The Trivium (which did not include history, something regarded as inferior by (eg) Aristotle), was designed to produce advocates in the law courts of Rome. The Quadrivium I'm less sure about: it may have been added as a step on the way to real learning, namely the degree in theology. None of this had anything to do with "landed scions" for whom the relevant training was in jousting, dancing, hawking, hunting, heraldry, and other courtly activities. The claim that the landed aristocracy needed university learning was put forward in the Renaissance by self-interested humanists (see J. H. Hexter's article "The Education of the Aristocracy in the Renaissance". Remember that scholars were not usually gentlemen!
Agreed that the purpose of the medieval university education was (from princes' and bishops' perspective) to produce well-educated judges and theologians to serve as minions, and (from students' perspective) upward mobility (or, if you were the scion of a landed family who could not hack it as a knight, avoid too much downward mobility).
The Trivium is a necessary foundation for both roles.
But the quadrivium--arithmetic, geometry, harmonics (i.e., not music practice and not music criticism), and astro-logy/nomy?
Maybe it would be better to call them: "arithmetic, geometry, trigonometry (i.e., sines, cosines, and waves), and dynamics (to the extent you can do dynamics before Newton/Liebnitz)? (All four done, from our perspective, badly: one tries to contemplate studying dynamic equations of motion without knowing calculus using only Roman numerals, and the mind does boggle…)
