The Sea in Brass

How many pulleys does a port need?

Kelvin's first machines in the 1870s summed eight constituents, then ten. The US Coast and Geodetic Survey's Tide-Predicting Machine No. 2, known as Old Brass Brains, summed 37 and made America's tide tables from 1912 until 1965. NOAA still lists 37 constituents for most ports. For Anchorage it lists 120.

So how many does a port really need? I added pulleys one at a time, largest first, which is the order a machine builder would choose, and after each one compared the machine with a year of the real sea: every hour of 2025 at each port. The error falls fast and then flattens. What it flattens onto is weather: wind, air pressure, river floods, none of which a pulley can predict. I count a port as having enough pulleys once the machine is within 1% of the best that every pulley together can do.

The Gulf ports, where the tide comes once a day, are done with 8 or 9: Kelvin's ten-pulley machine would have served them. Boston needs 17. Seattle and San Francisco need 24. Anchorage needs 90, two and a half Old Brass Brains. Its tide is nine metres of water pushed up the shallow Cook Inlet twice a day, and the shallows break each rhythm into overtones and combinations: quarter-day tides, sixth-day tides, all the way down to a twelfth of a day, each needing a pulley of its own. Choose Anchorage on the machine to see the wall of brass it takes.

Below, each port's error against the real sea (blue) settles onto its weather floor (the shaded band), while the error against the full machine (grey) keeps falling long after anyone could tell the difference at sea.

I guessed first, and guessed low

Before fetching any data I wrote down how many pulleys I thought each port would need to come within 5 cm of the full machine. I was right only for Pensacola. Everywhere else I was low, and at Anchorage I said 15 when the answer is 81.

port	my guess	answer
Boston	7	11
Eastport	9	13
San Francisco	6	10
Seattle	8	16
Anchorage	15	81
Pensacola	4	4
Honolulu	3	5
Woods Hole	6	7

I also guessed that the first pulley after M2 would be N2 or S2 at the two-tides-a-day ports, K1 at the mixed ones, and that Sa, the annual cycle, would make the top five at the Gulf ports. All of that held. At Key West Sa is the second-biggest pulley of all.

Three kinds of sea

A port's character is a contest between its pulleys. Where the half-day pulleys (M2, S2, N2) win, there are two nearly equal tides a day. Where the one-day pulleys (K1, O1) win, there is one. In between, the two tides of each day are unequal. Here is a month of each, drawn by the machine.

Boston's second-biggest pulley isn't S2, the sun, but N2, the moon's changing distance: 30.5 cm against 20.8. So Boston's biggest tides follow the moon's nearness more closely than full and new moon. The biggest of 2026 come on the days the moon is nearest, one to three days after a new or full moon.

Pensacola has almost no M2 at all (1.7 cm). Its two big pulleys, K1 and O1, are nearly equal (12.5 and 12.2 cm), and every 13.7 days, when the moon crosses the equator, they cancel and the tide almost stops. On 11 days of 2026 the range is under 10 cm; at its biggest it is 72 cm. Pensacola's third-biggest pulley is Sa, the annual cycle, at 11.3 cm. Sa is mostly not gravity at all but the seasons: the Gulf warms and its water expands each summer, and winds and rivers change.

Checking the brass

A tide machine is only as good as the angles its cranks are set to. Each crank's setting at any moment is a whole-number mix of five slow astronomical angles: the longitudes of the moon, the sun, the moon's perigee, the moon's node and the sun's perigee. On top of that each pulley gets a small correction, f for its radius and u for its angle, as the tilt of the moon's orbit swings over 18.6 years. I wrote all of this from my memory of Paul Schureman's Manual of Harmonic Analysis and Prediction of Tides (1958), whose conventions NOAA's predictions still follow, and then checked it against NOAA's own predictions: every six minutes of 2026, at all ten ports.

The first try came within about a millimetre at most ports. Before starting I'd written down that any error would probably be in M1, a minor pulley with a famously awkward convention, and M1 was where it was. Fitting the leftover difference at each pulley's own speed shows exactly which pulley is set wrong and by how much. Seven needed changing to match NOAA, all small or obscure, and most of them only in Anchorage's long list:

After those changes the machine and NOAA differ by 0.29 to 0.5 mm rms. NOAA rounds its predictions to the millimetre, which on its own makes 0.29 mm. The largest single difference anywhere in the year is 1.7 mm, at Anchorage, where the tide's range is nine metres.

Show the agreement port by port

port	pulleys	rms (mm)	worst (mm)	f, u drifting: worst (mm)	New Year step, 2026 (mm)	2027 (mm)
Boston	34	0.30	0.7	34	-16	+7
Eastport	34	0.32	0.9	62	-60	-1
Woods Hole	34	0.29	0.6	9	+1	-0
San Francisco	33	0.31	0.7	29	+3	-5
Seattle	34	0.38	1.0	63	-60	-78
Anchorage	120	0.50	1.7	164	-130	-63
Pensacola	30	0.29	0.6	12	+1	+5
Honolulu	31	0.29	0.6	10	-2	-11
Galveston	32	0.29	0.6	11	-0	+8
Key West	32	0.29	0.6	9	-4	-2

The predicted sea jumps at midnight on New Year's Day

One more thing fell out of the checking. The corrections f and u drift slowly all the time, with the moon's orbit. A brass machine couldn't follow that drift during a run, so its operator set each crank once for the whole year, using mid-year values. NOAA's predictions still do this. My machine matches them only when it does the same; letting f and u drift continuously, which is closer to the physics, misses NOAA by up to 16 cm at Anchorage.

That means a step. At 00:00 UTC on 1 January the settings change, and NOAA's predicted tide jumps: by 13 cm at Anchorage on 1 January 2026, 6 cm at Seattle and 6 cm at Eastport, under a centimetre at the Gulf ports. Here are NOAA's own six-minute predictions for Anchorage around that midnight, measured against the machine set for each year. Each line sits on zero, to the millimetre, on its own side of midnight.

No one at sea would notice. In a typical hour the weather moves Anchorage's water by 23 cm. But the step is there in the official numbers, left over from the brass machines.

The real sea had one more surprise for me. At nine of the ten ports, the water in 2025 stood 6 to 16 cm above the mean sea level the predictions are measured from, which is the average of 1983 to 2001. That gap is the sea's rise since then, plus any sinking of the land. The tenth is Anchorage, where NOAA measures mean sea level over 1997 to 2001 instead, and the water sat almost exactly on it.

The machines

William Thomson, later Lord Kelvin, designed the first tide-predicting machine in 1872, and Alexander Légé built it. The summing chain, running over and under the pulleys with a weight on its free end, was Beauchamp Tower's suggestion. A model for eight constituents was shown in 1873 and a machine for ten in 1876. One with twenty was built for India in 1879, and extended to twenty-four two years later. William Ferrel's American machine of 1882 read the times and heights of high and low water off dials. Its successor, Rollin Harris and E. G. Fischer's Tide-Predicting Machine No. 2, weighed 2,500 pounds and summed 37 constituents twice. One side summed the height. The other side, its cranks a quarter turn ahead, summed the height's rate of change, and an electric contact stopped the whole machine each time that sum reached zero, at every high and low water. Setting it up for a port took two and a half to four hours, and a year's predictions took eight to fifteen. In 1943 Arthur Doodson, at the Liverpool Tidal Institute, was sent harmonic constants for a place called only "Position Z" and asked for its tides through the spring of 1944. He predicted them on two machines, a Kelvin and a Roberts-Légé. Position Z was Normandy.

• Tide-predicting machine and Tide-Predicting Machine No. 2, Wikipedia
• Tide Predicting Machine No. 2, NOAA
• William Thomson's tide predicting machine, 1872, Science Museum Group
• D-Day: the starting point for modern UK oceanography, National Oceanography Centre