The Pythagorean theory (simple ratios are consonant) is a consequence of the design of most Western musical instruments. They're typically built around either a string or a column of air, which is constrained at both ends. The overtones are unavoidably integer multiples of the fundamental because of this constraint. If they were not then at least one end of the string would be oscillating, which is not possible because it is clamped in place.

Perception of consonance of a sound can be approximated by decomposing it into sine waves and calculating the sum of the values on the Plomp and Levelt curve (which is empirically measured and a consequence of human biology) for the intervals of all pairs of sine waves, weighted by amplitude. For instruments based around the harmonic series this results in standard Western harmony. There is nothing mystical about it, and the human brain does not have any ratio detector hardware. It's purely an artefact of a simpler underlying rule.

Other instruments, eg. tuned percussion, do not have this constraint. If you clamp a bar at one end and strike it you can get all kinds of inharmonic overtones. This means cultures with music based around these types of instruments (eg. Indonesian classical music) need completely different harmonic theory.

See http://sethares.engr.wisc.edu/paperspdf/consonance.pdf

This generalizes all existing theories of musical harmony, and explains things like stretched tuning in pianos (a consequence of non-ideal strings).