Twill direction, and how it is named

Denim has a diagonal on it. So does gabardine, so does the lining of a jacket, so does the twill tape on the inside of a collar. It runs one way rather than the other, it has a name, and the name turns out to be attached to something a weaver can choose and a matrix cannot see.

The convention is straightforward once stated. A twill whose line climbs from lower left to upper right is a Z twill, or a right-hand twill; one climbing from lower right to upper left is an S twill, or left-hand. The letters name the direction of the letter’s own diagonal stroke, which is the one mnemonic in this subject that survives being said out loud rather than drawn.

Denim is traditionally a right-hand twill. Left-hand denim exists, is softer, and is a deliberate choice made for a reason this essay works its way round to.

Nothing in the matrix knows #

Start with the negative result, because it sets the whole shape of the essay.

The two twills are related by a reflection of the draft. Reflect a binary matrix left to right and float lengths are unchanged, because a run of ones is still a run of ones read backwards. Interlacings are unchanged, because a change of face at one intersection is still a change of face. The connectivity digraph is relabelled and not otherwise altered, so the layer count is unchanged.

Every quantity this site computes from a weave is a function of the matrix that is invariant under reflection. Therefore no quantity this site computes distinguishes a Z twill from an S twill, and the figures print the counts side by side to make the point rather than to make a comparison.

That is a stronger statement than it looks. It says the difference is real in the cloth, visible to anyone, and entirely outside the model — the first properly clean case on this site of a property that is not in the draft at all. Colour is another, and so is anything to do with yarn. Handedness joins the list.

Then what does the direction do? #

Two things, and they are unrelated to each other.

The first is a matter of appearance and it involves the yarn. A spun yarn is twisted, and the twist has a direction of its own: S twist if the surface fibres lean the way the middle stroke of an S leans, Z twist if they lean the other way. Most commercial ring-spun yarn is Z-twisted, which is a fact about machinery rather than about fibre.

Now put the two handednesses together. The surface fibres of the yarn lie along a helix; the twill line lies along its own direction across the cloth. When the two run the same way, the light catches the fibres and the diagonal together and the line is subdued — the twill reads soft, the surface looks smooth. When they run opposite ways, the fibre helix cuts across the twill line and the diagonal is sharpened; the twill reads prominent and the cloth looks crisper.

With Z-twist yarn, therefore, an S twill gives the prominent line and a Z twill gives the quiet one. That is a genuine effect, it is what a designer is choosing when they specify a hand, and it involves no arithmetic on the draft whatever.

The second thing the direction does is affect how the cloth behaves under twisting and washing, because the twill line and the yarn twist together give the fabric a slight torsional bias. Left-hand denim is softer partly because the twill runs with the yarn twist rather than against it, so the yarns are less inclined to stand proud of the surface and the fabric fulls more readily in finishing.

The angle, which is rarely what it is called #

Here is where the arithmetic returns, and it corrects a very widespread piece of shorthand.

A twill is described as running at forty-five degrees. It usually does not.

The twill line advances a fixed number of ends for each pick. Call that the step — one, in almost every twill. Over one pick the line climbs a distance $1/n_2$, where $n_2$ is the picks per centimetre; over that same pick it moves $s/n_1$ across, where $n_1$ is the ends per centimetre. So the angle the line makes with the horizontal satisfies

$$\tan\theta = \frac{n_1}{s\,n_2}$$

and the familiar forty-five degrees requires $s\,n_2 = n_1$: a step of one end in a cloth set with as many picks as ends.

Denim is not set that way. A typical denim has a warp sett roughly twice its weft sett, because the warp is the system that shows and the system that has to survive wear. Put $n_1 = 2n_2$ and $s = 1$ into the formula and the twill line lies at sixty-three degrees, not forty-five.

The trade knows this and has names for it. A twill at more than forty-five degrees is a steep twill; below, a reclining twill. What the trade names as a construction is in fact a consequence: a steep twill is not a different weave, it is the same weave in a denser warp. Whipcord and cavalry twill are steep because they are warp-faced and warp-dense, and it is possible to steepen a twill further by stepping two ends per pick instead of one, which is a genuine change of draft.

This matters because the angle is one of the first things anyone measures on a cloth and it is routinely read as evidence about the weave. It is evidence about the weave and the setting together, and quoting it without the setting says nothing.

Naming the whole thing #

A twill is fully specified by three pieces of information and it is worth listing them because sources usually give two.

The sequence — how many picks the warp is on the face for and how many it is under, in order. Written 2/2, 3/1, 1/3, or in longer cases 2/1/1/2 and so on.

The step, which is nearly always one end per pick and is occasionally larger. A step of two on a repeat of eight gives a steeper line and, if it shares a factor with the repeat, something other than a twill entirely.

The hand, Z or S.

Three of those and the draft is determined. Note what is not in the list: the number of shafts, which is the repeat’s length and follows from the sequence; the angle, which needs the setting; and the balance, which also follows from the sequence.

The census is worth a moment. On a repeat of four there are exactly three distinct twills. On eight there are twenty-one. The numbers are small enough that a designer could work through them, and the fact that the trade uses perhaps a dozen of them repeatedly is not a failure of imagination — most of the rest are unbalanced in ways with no use, or are so nearly the same as a neighbour that the cloth cannot tell.

Why the step is nearly always one #

The step was listed above as a free parameter, and it is worth showing why it is so rarely used, because the reason is the satin theorem in a different suit.

A twill line stepping $s$ ends per pick visits the ends in the order $0, s, 2s, \dots$ modulo the repeat $n$. If $s$ and $n$ share a factor, the line does not visit every end. It visits a fraction of them, comes back to where it began early, and the repeat collapses.

What the collapse looks like depends on the sequence. A two-up-two-down twill stepped two ends per pick produces a draft with two pairs of identical adjacent ends — the picture is a rib rather than a diagonal, and two ends doing the same thing read in the cloth as one thick thread.

A three-up-one-down twill stepped two ends per pick does something worse, and it is worth putting on the page.

Two of the four ends are never reached by the line at all. One of them ends up on the face at every pick, which is a solid column on the point paper and a thread lying loose on the surface in the cloth — the crudest of the integrity failures, and the only one with a visible signature.

So the rule is the same rule as for satins: the step must be coprime with the repeat. On a repeat of four the usable steps are one and three, and three is just one in the other direction. On eight they are one, three, five and seven, which is why steep twills on eight shafts exist and steep twills on four do not.

How a loom is told which way #

The abstraction so far has been the matrix. It is worth one section on how a hand loom is actually told to weave a twill, because the two ways of reversing the hand are not equivalent on the loom even though they are equivalent on paper.

A shaft loom carries the warp ends on a set of frames. The threading says which shaft each end is drawn through; the lifting plan says which shafts rise on each pick. A twill on four shafts is normally threaded straight — end one on shaft one, end two on shaft two, and round again — and lifted in a rotating sequence.

Reversing the hand can be done either way. Reverse the lifting plan and the threading stays straight; reverse the threading and the lifting plan stays as it was. On paper the two produce mirror-image drafts and nothing else. On the loom they are different jobs: changing the lifting plan is a matter of reading the pedal tie-up backwards, and changing the threading means re-drawing several hundred ends through different heddles.

Which explains a small piece of workshop lore that otherwise looks arbitrary. A weaver who wants both hands in one cloth — a herringbone — reverses the threading, because the lifting plan has to serve both halves at once. A weaver who wants one hand or the other in separate cloths reverses the lifting plan, because it costs nothing. The choice is about the machine and not about the fabric, and the fabric cannot tell.

What the eye does with a diagonal #

The direction of a twill is one of the clearest cases of the site’s recurring theme: the properties easy to see are not the properties that need computing.

Handedness is visible instantly and to everybody. Nobody needs a test for it, and no test taken from the matrix can supply one. Meanwhile the layer count is invisible and needs a computation, and the float is nearly invisible and decides most of what the cloth does.

There is a further trap specific to the diagonal, and it has its own essay: the diagonal is not a thread. Nothing runs along the twill line. No yarn in a woven cloth is anything but straight-along-the-warp or straight-along-the-weft, and the line the eye follows is a sequence of separate floats belonging to different threads, aligned by the step. A reader who has followed that argument will notice that reversing the hand does not reverse anything about the yarns; it reverses the direction in which the floats are stacked.

The section, which is also unchanged #

One more invariance, because it is the one people find hardest to accept.

A cross-section taken along a warp end shows that end passing over three picks and under one, and it does that in a Z twill and in an S twill alike. Cut the cloth the other way and the same is true. The crimp is the same, the cover is the same, the thickness is the same, and a laboratory analysis that reported them would be unable to say which cloth it had.

The only physical difference is which neighbouring end is a step ahead and which a step behind, and that is a statement about the arrangement of ends rather than about any of them.

Where the letters came from #

The S and Z convention is not a weaving invention. It comes from spinning, where it names the direction of twist in a yarn, and it was borrowed for the twill line afterwards because the two things need to be compared and a shared vocabulary makes the comparison sayable.

That order of borrowing shows in the awkwardness of the fit. The letters describe the direction of a helix seen on a yarn held vertically, which is unambiguous; applied to a line on a flat cloth they need a further convention about which way up the cloth is being held, and the trade supplies it by fixing that the warp runs vertically. Turn a piece of twill through a right angle and its hand appears to reverse, which is not a paradox but a reminder that the naming has a hidden argument in it.

The older names, right-hand and left-hand twill, carry the same hidden argument and are more common in the English-language trade. Continental sources tend to prefer S and Z, and the two vocabularies coincide: right-hand is Z, left-hand is S. Where they do not coincide is in what “right-hand” was originally about — it referred to the direction the weaver’s hand moved when drawing the pattern, which is a piece of workshop history rather than a description of anything in the cloth.

Reading the setting off the angle #

The angle formula was used above to correct a piece of shorthand. It can also be run backwards, and run backwards it is the fastest measurement in this collection: it gives the ratio of the two setts off a photograph, without counting a single thread.

Rearranged, the sett ratio is

ends per centimetre ÷ picks per centimetre = step × tan θ.

A protractor on a scanned swatch and the step read off the draft, and the setting follows. On a denim at sixty-three degrees with a step of one, tan 63.4° is 2.00 and the cloth is set two ends to one pick. Nothing has been counted and nothing has been unravelled.

What that measurement is worth depends on where on the scale it lands, and the dependence is severe enough to change when the method is usable. Differentiating the angle with respect to the ratio gives a sensitivity of r/(1 + r²) radians per proportional change in r, which is largest at a square set and falls away on either side of it:

So an angle read to one degree gives the ratio to about three and a half per cent on a square-set cloth and to seven and a half per cent on a whipcord. The steeper the twill, the worse the angle is at reporting how it was set — which is unfortunate, because a steep twill is exactly the cloth whose setting somebody wants to know and whose threads are hardest to count.

What finishing does to the angle #

One more thing follows from the same formula, and it decides whether the measurement above is being taken on the cloth the loom made.

Both setts move in finishing, because relaxation shrinkage changes both. Shrinkage in length packs the picks closer, raising the picks per centimetre and making the line shallower. Shrinkage in width packs the ends closer, raising the ends per centimetre and making it steeper. The two act in opposite directions on one number, which is why the angle is far more stable than either sett.

Put ordinary figures in. A denim relaxing six per cent in length and three per cent in width has its picks per centimetre raised by a factor 1/0.94 and its ends per centimetre by 1/0.97, so the ratio falls by 0.97/0.94 — three per cent. Against the table above, three per cent of ratio at r = 2 is seven tenths of a degree.

That is the useful result, and it is a stronger statement than the sensitivity table on its own:

a twill’s angle is a loom-state quantity that survives finishing to under a degree, because the two shrinkages that would move it move it opposite ways and very nearly cancel.

So the angle can be quoted on a finished cloth and compared with a loom-state specification, which is not true of the sett, the thread count, the cover or the crimp — every one of which is a different number before and after finishing, often by ten per cent or more. Among the quantities that can be read straight off a cloth, this one is unusually honest about the cloth it came from.

The cancellation is not exact and it fails in one direction that is worth naming. A cloth finished with a deliberate width change — tentered wide, or over-fed and allowed to shrink hard — moves one sett without the other, and there the angle moves the full amount. A twill that arrives visibly steeper than its specification has usually been pulled narrow on the stenter, and the angle is the cheapest evidence of it.

Where the ladder goes next #

The natural next rung is what happens when a twill is made to change direction partway across: broken and herringbone twills, where reversing the hand produces a figured cloth and, in one of the two ways of doing it, a defect that no drawing announces.

The natural companion is the float, which is the quantity a twill’s sequence actually sets, and behind it the interlacings, which the sequence sets as well.

And for the property that the direction most resembles — real, visible, and outside the matrix — the case to compare is colour and weave, where a threading order the draft knows nothing about produces the pattern everybody sees.