Many mindfulness practices seem to direct attention at two place at once, to quiet the inner voice. Perhaps this relates to more than just speech, but to attention itself. George Gurdjieff's "The Fourth Way" deals with self remembering, and his pupil, P. D. Ouspensky, has a very vivid description in [1] of how focusing on two things at once leads to a changed state of consciousness, that seems like meditation, and comes from the saturation of the two streams of attention.

[1] https://en.wikipedia.org/wiki/In_Search_of_the_Miraculous