Matthew 2:11 — And when they were come into the house, they saw the young child with Mary his mother, and fell down, and worshipped him: and when they had opened their treasures, they presented unto him gifts; gold, and frankincense, and myrrh.

Gold, frankincense, Myrhh. Three gifts. (By the way, that’s the only basis for the tradition that there were three Magi.)
There are traditions all around the Magi — how many there were, what there names were, where they came from … And what those gifts meant. Do those gifts tell something about who Jesus is, or what was in His future, or something else? Does the carol have it right?

Born a King on Bethlehem’s plain, Gold I bring to crown Him again, King forever, ceasing never, Over us all to reign.
Frankincense to offer have I; Incense owns a Deity nigh; Prayer and praising, voices raising,Worshiping God on high.
Myrrh is mine, its bitter perfume Breathes a life of gathering gloom; Sorr’wing, sighing, bleeding, dying, Sealed in the stone cold tomb.

The carol really reflects the Western Christian tradition. The gifts, it is reckoned, were prophetic — Gold is for King Jesus, the King of Kings! Frankincense is for the God that Jesus was. Myrhh was used in embalming, and so foretold Jesus’s sacrificial death.
The Eastern church understands things just a little differently. The gifts, it has been suggested, reflected a choice that Jesus would make. If He took the gold, He would be choosing the royal road. If He took the Frankincense, it would be the priestly path. Myrhh would be the choice of the healer.
Both interpretations are interesting — but it’s hard to see that Matthew would have recognized either one. His view of the gifts would surely have been informed by his whole gospel viewpoint.
Harry Ironside provides a good description of Matthew’s point of view: “Matthew is in a very real sense the Jewish Gospel. This does not mean that it has no message for Christians, but rather that it is designed by the Holy Spirit to present Christ so as to make it clear to honest Jewish inquirers that He is the One of whom Moses and the prophets spake.”
Everything about the three gifts speaks about the temple. The Jews would recognize the association with divinity — and surely that is the significance of the three gifts. The temple vessels were Gold, the temple incense was made with frankincense and burned to invoke the presence of the Lord, and myrhh was an important component of the perfumed anointing oil that imparted holiness and could not be used outside the temple. The Jews must surely have recognized that the Magi were worshiping Jesus, Immanuel — God with us.
It’s easy to take the Christmas story and romanticize it — to build out from the elements of the accounts of Matthew and Luke. There’s really no need. If we look at the words they were given to speak, from their points of view, there is more than enough glory!