The way it seems to me, is that there is a dividing of the service into three parts, of scriptural hymnody, non-scriptural hymnody, and incense and intercessory prayers, and that the three parts aren't really intertwined the way they are in the Byzantine or Western rites, but each have their own section. Is my understanding correct?
I'm not sure what you mean by divided into Scriptural and non-Scriptural hymnody... The Agpeya doesn't have hymns, just chanted responses. In Vespers there are doxologies... If you mean in the Psalmody (vespers/midnight/morning praise), here's a bit about how it's organized:
"The office of Midnight Praise is deeply rooted in Christian liturgical history. The Roman governor Pliny wrote in his letter to the Emporer Trajan that the Christians gathered before sunrise to participate in antiphonal singing in which they sang 'a song to Christ as to a god.' Pliny tells us that the Christians disperesed after the first gathering before sunrise to come together again later for the Lord's Supper. Any visitor to a Coptic monasatry will realize that the monks are observing the same tradition to this day.
Coptic Midnight Praise basically consists of four "Canticles", the "Psali" and the "Theotokia". The "Canticles" are Biblical Old Testament hymns that are always the same. In contast, the "Psali" and the "Theotokia" change according to the day of the week, the liturgical season or hte feast.
The "First Canticle" is the hymn Moses and the Isralis sang after they crossed the Red Sea. To us Christians, it is a song of victory over Satan which we attained when we crossed the waters of the baptismal font.
The "Second Canticle" is a psalm of praise (Ps 136). History tells us that when the Emperor constantius sent his General Syrianus to arrest St. Athanasius, he found him sitting in the church. A deaocn was singing Ps 136, while the people chanted the response "For His mercy endures forever."
the "Third Canticle" is the song which the Three children sang in the firey furnace. It has been preserved for us in the deuterocanonical part of the Book of Daniel.
The "Fourth Canticle" is made up of three psalms of priase, Ps 148, 149, 150.
The "Psali" is a hymn of glorificatioin of the Lord Jesus. The response is generally a recital of the name of the Lord.
The "Theotokia" is a hymn honouring the "Theotokos" (the Mother of God). It is a wonderful theological treatise explaning many of the Old Testament "types" of the holy Virgin Mary. ..."
--Preface to Coptic Psalmody by Heg. Fr. Athanasius Iskander