One last recommendation of the Holy Father & Synod for fuller participation is "Biblically-inspired liturgical song". His words seem to suggest a hierarchy of preference and several categories of liturgical music which are acceptable:
1. Gregorian chant. Here, the Holy Father cites Sacrasanctum Concilium (¶ 116), which gives Gregorian chant "pride of place" in the liturgy, and the General Instruction of the Roman Missal (GIRM) which itself, in ¶ 41, cites SC. You can listen to several examples online from the Miles Christi schola cantorum.
2. "Songs handed down to us by the Church's tradition". There is a long tradition of beautiful and theologically-rich hymns in the Church. For example, St. Thomas Aquinas wrote several, including probably his best known, Tantum Ergo Sacramentum. Also, Fr. Frederick Faber wrote many English-language hymns in the 1800's, including Faith of Our Fathers and Sweet Sacrament (Jesus, My Lord, My God, My All).
3. "Songs which are of clear biblical inspiration and which express... the beauty of God's word." In this last category, we have other liturgical music - including modern - which draws on Scripture for inspiration. The Psalms seem the obvious choice, as they were the early hymns, though any book of the Bible is fair game, as "(a)ll scripture is inspired by God and is useful for teaching... and for training in righteousnes..." (2 Tim 3:16) It is notable, though, that the Holy Father gives some additional criteria. The songs must be used "at the times called for by the particular rite." The GIRM is the key reference here, providing instruction on which parts get preference (¶ 40) and on music during specific parts of Mass (e.g. ¶ 47-48)
Indeed, this is what is meant by physical "active participation" - our taking part in the Mass "by means of acclamations, responses, psalmody, antiphons, and songs, as
well as by actions, gestures, and bodily attitudes." Our minds and spirits must be actively engaged, but so should our bodies - by picking up the hymnal and singing!
There are many more articles on liturgical music. For a further breakdown of Church documents, as well as independent articles, I recommend stopping by Adoremus.