometry alone.

The dimensions are safe. Everything Munden’s constants govern is unchanged at constant K, so a fibre swap that holds the tightness factor holds the finished size, the cover and the extension available. That is what makes such substitutions attractive and it is the half that works.

Every force moves by the group above, which spans a factor of five across ordinary fibres — so the contact force, the modulus, the bending rigidity, the run resistance and the recovery force of a cuff all move together and by a great deal. A substitution that looks dimensionally neutral is a substantial change of handle.

And the fibre’s fineness is the lever, not its polymer. Two polyesters at the same modulus and a fivefold range of fibre fineness differ by five in yarn rigidity, which is far more than any fibre-to-fibre modulus difference in the table. A microfibre knit is limp because its fibres are fine and not because of what they are made of — which is the same statement, one level down, as this collection’s standing point that structure beats material.

The range the trade uses #

Thirteen to seventeen, and both ends turn out to have something under them.

At about thirteen the loop’s tightest bend reaches the curvature of one yarn wrapped hard round another, which is where the contact stops being a point and starts being an arc. Below it the fabric is looser than any wrap; above it, tighter.

At the top of the range the occupancy passes one and a half, which is a fabric with half again as much thread as its own footprint. Whether that is what stops a knitter going further is not established here — the machine’s own limits are certainly involved — and the correspondence is recorded rather than claimed.

What the picture cannot show #

Two loops at the same tightness factor, distinguished. They are the same drawing. A figure of the 12 tex cotton and one of the 30 tex wool are identical up to a scale factor, which is the point being made and also the reason there is nothing to look at.

Nor can any figure show the forces differing while the shapes do not. That is a fact about two numbers attached to one picture, and it has to be printed.

What breaks the collapse #

Three things take a fabric off the one-variable curve, and it is worth knowing which.

A different structure. A rib, an interlock, a tuck or a miss changes the topology, and the loop’s own arithmetic travels while the fabric’s does not. What a tuck costs is a statement about yarn accounting in a structure this ladder has not solved.

A finish that changes the loop length. Milling, felting and stentering all move the effective loop length in the fabric plane, which moves K itself — so a finished fabric is at a different tightness factor from the one it was knitted at, and specifying the knitted value says less than it appears to.

A yarn that is not round. The whole model enters the yarn through one diameter, which assumes a circular section. A flat or a heavily flattened yarn has a different effective diameter in the two directions and is not on this curve at all.

What was known before #

The tightness factor itself, since the middle of the last century, and its correlations with everything. Munden’s collapse of the dimensions, from 1959. The geometric argument for why the yarn count cancels out of the dimensions, offered afterwards as a rationalisation.

What appears to be new is the statement that the whole model has one variable — that the loop’s shape, not merely its dimensions, is a function of K alone — and the corollary that everything the trade correlates with K is correlated for one reason rather than several.

An index that is a ratio, and one that is a number #

There is a distinction worth drawing between the tightness factor and the other indices this collection handles, because it decides what each is good for.

K is dimensional as it is quoted — square root of tex per centimetre — and dimensionless only after the fibre’s density is divided out. So a tightness factor compared across two fibres is a comparison with a hidden constant in it, and the four per cent above is the size of that constant’s variation between cotton and wool.

A cover factor is the same shape of object on the woven side and has the same property: quoted as a sett times a root of a count, it carries the fibre’s density inside it too. Both work because the trade uses one fibre at a time and both mislead the moment two are compared.

The fix in both cases is the same and nobody wants it: quote the ratio of a diameter to a length, which is dimensionless and comparable, and lose the convenience of a number that can be read off a count and a machine setting. That trade-off is why neither trade has made the change in a century.

What a single variable does to a specification sheet #

A last practical consequence, and it cuts against the way knitted fabrics are usually described.

A specification that lists a count, a machine gauge, a stitch density, a fabric weight and a tightness factor is listing five numbers of which — for the geometry — one is free. The stitch density follows from the loop length; the loop length and the count give the tightness factor; the weight follows from the loop length, the count and the stitch density together.

That is not an argument for shortening the sheet, because the redundant numbers are the ones a machine can be set to and a bale can be checked against. It is an argument for knowing which of them is the one that decides, so that a fabric that fails a handle requirement is adjusted by the lever that moves handle rather than by one that moves nothing.

The lever is the loop length, and it is the yarn feed. Everything else on the sheet either follows from it or is a check on it.

Where the ladder goes next #

Out of the plain jersey. A rib folds before it extends, and the folding is a mechanism this ladder cannot compute — but the force it pulls back with is the loop’s own, and that is now available.