Reading the Timegrapher

A watchmaker does not judge a watch by looking at its hands against a clock for a day. A timegrapher does it in seconds. The watch sits on a microphone stand, the machine hears every tick, and it compares each one against a perfect clock running at the watch’s nominal beat. The result is drawn as a trace of dots, one per beat, and a trained eye reads three things from it at a glance. Tick Again puts one on the bench, because adjusting a watch against its trace is half of the real work.

Rate is a slope

If the watch beats exactly on time, every beat lands where the perfect clock expects it, and the dots make a flat line. If the watch runs fast, each beat comes a little earlier than expected, the error grows with every beat, and the dots climb. A slow watch’s line falls. The slope is the rate. Gaining 86.4 seconds a day is gaining a thousandth, one millisecond a second, so the line climbs 40 milliseconds across 40 seconds of trace.

public static func trace(_ w: Watch, count: Int) -> [Beat] {
    guard w.isRunning else { return [] }
    let bps = w.caliber.escapement.beatsPerSecond
    let now = Int((w.runningTime * bps).rounded(.down))
    let first = max(now - count + 1, 0)
    // The watch's beats come at its own pace against the clock.
    let pace = 1 + w.rate / 86_400
    return (first...now).map { k in
        let drift = Double(k - first) / bps * (1 - 1 / pace) * 1000
        let tock = k % 2 != 0
        return Beat(index: k, tick: !tock, offset: drift - (tock ? w.beatError : 0))
    }
}

The pace is the watch’s speed against real time: a watch gaining 86.4 seconds a day runs at 1.001. The model runs the watch at that pace, so its beats really do come at the rate shown, and the trace is the drift of beat k against where a perfect clock would have put it.

Beat error is a gap

The second thing on the trace is whether the tick and the tock are evenly spaced. If the balance’s rest point is not exactly on the lever’s centre line, the swing one way reaches the lever a little sooner than the swing back, so every tock comes slightly late. On the trace, the dots split into two parallel lines, the ticks on one and the tocks on the other, and the gap between them is the beat error in milliseconds. A watchmaker corrects it by turning the collet that holds the inner end of the hairspring, nudging the rest point back to the centre line, and that is exactly what the game asks of you: each nudge of the collet moves the beat error by 1.2 milliseconds, and below 0.8 it reads as in beat.

Amplitude from a sound

The third reading is the amplitude, and it is the clever one, because nothing on the stand can see the balance. A timegrapher gets it from the time between the unlock and the drop, the width of the lift window, which it can hear. The second article worked out the window from the amplitude; the timegrapher runs the same relation backwards:

That is why a timegrapher asks you for the lift angle. Tell it the wrong one and it reports the wrong amplitude, by a fixed proportion, even though it hears every beat correctly. Tick Again’s timegrapher does exactly this: it times the unlock to the drop in the beats the model produces and turns that into degrees through the lift angle set on it, so the wrong lift angle misreads there just as it does on a real machine.

What the tests hold

• A watch running a thousandth fast with a beat error of 4 milliseconds gives a trace of 200 beats whose ticks climb 39.6 milliseconds, give or take one, from first to last.
• Every tock lies 4 milliseconds below the line through the ticks either side of it, to a microsecond.
• With a beat error of 5 milliseconds the gaps between impulses alternate 0.205 and 0.195 seconds.
• The reading reports 18,000 beats an hour and an amplitude above 250 degrees for a healthy C1, and a stopped watch gives no reading at all.
• The amplitude it hears agrees with the balance’s swing to half a degree, falls with a tired spring, and set to a lift angle of 44 degrees instead of 52 reads short by that proportion.

So far the watch has been running at full swing. The next article is about where that swing comes from, and what happens to it as the spring runs down.