Tourbillon: A Watch Escapement That Spins to Average Out Gravity

The Problem: Gravity Pulls on a Balance That Is Never Perfectly Poised

A mechanical watch keeps time with a balance wheel swinging back and forth on a spiral hairspring. Ideally its rate depends only on the wheel's moment of inertia and the spring's stiffness. In practice no balance is perfectly poised: its centre of mass sits a tiny distance e off the pivot axis, at what watchmakers call the heavy point. With the watch lying flat, gravity acts along the balance staff and exerts no torque about it. Stand the watch on edge and the heavy point becomes a small pendulum bob riding on the wheel: gravity helps part of each swing and resists another, and the watch gains or loses.

Around 1800 this mattered. A pocket watch spent the day upright in a waistcoat pocket, usually pendant up, so the error never averaged away. Its size and even its sign depend on which way the heavy point faces and on how far the balance swings, which falls as the mainspring runs down, so no single regulator setting could remove it. Abraham-Louis Breguet (1747–1823), born in Neuchâtel and working in Paris, answered with a mechanism he called the tourbillon, “whirlwind”. Breguet's tourbillon patent was granted 26 June 1801 (7 Messidor, Year IX, in the French Republican calendar). He sold only about 35 tourbillon watches before his death in 1823.

How a Tourbillon Works, Step by Step

The tourbillon does not make the balance perfect; it keeps turning it so no heavy point faces one direction for long.

• The cage. The balance, hairspring, pallet fork and escape wheel ride in a light rotating frame, the cage or carriage, which in modern wristwatches typically weighs a fraction of a gram. The cage takes the place of the fourth wheel of an ordinary going train, and the balance usually sits on its axis.
• The drive. The third wheel meshes with a pinion on the cage and tries to turn it.
• The fixed wheel. The escape wheel's pinion, mounted off-centre in the cage, meshes with a stationary wheel screwed to the plate, concentric with the cage.
• The lock. To turn, the cage must roll the escape pinion around the fixed wheel, which forces the escape wheel to turn. While a pallet stone locks the escape wheel, the cage and whole train stand still.
• The step. Each time the balance swings through its centre, the fork unlocks the escape wheel. The fixed wheel pushes back on the escape pinion; that reaction drives the escape wheel and impulses the balance, and the cage advances one small step. The balance meters the rotation of its own carrier.

The numbers lock together. A balance at 3 Hz = 21,600 vibrations/hour (6 beats per second) makes 360 beats per minute. If the cage turns once every 60 s (6° per second), it steps 360° ÷ 360 = ~1° per beat. A common 15-tooth escape wheel advances half a tooth, 12°, per beat, so it must make 12 turns relative to the cage per cage revolution: a fixed wheel with 12 times the escape pinion's tooth count, such as 84 teeth against 7 leaves. This is an epicyclic train, with the fixed wheel as sun, the cage as carrier and the escape pinion as planet. A 60-second cage often carries the seconds hand.

The Governing Relation: Why One Turn Cancels the Error

Treat the balance as an oscillator with moment of inertia I and hairspring stiffness k, so ω = √(k/I). Its mass m has its centre a distance e off the pivot, with the heavy point at angle φ from straight down when the balance is at rest. In a vertical position gravity adds a torque −m·g·e·sin(θ + φ). For a swing θ = A·sin(ωt), averaging that torque over a cycle gives the fractional rate change

Δf/f ≈ (m·g·e / I·ω²) · cos φ · J₁(A)/A

where A is the amplitude in radians and J₁ is the first-order Bessel function. The error scales with cos φ, so it depends on which way the watch hangs; for a given balance it shrinks as 1/ω²; and J₁ crosses zero at A = 3.83 rad, so the poise error changes sign near 220° balance amplitude: with the heavy point down, the watch gains below about 220° and loses above it.

Worked example. Take an illustrative 3 Hz balance with m ≈ 0.1 g and I ≈ 1.0 × 10⁻⁹ kg·m² (10 mg·cm²), so ω = 2π × 3 ≈ 18.85 rad/s and k = I·ω² ≈ 3.55 × 10⁻⁷ N·m/rad. Let the centre of mass sit just e = 1 µm off the pivot. Then m·g·e = 10⁻⁴ kg × 9.81 m/s² × 10⁻⁶ m ≈ 9.8 × 10⁻¹⁰ N·m, and m·g·e / I·ω² ≈ 2.76 × 10⁻³.

• Heavy point down (cos φ = 1), amplitude 270° (4.71 rad): J₁(A)/A ≈ −0.060, so Δf/f ≈ −1.65 × 10⁻⁴. Times 86,400 s per day, the watch loses about 14 s/day.
• Same position, mainspring nearly run down, amplitude 200° (3.49 rad): J₁(A)/A ≈ +0.040, and the same watch now gains about 10 s/day.
• Heavy point up (cos φ = −1): both signs flip. Heavy point sideways (cos φ = 0): the first-order error vanishes.

Now let the cage turn once every 60 s. The heavy point sweeps through 360° each minute, so at 270° amplitude the instantaneous error swings between about −14 and +14 s/day. Because cos φ averages to zero over a full turn, the error over each revolution cancels to first order, whatever the imbalance or its starting direction. Breguet traded a fixed daily error for a harmless once-a-minute wobble.

Dial up or dial down, the cage axis is parallel to gravity and there is no poise torque to average, so only the vertical positions benefit. Nor is the flat-versus-hanging difference removed. Hanging, the balance pivots bear on their sides rather than their tips, friction rises and amplitude is typically tens of degrees lower, shifting the rate through isochronism error that no rotation in the vertical plane can average.

Real Hardware: Cages, Flying Tourbillons and the Wristwatch

Because the escapement starts and stops the cage on every beat, the cage's inertia is paid for in balance amplitude. Makers skeletonise cages of dozens of parts, cut them from titanium, aluminium alloys or thin steel, and poise the cage itself, or gravity would alternately help and resist the train once per turn.

• Breguet's originals were pocket watches, and some cages took several minutes per turn.
• The flying tourbillon (Alfred Helwig, Glashütte, 1920) supports the cage from one side only, with no upper bridge, leaving the mechanism fully visible.
• The wristwatch tourbillon. A few tourbillon wristwatch movements were built in the 1940s and 1950s for observatory competitions, but Audemars Piguet serialised the tourbillon wristwatch in 1986 with a thin self-winding model. By then quartz had won the accuracy race, and the tourbillon became a showcase of craft.
• Multi-axis tourbillons nest one cage inside another so the balance also tumbles through the flat positions; Jaeger-LeCoultre's Gyrotourbillon of 2004 turns on two axes. Other makers incline a single cage, and some spin faster, such as Greubel Forsey's Tourbillon 24 Secondes.
• Affordable tourbillons. Chinese makers such as Tianjin Sea-Gull now build tourbillon movements at a small fraction of Swiss prices.

Testing, Measurement and Specification

Rate is measured on a timegrapher, which picks up the escapement's ticks with a microphone and reports rate in seconds per day, amplitude in degrees (computed from the timing of sounds within each beat and the movement's lift angle) and beat error in milliseconds. Repeating the reading dial up, dial down and in vertical positions such as crown up gives the positional spread a tourbillon targets.

A tourbillon needs one extra precaution: measure over whole cage revolutions. A few-second reading captures only part of the sweep, so in a vertical position a short-window trace can rise and fall with a 60-second period. Averaged over full minutes, the vertical positions should agree closely.

The usual specification is ISO 3159, applied by the Swiss chronometer testing office COSC: movements run for 15 days in five positions at three temperatures (8 °C, 23 °C and 38 °C). For movements over 20 mm in diameter the criteria include a mean daily rate within −4/+6 s/day and a horizontal-to-vertical rate difference within −6/+8 s/day. A tourbillon earns no exemption. Before quartz, the proving ground was the observatory trial, at Kew in England and at Neuchâtel and Geneva, where finely adjusted tourbillon and karrusel pocket watches earned many top marks.

Failure Modes, Misuse and Misconceptions

• Energy cost. Starting and stopping the cage six times a second dissipates energy that would otherwise sustain the balance, so a heavy cage lowers amplitude and can push a watch across the 220° crossover as it runs down.
• Out-of-poise cage or eccentric fixed wheel. Either creates a once-per-revolution torque ripple, seen as a 60-second amplitude wobble; poor depthing between escape pinion and fixed wheel can stop the watch at the same point every turn.
• Shock. The cage pivots carry the whole regulating organ; a hard knock can bend them, and a flying tourbillon's cantilevered bearing is especially exposed.
• A stopped cage is a stopped watch. The cage is the fourth wheel, so dirt or a burr that jams it halts timekeeping.
• Service burden. Extra pivots to oil, and a balance regulated and put in beat inside a moving cage, make servicing slow and costly.

The biggest misconception is that a tourbillon cancels gravity or improves any watch. It only averages errors that depend on which way the balance faces within a vertical plane. A wristwatch is tilted, turned and laid flat all day, so its orientation errors already partly average out, while the cage costs amplitude. Its modern benefit is debated; a well-poised, well-adjusted conventional movement can match it on the wrist. The commonest misuse is marketing: cheap watches with an open-heart dial window showing an ordinary balance are often sold as tourbillons, though nothing in them rotates.

Tourbillon vs Karrusel

The mechanism most often confused with the tourbillon is the karrusel, patented in 1892 by Bahne Bonniksen, a Danish-born watchmaker working in Coventry, England. It too carries the escapement round on a rotating carriage, but the kinematics differ at the root.

• Tourbillon: the cage is the fourth wheel. Rotation and escapement are locked together through the fixed wheel, so the escapement releases every step; stop the cage and the watch stops.
• Karrusel: the carriage is geared to the third wheel pinion, while the fourth wheel's arbor passes through the carriage bearing and, instead of standing still, turns and drives the escape pinion in the normal way. The two are parallel branches of one gear-locked train, so almost none of the escapement's power passes through the slow carriage's rotation; holding the carriage still jams the third wheel and stops the watch.
• Speed: classic karrusels make one turn every 52.5 minutes, under 7° per minute, against 360° per minute for a 60-second tourbillon.

The slow karrusel still averages vertical-position errors, over most of an hour instead of a minute, which is ample for a pocket watch carried all day. Its creeping carriage stores almost no kinetic energy and its parts need not be as delicate, so it was cheaper to make, and karrusels scored very well at Kew.

The tourbillon compared with the escapements it is most often confused with

Design	How the carriage turns	Typical rotation	If the carriage is held
Fixed escapement (ordinary watch)	No carriage: balance and escapement are fixed to the plate	None; the heavy point faces the same way for as long as the watch hangs the same way	Nothing to hold; the positional error stays
Tourbillon (Breguet, patent granted 26 June 1801)	The cage is the fourth wheel; its escape pinion must roll around a fixed wheel, so the escapement releases every step	Once every 60 s (6° per second); ~1° per beat at 3 Hz	The watch stops
Flying tourbillon (Alfred Helwig, Glashütte, 1920)	Same kinematics, but the cage is cantilevered from one side with no upper bridge	Usually once every 60 s	The watch stops
Karrusel (Bahne Bonniksen, patented 1892)	Carriage geared to the third wheel pinion as a side branch; the fourth wheel is not fixed but turns and drives the escapement in the normal way	Classic: one turn every 52.5 minutes	The watch also stops (one gear-locked train), but almost none of the escapement's power passes through the slow carriage's rotation
Open-heart dial window	No carriage: an aperture in the dial simply shows an ordinary balance	None	Nothing rotates; often mislabelled a tourbillon