The family, by its coding
The Turk's head tube carries three codings, and the names matter (the app fixed its own labels in 1.0.9): an ordinary head codes over-1-under-1 along the cord; a Gaucho codes over 2 under 2 along the cord — the doubled-travel families; the herringbone is the column coding, a different animal that needs 2 × bights to be a multiple of its pattern or the pattern breaks at the seam.
What doubling does to the math
| the quantity | single pass | each doubling |
|---|---|---|
| the cords to tie | gcd(bights, leads) — one cord when they share no factor | the same cord continues: a doubling re-traces, it does not add cords |
| the crossings | (leads − 1) × bights | × the number of passes: a 2-pass knot runs double, a 3-pass runs triple |
| the coverage | passes × bights × cord ÷ (circumference × cos angle) | rides up with the passes — the mandrel fills fast at the 95% aim |
A worked doubling
The base: a 5-lead × 4-bight tube on a 25 mm mandrel. gcd(5, 4) = 1, one cord, tied as one — the base that takes the re-trace. Per pass the crossings are (5 − 1) × 4 = 16. Tie the ordinary (over-1-under-1) first: 16 crossings laid exactly once, the run list's way. Then the doubling: re-trace the same tube one pass over, coding over 2 under 2 along the cord — the tube now carries 32 crossings, and the coverage doubles. A second re-trace takes it to 48 and nears the mandrel's fill.
- The look: over-2-under-2 turns the plain diamond grid into the doubled weave a Gaucho reader expects — same geometry, doubled travel.
- The cord: the original cord keeps going; the working tails are the only addition. Cut from the single-pass estimate plus the re-trace's travel — the coverage estimate's passes field does it.
Mistakes worth avoiding
| the mistake | what it costs |
|---|---|
| calling the column weave a Gaucho | the column coding is the herringbone; the app's own labels were wrong until 1.0.9 — the sources settled it, the row coding is the Gaucho |
| re-tracing a base that ties as several cords | gcd > 1 ties the base in parts (a 9 × 6 knot is 3 cords); the re-trace then rides those parts unevenly — start from a one-cord base, gcd(b, l) = 1 |
| doubling past the mandrel's fill | each pass multiplies the coverage; past the 95% aim the tube refuses the next pass — count the coverage before counting the passes |
| tying the base to a different tension than the re-trace | the passes fight for the crossings; even tension per pass, re-snugged in order, is the whole craft here |
The app helps
The Grid draws the knot from bights × leads, switches the coding row or column, and prints the run list so every crossing is coded exactly once, doubled knots included; its rim rows can be locked over-1-under-1 while the body carries the family's coding. The bench's calculator carries the cord count, the crossings and the coverage with the passes field for the doublings.
Sources: the coding names and their 1.0.9 correction from the knot-grid sources, the gcd and the crossings rules from the app's unit-tested math (the same formulas the turk's head guide carries), the coverage estimate the app's own convention with its 95% aim.