"Summer is icumen in" is definitely in six parts, revolutionary as this may seem for the date.While the tune is written as a single line,there are clear Latin instructions indicating where the second and subsequent singers come in to make it into a round. This generates four lines going together. In addition there is a two part "pes"("foot" in Latin) which is a continuous bit which underpinsthe melody line. This adds two more parts, making six in all. And the whole thing sounds amazizngly modern for the date. There are all sorts of books around saying that spacemen landed and taught various ancient cultures new tricks like pyramid building. My candidate for a theory likethis would be "Summer is icumen in"...it stands to reason nobody could have written this music at that date. And in Reading!!