For the EF:
The only options are for
everything to be done musically, or
nothing. There is nothing “based on” the Propers; the musical parts either
are the Propers (Introit, Gradual, etc.), or else they are the Ordinary (Gloria, Credo, etc., as Chatter mentioned). (There are also parts reserved to the priest like the Pater Noster, which I’ll omit from consideration). Yes, there are choices. Nothing (except good taste) prohibits singing these to any melody you like. For instance, you might have the Propers chanted by a Gregorian schola, and the Ordinary sung by a different choir to a polyphonic setting by Palestrina. Or you can have the Ordinary chanted by a choir to the Gregorian melody of your choosing, and hustle through the Propers by singing them to a psalm tone – e.g., using the “
Rossini Propers,” which were very popular before the Council among congregations that either could not field a suitable Gregorian schola, or got bored sitting through interminable tracts and graduals and wanted to hurry things along. Or, if you cared to, you could sing polyphonic
Propers (these have been composed for many feasts), and chant the Ordinary
recto tono. And you can break things up more finely, too; for instance, singing a polyphonic Kyrie and Agnus Dei, but using Gregorian chant for the other parts of the Ordinary (polyphonic Credos are generally super-long; some composers don’t even bother setting it). It’s whatever you feel like. If Marty Haugen wrote a Latin Gloria, you could in theory sing it (assuming you would not be tarred and feathered).
The major exceptions to the above are:
- Additional motets (i.e., songs) can be sung during Communion. If it is a sung Mass, additional music must be in Latin. Also, nothing prevents singing a recessional hymn (in Latin or the vernacular), since the Mass is technically over by this point.
- At a said Mass, it used to be extremely common (varying by country; for example, especially popular in Germany) to sing vernacular hymns during the Mass. Since the Mass was mostly silent and inaudible, people would sing an Entrance Hymn, Offertory Hymn, and so on, just as is now customary in the OF (that is where it came from). Or for another example of a musical tradition, you can read the very good Wikipedia article on the Deutsche Singmesse.
For the OF:
To provide a comprehensive list of options would be very time-consuming. Suffice it to say that the Propers are very rarely used, though they are of course permitted; and that the Ordinary is generally sung (Gloria, Agnus Dei) or spoken (Kyrie, Credo) by the congregation.