Isn't it circular to say that we use periodic processes to define a time interval? Because in order to determine whether a process is periodic, i.e. whether the same changes always repeat at the same time intervals, we already need a measure of time.
I think what you're saying is that we need a "master clock" to begin with, to determine whether any other clock is precise or not.
In reality, no such master clock exists.
It used to be that we just said one spin of the Earth (i.e. a solar day) was a day by definition, without any regard for each day being of even length (as determined by any other kind of clock).
The vast majority of things were synchronised to the day cycle, so if some other clock were not measuring time like the spin of the Earth, then it is regularly resynchronised to force it to keep pace with the Earth on average.
That's still pretty much the essence of how man keeps time as a whole - it's how UTC works for example - but many in physics have gone off the rails for two reasons.
One, they think they've found the "master clock" in the form of atomic clocks, when in reality they have just deviated from talking about the solar-day standard to talking about their own ad-hoc atomic standard (which they falsely present to everyone as uniquely constant in a way that nothing else is).
Two, they've become neurotic about trying to iron out resynchronisation itself (attempting the impossible), rather than conceiving a system of physics which have conventions for resynchronisation which are useful and acceptable to society.
The measurement of time is fundamentally a very political matter, typically requiring a dictator to impose a common system - as with both the Julian and Gregorian calendars, for example.
If we do not understand periodicity in terms of time, then it can only mean that an object regularly returns to the same place without any information of the corresponding duration.
Yes, but the point of a clock is to define that duration. The information that one step of duration has elapsed, is the information that the clock provides.
If another clock doesn't correspond, that is because it is a different clock, and if you're second-guessing one clock with another, then you just haven't decided yet which clock you're going to use.
The traditional beauty of the solar day as a clock, is that its progression is under nobody's control, it is visible to everybody (who make up the large civil societies), it controls most things of basic biological importance, and there is nothing that seems like a bigger and more powerful signal to life on Earth.
But can't we then just as well take a spatially non-periodic movement as a clock? For example by saying that object x
moving from location y
to z
is defined as a second (regardless of whether it returns to y
at some point).
You can do that, but eventually you're going to encounter some inconvenient extreme, such as the distance of the clock from your current location (for example, if you just set something going out into space), or the lack of sufficient resolution on the clock (for example, if you use that "pitch drop" experiment as a clock, it's very difficult to tell one year from the next, let alone keep the time of day).
Hourglasses for example are generally made cyclical by regular resets (by inverting them), otherwise they would be quickly exhausted to a static condition, and thrown on the rubbish heap.
We actively look for periodic processes in the natural world as the foundation of time-keeping, precisely because they sustain.
