LH includes items other than psalms. We do have the treasury of other musical items that sometimes look like psalms, and sometimes look like something a bit different:
136. At morning prayer between the first and the second psalm a canticle from the Old Testament is inserted, in accordance with custom. In addition to the series handed down from the ancient Roman tradition and the other series introduced into the breviary by St. Pius X, several other canticles have been added to the psalter from different books of the Old Testament, in order that each weekday of the four-week cycle may have its own proper canticle and on Sunday the two sections of the Canticle of the Three Children may be alternated.
137. At evening prayer, after the two psalms, a canticle of the New Testament is inserted, from the letters of the apostles or the Book of Revelation. Seven canticles are given for each week of the four-week cycle, one for each day. On the Sundays of Lent, however, in place of the Alleluia Canticle from the Book of Revelation, the canticle is from the First Letter of Peter. In addition, on the solemnity of the Epiphany and the feast of the Transfiguration the canticle is from the First Letter to Timothy; this is indicated in those offices.
These assignments have not exhausted the listings of canticles from either the Hebrew or Christian Scriptures, but they do cover a good deal of ground, especially giving us regular glimpses of early Christian songs otherwise buried in proclaimed texts at Mass.
138. The gospel Canticles of Zechariah, of Mary, and of Simeon are to be treated with the same solemnity and dignity as are customary at the proclamation of the gospel itself.
Hence the dignity of standing as one does for the proclamation of the gospel at Mass, and of signing oneself.
139. Both psalmody and readings are arranged in keeping with the received rule of tradition that the Old Testament is read first, then the writings of the apostles, and finally the gospel.
Which is why the Morning Prayer canticle is inserted between the assigned psalms, and the Evening Prayer New Testament canticle comes after both psalms.
Any thoughts? Any missed Gospel canticles we should have brought to the party?
Catholic Sensibility 
With some of the NT canticles I can’t help wondering if there was a little stretching for the sake of uniform structure in calling them canticles at all. Some are pretty clearly of similar style to the psalms and OT canticles (and the canticles of Mary, Zachariah and Simeon are clearly meant to be such) but a few of the ones out of the epistles frankly look a bit odd laid out in verse form for this use. But then, there may simply not have been 28 felicitous choices there — the NT is much shorter on poetic interludes than the later OT.
Also the NT canticles are on a two-week cycle, not a 4.
Actually, the NT canticles are on a 1 week cycle.