Seventeen and a Half Turns

A pocket watch of C1’s kind is wound by turning its crown, and a full wind takes seventeen and a half turns. That number is not a setting in Tick Again; it falls out of three tooth counts and the length of a spring, and so does everything else about how the watch runs from full to stopped. This article follows the power from the crown into the spring and out again.

Crown to ratchet

Turning the crown turns the winding pinion on the stem, which has 16 teeth. That drives the crown wheel, 30 teeth, lying flat on the barrel bridge; the crown wheel turns the ratchet wheel, 40 teeth, which sits square on the barrel arbor and winds the spring. The crown wheel is only an idler: it changes the direction of the drive and passes the teeth along, so it drops out of the ratio. One turn of the crown is 16 over 40, four tenths of a turn of the ratchet.

let room = capacity - wound
guard room > 1e-9 else { return .full }
let ratchet = min(turns * w.ratchetPerCrownTurn, room)
let before = ratchetTurns
ratchetTurns += ratchet
crownTurns += ratchet / w.ratchetPerCrownTurn
wound += ratchet
let teeth = Double(w.ratchet.count)
return .wound(clicks: Int((ratchetTurns * teeth).rounded(.down) - (before * teeth).rounded(.down)))

The click, a small sprung pawl, rides over the ratchet’s teeth and stops it turning back, and every tooth it drops over is one click you hear and feel: four tenths of 40, so 16 clicks a turn of the crown. Turn the crown the other way and the model lets the drive slip at the winding pinion, ticking over its sixteen teeth without storing anything, as the keyless works of a real watch slip when the crown is turned back.

Two numbers there are estimates waiting on a watchmaker’s review: the winding pinion’s 16 teeth, and the seven turns the mainspring takes from let down to fully wound, which is an estimate for pocket mainsprings of the period. Everything else in this article is arithmetic from them.

Forty-two hours

Running, the spring lets down through the barrel, which the first article showed turning once every six hours. Seven turns at a sixth of a turn an hour is a power reserve of 42 hours. But a spring does not push equally hard all the way down. Fully wound it pulls hardest; through most of its run its pull falls slowly; near the end it falls away fast. The model writes that as a curve, and the swing the balance settles to follows it:

public static func springTorque(_ fraction: Double) -> Double {
    0.62 + 0.38 * pow(min(max(fraction, 0), 1), 0.3)
}

The shape is the honest part and the exact numbers are not: the curve is an estimate of a period mainspring and is marked for review. With it, C1 swings 270 degrees fully wound, about 251 at half wind, about 219 with a tenth left, and then falls off a cliff in the last few percent, where a short smooth tail takes the torque to nothing.

public var isRunning: Bool { isComplete && wound > 0 && amplitude >= caliber.escapement.liftAngle / 2 && !isHacked }

The swing does not jump to its new figure either. When a watch is wound, its amplitude rises towards the settled value with a time constant of 1.2 seconds, and when the drive is taken away it decays over 4, so a watch that is let down slows to a stop over a few seconds, the way a real balance does.

A short swing gains

The figure above reads one more number: the gain from the short swing. A lever watch’s balance is not perfectly isochronous, and in the model a short swing runs fast: it adds 0.08 seconds a day for every degree the swing falls short of 270, so a watch at 220 degrees gains four seconds a day more than it did fully wound. It is a small effect, and it is there because watchmakers regulate a watch at the amplitude it actually runs at, and a game about regulation should make that matter. The last article puts it together with everything else that moves a watch’s rate.

Catching up a shelf

In the game, finished watches hang on a shelf in the workshop and keep running in real time on their real reserve, which means a watch you wound on Monday has stopped by Wednesday. The model has to work out what happened while the app was closed, and stepping 18,000 beats an hour through a month would take a while. So it doesn’t: once the spring can no longer restart the balance and the balance is still, the rest of the absence is taken in a single step. A test simulates thirty days of a fully wound watch and requires it to finish in under half a second, stopped with turns left in the spring after a run of 42 hours, give or take an hour and a half.

What the tests hold

• A full wind is 17.5 turns of the crown and seven of the barrel arbor, and the next turn reports the spring full.
• The click drops sixteen times a crown turn, and turning back slips without storing anything.
• Fully wound, C1 is still running at 41 hours with an hour of reserve left, and stops by 43 having run 41.75 hours, give or take a quarter, with the seconds hand having made exactly one turn a minute all the way.
• The swing at half wind is below 270 and above 230, lower again with a twentieth left, and zero at empty.

Winding is the one thing you can do to a watch without opening it. The next article is about opening it, and the order in which a movement will let you take it apart.