Which musical interval spans eight notes?

The octave, and the reason it sounds like the same note higher is physical rather than cultural: the upper note is at exactly double the frequency, so its waveform lines up with the lower one every cycle. Octave equivalence appears in every musical tradition anyone has documented, which makes it about as close to a universal as music has.

Almost everything else is a compromise. Stacking pure fifths, which are a 3:2 ratio and the next most consonant interval, never lands back on an octave - twelve fifths overshoot seven octaves by a small amount called the Pythagorean comma. So no tuning can have both pure fifths and pure octaves, and every system in history has chosen where to hide the error.

Equal temperament hides it everywhere. Modern Western tuning divides the octave into twelve exactly equal steps, which makes every interval except the octave slightly out of tune and every key equally usable - which is what Bach's Well-Tempered Clavier is demonstrating. It is a deliberate trade of purity for freedom of movement, and musicians who play unfretted instruments or sing unaccompanied drift back towards the pure ratios without meaning to.