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Pneumatic clocks are not new (see link for a very fine
article on the subject) but this is a domestic version
that's powered by a bicycle pump. Once a week the
owner must deliver a fresh change of air to the clocks
reservoirs via a standard bicycle pump.
The modestly amount of stored compressed
sufficient to drive the tiny piston, and escape
mechanism that operates the clock.
De-luxe version comes in the form of a miniature of the
famous Popp clock of which there were once 7,800 on
the streets of Paris examples (link)
The French were always innovators of gloriously designed eccentricity [xenzag, Feb 22 2020]
||This could be an interesting constant-force energy source for a clock, so [+].
||Also in the Museum of Retro Tech there's a whole gallery of "Oddly powered clocks" including several powered by changes in atmospheric pressure.
||//constant-force energy source// I'm not sure how you plan
to keep the driving force constant, as it will tend to decline as
the pressure falls. That's not a disaster (most spring-powered
clocks and watches don't have a constant driving force), but it
will affect the timekeeping.
||You could always use a fusée, though, to recover a constant
||A slow moving wide area piston - or a diaphragm - driven by very low pressure air via a regulator, and gearing to a fusée as described. If the flow rate is low, there will be little energy lost in the regulator, and it can be valved by a cam on the fusée.