How to use this calculator
- Enter thread and leadscrew. Pitch in mm or TPI for each — the solver handles the 25.4 factor exactly when they differ.
- Read the train. Driver 1 goes on the spindle/stud side; a second pair appears only when compounding is needed. Idlers fill space without changing the ratio.
- Check the error output. Zero means exact. Anything else accumulates along the thread — judge it against your engagement length.
- Verify on the machine. Banjo clearance, gear mesh backlash and thread hand (each idler flips direction) are physical checks the arithmetic cannot make.
How it works
Per spindle revolution the work must advance one pitch; the leadscrew advances its own pitch per revolution of itself. The change gears bridge the two —
i = advance / P_leadscrew · i = (driver₁ × driver₂) / (driven₁ × driven₂) · 1 in = 127/5 mm exactly
Same-system jobs reduce to small fractions; cross-system jobs carry the prime 127. The solver does exact integer arithmetic on the reduced ratio, prefers the simplest exact train, and falls back to the closest approximation with its error stated. The thread itself comes from the thread pitch chart and tap drill calculator; the pass-by-pass depths from the threading infeed calculator; and the same worm-fraction thinking on the mill side lives in the dividing head indexing calculator.
Worked example
Verified against the live calculator
Cutting M1.5 on a classic 8 TPI leadscrew:
i = 1.5 × 8 / 25.4 = 60/127 → 60 driver : 127 driven, exact
One pair plus an idler — this is the 127 gear doing the only job it
exists for. Remove it (the no-127 set) and the best the solver can
assemble is 20:45 × 85:80 at −463 ppm:
each thread lands half a thousandth of a millimeter short, which a
ten-thread nut never notices and a 300 mm leadscrew certainly does.
Meanwhile the awkward-looking 11.5 TPI pipe thread
needs no special gear at all: 8/11.5 reduces to 16/23, and
80:115 cuts it exactly.
Frequently asked questions
How do you calculate change gears for thread cutting?
Ratio first: thread advance per spindle revolution ÷ leadscrew pitch. Then find gears whose driver ÷ driven (times a second pair if compounded) equals it exactly. M1.5 on an 8 TPI leadscrew needs 1.5 × 8 ÷ 25.4 = 60/127 — one 60-tooth driver into a 127-tooth driven, with any idler between.
Why do lathes use a 127-tooth gear for metric threads?
Because 1 inch = 25.4 mm exactly, and 25.4 = 127/5. Any metric-on-imperial ratio therefore carries a factor of 127 — a prime number, so no combination of smaller gears can replace it exactly. The 127 transposing gear makes metric conversion exact; everything else is an approximation.
How accurate are metric threads cut without a 127 gear?
From a plain every-5-teeth set, the best train for M1.5 on 8 TPI is 20:45 compounded 85:80 — 463 ppm short, about half a thousandth of pitch per thread. Fine for a nut that engages ten threads; wrong for a long leadscrew. Import lathes ship 63-tooth gears because 80/63 gets the same job to 125 ppm.
What about cutting 11.5 TPI pipe threads?
Fractional TPI is just another ratio: 8 ÷ 11.5 = 16/23, and 80:115 cuts it exactly from a by-5s set — no special gear needed. Enter any fractional TPI; the solver treats it as exact arithmetic, not a decimal approximation.
Method & assumptions
- Exact rational arithmetic on the ratio (decimals taken exact to six places; TPI through the exact 127/5 inch factor); the searched sets are generic every-5-teeth banks with one gear of each size, ±127.
- Real lathes ship different sets (many imports carry 57s and 63s; quick-change boxes cover the common pitches internally) — the required-ratio output is the universal number to check against your own bank.
- Single-start threads; multi-start leads enter the per-start pitch and index the starts separately.
- Banjo geometry, mesh clearance and thread hand (idler count) are machine-side checks; gear hobbing and helical-lead trains with machine constants are outside this screen.