The XXI Sunday after Pentecost and second Sunday of October is of semi-double rite. Again, like last Sunday, the Office is classically that of a semi-double 'green' Sunday.
At Vespers on Saturday the ferial psalms are sung and the antiphon at the Magnificat is Refulsit sol (for the Saturday before the 2nd Sunday of October). A commemoration is made of the preceding Office of St. Francis of Assisi and of SS Placid and Others. Unlike last Saturday the Suffrage of the Saints is not sung because the commemorated Office of St. Francis is of greater double rite.
At Mattins there are the usual nine psalms and nine lessons divided into three nocturns. The continuing history of the Machabees is read in the first nocturn. At Lauds there is a commemoration of St. Placid and the Suffrage of the Saints is sung. At Prime the Athanasian Creed, Quicumque, is sung as are the Dominical preces.
At Mass the Gloria is sung, the second collect is of St. Placid and the third collect is A cunctis. The Credo is sung and the preface is that of the Holy Trinity as used on Sundays.
In Vespers a commemoration of the following double rank feast of St. Bruno is made.
Before the 1911-13 reform the first, calendar Sunday of October (not to be confused with the first Sunday for Mattins although the two could be the same as they will be next year)was the feast of the Most Holy Rosary of the BVM. A decree of the Sacred Congregation for Rites (No. 4308) allowed for feasts previously kept on a Sunday still to be celebrated as Votive Masses on their former days. So all Masses (except the Conventual one in Cathedral and Collegiate churches) may be of the Holy Rosary with Gloria, 2nd collect of the Sunday (and in private Masses the 3rd collect of St. Placid), Creed, Preface of the Blessed Virgin and last Gospel of the Sunday. These Masses are celebrated in white vestments. Traditionally, the Rosary is prayed before Mass, with the Litany of Loreto and the prayer to St. Joseph, Spouse of the BVM. The Catholic Encyclopedia has an article here.
In the 'liturgical books of 1962' the first Sunday of October rather than the second is celebrated and so a different antiphon at the Magnificat is used (the one sung in the Old Rite last week) and first Vespers is without any commemorations or Suffrage. Mattins, as usual is stripped down to one nocturn of three lessons only with six lessons cast aside. At Lauds there is neither commemoration of St. Placid or the Suffrage. At Prime the Athanasian Creed is not sung and there are no preces. At Mess there is only one collect. Vespers are without any commemoration. St Placid is completely omitted this year.