In the Orthodox tradition, December 31 is the last day of the 7 days of Christmas, also known as the “Leavetaking of the Nativity of our Lord”. It is traditionally observed with Byzantine troparion and kontakion hymns.
Your Nativity, O Christ our God, / Has shone to the world the Light of wisdom! / For by it, those who worshipped the stars, / Were taught by a Star to adore You, / The Sun of Righteousness, / And to know You, the Orient from on High. / O Lord, glory to You!
Today the Virgin gives birth to the Transcendent One, / And the earth offers a cave to the Unapproachable One! / Angels with shepherds glorify Him! / The wise men journey with a star! / Since for our sake the Eternal God was born as a Little Child!
Here is a Byzantine hymn of the Nativity ( a troparian chant) by St. Romanos the Melodist, a Syrian composer of the 6th century, the golden age of Byzantine music.