What is the meaning behind the ringing of the bells during Mass, and why are they at the specific moments they are (especially the 3rd ring)?
The ringing of bells within the Mass has been around for many years. To expand this question to bells in general, the Church uses bells (including those within the bell towers of church buildings) as a means to calling people to attention. For example, one function of the bells in a bell tower is to call people to prayer – as in many cases the bells in bell towers will be set to peal in an extended toll (lasting about a minute) approximately 5-10 minutes before Mass is to begin at the given parish church/chapel/etc. Similarly, during the Mass, the ringing of the bells is to call attention to those who are meant to hear them (either the people, the choir, or sometimes both) to be ready to begin the next element of the Mass or be attentive to what is taking place.
Here at St. Benedict’s the bells are used extensively at all Masses. In Ordinary Form Masses, a bell is rung to signal the beginning of Mass (either as the ministers leave the sacristy to begin Mass as on weekdays, or when they are at the back of Church for beginning Masses on Sundays/Holy Days). This ring is meant primarily to let the choir know it is time to begin – though it also serves so that all who are present will hear it and know to stand for the beginning of Mass. During the Eucharistic Prayer, the server first rings the bells at the moment when the priest extends his hands over the bread and wine with his palms facing downward – an act of calling upon the Holy Spirit to come down from heaven that is referred to as the “epiclesis.” This single ring is meant to draw attention to the proximity of the moment of the consecration (which, in the Extraordinary Form Masses means that if the choir is still singing the “Sanctus,” they need to either finish it or stop after the first “Hosanna in excelsis” before starting the second half of this acclamation [that is, the “Benedictus”]). Then comes the two sets of three rings: one set at the elevation of the Body of Christ and the other at the elevation of the chalice with the Blood of Christ – calling to attention this moment when Christ literally appears for all to see in the sacrament. (Once again, in the Extraordinary Form there is added meaning: all of these rings during the Eucharistic Prayer also serve to draw attention to any who pass by an altar where a Mass is being offered that they ought to kneel in place until the end of the consecration before moving on – especially if these who are passing by are a priest and other ministers who may be making their way to or from the sacristy in route to/from a separate Low Mass at another altar in the same church). Finally, there is the ring that takes place as the priest drinks from the chalice – which alerts the choir to begin the Communion Antiphon or other chosen music selection for Communion.
In the Extraordinary Form, there are some additional rings that are not found in the Ordinary Form: 1 – when the priest unveils the chalice at the beginning of the
Offertory: to let the choir know to begin the Offertory Antiphon; 2 – during the moment when the priest says the threefold “Sanctus” they are rung three times (that is, the beginning of the “Holy, Holy, Holy,); and once during each recitation of the words “Domine, non sum dignus” (that is, the saying of the “Lord, I am not worthy…” prior to receiving communion, which the priest says alone three times before his communion, and then all the servers [and perhaps the people] say together three times prior to distribution of Holy Communion). In all of these cases, the bells serve to draw attention to particularly sacred moments, while also alerting the people to reverent attention or the choir to begin singing what is proper to the moment.