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The invention of the mechanical clock in the 13th century initiated a change in timekeeping methods from continuous processes, such as the motion of the gnomon's shadow on a sundial or the flow of liquid in a water clock, to repetitive oscillatory processes, like the swing of a pendulum or the vibration of a quartz crystal, which were more accurate. All modern clocks use oscillation.
Although the methods they use vary, all oscillating clocks, mechanical and digital and atomic, work similarly and can be divided into analogous parts. They consist of an object that repeats the same motion over and over again, an oscillator, with a precisely constant time interval between each repetition, or 'beat'. Attached to the oscillator is a controller device, which sustains the oscillator's motion by replacing the energy it loses to friction, and converts its oscillations into a series of pulses. The pulses are then added up in a chain of some type of counters to express the time in convenient units, usually seconds, minutes, hours, etc. Then finally some kind of indicator displays the result in a human-readable form.

Power source
This provides power to keep the clock going.
In mechanical clocks, this is either a weight suspended from a cord wrapped around a pulley, or a spiral spring called a mainspring.
In electric clocks, it is either a battery or the AC power line.
Since clocks must run continuously, there is often a small secondary power source to keep the clock going temporarily during interruptions in the main power. In old mechanical clocks, a maintaining power spring kept the clock turning while the mainspring was being wound. In quartz clocks that use AC power, a small backup battery is often included to keep the clock running if it is unplugged temporarily from the wall.
Oscillator
The timekeeping element in every modern clock is a harmonic oscillator, a physical object (resonator) that vibrates or oscillates repetitively at a precisely constant frequency.
In mechanical clocks, this is either a pendulum or a balance wheel.
In some early electronic clocks and watches such as the Accutron, it is a tuning fork.
In quartz clocks and watches, it is a quartz crystal.
In atomic clocks, it is the vibration of electrons in atoms as they emit microwaves.
In early mechanical clocks before 1657, it was a crude balance wheel or foliot which was not a harmonic oscillator because it lacked a balance spring. As a result they were very inaccurate, with errors of perhaps an hour a day.
The advantage of a harmonic oscillator over other forms of oscillator is that it employs resonance to vibrate at a precise natural resonant frequency or 'beat' dependent only on its physical characteristics, and resists vibrating at other rates. The possible precision achievable by a harmonic oscillator is measured by a parameter called its Q, or quality factor, which increases (other things being equal) with its resonant frequency. This is why there has been a long term trend toward higher frequency oscillators in clocks. Balance wheels and pendulums always include a means of adjusting the rate of the timepiece. Quartz timepieces sometimes include a rate screw that adjusts a capacitor for that purpose. Atomic clocks are primary standards, and their rate cannot be adjusted.
Synchronized or slave clocks
Some clocks rely for their accuracy on an external oscillator; that is, they are automatically synchronized to a more accurate clock:
Slave clocks, used in large institutions and schools from the 1860s to the 1970s, kept time with a pendulum, but were wired to a master clock in the building, and periodically received a signal to synchronize them with the master, often on the hour. Later versions without pendulums were triggered by a pulse from the master clock and certain sequences used to force rapid synchronization following a power failure.
Synchronous electric clocks don't have an internal oscillator, but rely on the 50 or 60 Hz oscillation of the AC power line, which is synchronized by the utility to a precision oscillator. This drives a synchronous motor in the clock which rotates once for every cycle of the line voltage, and drives the gear train.
Computer real time clocks keep time with a quartz crystal, but are periodically (usually weekly) synchronized over the internet to atomic clocks (UTC), using a system called Network Time Protocol.
Radio clocks keep time with a quartz crystal, but are periodically (often daily) synchronized to atomic clocks (UTC) with time signals from government radio stations like WWV, WWVB, CHU, DCF77 and the GPS system.
Controller
This has the dual function of keeping the oscillator running by giving it 'pushes' to replace the energy lost to friction, and converting its vibrations into a series of pulses that serve to measure the time.
In mechanical clocks, this is the escapement, which gives precise pushes to the swinging pendulum or balance wheel, and releases one gear tooth of the escape wheel at each swing, allowing all the clock's wheels to move forward a fixed amount with each swing.
In electronic clocks this is an electronic oscillator circuit that gives the vibrating quartz crystal or tuning fork tiny 'pushes', and generates a series of electrical pulses, one for each vibration of the crystal, which is called the clock signal.
In atomic clocks the controller is an evacuated microwave cavity attached to a microwave oscillator controlled by a microprocessor. A thin gas of cesium atoms is released into the cavity where they are exposed to microwaves. A laser measures how many atoms have absorbed the microwaves, and an electronic feedback control system called a phase locked loop tunes the microwave oscillator until it is at the exact frequency that causes the atoms to vibrate and absorb the microwaves. Then the microwave signal is divided by digital counters to become the clock signal.
In mechanical clocks, the low Q of the balance wheel or pendulum oscillator made them very sensitive to the disturbing effect of the impulses of the escapement, so the escapement had a great effect on the accuracy of the clock, and many escapement designs were tried. The higher Q of resonators in electronic clocks makes them relatively insensitive to the disturbing effects of the drive power, so the driving oscillator circuit is a much less critical component.
Counter chain
This counts the pulses and adds them up to get traditional time units of seconds, minutes, hours, etc. It usually has a provision for setting the clock by manually entering the correct time into the counter.
-In mechanical clocks this is done mechanically by a gear train, known as the wheel train. The gear train also has a second function; to transmit mechanical power from the power source to run the oscillator. There is a friction coupling called the 'cannon pinion' between the gears driving the hands and the rest of the clock, allowing the hands to be turned by a knob on the back to set the time.
-In digital clocks a series of integrated circuit counters or dividers add the pulses up digitally, using binary logic. Often pushbuttons on the case allow the hour and minute counters to be incremented and decremented to set the time
Indicator
This displays the count of seconds, minutes, hours, etc. in a human readable form.
The earliest mechanical clocks in the 13th century didn't have a visual indicator and signalled the time audibly by striking bells. Many clocks to this day are striking clocks which strike the hour.
Analog clocks, including almost all mechanical and some electronic clocks, have a traditional dial or clock face, that displays the time in analog form with moving hour and minute hand. In quartz clocks with analog faces, a 1 Hz signal from the counters actuates a stepper motor which moves the second hand forward at each pulse, and the minute and hour hands are moved by gears from the shaft of the second hand.
Digital clocks display the time in periodically changing digits on a digital display.
Talking clocks and the speaking clock services provided by telephone companies speak the time audibly, using either recorded or digitally synthesized voices.