I'm surprised no one has said Pange Lingua for Holy Thursday yet. That's my vote. (But then again, I don't participate in all that many processions in the space of a year...)
Pange Lingua is the answer for me too.. No matter how many Holy Thursdays one gets on this earth, I doubt its enough. I couldn't imagine someone being Catholic and not going to Mass on Holy Thursday (or more horrifying, finding themselves at a Parish that was doing something modern in Pange Lingua's place).
I am totally nerding out on this topic, Jeffrey. I've thought of it often, esp when I consider compositional fodder. I tend toward a love of medaeival hymns that sound like sacred pub songs. I hope I don't offend anyone with that idea. They just seem like part of a joyful, Christ-filled life. I find some hymns, like 'Salve, Mater', so irresistible that choir friends and I have been known to sing them in public. We are actually looking for flash mob venues... I digress. Here are my favorite-
Palm Sunday Hosanna filio David Gloria, laus, et honor
In addition to the above mentioned, I remember fondly, at the Vigil of All Saints, beginning in the garden at the Shrine of the Blessed Virgin, carrying the relics of the parish in procession to the church singing the Litany of the Saints.
O Redemptor for the blessing of oils on Holy wednesday (sometimes during quadragesima also used) (9th c) Beata viscera Maria (usually used at (immaculate) Conception) (by Perotin in the early 13th c) Dum Pater Familias (Pilgrims Hymn for St James the Apostle) (July 24) (12th c.)
and from the Byzantine rite, "Rejoice of Bethany" for Lazarus or Palm Sunday is an incredibly moving processional hymn.
Processional Hymns should have a short refrain that can be memorized. E.g. the Lourdes Hymn. 2 words, 5 syllables divided over 4 or 5 notes, works in any language! Here's an official source of liturgical Latin processional hymns: "Cantus ad Procesiones", an old book, published by McLoughlin Reilly.
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