[chriswarbo]

They're not tied to anything physical; but they are a mathematical sweet spot.

24ths and 60ths are useful fractions because 24 and 60 are highly composite numbers, i.e. they have many divisors. 60 is a particularly good choice, since it's a "superior highly composite" number; and so is 12, if we consider an AM or PM period, rather than a full rotation.

Such divisibility was important before the invention/discovery of general fractions (i.e. the rational numbers). Babylonian arithmetic used something akin to base 60 floating-point, which could represent halves, thirds, quarters, fifths, and their sums and products (but not e.g. sevenths, or reciprocals of higher primes; in the same way base 10 floating-point can't represent a third).

Minute, meaning "small", describes the shifting of the radix point in such a base 60 floating-point system. Second, as in the ordinal 2nd, describes a further shift of radix point; and likewise for thirds, fourths, etc. for ever-smaller increments.

[fph]

Weird that we chose these numbers because they have many divisors, and yet no one ever says "it's a third past 10", only "half past" and "a quarter past".

[perilunar]

Be the change you want to see: start using ‘a third past’ instead of ‘twenty past’ and see what happens.

[prollings]

Where I am from it is common to use "20 to" and "20 past" too.