In addition to Great Lent, the Byzantine tradition has three other fasts connected with major feasts. The liturgical year begins on September 1st, so the first of these is the fast of the Lord’s Nativity, which is often called “St Philip’s fast”, since it begins on November 15th, the day after the feast of the Apostle St Philip. This is very similar to the custom of the Ambrosian and Mozarabic Rites, which begin Advent on the Sunday after the feast of St Martin. Another fast is kept from the Monday after the feast of All Saints (which is celebrated on the Sunday after Pentecost, the Western day for the feast of the Holy Trinity) to the feast of Ss Peter and Paul; because of the variable date of Pentecost, this can run as long as 42 days, or as short as 8. The fast of the Dormition is kept from August 1-14, and is the strictest of the three.
One of the liturgical customs associated with these fasts is the celebration of the “Inter-Hours”, as they are called (in Greek Μεσώριον sing., -ια plur., in Church Slavonic Междочасїе sing., -їѧ plur.), a second Prime, Terce, Sext and None, which are said after the main Prime etc. Most Greek liturgical books appoint them to be said during the Nativity and Apostle fasts; the article about them on Russian Wikipedia says that they are also done during that of the Dormition. They are not said during Great Lent, since the Hours from Prime to None are lengthened by various other additions in that season. In point of fact, the Inter-Hours are now something of an archaism, in that they are associated with the practice of keeping some weekdays within the fasting periods as “aliturgical” days, i.e., days on which the Eucharist is not celebrated. This practice is still strictly observed for all the weekdays of Lent, but has apparently mostly fallen out of use for the other three fasts. Some sources indicate that the Inter-Hours are in practice celebrated in monasteries only on the first day of each minor fast, so effectively, twice or three times a year. (See this article on Academia for more details.)
The beginning of Psalm 45, the first Psalm of the Inter-Hour of Prime, in a Byzantine Psalter of the mid-10th century known as the Paris Psalter. (Bibliothèque nationale de France, Département des manuscrits, Grec 139; folio 119v, image cropped.)
Their structure is similar to that of the main Hours to which they correspond, but not identical. They begin with the same series of prayers said at the beginning of the other Hours, conveniently known as “The Usual Beginning.” However, when an Hour is said immediately after another, it starts with the very last part of the Usual Beginning, “Come let us worship…” Three invariable Psalms are then said: at Prime, 45, 91 and 92 (according to the numbering of the Septuagint); at Terce, 29, 31 and 60; at Sext, 55, 56 and 69; at None, 112, 137 and 139. A group of prayers called the Trisagion prayers are then said, which are repeated from the Usual Beginning (omitting the first two parts and the last part), and then a series of three chants called tropars, with the two parts of the doxology between them. These do not vary according to the day or season; the final tropar in any such group is always dedicated to the Virgin Mary.
At Prime:
Have mercy upon us, o Lord, have mercy upon us; for lacking all apology, we sinners bring to Thee this supplication, as to our Master: have mercy on us. Glory be…
Lord, have mercy us, for in Thee we have placed our trust, be not exceedingly wroth with us, and remember not our iniquities, but look (upon us) even now, as one merciful, and ransom us from our enemies; for Thou art our God, and we are Thy people, all of us the works of Thy hands, and we have all called upon Thy name. Both now and forever…
Open to us the gate of mercy, blessed Mother of God; as we hope in Thee, let us not err; may we be delivered through Thy urgent prayers, for Thou art the salvation of the nation of Christians.
At Terce:
God of our fathers, who dealest with us ever according to Thy goodness, put not Thy mercy away from us, but by their prayers, govern our life in peace. Glory be…
Thy Martyrs, o Lord, in their contests bore away the crowns of incorruption from Thee, our God; for having gained Thy strength, they threw down tyrants, and shattered the weak insolence of demons; by their prayers, o Christ our God, save our souls! Both now and forever…
Virgin Mother of God, Thou art the unconquerable fortress of Christians; for as we flee to Thee, we remain unwounded, and when we sin again, we cry out to Thee, “Hail, that art full of grace; the Lord is with Thee!”
At Sext:
Save, o Lord, Thy people, and bless Thine inheritance, granting victory to (our) kings over the barbarians, and preserving Thy citizenry (i.e., the members of the Church) through Thy Cross. Glory be… (This is also the tropar of the Exaltation of the Cross, and of the commemoration of the Cross celebrated on the Third Sunday of Lent; the traditional music of the Church Slavonic version is particularly nice.)
Be Thou prevailed upon, o Lord, by the pains of the Saints which they suffered for Thee, and heal all our ailments, we beseech Thee, that lovest mankind. Both now and forever…
By the prayer of all the Saints, o Lord, and of the Mother of God, give us Thy peace, and have mercy on us, as the only merciful one.
At None:
Thou who didst enlighten the things of the world through Thy Cross, and call sinners unto repentance, separate me not from Thy flock, o Good Shepherd, but seek me, Master of those who wander, and number me together with Thy holy flock, who alone art good and love mankind. Glory be…
Like the thief, I confess and cry out to Thee, o Good one: remember me o Lord, in Thy kingdom, and number me within it, who didst willingly accept sufferings for our sake. Both now and forever…
Come, let us all sing hymns to Him who was crucified for us, for Mary beheld Him upon the Cross and said, “Although Thou abidest the Cross, Thou are My Son and God.
There follows a series of elements also said at the other Hours except for Vespers and Orthros: Kyrie, eleison 40 times, the Prayer of the Hours, Kyrie, eleison 3 times, Glory be, a brief prayer to the Virgin (“Higher than the Cherubim…”) a conclusion said by the priestly celebrant, and then a very well-known prayer of St Ephraim the Syrian, accompanied by three prostrations. (In Lent this is added to all of the Hours.)
At the end of Prime, two longer prayers are said by the reader, at the others, just one. The prayers of the Inter-Hours of Prime and Terce are proper to them, but those of Sext and None are taken from the main Hour that precedes them. (When the Inter-Hours are said, the prayer of the main Terce, which is quite short, is repeated at the main Sext and None.) Like almost all prayers of the Byzantine Rite, these are not changed from one day or season to another. They are traditionally attributed to St Basil the Great.
Eternal God, light without beginning and everlasting, maker of all creation, fount of mercy, sea of goodness, unsearchable abyss of love for mankind, shine the light of Thy countenance upon us, o Lord. Shine in our hearts, o spiritual Sun of justice, and fill our souls with Thy rejoicing, and teach us ever to take thought of Thy matters, and speak forth judgments, and confess Thee without ceasing, our Master and benefactor. Guide the works of our hands towards Thy will, and help us on the way to do what is pleasing and welcome to Thee, that even through us, Thy unworthy servants, Thy all-holy name may be glorified, Father, Son and Holy Spirit, of one divinity and kingdom to which beseem all glory, honor and worship, unto the ages. Amen.
Thou who sendest forth the light, and it goeth, who makest the sun to rise upon the just and the unjust, the wicked and the good, who makest the morn, and enlighten all the world; enlighten also our hearts, Master of all. Grant us to please Thee in the present day, preserving us from every sin and from every wicked deed, delivering us from every arrow that flieth in the day, and every opposing power, by the prayers of our all-immaculate Lady, the Mother of God, of Thy immaterial ministering heavenly powers, and all the Saints that have been pleasing to Thee from the beginning of the world. For it is Thine to have mercy on us and save us, our God, and to Thee do we give glory, the Father and the Son and the Holy Spirit, now and every, and unto the ages of ages. Amen.
At Terce:
O Lord our God, who hath given Thy peace to men, and sent the gift of Thy All-holy Spirit to Thy Disciples and Apostles, and by Thy power opened their lips with tongues of fire, open Thou also the lips of us sinners, and teach us how we must pray and for what things. Govern our life, calm haven of those tossed by storms, and make known to us the way in which we shall go. Renew a righteous spirit within us, and by (Thy) governing Spirit, give support to what is liable to err in our thoughts, so that each day, being led on the way by Thy good spirit to that which is beneficial, we may be deemed worthy to obey Thy commandments and ever to remember Thy return in glory, that shall search through the deeds of men, and not be deceived by the corruptible delights of this world, but strengthen us to reach out for the enjoyment of the treasures that are to come, for Thou art blessed and praiseworthy in all Thy Saints, unto the ages of ages. Amen.