Octaves have a very straightforward mathematical basis. Double the pitch, you get an octave.
What is pitch? It is defined in hertz, the number of times the underlying vibration repeats per second.
If you have an octave, you take one pitch like 220, then you double it to 440. That is the next octave. It's clear that if you define the "underlying vibration" as a sound that oscillates twice (2hz) then the 440hz sound is also in a way a 220hz sound. Thus the octave is really just a recognition of that mathematical relationship between the two pitches.
The 5th too is very simple. It is 1.5x the root. So the the 8th of 440Hz is 880Hz, and the 5th is 660Hz. Western musical theory is amazingly fractional
Actually, most music produced on physical instruments in the west doesn't have such nice ratios. I suggest reading about equal temperament (or watching some Adam Neely videos) if you're interested in learning more about how our tuning system diverges from the Pythagorean ideal.
What is pitch? It is defined in hertz, the number of times the underlying vibration repeats per second.
If you have an octave, you take one pitch like 220, then you double it to 440. That is the next octave. It's clear that if you define the "underlying vibration" as a sound that oscillates twice (2hz) then the 440hz sound is also in a way a 220hz sound. Thus the octave is really just a recognition of that mathematical relationship between the two pitches.