Octaves traditionally served a number of purposes: they signalled the importance of a particular feast; and through the repetition of the texts for that feast, helped the laity to truly learn certain key texts of Scripture.
These days we tend to prefer to constantly seek novelty, and disdain the idea of learning things by heart. Yet constantly repeated texts help shape our mindset, forming us spiritually in a particular way of thinking.
And these traditional octaves also provide an opportunity, through the readings set set at Matins for the feast, to penetrate the mystery of the feast more deeply.
Accordingly, here are the readings that were previously set for the sixth day of the feast in the Benedictine office, courtesy of the excellent Divinum Officium website:
Homily by St Ambrose, Bishop of Milan.
Book ii. on Luke ii.
What are the gifts of the faithful and true? Gold to our King, frankincense to our God, and myrrh to Him Who died for us. The first is that whereof are made the royal honours of kings, the second is that mystic offering which is used in the worship of the Divine Power, and the third is that wherewith we pay respect to the dead, whose bodies it keepeth from corruption. My brethren, let us who hear and read these things, make offering out of what treasures we have albeit we have it in earthen vessels. 2 Cor. iv. 7. If we confess that all that we have, we have, not from ourselves, but from Christ, how much more should we confess that whatever we have is not our own, but Christ's?
The wise men out of their treasures presented unto Him gifts. Wilt thou know how pleasing to Him they were? The star appeared to them, but disappeared when it came near Herod. Then it appeareth to them again, leading them on the way that led to Christ. This star then was the way, and we know that Christ calleth Himself the Way. John xiv. 6. And truly also in the mystery of His Incarnation He is called a Star; as it is written There shall come forth a Star out of Jacob, and a Man shall rise out of Israel. Where Christ is, there is a Star; yea, He is Himself the bright and morning Star. Apoc. xxii. 16. And the light that leadeth to Jesus is His own.
Remark another point. The wise men came by one way and departed by another. They that had seen Christ, knew Christ, and they departed better than they came. There are two ways, the one which leadeth to destruction, the other which leadeth to the kingdom; the one is the way of sin, which leadeth to Herod; the other is Christ, the true Way, Who leadeth us home to the fatherland, from that journeying here, whereof it is said My soul hath long dwelt as an exile.