Twist Multiplier — English System

Twist holds a staple yarn together. Fibres in a spun yarn are only a few centimetres long, so nothing but friction stops them sliding past one another, and twist is what generates the transverse pressure that creates that friction. But how much twist is right depends on how thick the yarn is, and that is what the twist multiplier is for.

The reasoning is geometric. What the fibres actually care about is the angle they make with the yarn axis — the helix angle at the surface. Unroll one turn of the outermost fibre and it is the hypotenuse of a right triangle whose base is the circumference $\pi d$ and whose height is the axial advance of one turn, $1/T$. So $\tan\theta = \pi d T$. Hold the angle constant and $T$ must go as $1/d$. And since a yarn's diameter goes as the square root of its linear density, $1/d$ goes as the square root of the indirect count. Hence $T \propto \sqrt{N_e}$, and the ratio $TM = T/\sqrt{N_e}$ is a number that describes the yarn's character independently of its fineness: about 3 for a soft hosiery yarn, 4 for weft, 4.5 to 5 for warp, 6 and above for a hard voile or crepe.

On the attribution, which is worth being careful about. The square-root twist law is conventionally attributed to Koechlin (1828), though the papers repeating this do not cite a primary source and the original has not been located — including in peer-reviewed work, where the claim is passed along without a reference of any kind and the given name "Daniel Koechlin" is unsupported. A verifiable early statement in print is Ashenhurst's: T. R. Ashenhurst, A Treatise on Textile Calculations and the Structure of Fabrics, 1st ed., Bradford, 1884, Rule (14). If you need to cite the law, cite Ashenhurst.

The multiplier is bound to its units twice over, and this is the trap. $TM$ assumes turns per inch and it assumes the English cotton count. Change the length unit alone and the number is meaningless, because the count in the denominator has to change too. The metric factor $\alpha_m = T_{\text{m}}/\sqrt{N_m}$ is bound to the metre and to $N_m$; the tex factor $\alpha_{\mathrm{tex}} = T_{\text{cm}}\sqrt{\mathrm{tex}}$ has genuine dimensions of $\mathrm{tex}^{1/2}\,\mathrm{cm}^{-1}$, which no unit engine can carry. The three are related by ratios that fall straight out of the count constants and are not measurements at all:

\[ \alpha_{\mathrm{tex}} = 9.567\,344 \times TM, \qquad \alpha_m = 30.254\,599 \times TM, \qquad \frac{\alpha_m}{\alpha_{\mathrm{tex}}} = \sqrt{10} \ \text{exactly}. \]

The 9.567 344 is $\sqrt{590.5412474}/2.54$ — an inch turned into a centimetre and an $N_e$ turned into a tex — and it will reappear, unchanged, on the cover-factor page for exactly the same reason.

Twist is not free. Yarn strength rises with twist to a maximum and then falls, because beyond that point the fibres are inclined so far from the axis that each contributes only $\cos\theta$ of its tenacity along it. For cotton the maximum sits somewhere near 25 to 30° of surface angle, which is where a warp multiplier lands. Past it a yarn gets lean, hard, snarly and weaker. Direction matters too and is a separate specification: S or Z. Weaving a Z-twist warp against an S-twist weft changes how the twill line catches the light, and plying reverses the twist of the singles by convention so the ply is balanced and does not snarl.

Measure twist to ISO 2061:2015 — not ISO 2060, which is the linear-density skein method. The two numbers are one digit apart and get swapped constantly, including in otherwise careful specifications. ASTM D1422 covers the untwist-retwist method and D1423 direct counting.