Derekh Hashem, Part Four, On the Daily Order of Prayer

The correspondence of the sections of prayer to the worlds: And behold that it was according to this order that the divisions of prayer were ordained. Hence the three sections at the beginning are for the refinement of the three worlds - this world, the world of the angels and the world of the throne. And that [corresponds to] the sacrifices, the verses of song and the blessing over the Shema. Afterwards is the standing prayer, which corresponds to the world of divinity, to bring down the influences according to their distinctions. And there are three other parts afterwards to bring down the duration of flow to the worlds - one after another, until the end. And they are the sanctification of the order (kiddushah de'sidrah), the song of [the day of] the Levites and "There is none like our God." And after all this is Aleynu; and it is to return and affirm His Kingship, may He be blessed, over all the worlds, after they were all blessed.