This form of Christian chant – Gregorian chant – is, of course, of unquestionably Western provenance! As the notation on the linked YouTube page notes,
“Monks of the one of the Abbeys of the Solesmes Congregation sing this beautiful chant. The Te Deum is attributed to two Fathers and Doctors of the Church, St. Ambrose and St. Augustine, and is one the most majestic chants in the Liturgy of the Church. It is sung in traditional seminaries and monastic houses at the Divine Office and for Double feasts of the First Class, The Nativity, Easter, Corpus Christi, Epiphany, Pentecost and those which have an Octave. The solemn Te Deum is sung on all occasions of public [liturgical] rejoicing, in Traditional Catholic Churches.”
And in English translation, it is sung (or recited, in the Office of Morning Prayer) in not a few traditional Anglican churches, as well!
Nota Bene: The Abbey of Solesmes, under Dom Prosper Guéranger, was largely responsible for the rebirth and liturgical restoration of Gregorian Chant, in the 19th and early 20th centuries, and remains the mother-house of a Benedictine community to this day (with several interruptions along the way!). For more information, check out this fascinating brief account of its history, or the Abbey’s own history page.
Here is the Te Deum in traditional English translation:
Te Deum laudamus.
WE praise thee, O God; we acknowledge thee to be the Lord.
All the earth doth worship thee, the Father everlasting.
To thee all Angels cry aloud; the Heavens, and all the Powers therein;
To thee Cherubim and Seraphim continually do cry,
Holy, Holy, Holy, Lord God of Sabaoth;
Heaven and earth are full of the Majesty of thy glory.
The glorious company of the Apostles praise thee.
The goodly fellowship of the Prophets praise thee.
The noble army of Martyrs praise thee.
The holy Church throughout all the world doth acknowledge thee;
The Father of an infinite Majesty;
Thine adorable, true and only Son;
Also the Holy Ghost the Comforter.
THOU art the King of Glory, O Christ.
Thou art the everlasting Son of the Father.
When thou tookest upon thee to deliver man, thou didst humble thyself to be born of a Virgin.
When thou hadst overcome the sharpness of death, thou didst open the Kingdom of Heaven to all believers.
Thou sittest at the right hand of God, in the glory of the Father.
We believe that thou shalt come to be our Judge.
We therefore pray thee, help thy servants, whom thou hast redeemed with thy precious blood.
Make them to be numbered with thy Saints, in glory everlasting.
O LORD, save thy people, and bless thine heritage.
Govern them and lift them up for ever.
Day by day we magnify thee;
And we worship thy Name ever, world without end.
Vouchsafe, O Lord, to keep us this day without sin.
O Lord, have mercy upon us, have mercy upon us.
O Lord, let thy mercy be upon us, as our trust is in thee.
O Lord, in thee have I trusted; let me never be confounded.
Here is an English-language plainsong (chant) version: