(September 9, 2016 at 1:32 am)bennyboy Wrote:(September 9, 2016 at 1:12 am)Alex K Wrote: Looks deeper before you realize that you don't end up in the same place after going round once if you use pure intervals due to the pythagorean comma. A pity, really
Yep, and the math should be almost painfully obvious. How many iterations of (3/2)^n (a circle of perfect fifths) are every going to arrive at an even multiple of 2? When you then start taking each of those notes and their overtones (f/2, f/3, f/4 etc.), you quickly realize that any amount of modulation among keys will almost instantly lead to chaos.
Tuning systems require two things: (1) fudge frequencies to make the math work; (2) have people's tolerances built to accept the impure sounds.
There's the reason why tempered tuning was invented.
As much as Arky wants us to think that mathematically-perfect relationships are sonorous to the human brain, the fact is, they aren't -- which was the point of my mentioning blue notes. Mathematically imperfect, but a hell of a lot better than equal-temperament.
These are things a musician knows.
