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.
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".
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.