November 3, 2026 · Indumor
In this chapter
A few years ago a 1958 engine lathe came into the shop, 1.5 m between centers, with a 4 tpi lead screw and no quick-change gearbox: every pitch meant setting up the quadrant by hand. The first job was an M20×2.5, and the gear drawer had no 127-tooth gear in it. The oldest turner in the shop did not even look at the chart: he put the 50 and the 63 on as drivers, the 100 and the 80 as driven, cut the thread and the nut went on smoothly. The error of that train is 0.0125 %, a hundredth of a millimeter over 80 mm of thread. This chapter explains why it worked and how to work out the train for any pitch without guessing.
To cut a thread the carriage has to advance exactly one pitch per revolution of the work. That link is the lead screw, which takes its drive from the headstock through the quadrant (the banjo): a slotted plate carrying interchangeable gears. The ratio you need is always the same sum:
P: pitch, both in the same unit. A and C: driving gears (A on the stud that comes from the headstock). B and D: driven (D on the lead screw). The idler of the simple train does not enter the sum: it only sets the direction.
With three gears the idler only transmits and sets the direction; A and D fix the ratio. When the number cannot be made with two gears from the set, you build a compound train: B and C share a stud, and the ratios multiply. When the lead screw is in inches and the thread is metric, or the other way round, 25.4 enters the sum:
tpi: threads per inch. The 25.4 becomes the 127-tooth gear (127 = 5 × 25.4): in the denominator with an inch lead screw and in the numerator with a metric one.
| Workpiece pitch | 6 mm lead screw i | 4 tpi lead screw i | 8 tpi lead screw i |
|---|---|---|---|
| 1 mm | 1/6 | 20/127 | 40/127 |
| 1.5 mm | 1/4 | 30/127 | 60/127 |
| 2 mm | 1/3 | 40/127 | 80/127 |
| 2.5 mm | 5/12 | 50/127 | 100/127 |
| 3 mm | 1/2 | 60/127 | 120/127 |
| 8 tpi | 127/240 | 1/2 | 1 |
| 10 tpi | 127/300 | 2/5 | 4/5 |
| 12 tpi | 127/360 | 1/3 | 2/3 |
| 16 tpi | 127/480 | 1/4 | 1/2 |
| 20 tpi | 127/600 | 1/5 | 2/5 |
i = drivers / driven. With an inch lead screw the 127 shows up in the denominator of metric pitches; with a metric lead screw, in the numerator of threads per inch.
The classic set runs from 20 to 120 teeth in steps of 5, plus a 127-tooth gear. That gear converts exactly between inches and millimeters because 127 = 5 × 25.4: with it on the lead screw, a 1 mm pitch is 20/127 on a 4 tpi screw. When it is missing you replace it with a close fraction, and the best known is 80/63. The table lists the approximations with their error, which is so small it only matters on long precision lead screws.
| Train | Value | Stands in for | Error % | Comment |
|---|---|---|---|---|
| 127/100 | 1.27000 | 25.4/20 = 1.27 | 0.0000 | Exact: 127 = 5 × 25.4 |
| 80/63 | 1.26984 | 1.27 | -0.0125 | The usual one without the 127 |
| 47/37 | 1.27027 | 1.27 | 0.0213 | 47 and 37 tooth gears |
| 37/47 | 0.78723 | 100/127 = 0.7874 | -0.0213 | The inverse of the one above |
| 63/80 | 0.78750 | 100/127 = 0.7874 | 0.0125 | The inverse of 80/63 |
Typical set: 20, 25, 30 … 120 (in steps of 5) plus 127. An error of 0.0125 % is 0.01 mm over 80 mm of thread: any nut tolerance swallows it.
The table was computed with the set above, searching every combination for the smallest error and, among the exact ones, the simplest. Pitches from 1 to 6 mm come out with two gears on either lead screw; the fine ones (0.5 to 0.8) need a compound train. Before cutting, check that the gears clear each other and the plate: the 127 is big and sometimes has to go in position D rather than B.
| Pitch mm | 6 mm screw i | Train | Error % | 4 tpi screw i | Train | Error % |
|---|---|---|---|---|---|---|
| 0.50 | 1/12 | 20×25 / 75×80 | – | 10/127 | 20×25 / 50×127 | – |
| 0.60 | 1/10 | 20×25 / 50×100 | – | 12/127 | 20×30 / 50×127 | – |
| 0.70 | 7/60 | 20×35 / 75×80 | – | 14/127 | 20×35 / 50×127 | – |
| 0.75 | 1/8 | 20×25 / 50×80 | – | 15/127 | 20×30 / 40×127 | – |
| 0.80 | 2/15 | 20×25 / 50×75 | – | 16/127 | 20×40 / 50×127 | – |
| 1.00 | 1/6 | 20/120 | – | 20/127 | 20/127 | – |
| 1.25 | 5/24 | 25/120 | – | 25/127 | 25/127 | – |
| 1.50 | 1/4 | 20/80 | – | 30/127 | 30/127 | – |
| 1.75 | 7/24 | 35/120 | – | 35/127 | 35/127 | – |
| 2.00 | 1/3 | 20/60 | – | 40/127 | 40/127 | – |
| 2.50 | 5/12 | 25/60 | – | 50/127 | 50/127 | – |
| 3.00 | 1/2 | 20/40 | – | 60/127 | 60/127 | – |
| 3.50 | 7/12 | 35/60 | – | 70/127 | 70/127 | – |
| 4.00 | 2/3 | 20/30 | – | 80/127 | 80/127 | – |
| 4.50 | 3/4 | 30/40 | – | 90/127 | 90/127 | – |
| 5.00 | 5/6 | 25/30 | – | 100/127 | 100/127 | – |
| 6.00 | 1 | 20×50 / 25×40 | – | 120/127 | 120/127 | – |
Train a/b: A/D with any idler. Train a×c / b×d: A and C drivers, B and D driven. No error: exact train. For i = 1 (6 mm pitch on a 6 mm screw) two equal gears will do, 60/60.
With a 6 mm metric lead screw the 127 goes on as a driver. From 4 to 22 tpi every train is exact; from 24 up the set of distinct gears runs out and approximations appear.
| Thread tpi | Pitch mm | i | Train | Error % | Thread tpi | Pitch mm | i | Train | Error % |
|---|---|---|---|---|---|---|---|---|---|
| 4.0 | 6.350 | 127/120 | 127/120 | – | 16.0 | 1.588 | 127/480 | 20×127 / 80×120 | – |
| 5.0 | 5.080 | 127/150 | 20×127 / 50×60 | – | 18.0 | 1.411 | 127/540 | 20×127 / 90×120 | – |
| 6.0 | 4.233 | 127/180 | 20×127 / 45×80 | – | 19.0 | 1.337 | 127/570 | 20×127 / 95×120 | – |
| 7.0 | 3.629 | 127/210 | 20×127 / 60×70 | – | 20.0 | 1.270 | 127/600 | 20×127 / 100×120 | – |
| 8.0 | 3.175 | 127/240 | 20×127 / 60×80 | – | 22.0 | 1.155 | 127/660 | 20×127 / 110×120 | – |
| 9.0 | 2.822 | 127/270 | 20×127 / 60×90 | – | 24.0 | 1.058 | 127/720 | 20×30 / 40×85 | 0.0463 |
| 10.0 | 2.540 | 127/300 | 20×127 / 75×80 | – | 26.0 | 0.977 | 127/780 | 35×65 / 110×127 | 0.0175 |
| 11.0 | 2.309 | 127/330 | 20×127 / 60×110 | – | 27.0 | 0.941 | 127/810 | 20×30 / 45×85 | 0.0463 |
| 11.5 | 2.209 | 127/345 | 20×127 / 60×115 | – | 28.0 | 0.907 | 127/840 | 20×45 / 70×85 | 0.0463 |
| 12.0 | 2.117 | 127/360 | 20×127 / 80×90 | – | 32.0 | 0.794 | 127/960 | 25×50 / 90×105 | -0.0125 |
| 13.0 | 1.954 | 127/390 | 20×127 / 65×120 | – | 36.0 | 0.706 | 127/1080 | 20×25 / 50×85 | 0.0463 |
| 14.0 | 1.814 | 127/420 | 20×127 / 80×105 | – | 40.0 | 0.635 | 127/1200 | 20×25 / 45×105 | -0.0125 |
From 24 tpi up the set gives no exact train with distinct gears: use the gearbox, a duplicate gear (two 120s) or accept the error in the table, which is smaller than the thread tolerance itself.
The depth of a 60° external thread is 0.613 times the pitch, measured on the radius: the theoretical triangle of 0.866·P is never cut in full because the crest stays flat at P/8 and the root is rounded. The compound rest is swung to 29.5° (half the angle less half a degree) so the tool feeds in along the leading flank and the trailing flank only rubs: the chip comes off one side, the tool does not chatter and the finish is better. Feeding at an angle, the compound has to travel 15 % more than h to reach the root.
h: depth of the 60° external thread measured on the radius (ISO 965-1: d₃ = d − 1.2269·P). For an internal thread the depth is 0.541·P. Whitworth, 0.640·P; trapezoidal, 0.5·P plus the clearance.
| Pitch mm | h mm | Compound at 29.5° mm | Passes | Pitch mm | h mm | Compound at 29.5° mm | Passes |
|---|---|---|---|---|---|---|---|
| 0.50 | 0.31 | 0.35 | 4 | 2.50 | 1.53 | 1.76 | 9 |
| 0.75 | 0.46 | 0.53 | 4 | 3.00 | 1.84 | 2.11 | 10 |
| 1.00 | 0.61 | 0.70 | 5 | 3.50 | 2.15 | 2.47 | 12 |
| 1.25 | 0.77 | 0.88 | 6 | 4.00 | 2.45 | 2.82 | 13 |
| 1.50 | 0.92 | 1.06 | 6 | 5.00 | 3.07 | 3.52 | 15 |
| 1.75 | 1.07 | 1.23 | 7 | 6.00 | 3.68 | 4.23 | 17 |
| 2.00 | 1.23 | 1.41 | 8 |
Compound at 29.5°: compound travel needed to reach h (h / cos 29.5°). Passes for carbon steel with a carbide tool; decreasing, the last two 0.03 to 0.05 mm.
| Thread tpi | Pitch mm | h mm | h thou | Compound at 29.5° mm | Passes |
|---|---|---|---|---|---|
| 32 | 0.794 | 0.49 | 19.2 | 0.56 | 4 |
| 24 | 1.058 | 0.65 | 25.6 | 0.75 | 5 |
| 20 | 1.270 | 0.78 | 30.7 | 0.90 | 6 |
| 18 | 1.411 | 0.87 | 34.1 | 0.99 | 6 |
| 16 | 1.588 | 0.97 | 38.3 | 1.12 | 7 |
| 14 | 1.814 | 1.11 | 43.8 | 1.28 | 7 |
| 13 | 1.954 | 1.20 | 47.2 | 1.38 | 8 |
| 12 | 2.117 | 1.30 | 51.1 | 1.49 | 8 |
| 11 | 2.309 | 1.42 | 55.8 | 1.63 | 9 |
| 10 | 2.540 | 1.56 | 61.3 | 1.79 | 9 |
| 8 | 3.175 | 1.95 | 76.7 | 2.24 | 11 |
| 6 | 4.233 | 2.60 | 102.2 | 2.98 | 13 |
| 4 | 6.350 | 3.90 | 153.4 | 4.48 | 18 |
| Thread | Included angle | Compound | Remark |
|---|---|---|---|
| ISO metric, UNC and UNF (60°) | 60° | 29° to 30° (29.5° is usual) | The leading flank cuts; the trailing one only grazes and finishes |
| Whitworth and BSP (55°) | 55° | 27° a 27.5° | 55° tool, rounded crests |
| Trapezoidal Tr (30°) | 30° | 14.5° | Rough with a narrow tool, finish with the full form |
| ACME (29°) | 29° | 14.5° | Same as Tr; exactly half the angle |
| Internal thread (any form) | – | The same angle. swung the other way | The compound feeds toward the operator |
| Pitches under 1 mm, full-profile inserts | – | 0° (straight plunge) | Fewer passes, chips from both flanks |
The number of passes is indicative: it depends on the material, the rigidity of the setup and the tool. What does not change is the rule of shrinking them: a first pass of 0.3 mm, last ones of 0.03, and one or two spring passes at the end. Spindle speed comes from the charts in chapter 9, at a half or a third of the turning speed.
After every pass you return to the start and close the half nuts on the same thread. The thread dial, a marked disc driven by a small worm gear riding on the lead screw, tells you when to engage. The rules depend on the lead screw and the pitch, and there is one case where the dial is useless: an inch lead screw and a metric thread (or the reverse) share no line, so the half nuts stay closed the whole time and you bring the carriage back by reversing the machine.
| Case | Where to engage |
|---|---|
| Inch lead screw, even tpi (4, 6, 8, 10…) | On any line of the dial |
| Inch lead screw, odd tpi (3, 5, 7, 9, 11…) | On any numbered line (1, 2, 3 or 4) |
| Inch lead screw, half threads (4.5 or 11.5) | Always on the same two lines: 1 and 3, or 2 and 4 |
| Inch lead screw, quarter threads (4.75 or 5.25) | Always on the same line |
| Metric on an inch lead screw (with or without the 127) | Never: keep the half nuts closed and reverse the machine to come back |
| Metric lead screw, pitch listed on the dial table of the lathe | On the line the table gives (the metric dial changes gear with the pitch) |
| Metric lead screw, threads per inch | Never: half nuts closed and reverse |
Rules for the classic dial, with four numbered lines and four in between. Another lead screw or another dial gear changes them: on some lathes odd threads go on every other number. If the lathe carries its own chart, that chart rules.
For a pitch not in the tables, or for a different gear set, the calculator tries every simple and compound combination and sorts them by error. The threads cut with these trains are in chapters 4 and 7; the dividing head in chapter 12 uses the same idea of a gear train for differential indexing.
Searches every simple (a/b) and compound (a·c / b·d) combination with distinct gears, sorts them by error and shows the five best. On a lathe with a gearbox, the box ratio multiplies the quadrant ratio.
That old turner did one more thing before cutting: a dry pass with the tool clear of the work, then a barely visible scratch pass that he checked with the pitch gauge. That pass costs nothing, and it is the one that tells you the quadrant is set up wrong before the part is ruined.
Worked example · an M20×2.5 on a lathe with a 4 tpi lead screw
Ratio: i = 2.5 × 4 / 25.4 = 10 / 25.4 = 50 / 127. Simple train: A = 50 driver, D = 127 on the lead screw, with any idler. Without the 127: 50 × 63 / (100 × 80) = 0.39375 against 0.393701, an error of +0.0125 %, which over 80 mm of thread is 0.01 mm.
For exact conversion between inches and millimeters: 127 = 5 × 25.4. On an inch lead screw it goes in as a driven gear to cut metric pitches; on a metric lead screw it goes in as a driver to cut threads per inch. Without it, 80/63 is used, with an error of 0.0125 %.
From the ratio i = workpiece pitch / lead screw pitch, in the same unit. That fraction is made with two gears (i = A/D) or with four (i = A·C / B·D). For 2.5 mm on a 4 tpi lead screw: i = 2.5 × 4 / 25.4 = 50/127.
0.613 times the pitch, measured on the radius: an M12×1.75 is threaded 1.07 mm deep. With the compound at 29.5° the compound travel is 1.15 times more, 1.23 mm.
Sources: ISO 68-1 (basic profile of the metric thread); ISO 965-1 (d₃ = d − 1.2269·P); ASME B1.1 (unified inch threads); DIN 13; operating manuals of engine lathes with and without quick-change gearboxes (no brand).
We cut metric, inch and trapezoidal threads on the lathe, checked with nut and gauge, in our shop in Córdoba, Argentina.
© Indumor 2010–2026 — Industrial steel fabrication, Córdoba, Argentina
