The octave follows the universal laws of nature – which is what we too are subject to, so the theory – philosophically – is that by understanding these laws through studying the nature and character of the octave, it is helpful to our understanding or ourselves and the world around us.
In visualising an octave, I always saw it as a circle. I am content with the symbolism of the circle. For me it represents the notions of totality, wholeness, original perfection, the Self, the infinite.