[Waterluvian]

> composers tend to base their methods in mathematical principles such as group theory

I know many composers were and are very in tune with the mathematics of music. But the “tend to” makes me wonder: were most of them in tune, or is it that pleasant sounding music will inevitably display mathematical patterns?

[odyssey7]

"I am taking such pains with these things, not so much out of concern that the pupil might, if they were not explained, thoughtlessly accept and mechanically apply them, as to establish explicitly that these principles are not derived from aesthetics but from practical considerations. If what is known as aesthetics does in fact contain much that is merely practical handling of the material, and if what is known as symmetry is perhaps often not much more than an organization of the material that reveals a sensible regard for its properties, yet I consider it worthwhile to set down these observations. For the conditions of practicality can change, if we take a different view of the material and if the purpose changes. But aesthetics alleges it has discovered eternal laws."

-- Arnold Schoenberg, Theory of Harmony

[fanf2]

That sentence is specifically about composers of methods in change ringing, where the musical possibilities are tightly constrained by the rules and the physics of ringing large bells. Change ringing is based on permutations constructed by swapping adjacent pairs. (Except for the jump changes that the article is about.) For a “triples” peal on 7 bells the ringers are supposed to ring all 5040 possible permutations. It’s called “triples” because you can make up to 3 swaps at a time; one of the challenges (for example) is to construct a triples peal that only uses triple changes.