For those of us who live the Church's calendar - by saying the Divine Office, by attending Mass through the week - the joy of a great celebration can seem a little flattened by the ordinariness of the days after: back to the 'ordinary'; back to green vestments. This is why liturgical traditions have tended to give us a gentle let-down. In the Eastern traditions, very often the 'day after' is an associated celebration; a rendering of the same theme in a different key. Thus the Byzantines and the East and West Syrians keep December 26 as the commemoration of Mary, the Mother of God. Byzantine Calendars also note days as 'the leave-taking of such-and-such'. In our Western traditions, the Octave has performed a similar role. For a week after the Great Day the festival continues to colour our worship. Then, on the eighth day, which will be the same day of the week, we say farewell to the festival by celebrating it again ... even if at a reduced level.
It was unfortunate that the 'reforms' of the 1960s almost entirely eliminated the concept of the Octave. Out of the window went the practice of keeping a Sunday as the 'Sunday within the octave of-such-and-such'. We now have only the octaves of Christmass and Easter, except that in the Ordinariates we have the Pentecost Octave restored to us! Although: the discerning eye can see one or two shadows of the old Octave Days; the eighth day after the Assumption, August 22 is a Feast of our Lady; and in the Ordinariates November 8 is the Feast of All Saints of England and Wales. The old Octave Day of our Lady's Birthday, September 15 - also the day after Holy Cross Day - is, very neatly, the memorial of our Lady at the Cross.
The Octave of Christmass, January 1, has long been marked in the West by texts which take the Divine Maternity of Mary as their theme. As the Council of Ephesus insisted, she is celebrated as Theotokos: Mother of God. This is a safeguard of the Divinity of our Lord; He is God and so His mother is Mother of God. But the title alarms post-Christian folk-Protestants (who are mostly in fact left-wing Arians). Sometimes Anglicans have suggested that Godbirthgiver would be a better translation, as if birthgiver is a more natural word than mother. Byzantines appear unaware of any problem here: at the top of every ikon of our Lady among so many millions, are the words Meter Theou: 'Mother of God'.
Jesus was circumcised on the eighth day, and so, with the increasing growth of Biblicism, January 1 came to be called in the Middle Ages the Feast of the Circumcision, although the Lord's Circumcision was barely mentioned in texts, other than the Gospel. This changed when Cranmer got to work. The Novus Ordo did not behave foolishly in renaming the day 'Mary Mother of God' because that is what the ancient texts are in fact mostly about.
I know that some faithful Traddies number among their (many justified) criticisms of the Novus Ordo its abolition of the Feast of the Circumcision; but, well, if you don't entirely trust me (and why should you?) read Dom Gueranger's article for January 1.
Possibly because of squeamishness about the Ephesian Dogma, combined with a Victorian shyness about concentrating too explicitly on processes involving the virile generative organ, modern Anglicanism often restyles the day 'Name [or Naming] of Jesus'.
Ah! The season of Snowflakes!!