Warp or Weft Cover Factor from Setting and Count

One thread system's cover factor is threads per inch divided by the square root of the English cotton count. It is worth knowing that this is not a definition someone chose but a ratio that fell out of geometry: it is the thread diameter divided by the thread spacing, with the diameter replaced by the old rule $d = 1/(28\sqrt{N_e})$ inches. Fractional cover is $d/p = n/(28\sqrt{N_e}) = K_1/28$, which is why the cover factor runs from 0 to 28 and why 28 means the cloth is full. The full derivation, and the inclusion–exclusion argument that combines warp with weft, is on the cloth cover factor page.

The square root is doing real work and it repays a moment's thought. Take a cloth in $N_e$ 20 and reweave it in $N_e$ 40 — half the linear density, a yarn half the mass per unit length. How much closer must the sett be to keep the same appearance? Not twice: $\sqrt{2}$ times, about 41% more ends, because the finer yarn is thinner by $\sqrt{2}$ rather than by 2. It is the same square root that governs twist, appearing for the same reason — area goes as diameter squared.

Warp and weft are not symmetric in practice, even though the equation treats them alike. Ends per inch cost money once, at warping. Picks per inch cost money every single pick, because each one is an insertion and insertions are what a loom is paid for. So a designer who can reach a target cover with more ends and fewer picks has made the cloth cheaper without changing how it looks, and that is exactly why so many shirtings carry noticeably more ends than picks. The counterweight is that the warp sett is limited by the reed — a whole number of ends per dent, on a reed with a fixed dents-per-inch — and by jamming, where a warp set too close will not open a clean shed and end breaks climb steeply.

The tex-system equivalent, for anyone working from a metric specification, is $K_{\mathrm{tex}} = (\text{threads per cm})\sqrt{\mathrm{tex}}\,/\,10$, which reaches complete cover at 26.7. Convert between the two with $K_{\mathrm{tex}} = 0.9567\,K_{\text{english}}$ — not with the 26.7, which is a ceiling and not a conversion factor. That confusion is worth naming because it is so widespread, and because a 4.3% error in cover propagates into everything designed from it.