Might a modern pentivium add numbers in n-dimensional space (matrices / calculus), or a sextivium explorations and representations of infinity? Or would these fall under the subsequent study of philosophy? How about parallel/concurrent systems? It certainly seems to me as a layman that large swathes of research mathematics and theoretical physics are approaching subject areas traditionally ascribed to philosophy. I wonder what the latin word for GPU would be.
I'm finding the content and descriptions mostly interesting in a retrospective sense -- I'm not ent
advocating that the mediaeval curriculum be implemented in its original form today. At the same time, I find the notion of input, processing, and output to be quite useful, and one which might be considered as part of a modern curriculum.
For the maths progression of the quadrivium, there are a few other possible clusters or progressions. We've made some advances in maths as well.
I see a general structure of: quantitative notations and operations (arithmetic), solving for unknowns (algebra), geometry and trigenometry, integrals and derivatives, matrixes, differential equations and approximations (FFT). There's also logic (informal, formal, programmatic), programming, and artificial intelligence / machine learning. Genetic engineering seems also to fall somewhere on this axis.
Collectively, all of this falls under inforamtion acquisition, processing, storage, and transmission.
The trivium: grammar, logic, and rhetoric. Or in computing terms: input (parsing), processing, and output.
The quadrivium: maths, geometry, music, and astronomy. "What's the connection?" you ask?
Numbers.
Numbers in space.
Numbers in time.
Numbers in space and time.