People like me (is there anyone like me?) tend to nag you about the desireability of keeping the Pentecost Octave. But there was an observance this week which is much older than the Octave; indeedd, much older than Christianity itself: the Summer Ember Season.
The pagan Romans kept Feriae messis, Days of Harvest, connected with the corn harvest. In the ancient Liturgy of the (local) Church of Rome, which we are priviged to have received for our own, this ancient piece of local Romanita is preserved for us as the Ember Wednesday, Friday, and Saturday of the week following Pentecost. Like the other pagan agricultural festivals, this one was converted into a fast: S Leo regards fasting as particularly appropriate for the Ember Days this week "So that if, amid the joys of the festivities, negligent liberty and inordinate licence has made any presumption, this [fast] may discipline (castiget) it by the censure of religious abstinence".
So some elements in the Old Rite propers for these days precede the imposition of 'Spirit' themes by the Octave. Today's Gospel about the Bread of Life, for example, christianises the pagan celebration of the corn harvest. I will leave you to fish out your good old English Missals and find the evidence for this also in the readings for Friday and Saturday. (Remember that our forefathers in the Faith believed that healings from illnesses and exorcism of unclean spirits were closely related to fasting [see Mark 9:29 with the variant reading and Matthew 17:21], so also be on the lookout for pericopes about healings and exorcisms.)
In 455, the Arian Gaiseric was attacking Rome (he took it on the Ember Wednesday, secured thousands of potential candidates for the slave markets, and pillaged most of the basilicas ... it's the sort of thing heretics do). Pope S Leo wrote a number of collects on this occasion; one of them survives as the collect for the Ember Friday this week. Find it in your English Missal. You'll see the point of the phrase hostili nullatenus incursione turbetur. Relevant today?
See Willis 1964.